Handbook of Logical Thought in India 8132225767, 9788132225768

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Sundar Sarukkai Mihir Kumar Chakraborty Editors

Handbook of Logical Thought in India

Handbook of Logical Thought in India

Sundar Sarukkai • Mihir Kumar Chakraborty Editors

Handbook of Logical Thought in India With 220 Figures and 32 Tables

Editors Sundar Sarukkai Founder, Barefoot Philosophers Former Professor of Philosophy National Institute of Advanced Studies Bangalore, Karnataka, India

Mihir Kumar Chakraborty School of Cognitive Science Jadavpur University Kolkata, West Bengal, India Former Professor of Pure Mathematics University of Calcutta Kolkata, West Bengal, India

ISBN 978-81-322-2576-8 ISBN 978-81-322-2577-5 (eBook) https://doi.org/10.1007/978-81-322-2577-5 © Springer Nature India Private Limited 2022 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature India Private Limited The registered company address is: 7th Floor, Vijaya Building, 17 Barakhamba Road, New Delhi 110 001, India

Preface

Any volume, such as this handbook, reflects the times in which it is produced and is not a mere collection of received knowledge. Although knowledge, particularly philosophical knowledge, is often considered to be “timeless,” it is important to recognize the implicit and explicit presence of the temporal and material conditions under which knowledge is defined and produced. Today it is no longer possible to believe that knowledge in any domain is reducible to purely epistemological factors alone. There are important social factors that influence the nature of disciplines and knowledge, and what gets produced as such. Thus, a preface to this handbook has to reflect on these factors as it matters to the very meaning of terms like “Indian philosophy,” “Logic in India,” and “Indian logic” since not only are terms like “India/Indian,” but also those like “philosophy” and “logic,” have become widely contested. Since logic and philosophy accrued cultural values within dominant Western societies, colonial discourse consistently denied the capacity for logic or philosophy to the colonized societies. This is ironic considering that logic is arguably the heart of philosophy in the many traditions of philosophy in India. Moreover, while the terms “India” and “Indian” point to a geographic localization, the scholars in the field of “Indian philosophy,” visible in the English speaking world, the de facto world today of international academics, are not so geographically situated. Even when there are “Indian” scholars, many prominent ones among them are outside India. This sociology matters to the international face of this discipline and is also well reflected in the fact that well-established journals such as Journal of Indian Philosophy are located outside India. Routinely, handbooks and encyclopedias on Indian philosophy in English produced outside India have little representation of scholars working in India. This has led to a unique and poignant presence of the “international” as intrinsic to “Indian” philosophy. While this may seem to be a problem specific to the production of Indian philosophy in English language, there are major consequences of it including the inability to draw upon traditional scholars in these disciplines who are often uncomfortable in English or refuse to write in English. This leads to a skewed ownership of intellectual practices within cultures and raises the question of who can be spokespersons for these historical traditions.

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Preface

The collection of articles in this volume has to be seen within this socio-political background of international scholarship. Its strength and weakness (particularly in terms of representation) are a testament to these deep-rooted sociological problems related to writing about, and disseminating work, in Indian logic in particular and in Indian philosophy in general. What adds to the problem is the asymmetry in acknowledging and accepting the legitimacy of these traditions within “mainstream” philosophy which de facto has become “Western” philosophy. The constant struggle to make these philosophical traditions part of “mainstream” global philosophy is another important factor that influences the content, the style of writing as well as the rhetoric in this domain. This begins right from the title – “Logical Thought in India” – as a way to include the various logical traditions that were part of ancient and later philosophical traditions in India as well as point to some contemporary work in this field. We do this while being very conscious of the fact that “logic” as referred to in contemporary mathematics or computer science is markedly different from earlier traditions of logic. However, bringing these topics under one umbrella of “logic in India” is to indicate not just broader metaphysical commonalities but also the continued lived practices that evolve over time. Today, logic is primarily spoken of under the “modern” label but there is much in terms of its training that shares a common space with traditional training in philosophy. By doing this, we are not suggesting that the multiplicity of different logical traditions must be clubbed under one “mainstream” vision. Rather, we want to explore the possibility of placing these multiple offerings on the same plate without inbuilt biases on the merit of each. The challenge in such a comprehensive approach is that there is substantial material that could be part of this collection. Moreover, there are also a large number of books and articles that have already been published on many of these topics. Thus, we have attempted to balance the articles between introductory ones that discuss basic themes and those which offer some new ideas that have not had much visibility before. So while there are the standard inputs from traditions such as Nyāya, Buddhist, and Jaina logical schools, there are also some new approaches from the Tamil traditions as well as from the poetic, aesthetic, and linguistic traditions. There are also some absences, in spite of our best efforts to convince people to write on those topics. Typically, Indian philosophy is grouped under various schools. We believe that while this is useful for many purposes, it is also limiting in that readers tend to associate ideas and styles exclusively with one school. However, these traditions are not independent of each other in that there are themes that are common, arguments that are commonly debated, overlapping epistemological and metaphysical structures, and so on. Thus, in this handbook we have decided to group the articles not under the respective schools but distributed under the following headings: Texts, Fundamentals, Particularities, Language, Comparative, and Modern. Each of the section discusses some aspects of logic under these larger themes. However, they are not mutually exclusive but they are distributed in this manner in order to suggest a different way of understanding these logical traditions. The articles under the first section “Texts” provide a small sample of the wide variety of texts encountered in Indian philosophy. These texts often require codes of reading and interpreting. As is

Preface

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well known, some of the most important discussions in Indian philosophy are not to be found in the primary texts but in the various commentaries to primary texts. From this very small sample of texts discussed in this section, one can already note not just the different themes but also styles of reasoning. The second section has articles under the title of “Fundamentals.” These articles discuss some foundational issues associated with different traditions and give some idea of the fundamental principles of some of these traditions as well a brief account of their development. Even though there are overlapping foundational themes, there are also some unique features in each of these philosophical schools. The articles in the section “Particularities” provide samples of the type of uniqueness that characterize some of the philosophical approaches of these different schools. Following this section is the one under the heading of “Language.” Indian philosophy has had a sustained and very rich reflection on language. There are many innovative aspects in their approach, such as the creation of foundational ideas in linguistics as well as the attempt to create a “technical” language. The next section consists of articles that are largely about the comparative approach between different schools. While almost all of Indian philosophy should be seen as a comparative enterprise in that agreement and disagreement between schools can be seen as the core of philosophical practice, we can nonetheless identify a few limited themes to give a sample of this comparative process. We include articles that not only illustrate a notion of comparative between the earlier traditions but also that between these traditions and modern logic. The last section compiles articles under the heading of “Modern.” These articles are examples of the rich practices of logic in India, their contemporary interpretations, and those in the domains of mathematical logic and in computer science. In bringing these articles also into this fold, we are also problematizing a specific essential idea of “India” as something totally encoded in the past or located in some specific traditions. When this volume was first envisioned, it was in terms of the traditional grouping under the various schools. However, as the volume took shape, we realized that this would not be the best way to present the material in this volume. So although our section editors were responsible for grouping articles under each school, we have changed the final categorization of the sections. We are deeply grateful to the following scholars who were responsible for soliciting articles from their colleagues: Pradeep P. Gokhale (Chaps. ▶ 1, ▶ 4, ▶ 8, and ▶ 12), Jayendra Soni (Chaps. ▶ 2, ▶ 9, ▶ 16, and ▶ 31), Shrinivasa Varakhedi (Chaps. ▶ 23 and ▶ 30), Jonardon Ganeri (Chaps. ▶ 14, ▶ 20, ▶ 27, ▶ 34, and ▶ 35), Nirmal Selvamony (Chaps. ▶ 3, ▶ 10, ▶ 19, and ▶ 22), Amba Kulkarni (Chaps. ▶ 24, ▶ 25, and ▶ 26), and Mrinal Kaul (Chaps. ▶ 5, ▶ 11, ▶ 15, ▶ 18, and ▶ 29). We are also deeply thankful to all the authors of this volume who have spared their precious time and effort to contribute to this venture. Bangalore, India Kolkata, India November 2022

Sundar Sarukkai Mihir Kumar Chakraborty

Contents

Volume 1 Part I

Texts

.............................................

1

1

Buddhist Logic: Sample Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pradeep P. Gokhale

3

2

General Introduction to Logic in Jainism with a List of Logicians and Their Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jayandra Soni

23

3

Logic in nīlakēci and maṇimēkalai . . . . . . . . . . . . . . . . . . . . . . . . . Nirmal Selvamony

45

4

Introduction to Buddhist Logicians and their Texts . . . . . . . . . . . . Madhumita Chattopadhyay

67

5

Logical Proofs in the Śivadṛṣṭi of Somānanda John Nemec

91

Part II

Fundamentals

................

....................................

115

6

Logic in India . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Amita Chatterjee

117

7

Charvaka (Cārvāka) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ramkrishna Bhattacharya

139

8

General Introduction to Buddhist Logic . . . . . . . . . . . . . . . . . . . . . Joerg Tuske

161

9

Logic of Syād-Vāda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anne Clavel

183

10

Logic in Tamil Tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nirmal Selvamony

205

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11

Contents

Causal Reasoning in the Trika Philosophy of Abhinavagupta . . . . Mrinal Kaul

Part III

Particularities

....................................

227

255

12

Some Issues in Buddhist Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pradeep P. Gokhale and Kuntala Bhattacharya

257

13

The Nyāya on Logical Thought J. L. Shaw

...........................

281

14

Early Nyāya Logic: Pragmatic Aspects Jaspal Peter Sahota

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353

15

Āgama as Pramāṇa in Kashmir Śaivism Navjivan Rastogi

....................

373

16

Logical Argument in Vidyānandin’s Satya-śāsana-parīkṣā . . . . . . Himal Trikha

421

17

Jaina Theory of “ANUMĀNA” [Inference]: Some Aspects Tushar K. Sarkar

......

437

18

Pratyabhijñā Inference as a Transcendental Argument About a Nondual, Plenary God . . . . . . . . . . . . . . . . . . . . . . . . . . . . David Peter Lawrence

489

19

Logic in Tamil Didactic Literature . . . . . . . . . . . . . . . . . . . . . . . . . Thanga Jayaraman

519

20

The Logic of Late Nyāya: Problems and Issues . . . . . . . . . . . . . . . Eberhard Guhe

539

Volume 2 Part IV

Language

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569

21

Logical Aspects of Grammar: Pāṇini and Bhartṛhari . . . . . . . . . . Raghunath Ghosh

571

22

Logic in tolkāppiyam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nirmal Selvamony

595

23

Dependency of Inference on Perception and Verbal Testimony . . . Vinay P

617

24

Influence of Navya-Nyāya Concepts and Language in Vyākaraṇa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . B. V. Venkataramana

633

Contents

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25

Technical Language of Navya Nyāya . . . . . . . . . . . . . . . . . . . . . . . O. G. P. Kalyana Sastry

657

26

The Importance of Śābdabodha in Language Analysis Rajaram Shukla

.........

681

27

Early Nyāya Logic: Rhetorical Aspects . . . . . . . . . . . . . . . . . . . . . Keith Lloyd

695

28

Logic of Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reetu Bhattacharjee, Mihir Kumar Chakraborty, and Lopamudra Choudhury

719

Part V

Comparative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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29

Abhinavagupta on Śānta Rasa . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sthaneshwar Timalsina

803

30

Convergence and Divergence of Nyāya and Tattvavāda (Dvaita) Theories of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vaishnavi Nishankar

821

The Opponent: Jain Logicians Reacting to Dharmakīrti’s Theory of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Marie-Hélène Gorisse

847

32

The Catuṣkoṭi, the Saptabhaṇgī, and “Non-Classical” Logic . . . . . Graham Priest

869

33

Imperative Logic: Indian and Western . . . . . . . . . . . . . . . . . . . . . . Manidipa Sanyal

889

31

Part VI

Modern . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

929

34

Later Nyāya Logic: Computational Aspects . . . . . . . . . . . . . . . . . . Amba Kulkarni

35

The Logic of Late Nyāya: A Property-Theoretic Framework for a Formal Reconstruction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eberhard Guhe

955

A Few Historical Glimpses into the Interplay Between Algebra and Logic and Investigations into Gautama Algebras . . . . . . . . . . Hanamantagouda P. Sankappanavar

979

36

931

37

Algebraic Logic and Rough Set Theory . . . . . . . . . . . . . . . . . . . . . 1053 Mohua Banerjee

38

The Logic of Nonpersons Rohit Parikh

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1103

xii

Contents

. . . . . . . . . . . . . 1119

39

Philosophical Aspects of Constructivism in Logic Ranjan Mukhopadhyay

40

Logics of Strategies and Preferences . . . . . . . . . . . . . . . . . . . . . . . . 1135 Sujata Ghosh and R. Ramanujam

41

Synthesizing Skolem Functions: A View from Theory and Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1187 S. Akshay and Supratik Chakraborty

42

Terms and Automata Through Logic . . . . . . . . . . . . . . . . . . . . . . . 1223 Kamal Lodaya

43

Consequence-Inconsistency Interderivability in Paraconsistent Logics and Paraconsistent Set Theory . . . . . . . . . . . . . . . . . . . . . . 1255 Soma Dutta and Sourav Tarafder

44

An Introduction to Theory of Graded Consequence Mihir Kumar Chakraborty and Sanjukta Basu

. . . . . . . . . . . 1289

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1329

About the Editors

Sundar Sarukkai works primarily in the philosophy of the natural and the social sciences. He is the founder of Barefoot Philosophers (www.barefootphilosophers.org) and is currently a Visiting Faculty at the Centre for Society and Policy, Indian Institute of Science, Bangalore. He has authored Translating the World: Science and Language, Philosophy of Symmetry, Indian Philosophy and Philosophy of Science, What Is Science?, JRD Tata and the Ethics of Philanthropy, and coauthored two books with Gopal Guru – The Cracked Mirror: An Indian Debate on Experience and Theory and more recently, Experience, Caste and the Everyday Social. His book Philosophy for Children: Thinking, Reading, Writing is published in English, Hindi, Tamil, Kannada, Malayalam, and Telugu. His latest book is The Social Life of Democracy. He is the Series Editor for the Science and Technology Studies Series, Routledge. He was a professor at the National Institute of Advanced Studies until 2019 and was the Founder-Director of the Manipal Centre for Philosophy and Humanities from 2010 to 2015.

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About the Editors

Mihir Kumar Chakraborty School of Cognitive Science Jadavpur University Kolkata, West Bengal, India Former Professor of Pure Mathematics University of Calcutta Kolkata, West Bengal, India Mihir Kumar Chakraborty is currently Visiting Professor, School of Cognitive Science, Jadavpur University, India, and Adjunct Professor, Indraprastha Institute of Information Technology Delhi (IIIT-Delhi). He was formerly Professor at the Department of Pure Mathematics, University of Calcutta, India; Visiting Professor, Indian Institute of Engineering Science and Technology, Shibpur, India; Visiting Professor, Indian Statistical Institute, Kolkata, India; Guest Professor, Chongqing Southwest University, China; and Indian Institute of Science Education and Research Kolkata (IISER, Kolkata), Kalyani, West Bengal, India. His research interests are non-standard logics, fuzzy set theory and fuzzy logic, rough set theory and logics of rough sets, paraconsistent logics, foundations of mathematics, mathematics and culture, logic of diagrams, functional analysis, topology, and knowledge representation. He has widely toured various countries on academic visits and authored and edited several books in both English and Bengali. One of the founders of Calcutta Logic Circle (CLC) and Association of Logic in India (ALI), he has also been awarded fellowships by Indian Institute of Advanced Studies (IIAS), Shimla; West Bengal Academy of Science and Technology (WAST); Indian Council of Philosophical Research (ICPR); International Rough Set Society (IRSS); and Deutscher Akademischer Austauschdienst (DAAD).

Section Editors

Jonardon Ganeri Bimal K. Matilal Distinguished Professor of Philosophy Department of Philosophy University of Toronto Toronto, ON, Canada

Pradeep P. Gokhale Department of Philosophy Savitribai Phule Pune University Pune, India

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Section Editors

Mrinal Kaul Department of Humanities and Social Sciences (HSS) Indian Institute of Technology-Bombay (IITB) Mumbai, India

Amba Kulkarni Department of Sanskrit Studies University of Hyderabad Hyderabad, Telangana, India

Nirmal Selvamony Madras Christian College Chennai, Tamil Nadu, India Central University of Tamil Nadu Thiruvarur, Tamil Nadu, India

Jayandra Soni Department of Philosophy University of Innsbruck Innsbruck, Austria University of Marburg Marburg, Germany

Section Editors

xvii

Shrinivasa Varakhedi Kavikulaguru Kalidas Sanskrit University Ramtek, Nagpur, India

Contributors

S. Akshay Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India Mohua Banerjee Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India Sanjukta Basu Department of Philosophy, Rabindra Bharati University, Kolkata, West Bengal, India Reetu Bhattacharjee School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India Kuntala Bhattacharya Department of Philosophy, Rabindra Bharati University, Kolkata, West Bengal, India Ramkrishna Bhattacharya Pavlov Institute, Kolkata, West Bengal, India Mihir Kumar Chakraborty School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India Former Professor of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India Supratik Chakraborty Department of Computer Science and Engineering, Indian Institute of Technology Bombay, Mumbai, India Amita Chatterjee School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India Madhumita Chattopadhyay Department of Philosophy, Jadavpur University, The Center for Buddhist Studies, Kolkata, West Bengal, India Lopamudra Choudhury School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India Anne Clavel Department of Philosophy, University of Lyon, Lyon, France Soma Dutta Department of Mathematics and Computer Science, University of Warmia and Mazury in Olsztyn, Olsztyn, Poland xix

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Contributors

Raghunath Ghosh University of North Bengal, Darjeeling, West Bengal, India Sujata Ghosh Indian Statistical Institute, Chennai, India Pradeep P. Gokhale Department of Philosophy, Savitribai Phule Pune University, Pune, India Marie-Hélène Gorisse Faculty of Arts and Philosophy, Ghent University, Ghent, Belgium Eberhard Guhe Department of Philosophy, Fudan University, Shanghai, China Thanga Jayaraman Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India Mrinal Kaul Department of Humanities and Social Sciences (HSS), Indian Institute of Technology-Bombay (IITB), Mumbai, India Amba Kulkarni Department of Sanskrit Studies, University of Hyderabad, Hyderabad, Telangana, India David Peter Lawrence Department of Philosophy and Religious Studies, University of North Dakota, Grand Forks, ND, USA Keith Lloyd Faculty of English, English Kent State University Stark, North Canton, OH, USA Kamal Lodaya The Institute of Mathematical Sciences, Chennai, India Ranjan Mukhopadhyay Visva-Bharati University, Santiniketan, West Bengal, India John Nemec Indian Religions and South Asian Studies, Department of Religious Studies, University of Virginia, Charlottesville, VA, USA Vaishnavi Nishankar Research Scholar, Mumbai, India Vinay P Faculty of Vedanta, Karnataka Samskrit University, Bangalore, Karnataka, India Rohit Parikh CS, Math, Philosophy, Brooklyn College and CUNY Graduate Center, City University of New York, New York, NY, USA Graham Priest Departments of Philosophy, The CUNY Graduate Center, New York, NY, USA University of Melbourne, Parkville, VIC, Australia R. Ramanujam Instititute of Mathematical Sciences, Chennai, India Navjivan Rastogi The Department of Sanskrit and Prakrit Languages, Abhinavagupta Institute of Aesthetics and Shaiva Philosophy, University of Lucknow, Lucknow, India

Contributors

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Jaspal Peter Sahota Inavya Ventures Ltd, London, UK Hanamantagouda P. Sankappanavar Department of Mathematics, State University of New York, New Paltz, NY, USA Manidipa Sanyal University of Calcutta, Calcutta, India Tushar K. Sarkar Jadavpur University, Kolkata, West Bengal, India University of Waterloo, Windsor, ON, Canada O. G. P. Kalyana Sastry Department of Nyaya, National Sanskrit University, Tirupati, Andhra Pradesh, India Nirmal Selvamony Madras Christian College, Chennai, Tamil Nadu, India Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India J. L. Shaw Department of Philosophy, Victoria University of Wellington, Wellington, New Zealand Rajaram Shukla Department of Vaidic Darshan, Sanskrit Vidya Dharm Vigyan Sankay, Banaras Hindu University, Varanasi, India Jayandra Soni Department of Philosophy, University of Innsbruck, Innsbruck, Austria University of Marburg, Marburg, Germany Sourav Tarafder Business Mathematics and Statistics, Department of Commerce, St. Xavier’s College, Kolkata, India Institute of Philosophy and the Humanities, University of Campinas (UNICAMP), São Paulo, Brazil Sthaneshwar Timalsina Religious Studies, San Diego State University, San Diego, CA, USA Himal Trikha Centre national de la recherche scientifique, Paris, France Joerg Tuske Department of Philosophy, Salisbury University, Salisbury, MD, USA B. V. Venkataramana Department of Philosophy, Evening Sanskrit College, Karnataka Sanskrit University, Bangalore, Karnataka, India

Part I Texts

1

Buddhist Logic: Sample Texts Pradeep P. Gokhale

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Text I: Diṅnāga’s Nyāyapraves´akasūtram: Aphorisms Introducing the Theory of Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Text II: Chapter Two of Nyāyabindu: “On the Inference for Oneself” . . . . . . . . . . . . . . . . . . . . . . . . . . Text III: A Section from Hetubindu: On the Nature of a Valid Argument . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 4 13 17 21 21 22

Abstract

In this chapter, three sections from three different texts are chosen which discuss some of the core doctrines in Buddhist logic. The first section, namely, Nyāyapraves´akasūtram discusses Diṅnāga’s theory of inference which includes his theory of “inference for oneself” and “inference for others” and fallacies of inference including those of the thesis, reason, and instance. The second section chosen from Dharmakīrti’s Nyāyabindu deals with “Inference for oneself.” Dharmakīrti here tries to construct a comprehensive and tight framework of inference. Accordingly, hetu can be only of three kinds: svabhāva, kārya, and anupalabdhi. Dharmakīrti in this chapter also explains the 11-fold classification of anupalabdhi-hetu. The third section selected from Dharmakīrti’s Hetubindu discusses formal logical issues concerning the inference for others. Here Dharmakīrti contends that the statements of pakṣadharmatā and vyāpti are the only logically necessary and sufficient premises and that the statements of positive and negative concomitance

P. P. Gokhale (*) Department of Philosophy, Savitribai Phule Pune University, Pune, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_4

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entail each other such that only one of the two needs to be stated. This gives rise to two basic forms of inference which resemble Modus Ponuns and Modus Tollens.

Introduction In this chapter, three sections from three different texts are chosen which discuss some of the core doctrines in Buddhist logic. 1. The first section is Nyāyapraves´akasūtram, a short text which the tradition attributes to Diṅnāga. The text along with Haribhadrasūri’s Vŗtti and Pañjikā of Pārśvadeva was edited by A. B. Dhruva in 1968. It was then translated and commented upon in Hindi by Ranjan Kumar Sharma in 1999. There has been a controversy about the authorship of this text – Vidhushekhar Bhattacharya and M. Tubianski arguing that it is not the work of Diṅnāga but of some other author of Diṅnāga’s tradition (probably Śaṅkarasvāmin) and A. B. Keith (The Indian Historical Quarterly Vol. 6:1, 1928, pp. 16–22) countering their arguments and reclaiming the authorship of Diṅnāga. However, the importance of this text lies in the clear and precise way it presents the gist of Diṅnāga’s theory of inference. The translation of the text included here is authored by Pradeep Gokhale. 2. The second section is chosen from Dharmakīrti’s Nyāyabindu. Nyāyabindu is a systematic treatise which gives essence of Dharmakīrti’s theory of inference. The second and the third chapters of the text deal with “Inference for oneself” (svārthānumāna) and “Inference for others” (parārthānumāna), respectively. Here the chapter on svārthānumāna has been selected. The translation included here largely follows Mrinalkanti Gangopadhyay’s translation (as in Vinītadeva’s Nyāyabinduṭī kā, 1971) but at times deviates from the latter. 3. For the discussion of “Inference for others,” however, a section from Dharmakīrti’s another work, namely, Hetubindu, has been chosen. In Nyāyabindu, the account of inference has been given in an aphorismic and expository way. On the other hand in Hetubindu, it is given in an explanatory and argumentative way. The original text is not available in Sanskrit, but its restorations by Rahul Sankrityayan (included as Appendix 7 of the Hetubinduṭī kā, 1949) and Ernst Steinkellner (Dharmakī rti’s Hetubinduḥ, (Teil I), 1967) are available. Here an English translation by Pradeep Gokhale based on these restorations has been taken from Hetubindu of Dharmakī rti (A Point on Probans) by Pradeep P. Gokhale (1997), pp. 14 to 21. The author has edited and revised the translation while including it in this chapter.

Text I: Din˙nāga’s Nyāyapraves´akasūtram: Aphorisms Introducing the Theory of Reasoning Proof and refutation along with their fallacies are meant for the sake of others’ knowledge and perception and inference along with their fallacies are meant for one’s own knowledge.

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This is the collection of the topics discussed in this work. Out of them, proof means the statement of the thesis, etc., because the fact unknown to the inquirers is made known to them by the statements of thesis, probans, and instance. Out of them thesis refers to the property-bearer, which is desired to be itself proved as qualified by a well-known characteristic. Moreover, it should be uncontradicted by perception, etc. For example, “Word is eternal” or “Word is non-eternal.” Probans has three characteristics. What are these three characteristics? 1. Being the property of the thesis-subject 2. Existence in similar cases 3. Nonexistence in dissimilar cases What is the similar case and what is dissimilar case? Similar case is similar to the thesis by way of possessing the provable property as the common feature. For example, when Word is to be proved as impermanent, a thing such as a pot is a similar case. Dissimilar case is that where provable property does not exist. Whatever is permanent is seen to be a nonproduct like space. Here, the property of “being a product” or the property of “being produced through efforts” exists only in similar cases; it does not exist in dissimilar cases at all. Probans for proving impermanence, etc., can be explained in this way. [Diṅnāga here has stated the doctrine of the triple character of probans. Of course when he says that the probans should exist in similar cases, he does not mean that it should exist in all similar cases. What he means is that it should exist in at least a few similar cases. Though Diṅnāga does not make the point clear in this text, it is implied. Here Diṅnāga refers to two probanses for proving impermanence of Word: “being a product” and “being produced through efforts.” Out of them the first probans exists in all similar cases, whereas the second probans exists in a part of the class of the similar cases. Here the similar cases, namely, “the class of impermanent things” consists of the things like pot which are produced through efforts and also the things like lightning which is a natural product, not produced through efforts. In this way, the probans “being produced through efforts” presented for establishing impermanence of Word exists only in a few similar cases, not all. Even then it is a good probans.] Instance is of two kinds: one by similarity and the other by dissimilarity. Out of them, instance by similarity is that where the existence of probans in similar cases is shown. For example, whatever is a product is seen to be impermanent like a pot, etc. Instance by dissimilarity is that where the nonexistence of probans is stated where there is nonexistence of the provable property. For example, whatever is permanent is seen to be a nonproduct like space. Here, by the word “permanent,” “absence of impermanence” is stated. Similarly, the word “nonproduct” also indicates “absence of product-hood.” For example, existence is absence of absence. Hereby, we have stated thesis and other factors of proof.

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The statement of these three, that is, the thesis, the probans, and the instance, is called proof at the time of convincing others. For example, “Word is impermanent” is the statement of the thesis; “because it is a product” is the statement of the reason, that is, that of the property of the thesis-subject. “Whatever is the product is seen to be non-eternal like a pot, etc.” is the statement of positive concomitance in similar cases. “Whatever is eternal is seen to be a non-product, like space” is the statement of negative concomitance. Only these three are called limbs of inference. [In the above formulation of inference given by Diṅnāga, his use of the word “seen” (“dr¸ṣṭam”) is important. It implies that the positive concomitance (anvaya) and negative concomitance (vyatireka) refer to the observed world. In this sense, the inference as understood by Diṅnāga is a passage from “observed to unobserved.” In Dharmakīrti’s revised version of inference, the statement of concomitance is an observation-independent universal statement. Hence, Dharmakīrti drops the word “seen” in his formulation.] Fallacious thesis is that which is desired to be proved but which is contradicted by perception, etc. This is as follows: 1. 2. 3. 4. 5. 6. 7. 8. 9.

Contradicted by perception Contradicted by inference Contradicted by scripture Contradicted by public custom Contradicted by the own utterance The predicate term of which is unproved The subject term of which is unproved Both the terms of which are unproved The relation in which is well-known

1. “Contradicted by perception” is like the following: Word is inaudible 2. “Contradicted by inference” is like the following: The pot is eternal 3. “Contradicted by (one’s own) scripture” is like the following: When a Vaisesika would try to prove, “Word is eternal” 4. “Contradicted by public custom” is like the following: “Scull of human head is pious, because it is the limb of animal, like conch-shell and oyster-shell.” 5. “Contradicted by its own utterance” is like the following: “My mother is barren.” 6. “The predicate term of which is unproved” is like the following: A Buddhist argues to a Saṅkhya,“Word is perishable” 7. “The subject term of which is unproved” is like the following: A Saṅkhya argues to a Buddhist, “Soul is conscious.” 8. “Both the terms of which are unproved” is like the following: A Vaisesika argues to a Buddhist, “Soul is the material cause of pleasure, etc.” 9. “The relation in which is well known” is like the following: “Word is audible.”

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The statements of these amount to defective declaration (of the thesis) either because the statement is rendered impossible by way of denial of the nature of the provable property or due to superfluity of proving it. The fallacies of thesis have thus been stated. Fallacies of probans are: 1. Unproved probans 2. Inconclusive probans 3. Contrary probans Out of them, unproved probans is of four kinds as follows: 1. 2. 3. 4.

Probans unproved for both the debaters Probans unproved for one of the debaters Probans unproved because of doubtfulness Probans, the locus of which is unproved Out of them,

1. “Probans unproved for both” is like the following: When Word is to be proved impermanent, then “because Word is visible” is unproved for both. 2. “Unproved for one of the debaters” is like the following: When somebody argues against the one who accepts the manifestation of sound, “Word is non-eternal, because it is a product.” 3. “Unproved because of doubtfulness” is like the following: When combination of elements, about which there is a doubt that it is vapor, is being stated as a probans for proving fire. 4. “The locus of which is unproved” is like the following: When somebody argues against the one who accepts nonexistence of space (ether), “Ether is a substance because it is locus of qualities.” Inconclusive probans is of six kinds: 1. 2. 3. 4. 5. 6.

Common probans Uncommon probans Probans existing in a part of similar cases, but the whole of dissimilar cases Probans existing in a part of dissimilar cases, but the whole of similar cases The probans existing partly in both cases Probans concomitant with the contrary probandum Out of them

1. “Common probans” is like the following: Word is eternal, because it is knowable. Here, knowability is the inconclusive probans because it is common to both eternal and noneternal types of objects. It is inconclusive because it causes the

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doubt: Is Word impermanent because of knowability, like a pot, or it is permanent because of knowability, like ether? “Uncommon probans” is like the following: Word is eternal, because it is audible. This is because audibility being excluded from both eternal and noneternal types of objects, and because the thing cannot be excluded from both the categories, namely, eternal and noneternal, audibility becomes the cause of doubt as to how the Word becomes audible due to eternality or noneternality. “Probans existing in a part of similar cases, but the whole of the dissimilar cases” is like the following: Word is not caused by efforts, because it is impermanent. Here, that Word is not caused by efforts is to be proved. Lightening, ether, etc., are similar cases. Out of these similar cases, impermanence exists in the things like lightening but not in the things like ether. Again, (in this inference) that Word is not caused by efforts is to be proved. For that, the things like pot are dissimilar cases. In all these dissimilar cases, namely, pot, etc., impermanence exists. Hence, impermanence becomes inconclusive because of its similarity with lightening and pot. Here the doubt arises as to whether Word is made by efforts because it is impermanent like a pot, or it is not made by efforts because it is impermanent like lightening, etc. “Probans which exists in a part of dissimilar cases, but the whole of similar cases” is like the following: Word is made by efforts, because it is impermanent. Here, that Word is made by efforts is to be proved. Pot, etc., are its similar cases. Impermanence exists in all these cases like pot. Now, (in this inference) “the pot is made by efforts” is to be proved. Its dissimilar cases are lightening, ether, etc. Out of them, impermanence exists in one part, namely, lightening, etc., but not in ether, etc. Therefore, this probans is also inconclusive like the earlier one due to its similarity with lightening and pot. “The probans existing partly in both cases” is like the following: Word is eternal, because it is intangible (or having unlimited size). What is to be proved (in this inference) is that Word is eternal. Its similar cases are ether, atom, etc. Out of them, amūrta-hood (unlimited size) lies in one part, namely, ether, etc. It does not lie in atom. Now, again, eternality is to be proved (in this inference). Its dissimilar cases are pot, pleasure, etc. Out of them amūrta-hood exists in one part, namely, pleasure, etc., but it does not exist in pot, etc. Therefore, this probans too is inconclusive because of dissimilarity with pleasure and ether. [In the above example, the meaning of the term amūrtatva is not clear. Rather two meanings of the term seem to be mixed up in the above example. The word mūrta has two senses (1) Tangible: That which can be known through external senses. (2) The thing which has limited size. Amūrta is the opposite of mūrta. Now, Ether is amūrta in both the senses: It cannot be perceived through external senses and it has unlimited size. Atom (paramāṇu) is mūrta in the second sense because it has limited size. And Diṅnāga also treats atom as mūrta. (But it is not mūrta in the first sense. It is not perceptible through external senses. Diṅnāga seems to ignore this aspect.) But pleasure cannot be measured by size. So the question whether it is mūrta in the second sense does not arise. But it can be called amūrta in the first sense, because it cannot be cognized through external senses.]

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6. “Probans concomitant (also) with the contrary probandum” is like the following: Word is noneternal, because it is a product like a pot; Word is eternal, because it is audible like soundness. These two probanses taken together is one inconclusive probans, because they together cause a doubt. Contrary probans is of four kinds as follows: 1. The probans which proves opposite of the nature of the provable property. 2. The probans which proves opposite of a specific character of the provable property. 3. The probans which proves opposite of the nature of the property bearer. 4. The probans which proves opposite of a specific character of the property bearer. Out of them 1. “The probans which proves opposite of the nature of the provable property” is like the following: Word is permanent, because it is a product or because it is produced through efforts. This probans is contrary probans, because it exists only in dissimilar cases. 2. “The probans which proves opposite of a specific character of the provable property” is like the following: The things like visual sense organ are meant for something else, because they are composite, like the specific bodies, such as bed and seat. This probans proves the other-dependent character of visial sense organs, etc. But it also proves composite nature of that other, namely the self, because this probans is concomitant which both. 3. “The probans which proves opposite of the nature of the property bearer” is like the following: Existence is neither a substance not a motion nor a quality because it resides in a single substance and it also resides in qualities and motions, like specific universals. This probans proves that existence is not a substance, etc., but it also proves that existence is absence, because the probans is concomitant with both. 4. “Probans which proves opposite of a specific character of the property bearer” is like the following: The same probans in the same thesis-subject which is stated above proves “causing the experience of non-existence,” which is opposite of the specific characteristic of the property-bearer, namely, “causing the experience of existence” because it is concomitant with both. [Out of the four types of Contrary probans mentioned above, the first one is basic according to which the probans exists only in dissimilar cases and none of the similar cases. Elsewhere Diṅnāga refers to two subtypes of this contrary probans: (1) the probans which exists in a few dissimilar cases but in none of the similar cases and (2) the probans which exists in all dissimilar cases but in none of the similar cases. In fact, the two types of contrary probans mentioned above along with the first five types of inconclusive probans and the two types of good probans which make

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the total of nine types of good or bad probans constitute what Diṅnāga calls the wheel of probans (Hetucakra). The wheel of probans is a ninefold classification of probans, which is the property of the thesis-subject (pakṣadharma). That is, it fulfills the first condition of a good probans, but it may or may not fulfill the other two conditions. Hence, the property of the thesis subject (pakṣadharma): • May exist in all similar cases, only a few similar cases, or in no similar case • May exist in all dissimilar cases, only a few dissimilar cases, or in no dissimilar case The combinations of these possibilities yield nine varieties, which can be presented in tabular form as given in the Hetucakra-Table below. Position with respect to dissimilar cases Position with respect to similar cases I. Resident II. Nonresident III. Resident and nonresident

(a) Resident

(b) Nonresident

(c) Resident and nonresident

Ia: Inconclusive IIa: Contrary

Ib: Good

Ic: Inconclusive

IIb: Inconclusive IIIb: Good

IIc: Contrary

IIIa: Inconclusive

IIIc: Inconclusive

(Hetucakra: Table)]

Fallacious instance is of two kinds, one by similarity and one by difference. Fallacious instance-by-similarity is of five kinds, as the following: 1. 2. 3. 4. 5.

Where the probans is unproved Where the provable property is unproved Where both the properties are unproved Where positive concomitance is absent Where the converse of positive concomitance is present There

1. “Where the probans is unproved” is like the following: “Word is eternal, because it has unlimited size like an atom. Whatever is seen as having unlimited size is seen to be eternal like an atom.” In this example, eternality, which is the probandum, exists in atom, but the probans namely having unlimited size does not exist there, because atoms have limited size. 2. “Where the provable property is unproved” is like the following: “Word is eternal, because it is intangible like cognition. Whatever is seen to be intangible is seen to be eternal like cognition.” In this example, intangibility, that is the probans, resides in cognition, but eternality, that is, the provable property does not exist there, because cognition is noneternal.

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3. “Where both the properties are unproved” is of two kinds: real and unreal. Out of them, “like a pot” as instance is real and it is that in which both the properties are unproved. The second example would be “like space.” There the instance is unreal according to the one who does not accept reality of space, and it is also the one in which both are unproved. 4. “Where positive concomitance is absent” is like the following: It is that in which coexistence of probandum and probans is shown without positive concomitance, for example, “In a pot both product-hood and non-eternality are seen” 5. “Where the converse of positive concomitance is present” is like the following: It is where the debater says, “Whatever is seen to be non-eternal is seen to be a product,” when he should have said, “Whatever is a product is seen to be non-eternal.” “The fallacious instance-by-difference” is of five kinds like the following: 1. 2. 3. 4. 5.

Where absence of probandum is not there Where absence of probans is not there Where absences of both are not there Where negative concomitance is not there Where the converse of negative concomitance is present Out of them

1. “Where absence of probandum is not there” is like the following: “Word is eternal, because it has unlimited size unlike an atom. Whatever is seen to be non-eternal is seen to have limited size, for example atom.” In this example, the probans namely having unlimited size (amūrtatva) is absent from the atom, because atoms have limited size, but the probandum, namely, eternality is not absent from atoms, because atoms are eternal. 2. “Where absence of probans is not there” is like the following: “unlike motion.” Here eternality is probandum which is absent from motion, because motion is not eternal. But intangibility (amūrtatva), which is the probans, is not absent from the motion because motion is intangible. 3. “Where absence of both is not there” is like the following: “unlike space” – When this is presented as an instance to one who accepts existence of space. Both “eternality” and “having unlimited size” are not absent from space, because space is eternal and has unlimited size. 4. “Where negative concomitance is not there” is like the following: It occurs where the absence of probans and probandum in a dissimilar case is shown without indicating absence of probans and probandum (in general). For example, one says: “Tangibility and non-eternality both are seen in a pot.” 5. “Where the converse of negative concomitance is present” is like the following: “One says that whatever is seen to be tangible is seen to be non-eternal,” when one should have said “Whatever is non-eternal is seen to be tangible.”

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The statements of fallacious thesis, probans, and instance are called fallacies of proof. For the sake of one’s own knowledge, however, there are only two means of knowledge – perception and inference. Out of them, perception is that cognition which is devoid of mental construction, with respect to the object such as form (rūpa) without the construction of name, universal, etc. It is called pratyakṣa in the sense that it occurs with respect to the respective sense organ (akṣa). Inference is the cognition of an object based on the probans. We have said that probans has three characteristics. The cognition, which arises with regard to the inferable object from the probans of that kind, such as “There is fire here” or “Word is impermanent,” is inference. In both the cases, the cognition itself is the result, because it is of the nature of determinate cognition. It is also pramāṇa (means to knowledge), because it is cognized as having an operation. Constructive cognition which occurs regarding a different object (from what is given) is called pseudo-perception. That is, the cognition which arises in constructing mentally ‘a pot’ or ‘a cloth’ is a pseudo perception, because it does not have a unique particular as its object. The cognition caused by fallacious probans is pseudo-inference. We have stated various kinds of fallacious probans. The cognition which an ignorant person has with reference to the inferable object from the fallacious probans is a pseudo-inference. Refutation means pointing out a defect in a proof. The defects (possible) in a proof are: 1. Deficient-ness (incompleteness) or 2. The defect in thesis such as “contradicted by perception,” etc., or 3. Defect in probans, namely, “unproved probans” “Inconclusive probans,” or “contrary probans,” or 4. Defect in instance, such as “(the instance-by-similarity,) the probans in which is unproved.” Pointing it out and convincing the questioner (the judge) is refutation. Pointing out the defect in proof when the defect is not there is pseudo-refutation. For example, 1. 2. 3. 4. 5. 6.

To say that the proof is deficient (that is, incomplete), when the proof is complete To say that there is a defect in the thesis when the thesis is nondefective To say that here there is an unproved probans, when the probans is proved To say that the probans is inconclusive, when actually the probans is conclusive To say that the probans is contrary, when he probans is not contrary To say that the instance is defective, when the instance is not defective

These are the pseudo-refutations, because the opponents’ thesis is not refuted by them, because the thesis is faultless. We stop here. In this work, we have stated the categories only to give the initial direction. Whatever is justified or nonjustified with respect to this is explained well by us elsewhere. The text called “aphorisms introducing the theory of reasoning” is complete.

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Text II: Chapter Two of Nyāyabindu: “On the Inference for Oneself” [The second text has been chosen from Nyāyabindu, a lucid and concise work of Dharmakīrti on the theory of Pramāṇas. The work consists of three chapters. In the first chapter, Dharmakīrti introduces the notion of right cognition (samyak-jn˜ āna), that is, pramāṇa and then classifies it into two kinds, perception and inference. He then defines perception and also its fourfold classification. The Second and third chapter are devoted to the discussion of inference. The second chapter, which is now being presented in translation, deals with “inference for oneself” with its different aspects.] [2.1:Two Types of Anumāna] Inference is twofold – for one’s own sake (svārtha) and for the sake of others (parārtha). [2.2: Definition of Svārthānumāna] Of these, a svārthānumāna is knowledge which is produced through a reason property [Literally, mark (liṅga)] having three characteristics (trirūpa) and pertains to an “inferable object” (anumeya). Here [i.e., in the case of inference] also the arrangement (vyavasthā) with regard to the effect [produced by] an instrument of right cognition and [the instrument itself] is just the same as in the case of perception. [2.3: Trirūpaṁliṅgam: Triple Character of the Reason Property] The three characteristics of a mark are (1) its definite (eva)presence in the inferable object, (2) its presence only (eva) in similar cases (sapakṣa), and (3) its definite (eva) absence in dissimilar cases (asapakṣa). And [all these must be] properly ascertained. [2.4: Definitions: Anumeya (Pakṣa), Sapakṣa, Asapakṣa] An inferable entity here means a property-bearer (dharmin) – the property of which is “sought to be known” ( jijn˜ āsita). /5/ A similar case is an object which is similar (samāna) through the possession of the “general inferred property” (sādhya-dharma-sāmānya). A case which is not similar is dissimilar – [it can be] different from it, contrary to it or its absence. [2.5: Trī ṇiliṅgāni: Three kinds of Reason Property] There are only three kinds of the reason property, which have the three characteristics. [The three kinds of reason are] nonapprehension (anupalabdhi), own nature of a thing (svabhāva), and effect (kārya). [2.6: Anupalabdhi-hetu: Nonapprehension as Reason] Of these, a case of nonapprehension, for example, is: There is no jar on a particular spot somewhere, because there is non-apprehension (anupalabdhi) of “what fulfils the conditions of cognisability (upalabdhilakṣaṇa-prāpta).” The fulfillment of the conditions of cognizability consists in the presence of the totality of the conditions (pratyaya) of cognition (upalambha) other than [the object itself] and “a distinct nature (of the object)” (svabhāva-vis´eṣa). The distinct nature is that (by which) the object, if existent, is necessarily perceived – when all the other conditions of cognition are available. [2.7: Svabhāva-hetu: Own Nature as Reason] Own nature (of an object) is the reason in relation to an inferable property which exists merely on account of its own [i.e., of the reason] existence. For example: This is a tree, because it is a s´iṃs´apā. [2.8: Kārya-hetu: Effect as Reason] An effect (kārya) as reason is, for example, there is fire here, because there is smoke.

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P. P. Gokhale [2.9: Justification of the Three Kinds of Reason] Of these three, two [forms of the reasons] establish a positive entity (vastu) and the other is the cause for establishing negation (pratiṣedha). A thing can indicate [the existence of] another thing only when there is a “connection through one’s intrinsic nature” (svabhāva-pratibandha). Because there can be no invariable and necessary concomitance in the case of one that is not so connected [i.e., through one’s intrinsic nature]. Such a connection is [a connection] of the reason property with the inferable object and [it is established] on the ground that [the reason property] is actually identical with the inferable entity or that [the reason property] actually originates from the inferable entity. That is because an entity [i.e., the reason property] which is not identical in nature with it or which does not originate from it cannot be necessarily and invariably connected with it [i.e., with the inferable entity]. These relations of identity and causation belong respectively to [a reason property] which has an identical nature and that which is an effect, and that is why the existence of a positive entity is proved by them. The establishment of negation also is accomplished by the same nonapprehension as has just been mentioned. Because there can be no such nonapprehension when an entity is actually present. Because, otherwise in the case of objects which are “temporally, spatially or by nature inaccessible” (des´a-kāla-svabhāva-viprakṛṣṭa) and which do not fulfill the conditions of cognizability – there can be no ascertainment of an absence even when one’s own perception has ceased to function.

[Dharmakīrti here has tried to construct a comprehensive and tight framework of inference. Given any situation of inference, there are two objects – sādhya and hetu. Sādhya is known on the basis of hetu. For that there has to be necessary relation between the two such that hetu cannot be there without sādhya. Dharmakīrti calls this relation svabhāvapratibandha. It is a binding relation based on the own nature of things. Now there are two possibilities: sādhya is of the nature of existence or nonexistence. If it is of the nature of nonexistence, then it can be known through nonperception as hetu. If it is of the nature of existence, then it could be related with hetu in either of the two ways. They may be two distinct entities or two inseparable entities. If hetu and sādhya are two distinct entities, then the necessary relation between them of the form “hetu cannot be there without sādhya” is possible only if the relation is that of causal necessity. For example, smoke and fire are two distinct entities, but it is possible to infer the existence of fire from smoke. We can say that smoke cannot be there without fire only because smoke is caused by fire. If on the other hand, hetu and sādhya are inseparable entities, then they must be two aspects of one and the same thing such that the thing cannot have hetu aspect without having sādhya aspect. For example, s´iṁs´apā and tree are two aspects of one and the same thing such that a thing cannot be a s´iṁs´apā without being a tree. The issue of nonapprehension as hetu is more complex. Nonexistence of a thing can be inferred on the basis of its nonperception only if the thing is perceptible. If the thing is imperceptible, that is, if it is not remote in space and time but remote by its very nature, then its nonexistence cannot be established by nonperception. But Dharmakīrti also claims that nonexistence of a thing can also be established

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by existence of a thing which is incompatible with it. For example, fire and cold are incompatible with each other. So if one experiences fire which is capable of removing cold, one can infer that there is absence of cold there. Another point to be noted is that although Dharmakīrti talks of nonexistence as the object of inference, he does not mean thereby that nonexistence exists in the same way as a positive entity exists. When one sees no pot on the floor, one may describe the situation as “there is no pot on the floor.” At that time there is a negative description of floor, but no entity called “absence of a pot.” Hence, through nonapprehension, a negative usage (abhāvavyavahāra) is established according to Dharmakīrti.] [2.10: Explanation of Anupalabdhi-hetu] A negative usage (abhāva-vyavahāra) is established by nonoccurrence (nivṛtti) of the perception of a knower (pratipattṛ) referring to the past or the present provided the “reminiscent impression” (smṛti-saṃskāra) has not been obliterated (amūḍha). That is because an absence is ascertained exclusively on the basis of such [nonoccurrence of perception]. [2.11: Eleven kinds of Anupalabdhi-hetu] This [nonapprehension] is of eleven kinds according to the difference in the formulation. 1. [Svabhāvānupalbdhi]: The example of the “non-apprehension of the own nature of an object”: “Smoke does not exist here, because there is the non-apprehension of what fulfils the conditions of cognisability [i.e., of smoke which is a perceptible object].” 2. [Kāryānupalabdhi]: The example of the “non-apprehension of the effect”: “The efficient (apratibaddhasāmarthya) causes of smoke do not exist here, because there is no smoke.” 3. [Vyāpakānupalabdhi]: The example of the “non-apprehension of the pervader”: “A s´iṃs´apā does not exist here, because there is no tree.” 4. [Svabhāva-viruddhopalabdhi]: The example of “apprehension of something incompatible with the own nature of an object (svabhāva-viruddha)”: “Sensation of cold does not exist here, because there is fire.” 5. [Viruddha-kāryopalabdhi]: The example of “apprehension of an effect produced by something incompatible with the object”: “Sensation of cold does not exist here, because there is smoke.” 6. [Viruddha-vyāptopalabdhi]: The example of “apprehension of something pervaded by what is incompatible with the object”: “There is no certainty of destruction of even a produced entity because it (=the destruction) depends on other causes.” 7. [Kārya-viruddhopalabdhi]: The example of “apprehension of what is incompatible with the effect produced by the object”: “The efficient causes of cold do not exist here, because there is fire.” 8. [Vyāpaka-viruddhopalabdhi]: The example of “apprehension of what is incompatible with the pervader of the object”:“The touch inhering in ice does not exist here, because there is fire.” 9. [kāraṇānupalabdhi]: The example of “non-apprehension of the cause produced by the object”: “Smoke does not exist here, because there is no fire.” 10. [Kāraṇa-viruddhopalabdhi]: The example of “apprehension of what is incompatible with the cause of the object”: “The person does not betray any special symptom of cold – e.g., having erect hair, etc. – because there is a specific form of fire nearby.” 11. [Kāraṇa-viruddha-kāryopalabdhi]: The example of apprehension of the effect produced by what is opposed to the cause: “This spot is not characterised by a person who has the special symptoms of cold such as having erect hair and the like, because there is smoke.”

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P. P. Gokhale [2.12: Explanation of the 11-fold Classification] All these ten forms of formulation based on nonapprehension – beginning with kāryānupalabdhi – can be included indirectly into [the first form, namely,] svabhāvānupalabdhi. Although there is difference [in the nature of] the various forms of formulation, because they either affirm or negate something else, still, [such inclusion is possible] in an indirect way. [These formulations] are specially mentioned here – even under the discussion on inference for one’s own sake – because due to repeated consideration of the formulations one can oneself have the ascertainment of an absence in a similar way [i.e., just in any of the forms stated above]. In these various cases of nonapprehension which is the cause of a negative usage, the absence is said to be proved by either the apprehension of the incompatible, etc., or the nonapprehension of the cause, etc. [2.13: Significance of the Condition of Cognizability] In all these cases, it is to be understood that such apprehension or nonapprehension means only [the apprehension or nonapprehension of the incompatible or the cause] which actually fulfills the condition of cognizability. That is because, in the case of others, (that is, if the condition of cognizability is not fulfilled,) the existence and nonexistence of incompatibility (virodha) as well as the causal relation are not established. In the case of an inaccessible object, the nonapprehension, which is characterized by nonoccurrence of perception and inference, is only a source of doubt. That is because the nonoccurrence of pramāṇa does not necessarily prove the absence of an object.

[An Explanation of the 11-fold Classification of Nonapprehension As Dharmakīrt himself clarifies, among these eleven kinds svabhāvānupalabdhi is basic and all others are based on it. In fact, he could have said that svabhāvānupalabdhi and svabhāvaviruddhopalabdhi are basic. These two are governed by two basic principles: 1. The basic nonapprehension principle (svabhāvānupalabdhi) is: If an object A fulfills the condition of cognizability, but it is not apprehended in a particular place at a particular time, then A does not exist there at that time. 2. The basic incompatibility principle (svabhāvaviruddhopalabdhi) is: If A and B are two mutually incompatible objects, and A is apprehended at a particular place at a particular time, then B does not exist there at that time. These are the two basic forms of apprehension or nonapprehension which give the knowledge of absence of a thing. By applying other principles we get the remaining varieties: 3. The principle of cause as sufficient condition: If a cause has unobstructed power to produce the effect, then it can be called the sufficient condition (which is called efficient cause above) of the effect. In such a case, nonexistence of effect implies the nonexistence of its cause. This principle is used along with the basic nonapprehension principle in the second type of anupalabdhi-hetu (“Kāryānupalabdhi”) above. Similarly, this principle is used along with the basic incompatibility principle in the seventh type of anupalabdhi-hetu (“Kārya-viruddhopalabdhi”) above.

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4. The principle of pervasion: When there is pervaded-pervader relation (vyāpyavyāpaka-sambandha) between two objects, the pervaded object implies the pervader-object and the negation of the pervader-object implies the negation of the pervaded object. These can be called the assertive and the negative versions of the principle of pervasion. For example, between “being a Śiṁśapā”(=S) and “being a tree”(=T), S is the pervaded property and T is the pervading property. Here S implies T and negation-T implies negation-S. The negative version of this principle is used along with the basic nonapprehension principle in the third type of anupalabdhi-hetu (“Vyāpakānupalabdhi”) above. The assertive version of this principle is used along with the basic incompatibility principle in the sixth type of anupalabdhi-hetu (viruddhavyāptopalabdhi) above. The negative version of this principle is used along with the basic incompatibility principle in the eighth type of anupalabdhi-hetu (Vyāpaka-viruddhopalabdhi) above. 5. The principle of cause as a necessary condition: A cause in Indian philosophical tradition is generally understood as a necessary condition of the effect. Fire is the cause of smoke in this sense. In such cases, the effect implies the existence of its cause and nonexistence of a cause implies nonexistence of its effect. These can be called the assertive form and the negative form of the principle, respectively. In the fifth type of anupalbdhi-hetu above (“viruddhakāryopalabdhi”), the assertive form of this principle is used along with the incompatibility principle. In the ninth type of anupalabdhi-hetu above (“kāraṇānupalabdhi”), the negative form of this principle is used along with the basic nonapprehension principle. Similarly, the negative form of this principle is used along with the basic incompatibility principle in the tenth type of anupalabdhi-hetu (“kāraṇa-viruddhopalabdhi”) above. And lastly in the eleventh type of anupalabdhi-hetu (“Kāraṇa-viruddha-kāryopalabdhi”), both the assertive and negative versions of this principle are used along with the basic incompatibility principle.]

Text III: A Section from Hetubindu: On the Nature of a Valid Argument [The third text chosen for this chapter is a section from another work of Dharmakīrti, namely Hetubindu. The main focus of the work is the nature of reason – what makes the reason a sound reason, what the constituents of a sound reason are, and what the types of sound reason are. All these issues are important in the context of “inference for oneself” and Dharmakīrti duly discusses them in that context. They are included in the second chapter of Nyāyabindu, which was the previous section of this chapter. But an important question in the context of “inference for others” is:“How is a sound reason to be presented in an argument? What is a valid form of an argument (or what are the valid forms of arguments)?.” The following passage from Hetubindu deals specifically with this issue.]

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P. P. Gokhale [Three Kinds of Reasons] The reason possessing these distinctive characteristics, is only threefold; that is, it is only of three kinds: own-nature, effect and non-apprehension. For example, (1) Realness (is an own-nature kind of reason) when something is to be proved as impermanent; (2) Smoke (is an effect kind of reason when some region (is to be proved as) possessing fire. (3) Non-apprehension of a thing which fulfils the condition of apprehensibility (is a non-apprehension kind of reason) when the absence (is to be proved). (These are the only three kinds.) That is because the non-variance relation is limited only to the reason of these three kinds. The non-variance relation means the pervasion of the reason-property of the provable property-bearer (with the provable property) as has been stated above. It is called limited (Niyata) to this because it does not exist in any case other than the threefold reason. [Explanation of Own-nature as Reason (Svabhāva-hetu)] Among these (three kinds of reason), own-nature is the reason when the provable property is concomitant with mere existence of the reason-property. The provable property is in reality the own-nature of the reason, although the properties differ because of differences between their exclusions from different things. The qualification in the above definition namely, “concomitant with mere existence ...” is stated in response to the opinion of others, (and is not really required) because there will be no variability in the positive concomitance when the provable property is the own-nature of the probans. The reason (why the qualification is stated) is that the opponents accept as own-nature even that which is conditioned by some other object and which is also not implied by its (i.e., reason’s) very existence. By making the above qualification one points out that a thing of that kind (i.e., a thing that is conditioned by some other object) is not the own-nature of that (reason) and that the reason has a variable relation with such a provable property. For example, product-hood has (variable relation) with caused destruction. [Forms of the Argument from Own-nature:] The (argumentative) expression of it (=the own-nature kind of reason) is of two kinds, One is through similarity and the other is through difference. For instance; [An expression based on similarity:] All that is real is momentary. For example, the things like a pot. And sound is real. And [An expression based on difference:] There is the absence of realness where there is the absence of momentariness. And sound is real. The (argumentative) expressions (of the self-nature kind of reason) based on similarity and difference, which are characterised by the presentation of all-subsuming pervasion, have (thus) been stated.

[Note: The two basic forms of argument Dharmakīrti is introducing here are similar to Modus Ponuns and Modus Tollens, respectively. Of course in Dharmakīrti’s formulation only premises are stated and conclusion is not stated, because according to Dharmakīrti it follows necessarily from the premises and hence is redundant.] [Steps in an Argument: Praijn˜ā (Declaration) Is Redundant.] Here (in the above examples) we have not indicated statement of Declaration, because the content of a Declaration is cognized from the force of the statement of (1) the (reason, being a) property of the property-bearer and (2) the relation (of pervasion). [Objection:] How can the object of (inferential) knowledge be cognized if it is not indicated (by stating the Declaration)? [Answer:] Who indicates the object of (inferential) knowledge when one knows (inferentially) by oneself? When someone who sees smoke in the region recollects its

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pervasion with fire, he knows by the very force of it the content of the Declaration “Here there is fire.” Nobody indicates to him that there is fire there. Nor is it the case that he himself knows something (namely, the content of his Declaration) beforehand, because without an authentic means to knowledge, there is no reason for his knowledge (of the provable property). And in case he knows (the content of the Declaration beforehand), then redundancy of the reason follows. What is this sequence of knowledge such that after settling by oneself and from no source, that there is fire, one follows a reason for knowing it? Even if it (=the content of the Declaration) is stated by someone else, it gets washed away, as that is of no use. If the indication of the subject is the use of it, then (we ask,) what is the use of that being indicated? All this would be fair if the (inferential) knowledge is not possible otherwise. Therefore this fellow (i.e., the Naiyāyika) who in the case of his own knowledge knows the inferential object without anyone to present it (separately), requires a manifest price (viz. Declaration) when he sees us in need (to make an inference for others), (as he is) like a Brahmin performing seasonal rites. [For his own rites he needs no expenses but he demands positive fees from needy persons for whom he has to perform the rite]. Now even if we state (the Declaration), he knows the probandum (not from our statement, but) by following the mark (i.e., probans) known by himself. What is the difference between the two states of affair then? And we have seen that the (inferential) knowledge takes place even without Declaration from the argumentative expression based on similarity, etc. Which merely consists of the statement of (two things:) (1) (the reason being) the property of the provable property-bearer (2) the relation (of pervasion). What is the use of the Declaration then? A proof (i.e., inference for others) is meant for creating a determinate cognition in others like that which he (=the inferer) himself has. Now what is the reason for him to follow a novel order while conveying his piece of knowledge to others when he himself knows without the object of knowledge being presented? No purpose, therefore, is served by the statement of the object of knowledge, because (inferential) knowledge arises even without that. [Steps in an Argument: Upanaya and Nigamana serve no purpose.] The same (line of argument) relates (the alleged necessary steps in an argument, viz.) Application (Upanaya), Conclusion (Nigamana), etc., because the knowledge of the probandum arises only from this (i.e., from the statement of the probans’ being the property of the provable property bearer and pervasion). Leave your attachment to (the behaviour of) a naked master (“ḍiṇḍika-rāga”), shut your eyes and think whether the knowledge will arise from this much or not. If it will, what is the use of a series of (unnecessary) verbal constructions? Therefore, the argumentative) expression only of this size (i.e., the one consisting of the two steps) is preferable in a statement of proof.

[This is Dharmakīrti’s response to the Nyāya view according to which an argument should contain five steps: (1) Pratijn˜ ā (Declaration of the thesis to be proved, (2) Hetu (Reason), (3) Udāharaṇa (Instance and also the statement of pervasion according to later Naiyāyikas), (4) Upanaya (Application of instance or pervasion to the present case), and (5) Nigamana (Conclusion). Contrariwise according to Dharmakīrti only two steps are necessary and sufficient: (1) Pakṣadharmatā (The statement of the existence of the reason property in the provable property bearer) and (2) Vyāpti (The statement of pervasion). The exact significance of the term ḍiṇḍika-rāga is not clear. But probably what Dharmakīrti means is that while deciding whether a particular step in an argument is logically necessary or not one should not try to adhere to what one’s teacher says. One has to appeal only to one’s power of reasoning.]

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P. P. Gokhale [The order of the premises is logically insignificant] Here there is no restriction regarding the sequence (of the steps of an argument, which states that): the reason should be expressed first and then the instance. That is because in either case it (=the argument) causes (inferential) knowledge.

[This is Dharmakīrti’s response to the Nyāya view that the steps in an argument should be stated in the fixed order as: Pratijn˜ ā-Hetu-Udāharaṇa-Upanaya-Nigamana. If in a debate, a debater presents an argument by following a different order, then the debater, according to Nyāya, will commit an occasion of defeat called “aprāptakāla.” According to Dharmakīrti logically it makes no difference if you state pakṣadharmatā first and vyāpti next or vice versa. Dharmakīrti makes the same point while criticizing the Nyāya theory of occasions of defeat in his work Vādanyāya.] [The positive pervasion and negative pervasion are equivalent] There is no difference in the meaning (of the two forms, namely positive and negative,) in the statement of the relation (i.e., pervasion) also, because in both cases it conveys its (=reason’s) being the same thing as that (=provable property). And this is because, if the reason (in the inference based on own-nature) is not identical with the provable property, the existence of the latter will not conclusively follow from that of the former. Nor will the former be excluded when the latter is excluded. For example, (there is no conclusive existence of) the property of “being invariably related with effort,” when product-hood exists. Similarly, in the case other than that of the own-nature, when the effect is absent, that (= its other relatum, i.e., the cause) is not excluded. When, for example, there is exclusion of one out of the two (namely, the property of being invariably related with effort), the exclusion of the other (namely, product-hood) does not follow.

[Here Dharmakīrti is underlining the rule of transposition according to which the statements of the form “If A then B” and “If not-B then not-A” are logically equivalent. Similarly, he is pointing out that the two vyāpti-relations, namely, identity and causation are mono-directional or unilateral and not bi-directional or bilateral. For example, one can say that “being śiṁśapā” by its own-nature amounts to “being a tree,” but one cannot say that “being a tree” by its own nature amounts to “being a śiṁśapā.” Similarly “being invariably related to efforts” amounts to being impermanent, but being impermanent does not amount to being invariably related to efforts. (For example, lightning is impermanent, but it is not connected with anybody’s efforts.) That is why for going from anvaya to vyatireka or vice versa, we have to alter the order of the antecedent and the consequent. That is how the same mono-directional identity relation is expressed through anvaya statement as well as vyatireka statement.] [Any one type of pervasion should be stated, not the both] Therefore, even when one out of the two concomitances, namely positive and negative, as they were defined before, is expressed, it implies the other. Therefore, both should not be expressed in a single statement of proof, as it is redundant. When positive concomitance is proved through one (=provable property) being the very nature of the other (namely the reason property); that the former is absent when the latter is absent is also proved. Similarly, when that (i.e., negative concomitance) is proved, positive concomitance is also proved. [The possible objection of the Opponent]: The (separate) expression of negative concomitance is for indicating the rule that reason should be absent only from the cases where

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there is the absence of that (=the provable property) and not from the cases other than or contrary to the bearer of the provable property. [Answer]: This cannot be the justification for the (separate) expression of negative concomitance. Because, the cases other than as well as the contrary to the bearer of the provable property are (nothing but) dissimilar cases.

[In Dharmakīrti’s scheme, the two conditions, namely, “existence of Reason property only in sapakṣa” and “non-existence of it anywhere in vipakṣa,” when they are understood not as existential statements, but expressions of a necessary rule, imply each other. For that he is defining vipakṣa or asapakṣa as a wider concept, as an external negation of sapakṣa such that it includes something dissimilar to sapakṣa, something other than sapakṣa as well as something contrary to sapakṣa.]

Definitions of Key Terms Pakṣa

Sādhya (Probandum)

Hetu (Probans)

Sapakṣa (Similar cases) Vipakṣa or Asapakṣa (Dissimilar cases)

Anvaya-vyāpti (Positive pervasion or Positive concomitance) Vyatireka-vyāpti (Negative pervasion or Negative concomitance)

(1) Locus or property-bearer in which a property is inferentially claimed to exist. (2) Thesis Target property: The property which is inferentially claimed to be existent in Pakṣa. Reason property: A property which provides the reason or ground for the existence of Sādhya in Pakṣa. A set of property-bearers similar to Pakṣa in that they possess Sādhya. A set of property-bearers dissimilar to Pakṣa in that they do not possess Sādhya. Positive invariable relation between Hetu and Sādhya of the form: “Wherever there is Hetu, there is Sādhya.” Negative invariable relation between Hetu and Sādhya of the form: “Wherever there is absence of Sādhya, there is absence of Hetu.”

Summary The first section of the chapter throws light on how an inference when presented in the form of an argument consists of three aspects: thesis, reason, (probans) and instance. It also shows how one can distinguish between good thesis and bad thesis,

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good probans and bad probans, and between good instance and bad instance. Diṅnāga shows with illustrations that a probans in order to be a good probans should be the property of the thesis subject, should exist in at least a few similar cases, but it should not exist even in a single dissimilar case. The second section exposes Dharmakīrti’s theory of inference for oneself. According to Dharmakīrti, hetu which possesses three characteristics is only of three kinds: self-nature, effect, and nonapprehension. That is because hetu can have necessary relation with sādhya only in three ways. The third section indicates the way in which Dharmakīrti in his theory of inference for others was aware of the rules of formal logic which the inference has to adhere to. Dharmakīrti hints at the rules such as Modus Ponuns and Modus Tollens, Transposition between the statements of positive and negative concomitance and idea that no redundant steps should be presented in an argument.

References Dhruva, A.B., ed. 1968. Nyāyapraves´a. Baroda: Baroda Oriental Research institute. Gangopadhyay, Mrinalkanti, Trans. 1971. Vinī tadeva’s Nyāyabinduṭī kā (Sanskrit original reconstructed from the extant Tibetan version with English Translation and annotations). Calcutta: Indian Studies Past and Present. Gokhale, Pradeep P., Ed. and trans. 1997. Hetubindu of Dharmakī rti (A point on probans). Delhi: Sri Satguru Publications. Sankrityayan, Rahul, ed. 1949. Hetubinduṭī kā, G. O. series, no. CXIII. Baroda: Oriental Institute. (Appendix 7). Sharma, Ranjankumar, ed. 1999. Nyāyapraves´akasūtram. Sarnath: Central Institute of Higher Tibetan Studies. Steinkellner, Ernst. 1967. Dharmakī rti’s Hetubinduḥ, (Teil I). Wein: Herman Böhlaus Nachf.

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General Introduction to Logic in Jainism with a List of Logicians and Their Texts Jayandra Soni

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ages of Jaina Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lists of Jaina Logicians and Their Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Short List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kundakunda (Second to Eighth Centuries) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Samantabhadra (Fourth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Umāsvāti (Fourth or Fifth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Siddhasena Divākara (c. Fifth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Akalaṅka (Eighth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Mallavādin (Fifth to Sixth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vidyānandin (Ninth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hemacandra (1089–1172) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Māṇikyanandin (Eleventh Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prabhācandra (Eleventh Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Abhayadeva (Eleventh Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vādidevasūri (Twelfth Century) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Yaśovijaya (1624–1688, Wiley 2004: 239) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Long List . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In comparison to the work and research done in Brahmanism/Hinduism and Buddhism, studies in Jaina philosophy and particularly Jaina logic are relatively ignored areas. The Jaina contribution to the land of its origin, for example, to J. Soni (*) Department of Philosophy, University of Innsbruck, Innsbruck, Austria University of Marburg, Marburg, Germany e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_14

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literature, art, and architecture, is widely acknowledged, and one is hard pressed to find a reason for the scanty interest in its philosophy and logic. This chapter suggests a possible reason for this in the Indian tradition. At the same time, it is shown that Jaina philosophers and especially logicians have made significant contributions to the development of ideas throughout the history of Indian philosophy. A short list of 13 thinkers and their major works (annotated) is regarded as the minimum one needs for a study of the basic orientation to the development of Jaina logic over a period of 1,500 years. A long list of 43 thinkers with their major works related specifically to logic is also provided additionally. This long list too is a succinct and summary one, especially when it is known that in the Jaina tradition, a list can be drawn entailing, e.g., 352 authors and 951 works. Keywords

Jaina logic · Ages of Jaina logic · Anekānta-vāda · Syād-vāda · Naya-vāda · Kundakunda · Samantabhadra · Umāsvāti · Siddhasena Divākara · Akalaṅka · Mallavādin · Vidyānandin · Hemacandra · Māṇikyanandin · Prabhācandra · Abhayadeva · Vādidevasūri · Yaśovijaya

Introduction A brief look at published works and research done in Brahmanism/Hinduism and Buddhism shows that studies in Jainism are lagging behind, despite the fact that Jainism has been an influential part of Indian culture since ancient times. Of the many traditions, apart from the Brahmanical, that were extant around the fourth to fifth centuries BCE (like the Ājīvikas and several others), only Jainism and Buddhism have survived to this day. The Jaina treasures, for example, of literature, art, and architecture, are acknowledged contributions to the land of its origin throughout history, and one is hard pressed to find a reason for the scanty interest of this tradition. The Jaina contribution to philosophy generally and to logic in particular has hardly been taken seriously. One reason might well be because of Śaṅkara’s (eighth century) criticism, albeit unfounded, that the Jaina syād-vāda is a theory of “doubt” because the use of the word syāt (also syād or syān) according to him indicates uncertainty and that the Jainas are not in a position to make any clear and definite statements about an object of inquiry. Moreover, syād-vāda was seen as the only main aspect of Jaina philosophy as a whole, which therefore perhaps led to other areas being ignored as well (see “Syād-vāda is not Saṃs´aya-vāda,” in Soni 1996: 20–45). However, Jaina thought is well recorded, and its value in contributing to the history of ideas in India is clearly evident on the basis of the available material (e.g., introducing the concept of the mode or modification of a substance in the discussions about substance and its quality: dravya, guṇa, and paryāya; see Soni 1991). Syād-vāda is a part of the “theory of manifold perspectives” (anekānta-vāda), literally “perspectives that are not one-sided” or “many-sided perspectives,” and has

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been appropriately been seen as the “central philosophy of Jainism” (see Matilal 1981). It will be dealt with in detail by Anne Clavel in a contribution to this section of Jaina logic. Suffice it to make but a few remarks about it here. The sevenfold predication which constitutes syād-vāda involves demonstrating, for example, that a person is or can be a daughter and a mother, at the same time, but that only one aspect among several others is emphasized in a particular context. Not only in Śaṅkara’s time in the eighth century but also thereafter, this syād-vāda, which demonstrates that our knowledge is partial when we pass a judgment on an object of investigation, was regarded by others to be the entirety of Jaina philosophy. Other aspects of the Jaina position which constitute their theory of manifoldness as a whole were ignored (e.g., naya-vāda, the theory of particular standpoints, like the universal or particular standpoint when talking of a fruit or a mango). Further, Jaina epistemology and logic were also largely ignored. Only recently, especially since this century, has there been a keen interest in what has been called the Jaina multivalued logic. The Jainas themselves took cognizance of the faults that others inadmissibly leveled against them and hence mentioned them explicitly (doubt, contradiction, infinite regress, etc.) and demonstrated their inapplicability from the Jaina point of view (see Soni 2007a for various lists of faults leveled against the Jainas). Although Śaṅkara’s critique of Jaina philosophy had a lasting and scathing effect on Jainism in the Indian tradition, there have been earlier attacks on Jaina syād-vāda, for example, by Dignāga (fifth to sixth century) and particularly by Dharmakīrti (seventh century) in his chapter on inference for one’s own sake (svārthānumānapariccheda) of his Pramāṇa-vārttika, “commentary on the means of cognition.” Even ad hominem remarks are used, for example, calling the Jainas “shameless ones,” referring to the monks of Digambara Jainism, and “proud fellows” in a derogative sense. Some criticisms against Jaina philosophy also included the problem of particulars and universals, which occupied Indian thinkers quite early in their philosophical activity. Jaina thinkers like Akalaṅka and Vidyānandin took up the challenge and occupied themselves with it; the latter pointedly meets Dharmakīrti’s attacks against Jaina views (see Soni 1996: 20–22, and 1999: 245–158, and Gorisse’s contribution mentioned below in this section on Jaina logic). In any case, it is clear from some of the criticisms against the theory that the Jaina technical use of the word syāt was not recognized by the opponents, where the word is not the third-person optative of the same verbal root as, which in the third-person singular is asti; each of the predicates of the theory employs both syād and asti, and the opponents ignored the special use of syāt as an indeclinable particle, erroneously regarding both as finite verbs. With a meaning similar to something like “from one perspective,” the Jainas employed the word syāt in a single statement to implicitly say two things: that the predication is from one perspective and, at the same, it implies there can be another perspective. This basic form of the theory was then substantiated logically by the Jainas to account epistemologically for partial and perspectivistic predications of an object inquiry. The opponents were not ready to see that the Jainas expanded and systematized an attitude in Indian philosophical thinking which some others also employed (see the introduction in Shah 2002, especially p. ix and the footnotes).

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Indian logicians did not use symbols to express their abstract thinking, but they effectively used natural language in a technical way to present their arguments. Indeed, this way of implementing natural language presupposed a study of and acquaintance with how to employ the language for technical philosophical discussions of abstract ideas, as is the case with knowing the rules governing the use of the various symbols in logic. However, there have been successful attempts to apply symbolic logic to represent the logical arguments of the Indians, including the sevenfold predication of syād-vāda (e.g., in the works of pioneers like D.H.H. Ingalls, B.K. Matilal, Frits Staal, and recently by Jonardon Ganeri, to name but a few in this area). Piotr Balcerowicz formalizes it in Potter 2014: 29–51. As with other Indian systems of thought, Jaina logic is also a part of epistemology (prāmāṇya-vāda) in which logical issues are incorporated in the means of cognition or knowledge called “inference” (anumāna). Inference incorporates a “new” element that is not contained in another means of cognition such as sense perception, although perception may aid it. The classical example for inferring something new, something which perception alone does reveal, is inferring the existence of fire by seeing only the smoke. The model for the implementation of logic for valid inferential arguments was preeminently set up by the Nyāya school, and the Jainas heavily drew upon its methods and terminology. This is evident in one of the earliest presentations of Jaina logic, in Siddhasena Divākara’s c. fifth century “Introduction to [Jaina] Logic,” Nyāyāvatāra (see stanzas 18 and 22 for a reiteration of the Nyāya categories of the major and middle terms, sādhya and sādhana, reason or hetu, and argument fallacies or hetvābhāsa). We shall see in more detail in this section on Jaina logic how the Jainas used logic to criticize other views and to defend their own theory. This will be shown in the contributions by Marie-Hélène Gorisse, “The Opponent: Jaina logicians reacting to Dharmakīrti’s theory of inference” with special reference to Māṇikyanandin and Prabhācandra, and Himal Trikha: “An example for the use of logical principles in Vidyānandin’s Satya-s´āsana-parī kṣā.”

Ages of Jaina Logic In his well-known 1971 work Jaina Ontology, now a standard reference work, K.K. Dixit very conveniently divides the history of Jaina philosophical speculation into three so-called Ages of Logic (88–164), after having dealt with the “Age of the Āgamas” (12–87). The word “logic” in the “Ages of Logic” may be understood as the logic of the arguments by Jaina thinkers in different periods or ages, namely, their arguments both against non-Jaina views as well as those in support of their own position on philosophical issues. The ages are divided chronologically in terms of important texts by renowned thinkers. Dixit seems to want to clearly demarcate Śvetāmbara and Digambara contributions in the different ages. It is clear even from his classification that in the early phases of Jaina philosophical activity, the Digambaras took the center of the philosophical stage. Dixit’s work is useful for us as a starting point because of the basic orientation it provides and for the classification it gives.

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In the first period, Dixit (1971: 89) says that the important Śvetāmbara texts are Siddhasena’s San-mati (perhaps fifth century), Mallavādin’s Naya-cakra (fifth or sixth century), and Jinabhadra’s Vis´eṣāvas´yaka-bhāṣya (sixth to seventh century). And the first important Digambara texts are Kundakunda’s (eighth century?) three “essences” or sāras (Pan˜ cāstikāya, Pravacana, and Samaya) and Samantabhadra’s (fourth century) Āpta-mī māṃsā. The second age of logic according to Dixit is represented by the eighth-century Śvetāmbara scholar-monk Haribhadra with his Anekānta-jaya-patākā, his magnum opus, and his Śāstra-vārtā-samuccaya and two Digambara thinkers for the last of this stage, Akalaṅka (eighth century, Rāja-vārtika, Aṣṭa-s´atī , Laghī yas-traya, Nyāya-vinis´caya, Pramāṇa-saṅgraha, and Siddhi-vinis´caya) and Vidyānanda [= Vidyānandin] (ninth century, Tattvārtha-s´loka-vārtika and Aṣṭa-sahasrī ). The third stage is represented by four thinkers: (1) the Digambara Prabhācandra (eleventh century, Nyāya-kumuda-candra, a commentary on Akalaṅka’s eighth-century Laghī yas-traya, and Prameya-kamala-mārtaṇḍa, a commentary on Māṇikyanandin’s eleventh-century Parī kṣā-mukha), the Śvetāmbaras (2) Abhayadeva (also eleventh century, San-mati-ṭī kā), (3) Vādideva (twelfth century, Syādvāda-ratnākara), and the polyglot (4) Yaśovijaya (seventeenth century, Nayarahasya, Anekānta-vyavasthā, Nayopades´a, and Tarka-bhāṣā). The threefold division of the ages of logic, in contrast to the age of the Āgamas, is based on the view that certain tendencies characterize the ages of logic. These are based on the areas of philosophical activity of the thinkers mentioned (Dixit, 106): (i) (ii) (iii) (iv)

To vindicate the doctrine of anekānta-vāda To establish a particular doctrine of pramāṇas To evaluate the non-Jaina philosophical views To defend the traditional Jaina philosophical views

Dixit (p. 107) then summarily divides these “ages” of Jaina logic by grouping them thematically into three major parts, namely: (i) That related to the doctrine of anekānta-vāda (ii) That related to the doctrine of pramāṇas (iii) That related to the traditional Jaina philosophical views Dixit’s threefold division of the age of logic takes into account 12 thinkers and 25 works from about the fourth to the seventeenth centuries. The advantage of this classification is that it groups together a specific number of thinkers and texts in order to facilitate an overview of Jaina speculation on specific themes directly or indirectly related to Jaina ontology through arguments put forward by the thinkers. In the field of logic, the main points, as we saw, deal with a vindication of the theory of manifoldness (anekānta-vāda), the development of the doctrine of the means of cognition or knowledge (pramāṇas), and a defense of traditional Jaina philosophical views. That the scheme is practical may be seen from the fact that although Māṇikyanandin’s

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eleventh-century Parī kṣā-mukha is conspicuous by its absence, Prabhācandra’s Prameya-kamala-mārtaṇḍa in the third age is mentioned, which is a commentary on it. One could argue that the scheme is an oversimplification of 13 centuries of Jaina speculation and disregards a vast amount of speculation by other thinkers. This would no doubt be true. If thinkers are left out (e.g., Māṇikyanandin, just mentioned) and many important works ignored (like Vidyānandin’s Satya-s´āsana-parī kṣā and Āpta-parī kṣā), we certainly get a limited picture. In other words, Dixit’s work has to be consulted with caution and with exhaustive supplementation. Moreover, many of his remarks have to be carefully weighed in the light of their opinionatedness, as, for example, in the case of Prabhācandra (see Soni 2013).

Lists of Jaina Logicians and Their Work One can draw two specially selected lists of Jaina logicians for our purposes here, a short one of 13 thinkers and their works relevant for Jaina logic and a long one of at least 43 (see Trikha 2015: 425, for other lists of Jaina thinkers entailing 93 authors and 216 works and 352 authors and 951 works). Logic in Indian philosophy, as already pointed out, is dealt with under the rubric of epistemology in which inference in general, and especially what constitutes a “valid” inference, is included as an instrument or means of cognition or knowledge (see also Vidyabhusana 1971). Our short and long lists, which span a period from about the second to the seventeenth centuries, therefore includes noteworthy philosophical activity of specific thinkers concerning epistemology as well. Their arguments are relevant for logic in general. In our short list, it is necessary to include early thinkers and authors because many aspects of the development of Jaina logic presuppose an acquaintance with their work so as to be able to follow the arguments of later thinkers based on them. For example, Prabhācandra’s contributions are based on a treatment of Kundakunda’s seminal works, namely, Kundakunda’s four so-called essence or sāra-works. Moreover, insightful works, e.g., by Akalaṅka and Vidyānandin require a thorough grasp of Umāsvāti’s Tattvārtha-sūtra, and hence, Umāsvāti cannot be omitted, not only because the Jaina tradition accords a very special status. In this sense, the short list is the minimum one needs for a basic orientation to the development of Jaina logic over a period of 1,500 years. The list also includes thinkers who have given nuanced interpretations of the basic problems, in many cases affording a different perspective to broaden our understanding of the crucial issues, as, for example, in the work by Māṇikyanandin and Vādidevasūri. It must be noted that especially for the early thinkers, only their names have been passed on to posterity with hardly any biographical details, if at all. Even their dates in some cases are uncertain. However, the Jaina tradition has a well-recorded list of authors, and works from which a selection, related especially to logic and epistemology, is given here. Preference has been given only to the author and his selected works. In the development of ideas in Indian philosophy, different schools emerged on the basis of a basic text which a tradition came to regard as an authoritative work, generally a sūtra-text. Such works were regarded as an authentic and reliable source

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for the tradition concerned. Although the text itself remained by and large unaltered in the respective traditions, commentators freely argued about how the text is to be understood, without which the basic text is often hardly understandable. The same basic text was often commented on by many commentators with their special emphases and interpretations, with arguments justifying their position or expanding on earlier views. Flexibility in thinking was exploited in the commentaries to balance the rigidity of a basic text that a tradition respects as unalterable. Some initial commentaries were themselves commented upon in different periods, expanding the scope of the basic text and making it relevant for the specific period when a commentary to a commentary was written. This is the case, for example, with the basic works by Umāsvāti and Kundakunda. Since much of the development of Jaina logic has evolved out of their seminal works, they are, together with Samantabhadra, among the first three for the beginning of the history of ideas related to Jaina logic.

The Short List A short list of 13 Jaina logicians in chronological order with their works annotated (for many dates, see also Dundas 2002 under the names of the respective thinkers):

Kundakunda (Second to Eighth Centuries) Since Kundakunda’s dates vary from the second to the eighth centuries, unresolvable problems arise regarding possible influences on him and those whom he may have possibly influenced. He was also known as Padmanandin. Four works with the suffix -sāra or “essence” are regarded as his philosophical masterpieces with far-reaching consequences on the development of Jaina logic. The significance ascribed to them, particularly in the Digambara tradition, gives them an almost scriptural status. He wrote in the Prakrit language, and these four works have been commented upon in Sanskrit, including a Sanskrit translation of all the Prakrit stanzas, by the commentator Amṛtacandra (perhaps the tenth century) offering important insights into Kundakunda’s basic thought. There are many editions of these works, among them those published by the Shrimad Rajchandra Ashram in Agas, Gujarat, with translations in English and/or Hindi. The Pavayaṇa-sāra/Pravacana-sāra or the “Essence of Scripture” is written in three books or chapters. Chapter 1 is concerned with knowledge (which is said to take place through the sentient principle, jī va) and omniscience which the Jaina tradition accepts. Chapter 2 deals with the objects of knowledge, namely, substances which have to be always seen together with their qualities and modes or modifications. This chapter also deals with the manifestation of the sentient principle through its so-called application, technically called upayoga. The application of this intrinsic function of the sentient principle has direct implications and relevance for epistemology and logic (see Soni 2007b). Chapter 3 deals with conduct, particularly for

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ascetics, with interesting basic ideas for Jaina ethics. Ethical arguments can be derived from them, for example, regarding attachment to worldly pleasures (3, 4) and bondage (3, 19). Samaya-sāra or the “Essence of the Doctrine” deals with the basic categories of Jaina ontology and metaphysics, like the sentient principle ( jī va) and the insentient categories (altogether called ajī va), the influx of matter into the sentient principle, the resulting bondage, etc. The significance of this work for Jaina logic is the arguments with regard to the sentient substance ( jī va-dravya). Views on it are seen from the so-called practical or common point of view (vyavahāra) and from the “certain or definite” (nis´caya) point of view. Stanzas 31–32 of the text, for example, supply the basic Jaina view to help better understand the logical arguments based on them. The Pan˜ catthi-kāya-sāra/Pan˜ cāstikāya-sāra or the “Essence of the Five Entities” deals with the sentient and insentient categories, the principles of motion and rest, and space or ether. It is noteworthy that the category time (kāla) is omitted here in this work, because it is not an entity like the others. The category time is given a special and separate status. The text also supplies the basic Jaina position for the arguments justifying the acceptance in the tradition of the five entities. The Niyama-sāra or the “Essence of Restraint” deals with ascetic conduct for the purposes of liberation from bondage as the Jainas understand it. The work is related more to the Jaina ethical code of conduct than to Jaina logic, although relevant for the arguments related to ethics. The work also entails basic Jaina metaphysics and ontology. In addition to the seven basic categories found in Umāsvāti’s TS, Kundakunda includes two more categories, referring to human actions which bring merit or demerit (puṇya and pāpa).

Samantabhadra (Fourth Century) Samantabhadra, who belongs to the Digambara tradition, wrote two seminal works which have had far-reaching consequences for both Jaina philosophy as a whole and Jaina logic especially. The first is his Āpta-mī māṃsā (an “Investigation into [who is] a Reliable Teacher”). The work is also referred to as the Devāgama-stotra, a “Hymn about the Coming of the Gods,” because the text begins with the compound word devāgama referring to the gods who come when a Jina gives his first sermon. The Āpta-mī māṃsā is a relatively short text of 114 stanzas, divided into 10 sections in the form we have it. Jaini (1979: 84) summarizes the significance of the work for Jaina logic which “introduced a major point of controversy into Indian philosophical dispute. This controversy inspired several Jaina logicians to produce extensive commentaries on Samantabhadra’s work . . ..” The Jaina logicians referred to here are the commentators to this work, namely, by the Digambaras Akalaṅka and Vidyānandin. It is noteworthy that a commentary on the latter’s commentary was written by the seventeenth-century renowned polymath Yaśovijaya, a Śvetāmbara thinker, who, apart from showing respect to Digambara scholarship, adds a new dimension with his expertise on Navya-Nyāya (which developed around the

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thirteenth century, initiated largely by Gaṅgeśa’s Tattva-cintāmaṇi). Commentaries written at different periods to one and the same work are valuable because of their nuanced arguments based on ideas that were developed in the meantime. Samantabhadra’s second work important for Jaina logic is his Yukty-anus´āsana (“Instructions into Reason”) in 64 compact stanzas. In this work Jaina syād-vāda is exploited to justify the lines of arguments used in the tradition, and there are several examples demonstrating their logical consistency (e.g., in stanzas 40 about particulars and universals and 46–47 on syād-vāda and substance, quality, and mode).

Umāsvāti (Fourth or Fifth Century) Umāsvāti wrote the first all-encompassing work on Jaina philosophy in the Sanskrit language, the Tattvārtha-sūtra (also called Tattvārthādhigama-sūtra), “Mnemonic Rules on the Meaning of the Reals” (Dundas 2002: 86). Its significance and status can be compared with say of the Brahma-sūtra for Vedānta. No biographical details are available about Umāsvāti who is also called Umāsvāmin. Scholars have come to accept his dates as given above, although the traditional dating is a couple centuries earlier. Both the Digambaras and Śvetāmbaras regard him as belonging to their own tradition so that the work has a unique status for Jainism as a whole. Although there are two versions of the text depending on the tradition, there is no essential difference regarding basic Jaina metaphysics, ontology, and epistemology. The differences in the two versions pertain mainly to the rules concerning the ascetics. However, there is an unresolvable dispute concerning the first commentary to this work, because of the differing views between the two groups. The Digambaras maintain that Pūjyapāda wrote the first commentary called Sarvārtha-siddhi (“Establishing the Reals”) in the sixth century, and the Śvetāmbaras hold that Umāsvāti wrote an auto-commentary called Svopajn˜ a-bhāṣya (“Self-composed Commentary”). Whatever the case, it is obvious that as with Kundakunda’s work for the Digambaras, Umāsvāti’s work has also gained an almost scriptural status for both traditions. The significance of Umāsvāti’s Tattvārtha-sūtra for the whole of Jaina philosophy lies in the status it has achieved since early times. Practically all thinkers in both traditions see the development of Jaina logic directly based on the ideas contained in it. One of the curious facts is that Umāsvāti explicitly mentions naya-vāda but not syād-vāda. However, syād-vāda is regarded as being implicit in Chaps. 5 and 32 (5, 31 in the Śvetāmbara version) of the work which talks about so-called contradictory statements which nonetheless retain their validity depending on what is being emphasized in a particular context. The example that Pūjyapāda gives in his commentary to it is “Devadatta being a father, son, brother, etc.,” where the one or the other is important for a specific context. It has been already stated both these theories are part of the Jaina theory of manifold perspectives as a whole, with far-reaching consequences for the development of Jaina logic.

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All standard works on Jaina philosophy refer to the basic contents of Umāsvāti’s work, and hence no details are relevant here for the contents of his Tattvārtha-sūtra (see, e.g., the three entries on Jaina philosophy in the Routledge Encyclopedia of Philosophy (Soni 1998) and volume ten of the Encyclopaedia of Indian Philosophies (Potter 2007).

Siddhasena Divākara (c. Fifth Century) Siddhasena is also regarded as belonging to both the traditions and is perhaps the first thinker who deals specifically with issues in Jaina logic, writing in both Sanskrit and Prakrit. He is “portrayed as having been attached” to the legendary king Vikramāditya who is said to have given him the “epithet ‘Sun’ (divākara)” (Dundas 2002: 131). The Jaina tradition ascribes to him three works significant for Jaina logic. Siddhasena’s “Introduction to [Jaina] Logic” or Nyāyāvatāra, written Sanskrit, clearly shows how he adopted the method of the Nyāya school to serve the purpose of arguing in favor of Jaina views, to prove his own arguments and to show inconsistencies in other, non-Jaina, views. Among other things, his arguments concern explicitly defining fallacious examples based on the use of the technical term in Indian logic where the hetu or the probans is the middle term, in association with the major and minor terms. Siddharṣi (gaṇi) (tenth century), Abhayadeva-sūri (eleventh century), and Devabhadrasūri (twelfth century) commented on Siddhasena, adding significant aspects based on the development of logical arguments till their times, with special references to the logic in other schools. The commentaries serve as indispensable tools to better understand Siddhasena’s basic ideas. See Balcerowicz 2001 for an extensive study on the Nyāyāvatāra and its most important commentaries. Siddhasena’s Sammai-takka-sutta/Sanmati-tarka-sūtra/Sanmati-prakaraṇa or the “Mnemonics on Proper Understanding” is in three parts with a total of 168 stanzas. It is written in Prakrit and deals mainly with the means of knowledge, the objects of knowledge, and the standpoints. Usually the seven standpoints in Jaina logic also take into consideration the quality of a substance and its mode, but according to Siddhasena, the quality of a substance cannot be given a separate status that others (e.g., Umāsvāti) give it. Siddhasena omits it because Jaina scripture makes no explicit reference to the term quality as such. Siddhasena is also credited with having written what is simply called the “Thirtytwos,” Dvā-triṃ-s´ikā, 21 short compositions, each in 32 stanzas. This relatively ignored work is also significant for Jaina logic because, apart from eulogies to Mahāvīra, it contains a critique of Buddhist and Hindu schools (see Upadhye 1971, especially his introduction on “Siddhasena and his Works,” pp. *3–*72).

Akalan˙ka (Eighth Century) With Akalaṅka a landmark is set for a significant phase in Jaina epistemology and logic, one which directed future cogitations and gave Jaina philosophy an undeniable

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position as a noteworthy contester and contributor to the history of ideas in the Indian philosophical tradition. With the earlier thinkers like Kundakunda, Samantabhadra, and Pūjyapāda’s commentary, there is a clear Digambara line of philosophical acumen and achievement, extended now by Akalaṅka with unparalleled insight and precision until then. To say appropriately that he “established a reputation as the outstanding critic of the Buddhist logician Dharmakīrti of Nālandā” (Jaini 1979: 84) is to acknowledge him on the reputation of others. However, the Jaina tradition as a whole, and on its own, accord to him the renowned status for which he is known in defending basic Jaina ideas against attacks, from both within the tradition (e.g., for his views on Jaina epistemology and arguing for the special status of the quality and mode or modification of a substance) and outside it. Several works by him remain masterpieces for their lasting significance for epistemology and logic. The first work that needs to be mentioned is Akalaṅka’s Tattvārtha-rāja-vārtika (or simply Rāja-vārtika), a commentary on Umāsvāti’s Tattvārtha-sūtra, expanding in much greater detail the Sarvārtha-siddhi by Pūjyapāda, mentioned above. This commentary is valuable for the basic orientation to the Jaina Digambara position as a whole. At innumerable places Akalaṅka develops arguments justifying or clarifying the Jaina position (e.g., the commentary to TS 1, 6 on how knowledge is gained not only through the acknowledged different means but also through the standpoints or nayas). The work is a necessary source for Akalaṅka’s basic views which he develops further in other works. Akalaṅka’s Aṣṭa-s´atī, the “800 stanzas,” is a commentary on Samantabhadra’s Āpta-mī māṃsā mentioned above and is one of his earliest works relevant for Jaina logic and epistemology. The significance of the original work and this commentary is evident by the fact that Vidyānandin (ninth century) and Yaśovijaya (seventeenth century) commented on this commentary, adding intricate elements Jaina argumentation. The Laghī yas-traya, “Three Short Pieces,” with a total of 88 stanzas with Akalaṅka’s own gloss on them are (1) the Pramāṇa-praves´a, “an introduction to the means of knowledge”; (2) Naya-praves´a, “an introduction to the (Jaina) standpoints”; and (3). Pravacana-praves´a, “an introduction to scripture.” All the three were commented upon at length in the eleventh century by Prabhācandra in his Nyāya-kumuda-candra (see below), a landmark for the perceptive arguments he develops based on Akalaṅka’s basic ideas. The Nyāya-vinis´caya, the “Ascertainment of Logic,” deals with perception, inference, and Jaina scripture. The division into these three sections does not mean that Akalaṅka is giving a different classification of Jaina epistemological categories but that he is grappling with them in Buddhist and Brahmanic terms, perhaps to bring out the unique Jaina position. An eleventh-century Śvetāmbara scholar, Vādirājasūri, has shown deep respect to Akalaṅka by writing a voluminous commentary on it called Nyāya-vinis´caya-vivaraṇa (literally, “Uncovering” the Ascertainment of Logic) substantiating the basic ideas even more clearly. The Pramāṇa-saṅgraha, the “Collection of the Means of Knowledge,” is divided into nine sections dealing with logical and epistemological categories (sections 1 and 2); inference, including the “faults” (doṣas) of doubt, contradiction, reason, and

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infinite regress (sections 3–5); and rejection of some Buddhist views (section 6) with sections 7–8 dealing specifically with the Jaina categories and anekānta-vāda. This work and the previous two mentioned above have been published together in one volume as Akalaṅka-grantha-traya, “Three works by Akalaṅ ka.” (see Soni 2002). The Siddhi-vinis´caya, the “Ascertainment of Perfection,” continues with Akalaṅka’s ongoing concern with epistemological and logical issues. It was commented upon by Anantavīrya (tenth century). Akalaṅka’s works in the above list need to be thoroughly studied in order to determine the state of Jaina logic during his time and also for the development of his ideas which the commentaries bring to light.

Mallavādin (Fifth to Sixth Century) Although Mallavādin’s (Dvādas´āraṃ) Naya-cakram (“Twelve-Spoked Wheel of Standpoints”) has been lost to posterity, a commentary on it by Siṃhasūri (seventh century) called Nyāyāgamānusāriṇī , “Scrutinising Logic,” is extant. The text has been reconstructed from it as far as possible, because the commentator does not quote the original in full. The Jaina tradition as a whole and especially the Śvetāmbaras are fortunate that a Jaina scholar-monk and specialist, Muni Jambūvijaya (1923–2009), was the most competent person to edit it with critical notes, so that this milestone in Jaina logic is now accessible (see the reference below under = Jambūvijaya 1966–1988). The significance of the work for Jaina philosophy and logic is clear from a summary description of it (Dixit 1971: 114): “Nayacakra is written in the form of a marathon debate taking place between some seventeen disputants where the incoming one criticises the outgoing one before presenting his own case.” The case of this work shows how commentaries can be a help not only in restoring original works but also for the wealth of information they provide in understanding the text.

Vidyānandin (Ninth Century) This Jaina scholar-monk’s works related to logic can be conveniently divided into commentaries and independent works, with three in each group. Not much has been researched on this outstanding Jaina thinker who demands here a more detailed record of his masterpieces. Among his commentaries the Aṣṭa-sahasrī (the “Eight Thousand”) is a landmark in which the scholiast Vidyānandin brings up-to-date Jaina responses to the critiques leveled against Jainism. This is a commentary on Akalaṅka’s work Aṣṭa-s´atī which itself is a commentary on Samantabhadra’s Āptamīmāṃsā. In the seventeenth century, Yaśovijaya brought it again into prominence with his commentary on Vidyānandin’s commentary. The second noteworthy commentary by Vidyānandin is the Tattvārtha-s´lokavārttika on Umāsvāti’s Tattvārtha-sūtra. As the title indicates, it is a commentary written in both verse (s´loka) and prose (vārttika). There are several arguments

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developed in the entire work in defense of the Jaina position and a critique of other thinkers and schools. Among many significant topics for Jaina logic are, for example, his responses from the Jaina perspective to the logical objections raised by the Buddhist Dharmakīrti and the Mīmāṃsaka Kumārila Bhaṭṭa. The third commentary is Vidyānandin’s Yukty-anus´āsanālaṅkāra on Samantabhadra’s Yukty-anus´āsana. Among his chief concerns in this text are the arguments he develops for Mahāvīra’s teaching as being trustworthy. Vidyānandin’s first independent work for our purposes here is his Āpta-parī kṣā, an “Investigation into (who is) a Reliable Person.” Vidyānandin develops arguments against the reliability, for example, of Īśvara in the Nyāya school, Kapila in Sāṅkhya, and the Buddha. In proving the reliability of an Arahanta (a Jina, especially Mahāvīra), Vidyānandin also develops arguments against the existence of a creator god in this work containing 124 stanzas. The original plan of Vidyānandin’s independent work Satya-s´āsana-parī kṣā, an “Investigation into the True Teaching,” was originally to investigate 14 teachings of the different schools in Indian thought. However, only 11 have been completed, with the 12th incomplete, in the form available to us since 1920 when it was discovered. Sections 9–10, for example, investigate the teachings of the Nyāya and Vaiśeṣika schools. Among the few studies on this text, Trikha 2012 has done an in-depth analysis of Vidyānandin’s critique of the Vaiśeṣika school, bringing out the different levels of argument and their logical impact. Similar studies on the other schools need to be undertaken to bring out the value of the type of arguments Vidyānandin employs at that time. Vidyānandin’s independent work Pramāṇa-parī kṣā, an “Investigation into the Valid Means of Knowledge,” discusses various epistemological categories, including the technical term hetu or reason in Indian logic. Vidyānandin develops his arguments based on the typical Jaina position that for a means of knowledge or cognition to be “valid” (including preeminently inference), such a means has to have what the Jainas technically call samyag-dars´ana, a correct view (sometimes translated as correct “faith”) as its sign (lakṣaṇa). Together with samyag-jn˜ āna and samyak-cāritra (correct knowledge and conduct), samyag-dars´ana constitutes what the Jainas refer to as the three jewels which, together, make up the path to liberation from bondage (mokṣa-mārga). Vidyānandin’s independent work Patra-parī kṣā is an investigation of many schools showing how their views lead to contradictions or entail the fault of infinite regress. For this he draws pointedly on the logical terms of reason, middle and major terms (hetu, sādhana, sādhya). Dharmakīrti, Sāṅkhya, and Yoga are specifically mentioned and critiqued. The main point of this relatively ignored short and compact text is to vindicate the Jaina position of multiple views which do not entail any of the errors which other schools say they have. Vidyānandin closes a significant phase in the early development of Jaina logic and philosophy following in the Digambara line of thinkers like Samantabhadra, Umāsvāti, Siddhasena, and Akalaṅka. A thorough grasp of the scholastically produced argumentation in at least the works mentioned above can reveal further insights considering the status of Indian philosophy as a whole in the ninth century.

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In Vidyānandin we see clearly that to understand the intricacies of Jaina philosophy and logic, a knowledge of the other schools is also imperative.

Hemacandra (1089–1172) See Dundas 2002 (especially pp. 133–136) for the available biography on him and his importance “for establishing Śvetāmbara Jainism as a resilient and self-confident presence in western India” (p. 134). Among the several works of Hemacandra, his Pramāṇa-mī māṃsā, an “Investigation into the Means of Knowledge,” stands out for the encyclopedic nature of his examination of the other schools of philosophy and logic. His Anya-yoga-vyavacchedikā, the “Rescinder of other Systems” (Thomas 1960: 7) in 32 stanzas, was commented upon by Malliṣeṇa in his Syād-vāda-man˜ jarī which Thomas 1960 translates as “The Flower-Spray of the Quodammodo Doctrine” to bring out its logical significance. This has become one of the standard works on Jaina logic.

Mānikyanandin (Eleventh Century) ˙ Māṇikyanandin has left his mark as a notable Digambara thinker with his Parī kṣā-mukha the “Gateway to Investigation” in six sections with a total of 210 stanzas. He summarizes in it the basic issues in Jaina logic and epistemology, drawing on previous Digambara thinkers, particularly Akalaṅka. Section “Lists of Jaina Logicians and Their Work” is particularly concerned with logic. In 3, 14, for example, he says that the knowledge derived from inference is the knowledge of the major term which itself is based on the knowledge of the minor term. Inference also involves the technical term “pervasion” of elements in the major and minor terms in the inferential process. That the work contains a wealth of significant aspects for Jaina logic is evinced by the voluminous commentary on it by Prabhācandra (see below). The work with its commentary also became a standard one for basic Jaina logic in the Digambara tradition.

Prabhācandra (Eleventh Century) Commentaries highlight intricacies of an original text and as such are not restricted to the parsimony of the original text. They are also indispensable for a proper understanding of the text from within the tradition, apart from substantiating the original text as we have it. Commentaries are sometimes also an aid in reconstructing the original, as in Mallavādin’s case above. Prabhācandra is acknowledged in the tradition as a reliable commentator who unpacks the text, often indicating the intention of the author as he sees it. Two of his commentaries are noteworthy here. One is the Prameya-kamala-mārttaṇḍa (“The Sun [That Causes] the [Day] Lotus of Knowables [to Bloom],” Jaini 1991: 109), a commentary on Māṇikyanandin’s Parī kṣā-mukha, mentioned above.

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The other commentary is called Nyāya-kumuda-candra (“The Moon [That Causes] the [Night] Lotus of Logic [to Bloom],” Jaini 1991: 109) on Akalaṅka’s Laghī yas-traya, mentioned above. Whereas Amṛtacandra’s commentaries on Kundakunda “merely” expound the basic text, Prabhācandra is listed here separately because his commentaries expound the text in such a way that they can also been seen as independent works (see also Jaini 1991: 109 and Potter 2013: 53).

Abhayadeva (Eleventh Century) Abhayadeva’s name to fame is his Tattva-bodha-vidhāyinī -ṭī kā, a “Commentary Establishing the Knowledge of the Reals” (also called Vāda-mahārṇava, the “Great Ocean of Discussion”), a commentary on Siddhasena’s San-mati-tarka. We saw that the original work written in Prakrit was largely a text of mnemonics which is difficult to unpack without a commentary. Abhayadeva does it in Sanskrit, and his arguments against the other schools, implicitly or explicitly alluded to, reflect the Jaina view in defending its own position of manifold perspectives and standpoints as a basic philosophical attitude. Further, in the third part of the text, Abhayadeva clearly elucidates Siddhasena’s view that we can talk only of a substance and its mode, with the quality of a substance omitted, because the canonical texts do not mention it.

Vādidevasūri (Twelfth Century) This Śvetāmbara scholar from Gujarat is famous for his work which became a standard for medieval Jaina logic and epistemology, together with his own commentary on it: his Pramāṇa-naya-tattvālokālaṅkāra, the “Ornament-Lustre of the Means of Knowing and the Standpoints” together with his auto-commentary Syādvāda-ratnākara, the “Jewel-mine of Perspectives.” As is customary with such scholastic works, the opponents’ views are first put forward and then refuted from the point of view of the author and his tradition. The arguments are developed in a complex way requiring a keen understanding of all the systems of Indian thought, with a thorough grasp of the basic Jaina position.

Yaśovijaya (1624–1688, Wiley 2004: 239) This Śvetāmbara polymath and prolific writer is seen as “perhaps the last truly great intellectual figure in Jainism, whose fame rests largely on his learning combined with a mastery of sophisticated logical techniques” (Dundas 2002: 110). He is renowned not only for his mastery over the other schools of Indian philosophy but also over the Navya-Nyāya logic, referred to above under Samantabhadra. Further, Yaśovijaya draws on both Digambara and Śvetāmbara scholarship in bringing out clarity, for example, regarding the standpoints (nayas, see also Dixit 1971: 162).

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Ganeri (2008: 2–3) summarizes Yaśovijaya’s “intellectual biography” in three phases bringing out the outstanding relevance for his contributions to Jaina logic: “an apprenticeship in Varanasi studying Navyanyāya, a period writing Jaina philosophical treatises using the techniques and methods of Navyanyāya, and a time spent writing works with a markedly spiritual and religious orientation.” This special training earned him “the respectable title Nyāyaviśārada, ‘One who is skilled in logic’” (ibid., p. 3). There are several works attributed to him and we select but a few here. Yaśovijaya’s Jaina-tarka-bhāṣā or “A Manual of Jaina Logic” (Bhargava 1973: 33) deals with the means of knowing, the standpoints (naya), and the typical Jaina use of words in the technique called nikṣepa. This technique is employed when describing something through the use of specific terms, for example, of name, representation, substance, and its mode. The significance of the use of these words in philosophical discourse is to further emphasize the partial and restricted views when they are used to describe an object of inquiry. The aim is to demonstrate that no description can be all encompassing but that only certain elements are highlighted in a specific context. His Jaina Nyāya-khaṇḍa-khādya (a “Short Section on Logic”) and Jn˜ āna-bindu (a “Drop of Knowledge”) can also be mentioned with the “Manual” for reiterating Jaina epistemology and logic drawing on Navya-Nyāya logic to uphold the Jaina position of manifoldness of perspectives and standpoints. Another group of works dealing with specific aspects of Jaina anekānta-vāda are the following, each highlighting one or the other aspect in a nuanced way: Nyāyāloka, Nayakarṇikā, Nayopades´a, Nayarahasya, and Anekānta-vyavasthā. In all these works, Yaśovijaya defends Jaina views against the explicit or implicit attacks by other schools. Yaśovijaya’s Aṣṭa-sahasrī-tātparya-vivaraṇa-ṭī kā is a brief commentary on Vidyānandin’s work that deserves to be mentioned here for acknowledging the Digambara Vidyānandin for his perceptive insights; in his commentary, Yaśovijaya adds elements based on his training in Navya-Nyāya. How exactly the Jainas argued against so-called opponents is exemplified in three articles in this section on Jaina logic, dealing with specific authors and issues: Anne Clavel on the central aspect of Jaina logic, namely, the theory of manifold perspectives encapsulated in the seven statements of syād-vāda, Marie-Hélène Gorisse who draws on the arguments presented by Māṇikyanandin and Prabhācandra in their reaction to Dharmakīrti’s theory of inference, and Himal Trikha who deals with the use of logical principles in Vidyānandin’s Satya-s´āsana-parī kṣā. A long list of selected Jaina thinkers is also supplied below alphabetically with their most significant works on Jaina logic and epistemology without further detail. The aim of this list is to place on record that from the early beginnings of Indian philosophical activity, Jainism has continuously produced a host of reputable thinkers whose works need much further study. It is fortunate that even though the contribution of Jaina philosophy was not taken very seriously, the tradition itself clung onto what was handed down over the years from generation to generation. Indeed, much has been lost to posterity but what is available should encourage further research (see also Mahaprajna 1984: 159–161 and 162–177).

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The Long List The list of 13 thinkers given above can be expanded with an additional 43 thinkers, with only the names of the philosophers and selected titles of relevant works for logic. This list with a total of 56 thinkers is arranged alphabetically for quick reference. An asterisk (*) below indicates that the philosopher is mentioned in the short list above: *Abhayadeva, tenth to eleventh century Abhayatilaka, fourteenth century: Nyāyālaṅkāra-vṛtti, Tarka-nyāya-sūtra-ṭī kā, Pan˜ ca-prastha-nyāya-tarka-vyākhyā *Akalaṅka, eighth century Anantakīrti, ninth to eleventh century: Bṛhat-sarvajn˜ a-siddhi, Laghu-sarvajn˜ asiddhi Anantavīrya tenth century: Siddhi-vinis´caya-ṭī kā, the commentary on Akalaṅka’s work Āśādhara, 1188–1250: Prameya-ratnākara Bhāvasena, eleventh to thirteenth century: Vis´va-tattva-prakās´a Candrasena, twelfth to thirteenth century.: Utpāda-siddhi Candrasūri, twelfth century: Ankekānta-jaya-patākā-vṛtti-ṭippaṇaka, a commentary on Haribadra’s work Devaprabhasūri, twelfth to thirteenth century: Nyāyāvatāra-ṭippaṇa, a commentary on Siddhasena’s work Devasena, tenth century: Dars´ana-sāra Dharmabhūṣaṇa, fourteenth to fifteenth century: Nyāya-dī pikā, Pramāṇa-vistāra Guṇaratnasūri, fourteenth to fifteenth century: Tarka-rahasya-dī pikā Haribhadra, eighth century: Ankekānta-jaya-patākā, Yoga-dṛṣṭi-samuccaya, Ṣaḍ-dars´ana-samuccaya *Hemacandra, (1089–1172) Jinabhadragaṇin, fifth to sixth century: Vis´eṣāvas´yaka-bhāṣya Jinadattasūri, thirteenth century: Ṣaḍ-dars´ana-samuccaya-vṛtti, a commentary on Haribhadra’s work Jinapatisūri, thirteenth century: Prabodha-vāda-sthala Jineśvarasūri, twelfth to thirteenth century: Pramālakṣma saṭī kā Jñānacandra, fourteenth to fifteenth century: Ratnākarāvatārikā-ṭippaṇaka, a commentary on Ratnaprabhāsūri’s work Kumaranandin, eighth century: Vāda-nyāya *Kundakunda, dated second to eighth century *Mallavādin, fourth to fifth century Malliṣeṇa, fourteenth century: Syād-vāda-man˜ jarī *Māṇikyanandin, tenth to eleventh century Merutuṅga, fifteenth century: Ṣaḍ-dars´ana-nirṇaya Naracandrasūri, thirteenth century: Nyāya-kandalī *Prabhācandra, eleventh century Pradumnasūri, twelfth century: Vāda-sthala

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Pūjyapāda, sixth century: Sarvārtha-siddhi (“Establishing the Reals”) mentioned under Umāsvāti Rājaśekharasūri, fourteenth to fifteenth century: Ratnākarāvatārikā-pan˜ jikā Rāmacandrasūri, thirteenth century: Vyatireka-dvātriṃs´ikā Ratnaprabhāsūri, twelfth to thirteenth century: Ratnākarāvatārikā, a commentary on Vādidevasūri’s Syād-vāda-ratnākara *Samantabhadra, fourth century Samantabhadra Laghu, thirteenth century: Viṣama-pada-tātparya-ṭīkā Śāntisena, twelfth century: Prameya-ratna-sāra Śāntyācārya (twelfth century): Nyāyāvatāravārtika Siddharṣi, tenth century: Nyāyāvatāra-ṭī kā, a commentary on Siddhasena’s work *Siddhasena Divākara, fifth century Śivārya, fifth to sixth century: Siddhi-vinis´caya Somatilakasūri, fourteenth to fifteenth century: Ṣaḍ-dars´ana-samuccaya-ṭī kā, a commentary on Haribhadra’s work Śrīcandrasūri, twelfth century: Nyāya-praves´a (Haribhadra-vṛtti-pan˜ jikā) Śrīdatta, seventh century: Jalpa-nirṇaya Śubhacandra eleventh century: Jn˜ ānārṇava (Jaini 1979: 255, fn. 21) Śubhacandra, sixteenth century: Ṣaḍ-dars´ana-pramāṇa-prameya-saṅgraha Śubhaprakāśa,?: Nyāya-makaranda-vivecana Sukhaprakāśa,?: Nyāya-dī pāvalī -ṭīkā Sumati, eighth to ninth century: Sanmati-tarka-ṭīkā, a commentary on Siddhasena’s work *Umāsvāti/Umāsvāmin, fourth or fifth century Vādībhasiṅha, eighth to ninth century: Syād-vāda-siddhi *Vādidevasūri, (1086–1169) eleventh to twelfth century Vādirājasūri, eleventh century: Pramāṇa-nirṇaya, Nyāya-vins´caya-vivaraṇa (a commentary on Akalaṅka’s work) Vasunandin, eleventh to twelfth century: Āpta-mī māṃsā-vṛtti *Vidyānandin, ninth century Vimaladāsa, fifteenth century: Sapta-bhaṅgī -taraṅgiṇī *Yaśovijaya, eleventh century

Summary This chapter began with the point that studies in Jainism are lagging behind in relation to those in Brahmanism/Hinduism and Buddhism. It was suggested that one of the reasons for the lack of attention paid to Jainism early in the Indian tradition may be attributed to Śaṅkara’s criticism of syād-vāda, erroneously seeing it as a theory of doubt. The Jainas themselves were certain about the well-founded logic of their theory and defended it with all their philosophical acumen, including other aspects contained in their anekānta-vāda, like naya-vāda. Their defense is evinced in the masterpieces of major thinkers in the tradition who also put forward noteworthy arguments for this and other aspects of the philosophy, such as their views

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regarding a substance, its qualities, and the modes or modifications of the substance. The so-called ages of logic shows that the Jainas constantly contributed to the development of ideas in the Indian tradition. The short and long lists of thinkers and their works, which span a period of about 1,500 years, evince how these thinkers kept abreast of philosophical themes in each epoch, even drawing on earlier ideas and furnishing new and nuanced arguments.

Definition of Key Terms Ajīva

Anekānta-vāda Anumāna Digambara

Doṣa Dravya

Guṇa Hetu Hetvābhāsa Jīva Naya-vāda Paryāya

Pāpa Pramāṇa Prāmāṇya-vāda Puṇya Sādhana Sādhya Saṃśaya Saṃśaya-vāda Sūtra

Non-sentient principle(s): one of two substances in Jainism (a generic term for five categories; see jīva) The theory of many sidedness, manifoldness, or non-one-sidedness Inference, one of the means of cognition/ knowledge Literally “sky-clad,” one of the two major groups of Jainism whose monks are naked (see Śvetāmbara) Fault or error, especially one which an argument might have Substance, of which there are two in Jainism, jīva and ajīva (dravya is treated together with guṇa and paryāya) Quality of a substance (treated together with dravya and paryāya) The “reason” in an inference A fallacy in an inference Sentient principle, together with ajīva one of the two substances in Jainism The theory of standpoints The mode or modification a substance can have, like a pot made of gold or clay (treated together with dravya and guṇa) Demerit, refers to actions which are unmeritorious An instrument or means of cognition/knowledge The theory of cognition/knowledge, epistemology Merit, refers to actions which are meritorious The middle term in an inference The major term in an inference Doubt A theory of doubt Literally “thread,” a teaching, also refers collectively to a work containing these

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Syād-vāda

J. Soni

Literally “white-clad,” the second major group in Jainism; the ascetics are clad in white clothes (see Digambara) The theory of perspectives, denoted by the use of the word syāt (which changes to syād or syān in accordance with the rules for phonetic change in Sanskrit)

References Balcerowicz, Piotr. 2001. Jaina epistemology in historical perspective. Critical edition and English translation of logical-epistemological treatises: Nyāy^avatāra, Nyāy^avatāra-vivṛti and Nyāy^avatāra-ṭippana with introduction and notes. In 2 volumes, Stuttgart: Steiner. Bhargava, Dayanand. 1973. Mahopādhyāya Yas´ovijya’s Jaina Tarka Bhāṣā, with translation and critical notes. Delhi: Motilal Banarsidas. Dixit, K.K. 1971. Jaina ontology. Ahmedabad: L. D. Institute of Indology. Dundas, Paul. 2002. The jains. London: Routledge. Ganeri, Jonardon. 2008. Worlds in conflict. The cosmopolitan vision of Yaśovijaya Gaṇi. International Journal of Jaina Studies (Online) 4 (1): 1–11. Accessed 28th Oct 2015. Jaini, Padmanabh S. 1979. The Jaina path of purification. Delhi: Motilal Banarsidas. Jaini, Padmanabh S. 1991. Gender and salvation. Jaina debates on the spiritual liberation of women. Delhi: Munshiram Manoharlal. Jambūvijaya, Muni. 1966–1988. Dvādas´āraṃ Nayacakraṃ of Ācārya Śrī Mallavādi Kṣamās´ ramaṇa. With the commentary Nyāyāgamānusāriṇī of Śrī Simhasūri Gaṇi Vādi Kṣamās´ ramaṇa. Published in three volumes, Bhavnagar: Śrī Jaina Ātmānanda Sabhā. Mahaprajna, Yuvacarya. 1984. New dimensions in Jaina Logic, New Delhi: Today and Tomorrow’s Printers and Publishers (English rendering by Dr Natmal Tatia of “Jaina Nyāya kā Vikāsa”). Published under the auspices of Jaina Vishva Bharati, Ladnun, Rajasthan. Matilal, Bimal Krishna. 1981. The central philosophy of Jainism (Anekānta-Vāda), Series 79. Ahmedabad: L.D. Institute of Indology. Potter, Karl H., general ed. 2007. Encyclopedia of Indian Philosophies. Volume X. Jain Philosophy Part I, ed. Dalsukh Malvania and Jayendra Soni. Delhi: Motilal Banarsidass. Potter, Karl H., general ed. 2013. Encyclopedia of Indian philosophies. Volume XIV. Jain philosophy part II, ed. Karl H. Potter, and Piotr Balcerowicz. Delhi: Motilal Banarsidass. Potter, Karl H., general ed. 2014. Encyclopedia of Indian philosophies. Volume XVII. Jain philosophy part three, ed. Piotr Balcerowicz and Karl H. Potter. Delhi: Motilal Banarsidass. Shah, Nagin J., ed. 2002. Jaina Theory of Multiple Facets of Reality and Truth (Anekāntavāda). Delhi: Motilal Banasidass. Soni, Jayandra. 1991. Dravya, Guṇa and Paryāya in Jaina thought. Journal of Indian Philosophy, 19, 75–88. Netherlands: Kluwer. Soni, Jayandra.1996. Aspects of Jaina Philosophy. Madras: Research Foundation for Jainology. Three lectures on Jainism published on behalf of the University of Madras, Department of Jainism, by the Research Foundation. Annual Lecture Series 1994–95. Contents: “The Karma Theory and Jaina Ethics”; “Syādvāda is not Samśayavāda”; and Vidyānandin on Umāsvāti’s pramāṇa-nayair adhigamaḥ, (Tattvārthasūtra 1, 6). Soni, Jayandra. 1998. ‘Jaina Philosophy, Issues in’ (6000 words). In Routledge encyclopaedia of philosophy, ed. Edward Craig. Soni, Jayandra. 1999. Aspects of Jaina epistemology with special reference to Vidyānandin. In Approaches to Jain studies: Philosophy, logic, rituals and symbols, Centre for South Asian

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studies, in the series South Asian papers, ed. N.K. Wagle and Olle Qvarnström, 138–168. Toronto: University of Toronto. Soni, Jayandra. 2002. Epistemological categories in the Akalaṅkagranthatraya. In Śikhisamuccayaḥ. Indian and Tibetan studies, ed. Dragomir Dimitrov et al., 185–191. Wien: Arbeitskreis für tibetische und buddhistische Studien. Universität Wien. Soni, Jayandra. 2007a. Anekāntavāda Revisited — for doṣas. In Indica et Tibeta Festschrift für Michael Hahn zum 65. Geburtstag von Freunden und Schülern überreicht, ed. Konrad Klaus and Jens-Uwe Hartmann. Wien: Arbeitskreis für Tibetische und Buddhistische Studien Universität Wien. Soni, Jayandra. 2007b. Upayoga, according to Kundakunda and Umāsvāti. Journal of Indian Philosophy 35: 299–311. Soni, Jayandra. 2013. Prabhācandra’s Status in and Contribution to the History of Jaina Philosophical Speculation. Paper presented at the 15th Jaina Studies Workshop, SOAS (School of Oriental and African Studies), London. Published online here: http://www.soas.ac.uk/research/publications/journals/ijjs/file88721.pdf. Accessed 16 June 2015 Thomas, F.W. 1960. The Flower-Spray of the Quodammodo Doctrine. Śrī Malliṣeṇasūri SyādVada-Man˜ jarī, translated and annotated. Berlin: Akademie-Verlag (Delhi: Motilal Banarsidass, 1968). Trikha, Himal. 2012. Perspektivismus und Kritik. Das pluralistische Erkenntnismodell der Jainas angesichts der Polemik gegen das Vais´eṣika in Vidyānandins Satyas´āsanaparī kṣā. Publications of the De Nobili Research Library edited by Gerhard Oberhammer, Utz Podzeit and Karin Preisendanz, Volume XXXVI. Wien: Sammlung de Nobili ... Universität. Trikha, Himal. 2015. Trends of research on philosophical Sanskrit Works of the Jainas. In Sanmati. Essays felicitating professor Hampa Nagarajaiah on the occasion of his 80th birthday, ed. Luitgard Soni and Jayandra Soni. Bengaluru: Sapna Book House. TS: Tattvārtha-sūtra by Umāsvāti. _ Upadhye, A.N. 1971. Siddhasena Divākara’s Nyāyāvatāra . . . as well as the Text of 21 Dvātriṁsikās and the Sammaı¨-Suttam. Bombay: Jaina Sāhitya Vikāsa Maṇḍala. Vidyabhusana, Satis Chandra. 1971. A history of Indian logic (ancient, mediaeval and modern schools), Jaina Logic 157–224. Delhi: Motilal Banarsidass. Wiley, Kristi L. 2004. Historical dictionary of Jainism. Lanham, etc.: The Scarecrow Press.

Logic in nīlakēci and manime¯kalai ˙

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic in nīlakēci . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . uttikaḷ in the Debate Between nīlakēci and picācakaṉ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic in maṇimēkalai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proposition (pakkam, Pn) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reason (ētu, R) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unacceptable (acittam) Types of R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Non-unidirectional (aṉaikāntika) R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contrary (viruttam) Rs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Illustration (tiṭṭāntam, E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defective Es of the cātaṉmiyam Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Defective Es of the vaitaṉmiyam Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nayam/niyāyam and vātam: Types of Logic in the Texts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The present chapter attempts to unearth the salient features of logic in the Tamil epics, nīlakēci and maṇimēkalai. In the former text, we have a few verses which expound the fundamental validative criteria (aḷavaikaḷ) of the Jaina faith, and a surfeit of debates the heroine has with exponents of rival faith systems. These debates amply illustrate the techniques of debate called uttikaḷ explained in tolkāppiyam. Using these techniques, the heroine goes about demolishing rival arguments. The kind of debate she engages in is identified in the essay as “vātam.” Though Jaina philosophy is based on the multivalued logical concept of nayam, nīlakēci’s debate is based on the bivalent logic of vākai (vātam). In the N. Selvamony (*) Madras Christian College, Chennai, Tamil Nadu, India Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_34

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case of the Buddhist epic, maṇimēkalai, logical matters are dealt with primarily in two cantos (27 and 29). Unlike her Jaina counterpart, maṇimēkalai, the heroine, is content to listen to the tenets of rival faith systems and also to those of her own spiritual guru, aṟavaṇa aṭikaḷ. The exponents of Tamil logic are aḷavai vāti, the logician who explains all the ten aḷavaikaḷ of Tamil logic, and aṟavaṇa aṭikaḷ who deals with aḷavaikaḷ briefly, and inference elaborately. Though the latter speaks of the five-member Tamil syllogism-like text, he elaborates the proper forms and defects of the three-member counterpart adopted by the Buddhists. Similarly, though aṭikaḷ identifies nayam (based on multivalent logic) as a trait of Buddhist syllogistic text, the theory of inference he expounds is based on bivalent logic.

Introduction Both the Tamil epics, nīlakēci (hereafter, N) and maṇimēkalai (hereafter, MM (The present writer concurs with those scholars who date nīlakēci to fifth c ACE (cōmacuntaraṉār 21; Chakravarti 1984, 8) and maṇimēkalai to second or third c ACE (vēṅkaṭacāmi nāṭṭār and avvai turaicāmip piḷḷai 496–497, 529; mayilai 2002: 269–280, 160, 165, 87, 159; Krishnaswami Aiyangar 1928, xxvi; 80; Dikshitar 1978, 8, 11–19, 414; Champakalakshmi 1998, 84, 86; va. cupa. māṇikkam in tamiḻt tuṟai āciriyarkaḷ 1979, 17; Danielou xv-xvi; Balusamy 1965, 157–167) seek to establish and propagate their respective faiths, Jainism and Buddhism. The eponymous heroine nīlakēci who sets out to fulfill this objective is initiated into the doctrines of Jaina faith (by her guru, municcantiraṉ) before she goes about demolishing the arguments of her opponents. Though a considerable bulk of the text consists of the heroine’s debate with exponents of rival sects, we learn about Tamil Jaina logic mainly from her guru and from nīlakēci herself who (like her Buddhist counterpart, maṇimēkalai) probably represents the many female Tamil logicians female Tamil logicians of the past. It is true that logic is only a small part of what the guru teaches the disciple, and that it is the subject of only 3 verses out of the total 58 on Jaina philosophy, but the centrality of nayam in Jaina logic is affirmed by these verses on validative criteria. Such a criterion called nayam, based on multivalued logical (Burde 2012: 137–147, 2014: 54–56) is a standpoint of reality. Though Jaina faith is based on such a concept, proponents of this faith (like nīlakēci) seek to establish it in a verbal duel, which itself is based on two-valued logic. Based on the logical traditions of Tamil and Jainism, N’s logic is of the vākai type (vātam). Buddhism absorbs the multivalued logic of nayam and contains it in its divalent logic of truth and falsehood. Unlike her Jaina counterpart, maṇimēkalai does not challenge the exponents of other faiths to verbal battle. But such a response on her part is not due to the influence of the logic of nayam but because she had not yet been initiated into the intricacies of Buddhist logic and philosophy by her guru, aṟavaṇa aṭikaḷ. From him we learn most of the Tamil Buddhist logic enshrined in the text, though the indigenous nonreligious logic inherited from the primal society is expounded in a modified form by the aḷavai

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vāti in canto 27. The latter explains all the ten aḷavaikaḷ, whereas aṭikaḷ in the 29th canto deals with the doctrine of inference, the five inferential members, and the defective and proper forms of only three of them. To the Buddhists, only the threemember inferential text, earlier called kāṇṭikai (Selvamony 1990, 1996, 2000: 100– 119) in Tamil philosophical tradition, was a valid one.

Logic in nīlakēci In stark contrast to MM, N is more concerned with the application of logic than a theoretical discussion of the principles of the latter. However, three verses in the first canto called taruma uraic carukkam (canto on doctrinal discourse) mention aḷavaikaḷ. A Jaina preceptor (N 34) converts the demoness nīli to Jaina faith and instructs her in Jaina doctrines. He groups aḷavaikaḷ into two classes: kāṇṭal (perception) and kāṭciyil aḷavaikaḷ (non-perceptual validative criteria) and speaks of seven others which are not put under either of the two classes mentioned earlier (N 118–120). kāṇṭal or perception is of five types: aimpoṟi (perception through five senses), maṉam (sensing pleasure, pain, and memory), avati (telepathy, Bhaskar 1992, 49), maṉap pariyāyam (direct knowledge of the minds of others, especially. of their past births, Tamil Lexicon 1982), and kēvalam (perfect knowledge, especially of the past, present, and future, Tamil Lexicon 1982) (N 118). There are five types of kāṭciyil aḷavaikaḷ: niṉaivu (remembrance), mīṭṭuṇarvu (recognition), ūkam (tarka, Bhaskar 1992, 34, 56), aṇumai (inference, N, comm., cōmacuntaraṉār 1973 [hereafter, CC]), and ākama moḻi (authoritative text) (N 119). The additional validative criteria are vaippu (disposition), nayam (deixis), pukuvāyil (approach), uyir (soul), kuṇam (quality), and mārkkaṇai (margana 2018, conditions of the jīva or soul, Wisdom Library). vaippu is knowing an entity by disposing it in its proper relation to its name, form, kind of substance, and significance (verse of unidentified source, N 120, comm. CC). Knowing an entity by attributing a meaning to it without embracing or rejecting the other meanings is known as nayam. Citing a verse from a lost text, the commentator of N (120) tells us that nayam, a synonym of cuṭṭu (deixis), consists in pointing to one aspect of something by adopting one standpoint (rather than another) implying that from another standpoint, yet another aspect of the same entity could be pointed at. The third criterion, pukuvāyil consists in knowing an entity by adopting not only deixis but also other categories like relation, instrument, time, number, place, purpose, and event (verse of unidentified source, N 120, comm. CC). uyir or life, an existent substance, is also considered a validative criterion. As for kuṇam, there are 14 types, which are criteria that could validate our knowledge of the qualities (kuṇam) of the souls (uyir). mārkkaṇai is the condition of the soul ( jīva) and Jain texts speak of 14 such. Of all the validating criteria, the one that is unique to Jaina philosophy is nayam. Bhaskar conveys the same idea thus: “Jain logic proves the existence of a substance

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through Anekantavada and Nayavada” (6). Interestingly, the statement shows the connection between nayam and logic. Logic, known as nyaya, is “a solid medium to comprehend the reality by means of Pramana and Naya, which are collectively named as Yuktisastra or Nyaya (logic)” (Bhaskar 1992, 2). If so, nayam and pramana are also known as nyaya. As nayam itself is pramana or aḷavai (as the Tamil text tells us), nyaya might well be a variant of the Tamil term, nayam (Ta. nayam > nyaya), if we note that nayam (nayaṉ) meant niyāyam or justice (tirukkuṟaḷ 1949, 219). Besides mentioning nayam a couple of times in this sense, tolkāppiyam 1982, (hereafter, tol.; tol. III. 5. 41: 4; 6. 27: 3) also says that there are a few kinds of nayam (tol. III. 5. 41: 4). Unfortunately, the commentators’ glosses of the term nayam are not very helpful unless one sets the gloss against the original meaning. nacciṉārkkiṉiyar’s meaning “acatiyāṭik kūṟutal” or speaking playfully (cf. pālacuppiramaṇiyaṉ 2016, 245) and ilakkuvaṉār’s 1963, “tactful manner” (210) do not convey directly the original meaning, namely, “neither true nor false,” unless one interprets the playful or tactful as that which is neither true nor false. Though the commentators do not mention the kinds of nayam, a rare verse of unidentified source in N does: uṇmai nalliṉmai uṇmai iṉmaiyum uraikkoṇāmai uṇmai nalliṉmai uṇmai yiṉmaiyō ṭuraikkoṇāmai naṇṇiya mūṉṟum āka nayapaṅkam ēḻu (N 660, comm. CC). Translated, the verse reads: presence, absence, presenceabsence, and inexpressibility, presence inexpressible, absence inexpressible, and inexpressible presence-absence constitute aspects seven of nayam (adaptational translation by the present author). From N we learn that each nayam is a paṅkam (> pakkam) or part. The commentator glosses paṅkam as “paṅku, kūṟu” (N 660, comm. CC; kūṟu, part). Significantly, the term paṅku is a variant of the term, pāl (part; from paku, to divide > pakal, division, day, being a division of time > pāl, a division, part; paku > paṅku > Skt. saptapaṅki, seven parts). According to the ontology in tol., reality basically has three parts (pālkaḷ/paṅkukaḷ): oṉṟu pāl (one part), vēṟu pāl (other part), and oṉṟi uyarnta pāl (part emerging out of the uniting of parts, tol. III. 3. 2: 1–2). This theory of pāl/paṅkam implies that the whole reality could only be seen through its parts as exemplified in the story of the seven blind men who could see only a part of the elephant at a time. Significantly, this analogy itself is a nayam, later known as the niyāyam, namely, antakaja niyāyam (murukavēḷ 1960, 118, naṭēca kavuṇṭar 2004, 363). nayam is both a kind of validative criterion and a generic name for validative criteria. Hence mokkalaṉ, a Buddhist scholar, refers to nīlakēci’s validative criteria as “nayap piramāṇaṅkaḷ” (validative criteria of nayam, N 375). The major red herring he finds in Jainism is the concept of nayam by virtue of which the Jaina exponents attribute to one and the same entity both permanence and transitoriness, presence and absence, and identity and difference (N 376). The Jaina will aver that each validative criterion (disposition, deixis, approach, soul, condition of soul, and quality) will reveal different aspects of the same entity. Though municcantiraṉ does not introduce kāṇṭikai to nīlakēci, she is already aware of its nature, its use, and its members. Mark her references to the terms

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associated with it: pakkam (N 702: 5), kōḷ (N 414: 4; 707: 4; mēṟkōḷ, N 753: 2; 763: 3; kuṟi, N 870: 4), ētu (N 831; ētuppōli, N 367), and kāṭṭu (example; N 866: 2). Deploying all the logical resources at hand, she engages exponents of faith systems other than Jainism and wins them all in debate. To understand the logical nature of the debates, we can analyze the debate nīlakēci had with picācakaṉ, the materialist in the court of the king, mataṉacittu. Though debates with the Buddhists form the bulk of the text, the full force of Jaina logic is evident in the debate with the materialist too, as it is in the most characteristic logical aspect of the debate, namely, utti, the debating technique (tol. III. 9: 112). In fact, in the early days, logic itself was identified with utti and therefore known as Yuktisastra (Bhaskar 2). Much before nyaya came to mean logic, utti did so (Preisendanz 2010, 58–59). Therefore, we may do well to dwell on the uttikaḷ nīlakēci used in her debate.

uttikal in the Debate Between nīlakēci and pica¯cakan ¯ ˙ A close look at the uttikaḷ in tol. (III. 9. 112) shows how they facilitate oral delivery of a carefully constructed argumentative text combining aḷavai, elements of kāṇṭikai, and the traditional techniques of debate. When picācakaṉ opens the debate, he first states the position of his opponents (piṟaṉ kōḷ kūṟal, tol. III. 9. 112: 19; cf. vipakkam) and rejects the opponents’ view. Though this rejection is not based on arguments, it does convey his resentful attitude to his opponents’ position. Following this, he states his own position (taṉ kōḷ kūṟal, tol. III. 9. 112: 10; cf. sapakkam) and while doing so, he states again the view of his opponents (about the existence of the soul, N 10: 6). From nīlakēci’s arguments, we learn that picācakaṉ had compared the emergence of knowledge and pleasure as the effect of the combination of the elements to the effect of intoxication due to the combination of material ingredients of toddy. This is an evidence of the use of the utti, namely, “analogy” (oppak kūṟal, tol. III. 9. 112: 9). The debate comes to an end when picācakaṉ accepts nīlakēci’s position, which is the utti called “accepting what the opponent accepts” (piṟaṉ uṭaṉpaṭṭatu tāṉ uṭaṉpaṭal, tol. III. 9. 112: 11). Though the debate in the present case ends when the opponent accepts nīlakēci’s position, it need not in other cases, wherein the opponent does not accept the main contention but only provisionally does a part of the argument in order to refute the main contention. Before advancing her arguments against picācakaṉ’s position, nīlakēci understands the main contention of her opponent, namely, the nonexistence of soul, and tolkāppiyar calls this “understanding what is meant” (nutaliyatu aṟital, tol. III. 9. 112: 2). We may infer this from the arguments she advances. Then she organizes mentally her objections in the most effective order, and this may be the utti called, “ordering by effect” (atikāra muṟaimai, tol. III. 9. 112: 2), which also may be inferred from her argument. nīlakēci’s argument begins by focusing on the question of aḷavai. She targets the validity of the doctrinal position of picācakaṉ before zeroing in on his

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position itself. Accordingly, she challenges picācakaṉ’s rejection of the authoritative text as a validative criterion in arguments. In fact, she turns his rejection against him. The primary and original text (mutal nūl in tol. III. 9. 95; ākamam in MM) authored by a sage, regarded as the founder of a school of thought, has the force of truth (called āṇai). Quoting or invoking such a text is the utti called “speaking from authority” (āṇai kūṟal, tol. III. 9. 112: 15). The force of authority inheres in the word. This is evident in tol.’s definition of mantiram: what is uttered authoritatively by persons who could wield the power of the word (āṇaiyiṉ kiḷakkum, tol. III. 8. 171: 1; cf. maturaikkāñci 176 in (caṅka ilakkiyam 1963)). If the authority (āṇai) derived from the word itself and also from the primary text composed by a sage (muṉaivaṉ, tol. III. 9. 96) in tiṇai or primal society, in the state society depicted in the epics, it did from the doctrinal texts of each religio-philosophical school, say for example, The Kalpa Sutras (Jainism) or Dhammapada (Buddhism). Then, she restates his view in a concise manner, and this may be regarded as the use of the utti, namely, tokuttuk kūṟal or summing up (tol. III. 9. 112: 3). When nīlakēci draws out the implication of picācakaṉ’s rejection of textual authority, she employs the utti, namely, conveying what is implicit through what is explicit (vantatu koṇṭu vārātatu uṇarttal, tol. III. 9. 112: 7). When she points out that picācakaṉ’s notion of each element causing each sense leads to the notion that there is a sixth element which causes the soul, she draws out the implication of picācakaṉ’s words. The postulation of a sixth element appears to be the implication (eccam) of picācakaṉ’s view and such postulation is the utti known as “conveying the implication of what is said” (colliṉ eccam colliyāṅku uṇarttal, tol. III. 9. 112: 21). When she forces picācakaṉ to choose one of the three options he has suggested (N 10: 17), namely: 1. Each sense produces one type of knowledge. 2. Each sense produces five types of knowledge. 3. There is some other source of knowledge. and convey his decision, she adopts more than one utti. Firstly, she points out that he does “not focus on one thing” (orutalai aṉmai, tol. III. 9. 112: 14), which itself is a utti, and, secondly, that his lack of focus on a single theme causes “ambiguity in what he had said earlier” (muntu moḻintataṉ talaitaṭumāṟṟu, tol. III. 9. 112: 8). The latter could well be a utti in some cases, but here, as nīlakēci points out, it is a fault rather than a utti. Therefore, she exhorts him to adopt the uttikaḷ of “stating one thing” (orutalai moḻi, tol. III. 9. 112: 9) and “conveying the decision” (muṭintatu kāṭṭal, tol. III. 9. 112: 15). nīlakēci brings the debate to an end not by arriving at a conclusion but by wielding her intimidating supernatural power. She takes the form of a ghost and terrifies him into accepting her position. Such a debate is an example of the vātam type based on two-valued logic, the values being truth and falsehood. Now let us see how logic fares in MM.

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Logic in manime¯kalai ˙ In MM, logic-related matters are dealt with in two cantos – “canto on what was heard from religious exponents on their positions” and “canto on what she heard on the True Way in the disguise of a nun” – mainly in the form of principles. In the former canto, maṇimēkalai visits the city of vañci and hears the exposition of the doctrinal position of ten religio-philosophical exponents (Kanagasabhai). Of the ten positions, each of the following six – aḷavai, Saivism, brahmaism, nikaṇṭam, vaicēṭikam, and pūtam – is identified as “vātam” or verbal duel. The term, aḷavai in aḷavai vātam, the first system, is logic itself. However, though the term “vātam” is tagged to brahmaism, nikaṇṭam, and pūtam, no argument is advanced by the exponent of each of these schools. Each merely states his position. The briefest assertion, namely, “nāraṇaṉ is solace” (MM 27: 99), comes from the exponent who recited the epic of the ocean-hued one. With a mythological narrative (purāṇam) as its authoritative text, the exponent would do well to chant (ōtiṉāṉ, MM 27: 99) rather than debate it. A close contender is the expounder of brahmaism who affirms that this world is an egg laid by a deity (MM 27: 96–97) though this expounder is identified as a “vāti,” a debater. The term “urai” (utterance, commentary) expresses the nature of vedism. Vedists choose to expound the Veda rather than debate it and that is why it is necessary for cāttaṉār to name the position of Vedism, “vētiyaṉ urai.” If the school of Ajivaka is a textual tradition, Sankhya is a doctrinal school with religious fervor (as indicated by the word, “matam”). The first exponent met by maṇimēkalai, namely, aḷavai vāti defines aḷavai, which generically means logic and, specifically, epistemic validative criterion. Such logic embraced epistemology, rhetoric, what we today call “research methodology” and even ontology. The logician (aḷavai vāti), who might well be a representative of the indigenous logical tradition, chooses to use more local Tamil technical terms than their Sanskritic equivalents unlike aṭikaḷ who prefers the Sanskritic terms to the local ones. The aḷavaikaḷ of aḷavai vāti are: kāṭci/kāṇṭal (perception, MM 27: 14–24), karuttaḷavu (karuttaḷavai, inference, MM 27: 25–40), uvamai (analogy, MM 27: 41– 42), ākamam (authoritative text, MM 27: 43–44), aruttāpatti (inferring from what is akin to what is said, MM 27: 45–46), iyalpu (appropriateness in a context, MM 27: 47–48), aitikam (popular belief, MM 27: 49–50), apāvam (absence, MM 27: 51–52), mīṭci (implication, MM 27: 53–54), and uḷḷaneṟi (nature or what is natural in a situation, MM 27: 55–56). The defects associated with aḷavaikaḷ (piramāṇa āpācaṅkaḷ) are the following: cuṭṭuṇarvu, tiriyak kōṭal, aiyam, tērātu teḷital, kaṇṭuṇarāmai, aṟivaṟiyāmai, ilvaḻakku, uṇarntatai uṇartal, and niṉaippu. cuṭṭuṇarvu consists in considering only one aspect of an object to know what it is. For example, when one is in doubt whether what one sees is a tree stump or a human, one’s inference is based only on the appearance of the object, and not on its other aspects such as name, class, quality, and action. Further, the doubt (aiyam) in the mind of the viewer itself is a defect. On the other hand, without any doubt if one were to take the tree stump for a human, the defect is called, tērātu teḷital, which

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means “deciding without due consideration.” tiriyak kōṭal is misperception; taking one thing for another. The example adduced is regarding radiant seashell as silver. If one encounters a tiger and does not realize that the animal is at close quarters, the defect is kaṇṭuṇarāmai or “not knowing even when seen.” ilvaḻakku consists in presupposing the existence of nonexistent things like “hare horn.” Though it is a fact that fire is hot and there is chillness in fever, fire is not the remedy for fever. One presumes so because of regarding something inappropriate from what one knows or “uṇarntatai uṇartal.” niṉaippu is going by what others say. For example, if one regards a couple as one’s parents because others say so, it is defective knowledge. Of the ten aḷavaikaḷ, only two were postulated by āticinēntiraṉ, the earliest Buddhist teacher who probably was a Jaina at first (Danielou 1993, 152). He could have been a Jaina because Jainism is a much older faith system than Buddhism, which had borrowed several concepts from the former (Burde 2014: 43–45; Selvamony 2008: 45). According to Dasgupta 1932, Bhadrabahu’s Jaina text on logic predates the earliest Buddhist texts on logic (309) and mayilai cīṉi vēṇkaṭacāmi also acknowledges Jainism as the oldest religion in Tamil Nadu (1970, 40). Influenced by Jainism, āticinēntiraṉ accepted only two validative criteria, kāṇṭal and karuttaḷavu, and among the two, considered the latter of utmost significance in Buddhist logic. To aḷavai vāti, karuttaḷavu has two components: aṉumāṉam, the proposition, and aṉumēyam, the reason (which are also the members of kāṇṭikai). It is of three kinds: mutal, eccam, and potu. mutal (first) is inferring the effect from the cause as in the case of predicting rain (effect) from the rain clouds (cause). eccam (remainder) is inferring the cause from the effect, as when the rain in the highlands (cause) is inferred from the wet lowland (effect). potu is inferring from what is common to both factum probandum and factum probans, as in the case of inferring the presence of an elephant in the forest from the trumpeting that is heard (MM 27: 25–40). According to aṭikaḷ, karuttaḷavai (inference) is of three types: kāraṇam, kāriyam, and cāmāṉiyam (MM 29: 52–56), which correspond to aḷavai vāti’s mutal, eccam, and potu. Of these three, he recognizes only kāriyāṉumāṉam (eccam) or inferring cause from effect, as the valid one. If the effect is pakkam (proposition), cause is ētu (reason). Evidently, these are only two of the five members of kāṇṭikai, though aṭikaḷ’s variety is a three-membered one, as the members, upanayam and nikamaṉam, in his opinion, are subsumed by the member, illustration (MM 29: 109–110). Based on the concept of nayam, which here means, analogy as justification, upanayam (secondary nayam) particularizes the general rule stated in illustration. analogy is nayam because it was the predominant impartial device (acceptable to both parties) used to settle disputes and such dispute resolution was called niyāyam, a variant of nayam, which means naṭu or impartiality. Just as nayam links reason and proposition in the middle (naṭu/naṭai, iṭakikaḻi, vestibule as the connecting middle, Tamil Lexicon 1982, IV, 2145) of kāṇṭikai, upanayam walks the argument and, therefore, it is called naṭai (walkway, tol. III. 9. 104: 3). An equivalent of niyāyam, oppuravu combines the ideas of both justice (cf. tirukkuṟaḷ 1949, 219, comm.) and analogy (oppu, analogy + uravu, strength). The fifth member is nikamaṉam (muṭivu; cf. tol. III. 9. 104: 2).

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Considering the preeminent role of the concept of nayam (employed by the confidanté in tol. III. 5. 41: 4) based on analogy in kāṇṭikai, we might infer that originally argument was based solely on nayam (niyāyam or analogy as in mutumoḻi). Later on when the force of both ētu (reason; employed by personae in tol. III. 5. 12: 2; 4. 27: 4) and nayam were deployed together to establish the proposition, then emerged the five-part text, kāṇṭikai. Now let us see what MM says about the first three members of kāṇṭikai in some detail.

Proposition (pakkam, Pn) The proposition (hereafter, Pn; kōḷ [that which is “apprehended,” position], mēṟkōḷ [position adopted earlier], cūttiram [cūḻ, to surround + tiram, firmness ¼ cūttiram, what is firmly held after considering all sides of an issue]) has two parts: subject (hereafter, S; poruḷ; taṉmi; vicēṭiyam; e.g., hill) and predicate (hereafter, Pr; pakkam, tuṇi poruḷukku iṭam [locus of factum probandum], civañāṉa cittiyār 1946, 15; taṉmam; vicēṭiyam; e.g., fire). If Pn (kōḷ) is “There is fire in this hill,” “hill” is S (poruḷ/taṉmi), whose Pr, “there is fire,” is its pakkam (the phrase, “pakkamum taṉmiyum” [in vēṅkaṭacāmi nāṭṭār 1946, and avvai turaicāmip piḷḷai {hereafter, V&A 1946} 486] shows pakkam and taṉmi are two different entities). pakkam also particularizes the argument of kāṇṭikai unlike the latter’s Aristotelian counterpart (Kanagasabhai 2017, 226; Randle 2017). Though pakkam is Pr of Pn, often Pn itself is called pakkam (“side” > Skt. paksha) in Tamil because Pn lends itself for taking sides: one’s own side (i.e., what one asserts or self-side, called taṉ kōḷ [“self-assertion”], tol. III. 9. 112: 10) or capakkam [< svapakkam], and the opponent’s side (piṟaṉ kōḷ [“the other’s assertion”], tol. III. 9. 112: 19) or vipakkam (V&A 1946 479). For example, if A asserts “There is fire in the hill,” the opponent, B, is expected to disagree with A holding the view that “There is no fire in the hill.” The disagreement between A and B is essential for debate. When A and B agree on a Pn, as in the case of a Pn marked by appiracitta campantam (a Pr on which there is no disagreement between the contenders), there is no possibility of an argument, resulting in pōli or pseudo argument (Matilal 2017: 28; Raja 286). They could disagree because A and B could take different sides (pakkam1 and pakkam2) of one and the same proposition. The term pakkam needs closer attention. It derives from the verbal base, paku, to divide > pakku, break + am, nominal suffix ¼ pakkam, part (pakuti), side, place. pakkam is the part or side or place of the predication of the S that has to be concluded (pakkam tuṇiporuḷukk(u) iṭamām, civañāṉa cittiyār 1946, 15). When one asserts something, one can only assert a part (pakkam/pāl) of reality (poruḷ), never its whole, and hence tol. asserts that reality obtains as one part, the other part, and also as the united part (III. 3. 2: 1–2). The part of reality asserted is a Pr that occurs only in a definite locus, namely, pakkam. For this reason, an action itself, rather than its locus, is often metonymically referred to as “pakkam” (tol. III. 2. 7: 6; 11: 9; 14: 5; 16: 1, 2, 3, 6).

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Importantly, pakkam is marked by two nayaṅkaḷ: oṟṟumai (identity) and vēṟṟumai (difference) (MM 29: 112–121), a point that easily goes unnoticed as in Kandaswamy’s discussion of nayam in MM (2000: 40–42). Later, aṟavaṇa aṭikaḷ explains these nayaṅkaḷ (MM 30: 217–234). If oṟṟumai is knowing cause and effect as one and the same thing (speaking of rice grain as rice), vēṟṟumai is being able to see them separately (V&A 1946 565–566). When nayam is applied to the proposition of kāṇṭikai, oṟṟumai nayam consists in the identity of the poruḷ and pakkam (S and Pr), and vēṟṟumai nayam does in the inapplicability of Pr to all entities other than S. Accordingly, a Pn can assert only one Pr negating its opposite. For example, the Pn, “sound is impermanent,” is a valid one because it asserts impermanence by negating its opposite, namely, permanence. This means that vēṟṟumai nayam creates the condition for argument by enabling assertion of a Pr and rejection of its opposite. Such a conception of Pn shows two things: (a) how the Buddhist scholars have adopted the theory of (multivalued) nayam for their two-valued logic and (b) how this conception could be the source of Dinnaga’s trairupya (triple-character) concept of which he himself is not the author (Matilal and Evans 2012: 1). However, later, when aṭikaḷ speaks of the uses of nayaṅkaḷ, he says that these help us see how everything is related to each other ruling out the possibility of discrete entities and also understand how one thing could be both cause and non-cause for an effect. The latter may be evident in the case of the seed being both the sprout as well as not the sprout (V&A 1946 567). From what aṭikaḷ says about nayam, it is apparent that there does emerge space for multivalued logic in Tamil Buddhism. But this space remains a gap in a text which privileges bi-valued logic especially in the praxis of debate. A good Pn has to avoid nine defects (MM 29: 147; Raja 286). 1. appiracitta vicēṭiyam: S (such as the hill itself) is unknown to opponent (MM 29: 151–152; Aiyaswami Sastri 1972, 128; Raja 1998, 288; Hikosaka 1989, 140). 2. appiracitta vicēṭaṇam: Pr (such as presence of fire) is unknown to opponent (MM 29: 150–151; Aiyaswami Sastri 1972, 128; Raja 1998, 288; Hikosaka 1989, 140). 3. appiracitta upayam: Both S and Pr are unknown to opponent (MM 29: 152; Aiyaswami Sastri 1972, 128; Raja 1998, 288; Hikosaka 141). 4. When a contrary Pr (such as infertility) is attributed to S (one’s own mother), we have the defect of cuvacaṉa viruttam (MM 29: 149; the variant, “cuvavacaṉa viruttam” [contrary self-utterance] is found in V&A 491; Aiyaswami Sastri 1972, 128; Raja 288; Hikosaka 139). 5. Applying contradictory Pr (such as permanence) to S (such as a pot) is the defect, aṉumāṉa viruttam (MM 29: 148–149; Aiyaswami Sastri 128; Raja 288; Hikosaka 139). 6. appiracitta campantam is a pakkam (Pr) on which there is no disagreement between the contenders (MM 29: 153; Aiyaswami Sastri 1972, 129). For example, if the Buddhist is the opponent, the Pn, “sound is impermanent” will be defective because he is not likely to disagree with that view. V&A point out that

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cāttaṉār speaks only of appiracitta campantam, not piracitta campantam (Raja 289; Hikosaka 141) and that Dinnaga’s Nyayapravesa, which succeeded maṇimēkalai and adopted its ideas (V&A 1946 496–497; 491), wrongly denoted the name of this defect as piracitta campantam (see Hikosaka [85–86] for the question of authorship of Nyayapravesa). 7. “To take the reverse of” (Aiyaswami Sastri 1972, 128) what is perceived is pirattiyakka viruttam (MM 29: 148; perceptual error; Raja 287; Hikosaka 139). “Sound is inaudible” is such a defective Pn. 8. Arguing against one’s own authoritative text is the defect, ākama viruttam (MM 29: 150; error related to authoritative text; Aiyaswami Sastri 1972, 128; Raja 288; Hikosaka 140), which is evident if a Vaiseshika were to assert, “Sound is permanent” contradicting the view of his own ākamam. 9. Contradiction of common knowledge (such as not accepting that what is seen in the sky is the moon) is ulaka viruttam (MM 29: 149–150; error pertaining to common knowledge; Aiyaswami Sastri 1972, 128; Raja 288; Hikosaka 140). The first three are defects due to incomprehensibility, which is a textual flaw according to tol. (III. 9. 110: 8). The fourth and fifth defects arise due to uttering something that is universally meaningless. These two could be subsumed under the type of textual flaw, poruḷila kūṟal of tol. (III. 9. 110: 4; uttering that which is meaningless). The remaining four are defective application of aḷavaikaḷ such as pirattiyakkam, aṉumāṉam, ākamam, and ulakurai.

Reason (ētu, R) The second member of karuttaḷavu is ētu (hereafter, R; from e, interrogative pronominal stem [Gnana Prakasar 1999, 396] as in evvayiṉ [e + vayiṉ; which place? tol. II. 9. 32] > etu, which? > ētu, which? > Skt. hetu; cātaṉam; e.g., “Because there is smoke”), which means both reason and cause. For example, when we infer the presence of the elephant in the forest from trumpeting, the reason for such inference is trumpeting. Therefore, trumpeting is ētu. But the cause of trumpeting is the elephant. What is at dispute is not whether the elephant is the cause of trumpeting or not, but whether the inferer’s reason for affirming the presence of the elephant in the forest is the trumpeting heard in the forest or not. Such inference is based on the presupposition that only elephants (and not sound recording of trumpeting or any other thing) could trumpet ensuring the concomitance of elephant and trumpeting. A normative R has three characteristics: R must be invariably concomitant with Pr; R should apply to all similar instances (capakkam) and should not apply to all dissimilar instances (vipakkam; MM 29: 121–135; cf. Raja 285; Hikosaka 141–142). Defects of R arise due to unacceptability (acittam) of Pn or violation of the law of identity (when both p and not-p are true; aṉaikāntikam) or contrariety (viruttam) (MM 29: 191–192).

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Unacceptable (acittam) Types of R For the Pn, “sound is impermanent,” the R, “because it is visible” is futile because it is upayācittam (MM 29: 195–197) or unacceptable to both sides. If the Pn is “sound is impermanent,” the R, “because it is produced” is unacceptable (aṉṉiyatarācittam, MM 29: 198–202) to an adherent of Sankhya system because the latter is likely to understand R as “being produced by primal Matter itself.” When one is not able to tell whether the R for the fire in the hill is smoke or mist, such R is culpable of dubiety (cittācittam, MM 29: 203–206).

Types of Non-unidirectional (anaika¯ntika) R ¯ cātāraṇāṉaikāntika ētu (MM 29: 217–223; common non-unidirectional R) is that which applies to both one’s own Pr and that of the opponent. For Pn, “sound is impermanent,” R, “because it is known,” is defective as it applies to both permanence and impermanence. When R applies to S, but not to either one’s own Pr (capakkam) or that of the opponent (vipakkam), it is said to be acātāraṇāṉaikāntika ētu (MM 29: 223–230). If the Pn is “sound is permanent” and its R is “because it is audible,” audibility applies to S, sound. But, as permanence (capakkam) or impermanence (vipakkam) cannot be caused by audibility, audibility is not a valid reason to affirm permanence of sound or any other substance. When R applies partially to the proposition, and wholly to the opposite of the latter, such R is said to be capakka ēkatēca virutti vipakka viyāpi (MM 29: 231–242). In the Pn, “sound is not produced by any action because it is impermanent,” its R, impermanence, applies to one similar instance (capakkam) like lightning, but not to another such as ether (ākācam) as sound is a permanent quality of ether. Further, such R applies to all opposite instances (vipakkam) such as the pot. vipakka ēkatēca virutti capakka viyāpi (MM 29: 243–253) is R, which applies partially to the opposite (vipakkam) of the Pn and wholly to one’s own assertion (capakkam). In the Pn, “sound is produced by action because it is impermanent,” its R, impermanence, applies to the S, lightning, but not to the S, ether (vipakkam), of the opponent. It also applies to all Ss of capakkam such as the pot as well. As it applies to both one’s own assertion and also to the opposite assertion of the opponent’s, such R is defective. If R applies partially to both one’s own assertion and also to the opposite assertion, then, such R is upaya ēkatēca virutti (MM 29: 254–266). For the Pn, “sound is permanent because it is formless,” the R, formlessness, applies to one eternal S, ether, but not to another (eternal S), atom. Further, among those entities of vipakkam (impermanence) like pot and pleasure, R, namely, formlessness, applies to pleasure but not to pot. The defect of non-unidirectionality (aṉaikāntikam) is due to the uncertain relation between the R, formlessness, and the Pr (permanence) raising such unanswered questions like “Does formlessness cause permanence?” and “Are formless entities permanent like ether or impermanent like pleasure?”

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When “one and the same R” is adduced by both parties to arrive at opposite conclusions, such R is called virutta viyapicāri (MM 29: 267–274). The Pn, “sound is impermanent because it is produced by action,” is defective in capakkam and vipakkam instances. In the case of capakkam (impermanence), such as the pot, the impermanence of sound could be because it is produced by action. In the case of vipakkam (permanence) also, one can say that the permanent, generic audible nature of sound is due to the fact that it is produced by action.

Contrary (viruttam) Rs taṉmac corūpa viparīta cātaṉam (MM 29: 280–287) is R which establishes the opposite of what the Pr asserts. If the Pn is “sound is permanent because it is produced,” its R is defective because it asserts the opposite of what is affirmed, namely, impermanence. taṉma vicēṭa viparīta cātaṉam (MM 29: 288–302) is one that destroys the Pr’s unique quality (vicēṭam) necessary to establish the truth of the Pn. In the case of the Pn, “the soul is served by the perceptual organs because the soul is inseparable from the body,” the R, “inseparableness from the body,” destroys the true nature of the soul, namely, its non-dependence rather than dependence on the body. The unique “taṉmam” or Pr (non-dependence) of the S, namely, the soul, is destroyed by the R, bodily dependence. Unfortunately, cāttaṉār (through aṭikaḷ) depends a great deal on the example to explain this defective R (MM 29: 295–301), so much so one wonders whether his explanation applies to R or E. Therefore, the explanation given here avoids using the member, namely, example altogether. taṉmic corūpa viparīta cātaṉam (MM 29: 303–318) is R, which destroys the unique feature of S. The R of the Pn – “bhava is neither substance nor quality nor action,” “because it is not the existent reality (uṇmai) of substance, quality, and action” – is defective because it destroys the unique nature of the S (taṉmi), namely, bhava. The unique nature (corupa) of the S (taṉmi), bhava is existence, which is possible only through entities such as substance, quality, and action. By denying that bhava is the existent reality of substance, quality, and action, the R renders bhava, abhava. taṉma vicēṭa viparīta cātaṉam (MM 29: 319–325) is R, which negates the special quality of S. The R of the Pn, “bhava is not the existent reality (uṇmai) of substance, quality, and action,” “because it is neither substance, quality, nor action,” is defective because it destroys the unique nature of S (taṉmi), namely, bhava. In this type of R, the R of the previous type becomes the Pn.

Illustration (titta¯ntam, E) ˙˙ Normative E combines both invariable concomitance and example. Invariable concomitance between R (the presence of smoke) and Pr (fire) is called “aṉṉuvayam” (MM 29: 88; uṭaṉikaḻcci, V&A 1946 482). The Pr, “there is fire,” and R, “because

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there is smoke,” are connected by the following aṉṉuvayam: “Wherever there is smoke, there is fire.” The concomitance is invariable because the connection between R and Pr is universal, necessary, and logical (Stcherbatsky 2008, 325). The connection between “being made” (R) and “sound” (S), on the other hand, is real and factual (Stcherbatsky 2008, 325). It is a fact (and reality) that sound is something that is made. E is of two types: cātaṉmiyam and vaitaṉmiyam (MM 29: 136–142). cātaṉmiyam is that which has the Pr of S and R (V&A 1946 523). If the Pn is “there is fire in this hill” and the R is “because there is smoke,” then the E will be “wherever there is smoke there is fire (aṉṉuvayam), like the fireplace” (example). vaitaṉmiya tiṭṭāntam is one which has the opposite Pr of S and R (V&A 1946 523). If the Pn is “there is no fire in this mountain” and the R is “because there is no smoke,” the E will be “wherever there is no smoke, there is no fire, like a lake.”

Defective Es of the ca¯tanmiyam Type ¯ E in which R is not adequately present (cātaṉa taṉma vikalam, MM 29: 339–348): Let the Pn be “sound is permanent” and its R, “because it is formless.” If the E of such Pn is “whatever is formless is permanent, like the atom,” then the atom (example of E) is both permanent and has form and contradicts the R, and therefore, the E is defective. E in which the Pr of S is not adequately present (cāttiya taṉma vikalam, MM 29: 349–358): If the Pn is “sound is permanent,” and its R, “because it is formless,” and the E, “whatever is formless is permanent, like the mind,” the E is defective as it will be marked by the formlessness of R and lack the permanence of the Pr. E in which both R and Pr are not adequately present (caṉṉāuḷḷa upaya taṉma vikalam, MM 29: 363–372): If the Pn is “sound is permanent” and its R, “because it has form,” and the E, “whatever is formless is permanent, like the pot,” then the pot (E) will lack both the permanence of Pr and formlessness of R. Nonexistent example lacking both Prs of Pn and (acaṉṉāvuḷḷa upaya taṉma vikalam, MM 29: 373–384): If the Pn is “sound is impermanent,” and its R, “because it is formless,” and E, “whatever has form is impermanent, like the ether,” the E, which lacks both the impermanence of Pr and the form of R, will be defective if the opponent is one who denies the existence of the ether itself. aṉaṉṉuvayam (lack of invariable concomitance; MM 29: 385–392): If the Pn, “sound is impermanent because it is made,” is followed by an improper E such as “impermanence and the quality of being made are found in the pot,” without the aṉṉuvayam, namely, whatever is produced is impermanent,” such E is marked by the defect, aṉaṉṉuvayam. If the Pn and its R are “sound is impermanent, because it is produced,” the aṉṉuvayam, namely, “whatever is impermanent is produced,” is absurd

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(viparītāṉṉuvayam, MM 29: 393–401) because it reverses the order of the proper aṉṉuvayam, “whatever is produced is impermanent.”

Defective Es of the vaitanmiyam Type ¯ cāttiyāviyā virutti (lacks Pr of S; MM 29: 402–412): Consider the Pn, “sound is permanent, because it is formless.” If its E is “whatever is not permanent is not formless, like the atom,” it will be defective because it will have the Pr of permanence but not the formlessness of R. This is so because the atom has form and is permanent. cātaṉāviyā virutti (incompatible R; MM 29: 413–423): If the Pn is “sound is permanent because it is formless,” the E, “whatever is not permanent is not formless, like action,” is defective because only the opposite of Pr, namely, permanence, will apply to action, not the opposite of formlessness. This is so because action is formless and impermanent. uṇmaiyiṉ upayāviyā virutti (lacks Pr of S and R of existent example; MM 29: 429–440): If the Pn is “sound is permanent because it is formless,” the E, “whatever is impermanent is not formless, like the ether,” is defective because the opposites of both permanence of Pr and formlessness of R do not apply to the ether. This is so because the ether is both permanent and formless. iṉmaiyiṉ upayāviyā virutti (lacks Pr of S and R of nonexistent example; MM 29: 440–449): If the Pn is “sound is impermanent because it has form,” the E, namely, “whatever is not impermanent, is not formless, like the ether,” is defective to one who denies the existence of the ether, as both the asserted qualities, impermanence of Pr and having form, will not apply negatively as well as positively. avvetirēkam (non-opposite E; MM 29: 450–456): In the Pn, “sound is permanent because it is not produced,” the opposite E should be “whatever is not permanent is not non-produced, like the pot.” But instead of such an E with a proper aṉṉuvayam, a non-opposite (avvetirēkam) E such as “the pot is produced and impermanent” (lacking aṉṉuvayam) is defective. viparīta vetirēkam (inverted opposite E; MM 29: 460–468): If the Pn is, “sound is permanent because it has form,” the E, “whatever has no form is also not permanent,” is defective. The proper negative E will be: “whatever is not permanent is formless” (Raja 296). Though the five members of karuttaḷavai (kāṇṭikai) are mentioned by aṭikaḷ, the fallacies associated only with three members (Pn, R, and E) are explained by him. This is because only the three-member karuttaḷavai was favored by the Tamil Buddhist scholars like aṭikaḷ despite his description of the five-member kāṇṭikai in his discourse to his ward and the wide prevalence of the latter in the Tamil tradition. However, the three-member karuttaḷavai, according to aṭikaḷ, helps estimate the worth of one’s own philosophical position as well as that of others and tell truth from falsehood (MM 29: 470–473).

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Truth has to be the opposite of falsehood in order to validate logic. So, it has to be ēkāntika (assertion of either p or not-p), a kind of logical reductionism the Nyaya system expounded (Matilal 1999: 4) probably due to the influence of logic as vātam. Similarly, what is asserted must be either citta (real) or acitta (unreal). It cannot be both. If it is cittācittam (both citta and acitta at the same time), it is nothing but defective. In short, what we find in MM is two-valued logic.

nayam/niya¯yam and va¯tam: Types of Logic in the Texts Evidently, MM inherited its logic from tol.as well as from the philosophical school known as Nyaya, the earliest systematic logic in Sanskrit (Vidyabhusana 1978, 42). The influence of the latter is particularly evident in the identification of inference (karuttaḷavai) with kāṇṭikai (syllogistic text in Tamil tradition). Such identification is first found in Gotama’s 2016, Nyaya Sutras wherein the proposition of a syllogistic argument is treated as the statement of inference that requires proof (sutra 33) and, therefore, sutra 32 that mentions the five members of the syllogistic argument is placed after sutra 31, which speaks of a tenet that requires proof, in short, an inference. Unlike such treatment of the syllogistic argument in Nyaya, in tol. kāṇṭikai is treated as a kind of text (nūl). In MM, nyaya is also closely related to the concept of nayam expounded in tol. and Jainism. According to cuvāmināta tēcikar, nyaya originated from tol. and tirukkuṟaḷ (murukavēḷ 1960, 117). Let us consider the Tamil term niyāyam. As a variant of nayam (Ta. nayam > Pkt. nyaya [cf. Ta. pakuti, division of being > Skt. prakriti, primal substance; pa > pra] > Ta. niyāyam; nallapōlavum nayava pōlavum [puṟanānūṟu 58 in caṅka ilakkiyam] wherein nayam means niyāyam, katiravēl piḷḷai 1923, 1347) niyāyam originally meant naṭu or the middle (Winslow 1979, 672), which implied impartiality. naṭu is impartiality as it is neither of the two sides of the contending parties as in vātam. When nayam means naṭu, it means both impartiality (akanānūṟu 71: 1, 3; and kuṟuntokai 143: 4, 5, in (caṅka ilakkiyam 1963)) and non-monistic non-duality, which is neither p nor not-p; neither truth nor falsehood (tol. III. 5. 41: 4; “naṭuvāka nōkku” consider the middle, N 644: 4). When niyāyam means naṭu, it also means both impartiality and non-duality. In the post-caṅkam age, niyāyam was also used as a synonym of the earlier term mutumoḻi (tol. III. 8. 170; vaittiyanāta tēcikar 1974, 329–332), which referred to a primitive form of kāṇṭikai. Consequently, niyāyam itself came to mean kāṇṭikai (Vidyabhusana 1978, 42) and its variant, mutumoḻi. The latter was often a proverb, which explained something analogically. Analogy performed the function of ētu of kāṇṭikai – rationalizing or clarifying a difficult situation or answering a question or solving a problem. Such a solution or answer or clarification is niyāyam because it sorts out an issue without taking either of the two contending sides. In short, niyāyam was an impartial analogical utterance that sought to sort out an issue of two contending parties. For example, akaram mutal niyāyam or “the originariness of the phoneme /a/” is a niyāyam because it is an

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analogy between the originary speech sound and god as the creator of the universe (naṭēca kavuṇṭar 2004, 17–18). Even as the speech sound /a/ precedes all sounds, it also in-dwells in all of them just like god does in the whole world. akaram is niyāyam or nayam because it is neither wholly precedent nor non-precedent. Later, niyāyam meant not only analogy, and the “syllogistic” member, E, but the entire syllogistic text itself (murukavēḷ 1960, 144). As an analogy, it affirms a truth in a debate without taking either side of the contention. In Sanskrit sources, nyaya basically means “a system of philosophy delivered by Gotama 2016, or Gautama . . . is perhaps so called, because it ‘goes into’ all subjects physical and metaphysical according to the . . . syllogistic method treated of in one division of the system” (Williams 1964, 572). Williams’s explanation of the term shows how the meaning of nyaya in Sanskrit sources is quite different from its Tamil counterpart. Though MM was influenced by the Nyaya school, cāttaṉār was not persuaded to use the term nyaya, probably because it was not used in Tamil in the sense in which it was in non-Tamil texts. Neither did it denote “going into physical and metaphysical subjects,” nor was it free of the sense of non-duality in the Tamil context. Though the idea of non-duality existed in Tamil Buddhism in its concept of nayam (MM 30: 217–234; 29: 112–119), the kind of logic Tamil Buddhists were familiar with was a two-valued one and the term vātam described it better than niyāyam did. Evidently, MM identifies several exponents of faith (camayak kaṇakkar) as vāṭikaḷ, though its heroine does not engage any opponent in a vātam. In the case of N, it not only mentions the term vātam but also exemplifies the genre by constructing its plot (for the most part) with the debates between the heroine and the exponents of faiths other than her own. Therefore, we may say that vātam is the kind of logic that is common to both the texts. tol. calls this genre vākai and explains it as one in which each side asserts its belief (koḷkai) in an analytical form (III. 2. 15), which could imply the use of the argumentative text such as kāṇṭikai (III. 9. 103– 104, commentary, iḷampūraṇar; 101) and utti or technique of debate (III. 9. 112). Interestingly, one of the terms for proposition of kāṇṭikai, namely, kōḷ, is also the stem of the word, koḷkai, meaning “tenet” (koḷ to grasp > kōḷ what is grasped [such as a tenet], grasping/koḷkai tenet). Without a kōḷ/koḷkai, vātam (also vākai) is not possible. As we know, vātam is verbal duel, which was a part of tiṇai society, especially its social aspect called puṟam. In fact, combat is the characteristic praxis of puṟam life. So, both combat and verbal duel (vātam) were typical practices of the primal society. In stark contrast, the central personae of akam (the domestic aspect of tiṇai society) did not contend with each other as did their counterparts in puṟam. Their objective was union based on love (aṉpu; aṉpoṭu puṇarnta aintiṇai, tol. III. 3. 1: 2). If so, both union (of akam) and contention (of puṟam) were the two sides of the tiṇai coin. As there are evidences, in tiṇai society, of the use of nayam, based on multivalued logic, and the practice of contentious debate based on two-valued logic, we might say that the primal (tiṇai) society employed both logics, the two-valued and the multivalued.

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Contentious debate became the norm in the Tamil state society when the state itself sponsored religious polemic (as the Tamil epics testify). Though multivalued logic survived in the form of theory in both Jainism and Buddhism, it did not translate itself into popular practice at least in encounters of faith. This is evident in the 29th canto wherein aṟavaṇa aṭikaḷ focuses on the aḷavaikaḷ, namely, inference, reduces inference to the Tamil syllogistic text, and gives an elaborate account of the true and false forms of the three members – proposition, reason, and example. The rationale of these defective forms is two-valued logic. Though multivalued logic was the basis of Jaina philosophy, N is not based on such logic; its debate is not niyāyam (which was nayam-oriented) but of the vātam type based on two-valued logic. In the praxis of nīlakēci’s vātam, most of the uttikaḷ explained in tol. are exemplified. Though uttikaḷ were used both in vātam and niyāyam, they seem to have played a great role in fashioning the armory of logic by lending such technical terms as taṉ kōḷ (sapakkam) and piṟaṉkōḷ (vipakkam). True, maṇimēkalai does not practice vātam, but the bivalent logic on which the latter is based underlies aṭikaḷ’s exposition of Buddhist truth.

Definitions of Key Terms aḷavai ētu eṭuttukkāṭṭu karuttaḷavai kāṇṭal/kāṭci kōḷ/mēṟkōḷ nayam

pakkam pōli upanayam

utti vākai/vātam/vātu

Validative criterion; logic Reason; cause Illustration Inference Perception Proposition Teleologically positive middle between truth and falsehood; non-bivalent logic; also known as cuṭṭu (deixis, nīlakēci 120) or what is pointed at, which is neither truth nor falsehood about the entity pointed at, but a middle aspect Predicate of proposition; sometimes denotes the proposition itself Defective form of members of kāṇṭikai (upa, secondary + nayam [analogical mutumoli or proverbial language as] justification ¼ upanayam) Application, the fourth member of the five-membered kāṇṭikai; so called as it follows the nayam or the analogical member (as justification). If nayam as the example applies the universal principle (such as “wherever there is smoke there is fire”) to a particular case, such as “the fireplace,” upanayam also performs a similar function by applying reason (presence of smoke) to a particular subject (the hill). Technique (of debate) Logic as verbal duel

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Summary Points • nīlakēci contains several debates between the eponymous heroine and the Buddhists and exponents of the other faiths. Among all these debates that employ the debating techniques called uttikaḷ, the one between the heroine and the materialist is taken as a sample to show how these techniques are employed in the debate. – Though nayam, a concept based on multivalued logic, is central to Jaina logic, nīlakēci’s debate is based on two-valued logic. • In maṇimēkalai, aḷavaikaḷ discusses the ten aḷavikaḷ, the defects pertaining to them, and the three kinds of inference. – maṇimēkalai’s guru, aṟavaṇa aṭikaḷ, discusses inference elaborately, especially the three members of what Tamil scholars knew as kāṇṭikai, and briefly the concept of nayam. – Though the logic in maṇimēkalai accommodates the concept of nayam, its logic remains two-valued.

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Introduction to Buddhist Logicians and their Texts Madhumita Chattopadhyay

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plan of the Chapter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nāgārjuna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pre-Dignāga Buddhist Texts: Vasubandhu (Vādavidhi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dignāga: Nyāyamukha, Hetucakradamaru, and Pramāṇasamuccaya . . . . . . . . . . . . . . . . . . . . . . . Śankarasvāmin: Nyāyapraves´a . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dharmakīrti: Nyāyabindu, Hetubindu, Pramāṇavārttika, and Vādanyāya . . . . . . . . . . . . . . . . . . Sanskrit Commentators of Dharmakīrti: Dharmottara, Vinītadeva, Arcaṭa, and Prajñākaragupta . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Tibetan Commentators of Dharmakīrti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Post-Dharmakīrti Logicians: Jñānaśrīmitra, Ratnakīrti, Mokṣākaragupta . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

From the origin of Buddhism in the sixth century BC to its expansion into four schools, logic was not looked upon as an independent branch of study within philosophy, though scattered uses of logical concepts and arguments were not rare in early Buddhist literature. Since the middle of fifth century AD, Buddhist logic developed into an independent area of study. This was possible because of the contributions of some great Buddhist thinkers who tried to develop logic not only as an independent study but also as a formal system where one can validly infer an unapprehended phenomenon on the basis of an apprehended phenomenon. Rules were also developed for turning debate into a rational enterprise. Pioneering works in the field of logic were done by Dignāga, who was considered M. Chattopadhyay (*) Department of Philosophy, Jadavpur University, The Center for Buddhist Studies, Kolkata, West Bengal, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_1

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to be the father of Buddhist logic, and Dharmakīrti. Later Buddhist thinkers were more or less engaged in the discussion of logical issues dealt by these two great thinkers and tried to defend the Buddhist position against the attacks of the opponents, both of Brāhmaṇical and non-Brāhmaṇical schools. It is in course of such counterattacks and defenses that Buddhist logic developed to have a position of its own in Indian philosophy. Keywords

Catuṣkoṭi – Tetralemma – Enumeration of all logically possible alternatives regarding a particular concept and then showing that each of them leads to absurdity · Anumāna – Inferential cognition or inference – A form of knowledge employed to provide information regarding an unknown phenomenon on the basis of a known phenomenon · Hetu or sādhana – Probans – That known phenomenon on the basis of which the unknown phenomenon is apprehended · Sādhya – Probandum – The object which was unknown beforehand becomes apprehended on the basis of the probans · Avinābhāva – Invariable concomitance – The necessary relation between the probans and the probandum such that in the absence of the probandum the probans cannot occur · Dūṣaṇa – fallacies – defects of inference due to any of the factors involved in the inference process · Vāda – Debate – Dialogue holding between two parties to establish a particular point leading to victory or loss

Introduction The history of thought in India may be classified into three periods – the preclassical period (up to the fourth century BC), the classical period (up to the tenth century AD), and the scholastic period (from the eleventh century on). In the preclassical period, philosophical inquiry was mainly centering round the Vedic study and justifying the practice of Vedic rites and rituals. Buddhist philosophy came into existence in the classical period challenging the authority and supremacy of the Brāhmaṇas. Realizing that the only way to salvation is through enlightenment regarding the real nature of entities, the task of Lord Buddha and his followers was to show the untenability of the opponent’s view, be that of the Cārvākas, of the Jainas, or of the Brāhmaṇa tradition. For the refutation of the opponent’s views and establishing their own views, Lord Buddha and his followers did not rely on any scripture or authority but presented arguments based chiefly on ample illustrations (dṛṣṭānta) and not on universal concomitance. It was in such arguments that the seed of Buddhist logic was sown. From the origin of Buddhism in the sixth century BC to its expansion into four philosophical schools, no systematic treatment of logic by the Buddhist thinkers was noticed. There were only a few haphazard references to logic in course of discussion on other subjects like philosophy, religion, etc. The works of

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Nāgārjuna, Maitreya, Asaṇga, and Vasubandhu contained discussion on logic, no doubt, but that did not establish logic as an independent branch of study within philosophy. With 450 AD began a period when logic was completely differentiated from general philosophy, and there were a good number of Buddhist thinkers who concentrated mainly on that branch of learning. The earliest known writer of this period was Dignāga. The time of Dignāga has been thought to be before 557–569 AD when two of his works had been translated into Chinese. His works set the framework within which subsequent Buddhist thinkers discussed philosophical issues pertaining to inference and debate. Reference may here be made to Śaṇkarasvāmin (about 550 AD) who wrote a brief manual of inference for the Buddhists, called the Nyāyapraves´a. A bit later, around 600–660 AD, Dharmakīrti, the greatest Buddhist logician, developed his own views on inference and debate within the conceptual framework founded by Dignāga. Vasubandhu in his discourse on logic had pointed out that inference was based on a formal relation, vyāpti which he described as “avinābhāva,” i.e., as a necessary relation. But the question, wherein lies the necessity of such a vyāpti relation, had not been dealt with by Vasubandhu. Dharmakīrti systematically investigated into this problem and identified that this necessity consists in the relation of causality (tadutpatti) and essential identity (tādātmya). In his different texts like Pramāṇavārttika, Pramāṇavinis´caya, Hetubindu, and Nyāyabindu, Dharmakīrti concentrated on the probans and its relation with the probandum. Like Dignāga, Dharmakīrti did not admit the existence of general concepts or universals and reduced them to difference from others. In his logic, the notions of tatpariccheda (possessing a particular characteristic) and atadvyāvṛtti (difference from others) were considered to be the two sides of the same coin. This atadvyāvṛtti, also technically known as apoha, is nothing but a mental construct having its root in the objective reality. Among the later Buddhist thinkers of the Mādhyamika and Yogācāra schools, Śāntarakṣita, Kamalaśīla (both belonging to the eighth century), Dharmottara (730–800 AD), Jñānaśrīmitra (tenth century), Ratnakīrti (eleventh century), and Mokṣākaragupta (late eleventh century) need special mention. These later logicians did not possess much originality of thinking as the two great thinkers Dignāga and Dharmakīrti. They were more or less engaged in the discussion of epistemological and logical issues which had been dealt with by Dignāga and his followers. Their contributions, however, consisted in the fact that they tried to defend the theories of Dignāga and Dharmakīrti against the criticisms raised by the opponents of the Brāhmaṇical as well as non-Brāhmaṇical schools.

Plan of the Chapter In this chapter, an attempt is made to introduce the Buddhist logicians and their main logical texts in a brief manner so that the reader can get an idea of the development of Buddhist logic. This chapter is divided into eight sections:

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1. 2. 3. 4. 5. 6.

Nāgārjuna: Vigrahavyāvartanī and Vaidalyaprakaraṇa Pre-Dignāga Buddhist texts: Vasubandhu (Vādavidhi) Dignāga: Nyāyamukha, Hetucakradamaru, and Pramāṇasamuccaya Śankarasvāmin: Nyāyapraves´a Dharmakīrti: Nyāyabindu, Hetubindu, Pramāṇavārttika, and Vādanyāya Sanskrit commentators of Dharmakīrti: Dharmottara, Vinītadeva, Arcaṭa, and Prajñākaragupta 7. Tibetan commentators of Dharmakīrti 8. Post-Dharmakīrti logicians: Jñānaśrīmitra, Ratnakīrti, and Mokṣākaragupta

Nāgārjuna In the galaxy of Buddhist philosopher Nāgārjuna outshines all in respect of his contribution to philosophy. Because of the vastness of his knowledge and extraordinary skill in respect of reasoning, he has often been revered as the “second Buddha,” and even within 100 years of his death, his images began to be worshipped in the several temples of South India. Several stories and myths have grown up or sprang up centering his life. His time is generally accepted to be the earlier part of the first century AD. The major contribution of Nāgārjuna in Buddhism is that it is under his influence that the prajñāpāramitā become popular in the northern part of India. He was the first systematic proponent of the Mādhyamika school of thought. In order to establish the essenceless-ness of all entities from the ultimate standpoint, Nāgārjuna in his two major texts, Mūlamadhyamakas´āstra and the Vigrahavyāvartanī , developed a special method of logic which is often referred to as catuṣkoṭi or tetralemma. The method consists in enumerating all the theoretically or logically possible alternatives regarding a particular concept and then showing that each of these logically possible alternatives ultimately lead to some kind of absurdity. In short, Nāgārjuna’s method involves two steps – (1) enumeration of all the possible alternatives and (2) pointing to the absurdity or contradictory nature of each of these alternatives. As all the possible alternatives lead to absurd consequence, it follows that the original position or view is not tenable. This second part of Nāgārjuna’s method is comparable to the reductio ad absurdum form of argument and is known as prasaṇga. This prasaṇga method is later on followed by most of the Mādhyamika thinkers like Buddhapālita and Candrakīrti. Thus Nāgārjuna’s way of argument is a unique method based on two strategies, namely, the method of enumeration and the reductio ad absurdum. In short Nāgārjuna’s contribution to logic consists in his systematic application of the tetralemma to develop the assertions made in the Prajñāpāramita literature of the early Mahāyāna Buddhism about the emptiness of all entities in consonance with the notion of avyākṛtavastu or “unexplicated points” found in the early Buddhist literature.

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Vigrahavyāvartanī The texts Vigrahavyāvartanī and Vaidalyaprakaraṇa may be regarded as supplementary to Nāgārjuna’s magnum opus Mūlamadhyamakas´āstra. Nāgārjuna’s views on epistemology and philosophy of language can be found in these texts, and these topics are not so elaborately discussed in the former texts, from the stand point of logic. The importance of the text Vigrahavyāvartanī lies in the fact that it gives an insight into the kind of philosophical debates that used to take place in ancient India in the first and second century AD. It belongs to a group of six works known as the Yukti-corpus which the Tibetans refer to as a collection of six texts dealing with reasoning (rigs pa’i tshogs drug). These six texts are Mūlamadhyamakakārikā, Vigrahavyāvartanī , Yuktiṣaṣṭikā, Sunyatāsaptati, Vaidalyaprakaraṇa, and Ratnāvalī . The text Vigrahavyāvartanī consists of seventy verses, and the topics discussed here may be classified into the following: (1) the status of the theory of emptiness; (2) epistemological issues, especially regarding the relationship between valid cognition (pramā) and the sources of the valid cognition (pramāṇa); and (3) semantic issues like empty terms (i.e., terms which do not refer to any corresponding objects) and negative terms like “not.” The style in which the text is composed is in the form of a debate, where the objections raised by opponents have been stated at once and the answers have been given by Nāgārjuna. In short, the text seems to be divided into two parts – one part containing the objections against the Mādhyamika theses and the other part containing the counterparts or refutations of those counterarguments. This specific structure has led some modern thinkers to regard the text to be written by two different writers. While making clear the notion of emptiness through different arguments and refuting the position of the opponents, Nāgārjuna makes it clear categorically that when he asserts that everything is empty, he is not thereby making any sort of selfrefuting assertion; in fact, he does not have any position of his own to establish. Thus, discussion made in this regard may be looked upon as similar to the discussion of liar paradox, specially of the Eubulidean form, found in modern Western logic and the way he refutes the charge of self-refutation, may be considered as an example of a type theory. Vaidalyaprakarana Vaidalyaprakaraṇa ˙is one among the six main treaties on logic by Nāgārjuna. The term “vaidalya” is derived from the root √dal with the prefix “vi,” which has as one of its meaning the sense “to tear into pieces” or “tearing.” In the context of the present treatise, the title signifies that the main purpose of the book is to demonstrate the absurdity of the 16 categories or padārthas admitted by the Naiyāyikas. The text Vaidalyaprakaraṇa is divided into 73 sections. Refutation of the 16 categories is not given equal stress. Depending on the importance of the categories, length of the discussion varies. That is, to refute some categories only one section is devoted, while there are some categories to refute which 20 sections are devoted.

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Through the refutation of the 16 categories admitted by the Naiyāyikas, Nāgārjuna actually tries to point out that it is impossible to assert anything regarding either the existence or the nonexistence of the categories. In other words, these categories are not actually what they seem to be, that is, existent and true realities; rather they are mere conventions and creations of the human mind.

Pre-Dignāga Buddhist Texts: Vasubandhu (Vādavidhi) Vasubandhu Vasubandhu, a great Indian Buddhist scholar, is well known for his magnum opus Abhidharmakos´a, a compendium of Abhidharma philosophy, and two small treatises Viṃs´atikā and Triṃs´ikā upholding the Yogācāra philosophy. But his contribution cannot be regarded to be confined to these three texts only. Another major contribution of him consists in his writings on logic, which had influences on Dignāga, and in that sense may be regarded as marking the dawn of Indian formal logic. Vādavidhi Of the different texts written by Vasubandhu on logic, the Vādavidhi is the earliest one and from the title itself it can be said that this book was concerned with the formulation of rules (vidhi) to make valid flawless arguments in the context of debate (vāda). Though rules for good argumentation were also provided in earlier texts, e.g., in the Nyāyasūtra of Gautama or in the works of Asaṇga, the text Vādavidhi exhibits its excellence over others in this regard for providing a complete logical structure of an argument in judging its validity. The uniqueness of Vādavidhi can be exhibited through the following facts. First, while the earlier texts followed the five member syllogism schema, in the Vādavidhi the redundancy of the last two members were pointed out and instead of the five members, for the first time three-member syllogism/argument structure was emphasized and this structure was followed by later logicians. The three members of the syllogism were the thesis, the reason/ justification, and the example, that is, the statement exhibiting the relation between the probans and the probandum. Secondly, in the explanation of the relationship between the probans and the probandum, the novelty of Vasubandhu is evident again. In the earlier texts, this relationship was presented in terms of mere regular coexistence (sāhacarya), in the form whenever there is A there is B or whenever there is absence of B there is absence of A. In the Vādavidhi this relationship was presented as invariable concomitance (avinābhāva) between two events. That is, the event of the probans can occur only if the event of the probandum occurs. It has been categorically said that the statement of such invariable concomitance between the probans and the probandum is necessary for the validity of an inference – schema. Thus the necessary character of invariable concomitance between the probans and the probandum, rather than mere concomitance, was first emphasized by Vasubandhu, which had been taken up by Dharmakīrti later. Thus presenting an argument in terms of three members and defining the relationship between the probans and the probandum as invariable concomitance and not

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as regular coexistence are the two important features of Vādavidhi. The criterion for a valid inference structure has been precisely stated without omitting any essential aspect. In that respect the text Vādavidhi opens a new area in Indian logical tradition.

Dignāga: Nyāyamukha, Hetucakradamaru, and Pramānasamuccaya ˙ Dignāga Ᾱcārya Dignāga (beginning of the sixth century) has often been regarded as “the father of medieval logic” because of the distinctness of his works from those of his predecessors in respect of matter and manner of the treatment on the subject. His opus magnum the Pramāṇasamuccaya is a logico-epistemological treatise dealing mainly with the pramāṇa or valid cognition and its means. The significant contribution of Dignāga in respect of logic lies in the fact that he consolidated and systematized the insights found in the works of his teacher Vasubandhu into a formal basis. First he made the distinction between inference for oneself and inference for others the fundamental basis of his treatment of inference. Secondly, he made the threefold characteristics of reason or a hetu (trirūpahetu) as recognized by Vasubandhu more precise by emphasizing on the Sanskrit particle “eva.” Thirdly and perhaps most strikingly he devised the “wheel of reason” (hetucakra), a three-by-three matrix setup to classify the pseudo-reasons in the light of the last two characteristic feature of the trirūpahetu. It is possible to conceive of the nine reasons or probans which are present in or absent from the similar instances (sapakṣa) and dissimilar instance (vipakṣa) either partly or wholly. The probans which are wholly or partly present in the sapakṣa but wholly absent from the vipakṣa cases are considered as valid and their opposites are contradictory or uncertain. Another important contribution of Dignāga in respect of logic was in respect of example or the dṛṣṭānta of an inference. Before the time of Dignāga, an example was considered to be simply a familiar case which was cited only to help the understanding of the listener. Dignāga converted an example into a universal proposition, i.e., a proposition which expresses the invariable relation between the probans and the probandum. This example may be either positive/homogenous or negative/heterogeneous, and an inference can become invalid if this reason is defective. An important consequence of the trirūpahetu or triple character of the probans led Dignāga to his notion of “apoha,” often translated as “exclusion theory”/ “differentiation theory” of meaning. According to Dignāga words or utterance of words behave as inferential signs because their meaning can be determined only by excluding any dissimilar object where the intended meaning does not belong. To be precise, the meaning of such common nouns like “cow,” “water,” etc. arises in the mind of the hearer from their utterance, not because there are positive universals like cow-ness, water-ness, etc. but because of the exclusion of complement classes of non-cows, non-water, etc. Such notion of differentiation or exclusion is thus a substitute for the objective universal admitted by the realist school like the Nyāya.

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Pramānasamuccaya ˙ The major work of Dignāga is the Pramāṇasamuccaya. This book in a sense constitutes the very basis of Buddhist as well as medieval Indian logic. In the introductory verse of the text, Dignāga, after offering his prayer to the teacher Sugata, states that the purpose of this text was to be a collection of his own views on valid cognition, which were scattered in different texts. The text consists of six chapters, all of which deal with different fundamental epistemological questions. The first chapter entitled Pratyakṣapariccheda deals with the Buddhist definition of valid cognition, classification of valid cognition into perception and inference, objects of these two varieties, definition of perception and classification of perception, and identity of (pramā) valid cognition and the means of such cognition (pramāṇa). The second and third chapters deal with the nature and necessary constituents of two varieties of inferential cognitions – inference for oneself and inference for others. In the second chapter there are important discussions on the probans and the invariable concomitant relation or vyāpti among other things, while in the third chapter the main topics of discussion are with the constituent statements like thesis (pratijn˜ ā), probans (hetu), and example (dṛṣṭānta). In the course of that, discussion on the different fallacies of thesis and probans have been made. The fourth chapter deals exhaustively with the nature of an example as also with different fallacies arising thereof. For Dignāga the term “Dṛṣṭānta” stands for actual example as also the third statement in a proof which actually is a statement of the relation of pervasion along with the example. The fifth chapter deals with the examination of interpretation of linguistic signs (apoha parī kṣā). A verbal sign is not different from the inferential mark or the probans in the sense that both function by excluding entities incompatible with what is signified. In the last chapter of the text entitled “Jāti” (futile rejoinder), Dignāga has discussed (dūṣana) refuting the opponents by pointing to defects in their arguments. The text Pramāṇasamuccaya had exerted immense influence on both Buddhist and non-Buddhist thinkers in respect of logic and epistemology. Śaṇkaraswāmi, a disciple of Dignāga, composed Nyāyapraves´a, a summary exposition of Dignāga’s views. The famous commentary on Pramāṇasamuccaya had been composed by Jinendrabuddhi entitled Vis´ālāmalavatī. Dharmakīrti also wrote a commentary on Pramāṇasamuccaya, entitled Pramāṇavārttika, which itself had superseded the original text in respect of the content and also the arguments employed for establishing the views of Dignāga. Dignāga himself wrote a commentary on the Pramāṇasamuccaya. The original Sanskrit text is lost, though the Tibetan version of it is available and it has been the project of modern Buddhist scholars to restore the text in its original form from the Tibetan version. Nyāyamukha The Nyāyamukha of Dignāga has been regarded by G. Tucci as the oldest Buddhist text on logic. It is argued by scholars that this text was composed later than the Pramāṇasamuccaya and its auto-commentary, since passages from the latter two

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texts have been found to be quoted in the former. The Nyāyamukha, in a sense, contains the essence of the text Pramāṇasamuccaya; the difference between the two lies in the fact that in the former discussions on epistemology do not get much importance while the latter is wholly dedicated to epistemological discussions; on the contrary the text Nyāyamukha is basically intended to ascertain the real nature of the argument offered to establish a thesis as well as its refutation, a vāda. That is, the main concern of this book is with logic rather than with epistemology. This book unfolds one after another the different constituents of an argument, which according to him are three – the thesis, the probans, and the example. Along with presenting the nature of each of them, Dignāga points out with suitable example the fallacies which would occur if any of them were defective. For example, after discussing the thesis or the pakṣa, there occurs a discussion on the pakṣābhāsa. Similar is the case with hetu and dṛṣṭānta. After completing his discussion on the three constituents and their fallacies, Dignāga proceeds to discuss the refutation which consists in showing that the arguments provided by the opponent is vitiated by any of the logical error already enunciated by him. The fallacies of such refutation referred to as dūṣaṇābhāsa are also discussed in this context. In short, this book gives a primary idea of the logic by which one can establish one’s thesis and at the same time refute the view of the opponents who challenge that thesis. In that sense the title of the text Nyāyamukha is very appropriate. Hetucakradamaru The Hetucakradamaru is a small treatise on logic by Dignāga which deals exclusively with the threefold characterization of the probans (hetu). A valid reason or probans is one which is present in the locus and in all the homogeneous or similar instances and is not present in any of the heterogeneous or dissimilar cases. Regarding the presence, absence, and both presence and absence of the probans in similar and dissimilar instances, nine alternative combinations are possible. Of these nine possibilities, only two are considered to be valid while all the remaining seven are invalid because of the occurrence of different fallacies, like being either too broad or too restrictive or contradictory. The importance of this book consists in pointing out the necessity of all the three characters of the probans in establishing the probandum.

Śankarasvāmin: Nyāyapraves´a Śankarasvāmin Śankarasvāmin was a pupil of Dignāga. Much is not known about his life except that he was a native of Southern India. His time period is thought to be the middle of six hundred AD. However, his importance in Buddhist logic lies in the fact that it was through Śankarasvāmin along with ten other great scholars that the logic of Dignāga was handed to Śīlabhadra, who was the head of Nālandā University. The most famous work of Śankarasvāmin was Nyāyapraves´a.

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Nyāyapraveśa From the title Nyāyapraves´a, it is evident that the text was intended for introducing Buddhist logic. In that respect the content of this book came closer to the Nyāyamukha. In the introductory verse of Nyāyapraves´a, the author points out clearly that three propositions which constitute an inference as well as the fallacies involved in such inferences, means of refutation (dūṣaṇa) and fallacies in such refutation (dūṣaṇābhāsa), are pertinent for convincing others regarding matters unknown to them. As contrasted with that, perception and inference together with their fallacies are relevant for one’s own understanding. Though he mentions both the categories in the introductory verse, in the book his main thrust has been on the first one, namely, on those which are necessary for informing another person a new fact. As such he has discussed in details the three propositions constituting the inference, the triple character of the probans, two varieties of examples (dṛṣṭānta), and their defects. He has classified fallacies under three heads – those relating to the thesis, those related to the probans, and those related to the example and has made detailed discussion on each of them. He has made a brief presentation of the definition of perception and that discussion is made in consonance with the definition offered by Dignāga. Because of the simplicity of language and lucid way of presentation, this text gained popularity among students of both Buddhist and Jaina tradition, and Jaina logician like Haribhadra Sūri wrote a commentary, entitled Nyāyapraves´avṛtti.

Dharmakīrti: Nyāyabindu, Hetubindu, Pramānava¯rttika, ˙ and Vādanyāya Dharmakīrti The logico-epistemological dimension of Buddhism which started with Dignāga, reached its highest point with Dharmakīrti who played the central role of importance in Indian epistemological tradition. The importance of this great logician can be understood from the fact that the influence of his works is noticed in later works of both Buddhist and non-Buddhist thinkers. When Dignāga’s arguments to establish Buddhist philosophy by refuting the views of the Brāhmaṇical schools were strongly criticized by the Naiyayika Udyotakara or the Mīmāṃsaka Kumārila, it fell upon Dharmakīrti as a task to establish the Buddhist position on strong logical foundations. This he did by even criticizing the views of some Buddhist logicians like Ῑśwarasena whose views he did not like. The following seven texts have been regarded to be written by Dharmakīrti: (1) Pramāṇavārttika, (2) Pramāṇavinis´caya, (3) Nyāyabindu, (4) Hetubindu, (5) Vādanyāya, (6) Sambandhaparī kṣā, and (7) Santānāntarasiddhi. In addition to these seven, Dharmakīrti wrote an auto-commentary on the Svārthānumānapariccheda, a chapter on inference for oneself of the Pramāṇavārttika, and also on the Sambandhaparī kṣā. According to Bu-ston all these seven treatises are actually commentaries on the Pramāṇasamuccaya of Dignāga although Dharmakīrti himself considered only the first one to be a

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commentary on Pramāṇasamuccaya. As a matter of fact, all the other treatises deal with some or other epistemological issues and the text Pramāṇasamuccaya had as its core concern different epistemological issues. So, it is quite natural that all the works of Dharmakīrti are related with the works of Dignāga. Though Dharmakīrti in his Pramāṇavārttika intended to write a commentary on Dignāga, it is an irony of fact that his works superseded the original work in respect of merit. The seven treatise of Dharmakīrti have often been compared to that of a human figure, the first three constituting the body and the remaining four as constituting the limbs of the body. The first three are exclusively concerned with different basic epistemological issues, like the nature of valid cognition, its varieties, their objects, and the question of the identity of sources of valid cognition (pramāṇa) and the resultant cognition (pramā). The remaining books do not address these general issues but focus on one or other particular topics, for example, Hetubindu is an investigation into the nature of probans (hetu), and Sambandhaparī kṣā deals with the nature of relation. Vādanyāya is a treatise on the art of debating, that is, on the logic to be followed by proponents and opponents who wish to establish their own position or refute others, respectively. So these three deal exclusively with logical topics. The seventh text Santānāntarasiddhi deals with the question of admitting others’ minds in the framework of idealism. In respect of logic, the contribution of Dharmakīrti consists in the fact that he first provided reasons to account for the necessity aspect in invariable concomitance relation or vyāpti in terms of causality and essential identity. Thus giving epistemology a pragmatic orientation and making all the varieties of valid cognition invariably related with reality and explaining such invariable connection in terms of fundamental logical principles constitute the uniqueness of the logical system of Dharmakīrti. Nyāyabindu The text Nyāyabindu of Dharmakīrti is one of the most well-known popular books on Buddhist logic which is recommended for anyone wishing to pursue studies on the subject. In spite of the presence of a number of treatises on this area of Buddhism, including texts like Pramāṇasamuccaya of Dignāga or the magnum opus of Dharmakīrti the Pramāṇavārttika or the Pramāṇavinis´caya, the importance of this Nyāyabindu lies in the fact that it presents the principles of logic in a brief and interesting manner. As such this book is essential for anyone who is interested for a precise understanding of Buddhist logic. Because of the importance of this text, several commentaries and subcommentaries had been written on it. Unfortunately only the commentary by Dharmottara and his followers are available in Sanskrit, but there were other Sanskrit commentaries which have become extinct though some of them are available in Tibetan version. The earlier commentary on Nyāyabindu was Nyāyabinduṭī kā by Vinītadeva, which is extinct in Sanskrit. There was another commentary on the Nyāyabindu by Ᾱcārya Śāntabhadra which is also not available in Sanskrit. But the existence of these two commentators is known from the fact that the later commentators have referred to and refuted their views. Some commentators like Durveka

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Miśra have mentioned Vinītadeva and Ᾱcārya Śāntabhadra together which indicates closeness and similarity of the philosophical standpoints of the two commentators. The most well-known commentary on Nyāyabindu is the commentary by Dharmottara in Sanskrit and this Sanskrit original is still available and it was translated into Tibetan from Sanskrit by Jñānagarbha. There had been several subcommentaries on the commentary of Dharmottara. Of them, one is by Mallavādin which is known as Dharmottaraṭippaṇa, and another is by Durveka Miśra which is known as Dharmottarapradī pa. In his Ṭippaṇa, Mallavādin mentioned the views of other commentators prior to Dharmottara but did not mention the views of other sub-commentators while Durveka Miśra had mentioned the views of other sub-commentators on Dharmottara’s Nyāyabinduṭī kā. In addition to the abovementioned commentaries and subcommentaries on Nyāyabindu, mention may be made to two other treatises which were written on Nyāyabindu, not to explain the necessity of each and every term of the text but to explicate the original content. One such treatise is named Nyāyabindu-pūrva-pakṣasaṇkṣepa, written by Kamalaśīla. From the title it is evident that the main purpose of this treatise was to present before the reader in a nutshell the view of all the pūrvapakṣa-s (earlier thinkers) which had been refuted by Dharmakīrti in the Nyāyabindu. The other treatise is named Nyāyabindupiṇdārtha, composed by Jinamitra. The title of this treatise suggests that it is a summary exposition of the entire text Nyāyabindu. Both these texts were written in Sanskrit, though those original texts are not available. The Tibetan translations of the two, however, are available. In the very first Sūtra of Nyāyabindu, Dharmakīrti declares that the basic objective of this text is the exposition of valid cognition or samyagjn˜ āna since such valid cognition is the cause of the attainment of all human goals. Such valid cognition is classified under two heads – perception and inference. The basis of such classification is that there are two types of objects of knowledge – unique particular (svalakṣaṇa) and universals (sāmānyalakṣaṇa). Unique particulars can be apprehended by perception alone and the sāmānyalakṣaṇa can be apprehended by inference alone and not by perception. Perception is indeterminate while inference is determinate character. In this chapter Dharmakīrti has clearly pointed out why these two types of objects alone are admitted as objects of cognition and how the Buddhists do explain the identity of cognition (pramā) and the sources of cognition (pramāṇa). Dharmakīrti classifies inferences into two varieties – inference for oneself and inference for others. Accordingly the three chapters of Nyāyabindu are devoted, respectively, to perception, inference for oneself (svārthānumāna), and inference for others (parārthānumāna). In the chapters on inference, Dharmakīrti has presented the threefold characteristics of the probans, classification of probans into three varieties – probans as essence, probans as effect, and probans as non-apprehension. Probans as non-apprehension is classified to be of 11 varieties. In addition to this discussion on probans, Dharmakīrti has also explained the nature of vyāpti or invariable concomitance relation, and the necessary character of this relationship is accounted for by regarding relation of identity and causality as the only two valid sources of establishing vyāpti. Discussion on inference is completed

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with discussion of the fallacies which are classified into three – fallacies with regard to the thesis (pakṣābhāsa), fallacies with regard to the probans (hetvābhāsa), and fallacies with regard to the instances (dṛṣṭāntābhāsa), which are offered to illustrate the invariable connection. Since in this book, the thing in itself is admitted as one variety of objects of cognition (prameya), question arises as to which school of Buddhism this text Nyāyabindu stands for. Though Dignāga and Dharmakīrti have shown their affinity to the Yogācāra, Vijñānavāda tradition, which believes in the nonexistence of external objects, in the Nyāyabindu this belief, is not adhered. The way the definition of thing in itself has been presented indicates that it is something existent outside. Hence this treatise is regarded not to belong to the Yogācāra tradition. Accordingly some commentators have held that in this treatise Dharmakīrti has left the ideology of the Yogācāras and accepted the position of the Sautrāntikas. Some commentators have tried to show that some of the Sūtras of the Nyāyabindu indicate an attempt to combine the views of both the Yogācāra and Sautrāntika schools. An attentive explanation is given by some commentators who argue that there is nothing contradictory on the part of Dharmakīrti to uphold the Yogācāra standpoint in his Pramāṇavārttika and Sautrāntika standpoint in the Nyāyabindu. Their point is that when discussion on pramāna (knowledge) and prameya (object of knowledge) is made, that discussion is made from the practical point of view, vyāvahārika standpoint and not from the ultimate standpoint or pāramārthika point of view. The ultimate reality is apprehended through direct insight (svajn˜ ā) and does not involve the categories of pramāṇa, prameya, etc. So, though the Pramāṇavārttika deals with the ultimate standpoint, in the Nyāyabindu the vyāvahārika or the empirical point of view predominates. Having pointed out the inadequacy in the analysis of pramāṇaprameya done by other schools and pointing out their real nature from the Sautrāntika standpoint, it became easier for Dharmakīrti to regard the Sautrāntika treatment as unsatisfactory from the ultimate point of view. In the Pramāṇavārttika Dharmakīrti has refuted the existence of external objects and established the ultimate reality of consciousness alone. Hence it seems that it is only to represent the empirical point of view that Dharmakīrti has presented the Sautrāntika position in the text Nyāyabindu, though he was an ardent follower of the Yogācāra tradition regarding his ultimate position. Hetubindu The term “Hetu” is used in different senses in Indian philosophy. Sometimes it is used in the sense of reasoning (Nyāya), for example, when the expression Hetuvidyā is considered to be synonymous with or equivalent to Nyāyavidyā or Tarkavidyā or even to Ānvī kṣikī . The term “hetu” has, as its sense, reasoning. But this is not the sense in which the term “hetu” is used in the title “Hetubindu.” There the term “hetu” refers to the probans which is one of the terms of an inference. The word “bindu” in Sanskrit means “drop.” So, by using this term “bindu” in the title, it is indicated that the text Hetubindu presents like a drop a concise exposition of the nature of probans which is as vast and deep as the great ocean. The title may alternatively be understood as standing for a short treatise which embodies only a part of the

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elaborate treatment of the topic on probans as found in the Pramāṇavārttika. In whatever way the title may be understood, it is a fact that Dharmakīrtī has presented his own views regarding the nature of the probans in the text Hetubindu. In short, the text Hetubindu is a short treatise dealing exclusively with probans which plays the fundamental role in the process of inference. Regarding the subject matter, this text may be regarded as similar to the Hetucakra of Ācārya Dignāga, for the Hetucakra also deals with discussion of the nature of probans which is a member of the process of reasoning. The text Hetubindu does not start with any invocation to Buddha or to any other divine being as is the case with auspicious verse (mangalācaraṇa) of other Mahāyāna Buddhist texts but begins with a verse accompanied by an introductory sentence. The verse, however, actually is a part of the Pramāṇavārttika, which states in very clear terms the characteristic features of the probans, the types of the probans, and also the consequences which would arise in the case of a defective probans; the rest of the text composed in prose is actually an elaborated exposition of this introductory verse. The subject matter of Hetubindu is mainly concerned with inference for oneself (svārthānumāna) and so the text may be regarded as a short treatise on inference for oneself. The introductory verse of Hetubindu states that “the probans is that which subsists in the subject (pakṣa) of the inference and is pervaded by the probandum (sādhya) which also subsists in the same subject. Such a probans is of three types only, because invariable concomitance (avinābhāvaniyama) holds exclusively in these three. What are other than these are pseudo-probanses (hetvābhāsa).” Arcaṭa, the chief commentator of Hetubindu, has pointed out two alternative lists of topics from this verse. The first list contains three topics – (1) definition of probans, (2) number of the varieties of probans, and (3) reasons for restricting the number of probanses. The second list points to six topics – (1) nature of probans, (2) restriction of the number of types of probans, (3) restricting probanshood to the three types only, (4) reason for the numerical restriction and the restriction of probanshood, (5) statement of the implication, and (6) reason for not standing the definition of the pseudo-probans. Besides these main topics, other important topics of Buddhist philosophy have been clearly discussed. These include topics like refutation of the doctrine of universals (sāmānya), introduction of the notion of apoha as a substitute for uncaused destruction (nirhetukavinās´avāda), establishment of indeterminate perception as the only source of valid cognition, discussion of relation between cause and effect, nature of auxiliary conditions (sahakāritva), and the nature of absence (abhāva). Though these topics have been discussed elaborately in other works of Dharmakīrtī as, for example, in the Pramāṇavārttika, Dharmakīrti had introduced briefly along with the refutation of opponents’ views adding sarcastic remarks for them. The importance of this text can be judged from the great influence that it has imparted on later writers, both of the Buddhist and non-Buddhist traditions. There is a detailed and elaborate commentary on Hetubindu, known as Hetubinduṭī kā by Arcaṭabhaṭṭa, and a subcommentary entitled Āloka by Durveka Miśra. Both the commentary and subcommentary are available in Sanskrit.

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Pramānavārttika The text˙ Pramāṇavārttika occupies a central position in the logico-epistemological tradition of Buddhism. Basically it was intended to be a Vārttika, i.e., commentary, on the Pramāṇasamuccaya of Dignāga, but it contained more than what was discussed in the original text. The text is written in metrical verses and the total number of verses in this text is 1454. The text Pramāṇavārttika is divided into four chapters: one chapter dealing with perception (pratyakṣa), one dealing with inference for oneself (svārthānumāna), one dealing with establishment of pramāṇa (pramāṇasiddhi), and the other dealing with inference for others (parārthānumāna). But regarding the order of the chapters, there is a controversy among the thinkers. According to some the book starts with the chapter on establishment of pramāṇa (pramāṇasiddhi), while according to others the first chapter is inference for oneself. Those of the second group justify their argument on the ground that Dharmakīrti himself wrote an auto-commentary on it. Those of the first group, on the other hand, believe that the importance of Pramāṇavārttika lies not in its discussion of logic, but in its discussion of the fundamental reality of Mahāyāna in the form of Buddha and his specific qualities. The basic aim of the text is to present with logical vigor the Dharmakāya, Svabhāvakāya, and Jn˜ ānakāya of the Buddha, before those who challenge the authority of the Buddha. Whatever reason may be there to account for the difference in the arrangement of chapters, the importance of the text in the area of Buddhist logic lies in the fact that this book contains an elaborate analysis of the invariable concomitant relation (avinābhāva) and attempt to explain the necessary chapter of this relation in terms of causality or essential identity. Other important topics of logic are also discussed here elaborately. For example, the chapter on inference for others focuses on (i) the definition and function of such an inference, (ii) definition of thesis, and (iii) also on the notion of the probans (hetu). Incidentally it may be noted that although the Pramāṇavārttika was aimed to be a commentary on all the verses of the Pramāṇasamuccaya, in the chapter on inference for others, Dharmakīrti has commented on only first eight verses of the Pramāṇasamuccaya. Regarding the question to which school of Buddhism does Pramāṇavārttika belong, it is unanimously admitted by all that in this text, Dharmakīrti had defended the Yogācāra position very strongly. From the Yogācāra standpoint existence of the external objects are admitted only empirically from the standpoint of saṃvṛti-satya, and not from the ultimate standpoint, i.e., pāramārthika-satya. Hence the discussion on pramāṇa, prameya, etc. as done in logic or epistemology is carried out from the empirical standpoint. After refuting the discussion on pramāṇa and prameya as made by other schools, Dharmakīrti tries to reveal their real nature from the Sautrāntika perspective. After establishing the Sautrāntika standpoint, he showed that the reality of such external objects cannot be admitted from the ultimate point of view. So, from that point of view none of them can be admitted and what remains to be admitted is the ultimate reality of consciousness (vijn˜ aptimātratā) alone. In short, in the Pramāṇavārttika his affiliation to the Yogācāra-Vijñānavāda trend, which had been ultimate position, has been expressed.

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Vādanyāya Of the other logical treatises composed by Dharmakīrti, Vādanyāya is an important one. Etymologically the term “Vādanyāya” means the nyāya or logic of debate (vāda). The title is appropriate since the main thrust of the book is to discuss the rules of victory and defeat in debate. Considering a debate between two parties to be a rational enterprise, need is felt to formulate the rules by which one can judge the soundness, rightness, or wrongness of the arguments of the parties involved before declaring the winner and the loser in the debate. Dharmakīrti’s notion of vāda differs from that of the Naiyāyikas. For the Naiyāyikas vāda stands for a friendly discussion between a master or a guru and his disciple (s´iṣya) or between two co-disciples, and in such situation the question of victory and defeat does not arise. But what Dharmakīrti regards as vāda is a real situation of debate between two parties where one party tries to establish his own position by refuting the other. Naturally the question of winning and losing becomes important in such situation. Dharmakīrti developed his own logic of debate by pointing out that the points of defeat for the opponent (prativādin) cannot be the same as that of the proponent (vādī ), since the tasks of the two are not the same. The task of the vādī is to offer arguments in order to establish his own position, whereas the task of the opponent is to point out the defects in the arguments of the proponent to refute their position. So, it would be wrong to regard misapprehension and non-apprehension to be the same in both cases. For the proponent, non-apprehension will mean failure to present a sound argument or failure to justify the argument offered, and misapprehension will mean presentation of fallacious or redundant or irrelevant arguments to establish his own position. On the other hand for the opponent non-apprehension will mean inability to find the fault in the faulty arguments of the proponent and misapprehension would mean consideration of a non-fault as a fault. In the case of the proponent, the nigrahasthāna would therefore be asādhanāngavacana, failure to establish one’s own position, while for the opponent, it would be adoṣodbhāvana. Thus exposition of the nature of nigrahasthāna as found in Vādanyāya is not totally different from that found in the Nyāyasūtra; the difference lies in the precise way of presentation. The point in which the text Vādanyāya is critical of the Nyāya view is in respect of the classification of such points of defeat.

Sanskrit Commentators of Dharmakīrti: Dharmottara, Vinītadeva, Arcata, and Prajñākaragupta ˙ Arcata ˙ the different commentators of Dharmakīrti, Arcaṭa occupies an important Among position. From Tārānātha’s History of Buddhism, it is known that Arcaṭa was a Kashmīri and he was a Brāhmaṇa. Another name of Arcaṭa was Dharmākaradatta. Rahul Sankrityayan did not admit that Arcaṭa and Dharmākaradatta were one and the same person. Durveka Miśra in his subcommentary Āloka on the Hetubinduṭīkā

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mentions the two names as referring to the same person. He used the title Bhaṭṭa along with the name Arcaṭa and the title Bhadanta along with the name Dharmākaradatta. Since the title Bhadanta is generally applied to a Buddhist monk, it seems that Arcaṭa in his later life became a Buddhist monk and then he was known as Dharmākaradatta. From the statement of Tārānātha it is known that Bhadanta Dharmākaradatta was the teacher of Dharmottara. Much is not known about the personal life of Arcaṭa from his own statement as well as from the statement of Durveka Miśra; it can be known that Arcaṭa composed the following texts Kṣaṇabhaṅgasiddhi and Pramāṇadvitvasidhi and wrote a commentary on Hetubindu of Dharmakīrti. From the titles of the texts, the content of them can be easily known. The text Kṣaṇabhaṅgasiddhi has as its objective establishing the momentary character of all objects. The text Pramāṇadvitvasiddhi is aimed to show that the number of pramāṇas is restricted to two only. His literary style was very lucid and his penetrating philosophical insight is exhibited in his exposition of other’s positions both of Buddhist and non-Buddhist schools. Since he wrote a commentary on Dharmakīrti and he was the teacher of Dharmottara, it can be inferred that his period falls between the last part of the seventh century and first part of the eighth century AD.

Dharmottara Of all the commentators of Dharmakīrti, the most well known was Dharmottara. According to the report of Bu-ston, Dharmottara was the student of Vinītadeva but from the observations made by Tārānātha, it can be known that his teacher was Dharmākaradatta. He belonged to the lineage of Pramāṇavārttika in between Jñānaśrīmitra and Saṃkarānanda. As regards his time information may be gathered from Kalhaṇa. According to him, King Jayāpīḍa, one of the grandsons of Lalitaditya, saw in a dream that a sun was rising in the west and he invited Dharmottara to his court for teaching. Historical evidences report that the reign of Jayāpīḍa is known to be between 770 AD and the end of that century. His court had also other literary personalities like Kṣīra, Dāmodaragupta, and others. From this it can be said that the time of Dharmottara was the latter half of the eighth century AD. That Dharmottara lived in the eighth century is indicated by another fact. Dharmottara is a student of Dharmākaradatta, a contemporary of Prajñākarakara and a student of Śākyamati who was a student of Devendramati, a direct student of Dharmakīrti. Allowing a gap of 25 years among a teacher and his student, there is a difference of 100 years between Dharmakīrti and Dharmottara. Since the period of Dharmakīrti is believed to be seventh century, Dharmottara can be said to belong in the eighth century AD. Though Dharmottara did not write any commentary on Pramāṇavārttika, the magnum opus of Dharmakīrti, he commented upon two other texts of this great logician – Pramāṇavinis´caya and Nyāyabindu. The former commentary is known as Pramāṇavinis´cayaṭī kā, which Durveka Miśra has mentioned as Vinis´cayaṭī kā. The original Sanskrit text however is extinct, though Tibetan translation of it by

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Parahitabhadra, a Kashmir Pundit, and Blo-etn-sesa-rab is available. But the most well-known commentary of Dharmottara is that on the text Nyāyabindu of Dharmakīrti. Apart from writing commentaries on texts of Buddhist logic, Dharmottara himself had composed a few texts on Buddhism. One of them is Apohaprakaraṇa or Apohanāmaprakaraṇa. This text deals with the notion of apoha or exclusion which the Buddhists uphold as the referent of common noun and play the role similar to that of universal of the Nyāya Vaiśesika systems. Another well-known text of Dharmottara is Paralokasiddhi, which is aimed to prove the existence of a world after death, that is, a world different from the present one. The third text known to be written by Dharmottara is Kṣaṇabhangasiddhi, the main topic of which was to establish the doctrine of momentariness of the Buddhists. Most of the original Sanskrit texts of Dharmottara are lost; however, the Tibetan translation of these texts is available.

Vinītadeva Among the other noted scholars who wrote commentaries on Dharmakīrti, mention should be made to Vinītadeva. According to Tārānātha’s History of Buddhism in India, Vinītadeva lived during the time of King Lalitachandra who was the son of King Gopīcandra and Vinītadeva was a teacher at the Nālandā University. He was a student of Prajñākaragupta and his time is supposed to be the latter half of the eighth century. He wrote commentaries on the famous texts of Dharmakīrti like Nyāyabindu, Hetubindu, Vādanyāya, and Sambandhaparī kṣā and Santānāntarasiddhi. The Sanskrit originals of these commentaries are lost; however, the Tibetan translations of all these commentaries, which were made with the help of Indian Scholars and Tibetan interpreters, are available. In addition to writing commentaries on the texts of Dharmakīrti, Vinītadeva also wrote a commentary on the Ālambanaparī kṣā of Dignāga. Apart from these commentaries, Vinītadeva composed a text named Samayabhedoparacana-cakra, which dealt with the history of the 18 sects of early Buddhism. Prajñākaragupta The other well-known commentator of Dharmakīrti is Prajñākaragupta. He was also known as Alaṃkāra-upādhyāya. He had been a disciple of both Brāhmaṇa Śaṃkarānanda and Yamāri. He lived during the time of King Mahāpāla. According to Pandit S.C. Vidyabhushan, Prajñākaragupta was a lay devotee and was different from another Buddhist thinker of a more or less similar name – Prajñākaramati. This latter person was a monk and belonged to the University of Vikramaśīlā and lived during the period of King Canaka, who was the maternal uncle of King Mahāpāla. Prajñākaragupta had composed an elaborate commentary on the Pramāṇavārttika of Dharmakīrti which is known as Pramāṇa-Vārttikā-laṇkāra, in which he made a detailed analysis of the text. Another text written by Prajñākaragupta is Sahāvalambanis´caya. This text is basically on epistemology, dealing exclusively with a particular point, namely, joint occurrence of the ascertainment of an object and knowledge of the object. The original Sanskrit version of this text is no longer

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existent. But the Tibetan translation of it, made by the Nepalese Pandita Śāntibhadra and the Tibetan interpretation Śākya-hod, is available.

Tibetan Commentators of Dharmakīrti Most of the philosophical discussions found in Tibetan tradition since the eleventh century onward exhibit the strong influence of Dharmakīrti’s thoughts, specially his thoughts on logic and epistemology. This influence was to such an extent that Dharmakīrti’s terminologies were used even in course of discussion of the Mādhyamika thought. The two fundamental concepts of Dharmakīrti, namely, the concept of valid cognition (pramāṇa) and differentiation apoha, had been very frequently used in the case of debates taking place in the monasteries. In course of such debates, the logical rules that were followed were no other than those formulated by Dharmakīrti. The person who was responsible for the establishment of Dharmakīrti’s epistemology and logic in Tibet was Ngok Lo-dza-wa (1059–1109). The contribution of Ngok to Tibetan Buddhism was that he was the first person to emphasize on the philosophical aspect of Buddhism in Tibet which was formerly looked upon as a mere religion. Ngok made extensive translations of Dharmakīrti’s works. In addition to that he wrote commentary on the Pramāṇavinis´caya and paved the way for the tradition of logic and epistemology following the lines of Dharmakīrti. This task of establishing the logico-epistemological tradition got a strong footing by Cha-ba (1182–1251) who with his acute and original intellect made further developments in this regard. On the basis of all the epistemological works, Cha-ba composed a book Tshad ma sde bdun yid gi mun sel (Clearing of Mental Obscuration with Respect to the Seven Treaties on Valid Cognition) in which he presented in summary form the epistemological views of Dharmakīrti. The legacy of upholding the epistemological tradition of Dharmakīrti that was established by Ngok and Cha-ba continued in the thirteenth century in the hand of the fourth of five great Sagya masters, Sagya Paṇdita. Sagya Paṇdita pointed to the importance of Dharmakīrti by showing his necessity in understanding Buddhist account of knowledge as well as other aspects of Buddhism. Sagya Paṇdita stressed on critical acumen or prajṅā as central to Buddhist path and the study of epistemology as helpful to its development. And in this regard he put much emphasis on the study of Pramāṇavārttikakārikā as the text for understanding Dharmakīrti’s epistemology. He also composed a text entitled Tshad ma rigs gter (Treasure on the Science of Valid Cognition). In this text, he mainly depended on the words of Dharmakīrti himself as reflected in his different texts and not on the words of different commentators, though the views of the commentators were presented in a summary form. According to the available sources, Sagya Paṇdita studied with Sagya Śīlabhadra of Nalanda University for a long time, and on the basis of such study, he realized that the Tibetan interpretation of Dharmakīrti made so far corresponded partially to the actual thoughts of Dharmakīrti. That is why while writing his Tshad ma rigs gter he did not rely on the commentaries for understanding

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Dharmakīrti. In this book he criticized the realist interpretation of Dharmakīrti and attempted to reintroduce real universals in Buddhist epistemology. Though this approach of Sagya Paṇdita met with strong opposition initially, the situation changed around fifteenth century, mainly under the influence of Yak-dön. Yak-dön severely criticized the attempts of the followers of Cha-ba to read a realist interpretation of Dharmakīrti’s thoughts so as to believe in the existence of common sense objects and universals, as knowable by the sources of knowledge. To point to the incorrectness of those approaches Yak-dön wrote a commentary on Sagya Pandita’s Tshad ma rigs gter and his auto-commentary. Thus among the Tibetan commentators of Dharmakīrti, two completely conflicting attitudes can be noticed – one ascribing a negative anti-realistic trend to Dharmakīrti and the other a realistic one to his epistemology. But later on during the fourteenth century onward, a new trend was noticed among Tibetan Buddhist interpreters of Dharmakīrti. They tried to point out that Sagya Paṇdita’s views were quite compatible with the realist interpretation put forward by Cha-ba. This group of thinkers had as their leader Dzong-ka-ba, also known as Tshong kha pa, the founder of the Geluk school. Among the followers of Tsing kha pa, mention may be made of Gyel-tsap, Kay-drup, and Ge-dün-drup in this regard. All of them attempted to show that Sagya Paṇdita’s refutation of realism was mostly done on the basis of philosophical arguments. On the contrary if the common sense argument is accepted, there is no problem to provide a realistic interpretation to Dharmakīrti. In short, in the fourteenth and fifteenth centuries, when Dharmakīrti’s works lost their popularity in India, his thoughts were given new interpretations in Tibet so as to make them compatible with the Buddhist thoughts prevalent in Tibet during that time.

Post-Dharmakīrti Logicians: Jñānaśrīmitra, Ratnakīrti, Moksākaragupta ˙ Jñānaśrīmitra Of the four great erudite scholars respected as the gatekeepers (dvārapaṇdita) of the famous Vikramaśīlā Mahāvihāra, Jñānaśrīmitra was the second. Jñānaśrīmitra belonged to an age when Buddhist philosophy attained its maturity and so it was possible for him to refer to the views of his predecessors in his texts. Though his main concern was interpretation of the views upheld by Dharmakīrti, his competence of the philosophy of the entire Mahāyāna tradition is evident from his different writings. His exposition, therefore, stood as a bridge between the ancient and later Buddhist thoughts. In one of his texts Sākārasiddhi he provided a long list of the teachers of the school and sub-school he belonged to. There he mentioned the names of Maitreyanātha, Asaṇga, Vasubandhu, Dignāga, Dharmakīrti, and Prajñākaragupta. Thus it is evident that he belonged to the Yogācāra-Vijñānavāda tradition of Maitreyanātha and the sub-school of Pramāṇavārttika led by Prajñākaragupta. Jñānaśrīmitra, nowhere, had explicitly stated anything regarding his early life, nor about his time, but on the basis of the evidences obtained so far from other works of the Buddhist and non-Buddhist schools, it is now accepted that

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the philosophical career of Jñānaśrīmitra fell in the first half of the eleventh century AD. Jñānaśrīmitra composed twelve texts, namely, (1) Kṣaṇabhaṇgādhyāya, (2) Vyāpticarcā, (3) Bhedābhedaparī kṣā, (4) Anupalabdhirahasya, (5) Sarvas´abdābhāvacarcā, (6) Apohaprakaraṇa, (7) Ῑs´varavāda, (8) Kāryakāraṇabhāvasiddhi, (9) Yoginirṇayaprakaraṇa, (10) Advaitabinduprakaraṇa, (11) Sākārasiddhis´āstra, and (12) Sākārasaṇgrahasūtra. From the very title of these texts, it can be seen that they were composed on a variety of subjects, some of which were metaphysical like proving the momentariness of objects or discussing the Buddhist concept of causality or refuting the notion of an omnipotent godhead, while some were epistemological, discussing the relationship of knowledge and object (e.g., in Bhedābhedaparī kṣā) or showing the identity between the two (as in Advaita-bindu-prakaraṇa) that some dealt with the theory of meaning (e.g., Apohaprakaraṇa). But there were some texts in which he has dealt with topics of logic exclusively. Mention may be made of texts like the Vyāpticarcā, where the nature of invariable concomitance or vyāpti is discussed. Quoting the views of other logicians like Trilocana and Vācaspati and criticizing them, he tried to establish the position that the concomitance is in the form of essential identity and such invariable concomitance forms the basis of inferential cognition. In two other texts, namely, in Anupalabdhirahasya and Sarvas´abdābhāvacarcā, Jñānaśrīmitra discussed at length the nature and varieties of the probans as non-apprehension (anupalabdhi) which the Buddhists have admitted as the only way to obtain the knowledge of the absence (abhāva) of entities. Though these three texts of Jñānaśrīmitra deal exclusively with topics which fall within the scope of logic, his logical acumen is not confined to them only. Even in other texts where he had discussed other topics like momentariness or the existence of god, he had applied the subtleties of logic to refute the views of the opponents and to establish his own thesis. So, there is nothing exegetical to regard him as the great gatekeeper of the Buddhist tradition.

Ratnakīrti Among the latter thinkers who made noteworthy contributions to Buddhist logic, mention must be made to Ratnakīrti. Ratnakīrti was respected as a great scholar (mahāpaṇdita) and a practitioner of tantra. A detailed account of his personal life is not known except that he was a direct disciple of Jñānaśrīmitra and was associated with both Somapuri and Vikramaśīlā universities. Though he himself has nowhere stated his time, it is, however, inferred from the different sources available that he belonged to the first half of the tenth century AD. Regarding his academic lineage it can be said that he belonged to the tradition of VasubandhuDignāga-Dharmakīrti-Prajñākaragupta. His regard and respect for Dharmakīrti is evident from the fact that he considered him to be the only preceptor of the world (bhuvanaikaguru). Ratnakīrti took as his aim the task of presenting the rather difficult and elaborate works of Jñānaśrīmitra in a rather simple, lucid, and concise way. Besides Jñānaśrīmitra, he often had quoted passages from the works of Asamga, Maitreyanātha, Vasubandhu, Dignāga, Dharmakīrti, and Prajñākaragupta.

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A number of nine books had been identified as being composed by Ratnakīrti. His works may be classified under three heads on the basis of their aims – (1) those having refutation of the opponent’s position as their aim. To this group belong texts like Īśwarasādhanadūṣaṇa, Sthirasiddhidūṣaṇa, and Santānāntaradūṣana; (2) those aiming to establish the Buddhist views, like the Sarvajn˜ asiddhi, Kṣaṇabhangasiddhi, Apohasiddhi, and Citrādvaitaprakās´avāda; and (3) those aiming to establish the Buddhist theories by refuting the opponent’s views like Vyāptinirṇaya and Pramāṇāntarabhāva. Though the last two compositions have topics of logic and epistemology as their direct concern, his deep sense of logic is evident in each and every composition, whether it be for the purpose of refuting the opponents’ position or for establishing his own views.

Moksākaragupta After ˙the demise of Ratnakīrti, of those thinkers who tried to propagate Buddhist logico-epistemological traditions among the people, mention must be made to Mokṣākaragupta. Mokṣākaragupta belonged to the traditions of Dignāga and Dharmakīrti. His magnum opus is Tarkabhāṣā which is one of the popular texts of Buddhist logic after the Nyāyabindu of Dharmakīrti. Like the text Nyāyabindu, Tarkabhāṣā also contains three chapters dealing with perception, inference for oneself, and inference for others, respectively. This text apparently looks like a summary of Dharmottara’s commentary Nyāyabindutī kā; but the novelty of Mokṣākaragupta lies in the fact that he has presented the views of later Buddhist thinkers like Ratnākaraśānti, Ratnakīrti, etc. He also had presented the objections of non-Buddhist thinkers like Trilocana and others and tried to refute their objections from Buddhist perspective. His knowledge of Buddhism in general is evident from the fact that he had presented different important theories of Buddhist philosophy like the theory of momentariness, the existence of an omniscient being, and theory of apoha in the context of his discussion on logic. The detailed account of his private life is not available. But from different available sources, it is known that his time period is eleventh century AD.

Concluding Remarks History of the development of Buddhist logic shows that Buddhist logic continued up to the eleventh century AD and came to an end along with the decline and fall of Buddhism in India. But the influence that Buddhist logic left on later Indian logic continued for long and truly speaking was much stronger than the influence of Brāhmaṇic logic. Traditionally in the Nyāya system based on the Nyāyasūtra of Gautama, 16 categories were recognized while in the Buddhist logic only one category of pramāṇa was admitted. The Nyāyasūtra did not contain any reference to the invariable concomitance relation or vyāpti between the probans and the probandum as the basis of inferential cognition. Buddhist logicians, Vasubandhu,

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Dignāga, and Dharmakīrti particularly, gave much emphasis on this vyāpti relation and tried to explore its necessary character. Such discussions of inference and specifically of vyāpti left great influence on later Indian thinkers. After the decline of Buddhism, when there was revival of Brāhmaṇism, Brāhmaṇic logic became tinged by the colors of Buddhist logic and this gave birth to a new system of Brāhmaṇic thinking – the Neo-Naiyāyikas or the Navya Naiyāyikas who were engaged in the subtle analysis of vyāpti relation. Thus Buddhist logicians contributed significantly in developing a systematic formal system of logic in India.

References Anacker, S. 2002. The seven works of Vasubandhu: The Buddhist psychological doctor. Delhi: Motilal Banarsidass Pub Ltd. Reprint 2002. Bhattacharyya, K. 1978. The dialectical method of Nāgārjuna (Vigrahavyāvartanī ), text critically ed. E.H. Johnston and A. Kuns. Delhi: Motilal Banarsidass Pub Ltd, Burton, D. 2001. Emptiness appraised: A critical study of Nāgārjuna’s philosophy. Delhi: Motilal Banarsidass Pub Pvt Ltd. Chattopadhyay, M. 2002. Ratnakī rti on Apoha. Kolkata: Center of Advanced Study in Philosophy, Jadavpur University in collaboration with Maha Bodhi Book Agency. Chi, R.S.Y. 1984. Buddhist formal logic: A study of Dignāga’s Hetucakra and K’uei-cgi’s great commentary on the Nyāyapraves´a. Delhi: Motilal Banarsidass Pub Pvt Ltd. Dalsukhabhai Malvania, Pt. 1955. Pt Durveka Mishra’s Dharmottarapradī pa. Patna: Kashi Prasad Jayasaval Research Institute. Dreyfus, G. 1997. Recognizing reality: Dharmakī rti’s philosophy and its Tibetan interpretation. Albany: State University of New York Press. Dunne, J.D. 2004. Foundations of Dharmakī rti’s philosophy. Boston: Wisdom Publication. Joshi, Lal Mani. 1977. Studies in the Buddhistic culture of India. Delhi: Motilal Banarsidass Pub Pvt Ltd. Kajiyama, Y. 1998. An introduction to Buddhist philosophy: An annotated translation of the Tarkabhāṣā of Mokṣākaragupta. Wien: Arbeitskreis für Tibetische und Buddhistische Studien Universität. Katsura, S. 2008. Nāgārjuna and the Tetralemma. In Buddhist studies: The legacy of Godjin M. Nagao, ed. Jonathon A. Silk. Delhi: Motilal Banarsidass Pub Pvt Ltd. Kellner, Birgit. 2007. Jn˜ ānas´rī mitra’s Anupalabdhirahasya and Sarvas´abdābhāvacarcā: A critical edition with a survey of his Anupalabdhi theory. Wien: Arbeitskreis für Tibetische und Buddhistische Studien Universität. Mangala, M. 1988. Chinchore–Vādanyāya: The Nyāya-Buddhist controversy. Delhi: Sri Satguru Publications. Pradeep, P. 1993. Gokhale–Vādanyāya of Dharmakī rti: The logic of debate. Delhi: Sri Satguru Publications. Pradeep, P. 1997. Gokhale–Hetubindu of Dharmakī rti (A point on Probans). Delhi: Sri Satguru Publications. Shastri, Swami Dwarikadas, ed. 1972. Vādanyāyaprakaraṇa of Acārya Dharmakī rti with the commentary Vipan˜ citārthā of Acārya Santarakṣita and Sambandhaparikṣā with the commentary of Acārya Prabhācandra. Varanasi: Bauddha Bharati. Sukhlalji Sanghavi, Pt. and Muni Shri Jinavijayji. 1949. Hetubinduṭī kā of Bhaṭṭa Arcaṭa with the sub-commentary entitled Āloka of Durveka Mis´ra. Baroda: Baroda Oriental Institute. Thakur, Anantalal, ed. 1959. Jn˜ ānas´rī mitranibandhāvalī (Buddhist philosophical works of Jn˜ ānas´ rī mitra). Patna: Kashi Prasad Jayaswal Research Institute.

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Tilleman, T. 2000. Dharmakī rti’s Pramāṇavārttika; An Annotated Translation of the fourth chapter (parārthānumāna). Vol. 1. Wien: Verlag der österreichischen Akademie der Wissenschaften. Tola, F., and C. Dragonetti. 1995. Nāgārjuna’s refutation of logic (Nyāya). Delhi: Motilal Banarsidass Pub Pvt Ltd. Tucci, G. 1930. The Nyāyamukha of Dignāga: The oldest Buddhist text on logic after Chinese and Tibetan materials. Heidelberg. Vidyabhushana, S.C. 1988. A history of Indian logic. Delhi: Motilal Banarsidass Pub Pvt Ltd. 1988 reprint. Westerhoff, J. 2009. Nāgārjuna’s Madhyamaka: A philosophical introduction. New York: Oxford University Press. Westerhoff, J. 2010. The Dispeller of Disputes: Nāgārjuna’s Vigrahavyāvartanī . New York: Oxford University Press.

Logical Proofs in the Śivadrsti ˙ ˙˙ of Soma¯nanda

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John Nemec

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 The Syllogism of Somānanda’s Settled Opinion (siddhānta) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 When the Opponent Assumes the Nondualism for Which Somānanda Argues . . . . . . . . . . . . . . . 96 When the Opponent Must Accede to the Logic of Non-dual Śaivism . . . . . . . . . . . . . . . . . . . . . . . . . 98 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

Abstract

As is well known, Somānanada was a Brahmin of the Kashmir Valley who flourished circa 900–950 CE and who is the founding author of the famed Śaiva philosophical tradition known as the Pratyabhijñā or “Recognition” school. With the present chapter is pursued the somewhat modest concern of exploring various occasions when Somānanda deploys logical argumentation in his magnum opus, the Śivadṛṣṭi (ŚD). Three ways in which logic is deployed in the ŚD are examined in what follows. First is charted a syllogism appearing in the fourth chapter (āhnika), which is meant to prove the unity of all in the form of Śiva-asconsciousness. Following this, the present essay examines arguments Somānanda develops to illustrate how his opponents’ views can be understood to cohere logically only if they are understood implicitly to assume the existence of the ontological nondualism described by him. Finally, instances are examined in which Somānanda develops arguments with philosophical opponents, where the opponent is forced to accede to the nondualism for which he argues. Altogether, the present chapter illustrates the ways in which, for Somānanda, logical proofs offer more than mere rhetoric but are dispositive of the nature of reality itself. J. Nemec (*) Indian Religions and South Asian Studies, Department of Religious Studies, University of Virginia, Charlottesville, VA, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_47

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Keywords

Somānanda · Pratyabhijñā · Dharmakīrti · Śivadṛṣṭi · Pramāṇavārttika · Nondualism Abbreviations

conj. ĪPK ĪPKVṛ ĪPV ĪPVV KSTS PV ŚD ŚDVṛ

conjecture Īśvarapratyabhijñākārikās Īśvarapratyabhijñākārikāvṛtti Īśvarapratyabhijñāvimarśinī Īśvarapratyabhijñāvivṛtivimarśinī Kashmir Series of Texts and Studies Pramāṇavārttika Śivadṛṣṭi Śivadṛṣṭivṛtti

Introduction What place does logic hold in a tradition that makes a strong claim for the unity of all of existence as a non-dual, dynamic Śiva-as-consciousness? What is the status of logic itself in the context of such an ontological and epistemological nondualism? How can logic be deployed to know the reality that the tradition claims as true and ever-present? To what uses may logic be put? Scholars have long understood that the Pratyabhijñā offered a privileged place to logical discourse from the time of Utpaladeva (Torella 1994: xx–xxi, xxx) and have shown more recently that a sophisticated hermeneutic in Utpaladeva’s and Abhinavagupta’s writings ordered the tradition’s understanding of revelation (āgama) as it stands in relation to logical argumentation (yukti), such that the latter depended on the former but in practical terms was privileged over the same on the grounds that only it could bring those who did not already have faith to understand the legitimacy of the scripturally expressed views regarding the unity and ubiquity of Śiva (Ratié 2013). Still elsewhere, a cogent argument has been made for the philosophical oeuvres of Utpaladeva and Abhinavagupta as prosecuting a transcendental argument for the existence of God, Śiva (Lawrence 2018). The present chapter maps a somewhat more modest concern. It seeks to explore some of the instances in which Somānanda deploys logical argumentation in his magnum opus, the Śivadṛṣṭi (ŚD). As is well known (Nemec Forthcoming-a; Nemec 2011: 19–24), Somānanada was a Brahmin of the Kashmir Valley who flourished circa 900–950 CE and who is the founding author of the famed Śaiva philosophical tradition known as the Pratyabhijñā or “Recognition” school, the contributions to Indian thought of which feature, in particular, the writings of Somānanda’s disciple, Utpaladeva (fl. c. 925–975), author of the Īśvarapratyabhijñākārikās (ĪPK) and two auto-commentaries thereon, the Īśvarapratyabhijñākārikāvṛtti (ĪPKVṛ) and the

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longer Īśvarapratyabhijñākārikā-vivṛti or -ṭīkā, and also those of his great-grand disciple, Abhinavagupta (fl. C. 975–1025), the polymathic poet, aesthetician, tantric exegete, and philosopher who authored two sub-commentaries on Utpaladeva’s Pratyabhijñā writings, the Īśvarapratyabhijñāvimarśinī (ĪPV) (commenting on the ĪPKVṛ) and the Īśvarapratyabhijñāvivṛtivimarśinī (ĪPVV) (commenting on the Īśvarapratyabhijñākārikā-vivṛti). As also is well known, the Śivadṛṣṭi was supplanted at a relatively early date by the ĪPK and, in particular, the ĪPV, which was often received as the normative expression of Pratyabhijñā philosophy. The Śivadṛṣṭi nevertheless captures our attention not only for its intrinsic value as a philosophical and theological expression of Śaiva nondualism but also because it contains the seeds of thought that were developed more thoroughly, or at least in a more broadly understood manner, in the writings of Somānanda’s disciple and great-grand-disciple. This author does not yet detect the kind of sophisticated meta-discourse around logical argumentation and scriptural revelation that, as noted above, have been identified in the writings of Utpaldeva and Abhinavagupta, though much remains to be deciphered of the contents of the Śivadṛṣṭi, a famously difficult work of seven chapters, only half for which any commentary survives. At a minimum, however, one can see Somānanda deploying logical argumentation extensively to define the nature of reality (as a non-dual Śiva-as-consciousness), so too to overturn the arguments of an array of philosophical opponents, particularly in the later chapters of his magnum opus. Simply put, logic in the Śivadṛṣṭi functions to confirm or explain the nature of reality and to determine dispositively who understands just that, and the purpose of the present essay is to chart just such uses of logic by Somānanda. Three ways in which logic is deployed in the Śivadṛṣṭi are examined in what follows. First, Somānanda makes an ontological claim that stands as a logical ground of most all his philosophical arguments, namely, that the multiplicity of phenomena appearing in quotidian experience – in and as the world as it is regularly cognized and understood – itself furnishes evidence for the existence of an underlying unity, the oneness of all existence in the form of Śiva-as-consciousness. This syllogistic argument is examined in section “The Syllogism of Somānanda’s Settled Opinion (siddhānta),” below. Following this are charted two distinguishable if related modes in which Somānanda deploys logical argumentation, both of which serve to reinforce the fundamental and comprehensive ontological claim of the work by challenging the philosophical positions of opposing schools of thought. First, the present essay examines arguments he develops that illustrate how his opponents’ views can be understood to cohere logically only if they are understood implicitly to presume the existence of the ontological nondualism described by him. Each suggests an opponent whose views can only be made sense of logically on the basis of the existence of a Śaiva non-dual ontology, with the opponent’s reliance thereon being enacted implicitly and unwittingly. These have been reviewed at length by the present author in a forthcoming book manuscript (Nemec Forthcoming-b) and will be mentioned only in brief here in section “When the Opponent Assumes the Nondualism for Which Somānanda Argues,” below.

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Second, Somānanda can be seen to force his opponents to accede to his view of reality in the course of explaining their own positions. In these instances, Somānanda hypothesizes that his pūrvapakṣin opponents might try to rescue their positions in one way or another; yet, while the logic his opponents deploy in doing so may be deemed to be sound, the positions taken force these opponents effectively to adopt Somānanda’s own Śaiva position. That is, logic dictates that the opponent’s view can be rescued only by way of adopting the Śaiva position. A review of these arguments will offer some detail to Somānanda’s self-understanding of his own nondualism and are charted in section “When the Opponent Must Accede to the Logic of Non-dual Śaivism,” below. Of particular note in this latter group is a heretofore underexamined passage of the sixth chapter of the Śivdṛṣṭi, which challenges the Buddhist epistemologist Dharmakīrti (fl. c. 600–660) on his understanding of cognitions as utterly discrete and momentary phenomena, this by pointing to intrinsic logical contradictions and impossibilities attendant on maintaining that very momentariness of cognitions in the course of explaining cognitive errors whether incidental, as a false cognition of what is not present, or more fundamental, in particular what Somānanda’s Buddhist opponent claims to be a false understanding of the subjective-objective dimensions of cognition as real. Appearing in the part of the ŚD for which no premodern commentary is known to survive, this is a passage worthy of the reader’s considered reflection, this in particular given the central place Utpaladeva gave to a critique of the Buddhist epistemologists in his massively influential articulation of Pratyabhijñā. More simply put, the exemplar in question offers one further piece of new evidence for the as-yet little-understood contributions of the ŚD to the Pratyabhijñā tradition of philosophical debate with the Buddhist epistemological school, a tradition of debate that flourished following the production of the ŚD and in modes that supplanted Somānanda’s text with such overwhelming success.

The Syllogism of Soma¯nanda’s Settled Opinion (siddha¯nta) The settled opinion (siddhānta) for which Somānanda advocates is fundamentally an ontological one. And his proposition, that all things in the universe are in their very nature identical with the one and unitary Śiva, who exists in the form of consciousness, is presented in the fourth āhnika of the Śivadṛṣṭi on the order of a formal syllogism that is structured as follows: 1. All phenomena in the world are possessed of Śiva-nature (ŚD 4.1ab), 2. Because all phenomena in the world are possessed of causal efficacy (ŚD 4.1c-d). 3. Whatever has causally efficacy exists as a power or capacity that is controlled by an agent – what is Śiva-nature. This is so in the manner that a king directs those who serve him, for example, or the god of the dualist Śaivas deploys the power of illusion (māyā) and karmic equanimity respectively to create the world and to grace some among those dwelling within it with liberation (ŚD 4.4).

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4. Since all phenomena in the world are, indeed, possessed of causal efficacy, what is the mark of Śiva-nature (this inasmuch as the phenomena are identified ontologically with Śiva by virtue of being his very powers or śaktis) (ŚD 4.5), 5. Therefore, all phenomena in the world are powers or capacities of one who controls them – Śiva – , which is to say they are possessed of Śiva-nature (ŚD 4.6-7ab). The present author has translated and explained these passages of the heretofore understudied fourth āhnika of the ŚD in a forthcoming book (Nemec Forthcoming-b). It is evident the verses in question furnish all the five components of a classical, formal syllogism as defined at Nyāyasūtra 1.1.32 (pratijñāhetūdāharaṇopanayanigamanāny avayavāḥ), though they are nowhere identified as such either in Somānanda’s source-text (mūla) or Utpaladeva’s commentary thereon, the Sivadṛṣṭivṛtti (ŚDVṛ). Thus, (1) the thesis (pratijñā) is that all the many things apparent in the universe are ontologically unitary, one. (2) The reason (hetu) is that all are capable of producing results or effects in the world, which (3) requires an agent who directs or controls those causes, as is exemplified (the udāharaṇa) by a king whose subordinates act on his direction, or the like. And (4) inasmuch as all things in the world are causally efficacious (this being the “application” or upanaya), (5) one may conclude (nigamana) they are causally efficacious entities controlled by the one possessing them as powers, that is, by Śiva himself. Somānanda warrants this syllogism at ŚD 4.1d-3, this on the claim that the very existence of multiple capable entities in the world demands the existence of an organizing agent who commands and controls them, lest all things be themselves utterly independent. Action on Somānanda’s view, and contra his Buddhist interlocutors, is of necessity predicated on agentive intention, for, if each capable entity that may be witnessed to exist in quotidian life is to be kept to a discrete and designated sphere – to a circumscribed scope of action (lest everything be capable of doing anything) – then it follows that only the existence of a single, overarching and fully independent controller may guarantee their mutual relations, including not only that the range of their respective capacities is properly circumscribed but also that they may be mutually coordinated and set in motion. This, Somānanda argues, is to say that the very multiplicity of capable entities can only be understood to exist when said entities are set in and partake of a unitary, organized structural matrix, such that superior capacities bind or restrain subordinate ones, all in a single and controlled hierarchy. Without an underlying unity, the existence of multiple capacities – multiple capable entities – would engender chaos unseen in quotidian experience. The evident multiplicity of the world is evidence for unity, he thus claims, for there could be no other manner logically to understand a plural number of powers or capacities to function in a coordinated manner. Underscoring the realism of this position is the fact that Somānanda’s logical claim is not merely that multiplicity can only exist by way of unity but also that the very evidence for unity is the readily apparent multiplicity in the universe, which one experiences – knows – habitually in quotidian experience! For the fact that everything that is knowable performs some action and effects some result – at the very

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least, anything knowable in consciousness is capable of making itself thus known – suggests not only that all that can be experienced is real, but that all phenomena require a place in the aforementioned structural hierarchy and must be associated with a single agent who orders and controls them. Logic, he claims, proves that the multiplicity known ostensively is the very evidence of the ontological nondualism of all as Śiva, so too epistemological nondualism inasmuch as Śiva is a non-dual consciousness.

When the Opponent Assumes the Nondualism for Which Soma¯nanda Argues This, what is a theological claim made by way of this logical claim, is in fact grounds for a prolonged thread of arguments prosecuted in the fourth chapter of the ŚD, where the positions of the Buddhist Epistemologists, the Mīmāṃsā, the Sāṅkhya, and the realist Nyāya and Vaiśeṣika schools are interrogated in the course of reinforcing the terms of the syllogism identified above. As the arguments against Somānanda’s various philosophical interlocutors from the fourth āhnika of the ŚD have been examined in detail by the present author (Nemec Forthcoming-b), it is hoped that it will suffice here briefly to allude to these arguments, as follows. First, the logic of Somānanda’s siddhānta implicitly and explicitly challenges the claim of his Buddhist Vijñānavādin opponents that causality could function in an impersonal manner and by being grounded in an ontology recognizing a multiplicity of phenomena only related by causally contingent events. Indeed, the entire core ontology of Dharmakīrti’s Pramāṇavārttika (PV) is challenged in that Somānanda rejects the notion, expressed (in the course of explaining the unreality of universals) at PV (pratyakṣapariccheda) 3.44 (vastumātrānubandhitvād vināśasya na nityatā | asaṃbandhaś ca jātīnām akāryatvād arūpatā ||), that everything that is real necessarily ceases to exist. Instead, Somānanda argues not only as noted above, claiming that disparate appearances, disparate capable phenomena, could not be coordinated and could not interact in the absence of the existence of an organizing matrix, directed by an agent who possesses and directs and orders capable phenomena as capacities, but he also argues that everything is always and ever existent, if not always perceptible (for reasons explained in detail in the course of Somānanda challenging a Sāṅkhya articulation of the satkāryavāda). Somānanda argues, in a word, that none of the philosophical positions of the Buddhists addressed by him is logically supportable unless a permanent, if dynamic, unity – the one mapped in the syllogism found at the beginning of the āhnika, a unity embodied in the person of an ever-existent, powerwielding god, Śiva – is implicitly assumed to exist. In the fourth āhnika of the ŚD, the Mīmāṃsā presents as an apparent ally. As noted by the present author (Nemec 2019; Nemec Forthcoming-b), Somānanda seems to accede to his Mīmāṃsaka counterparts’ understanding of the nature of the relation of word (śabda) to thing denoted thereby (artha) – namely, that the relation or saṃbandha thereof is real and natural, not fabricated – but he does so only

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on the understanding that the same view implicitly assumes the existence of a unifying and unitary ontological reality of Śiva-as-consciousness to explain the same. (This, however, is evidently presented as a confirmation and not a criticism of the Mīmāṃsaka interlocutor.) Indeed, Somānanda argues that certain plainly observable capacities of language demand the existence of all as Śiva, as follows. First, one can see language used effectively to refer to unseen or unreal entities, the latter instance of which is particularly problematic if one understands the saṃbandha of word and object to be real, the logical implication being that unreal objects denoted by speech would themselves have to be real in order to partake (as relata or saṃbandhins) in their very relations with the words that denote them. A unity of all existence is therefore required to allow for such a possibility, reality and unreality being comprehensible, one may implicitly understand Somānanda to argue, only on the terms outlined above: a connection of a word to an “unreal” entity poses no problem when all of existence is understood to be the unitary and dynamic Śaiva in the form of consciousness, the existence of an illusion, error, or unreal object being explicable inasmuch as it appears in consciousness as a form or object thereof. Second, Somānanda similarly suggests that the same eternally existent relation could not be countenanced in instances where novel associations of word and meaning are fashioned, as when naming a newborn child or applying the word “cow” to a newborn calf, for example. It is the ontological unity of such names and objects, Somānanda argues, that necessarily serves to guarantee the Mīmāṃsaka understanding of language as regards denotation in such instances. Even linguistic multiplicity is sensible only under the condition of an epistemo-ontological nondualism of Śiva as consciousness. A detailed account of the satkāryavāda in ŚD 4, in turn, serves to support the thesis (pratijñā) of the syllogism that opens that āhnika by offering another reason (hetu) for the same. It defines the very manner in which a causal process, a procedure for the manifestation of apparently multiple entities in the universe, can be accounted for in the context of an ontological nondualism. The Sāṅkhya causal theory, even if it, like the Śaiva one, may properly be labeled a satkārya doctrine, fails to account for the appearance of the world as it is experienced in quotidian life, as a world where entities appear as if “manifested” and “destroyed” and cannot always be seen to be present. Causality must be grounded as much in explaining the non-manifestation of entities as in explaining how the multiple entities of the universe may be produced, what is that for which, Somānanda and Utpaladeva argue, only a theory of all as Śiva-as-consciousness can account. Non-manifestation must be accounted for, that is, for otherwise the doctrine that the effect is existent prior to the application of what causes it to be manifested, the satkāryavāda (so understood), could not stand, for everything would ever and always be perceptible, and everyday experience proves so much is not the case. More specifically, and as has been established in previous scholarship (Nemec 2012), what can account for any manifestation of duality is in fact constituted by an absence, a non-cognition of the ever-present non-duality (abhedākhyāti). Thus Somānanda says the following (ŚD 4.56 cd-57ab) in replying to an opponent’s objection, which suggests that Śiva-nature cannot be ubiquitous, because if he exists in the form of the effect he is destroyed when the effect is destroyed, and if Śiva is not present in the effect, then he is not ubiquitously present:

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J. Nemec naivaṃ yato hi bhāvānāṃ vināśe ’smāsu neṣṭatā || 4.56 || aṃśābhivyaktitā nāśo na nāśaḥ sarvalopitā | [Somānanda’s Reply:] This is not so, since we do not maintain that the entities are destroyed [at all]. For destruction is [nothing more than] the fact that a [different] part [of Śiva-nature] is manifested; it is not the case that destruction is a complete elision [of the “destroyed” entity in question].

Utpaladeva’s ŚDVṛ explains the matter in part as follows: asmaddarśane hi mauleḥ kaṭakotpādakāle ’ṃśasya hemamātrasyābhivyaktiḥ prāktanasya maulipariṇāmakṣaṇasyeti nāśa ucyate na tu sarvāṃśādarśanam. For in our philosophy, at the moment when the bracelet [made of gold] is produced from the [golden] diadem, there occurs a manifestation of a part [that is the bracelet] that is nothing but gold; it is the manifestation of the preceding moment, that of the transformation of the diadem [into the bracelet], that is spoken of as destruction, but it is not the case that every part [of the gold] ceases to appear.

In sum, only a unity of all as Śiva, with the possibility of the absence of a cognition of the same, can explain the appearance and disappearance, the alternating cognizability and non-cognizability, of objects that exist prior to their being produced as effects. Finally, Somānanda clearly indicates that he opposes the dualism of Nyāya and Vaiśeṣika philosophical formulations precisely because they cannot account for the manner in which individual entities may logically be understood to stand in mutual relation. For indeed – and as argued above –, he understands the appearance of duality in precisely the inverse manner as does his realist interlocutor: apparent duality, on his understanding, proves unity. Unity, in turn, is presupposed but may be subjected to a sort of (metaphorical) dissection by way of a witnessing of one or another of its aspects or parts (aṃśas). Reality is not built up from discrete entities into complex wholes, but rather the opposite: it is a single unity that comes to be known in real and discrete parts. Indeed, only by presupposing unity can any mereological analysis cohere logically. In a word, Somānanda prosecutes a thorough critique of his dualist, realist interlocutors in an effort to support the warrant that justifies the core argument of the fourth āhnika, leading him logically to claim, as he does, that the appearance of multiplicity in fact serves to establish the very existence of the one, unitary Śiva.

When the Opponent Must Accede to the Logic of Non-dual Śaivism Let us turn our attention now to three occasions when Somānanda finds his opponents inadvertently acceding to the logic of his own nondualism. In each, Somānanda hypothesizes that a demand for logical coherence in the given

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opponent’s position requires that same opponent to accept his own formulation of Śaiva nondualism. First, at ŚD 2.57ab, Somānanda says the following: kiñcit paśyati vā sūkṣmaṃ tad asmaddarśanānvayaḥ | Alternatively, she [i.e., paśyantī] sees some subtle entity, in which case you adopt our point of view.

The context of the present comment, which, as we shall see, suggests that agent and object of cognition are identical, is one in which Somānanda challenges the philosophical views of the Grammarians (the vaiyākaraṇasādhus, as he refers to them at ŚD 2.1c) – that is, in particular, Bhartṛhari. This assault famously consumes the entire second āhnika and begins at ŚD 2.2 with Somānanda summarizing his opponents’ position: ity āhus te paraṃ brahma yad anādi tathākṣayam | tad akṣaraṃ śabdarūpaṃ sā paśyantī parā hi vāk || 2.2 || They say the following. The supreme Brahman, which is beginningless and endless, imperishable, whose form is speech, is paśyantī, supreme speech.

Somānanda shares with Bhartṛhari and the Grammarians their conception of all as a non-dual consciousness, but he argues an implicit dualism is imputed in their formulation of the same: he asks how what Bhartṛhari refers to as paśyantī can be said to be the ultimate when by its very definition it involves a distinction of the seer from what is seen. Thus, ŚD 2.20cd-21ab: vartamānasamārūḍhā kriyā paśyantyudāhṛtā || 2.20 || dṛśiḥ sakarmako dhātuḥ kiṃ paśyantīti kathyatām | The verbal form “paśyantī” is in the present tense, [and] the verbal root “to see” is transitive, so, do tell, what does paśyantī see?

As Utpaladeva’s commentary on ŚD 2.57 clarifies, Somānanda’s position is that the Grammarians can attempt to navigate a way out of the implicit dualism of the etymological meaning of paśyantī, this by suggesting that the term in fact refers to a single entity, one with agency, which is both seer and seen, and which does not “take in,” as it were, external objects of sight. But this would be to accept Somānanda’s philosophical position: yadi vā bhavadbhir nānyan nīlapītādi sthūlaṃ vibhaktasvarūpaṃ dṛśyaṃ paśyaty api tu dṛśyabhuvam anāpannaṃ draṣṭāram eva kiñcid avikalpyaṃ sūkṣmam apṛthagrūpaṃ paśyati, na ca na (na ca na conj.; na ca Nemec 2011: 334, KSTS edition, Torella 2014: 569) paśyati dṛśyārthamayadraṣṭṛrūpaiva (dṛśyārthamayadraṣṭṛrūpaiva Nemec 2011: 334; dṛśyārthamayadṛṣṭṛrūpaiva KSTS edition, dṛśyārtham adraṣṭṛrūpaiva Torella 2014: 569)

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satī prakāśate paśyantīti kathyate. tato ’smadīyadarśanānugamaḥ syāt paśyantyarthaḥ kriyākartṛkarmakālavibhāgātmā tyakto bhavet. Alternatively, you might argue that she [i.e., paśyantī] does not see a distinct, coarse visible object whose nature is differentiated [from her], what is blue, yellow, or the like, but she instead sees a certain inconceivable, subtle entity not different in form [from her], what has not acquired the nature of a [distinct] visible object, [but] is nothing but the seer (this interpretation of dṛśyabhuvam anāpannaṃ follows that of Torella 2014: 569); and [yet] it is not the case that she does not see, she being (i.e., inasmuch as she is) what appears as the very form of an agent of seeing who consists in the visible objects [that are cognized], [and thus] is called “seeing” (Torella 2014: 569 here offers a somewhat cavalier emendation based entirely in semantics, though it renders the Sanskrit syntactically awkward and semantically insufficiently precise). [Reply:] For that reason, you must accept our point of view. You [therefore] must abandon the [literal] meaning of [the word] paśyantī, which refers to the separation of action, agent, object, and time.

Paśyantī, “seeing,” must be explained to its literal meaning, but logic dictates that even to attempt coherently to do so requires the adaptation of an understanding of the nature of awareness that not only contradicts the implicitly dualistic semantics in play but also amounts to the adoption in full of Somānanda’s Śaiva ontology, which alone can account for any cognitive awareness so subtle as to be utterly non-dual in nature. Elsewhere – this is the second instance in which an opponent is forced to accede to the logic of Somānanda’s nondualism – , the apparent duality of the everyday world is shown not to be explicable by other non-dualistic traditions. The logic is straightforward: it is either the case that multiplicity must be explained away so as to allow only for the existence of a single unitary being, in which case the apparent multiplicity found in quotidian experience is an illusion or an error, what must be distinguished from the one, divine being and thus effects a duality that differentiates what is real from unreal; or, multiplicity is preserved, but its very existence destroys any ontological nondualism for its very multiplicity – unless, that is, one accedes to Somānanda’s explanation of the real existence of the many that is witnessed in the world of quotidian experience as one. The opponents addressed in the relevant section of the ŚD are adherents of the Pañcarātra and Vedāntins, who maintain a host of mutually distinguishable views too numerous to detail in the present essay but which have been examined already by Dyczkowski (1987: 232, fn. 96) and the existence of which were first noted, in particular for a certain prevalence of illusionism in the Vedāntic positions, by Sanderson (1985: 210, fn. 41). It is a Vedāntin opponent who is criticized in the passage to be reviewed here below. Context for the passage we wish to highlight is offered in the opening verses of the ŚD’s sixth āhnika, as follows: atha śakteḥ śaktimato na bhedo dravyakarmavat | sthāpito dravyato bhinnā kriyā no na ca nāsti sā || 6.1 || evaṃ tathā śaktimataḥ śaktasya samavasthitā | jagadvicitratā śaive na punar darśanāntare || 6.2 ||

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Now, no [ontological] distinction is established [in our view] between the power and the possessor of the power just as [we make no such distinction] between substance and action. Action simply is not (na u) differentiated from substance, nor is it the case that it [i.e., action] does not exist. And it is in this way that it is in Śaivism [alone] and not in any other philosophical system that the variegated nature of the universe is firmly established for [none other than] the empowered possessor of power.

In brief, the present passage, for which neither Utpaladeva’s nor Abhinavagupta’s commentary survives (the latter being lost altogether, excepting for the brief quotation thereof in Abhinavagupta’s Parātriṃśikāvivaraṇa), suggests that the ontological distinction between substance and action noted in other traditions, such as the Nyāya and Vaiśeṣika, is erroneous. It also states that the variegated nature of the world of multiple entities is real and can only be explained by Somānanda’s brand of Śaivism, variegation belonging to, being the nature of, the empowered possessor of powers, Śiva himself. To be is to do, and vice versa, this per Śaiva nondualism and no other philosophical school, and the one who exists and acts is the empowered Śiva and no other, who is, as a result, the universe itself. It is immediately following this passage that the somewhat extensive arguments against the Vedānta, and the Pañcarātra, are offered. ŚD 6.3 suggests there are various views of variegation to examine, but the one here placed in question is brought to bear in the next verse. The criticism begins with Somānanda questioning how his various Vedāntin opponents could come to understand Brahman to become variegated (citra): yatra brahmocyate citraṃ kaiścid vedāntavādibhiḥ | ekasya citratā kena hetunā brahmaṇo bhavet || 6.4. || Where certain propounders of the Vedānta say that Brahman is variegated, [we must ask:] by what cause could the variegation of the unitary Brahman come into being?

The problem Somānanda identifies is in the mode in which his opponent explains how a singular Brahman could make itself multiple, or even apparently multiple; at question, therefore, is the nature of the relationship of the unitary Brahman to the myriad forms that make up the phenomenal world. Somānanda considers the possibility that the multiple entities of the universe, mutually differentiated as they are, are mutually distinguishable by their very natures. This suggests there is no cause of the production of variegation, but that it is always present, a possibility he rejects because so much would require there to be no underlying unity of phenomena: tathāvidhā vibhinnās te sarvadā nijabhāvataḥ | vibhinnā eva te naikyaṃ mṛtpiṇḍāt prāgavasthiteḥ || 6.5 || ghaṭādīnāṃ. . . [Objection:] Those sorts of entities [that make up the variegated universe] (tathāvidhāḥ) are manifold at all times by their very natures. [Somānanda’s Reply:] They are [therefore] truly

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manifold (vibhinnā eva); pots, etc., [for example,] would not be possessed of a unity derived from their [shared] previous state of being, from [their previous existence as] the ball of clay.

Here expressed, then, is a fundamental question regarding the relationship of the diverse world of phenomena to the divine agent that is said to be one. At question is the relationship of multiplicity to unity, with the integrity of the latter being challenged by the existence of the former. Somānanda suggests that it is not possible for his Vedāntin opponents to imagine an intrinsic difference in the entities in question, because there would thus be no way to find an implicit or underlying unity in them. But couldn’t they have a double nature, both unitary and manifold? This would suggest there is no need to account for the cause of multiplicity, but also that the ever-existent mutual differentiation of entities does not threaten the existence of any underlying unity. Somānanda telegraphs the matter as follows (ŚD 6.6a-b): . . .dvirūpatvaṃ na ghaṭādeḥ sadā sthiteḥ | [Opponent’s Objection:] They [i.e., the pot, etc.,] are possessed of a double nature (dvirūpatvaṃ). [Somānanda’s reply:] Not so, because the pot, etc., would continue to exist in perpetuity (sadā sthiteḥ) [as manifold entities].

The evident implication of this argument is this, that the variegated entities, the pot, etc., would perpetually exist as a pot and the like. On Somānanda’s view, it is not possible for the multiplicity to be real by dint of such diverse entities having such a dual nature, for in claiming the nature of the pot, for example, is that of the clay of which it is fashioned and a discrete entity (as a pot) leads to the eternal perdurance and not the ultimate dissolution of distinction or difference – in a word, of multiplicity. There would ultimately be no possible recourse to the unity of Brahman. If the manifold nature of the universe cannot be explained as perpetually existent, then it must be created; there must be a cause thereof. It is to the possibility that his Vedāntin opponent will attempt to account for this cause that Somānanda turns next, at ŚD 6.6c-d: nimittakalpanā kalpyā tarhy avidyā. . . [Opponent’s Objection:] There is a forming [of the manifold entities] by a cause (nimittakalpanā). [Somānanda’s Reply:] In that case, nescience (avidyā) must be imagined [by you as that cause].

With this tersely conveyed turn in the argument Somānanda anticipates his opponent will argue that a(n instrumental?) cause indeed can be conceived of for the variegation, to which he replies that it cannot but be “nescience,” avidyā, this of course being an item of prominent place in Vedāntic thought. The manifold entities that make up the universe thus are not eternally present but effected. Yet the implication of Somānanda’s reply is evidently this, that such a cause as avidyā

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inevitably leads one to understand the world of variegation to be, like its cause (i.e., avidyā), something unreal, a nescience. After all, Somānanda himself criticized his Grammarian opponents (at ŚD 2.22cd-23) on the hypothetical that they would claim paśyantī could see entities as external on the basis of an association with avidyā, part of the criticism being that it is difficult to countenance a connection of the real paśyantī with something unreal, such as avidyā. Utpaladeva’s ŚDVṛ with the following made the matter explicit, namely that avidyā by the very meaning of its name must be understood not to be real: avidyā hy avidyātvābhyupagamād evāsatyā, na cāsatyayā saṃbandho rūpaśelṣamayo yuktaḥ (“For avidyā is unreal only inasmuch as it’s nature as ‘non-knowledge’ is accepted, and it is not logically possible for there to be a connection that consists in a union of form with an unreal [avidyā].”). Here too, it appears, the conception of any manifestation by avidyā is doubted for the fact that the cause itself imputes non-existence, an illusionism or outright error. Somānanda thus hypothesizes next that his Vedāntin opponent might deny such a claim and argue instead that the variegated universe is not unreal but real, truly brought to be by its cause: . . .tadudgamāt || 6.6 || bhāvā bhaveyus tat prāptā hy asmākaṃ sarvasatyatā | [Vedāntin Opponent:] The entities may [in fact] come into being, this because it [i.e., avidyā] arises. [Somānanda’s Reply:] Therefore (tat), our own [view], the fact that everything is real, has been arrived at [by you].

In other words, if Brahman itself is understood to effect a real universe that is simultaneously one with itself, this with a power or instrument that, logically speaking, cannot be separated from itself if unity is to be preserved, then the position articulated becomes one identical to that of the Śaivas. The real manifestation of real effects that are nevertheless identical with their cause, which in turn is identified with the causal agent, all being consciousness – this is precisely the Śaiva view. But it is not that of the Vedāntins if avidyā can be at all what the term for it quite literally denotes. And yet, logic dictates they must accede to the Śaiva view of the matter if nondualism is to be preserved. Third and finally, we offer an important, heretofore little examined passage of the ŚD that presents new evidence of Somānanda’s engagement with, in particular, the Buddhist epistemologist Dharmakīrti. Somānanda opens his treatment of the views of his Buddhist opponents at ŚD 6.32cd, this by distinguishing realist from idealist traditions, and, as the following suggests, he rather understands and classes the schools in ontological terms: ye bāhyavādino bauddhās te bhedaṃ samupāśritāḥ || 6.32 || Those Buddhists who maintain the doctrine of [real] external [entities] (bāhyavādinaḥ ¼ bāhyārthavādinaḥ) resort to dualism.

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This is all Somānanda says here of the externalist Buddhists, who evidently merit no further discussion and this simply because they espouse a dualism, which of itself disqualifies them from serious consideration. Now, Caturvedi (Chaturvedi 1986: 227) suggests that Somānanda has a Vaibhāṣika opponent in mind, here, but it is also possible that a Sautrāntika opponent is addressed; for one must note that the ĪPK and its commentaries treat just such a Bāhyārthavādin with some detail. Specifically, Utpaladeva understands a Sautrāntika opponent to argue, contra the Vijñānavādin, that the latter must understand real objects to exist externally to consciousness, for it is the only logical inference to draw from the appearance of variegation in consciousness: An effect must reflect the nature of its cause, so multiplicity must present itself as a result of a multiplicity of causes. (See ĪPK and ĪPKVṛ ad 1.5.4; cf. Ratié 2011: 372.) This author regrets that there is no space here to summarize Utpaladeva’s treatment of these arguments, which is substantial. Somānanda for his part quickly moves to another ontological concern in initiating what is of present interest, a sustained critique of the Vijñānavādins, one requiring them to accede to the logic of Somānanda’s Śaiva nondualism. The critique is begun in earnest at ŚD 6.33. In invoking the name of his opponent Somānanda raises the question of the possibility of ontologically dissimilar entities or phenomena appearing in mutual relation, what is partially analogous to the impossibility, according to him, of material (mūrta) and immaterial (amūrta) entities finding ground for mutual interaction (at least in the absence of the existence of a Śaiva non-duality of all as consciousness), this as explained in the realist Naiyāyika and Vaiśeṣika schools (about which see Nemec Forthcoming-b): vijñānavādināṃ jñānaṃ satyaṃ pratyety asatyatām | bahiḥ kathaṃ na hy asatyaṃ satyād bhavitum arhati || 6.33 || The cognition of the Vijñānavādins, which is real, cognizes unreality (asatyatām) [that appears as if] externally [to itself]. How is this possible? For it is not the case that that which is unreal is able to come to exist as the product of that which is real.

Somānanda suggests, that is, that it is logically impossible to claim that that which is unreal can be produced by that which is real; and so, he further asks, how is it possible for this idealist school to suggest that the objects of experience can appear as external to consciousness even while they are held in fact not to be thus? As Chaturvedi in his Sanskrit commentary treats the matter (Chaturvedi 1986: 227), the Vijñānavādins – the following verses of the ŚD will clearly indicate that Somānanda here has Dharmakīrti in particular in mind, he being quoted in ŚD 6.39, as Torella 1994: xxii, fn. 28 noted, already, some time ago – suggest that nothing appears that is not apparent in, comprised of, consciousness; and, therefore, because this is all that may appear, no ontological difference between what is real and unreal may be maintained:

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tatraiva vijñānavādināṃ mate satyaṃ jñānaṃ bahiḥ ¼ bāhyaṃ jagadrūpeṇa asatyatāṃ pratyeti. satyaṃ jñānam asatyasvarūpeṇa bahirjagadākāreṇa bhāsate. Among these [Buddhist schools of thought], viz., in the thinking of the Vijñānavādins, a real cognition cognizes as external an unreality in the form of the world. [I.e.:] A real cognition appears in the form of the external world, whose nature is unreal.

Chaturvedi supports this interpretation of Somānanda’s intended argument by suggesting he here has PV (pratyakṣapariccheda) 3.335 in mind, that exceedingly well-known verse and doctrine (the sahopalambhaniyama), which Chaturvedi cites to illustrate the fact that no difference exists between an object – “blue” for example – and the consciousness thereof: darśanopādhirahitasyāgrahāt tadgrahe grahāt | darśanaṃ nīlanirbhāsaṃ nārtho bāhyo ’sti kevalam || PV 3.335 || Because there is no apprehension of that [object] which is devoid of that attribute that is perception, [and] because there is an apprehension [of the object] when that [perception] is apprehended, it is perception that has the appearance of “blue;” no external object exists that is distinct [from the perception]. (This translation is adapted from the French-language rendering of Ratié 2011: 346; cf. ibid. 347, fn. 87.)

As Chaturvedi sees it, then, Somānanda wishes to ask how a Vijñānavādin can explain the manner in which that which appears in consciousness can erroneously appear as if it were external to consciousness. So much should not be possible if what is unreal is not to be produced by what is real. Next, at ŚD 6.34ab, the Buddhist offers a retort, which is (as is Somānanda’s wont) laconically expressed: kāmināṃ katham etac ced uktā prāg asya satyatā | Objection: how is this so [i.e., how is it the case that it is impossible for something unreal to be produced by something real] for those possessed of desire[, etc.]? [Reply:] The reality of this [cognition, supposedly “erroneous,” belonging to those possessed of desire, etc.] was stated [by me], earlier.

Given that this half-verse refers to the fact that “the reality of this was stated, earlier” (uktā prāg asya satyatā), it is clear that Somānanda here makes an internal reference to an earlier part of the ŚD. And the context of this half verse demands a treatment of error, this because etat (ŚD 6.34a) must be taken to refer to “this [state of affairs, namely, that it is not the case that that which is unreal is able to come to exist as the product of that which is real (¼ ŚD 4.33c-d: na hy asatyaṃ satyād bhavitum arhati)].” For these two reasons, this author proposes that Somānanda refers to the quality of what are classed by the Buddhists as falsely-cognized entities, which can result from certain emotional states or the like. These were mentioned first at ŚD 1.45cd-46ab:

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dṛśyante ’tra tadicchāto bhāvā bhītyādiyogataḥ || 1.45 || tatra mithyāsvarūpaṃ cet sthāpyāgre satyatedṛśām | By his will, entities are seen here as a result of being associated with fear, etc. If you argue that they have an erroneous nature, I will establish the reality of such entities later on.

As has been known for some time (see Gnoli 1957: 22), Somānanda here refers to PV (pratyakṣapariccheda) 3.282: kāmaśokabhayonmādacaurasvapnādyupaplutāḥ | abhūtān api paśyanti purato avasthitān iva || (The KSTS edition, post correctionem, reads kāma- for bhīti- (ŚD 1.45d), but all the manuscripts so-far examined by the present author carry the latter reading, which therefore is preserved.) Utpaladeva’s ŚDVṛ adds some context to this statement in a passage that evinces no variant readings in any of the manuscripts examined thus far by the present author. It is recorded as follows: kāmaśokabhayādiyogāc ca te te bhāvāḥ puraḥ sphuranto dṛśyante. tatra bhagavadiccaiva kāraṇaṃ tāvaty aṃśe bhāvanāvaśād īśvaratāveśāt. na ca tatra caurādayo mithyārūpā bhānti yato ’gre satyatvam īdṛśānām avaśyaṃ sthāpanīyam. Moreover, by association with desire, sorrow, fear, etc., various entities are seen appearing in plain sight. It is the Lord’s will alone that is the cause of these, because the force of the [mental] cultivation [one has] toward such an entity results from [the individual] being penetrated by a state of being Īśvara; nor do they, i.e., the thieves and so on, appear in an erroneous form, as the reality of such [entities] will be definitively proven later on.

Thus, the present author concludes that the kāmins mentioned at ŚD 6.34a are those who are taken by desire, experience the apparitions of thieves, are afraid, and the like. (Of course, the survival of the ŚDVṛ on this passage would likely have aided in our interpretation of the same.) Somānanda therefore may be understood here to argue that the Buddhist cannot escape the problem to hand – that of the impossibility of something unreal being produced by what is real – by claiming that everyone, whether Śaiva or Buddhist, understands unreal appearances to come into one’s consciousness: the Śaivas have proven, so far as Somānanda is concerned, that everything is real qua the fact of appearing in/as consciousness; no unreality (i.e., an illusion or true error of cognition) is produced by what is real (i.e., by consciousness itself). (The argument for this position, moreover, is detailed in the fourth āhnika of the ŚD and examined in detail in Nemec Forthcoming-b.) What comes next is somewhat laconic (again) and admittedly somewhat mercurial in its meaning. At stake in ŚD 6.34cd-35 is a discussion of the nature of the agent and instrument of cognition. Let us read these lines of text first and following this interpret them in light of what precedes them. jñānasya karaṇatvena kartrapekṣā prasajyate || 6.34 || kartṛtve jñaptimātratve ākāṅkṣā karaṇe bhavet | dvirūpatve viruddhatvasakramatvaṃ pravartate || 6.35 ||

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[Somānanda: If your view of erroneous cognitions were correct, then:] What is occasioned (prasajyate) is that the cognition [of something held to be unreal] is in need of an agent [of cognition who deploys it, the kāmin, e.g.] as the instrument [of seeing the unreal cognition]. When the agency [as cognizer] exists as the very nature of the simple cognition, there arises a need for an instrument [by which the cognition sees “erroneously” and not accurately]. If [you argue that] there is a dual nature [to the cognition or jñāna, as both agent and instrument of knowing], [then] the result is that it is [self-]contradictory or is sequential in nature [and therefore not momentary].

This argument depends on the understanding that the cognition or jñāna, in the Buddhists’ view, is momentary and singular in nature. A problem arises in defining the nature of the source of the error in question. There must be both a cognition and something “added” that allows for the erroneousness to be introduced. The agent of knowing presents as the source of just this, given the nature of the source of cognitive error identified in PV 3.282, namely, the emotional states of the cognizers themselves. Consider, for example, what Manorathanandin’s commentary says about PV 3.282: bhāvanābhavaṃ kathaṃ spaṣṭam ity āha – kāmaś ca śokaś ca bhayaṃ ca tair unmādāś caurasvapnādayaś ceti kāmaśokabhayonmādacaurasvapnādibhir upaplutā bhrāntās te ’bhūtān apy arthān bhāvanāvaśāt purato ’vasthitān iva paśyanti, yasmāt tadanurūpāṃ pravṛttiṃ ceṣṭante. Regarding the question as to how the coming into existence of a [mental] cultivation (bhāvanā) is perceived, he says: Those who are mad with these, viz., desire and sorrow and fear, as well as those whose dreams are of thieves, etc. – it is those who are afflicted with [such] desire, sorrow, fear, madness, and dreams of thieves, etc., are [thus] perplexed, who see objects appearing as if in front of them even though they do not exist, this due to the force of [their] [mental] cultivation. Because of this, they perform the behavior that is befitting of as much.

Thus, the present author understands Somānanda to argue that the erroneous cognition is dependent on an agent of cognition to see erroneously at all, possibly meaning that Somānanda argues that on this view the cognition is corrupted by an agent who is in some sense distinguishable from it (what would not accord with the Buddhist’s own understanding of the nature of cognition). On the other hand, if the Buddhist understands the cognition itself to cognize what is on their view in fact an error, then they lack an explanation for what instrument triggers the error in cognition, for after all a cognition could see correctly and not perceive what is in fact not present, but it is the “excess” of the erroneous cognition that must be explained. Something must lead the error to be cognized. Finally, if the Buddhist opponent claims that in fact the cognition or jñāna functions as both kartṛ and karaṇa – if the cognition plays the role of both the agent of the action that is cognition and the instrument of the same such that it cognizes erroneously – then one of two faults would arise. Either the jñāna would be possessed of mutually contradictory natures – seeing what is in fact present and what fear or the like causes it erroneously to see, therefore being both real and unreal; or

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the same cognition, being both agent and instrument (cognizing in distinct ways, one correctly, one erroneously), would have to exist in more than a single moment of time, viz., sequentially as one and then the other of the two modes of the act of cognition. It is impossible, by the argument of viruddhadharmasamāveśa – the impossibility of contradictory properties belonging to the same object (note that Somānanda examines Dharmakīrti’s own reference to a form of contradiction that is the co-presence of mutually contradictory forms, such as existence and nonexistence, about which see Nemec 2012: 231, fn. 17; Nemec Forthcoming-b) – , for the cognition simultaneously to be both the instrument of seeing and the agent of that act in question, the latter producing what is in fact unreal, the former what is real. On the other hand, Somānanda’s Buddhist opponent cannot understand the jñāna to be possessed of a dvirūpatva by virtue of changing over time, for this involves the jñāna perduring beyond its mere (supposed) momentariness: sakramatva necessarily contradicts momentariness (kṣaṇikatva). So much, then, brings us to the moment when Somānanda compels his opponent to acquiesce to his own philosophical position. ŚD 6.36ab reads as follows: akramatve ’smanmatatvaṃ tadaikyaṃ pariniṣṭhayā | If it [i.e., the jñāna] is not possessed of a sequence (akramatve) [and yet is held simultaneously to be both the kartṛ and karaṇa], you have adopted our view of it (asmanmatatvam): its unity [i.e, that of the cognition] (tadaikyam) [N.B.: It is possible that tadaikya is not a compound but a post-sandhi conjunction of tadā and aikya.] would exist fundamentally (pariniṣṭhayā) [in your view, if you accept as much].

If the opponent understands a consciousness to exist that is both agent and object of cognition simultaneously, and also is such without any sequence existing in its nature, then they accept the Śaiva view. A jñāna understood as such – as that which is sequence-less, simultaneously both agent and object – would be fundamentally one – precisely that for which Somānanda argues. And for Somānanda, such an entity is and must be not simply one, but also utterly real. It is, in fact, all that exists. And it exits in perpetuity, not as a momentary instant of consciousness/cognition. Evidently with this in mind, Somānanda goes on immediately to challenge the momentariness doctrine of his Buddhist opponent: na cāpi kṣaṇabhaṅgitve yuktā vaināśikī sthitiḥ || 6.36 || And it also is not the case that, when it [i.e., the cognition] perishes in an instant (kṣaṇabhaṅgitve), a state that is its perishing is logically tenable [for it].

Chaturvedi suggests that the destruction of what is utterly momentary presents a contradiction in terms, for the jñāna would in a single moment both exist and not exist: to be destroyed, it must be “present” as it were, at its own demise, which must not occur in the moment when it acts as karaṇa and/or kartṛ, and the two states (as agent and/or instrument of action; as the entity that is destroyed) are mutually contradictory in nature. Chaturvedi’s Sanskrit commentary (1986: 228)

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says the following: kṣaṇabhaṅgitve ¼ jñānasya kṣaṇikatve vaināśikī ¼ naśvarā sthitiḥ na yuktā ¼ asaṅgatā naśvaratvaṃ sattvaṃ ca sahaiva varttetām iti sarvathā asaṅgatam ity arthaḥ. The jñāna cannot be truly momentary, as the Vijñānavādin would have it; it cannot be both that which perishes and that which functions in the act of cognition in a single moment, for then existence and destruction would co-exist. Again, the law of self-contradiction proscribes the Buddhist position. Following this, Somānanda challenges the very complexity in the cognition, as it involves both a subjective or apperceptive and an objective or perceptive moment. Specifically, he suggests that both cannot occur in a truly momentary jñāna simultaneously: ahaṃ vedmi sa māṃ vetti na kartṛkaraṇāditā | jñānasyaikakṣaṇe yuktā taddvitvena kṣaṇakṣayaḥ || 6.37 || “I see”; “he sees me” – to be the agent, the instrument, etc., in [only] a single instant is [a state (sthitiḥ, by anuvṛtti from ŚD 6.36d) that is] not logically possible (yuktā) for the [momentary] cognition. [And:] There is a destruction of the moment [i.e., the fact that the cognition exists in a single moment] (kṣaṇakṣaya ¼ kṣaṇikatvakṣaya, though Chaturvedi (1986, 229) plausibly suggests kṣaṇakṣaya should be understood to mean kṣaṇabhaṅgavādakṣaya) as a result of it [i.e., the cognition] being two-fold.

Here, the general concern, at least, is evident enough: it is not possible for a single, truly momentary jñāna to function in a single moment as more than one of the factors of action or kārakas. The specifics of Somānanda’s objection, moreover, are spelled out in what immediately follows: vibhāgakālagrahaṇakālayor bhinnakālatā | vinaṣṭatvāt phalaṃ kasya kramāt karmaphale yataḥ || 6.38 || A temporal distinction [necessarily] exists [for the cognition] when there is a moment of the separation [of subjective and objective dimensions of the cognition] and a [subsequent] moment of the apprehension [of the contents of a cognition with such subjective and objective dimensions]. Because of the fact that it [i.e., the cognition] is destroyed [immediately after coming into being, this because, according to you, it is momentary in nature], [we ask:] to whom (kasya) does the result [of the cognition] (phalam) belong, since the action [of cognition] and the result [of the act of cognition] (karmaphale) exist sequentially (kramāt)?

The argument is this, that the single jñāna cannot be real and come erroneously to be seen as consisting of the agentive and objective dimensions of awareness in one and the same moment of time. So much can only occur in a sequence, otherwise the cognition would be real and unreal in a single moment of time. It cannot be innately complex, subjective and objective, etc., in nature, since these are understood to be seen in error, according to the Buddhists, and that which is real cannot be innately unreal. Again, all this requires a sequential nature if it is to exist as described, and that sequential nature cannot be admitted to by the Buddhists if the single cognition is to be momentary in nature; thus, it is not possible to explain the effect of the

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cognition, which must cross those moments of existence to produce subject-object distinctions as only apparent but ultimately unreal. Now, the matter Somānanda brings to question is addressed at Pramāṇavārttika (pratyakṣapariccheda) 3.354: avibhāgo ’pi buddhyātmā viparyāsitadarśanaiḥ | grāhyagrāhakasaṃvittibhedavān iva lakṣyate || 3.354 || Although undivided, the nature of cognition is perceived as if it were possessed of the divisions of object of cognition (grāhya), agent of cognition (grāhaka), and the consciousness [itself] (saṃvitti) by those whose perception is distorted.

Dignāga himself already articulated the idea that the pramāṇa and its phala were in fact really one, singular, for example at verse 10 of the pratyakṣapariccheda of the Pramāṇasamuccaya: yad ābhāsaṃ prameyaṃ tat pramāṇaphalate punaḥ | grāhakākārasaṃvittyos trayaṃ nātaḥ pṛthak kṛtam || 10 || The appearance is the object of cognition, while the natures of the means of cognition and the result are present [respectively] in the form of the apprehending subject and the consciousness [itself]; hence, the three are not in fact [truly] distinct. (The present translation is based on the rendering of Ratié 2011: 43–44, fn. 17.)

Now, that Somānanda has PV (pratyakṣapariccheda) 3.354 here in mind is made explicit in ŚD 6.39, where he cites that verse, as follows: bhedavān iti lakṣyatve dṛṣṭānto ’sti na tādṛśaḥ | grāhyagrāhakasaṃvittibhedavān iva lakṣyate || 6.39 || (grāhyagrāhakasaṃvittibhedavān conj.; grāhyagrāhakasaṃvitter bhedavān KSTS edition) If [you argue that] it [i.e., the nature of the cognition] is [merely] perceived as something that is divided (bhedavān iti lakṣyatve) – [for you say:] “it [i.e., the nature of that cognition] appears as if possessed of the divisions of object of cognition (grāhya), agent of cognition (grāhaka), and the consciousness [itself] (saṃvitti)” – , [we reply:] no example of such a thing exists.

According to the Hindi-language translation of Chaturvedi 1986: 229, the idea here expressed is this, that nowhere in consciousness do the grāhya and the grāhaka not appear as distinct: nowhere do they appear only as if separate, because they always simply present themselves as mutually distinct entities. The problem, moreover, must also involve the difficulty of explaining how in a single moment a cognition that is real, on the one hand, can be false, on the other, the latter for presenting in the form of what is not, in fact, present. But what the present author initially found puzzling, here, was Somānanda’s reply to the objection he attributes

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to his Buddhist interlocutor, namely, the notion that this claim knows of no exemplar analogous to it. What, precisely, did Somānanda here have in mind? The answer is found by simply reading the immediately following verses of the Pramāṇavārttika. Indeed, it is evident Somānanda wrote this section of the ŚD (and various other parts of it) with the pratyakṣapariccheda of the PV in front of him or front of mind, for the verses following seek precisely to furnish examples of the type of misperception identified. PV 3.355–357 say the following: mantrādyupaplutākṣāṇāṃ yathā mṛcchakalādayaḥ | anyathaivāvabhāsante tadrūparahitā api || 3.355 || tathaiva darśanāt teṣām anupaplutacakṣuṣā | dūre yathā vā maruṣu mahān alpo ’pi dṛśyate || 3.356 || yathānudarśanaṃ ceyaṃ meyamānaphalasthitiḥ | kriyate ’vidyamānāpi grāhyagrāhakasaṃvidām || 3.357 || For example, for those whose eyes are afflicted by mantras, etc., clay shards and the like appear absolutely erroneously [i.e., as something other than they are], even though they are devoid of that [erroneous] form [in which they appear], this inasmuch as they are [simultaneously] seen exactly as they are (tathaiva) by the unafflicted eye; or to offer another example (yathā vā), something although it is small appears to be large at a distance in the desert; and [in a similar manner] this condition (-sthitiḥ) as the object known, the means of knowing it, and the result is produced of the object of awareness, the agent of awareness, and the consciousness in accordance with how they are witnessed (yathānudarśanam), even though it [i.e., the condition or sthiti in question] does not exist.

Clearly Somānanda does not accept these examples as sufficient to explain the erroneous perceptual phenomenon in question, what thus brings us back to Chaturvedi’s interpretation of ŚD 6.39, though this author must admit that Somānanda’s precise logic here remains somewhat elusive to him.

Conclusion The treatment of Dharmakīrti’s philosophy continues past this point of ŚD, āhnika 6, though this author’s present analysis must end here, highlighting as it does the place where Somānanda suggests that acceding to his own Śaiva non-dualistic position could rescue a logical coherence for his opponent’s doctrine. Many other examples of what is offered in the present essay could be adduced, as well, from reading across the ŚD the various places where Somānanda engages philosophical interlocutors of various doctrinal identities, probing and debating their particular conceptions of the nature of reality by way of arguing the logical coherence of the same. Though it prosecutes a somewhat modest ambition, what the present survey of such arguments makes clear, it is hoped, is the fact that, for Somānanda as for virtually all philosophers, logical coherence is not only a sine qua non for any dispositive statement regarding the nature of reality, but – more than this – logical

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coherence itself can and does correspond with the nature of reality, such that reasoned argument can display, explain – prove – the very same. While the present author, to date, has not yet identified any place within the ŚD that offers a meta-explanation for this fact – an argument for the reasons one may be assured that argumentation, properly executed, is dispositive of reality – it is a simple fact that Somānanda evidently presumes as much to be the case. Logic can prove and does prove, on his view, that only the acceptance of the existence of a real, dynamic, and utterly singular Śiva-as-consciousness, a total nondualism, can possibly be real. Only such an ontological reality allows one logically to account for the existence and awareness of the multiple universe known in quotidian experience, just as the same (ontological and epistemological) nondualism alone can account for any capacity to speak and convey knowledge about the same. That reality is thus, for Somānanda, is evident in experience if properly observed and realized; but so too is it known and knowable by way of reasoned argument: the intellectual arguments of Somānanda’s philosophical proofs reveal reality accurately, because reality cannot be thus unless it accords with logic proofs for the same. Philosophy and reality, discourse and what is described thereby, mutually correspond, and what Somānanda’s logical proofs establish is no mere theory, but they are dispositive of the very fact that all one has ever seen is and has been the one dynamic, unitary Śiva-as-consciousness.

Bibliography Chaturvedi, Radheshyam, Trans. 1986. The Śivadṛṣṭi of Śri Somānanda Nātha. Varanasi: Varanaseya Sanskrit Saṅsthan. (Hindi with Sanskrit commentary). Dyczkowski, Mark. 1987. The doctrine of vibration: An analysis of the doctrines and practices of Kashmiri Shaivism. Albany/New York: SUNY Press. Gnoli, Raniero. 1957. Śivadṛṣṭi by Somānanda: Translation and commentary, chapter I. East and West 8: 16–22. Lawrence, David. 2018. “Pratyabhijñā Inference as a Transcendental Argument about a Nondual, Plenary God.” In Handbook of logic in Indian thought, edited by S. Sarukkai. https://doi.org/10. 1007/978-81-322-1812-8_29-1. Nemec, John. 2011. The ubiquitous Śiva: Somānanda’s Śivadṛṣṭi and his tantric interlocutors. New York: Oxford University Press. Nemec, John. 2012. The two Pratyabhijñā theories of error. Journal of Indian Philosophy 40 (2): 225–257. Nemec, John. 2019. Somānanda on the meaningfulness of language. Indo-Iranian Journal 62 (3): 227–268. Nemec, John. Forthcoming-a. Somānanda. In The encyclopedia of philosophy of religion, ed. Stewart Goetz and Charles Taliaferro. Oxford: Blackwell Press. Nemec, John. Forthcoming-b. The ubiquitous Śiva Volume II: Somānanda’s Śivadṛṣṭi and his philosophical interlocutors. New York: Oxford University Press. Ratié, Isabelle. 2011. Le Soi et l’Autre: Identité, différence et altérité dans la philosophie de la Pratyabhijñā, Jerusalem Studies in Religion and Culture 13. Leiden and Boston: Brill. Ratié, Isabelle. 2013. “On Scripture and Reason in the Pratyabhijñā.” In Scriptural authority, reason and action: Proceedings of a panel at the 14th world sanskrit conference, Kyoto, September 1– 5, 2009, edited by Vincent Eltschinger and Helmut Krasser. 375–454. Beiträge zur Kultur- und Geistesgeschichte Asiens 79. Vienna: Verlag der Österreichischen Akademie der Wissenschaften.

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Sanderson, Alexis. 1985. Purity and power among the Brahmins of Kashmir. In The category of the person: Anthropology, philosophy, history, ed. M. Carrithers, S. Collins, and S. Lukes, 190–216. Cambridge, UK: Cambridge University Press. Shastri, Kaul Madhusudan, ed. 1934. The Śivadṛṣṭi of Somānandanātha with the Vṛtti by Utpaladeva, KSTS. Vol. 54. Pune: Aryabhushan Press. Torella, Raffaele. 1994. The Īśvarapratyabhijñākārikā of Utpaladeva with the author’s Vṛtti: Critical edition and annotated translation. Rome: IsMEO. Torella, Raffaele. 2014. Notes of the Śivadṛṣṭi by Somānanda and its commentary. Journal of Indian Philosophy 42 (5): 551–601.

Part II Fundamentals

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Logic in India Amita Chatterjee

Contents Western Versus Indian Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Principle of Noncontradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Modern Indian Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter aims at providing a broad overview of Indian logic pursued in the classical Indian philosophical systems as well as in contemporary India. It will be seen that traditional systems of logic do share some features, yet diverge from each other due to the differences in their ontology and practice. During the colonial period, logic was pursued mainly within a comparative framework. However, the perception of Indian logicians changed entirely as they came into contact with modern symbolic logic. The chapter is divided into five sections. The first section deals mainly with the distinction between Western and Indian conception of logic. The second section provides the structure of inference in classical Indian philosophical systems. The third section analyzes the concepts of negation available in the Nyāya, Buddhist, and Jaina logic and links different concepts of negation to the metaphysical presuppositions of the systems. The fourth section lays bare the nature of contradiction as conceived in these systems, and the final section gives a brief account of the developments in Indian logic during the nineteenth and the twentieth centuries.

A. Chatterjee (*) School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_39

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Western Versus Indian Logic Logical culture in the West was thought to be monolithic and inextricably intertwined with the development of the concept of reason. Kant wrote in the preface to the second edition of his first critique that the sphere of logic which “provides strict proof of the formal rules of all thought” is said to be “precisely delimited” and “complete” because logic is obliged to abstract from “all objects of knowledge and their differences, leaving the understanding nothing to deal with save itself and its form.” Though Kant never claimed to have gone beyond Aristotelian logic, he in fact anticipated many features of the modern formal logic developed by de Morgan, Boole, Venn, Peano, Frege, Russell, Whitehead, and others in the nineteenth and the twentieth centuries. In course of time this topic-neutral formal discipline, which belongs to the a priori realm of theoretical knowledge, assumed a near hegemony in the assessment of natural argumentation. Any rational being raised in this universalistic rational culture was supposed to argue in accordance with the rules of this classical logic. Logic was considered to be unitary and supposedly the common core of logic gets manifested in all cultures across board. Hence, if anyone failed to reason in accordance with the rules of logic he was considered to be irrational and a lesser human being. Though serious scholars of Indian philosophy and culture starting from William Jones to Jonardon Ganeri and other contemporary researchers thought otherwise, there was a strong opinion amongst some stalwarts of philosophy, e.g., Hegel, Husserl, Heidegger, and Rorty, that philosophy including logic being a European phenomenon, whatever ancient Indians and Chinese were doing could not be philosophy or logic, in spite of the vigor and rigor of their endeavor. The point that we want to put forward against this view is that neither oriental mind nor oriental culture is wanting in logic; rather these are guided by some alternative logics. As opposed to the general idea that logic is culture-invariant, we maintain that it is culture dependent. In fact, the second half of the twentieth century witnessed a breach in the universal logical tradition and culminated in the proliferation of alternative systems of logic. Some of these systems are just conservative extensions of the classical logic and the rest are marked by revisions, moderate, or radical. Logical culture has been prevalent in the classical Indian theoretical tradition for more than 2000 years. In fact, logic was a part of public and private life, rural and urban life, scientific and religious forms of life. “The wide surface-level gulf separating the logical traditions of ancient Greece and West on the one hand and that of India on the other has served as a deterrent in the search for the unity of logic as a trans-cultural theoretical discipline” (Pahi 2008). To make the nature of Indian Logic clear, we shall first try to show the main points of difference between the Western logic and Indian logic (Mohanty and Chatterjee 2009). The roots of Indian logic can be traced to two distinct traditions – the vāda (dialogical) and the pramāṇa (epistemological) traditions. Logic in India was never developed as an independent discipline but was embedded in epistemology or Pramāṇa theories. One of the important sources of Indian logic being the debating or the dialogical tradition, pattern/structure of explicit inference reflects this legacy. Though historical roots and trajectories of Indian and Western logical traditions were

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far apart, superficial similarities observed among two logical traditions led to some concerted efforts of translating Indian logic to syllogistics or the first-order predicate logic. But that effort was considered wrong-headed for more than one reason to which we shall come later. Let us concentrate first on the distinctive features of traditional Indian systems of logic. No Indian philosophical system, unlike Western logic, takes a purely formal approach to inference or inferential knowledge. Yet, we consider these theories of inference as logic insofar as these are theories of human reasoning and tell us how to distinguish good arguments from bad arguments, acceptable arguments from unacceptable ones. According to Wilfrid Hodges (2001), “in its first meaning, a logic is a collection of closely related artificial languages. . .In its second but older meaning, logic is the study of sound argument.” Indian theories of inference are compatible with the second meaning of logic. While the mainstream western logic primarily developed as a deduction-centric discipline and revolved around “the consequence relation,” logic in India has been mainly pervasion (vyāpti)-centric. Pervasion or vyāpti is the relation of universal concomitance between the ground of inference and the thing to be inferred. The notion will be made clear when we shall describe the structure of an inference. In the Indian logic, syntax always remains hyphenated with semantics. For inference as an accredited source of knowing the world, validity is not enough; soundness and epistemic progress also need to be guaranteed. In Indian theories of inference, the notion of validity/invalidity of an inference usually presupposes a host of background information and often essentially hinges on them. Two necessary corollaries of this stance are: (a) no constituent of an inference can have zero information content like a tautology and (b) validity of an inference cannot be delinked from soundness. Hence, an inference is valid only if it yields a true conclusion. One may point out that even in the West the main concern of older logicians was to discover the rules of sound argument. Therefore, the logical enterprises in the premodern world were not so different, be it in India or in the West. Here, the contrast is being drawn between the classical Indian logic and the modern logic, based on the developments in the nineteenth century mathematics, which “incorporates a theory of relations and thus goes beyond traditional syllogistics; it aims at developing a type of science that occupies the most fundamental place in the system of scientific disciplines. . . more traditional philosophical projects were largely eclipsed by changes within the field of philosophy, and by the increasing emphasis on the role of mathematics for modern logic” (Ziche 2011). In the Indian theories of inference, we find elaborate discussions on how inference results from a number of cognitive states and what conditions give rise to cognitive certainty. So, inference as a cognitive process admits of causal analysis. Here, the relation between premise and conclusion is viewed not as an abstract logical relation but as a psycho-cognitive relation of causal sequence. The question that Indian logicians addressed was how ordinary human beings reason in real life situations and avoid arriving at erroneous conclusions as opposed to the question raised in Western logic – how ideal rational beings reason under ideal conditions. This stance may make Indian logic vulnerable to the charge of psychologism, which

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both Frege and Husserl wanted to avoid. We can say the following in defense of Indian logic. Granting that in India logic was never separated from epistemology or psychology, we would like to argue that here psychologism never led to subjectivism because Indian logicians were dealing with psycho-causal conditions that apply to all cognitive agents. Besides, unravelling the psycho-causal conditions underlying an inferential process helps us build a viable model of mental reasoning. All Indian logicians adopted a grammar-based model of logical analysis, while in Western logic the geometrico-mathematical model is in use. But interestingly, this grammar-based model has led Indian theorists to some of the insights of mathematical logicians regarding logical connectives. It is true that the Naiyāyika-s, the most famous logical school of classical India, did not always stick to the Grammarian’s insights when they developed their formal language. While defining a language of properties and relations, they scratched beneath the grammatical surface expressed through ordinary language and arrived at logical correlates of their metaphysical categories. Western formal logic is extensional; Indian logic, it has been said, is basically logic of properties and hence intensional. Two properties having the same extension are found to possess different senses and would fail the substitutivity condition, e.g., potness (ghaṭatva) and the property of having a conch-like neck (kambugrī vādimattva) are extensionally the same but differ in meaning. However, we must remember what Indian logicians mean by “property” is somewhat different from its meaning in English. The term “property” here signifies any locatee, be it an abstract property or a concrete object, which resides in a locus. The basic combination in Indian logic is not a straightforward subject–predicate proposition but a Sanskrit sentence of the locus–locatee model, e.g., “a has f-ness.” As the sentences of the form “a has f-ness” can be easily correlated with the sentences of the form “a is f,” it is possible to read such logic of properties extensionally, thinks Matilal (1998) and his followers. But there are others like Guhe (2008) and Oteke (2009) who have showed that Nyāya logic should not be interpreted as a fragment of first-order logic because of the intensional nature of the property, “being a locus of”; one needs to go beyond the framework of first-order logic. The content of episodic cognition is the most suitable candidate to be the truthbearer. Indian logicians generally view the content of cognition as a relational complex referring to a complex object and not to a fact. Hence, it is always debatable whether the content of cognition should be represented by a proposition or just a complex term. Though ontological and epistemological commitments underlying different Indian logical systems are diverse and there are controversies centering the number of constituents (avayava) in an inference, yet every system accepts at least three inferential components. These are sādhya (the provable property, the probandum, the signified), hetu (the ground of inference, the reason, the sign, the probans), and pakṣa (the locus of inference). Finally, Indian theories of inference cannot be neatly categorized as deductive or inductive in the standard sense. Ancient Indian logicians were in fact trying to formulate conditions of human reasoning in general. Though they were trying to

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determine under what conditions an inferential leap from the known to the unknown would be warranted (Sarkar 1997), the nature and structure of inference cannot be identified with deduction or induction or abduction per se. In spite of sharing the above-mentioned framework, different Indian logical systems diverged from each other because of their difference in basic belief systems and metaphysics. Thus, within the logical culture of India, there emerged several subcultures of logic. For example, logic developed by the Nyāya School belonging to the orthodox tradition (Jha 1984) possesses features different from those logical systems belonging to the heterodox tradition, viz., Buddhist schools (Stcherbasky 1962). Even within the heterodox tradition, the presuppositions, nature, and practice of the Buddhist and the Jaina logic are entirely different. Though all systems of Indian philosophy except the materialist Cārvāka school developed their own theory of inference in accordance with their metaphysical presuppositions, we shall confine our discussion to the radical skepticism of the Madhyamika Buddhism, doctrinal realism of the Naiyāyika-s, and the pluralism of Jainism. These subcultures have many points of difference. However, we shall try to capture their differences mainly in respect of negation and contradiction.

The Structure of Inference Since fourth century AD, following the Buddhist logicians, all systems of Indian logic divide inference broadly into two types. Svārthānumāna (SA) or inference-foroneself deals with the psychological conditions, i.e., causally connected cognitive states leading to one’s own inferential cognition, while Parārthānumāna (PA) or inference-for-others essentially deals with the proper linguistic expression of this inference with a view to communicating it to others (Viśvanātha 1988). SA which is a process of mental reasoning par excellence consists of four steps in the Nyāya system, each of which is a state of cognition causally connected with the immediately preceding state. The process can be best explained with their typical example. A person first sees that (a) the hill (pakṣa/the locus of inference) possesses the smoke (hetu/the ground of inference/probans). This is a perceptual cognition which reminds him that (b) wherever there is smoke there is fire (sādhya/the provable/probandum), as he has always observed in a kitchen. The first step is technically called pakṣadharmatājn˜ āna, meaning the probans is known to be present in the locus of reasoning. The second step (known as vyāptijn˜ āna) is memory or a recollective cognitive state of the universal concomitance between smoke and fire. Then (a) and (b) are combined to produce a complex form of cognition called “parāmars´a” or “consideration” of the form (c) the hill possesses smoke pervaded by fire and then follows the conclusion (d) therefore, the hill possesses the fire. According to the Naiyāyika-s, PA has five constituents (avayava-s) arranged in the order pratijn˜ ā or assertion, hetu or reason, udāharaṇa or example, upanaya or application, and nigamana or conclusion. The typical example of a full-fledged inference for others is the following.

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Though the conclusion of a PA appears to be the same as the thesis proposed in the first step, these two perform two different tasks. The first step just presents the thesis while the conclusion declares that what is to be proved has been proved. The second step asserts that the hill is the locus of smoke and thus provides the ground of the final assertion in the conclusion. The third step, example or udāharaṇa, has two parts – a law-like generalization and an exemplar that instantiates the generalization. The fourth step asserts that the hill which is different from the kitchen is a locus of smoke in a nondeviant sense. All these steps warrant the conclusion or the fifth step. So, there is no redundancy involved in the five-membered inference for others. Sarkar (1997, 358) shows that necessary reliance on an exemplar brings out another unique feature of inference a la Nyāya. To quote from him, “it means that (i) all inferential justifications must have some observational basis and (ii) because of this self-imposed inflexible fact-compliance requirement, a Naiyāyika is prevented from entertaining a law-like generalization purely hypothetically. This . . . acts as an impediment to developing an abstract, formal approach to theory of inference. . .” According to the tradition, the first step is said to be generated by verbal cognition, the second is established by inference, in the third step, example is acquired through perception, and the fourth step is based on cognition of similarity. Since these four steps are established by four sources of true cognition admitted in the Nyāya School, the Naiyāyika considers this five-membered argument as the demonstration par excellence (parama-nyāya). Pervasion (vyāpti) is the relation of invariable concomitance of the ground of an inference (hetu) and the signified or the thing to be inferred (sādhya). Without the knowledge of this relation, it is not possible to infer. In a valid inference, “The hill has fire because it has smoke,” the sādhya is fire, the hetu is smoke, and pakṣa or the locus is the hill. If there is no universal concomitance between smoke and fire, by perceiving smoke, we shall not be able to infer fire in a new locus. In order that the relation of universal concomitance obtains between the sign and the signified, the sign needs to fulfill certain conditions, which also makes the sign legitimate (sadhetu). A sign is legitimate if and only if it possesses the following five features, viz.: 1. 2. 3. 4.

It is present in the locus of reasoning (pakṣa-sattva) It is present in a similar location (sapakṣa-sattva) It is not present in any dissimilar location (vipakṣa-asattva) It is not associated with the contradictory of the signified in the locus (abādhitatva) 5. If another sign tending to prove the contradictory of the signified is not present in the locus (asatpratipakṣitatva)

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These five features provide the truth conditions of the cognitive states involved in SA; (1) is the truth condition of pakṣadharmatājn˜ āna, (2) and (3) are the truth conditions of vyāptijn˜ āna disjunctively, and thus become the truth conditions of parāmars´ajn˜ āna too; while (4) and (5) have a direct relevance to the truth of the conclusion. The violation of any of these conditions results in unsound/fallacious arguments (Annaṃbhatta 1976) known as asiddha (unestablished), viruddha (hostile), savybhicāra (deviating), bādhita (contradictory), and satpratipakṣa (counterbalanced), respectively. All these defects of probans can be shown to be present in one nonveridical inference, e.g., “the lake has fire because it has potness.” In this example, the lake is the inference-locus, fire is the probandum, and potness is the probans. It violates the first condition, because the probans potness is not present in the locus of reasoning, the lake. It goes against the second condition, because potness is present only in pots but absent in various loci of fire, hence the probans is opposed or hostile. A more familiar example of this type of fault is: sound is eternal as it is an effect.) The inference under discussion is also vitiated by the defect due to a deviating probans. Here, the probans potness, which is present only in pots, can easily reside in a locus which is characterized by the absence of fire. That shows that potness is not invariably concomitant with fire, the probandum. In this example, the probans potness becomes contradictory and hence illegitimate, if the lake does not possess fire. Again, it is easy to show the possibility of the existence of an alternative probans, say, water, capable of proving the absence of fire in the lake, thus counterbalancing the force of the original probans and preventing the conclusion. All these defective probans are faulty because they somehow block the conclusion of the inference. Thus, it is obvious that the psychological conditions of inference are related to its conditions of validity in such a way that the fulfillment of the former guarantees the fulfillment of the latter. Thus the theory of reasoning, which began as a description of psychology of proof as well as a way of knowing, was transformed into a logical theory, not as a formal rule-driven axiomatic theory but as a theory of ordinary reasoning. Both Buddhist (Dharmakīrti 1955) and Jaina logicians admit the abovementioned distinction between SA and PA, they only differ with respect to the number of constituents of inference. Consequentially, their views on the nature and number of fallacies also vary.

Concept of Negation We have already mentioned that in ancient India inference was used as a part of the debating tradition. There are debates in the later Vedic texts, in the Buddhist and the Jain scriptures. I shall, however, remain confined to the Kathābatthu (Aung and Davids 1915) which supposedly contain a report of the Buddhist Council held around 255/155 BC. The text records topics of debate and various types of argument that a Buddhist monk can offer. The ancient text mentions two ways of refuting any motion, the underlying structure of which are given below. We have used the

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schematic representation of the examples available in the ancient text following Matilal (1998). 1. The Way Forward (anuloma) Opponent (O) Proponent (P) O P

: Is (A is B)? : Yes. : Is (A is C)? : No.

Rejoinder: O

: If A is B, then A is C; – Therefore – Not both: (A is B) and not (A is C); – Therefore – If not (A is C), then not (A is B). Thus the proponent’s thesis, A is B, is refuted and the debate is won by the opponent. The Way Backward (Pratiloma):

O P O P

: Is it not the case that A is not B? : No it is not. : Is it not the case that A is not C? : No it cannot be asserted that A is not C. Rejoinder:

O

: If A is not B, then A is not C; – therefore – Not both: (A is not B) and not (A is not C); – therefore – If not (A is not C), then not (A is not B).

Thus, the proponent’s denial of A is B is shown untenable. Interestingly, in the above representations, we find that even in the second century BC the Buddhist monks were acutely aware of untenability of inconsistency in an argument. They have used the well-known definition of material implication, “if p then q” ¼ def. “not both p and not q,” and the law of contraposition and modus ponens implicitly. If we look carefully, it also becomes obvious that there is an interchange between the object level and meta-level negation, i.e., between a negation attached to a nested sentence or a term and a negation attached to the principal assertion. This is striking because this kind of interchange is feasible only in a 2-valued system. (Maybe, this is due to utter conflation between object-level and meta-level negation on the part of the Buddhist monks, though there is ample

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evidence that the Grammarians were aware of this distinction.) But the Buddha himself used the well-known tetralemma, made famous by Nāgārjuna, the famous nihilist stalwart of the Mādhyamika School. When asked about the nature of postmortem status of a liberated soul, the Buddha usually kept mum but occasionally replied that none of the following alternatives can be upheld: 1. 2. 3. 4.

The liberated soul continues to exist after death The liberated soul does not exist after death The liberated soul both exists and does not exist after death The liberated soul neither exists nor does not exist after death

Nagarjuna used this tetralemma (catuṣkoṭi) to buttress his radically skeptical metaphysics. A radical skeptic can uphold his position provided he sticks to his “refutation only” stance without falling into any inconsistency. Nāgārjuna interprets negation to ensure that. Nāgārjuna says that joint refutation of all the four positions does not lead him to logical contradiction nor does commit him to any thesis of his own. Following Matilal (1998), we can interpret Nāgārjuna’s tetralemma as an example of illocutionary negation. It is obvious that (a) ~ ├ (Ex) (x is F) and (b) ~ ├ ~ (Ex) (x is F) are not contradictory. One who asserts both (a) and (b) actually tries to avoid any false knowledge claim. Let us try to be a little more explicit. A. B. C. D.

Nāgārjuna negates the thesis p. Nāgārjuna negates the thesis ~ p. Nāgārjuna negates the thesis (p & ~ p). Nāgārjuna negates the thesis (p v ~ p).

Nāgārjuna’s Neg. is a meta-level negation like the illocutionary negation. Negation operator, employed here, may be interpreted as nontruth-functional. For, under truth-functional interpretation Neg. (p & ~ p) and Neg. (p v ~ p) can never have the same truth value. Besides, Nāgārjuna maintains that all four theses are to be rejected as meaningless, but their negations are not true. Hence, by proffering this tetralemma, he is not committing himself to any position of his own. Rather, he strictly declared that the negation of a philosophical thesis is no philosophical thesis. The upshot of the whole discussion is that the logic of the four-cornered negation was necessitated by the world view of the Mādhyamika School who made a sharp distinction between conventional truth and transcendent truth, denied the finality of any conventional truth and upheld that objects of our experience are totally devoid of essence or nature (niḥsvabhāva). Just as the four-cornered negation of Nāgārjuna, interpreted as illocutionary negation is intimately connected with the Mādhyamika ontology, to understand the Nyāya (especially the Neo-Nyāya) concept of negation (Matilal 1968), we need to understand their ontology of absence. Absence, they point out, is not merely a logical or linguistic operator, it is as objectively real as a positive entity is. Four types of absence are admitted in the system: (1) mutual absence or difference (anyonyābhāva), e.g., a jar is not a pen and vice versa; (2) absence of not-yet type

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(prāgabhāva), e.g., absence of a bread in flour before it is baked; (3) absence of no-more type (dhvaṃsābhāva), e.g., absence of a vase in its broken pieces, and (4) absolute absence (atyantābhāva), e.g., absence of color in air. So, an absence is always of something and that something is called the counterpositive or the negatum (pratiyogī ) of that absence. Consider the absence of smoke in a lake. Smoke is the negatum (pratiyogī ) of the absence of smoke and pratiyogitā or the relation of negatum-hood is the relation between an absence and its negatum. Here, the lake is the locus (anuyogī ) of the absence. Hence, anuyogitā or the relation of locus-hood that connects the absence in question with its locus. So in Nyāya system, the negation is mainly of two forms: (a) x is not y and (b) x is not in y or absence of x in y. The Naiyāyika-s subscribe to two-valued logic. Hence, for them, if S is P or if S is in P is true, then S is not P/ non-P or S is not in P is bound to be false. The Nyāya view of negation is quite in keeping with their realist and pluralist ontology. Their world contains innumerable entities that are in principle knowable and nameable. Each such entity has an intrinsic nature (svabhāva). Many of these entities are eternal, and even those that are noneternal, are stable, i.e., nonmomentary (akṣaṇika). Many of these entities are mutually related, and these relations are as real as their relata. The Jaina-s, on the other hand, believes that reality is many sided. Each real object is complex and possesses infinite facets. It is impossible for any finite being to comprehend the reality in its fullness. The flip-side of this theory is that all worldviews are partial and incomplete; no worldview can be entirely wrong. Each has some element of truth in it. If every assertion contains some element of truth in it, then how is one supposed to decide on the truth of any given sentence? “The difference between Buddhism and Jainism in this respect lies in the fact that the former avoids by rejecting the extremes altogether, while the latter does it by accepting all with qualifications and also by reconciling them” (Matilal 1998). Since the truth of a judgment is never one-sided, no judgment claims absolute truth or falsity. As every predication is relative to a particular point of view, the Jaina-s attach a prefix “syād” (Maybe) to every assertion thus giving rise to a sevenvalued logic (saptabhaṅginaya) in the following manner. • • • •

Maybe, it (some object) exists Maybe, it does not exist Maybe it exists, maybe it does not exist (a case of successive assertion and denial) Maybe it is nonassertible/inexpressible (a case of simultaneous assertion and denial) • Maybe it does exist, maybe it is nonassertible/inexpressible • Maybe it does not exist, maybe it is nonassertible/inexpressible • Maybe it exists, maybe, it does not exist, maybe, it is nonassertible/inexpressible. Different logicians have given different interpretations to this sevenfold predication. Graham Priest (2008), for example, has developed a many valued propositional logic corresponding to the Jaina theory of predication and a system of paraconsistent logic which can be supported by the Mādhyamika conception of fourcornered negation.

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The Principle of Noncontradiction Let us now see how these three systems deal with the Principle of Noncontradiction (Staal 1962). Aristotle stated this Principle as follows. It is impossible for the same thing to belong and not to belong at the same time to the same thing and in the same respect.

The Nyāya concept of inconsistency naturally comes very close to the Aristotelian notion as the Naiyayika-s like Aristotle admit a two-valued logic. According to the Nyāya School, an assertion is inconsistent if two contradictory predicates p and not-p are asserted of the same object X at the same time. To avoid contradiction, they introduce a technique of delimiting the predicates (Sen and Chatterjee 2010), if in a particular situation, inconsistent predicates are ascribed to the same object simultaneously. When, for example, a monkey sits on a tree, the tree may very well have contact with that monkey in respect of one of its branches; while the same tree in respect of its roots may simultaneously harbor the absence of that contact. In such cases, the locushood resident in the tree is said to be delimited (avacchinna) by different delimiters (avacchedaka) – the tree, as delimited by its branch is the locus of contact with monkey, and this is in no way opposed to the fact that the same tree, as delimited by its roots, is the locus of the absence of monkey contact. There would be a contradiction if the tree would have been a location of a contact and the absence of that contact in respect of the same delimiter. We have seen that Nāgārjuna would resist assigning truth to both (P& ~ P) and its denial ~ (P& ~ P); hence, he is not also going against the principle of noncontradiction. The Jaina-s would say that two sentences are inconsistent if their epistemic contents are incompatible. For example, Let C1 C2

¼ A pot is on the floor, and ¼ A pot is not on the floor.

C1 and C2, however, need not be incompatible, because the person who holds C2 may be referring to the pot (a different one) which he saw here yesterday, while C1 has as its epistemic content the pot which is now on the floor. Even a slight difference in any component of an epistemic content can change its identity. So, it can very well be the case that C1 and C2 do not share the same epistemic content. Therefore, they are not absolute incompatibles. The Jaina-s uphold that epistemic content of any piece of cognition may remain unspecified in four different respects, viz., dravya (substantiality), kṣetra (location), kāla (temporality), and bhāva (features). Each of which again admits of an infinite number of variations. So, any two assertions like C1 and C2 cannot be shown to be incompatible so long as even one dimension of their respective content remains unspecified. C1 and C2 will be inconsistent if the same pot characterized by all four features is and is not at the same location. Exact specification of all dimensions and degrees of a specific

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content, however, is not possible in normal cases as human knowledge is limited. Hence no epistemic claim is ever wholly untrue. “Partial truth is a sufficient condition of truth assignment for the Jaina-s” (Schang 2010). Not only that, the sevenfold logic avoids self-contradiction as Priest (2008) has pointed out “No yes-no answer occurs in the Jaina question-answer game, consequently: two different questions can result in the same answer or not, but no single question can be answered by ‘yes’ and ‘No’ at once. This is the gist of self-contradiction, and even the third basic predicate of inexpressibility does not state it because non-distinction does not mean an internal co-existence of opposite properties. These cannot co-exist by definition.” This logic is the logic of partial knowledge. The Jaina-s, however, maintain that a kevalin or an omniscient being can have perfect and complete knowledge and has offered a unique existence proof for such omniscient being. To establish this point, they admit that the graded series of incomplete knowledge is strictly linearly orderable and possesses a maximal upper bound which can be accessed by an actual omniscient being (Sarkar 1997). It appears that the Jaina-s would admit absolute truth in the realm of perfect knowledge and this aspect of Jaina logic makes an out and out relativistic account of it questionable. It is interesting to note that in none of the three logical subcultures of Indian logical tradition the Principle of Noncontradiction has been given up, though negation operator has been interpreted differently in different philosophical schools in accordance with their ontological presuppositions – thus showing that logic is not ontology-neutral. They have just devised different ways of upholding the principle in view of their respective metaphysical commitment and soteriological end. However, despite large-scale differences in the metaphysical and background beliefs leading to divergent logical practices, all schools of Indian logic admit that to arrive at a sound conclusion we have to be sure about the absence of any counter-example to the generalization on the basis of which we are supposed to take our inferential leap. This can be acknowledged as a mark of minimal rationality which Indian logicians shared with all ancient systems of logic.

Modern Indian Logic Mahamahopadhyaya Satis Chandra Vidyabhusana (1921) in his History of Indian Logic mentioned three principal phases of development of Indian Logic – ancient, medieval, and modern. According to him, the representative text of the ancient phase is Gautama’s Nyāya-sūtra (First century AD), of the medieval phase is Dinnāga’s Pramāṇa-samuccaya (Fourth century AD), and the most salient representative work of the modern phase is Tattva-cintāmaṇi (TCM) of Gangeśācārya (thirteenth century AD), the founder of the Neo- Nyaya school of logic. A large number of commentaries and subcommentaries were written on all these texts giving shape and maturity to Indian systems of logic. We have already discussed the first two phases. Let us, therefore, take a brief look at the third phase. Neo/Navya-Nyāya originated in Mithila led by Gangeśa (Gangeśa Upādhyāya 1892), who integrated and popularized the special technique of subtle argumentation

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and developed a higher-order technical language with a view to making perspicuous, the structure of our cognitive report of objective realities. The tradition (Potter and Bhattacharyya 1993; Bhattacharyya 2009) was carried forward by Vardhamāna, Yajñapati, Upādhyāya, and Pakṣadhara Miśra, among others. From Mithila, Navya-Nyāya traveled to Navadvīpa, in Bengal. The famous early exponents of Navya-Nyāya in Navadvīpa are Pragalbha Miśra, Narahari Viśārada and Vāsudeva Sārvabhauma, Vāsudeva Sārvabhauma’s disciple, the unorthodox logician Raghunatha Śiromaπi (sixteenth century), wrote a commentary on TCM entitled Dīdhiti, in which he went far beyond Gangeśa by introducing changes in NavyaNyāya metaphysics and epistemology. Subsequent prominent proponents of NavyaNyāya in Bengal – including Bhavānanda Siddhāntavāgīśa, Mathurānātha Tarkavāgīśa, Jagadīśa Tarkālaṃkāra, and Gadādhara Bhattācāryya – wrote sub-commentaries on Dī dhiti, which contributed to the fullest development of Gangeśa’s technique of reasoning. The fame of Navadvīpa Naiyāyikas spread all over India, and scholars from other schools too adopted the Navya-Nyāya language. This highly technical language became the medium for all serious philosophical discussion by the sixteenth century, irrespective of the ontological, epistemological, and moral commitments of the discussants. Besides, Navya-Nyāya language was used in some important treatises on the Hindu law of inheritance, on Rhetoric and Grammar, and even on Aesthetics. That is, Mastery of Navya-Nyāya language became a hallmark of intellectual accomplishment all over India. Unfortunately, the craving for precision by the Navya-Naiyāyika-s reached such a height that they became gradually alienated from the masses, and their techniques reached obsolescence outside a very elite Sanskritist circle by the end of the eighteenth century. In the second decade of the nineteenth century, Indian logic is said to be discovered by the famous Orientalist and mathematician H. T. Colebrooke. He read Gotama’s Nyāya-sūtra with commentaries and glosses on it under the guidance of the traditional pundits and reported about his discovery at a public meeting of the Royal Asiatic Society in 1824. Prior to Colebrooke’s declaration, western scholars were blissfully ignorant about the scientific and logical pursuits of the ancient Indians, though they knew and acknowledged the contribution of Indian scholars in mathematics and astronomy. Most western scholars had thought that Indian mind was not logical at all. So, Colebrooke’s discovery is considered as a landmark in the history of ideas. Colebrooke (1824) showed that reasoning or inference proper forms part of the Nyāya theory of Evidence or Proof (pramāṇa). He meticulously observed the standard form of the argument and christened this argument “syllogism” following Aristotle because he found a lot of similarity between syllogism and the Nyāya inference. After Colebrooke, Indologists were engaged in the debate whether the structure of the Nyāya inference can at all be translated in terms of syllogistic. Though the name “Nyāya syllogism” stuck to the five-limbed inference, soon scholars found more dissimilarity than similarity between inference patterns of the West and the East. That is why, Max Mūller (1853) wrote, “Nevertheless, it would be wrong to call the Nyāya, Logic, in our sense of the word. The Nyāya, as well as the other systems, has for its highest object the solution of the problem of existence, and only as a means towards accomplishing this object, does it devote particular attention

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to the instruments of knowledge – and, as one of them, to syllogistic reasoning.” But after a few paragraphs, he observed, “Even terms as conclusion or syllogism are inconvenient here, because they have with us an historical coloring, and throw a false light on the subject.” However, in spite of Max Mūller’s warnings, most scholars, including Indian stalwarts like S. C. Vidyabhusana, B. N. Seal, S. C. Chatterjee, D. M. Datta, S. Radhakrishnan, and M. Hiriyanna – all endorsed this comparison with syllogism. With the development of mathematical logic by Russell, Lukasiewicz, and others, some new interpretations of the Nyāya argument came into vogue. Stanislaw Schayer (Ganeri 2001) was not ready to force Indian inference onto the Procrustean bed of the authentic Aristotelian syllogism under any circumstances. He pointed out for the first time that the Nyāya argument should rather be interpreted within the frame of the First-Order Predicate Logic. He symbolized the five-step argument as follows. 1. Thesis 2. Reason 3. Statement of pervasion 4. Application 5. Conclusion

Fa Ga (x) (Gx !Fx) Ga !Fa Fa

There is fire on a (¼ on this mountain). There is smoke on a. For every locus x: if there is smoke in x then there is fire in x. This rule also applies for x ¼ a. Because the rule applies to x ¼ a and the statement Ga is true, the statement Fa is true.

Like Colebrooke, Schayer too offered a path-breaking interpretation which influenced the views of later scholars: Both D. Ingalls (1951, 1955) and I. M. Bochenski (1961). Staal, Sibajiban Bhattacharyya, and B. K. Matilal too developed their interpretations along this line. J. F. Staal (1973) was astute enough to understand that terms in an Aristotelian syllogism are related by one single relation, i.e., the relation of belonging to, but in the so-called Indian syllogism three terms (pakṣa, hetu and sādhya) are related by two relations. The relation between reason (hetu) and the thing to be inferred (sādhya) is that of pervasion while the relation of both these terms with the locus of the argument is that of occurrence. As Staal put it, “Since such a relation relates x to y, it is a two-place relation, which may therefore be written as A (x, y)” which can be read either as x occurs in y or y is the locus of x. So a sentence, “if (the hetu) smoke occurs on a mountain (pakṣa), then fire (the sādhya) occurs on that mountain (pakṣa)” is to be symbolized by A (h, p) ! A (s, p). Though Sibajiban Bhattacharyya (1987) and B. K. Matilal agreed more or less with Staal, they offered refined interpretation of their own. Following the texts of the New School of Nyāya, Sibajiban Bhattacharyya has laid bare the structure of a Nyāya argument vis-à-vis Aristotelian syllogism. Let us therefore look at Bhattacharyya’s analysis. To compare the standard Nyāya argument with Aristotle’s syllogism, Bhattacharyya takes the last three steps and gives exact English rendering of the Sanskrit terms. Let us look at the examples he has given.

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N.N.1. (The) hill (is) fire-possessing/because of smoke. N.N.2. This (is) fire/because of heat. In both the arguments, identifying the hetu and the pakṣa is simple. The term which is suffixed by the fifth case ending in Sanskrit or prefixed by “because of” in English is the ground/reason/hetu of the argument. The locus of the hetu is the pakṣa. But identifying the sādhya is tricky. In N. N.1., fire (not the word “fire”) is the sādhya, though in N.N.2 it is the property of being fire or fireness (neither fire, nor the word “fireness”). The Navya-Naiyāyika-s have given us a thumb rule for identifying the sādhya of any argument. The rule is in two parts, to be applied depending on the case at hand. (i) Drop the suffix “– possessing” when it occurs in the second term of the conclusion (Check N.N.1 now); (ii) add the suffix “ness” to it when the suffix “– possessing does not occur” (Check this part against N.N.2). By applying this rule, N.N.1 and N.N.2 may be fleshed out as follows. N.N.1.: Whatever is smoke-possessing is also fire-possessing The hill is smoke-possessing Therefore, the hill is fire-possessing. Of course, the argument can be more elegantly paraphrased in English as Whatever possesses smoke possesses fire The hill possesses smoke Therefore, the hill possesses fire. N.N.2.: Whatever is heat-possessing is fireness-possessing This is heat-possessing This is fireness- possessing. Or, Whatever possesses heat possesses fireness This possesses heat Therefore, this possesses fireness. According to Navya-Nyāya, “ness” is an abstraction operator and “– possessing” is a concretization operator. There is an interesting relation between these two operators. The Navya-Naiyāyika-s express this relation by the following equation: a-possessingness ¼ a-ness-possessing ¼ a. (dhūma- vat-tva ¼ dhūma-tva-vat ¼ dhūma). Application of this rule makes the difference between the Nyāya argument and a syllogism palpable. N.N.1*. Whatever possesses humanity possesses mortality Socrates possesses humanity Socrates possesses mortality

syllogism All men are mortal Socrates is a man Socrates is mortal

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It is therefore evident that three terms in the Navya Nyāya argument are: humanity, mortality, and Socrates, while in the syllogism three terms are: man, mortal, and Socrates. The rule of identifying the sādhya of an argument can be used more widely for determining the predicate of any sentence. Consider, for example, the sentence, “Socrates is wise.” According to Navya-Nyāya, it should be wise-ness or wisdom by the second part of the rule, since it does not contain “– possessing.” But this sentence can also be transformed into “Socrates is wisdom-possessing.” And in the latter case, we are supposed to drop “–possessing” and we have once again wisdom as the predicate of this sentence. Another important difference between the Nyāya argument and a syllogism lies in the fact that while in a syllogism the major and the minor premise together necessarily imply the conclusion, in a Nyāya argument a third premise is required as a necessary condition to arrive at the conclusion. So, the Navya-Naiyāyikas maintain that the fully fleshed out inferential form is as follows: 1. Smoke is pervaded by fire. (“whatever possesses smoke possesses fire” reformulated in terms of pervasion) 2. The hill possesses smoke 3. The hill possesses smoke pervaded by fire 4. Therefore, the hill possesses fire. Against the Mīmāṃsakas, the Naiyāyikas point out that (3) is not a conjunction of (1) and (2). (3) is weaker because the conjunction of (1) and (2) implies (3), but (3) does not imply this conjunction. The necessity of introducing (3) becomes obvious if we look at the generalized form of the premises. 1*. Something is pervaded by fire 2*. The hill possesses something 3*. Therefore, the hill possesses fire. To have a sound and valid inference, it is necessary that “something” in (1) and (2) stand for the same object, (1) and (2) must be combined to form one complex judgment, to represent “consideration” (parāmars´a) (3), “The hill possesses something pervaded by fire.” Otherwise, one could have the following argument: Smoke is pervaded by fire. The hill possesses light. Therefore, the hill possesses fire. According to the Naiyāyika, (3) is both necessary and sufficient for deriving the conclusion because then we need not even look at the substitution instance of something. It is therefore evident that the Nyāya argument should not be interpreted either as a syllogism or as an argument of the modern predicate logic.

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Around the fifth decade of the twentieth century Indian academics were exposed to Russell’s Principia Mathematica as well as the Natural Deduction systems. Under the influence of Western formal logic and semantics philosophers, mathematicians and computer scientists came forward to pursue study and research in logic. Philosophers’ pursuit of logic followed three different trajectories. One group was engaged in reinterpreting tenets of traditional Indian schools of logic in the light of Set Theory and First Order Predicate Calculus; another group continued research in the area of formal logic – axiomatic systems of predicate logic, modal logic, and its variations, etc., and still another group was busy addressing deep-rooted philosophical problems through conceptual and logical analyses, thus contributing significantly to philosophical logic. The endeavor of the first group fell squarely in the area of comparative philosophy. All of them believed that apt symbolic representation of classical Indian logic would unravel the nature and structure of arguments available in Indian philosophical systems. For example, Navya-Nyya logic received various semantic interpretations in the hands of its modern interpreters. While Sibajiban Bhattacharya used Set theoretic tools and Predicate calculus in his reconstruction of Navya-Nyaya logic, Matilal (1998) suggested a Boolean semantics for some fragments of Navya Nyaya. Jonardon Ganeri (2004) offered a graph-theoretic semantics and Ganeri (2008) developed a set theoretic semantics. However, all of them encountered some problems in their attempt at reinterpretation of Indian logic by using tools of Western formal logic. Their predicament has been summed up by Ganeri (2001) as follows. “. . .any comparative project is liable to catch the Indian theory in a double-bind: either Indian logic is not recognized as logic in the western sense at all; or if it is, then it inevitably appears impoverished and underdeveloped by western standards. The only way to escape this dilemma is to reclaim for Indian logic its own distinctive domain of problems and applications, to see how it asks questions not clearly formulated elsewhere, and in what way it seeks to solve the problems it sets for itself.” Among the mathematically oriented philosophers, three names deserve special mention, Anjan Shukla, Viswambhar Pahi, and A. P. Rao. Anjan Shukla (1965, 1966) started his logical pursuit by solving an open problem in Alonzo Church’s Introduction to Mathematical Logic, Volume I, by developing a two-valued propositional calculus with independent axioms and rules with implication and converse implication as primitive connectives; in an alternative formulation of the same solution he used implication and nonequivalence as primitive connectives. Shukla (1967, 1970, 1971) also reduced the number of variables from two to one in one of the axioms of Lewis’s modal system S1. He solved the decision problem for the modal propositional calculus S1 and showed that it has the finite modal property. It was an open problem for 26 years which Shukla solved. His contributions to philosophical basis of modal logics were globally acknowledged. He constructed two modal propositional calculi which are closest approximations to “the true logic” obtained till 1970 and formulated two sequences of infinitely nonequivalent modalities in S6. His published works were cited by Prior, Cresswell, Pledger, Leberberg, Wronski, Zeman, Thomason, and others. Pahi taught in Rajasthan University and was mainly interested in extensions and matrices of implicational calculi

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(Pahi 1967), Independence Proofs in Propositional Logics, and problems related to Decidability besides problems of philosophical logic (Pahi 1987). A. P. Rao researched mainly in the area of Free Logical systems. He presented in a novel way the Classical Theory of First Order Logic (Rao 1970) from the point of view of meta-mathematics, especially for Indian academics who were not exposed to basic concepts of pure mathematics. The third group confined their logical exercises to philosophical logic. While discussing relevance of modern logic to philosophy, these philosophers offered two diametrically opposite viewpoints. One group of philosophical logicians upheld that all philosophical disputes could be settled with the help of logic alone. In fact, no significant philosophy could be done without a deep and extensive acquaintance with modern logic. Philosophers belonging to the other group emphatically pointed out that the activity of philosophizing cannot neatly be defined with the clarity and rigor of a formal system. To formalize philosophy is to trivialize it, mechanize it, affecting its creativity and growth. A compromise position was to uphold that logic includes both mathematical and philosophical logic (Sen 1980). Contributions in this area were truly heterogeneous. We find that logic-minded philosophers ventured into various different issues in logic-philosophy interface. They were particularly interested in the concepts of entailment, logical form, necessity and possibility, quantifiers and variables, different problematic conditionals and disjunctions, nature of the laws of thought, reference and truth, proper names and definite descriptions, regimentation of natural language, concepts of identity, consistency, justification of deduction, problems of induction and probability, different kinds of paradoxes, including the Liar paradox, the Sorites paradox, the paradox of material implication and their logical solutions, foundational issues and unsolved problems of relevant logics, deontic logics, and epistemic and doxastic logics. They published meta-level discourses on Aristotelean, Kantian, Hegelian, Fregean and Husserlian logic, dynamic systems, and different nonstandard systems of logic. Of course, this list is not an exhaustive one and includes only random examples. Though they were well aware of nonclassical many valued systems of logic, most of them concurred with Quine that bivalent logic would be the last thing to be given up (Sen 2000) and stuck to the classical laws of thought. Mathematicians, however, quickly proceeded to many-valued and nonstandard systems of logic, questioning the inviolability of the laws of thought. Naturally, they picked up research in nonmonotonicity and imprecision (Chatterjee 1994); some focused on para-consistent logics, fuzzy logics, Rough Set Theory, and Rough logic, etc. These researches highlighted the possibility of different concepts of consequence as well as of truth of which the notion of graded consequence (Chakraborty and Dutta 2019) and the correlated notion of Soft Truth deserve special mention. All these culminated into a bold statement on pluralism in mathematics and logic. A good number of researchers from mathematics dealt in algebraic semantics and logics, and were strongly drawn toward different kinds of algebraic structures. We find contributions in the area of Boolean algebra (BA) which provides the algebraic model of classical propositional logic. Some have looked into the algebraic models of modal logical systems through BAs with operators, in particular, topological BAs.

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Again, some scholars focused on Quasi-BAs having even more general structures than BAs which included all the axioms of BA except the law of excluded middle and the law of noncontradiction (Ghosh and Chakraborty 2004). Introduction of topological Quasi-BAs in connection with the semantics of Rough Set Theory by some Indian researchers has been recognized as a genuine move forward in the literature of Algebraic logic (Banerjee and Chakraborty 1996). In recent times, Logic research in India has also received a boost in the hands of computer scientists (Ditmarsch et al. 2011). Logic has become an essential tool in developing programming languages for computers, in natural language processing, and in knowledge representation techniques for artificial intelligence. Theoretical computer Scientists are found to depend on mathematicians in the areas of combinatorics, graph theory, and number theory, and on logicians in the areas of model theory and proof theory in devising algorithms and solving the problems of complexity and completeness in connection with real-time computations. Logic in contemporary India is no longer confined to the ancient systems of philosophy or their comparison with modern logic, but has become continuous with the mainstream of logic pursued all over the world.

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Charvaka (Ca¯rva¯ka)

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . From Mythology to Logical Thinking to Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Idealism and Materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Materialism? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Materialism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Skepticism and Agnosticism in the Vedas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Indian Scenario: Pre-Vedic (Harappan), Vedic, Puranic, and Local Indigenous Cults . . . Evidence of the Upaniṣads re: Ontology and Epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of Natural Elements in India and Greece . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Heterodox and Heretical Doctrines: Ājīvikism, Buddhism, Jainism, etc. . . . . . . . . . . . . . . . . . . . . . . The Āstika and the Nāstika Systems of Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Old (Pre-Cārvāka) Materialist Schools Before the Eighth Century CE. Maṇimēkalai – Bhūtavāda and Lokāyata . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Lokāyata Before and After the Fourth Century CE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . New (Cārvāka) Materialism – The Base Text and Its Commentators in and around the Eighth Century CE and After . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Philosophical Outlook of the Cārvāka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Aphorisms and Pseudo-aphorisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satirical Epigrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cārvākas, the Buddhists, and the Jains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Cārvākas Against Caste and Gender Discriminations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disappearance of the Cārvākas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A. Aphorisms and pseudo-aphorisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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R. Bhattacharya (*) Pavlov Institute, Kolkata, West Bengal, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_43

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R. Bhattacharya

Abstract

The Charvaka (Cārvāka)-s are the last known materialists in India. They appeared in or around the eighth century CE and, for some unknown reason, disappeared after the twelfth century CE. The earlier materialists flourished at least from the time of the Buddha (6th/5th BCE). They were found both in north and south India, bearing such names as Bhūtavāda, Lokāyata, etc. They believed perception to be the only means of knowledge. The Cārvākas, however, admitted inference to some extent as the second source, insofar as the inference was based on or followed from perception. It was in connection with the materialists’ denial of the Other World that their opponents developed their own form of arguments. The materialists were challenged by the opponents with the help of inference by analogy and verbal testimony. The Veda was the ultimate source of such testimony. The Buddhists and the Jains, too, were at one with the materialists in denying the status of the Veda. Thus, the philosophical systems of India were divided into two opposing camps – the āstikas (affirmativists) and the nāstikas (negativists). The contribution of the Cārvākas in particular to the logical thought in India lies in the distinction made by them between two kinds of inference: the first based on perception or verifiable facts, and the second, arguments based on scriptures. Their atheism and satire against vedic sacrificial rites made them stand out as heretics. Their opponents also complained that they were opposed to caste and gender discriminations. These traits also made the Cārvākas appear as the ultimate rationalists in pre-modern India. Keywords

Analogy or comparison · Atheism · Caste and gender discrimination · Consciousness · Elements or bhūtas · Inference · Materialism · Means of knowledge · The other world · Perception · Rationalism · Rebirth · Repeated observation · Spirit · Veda · Verbal testimony

Introduction Cārvāka (pronounced Char-va-ka, ‘ch’ as in ‘Charles’ in English) is a proper name first found in the Mahābhārata, the great epic of India, Book 12.39 (critical edition); 12.38 (vulgate). However, the ascetic bearing this name has got nothing to do with philosophy, let alone materialism. It is from the eighth century CE that we read in the Sanskrit philosophical texts of the Cārvākas who were materialist philosophers. They are mostly referred to in the plural, suggesting a whole school or at least a group. The members of this group were opposed to the concept of the Other World (the future state) and whatever followed from such denial, such as God, the immortal and incorporeal soul, the omniscient beings, heaven and hell and the like. In short, materialism in India appeared as the negation of theism, religion, belief in the supernatural, and the like. The Cārvākas were not the first to declare their views in

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clear, unambiguous terms. In fact, they were the last known group that continued to exist and grow up to the twelfth century CE. Thereafter all the works of this school, both the base text and commentaries on them, completely disappeared from the face of the earth. Some scholars have claimed Tattvopaplava-simha (The Lion to Upset all Principles), a polemical work by Jayarāśibhatta, to be an authentic Cārvāka work, but there are opponents of this view too. It is therefore better to leave it out of this discussion. In any case, such was also the fate of the Cārvākas’ predecessors, the pre-Cārvāka materialists of both north and south India. Their existence is testified in the Buddhist and Jain canonical and para-canonical works, as also, although indirectly, in the Upaniṣads, the source books of the brahmanical systems of thought. It is rather strange that, although the views of the deniers of all tenets of idealist philosophical systems, are known, no original works or commentaries there on have survived. They seem to have disappeared even before the fourteenth century CE. Nevertheless, a few fragments found scattered in the works of the opponents of materialism contain some extracts from the primary and secondary sources of the Cārvākas. Their opponents comprise Nyāya, Vaiśeṣika, Vedānta, Mīmāṃsā, and other non-brahmanical schools, such as Buddhism and Jainism. Not unlike the Presocratics, the works of the materialists of different kinds, too, have been referred to or quoted or paraphrased in the works that seek to refute their views. On the basis of all this their philosophical doctrine has been reconstructed. Thus, though the word, Cārvāka, originally signify the materialists of the eighth century and beyond, it is used as a generic name to refer to all materialists in India, right from the time of the Buddha down to the Cārvākas who flourished between the eighth and the twelfth centuries. Since they were condemned to be essentially argumentative, their opponents, too, had to develop several tools of debate in order to controvert them. Right from the time of the Buddha and Mahāvīra, attempts at refuting and denouncing the materialists continued for several centuries together. In the course of these clashes of arguments and counter-arguments, Logic in India gradually took shape. The materialists were known in Sanskrit by different names (Bārhaspatya, Bhūta-caitanya-vādin, Bhūta-vādin, Cārvāka, Dehātma-vādin, and Laukāyatika/Lokāyata/ Lokāyatika, etc.) at different periods and various parts of the land. They played a vital role in the formation of logical thought in south Asia. Materialism of India was known to China by its name, Lokāyata (Huan Xinchuan 1981, 179–186), and it was through the Buddhist works, both of the Hīnayāna (Small Vehicle) and the Mahāyāna (Great Vehicle) varieties, that the name “the doctrine of annihilation” (ucchedavāda) was communicated to the Buddhists all over the world. The doctrine of annihilation is but another name for proto-materialism.

From Mythology to Logical Thinking to Philosophy The transition from mythology to logical thinking and then to philosophy has not been the same or at least similar everywhere in the world. There were developed pre-Vedic urban centres in north-west and west India in the third millennium BCE. But in the absence of any written records, the intellectual aspects of the Indus Valley

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Civilization cannot even be conjectured. The first three sections of Vedic literature, the Saṃhitās, the Brāhmaṇas, and the Āraṇyakas (1500 BCE and after), were mainly preoccupied with sacrificial rituals and their muttering or singing of magic spells known as mantras. However, in order to provide some relief in the performance, the priests used to indulge in challenging other priests to answer riddles called brahmodyas (Sternbach 1975, 16–22) and argue about all sorts of questions. The exchange of arguments was called vāko-vākya (Vidyabhusana 1920/1988, 45). Such practice of holding informal debates was prevalent during the performance of Yudhiṣṭhira’s Horse Sacrifice (Mahābhārata critical edition, 14.87.1; vulgate 14.85) as also in the post mortem rites in the nineteenth century (testified in Tekchand Thakoor (the pen name of Peary Chand Mitter)’s novel, The Spoilt Child, chapter XX). There were also sceptics and even deniers of the cult of sacrifice even in the early Vedic times, but their doubts and denials did not form a cluster of thought that can be called philosophy proper. The oral tradition embodied in the Upaniṣads (600 BCE and after) may therefore be very conveniently taken as the point of departure. Logic and Philosophy in their rudimentary forms started to develop during the time of the Upaniṣads. The struggle between sacrificial ritual (karma) and knowledge ( jñāna) in the Īśā Upanisad (verse 2) marks one such moment of transition, when a compromise is made between the two, without denying the importance of either. It is followed by another question, whether or not there is life after death and the existence of the Other World (paraloka). This debate forms the focus of the Katha Upanisad (1.1.20). In the longer Upaniṣads, such as the Brhad-āranyaka and the Chāndogya, contests between the sages concerning the character of the Universal Spirit (Brahman) are reported. The identity of the self (ātman) and its relation to Brahman, etc., are quite prominent in the Upaniṣads. The Śvetāśvatara (1.2) mentions different claimants, including Time, Own Being (of every object), Destiny, Accident (Natural), Elements, etc., for the title of the first cause ( jagat-kārana). The emergence of a number of itinerant thinkers and their followers is recorded in the Maitrī Upaniṣad (7.8). They are denounced as non-Vedic (a-vaidika) (7.10) and their views are called negativism (nāstikya) (6.5). A large numbers of gurus and their doctrines are mentioned in the Long Discourses of the Buddha. The Upaniṣads do not speak in one voice: they do not follow a uniform or homogenous line of thought. They retain all sorts of speculations, dissimilar and even contradictory, prevalent among princes, sages, and commoners alike. Such an amalgam of views was redacted long after they had taken shape. In spite of such limitations, it will be proper to start from the late stage of the Vedic era.

Idealism and Materialism The word, materialism, as the name of a distinct approach to nature, both animate (including humans) and inanimate in the context of philosophy is found relatively recently. The first recorded occurrence of “materialism” in English, is presumably borrowed from French; it dates from the mid-eighteenth century (as noted in the

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Oxford English Dictionary). Idealism, the opposite of materialism, too, is not found in the philosophical context in English before the late eighteenth century. Other senses of these terms appear even later, in the nineteenth century only. The systems of philosophy that can be branded so, however, existed long before the names were coined. The terms have acquired different significations in philosophy and everyday use. A clear distinction between the vulgar and technical senses of these two words is maintained throughout this discussion.

What Is Materialism? Materialism considers matter to be the primary component of all things, living or non-living, that exist independent of the knower, human or any other animal. Every object, whether living or nonliving, has a nonliving material substratum. This is primary. Other attributes of all higher living beings, such as senses, consciousness, intelligence, etc., are secondary and presuppose the prior existence of matter. Materialism does not consider consciousness itself to be material. Nor does it claim (as many of its opponents wrongly declare) that matter (body) and consciousness are one and the same. Materialism simply asserts the primacy of matter over consciousness. Conversely, idealism holds that consciousness appears first, matter, next. Matter, it has been observed, is simply the name for what exists objectively, with the one proviso that mind, thought, consciousness are its products. All further questions as to the nature of matter, its structure or composition, the relation of mass, energy, space, time, etc., are not primarily philosophical, but are to be resolved by the natural sciences themselves. (Selsam and Martel 1987, 45).

Early Materialism The world history of philosophy bears out that materialism emerged in India, China, and Greece independent of one another. Nevertheless, the fundamental similarity in their approaches justifies the use of the term “materialism” to all the three manifestations. However, every country, depending on the peculiarities of her tradition, betrays some special features that are not to be found in others. For example, among the seven issues noted below, the first five are common to all materialist traditions everywhere and at all times, while. Nevertheless, the last two are specifically Indian: (i) Matter is the first cause ( jagat-kāraṇa); it precedes consciousness. (ii) Consciousness (variously rendered as self, spirit, or soul) ceases to exist after the death of the body. (iii) There is no Other World, that is, heaven and hell. (iv) There is no rebirth or reincarnation (metempsychosis). (v) Verbal testimony (āptavākya) or word (śabda) is not a valid instrument of cognition (pramāṇa); perception is the first and the best instrument.

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(vi) Performance of sacrificial rites (yajña) and post-mortem rites for the dead ancestors (śrāddha) is useless [It follows from the above five]. (vii) No benefit follows from paying donations (dakṣinā) and gifts (dāna) to priests and brahmanas [It follows from the first feature].

Skepticism and Agnosticism in the Vedas The Vedas, particularly the Saṃhitās, are widely revered and accepted as a selfrevealed work (so admitted by the Mīmāṃsā schools) or composed by God Himself (according to the Nyāya school). Yet, since very early times, we hear of strong dissident voices, challenging the authority of the texts as infallible. Yet there is no evidence that skepticism or infideism contributed to the birth of materialism in India. So far as the literary sources are concerned, they are either theological (as the Vedic literature) or quasi-historical secular works, itihāsa-purāṇa, like the Rāmāyaṇa and the Mahābhārata. In the Upaniṣads we find a strong idealist bent, permeating the philosophical issues concerning the self, and a definite assertion of life after death, of heaven and hell, and of rebirth. Materialism is said to be associated with the demons (asuras, literally non-gods). In any case, the Upaniṣadic philosophy of ātman and brhaman, replace both the Vedic sacrificial cult and the primitive materialism expounded by Uddālaka Āruṇi (see below). Materialism as a full-fledged philosophical doctrine does not appear before the Common Era. Even then, the doctrinal aspect seems to be of less importance than its anti-theological and anti-fideist nature.

The Indian Scenario: Pre-Vedic (Harappan), Vedic, Puranic, and Local Indigenous Cults There is no written record of anything significant in the remains of the oldest known centers of civilization in northwest India. For convenience’ sake, the word “Harappan” is generally used to designate the vast area extending from Mohenjodaro (now in Pakistan) to Lothal (Gujarat, India). However, unlike Mesopotamia and Egypt, no papyrus or any other written document is yet to be found which would shed light on the scientific achievements of the civilization. The architecture and town planning exhibit a very high caliber both of engineers and builders. That is all that can be said about the First Urbanization India (around 2500 BCE. For details, see Chattopadhyaya 1986, Chaps. 4, 7, 11 and 12). So much is being said about town planning, sewerage, etc. of the Harappan Civilization for the simple reason that they are all evidence of logical thought in practice. Religion had its own place in the cities, and superstition, too, was most probably utilized to keep the working people submissive. There are some cult items among the ruins, but nothing definite can be said about the patterns of belief and/or worship of the Harappans. Altogether, what has come down is a set of dumb witnesses of an

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advanced culture in which logical thinking must have had its place along with the irrational faith in rituals. The question of the arrival of the vedic people from the north via Iran is still hotly debated. However, one thing is certain: the vedic people were village dwellers performing sacrificial rituals of their own, owing practically nothing to the indigenous inhabitants of northwest India. Gradual intermixture of the people led to a mixture of culture (technically known as “acculturation”) over time. This inevitably led to the birth of a new set of beliefs and rituals. It was much later – unfortunately no definite date can even be suggested – that serious misgivings, doubts, and finally denials of all the existing ritual practices appeared. One such doubter called Kautsya challenged the vedic rituals and their viability on the ground of being nonsensical (For some instances see Radhakrishnan and Moore 1957, 34–36, 227 n1; Sarup 1984, 78–81; Del Toso 2012, 138–141). Scepticism, doubt concerning the widely accepted views regarding the origin of the universe and such fundamental issues are first manifested in a hymn called the “Nāsadīya Sūkta” in the Ṛgveda, Book 10 hymn 129 stanzas 1–7. Four of them are quoted below: There was neither non-existence nor existence then. There was neither the realm of space nor the sky which is beyond. What stirred? Where? In whose Protection? Was there water, bottom-lessly deep? 1 There was neither death nor immortality then. There was no distinguishing sign of night nor of day. That one breathed, windless, by its own impulse. Other than that there was nothing beyond. 2 ... ... ... Who really knows? Who will here proclaim it? Whence was it produced? Whence is this creation? The gods came afterwards, with the creation of this universe. Who then knows whence it has arisen? 6 Whence this creation has arisen – perhaps it formed itself, or perhaps it did not – The one who looks down on it, In the highest heaven, only He knows Or perhaps he does not know.7 (https://www.sanskritimagazine.com/indian-religions/hinduism/the-hymn-of-creationnasadiya-sukta/)

Two other hymns in the Ṛgveda also expressed doubt (1.164 and 10.190). However, the most effective doubt evolved round the question of life after death. Whether there was any Other World and rebirth is voiced in the question of Naciketas to Yama, the lord of the Dead in the Kaṭha Upaniṣad throughout the later period this question appeared again and again. It divided up the thinkers into two groups: those who believed in the existence of the Other World and those who did not. Accordingly, they were known as āstikas (the assenters) and the nāstikas

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(dissenters). These were the original meaning of the words. Only the materialists were then regarded as dissenters, with the Buddhists and the Jains joining hands with the brahmanical people in condemning the materialists solely on this ground. In the course of time, however, a second point of difference arose: whether one believed in the Vedas as an infallible source of knowledge or not. This point alienated the Buddhists and the Jains from the assenters as they did not believe in the authenticity of the Veda. This is how Dharmakīrti (1943), a Buddhist philosopher, branded the brahmanical thinkers as follows: Acceptance of the authority of the Veda and of someone as the creator, the desire of getting merit through the (holy) dip, the vanity of (higher) castes, and torturing the body to redeem the sins – these are the five marks of stupidity (Quoted in Rahul Sankrityayan 1978, 8. Translation amended).

The Jains, too, were strictly vegetarian and satirized the vedic priests and their followers for eating flesh and drinking wine (particularly in the Sautrāmaṇī sacrifice). Thus they were at one with the Buddhists and the materialists. Their target was the annual ritual called Śrāddha, the post-mortem rites for the dead ancestors. In this way the materialists belonged to the same camp as the Buddhists and the Jains as the three were opposed to the vedic sacrifice-oriented rituals.

Evidence of the Upanisads re: Ontology and Epistemology ˙ Mention has already been made of the Kaṭha Upaniṣad. It has been most probably composed to uphold the view of the immortality of the soul, rebirth, and the justice meted out by Yama against the nonbelievers in the Other World. Yama is the Lord of the dead and their abode called Naraka, equivalent to the Hell in other religious systems. Another Upaniṣad called Bṛhad-āraṇyaka contains the conversation between the sage Yājñyavalkya and his wife, Maitreyī: It is like this. When a chunk of salt is thrown in water, it dissolves into that very water, and it cannot be picked up in any way. Yet, from whichever place one may take a sip, the salt is there! In the same way this Immense Being has no limit or boundary and is a single mass of perception. It arises out of and together with these beings and disappears after them—so I say, after death there is no awareness. (Patrick Olivelle’s translation) This was of course not Yājñyavalkya’s own view. He said only to test his wife. Naturally she was confused. The sage then made his point clear. It is interesting to observe that Uddālaka Āruṇi uses the same simile (an instance of inference by analogy in the Chāndogya Upaniṣad). Uddālaka asks his son: “Put this chunk of salt in a container of water and come back tomorrow.” The son did as he was told, and the father said to him: “The chunk of salt you put in the water last evening—bring it here.” He groped for it but could not find it, as it had dissolved completely. “Now, take a sip from this corner,” said the father. “How does it taste?”

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“Salty.” “Take a sip from the center.—How does it taste?” “Salty.” “Take a sip from that corner.—How does it taste?” “Salty.” “Throw it out and come back later.” He did as he was told and found that the salt was always there. The father told him: “You, of course, did not see it there, son; yet it was always right there. “The finest essence here—that constitutes the self of this whole world; that is the truth; that is the self (ātman). And that’s how you are, Śvetaketu.” (6.13.1–3)

Concept of Natural Elements in India and Greece The separation of earth, air, fire, and water, as the basic constituent elements of all natural objects comprehensible by the five senses, marks the point of departure from mythology to philosophy. As in Greece, so in India, these four basic elements constitute the basis of many philosophical speculations, both in the idealist or materialist. All scientific speculations, whether in the field of natural sciences or of medicine, accept the concept of the elements called bhūtas or mahābhūtas that are at the basis of all phenomena. Unlike the Greeks, however, Indian speculators spoke also of a fifth element, sky (ākāśa, vyoma), or emptiness (śūnya). All philosophical systems except the Cārvāka/Lokāyata adhered to the five-element formula, corresponding to the five senses. Thus, earth corresponds to smell, air to touch, fire to heat, and so on. Even the earlier, pre-Cārvāka materialists had adopted it, which is one of the points on which the Cārvāka/Lokāyata differs from them. The medical compilation called the Caraka-saṃhitā is at bottom materialistic, but, unlike the Cārvāka/Lokāyata, it adheres to the five-element theory. In any case, one conclusion is inescapable: the material basis of all objects was universally accepted to be these four or five elements. All realist philosophical systems too accepted these elements as archē, whether or not there was a creator god. The doctrine of atomism too was molded to fit in this theory of elements. Besides the elements, own-being (svabhāva) was also claimed to be the first cause. In the works of later philosophers the doctrine of own-being is made to be associated with the Cārvāka/Lokāyata, although originally this doctrine was distinct from that of the elements (as in Śvetāśvatara Upaniṣad 1.2). The word, svabhāva, too was later interpreted in two diametrically opposite ways: svabhāva-as-accident and svabhāva-as-causality.

Heterodox and Heretical Doctrines: A¯jīvikism, Buddhism, Jainism, etc. The sixth/fifth century BCE in India, China, Greece, and Iran saw a great philosophical upheaval. Leaving aside other countries, India alone witness a breakdown of Vedic orthodoxy. The existence of no fewer than 52 itinerant preachers is attested

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by the Buddhist texts, such as “The Brahmajāla Sutta” in the Long Discourses (Dīgha Nikāya), but only six are named. One of them was known as Ajita of the hair-blanket. He preached proto-materialist doctrine, denying everything that was to be denied, particularly the Other World, of paying oblation to the gods, the inefficacy of gifts (dāna), etc. The Atharvaveda was a product of the acculturation between the Vedic and the non-vedic indigenous people. This Veda was for a long time not recognized as a Veda at all. There were difference of opinion even among the Vedists, such as Manu, the law-giver, opposing the entry of the Atharvaveda in the Trayī (the three Vedas, Ṛk, Sāman and Yajus only); Kumārila and Jayanta, two philosophers belonging to Mīmāṃsā and Nyāya schools respectively, supported the claim of the Atharvaveda, and thereby upheld the concept of Chaturveda (pronounced Chatur-veda, four Vedas) instead of three. For details, see Dipak Bhattacharya, “Trayi, triads and the Vedas” (forthcoming) and his Introduction to the Atharvaveda of the Paippalāda school, particularly vol.4. Thus, the so-called orthodoxy around the Veda was already in a jeopardy. The rise of Buddhism and Jainism dealt a blow to the concept of the Veda itself.

The A¯stika and the Na¯stika Systems of Philosophy Nāstikya is derived from nāstika, which literally means “one who says (or believes) that (it) does not exist.” The opposite word, āstika, similarly signifies “one who says (or believes) that (it) exists.” Originally it was the assertion and denial of the existence of the Other World, that is, life after death. In the course of time āstika and nāstika came to suggest the upholder and defiler of the authority of the Veda, the most sacred book of the brahmanical people, the theist and the atheist, and similar affirmation and denial of any doctrine (see Bhattacharya 2009, 227–232). Originally the negativistic systems were counted as three: (1) the Buddhist, (2) the Jain, and (3) the materialists. But then, perhaps to pair the number six of the affirmativistic systems (namely, (1) Mīmāṃsā (also called pūrva-Mīmāṃsā), (2) Vedānta (also called uttara- Mīmāṃsā), (3) Sāṃkhya, (4) Yoga, (5) Nyāya, and (6) Vaiśeṣika), the negativistic systems were also counted as six: the four schools of Buddhist philosophy (viz., Madhyamaka (or Mādhyamika), Yogācāra, Sautrāntika, and Vaibhāṣika), along with the Jain, and the materialists. Of these four, the first two were idealist, preaching the doctrine of emptiness (śūnyavāda), and that of the momentariness (kṣaṇikavāda), while the last two, realist. Madhusūdana Sarasvatī (seventeenth century CE), in his Prasthānabhedaḥ 1977, 1, first identifies these as “six [negativist] philosophies” (ṣaḍ [nāstika] darśanāni). Cimanabhaṭṭa (1923, 89–90) repeats it, emphasizing their anti-Vedic character. See also Radhakanta Deva, Śabdakalpadruma 1836 Śaka, s.v. nāstika. “The six systems of Indian philosophy,” however, refer to the affirmativist systems only. For details see F. Max Müller 1899/ 1971).

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Two Old (Pre-Ca¯rva¯ka) Materialist Schools Before the Eighth Century CE. Manime¯kalai – Bhūtava¯da and Loka¯yata ˙ What has been seldom noticed – and even when noticed, never studied in detail – is the upheaval of the philosophical scene in India at a point of time between the fourth and the seventh century. It is to be found in the Tamil epic Maṇimēkalai. Six systems are said to be accepting logic, namely, Lokāyata, Buddhism, Sāṃkhya, Nyāya, Vaiśeṣika, and Mīmāṃsā (27.77–85). The poem also provides evidence of the existence of at least two pre-Cārvāka materialist doctrines in south India, Lokāyata and bhūtavāda (an exact rendering of “materialism”). The exponent of bhūtavāda declares that in spite of several points of similarity, there are some differences between the two (27.265–77). As there is no mention of four elements instead of five (earth, air, fire, water, and space or emptiness), it is logical to assume that bhūtavāda was a five-elementalist (bhūta-pañcaka-vādin) doctrine. With the appearance of the Cārvākas, a new doctrine of four-elements (bhūta-catuṣṭaya-vāda) was introduced, perhaps for the first time. In the exposition of his doctrine the bhūtavādin in the Maṇimēkalai brings out the basic difference between old (pre-Cārvāka) materialism and new (Cārvāka) materialism. They are as follows: (a) Instead of five elements (including ākāśa or vyoma, space) as their principle (tattva), the Cārvākas spoke of four, excluding space, presumably because it was not amenable to sense-perception. (b) The followers of bhūtavāda believed in two kinds of matter: lifeless and living. Life originates from living matter, the body from the lifeless. The Cārvāka/ Lokāyatas did not believe in such duality; to them all beings/entities were made of the same four basic elements. (c) There was another domain in which the two differed more radically. Some of the pre-Cārvāka materialists were accidentalists (yadṛcchāvādins); they did not believe in causality. On the other hand, the Cārvākas appear to have endorsed causality; they adopted the doctrine of svabhāva-as-causality rather than the opposite one, namely, svabhāva-as-accident. (d) The Cārvākas admitted the validity of inference insofar as it was confined to the material and perceptible world (hence verifiable). They, however, made a clear distinction between the lokasiddha (commonly accepted) and tantrasiddha (following from scripture) hetu-s (Udbhaṭa quoted in the SVR, 265–266). Jayantabhaṭṭa used another set of terms to designate the same distinction: “The well-versed ones say that (in fact) there are two kinds of inference, ‘some in case of which the inferential cognition can be acquired by one self’ (utpanna- pratīti), and ‘some in case of which the inferential cognition is to be acquired (on somebody else’s advice)’ (utpādya- pratīti) [The former kind is valid, but the latter kind is not].” (Jayantabhaṭṭa Ch. 2, I: 184. Jayanta seems to have taken it from some commentary, most probably by Udbhaṭa, on the Cārvākasūtra). However, it should not be extended to the invisible and unverifiable areas, such as the imperishable soul, god, omniscient persons (admitted by the

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Buddhists and the Jains as well), the outcome of performing sacrifices called apūrva (as claimed by the Mīmāṃsa philosophers), etc., while some of the old materialists rejected inference as such as a means of knowledge, and clung to perception alone. Purandara, a Cārvāka philosopher, stated categorically: “The Cārvāka-s, too, admit of such an inference as is well-known in the world, but that which is called inference [by some], transgressing the worldly way, is prohibited [by them].” Eli Franco says, “This is probably the most significant contribution of Purandara to the Lokāyata school (at least it is the one for which he is remembered) [...]” (Franco 1991, 159). The most detailed and pointed challenge to inference as an independent means of knowledge has been provided in the Sarva-darśana-saṃgraha, Chap. 1. It runs as follows: Those who maintain the authority of inference accept the sign or middle term as the cause of knowledge, which middle term must be found in the minor and be itself invarriably connected with the major. Now this invarriable connection must be a relation destitute of any condition accepted or disputed; and this connection dose not posses its power of causing inference by virtue of its existence, as the eye, etc., are the cause of perception, but by virtue of its being known. What, then is the means of this connection’s being known? We will first show that it is not perception. Now perception is held to be two kinds, external and internal (i.e. as produced by the external senses, or by the inner sense, mind). The former is not the required means; for, although it is posssible that the actual contact of the senses and the object will produce the knowledge of the particular object thus brought in contact, yet as there never can be such contact in the case of the past or the future, the universal proposition, which was to embrace the invariable connection of the middle and major terms in every case, becomes impossible to be known. Nor may you maintain that this knowledge of the universal proposition has the general class as its object, because if so, there might arise a doubt as to the existaence of the invariable connection in this particular case (as, for instance, in this particular smoke as implying fire). Nor is internal perception the means, since you cannot establish that the mind has any power to act independently towards an external object, since all allow that it is dependent on the external senses, as has been said by one of the logicians. ‘The eye, ctc., have their object as described; but mind externally is dependent on the others.’ Nor can inference be the means of the knowledge of the universal proposition, since in the case of this inference we should also require another inference to establish it, and so on, and hence would arise the fallacy of an ad infinitum retrogression. (Chattopadhyaya and Gangopadhyaya 1990, 250–251).

Loka¯yata Before and After the Fourth Century CE Materialism in India has been known at different times by different names. One name that has stuck is Lokāyata in Sanskrit and Pali, and Logāyata in Prakrit. Earlier scholars tended to look at this word as always referring to materialism. Right from the Science of Polity by Kauṭilya (Kauṭilīya Arthaśāstra) (c. fourth century BCE) down to the present day, the word occurs in diverse texts, philosophical and non-philosophical. A little reflection, however, shows that in some of these sources

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Lokāyata does not mean a system of philosophy which was a target of abuse indulged in by all orthodox Veda-abiding thinkers. The word, lokāyata, does not occur in the Vedas and its ancillary literature, or anywhere in the Pali Three Caskets (Tipiṭaka). In Kauṭilya’s work Lokāyata is associated with Sāṃkhya and Yoga, two systems known to be very much pro-Vedic in the later philosophical tradition of India. But then it is not known whether by Sāṃkhya, Kauṭilya means the atheistic Epic Sāṃkhya (found in the Book of Tranquillity, Śāntiparvan in the Mahābhārata) and what he means by Yoga: Nyāya, Nyāya-Vaiśeṣika, or the system of philosophy attributed to Patañjali or something else (For details see R. Bhattacharya 2009/2011, Chap. 10, 131–136). The context suggests that Lokāyata stands for logic-based philosophy. All Pali and Buddhist Sanskrit texts, whether belonging to the Great Vehicle (Mahāyāna) or the Less Vehicle (Hīnayāna), carry only one meaning: disputatio, the science of disputation (vitaṇḍa(vāda)-sattham) as recorded by all commentators of the Buddhist canonical and para-canonical texts as well as by the ancient lexicographers. In the “Discourse on the Fruits of Being a Monk” and elsewhere (e.g., the Jātaka tales), the doctrine of Ajita Kesakambala is called “the doctrine of annihilation,” Ucchedavāda not Lokāyata. Pali sources, however, speak of seven kinds of ucchedavāda (“Brahmajālasutta,” 1.3.84–91. Dīghanikāya 1958, 30–32). But nothing definite is known about the other kinds than that of Ajita’s. All these doctrines mentioned in the “Brahmajālasutta” are part of the 62 heresies (diṭṭhiyo). Neither the text nor its commentators are of any assistance in knowing about such doctrines as eternalism (sassatavāda), whose adherents held that the self and the universe are eternal. This doctrine too is of four kinds. There are references to 16 kinds of the doctrine which held that the spirit is conscious after death and eight kinds of its opposite doctrine which held that the spirit is unconscious after death. We are further told that there were eight kinds of yet another doctrine which held that the spirit is neither conscious nor unconscious after death. There is, however, another term in Prakrit to suggest the earliest form of materialism: “the doctrine of (it does) not exist” (natthikavāda and nahiyavāda). See Jamkhedkar, 1984, 184. In Sanskrit works we have the word nāstika which is explained in two different ways: first as a doctrine preaching the non-existence of the Other World, and later, as the doctrine that denies or defiles the Veda. Cf. Manu-smṛti 2.11: “The nāstika is a defiler of the Veda” (nāstiko vedanindakaḥ), vol. 2, 1975. Although nāstika is admitted to be a synonym of cārvāka by the Buddhist and Jain authors and lexicographers, to the Vedists it would mean not only the Cārvākas but the Buddhists and the Jains as well, for all the three were non- or anti-Vedic in their outlooks. Several schools of materialists, some believing in the existence of five elements, some in four (space or Aether is not admitted) are known from The Book Composed against the [heretical] Sūtras (Sūtra-Kṛtāṅga-sūtra), a Jain canonical text. All of them are said to believe in the extinction of the soul or the spirit after the death of the body (See 1978, 1–10; English translation by Jacobi 1895, II: xiii). In the Maitrī Upaniṣad and the Padma Purāṇa, there is a story that Bṛhaspati, the preceptor/chaplain of the gods, intending to deceive the demons, created a system of philosophy. Bārhaspatya, “associated with Bṛhaspati,” is the fourth name given to the materialist. In some eighth-century works, all the four names, Bārhaspatya,

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Cārvāka, Lokāyata, and Nāstika, have been used interchangeably, suggesting materialism (Jinendrabuddhi 2005, 24: atha vā cārvākaṃ pratyetaducyate . . .. Haribhadra 1969, 80a: lokāyatā vadanty evam, etc., but in 85d: cārvākāḥ pratipedire. Kamalaśīla, Pañjikā to Śāntarakṣita’s Tattvasaṃgraha 1968, 1981 Chap. 22 entitled “Lokāyataparīkṣā,” uses the names Cārvāka and Lokāyata interchangeably. See TSP, II: 639,649,657,663,665, also II: 520 (bārhaspatyādayaḥ), 939 (lokāyataḥ) and 945 (lokāyatam). In Hemacandra’s Sanskrit dictionary (twelfth century), all the four names are treated as synonymous. Besides these four names. Śīlāṅka, the Jain commentator, uses another term, “the doctrine of the identity of the soul and the body” (tajjīva-taccharīra-vāda). This is one of the early records of this name, found only in Jain sources. Two other names are often found in later works: dehātma-vāda, the doctrine that claims that the body and the spirit are the same, and bhūta-caitanya-vāda, the doctrine that makes the elements and consciousness appear as one (See Śaṅkarācārya on Brahmasūtra 1988, 1.1.1; Vyomaśiva 1983–84, vol. I:155).

New (Ca¯rva¯ka) Materialism – The Base Text and Its Commentators in and around the Eighth Century CE and After After the Upaniṣads and stray references to proto-materialist ideas comes the time of the formation of the philosophical systems, each having a set of aphorisms (sūtras) and its commentaries and subcommentaries. Broadly speaking, the brahmanical position was unsparing in denouncing all the three of its non-Vedic adversaries, whereas the materialists had to put up a lone battle against all philosophical systems, whether Vedic or non-Vedic, but mostly against Buddhism, Jainism, Nyāya, and Vedānta. While all the writings, the base text/s and the commentaries, of the pre-Cārvāka materialist schools are as yet unavailable, some fragments of the new, Cārvāka school have survived. They can be divided into three kinds: (a) aphorisms (sūtras) and pseudo-aphorisms, (b) extracts from commentaries to the aphorisms, and (c) verses attributed to the Cārvākas. For details see Bhattacharya 2009/2011, 69– 104. The book of aphorisms was most probably compiled by Purandara, who is also credited with writing a short commentary (vṛtti) (Bhattacharya 2009/2011 Sen, 82, 90). Besides the aphorisms which can be safely admitted as genuine, some others are of dubious authenticity. The distinction is made on the basis of the fact that some aphorisms are found quoted in several works of the opponents with more or less the same wording. Those which occur only once are labelled as pseudoaphorisms. Aphorisms form the basis of all classical philosophical systems (excepting Buddhism, Jainism, and the like), whether Vedānta or Mīmāṃsā, Nyāya or Vaiśeṣika. Therefore such collections of aphorisms are rightly called base (mūla) text. All philosophical systems, excepting those mentioned above, developed round the commentaries (in some cases sub-commentaries, too) written by the adherents of every system. The case of the Cārvāka is no different. There was apparently a base text, probably the Paurandarasūtra, and an auto-commentary, Paurandarīyavṛtti.

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Names of four more commentators are found (with extracts from their works quoted) in other sources. They are Bhāviveka, Kambalāśvatara (“blanket-mule,” most probably a nickname), Aviddhakarṇa (“whose ear is not pierced,” probably another nickname), and Udbhaṭa-bhaṭṭa (Bhaṭṭodbhaṭa, most probably of Kashmirian origin, flourished in the ninth century). The Cārvāka system, very much like other systems, did not remain unaltered. It had its development. This is borne out by the interpretation of some of the aphorisms offered by some commentators in or before the eighth century. Kamalaśīla, Pañjikā on Tattvasaṃgraha verse 1864 mentions two approaches to the interpretation of an aphorism which contained no verb. One commentator supplied the verb “is born” after the subject, so that it reads “consciousness is born of these (elements),” while another commentator explained the same aphorism as “consciousness is manifested from these (elements).” Since Kamalaśīla uses plural in case of both, it is not clear whether he means two individual commentators (the plural being honorific) or two groups of commentators (each group having some adherents of its own). As no names are mentioned, it is impossible to decide what Kamalaśīla had in mind. Udbhaṭa is the last known commentator. Jayanta and Cakradhara both speak of the “old (cirantana) Cārvāka” like Bhāviveka and the recent one like Udbhaṭa. Udbhaṭa goes against the literal meaning of the aphorisms; he twists the meaning of some words which can be made, almost under duress to conform to the meanings preferred by Udbhaṭa. In many respects, this last of the commentators may be called a revisionist among the Cārvākas. In his way of interpretation the fundamental nature of the materialism is so compromised as to reach the threshold of idealism. Nevertheless, there is one point that unites the three commentators Purandara, Aviddhakarṇa and Udbhaṭa. They all assert that, although inference based on perception can provide true knowledge, inference based on authority (āpta) and verbal testimony (śabda or āptavākya) are inadmissible. Hence, any statement concerning the existence of heaven and hell, god, an omniscient person (admitted by both Jains and the Buddhists), etc. are open to question. Nevertheless, according to all of them, inference per se is not an independent instrument of cognition. Aviddhakarṇa and Udbhaṭa between themselves provide a number of arguments, both subtle and to the point, to establish this view. As an instance of the logical acumen of the materialists in India, we quote below a passage in which the claim of inference as an independent means of knowledge is sought to be refuted from all possible angles. The anonymous author says: When we say that the ascertainment of the form “fire is in the hill” establishes fire as characterised by the hill, it is not stated what sort of fire is being spoken of in these sentences. Unless specially stated, expression “fire” may stand either (i) for all instances of fire, or (ii) for some unexpected fire, or (iii) for the fire that is known before, or (iv) for the fire that is in the hill. The statement made above dose not also state definitely as to where exactly fire is being established. The fire may be established either (i) everywhere, or (ii) in some unspecified place, or (iii) in some location previously known to us, or (iv) in the hill etc. that are now perceived. Now each of the four possibilities about the fire may be established by anumāna can be combined with each of four possibilities that hold about the location where the fire

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may be established. Thus, we have in all 4
4 ¼ 16 possibilities, and due to some defect or other, none of them can be admitted. . .. 1. Every Fire (as related to the hill) is present everywhere. 2. Every such Fire is present at some unspecified place. 3. Every such Fire is present in the location of the fire that has been previously known to us. 4. Every such Fire is present in the places that are being perceived now, e.g., the hill. 5. Some unspecified fire is present everywhere. 6. Some unspecified fire is present at some unspecified place. 7. Some unspecified fire is present in previously perceived location (e.g., the kitchen). 8. Some unspecified fire is present in the hill before us. 9. Some previously known fire exists everywhere. 10. Some previously known fire exists at some unspecified places. 11. Some previously known fire exists at some previously perceived location. 12. Some previously known fire exists in the hill that is being perceived. 13. The fire present in the hill exists everywhere. 14. The fire present in the hill exists at some unspecified places. 15. The fire present in the hill exists at some previously perceived location. 16. The fire present in the hill exists in the hill. Now among these alternatives, (1) is inadmissible, because it is contradicted by perception. The same is true for (2), (3), (4), and (5). (6) is a case of proving what is already established, i.e., admitted by everyone. (7), (8), (9), and (10) are contradicted by perception. (11) is again a case of siddha–sādhana, i.e., establishing what is already established. (12), (13), (14), and (15) are contradicted by perception. (16) is once again a case of establishing what is already established. Thus none of the alternatives that may be considered here has been found to be tenable (Sen 2010, 19–20).

The Philosophical Outlook of the Ca¯rva¯ka Here are the aphorisms that are found quoted in the works of the opponents of the Cārvākas over and over again, but some are found once only.

The Aphorisms and Pseudo-aphorisms I. Materialism I.1We shall now explain the principle. I.2 Earth, water, fire, and air are the principles, nothing else. I.3Their combination is called “body”, “sense” and “object”. I.4 Consciousness (arises or is manifested) out of these.

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I.5 As the power of intoxication (arises or is manifested) from the constituent parts of wine (such as flour, water and molasses). I.6 The self is (nothing but) the body endowed with consciousness. I.7 From the body itself. +I.8 Because of the existence (of consciousness) where there is a body. I.9 Souls are like water bubbles. The doctrine of inherent nature (lit. own being) II.1 The world is varied due to the variation of origin. II.2 As the eye in the peacock’s tail. The doctrine of the primacy of perception III.l Perception indeed is the (only) means of right knowledge. III.2 Since the means of right knowledge is to be nonsecondary, it is difficult to ascertain an object by means of inference. The doctrine of the denial of rebirth and the Other World IV.l There is no means of knowledge for determining (the existence of) the Other World. IV.2 There is no Other World because of the absence of any Other-Worldly being (i.e., the transmigrating self). IV.3 Due to the insubstantiality of consciousness (residing in the Other World). The doctrine of the uselessness of performing religious acts V.1 Religious act is not to be performed. V.2 Its (religion’s) instructions are not to be relied upon.

The style of composing the aphorisms is not different from that of the other philosophical systems. Of these, one aphorism (I.5) is worth noting. It replies to an objection. Does any random combination of these elements as flour, water, and molasses (as mentioned in I.5) produce the intoxicating power? The Cārvākas said: No, only a particular kind of transformation of these elements, pariṇāma-viśeṣaḥ, can produce such an effect. None of these is intoxicating in itself, but the total effect of a particular transformation results in producing the power of intoxication. Thus, a particular kind of combination of these elements, and no other, is an essential condition. Only one kind of combination would yield the desired result. This is what is called upamāna in Indian logic, comparable to inference by analogy in Western logic. As the aphorism shows, it was meant to be understood as a simile; the actual object of comparison (upameya) is consciousness being born or produced from the four natural elements, earth, air, fire, and water. The momentous question of how consciousness can come out of a body admittedly consisting of unconscious elements, was sought to be answered with the help of a simile. The Cārvāka/ Lokāyatas are often accused of being pramāṇaikavādins, they are alleged to have accepted only one means of knowledge, namely, perception. But in the very aphorisms forming their base text, reveals that inference by analogy had its place in the materialist system. Similes based on analogy are as much part of the Cārvāka/Lokāyata sayings as any other system of logic and philosophy in pre-modern India.

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The simile in question – nonintoxicating elements giving rise to the power of intoxication – thus represents more than it appears. Elsewhere too, similar similes are employed to prove the existence of the immortal soul. Kumāra Kassapa employs a series of similes to refute the materialist’s claim that consciousness perishes with the death of the body. The duologue between Pāyāsi (Paesi in Māgadhi Prakrit), the king/governor, and Kumāra Kassapa, a Buddhist monk in the “Pāyāsisūttanta” of the Long Discourses (Dīghanikāya) shows how comparisons could be so employed by both the parties: the materialists and the idealists, in order to prove the same truth.

Satirical Epigrams The verses that are attributed to Bṛhaspati or the Cārvākas are mostly of the nature of what was called “sung while intoxicated,” pramatta-gīta in Patañjali’s Mahābhāṣya (1972, 18–19) and Śabara’s commentary on the Mīmāṃsāsūtra, 2.2.26 (110) and 3.1.17(159) as opposed to the pro-vedic verses called bhrāja. Such verses might have been composed by the Buddhists and the Jains as well and are not rare by any means. Only three of the ten and half verses quoted by Sāyaṇa-Mādhava may be said to reflect the materialist view proper; others might have originated from the Buddhist or Jain circles as well. The verses run as follows: There is no heaven, no final liberation, nor any soul in another world. Nor do the actions of the four castes, orders, etc., produce any real effect. While life remains let a man live happily; nothing is beyond death. When once the body becomes ashes, how can it ever return again? If he who departs from the body goes to another world, How is it that he comes not back again, restless for love of his kindred? (Sāyaṇa-Mādhava 1978, 13–15)

The Ca¯rva¯kas, the Buddhists, and the Jains The Buddhists and the Jains share some of the views of the Cārvākas. All the three were branded as “negativists” by the Vedists of all sorts. They were opposed more particularly to the performance of annual rituals for the departed ancestors (śrāddha) and sacrificial rites (yajña) with a view to fulfilling one’s heart’s desire, both in this world and the next. But the opposition of the two religious communities on the one hand and the Cārvākas on the other arose from different reasons. Both the rites involved slaying of animals, which was anathema to the doctrine of non-injury (ahiṃsā) of the Buddhists and the Jains. The Cārvākas too were opposed to the performance of post-mortem rites, for they regarded them as useless (since there can be no life after death) and no benefit can accrue from the performance of yajñas, for there were no gods to grant the sacrificers’ prayers. In spite of all this, the two religious communities clung to the idea of rebirth, after-life (paraloka), and the mysterious effects of karman and adṛṣṭa. The Cārvāka/Lokāyata stands apart from all other philosophical systems in India that strove against one another for centuries. They

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denied all that was axiomatic to others, specially the infallibility of the Veda and/or the doctrine of karman and rebirth. Since the Cārvākas did not consider philosophy to be a means of emancipation from the cycle of rebirth (mokṣa, mukti, or nirvāṇa) but viewed it as a practical guide to life, they incurred the wrath of all who believed in the Other World and rebirth. The Cārvākas preached an outlook that was life-asserting, while others proposed a negative attitude towards earthly existence. The Cārvākas did not think in terms of the four aims of life (puruṣārthas), namely, religious merit (dharma), wealth (artha), pleasure (kāma), and freedom (mokṣa). This too marks them apart from others.

The Ca¯rva¯kas Against Caste and Gender Discriminations What the opponents of the Cārvāka make them say regarding caste (varṇa) and women deserves attention. They are represented as opposed to caste discrimination and in favor of equality of women and men. This representation (censorial in intention) is borne out by the heretical views attributed to Kali, personification of the Iron Age, in Śrīharṣa’s Life of Naiṣadha (Naiṣadha-carita): Since purity of caste is possible only in the case of purity on each side of both families of the grandparents, what caste is pure by the purity of limitless generations? Fie on those who boast of family dignity! They hold women in check out of jealousy; but do not likewise restrain men, though the blindness of passion is common to both! Spurn all censorious statements about women as not worth a straw. Why dost thou constantly cheat people when thou, too, art as bad as women? (17. 40, 42, 58)

That this is not an isolated case or a mere figment of Śrīharṣa’s imagination is borne out by similar representations found elsewhere. In Kṛṣṇamiśra’s (eleventh century) allegorical play, Prabodhacandrodaya (Rise of Moon-like Intellect), Mahāmoha (Great Delusion), an avowed materialist, declares: If the bodies are alike in their different parts, the mouth, etc., how can there be a hierarchy of castes? 2.18ab. For details see R. Bhattacharya 36 (2010):37–42.

Disappearance of the Ca¯rva¯kas One of the many mysteries in the history of the philosophies in India is the disappearance of several religious and philosophical systems without leaving any trace whatsoever. The Cārvākas and their predecessors belong to those whose existence is attested but nothing of substance remains for the present generation to gather a more or less comprehensive knowledge of their doctrines. However, like the Ajīvikas, various stages of development of materialist thought in both north and south India can be and have been partially sketched by both Indian and European scholars since the first half of the nineteenth century. The fragments were discovered sporadically;

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their agreement with one another could be proved after the publication of new manuscripts. For instance, the discovery of Cakaradhara’s Granthibhaṅga, a commentary on Jayantabhaṭṭa’s Nyāyamañjarī, helped clear the identity of the “cunning Cārvāka” and “the well-educated Cārvāka” in the text. Instead of referring to two different persons, Jayanta, it is now known, was meaning one commentator only, that is, Udbhaṭabhaṭṭa. His name was found in other sources as well, and thus several fragments of his commentary are at present available. More intensive research and further discovery of new manuscripts will surely facilitate the task of reconstructing the views of the Cārvākas.

Definition of Key Terms Caste (varṇa) discrimination

Inference

Materialism

Perception

Rationalism

The Indian society guided by the Vedas, the sacred books and religious law-books, promoted the inviolable division of four castes, the Brahmana (the priest class), Kṣatriya (the warrior class), Vaiśya (the trading and agricultural class), and Śūdra (the working class). The hierarchy was considered to be ordained by God Himself. It is a means of knowledge to draw conclusions from already established conclusions, particularly in such cases where perception cannot be resorted to. The philosophical views that consider matter to be primary and consciousness, secondary that arises out of matter. Consequently, consciousness perishes with the death of the body. The five senses of humans, namely, sight, hearing, smell, touch, and taste, are accepted as the most definite means of knowledge, and superior to all others, such as inference, verbal testimony, and comparison. It is an approach that does not rely on any religious faith or dogma but prefers to be guided by reason in all walks of life (in the eighteenth century the meaning was different; it was then opposed to empiricism).

Summary Points • Priority of matter over consciousness. • Denial of the Other World, heaven and hell, etc. • Denial of the rebirth, karman.

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• Denial of the efficacy of religious rites, performance of sacrifices to the gods and post-mortem rites of the ancestors. • Denial of the existence of immortal and incorporeal soul. • Denial of the special status of the Veda as the ultimate source of all knowledge. • Denial of the existence of God or gods (atheism).

Appendix A. Aphorisms and pseudo-aphorisms I. bhūtavāda I.1. athātastattvaṃ vyākhyāsyāmaḥ I.2. pṛthivyāpastejovāyuriti tattvāni I.3. tatsamudāye śarīrendriyaviṣayasaṃjñāḥ I.4. tebhyaścaitanyam I.5. kiṇvādibhyo madaśaktivat I.6. caitanyaviśiṣṭaḥ kāyaḥ puruṣaḥ I.7. śarīrād eva +I.8. śarīre bhāvāt I.9. jalabudbudavajjīvāḥ II. svabhāvavāda II.l. janmavaicitryabhedājjagadapi vicitram II.2. mayūracandrakavat III. pratyakṣaprādhānyavāda III.l. pratyakṣam (ekam) eva pramāṇam III.2. pramāṇasyāgauṇatvād anumānād arthaniścayo durlabhaḥ IV. punarjanma-paraloka-vilopavāda +IV.l. paralokāsiddhau pramāṇābhāvāt IV.2. paralokino ‘bhāvāt paralokābhāvaḥ +IV.3. paralokicaitanyaṃ niravayavatvāt V. vedaprāmāṇya-niṣedhavāda +V.l. dharmo na kāryaḥ +V.2. tad upadeśeṣu na pratyetavyam

References Bhattacharya, Ramkrishna. 2009/2011. Studies on the Cārvāka/Lokāyata. Firenze/London: Società Editriche Fiorentina/Anthem Press. Bhattacharya, Ramkrishna. 2020. More Studies on the Cārvāka/Lokāyata. New castle upon Tyne: Cambridge Scholars Publishing. Bhattacharya, Dipak. 2016. Introductions. In Paippalāda-Samhita of The Atharvaveda, vol. 4. Kolkata: The Asiatic Society. Bhattacharya, Dipak. Trayi, triads and the Vedas (forthcoming).

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General Introduction to Buddhist Logic Joerg Tuske

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nāgārjuna’s prasaṅga Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Objects Are Empty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concepts Are Empty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Are Nāgārjuna’s Four Alternatives? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Is It Possible to Reject All Four Alternatives? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Diṅnāga on Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dharmakīrti on Diṅnāga on Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meaning as Exclusion: apoha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Case for apoha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Header Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter discusses some key concepts of Buddhist logic from different traditions. The first part deals with Nāgārjuna’s prasaṅga technique and his claim that reality is empty (s´ūnya). The idea that the claim that everything is empty, if true, might be regarded as empty itself is discussed. The second part provides an overview of Diṅnāga’s view on inference and his theory of the three conditions of an inferential sign (hetu). This theory is compared to the Nyāya view on inference and the question whether the second of Diṅnāga’s conditions is redundant is raised. The third part explores Dharmakīrti’s developments of Diṅnāga’s view on inference, particularly his introduction of the particle eva

J. Tuske (*) Department of Philosophy, Salisbury University, Salisbury, MD, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_2

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and the three types of hetu. The final part of this chapter presents an outline of Diṅnāga’s apoha (exclusion) theory of the meaning and function of concepts and words. Keywords

anupalabdhi-hetu (reason based on nonobservation) · apoha (exclusion) · catuṣkoṭi (tetralemma) · eva (only exactly really) · hetu (inferential sign reason) · kārya-hetu (reason qua effect) · nis´citam (determinately, necessarily) · pakṣa (locus of inference) · paryudāsa (nominally bound) · prasajya-pratis´edha (verbally bound) · prasaṅga technique (technique showing the “consequences”) · sādhya (thesis-to-be-inferred) · sapakṣa (homologue or place which is known to possess the sādhya already) · svabhāva-hetu (reason based on natural reason) · s´ū nya (empty) · vipakṣa (heterologue or place which is known not to possess the sādhya)

Introduction The term “logic” played an important role in Indian philosophical thinking, even though there is no equivalent to this word in Sanskrit or Pāli. Indian thinkers were clearly concerned with ideas of inference and the meaning of language, all of which play a large role in the study of logic, as defined by Western philosophers. What makes logic “Buddhist” is the fact that many of the Buddhist thinkers were developing ideas based on their views about metaphysics and epistemology in opposition to other Indian thinkers, for example Naiyāyikas. As with most themes in Indian philosophy, it is difficult to find one common narrative within Buddhist logic. However, several contributions by Buddhist thinkers stand out: (1) the prasaṅga method of Nāgārjuna, (2) the theory of inference developed by Diṅnāga, (3) the development of the theory of an inferential sign by Dharmakīrti, and (4) the apoha theory of the meaning of concepts and words by Diṅnāga and Dharmakīrti. In this chapter, an introductory outline of these four key developments in Buddhist thinking will be given.

Nāgārjuna’s prasan˙ga Technique The prasaṅga technique developed by Nāgārjuna (ca. 150 CE) is a logical tool to establish his claim that all concepts and objects are empty or void (s´ūnya). He was aware that this claim was open to misinterpretation. In his work Mūlamadhyamakakārikā (MMK) he says: By a misperception of emptiness A person of little intelligence is destroyed. Like a snake incorrectly seized

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Or like a spell incorrectly cast. (MMK 24.11)

First, before analyzing the claim that concepts are empty or void, the probable meaning of Nāgārjuna’s claim that objects are empty or void will be discussed. This is necessary for understanding the prasaṅga technique.

Objects Are Empty In his work Vigrahavyāvartanī (VV), Nāgārjuna claims that: Whether in the causes (hetu), in the conditions (pratyaya), in the combination of the causes and the conditions (hetupratyayasāmagrī ), or in a different thing, nowhere does exist an intrinsic nature of the things, whatever they may be. On this ground it is said that all things are void (s´ūnyāḥ sarvabhāvāḥ). (VV 1)

These statements can be paraphrased as: 1. An object is empty or void if and only if it lacks any intrinsic nature (svabhāva). Nāgārjuna also says: That nature of the things which is dependent is called voidness, for that nature which is dependent is devoid of an intrinsic nature (yas´ ca pratī tyabhāvo bhavati hi tasyāsvabhāvatvam). [. . .] Those things which are dependently originated are not, indeed, endowed with an intrinsic nature; for they have no intrinsic nature (ye hi pratī tyasamutpannā bhāvās te na sasvabhāvā bhavanti, svabhāvābhāvāt). – Why? – Because they are dependent of causes and conditions (hetupratyayasāpekṣatvāt). If the things were by their own nature (svabhāvataḥ), they would be even without the aggregate of causes and conditions (pratyākhyāyāpi hetupratyayam). (VV 22)

This statement can be rendered as: 2. An object has self-nature if and only if it is independent (i.e., without causes and conditions) of other things. This is in part an “antiessentialist” doctrine. What is meant by this is that it is a rejection of the idea that objects possess fixed, eternal essences. But it is also more than that: it claims that objects have no intrinsic properties whatsoever. A property that belongs to an object independently of its relation with other things is an “intrinsic property.” A four-legged chair, for example, has the intrinsic property of “being four-legged,” and it might have the relational property of being to the left of the table. Nāgārjuna, however, claims that all properties are relational. Nāgārjuna’s system is called Mādhyamika or the “doctrine of the middle way.” “Middle way” means steering an alternative between two extreme positions which are called “eternalism” and “annihilationism” (or “nihilism”). An eternalist is an essentialist, who claims that each object exists independently of any others by

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having an immutable essence. An annihilatonist claims that objects do not exist at all. At first glance, annihilationism might not sound like a feasible position. It is based on the idea that it is seemingly impossible to establish a grounding for the objects of the five senses. Everything one perceives can, at least in theory, be divided up further. Hence, there is no final basis to reality. The annihilationist claims that this makes reality nonexistent. Nāgārjuna adopts a middle position between these extremes: objects exist, not by self-subsisting but by having a nature determined by their relations to other objects. The Sanskrit word s´ūnya which is translated as “empty” or “void” has another technical sense. It was used by the early Indian mathematicians as the name for the numeral zero. The point of introducing a zero into the counting system is that it serves as a place marker. The value of the other numerals like 1 becomes dependent on their place with regard to the zero. Take for example 10 or 100. Likewise, a zero only has a value in relation to the other numerals. Matilal points out that the analogy of a place marker works well if the emptiness thesis is understood as saying that an object’s or concept’s properties, including those that determine its nature, are relational. To say that a concept is s´ūnya means that it is like the zero because it has no absolute values of its own but has a value only with respect to a position in a system. (Matilal 2005, p. 118)

So, according to the doctrine of emptiness none of the observed objects in the world have any true nature which can be grasped by anyone. Objects are only objects in relation to other objects. Another way the doctrine of emptiness works is by application to concepts. It was just shown how it applies to objects but of course in order to think and talk about objects, one needs concepts. So, the next claim is that concepts are empty.

Concepts Are Empty The idea is that every concept, statement, or even philosophical theory is empty. A natural way to interpret this, in the light of the doctrine about objects, is: 3. Statements/theories/concepts have no independent semantic properties. This means that theories are empty in that the terms and concepts involved in them fail to refer at all. One example of a scientific theory which was held to be true at some point but which turned out not to refer to anything was the phlogiston theory in chemistry. Phlogiston is the putative name of a substance emitted during combustion. It turned out however that there is no such substance. The term phlogiston does not refer to anything just like the description “The present king of France” does not refer to anything when uttered in 2017. The fact that a statement is empty means that it lacks a referent. However, this does not mean that it does not have a functional role in language. Nāgārjuna wants to say that all statements are like statements with

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nonreferring expressions. This reading is supported by Nāgārjuna’s general philosophical prasaṅga technique. This technique works in the following way: one takes a hypothesis p and shows that it leads to a contradiction. In addition to that one shows that the negation of p also leads to a contradiction. If this can be done then one can infer that the concepts in this hypothesis fail to refer. Example 1: “A Square Circle Is Circular”

The idea of a square circle is contradictory, so to say that it is circular is contradictory as well. However, even if one said that a square circle is not circular one would still face the contradiction because of the term “square circle.” It is nonreferring. ◄

Example 2: Origination

Nāgārjuna argues that the hypothesis that objects self-originate, i.e., that they do not have a cause outside of themselves, leads to a contradiction. If they did originate solely from themselves, then their existence at an earlier time would be the cause of their existence at a later time. But that means that they can never come into existence. They would have to have always existed. However, at the same time Nāgārjuna wants to say that the hypothesis that objects originate from other causes than themselves leads to a contradiction as well. Nothing depends solely on the existence of something else. If both of these theses can be shown to be contradictory, and there is no third option, then the whole concept of causation or origination becomes suspect. Nāgārjuna nevertheless does not deny that this kind of statement might be useful for this discourse. ◄ In general, his attitude toward philosophical theories seems to resemble a theory called instrumentalism in philosophy of science. The instrumentalist claims that the theoretical terms in a scientific theory (e.g., “electron”) do not refer to anything real, but the theory as a whole is useful in that it allows one to make predictions based on observational data. George Berkeley for example criticized Newton for talking about forces as if they were entities in the world when really they are mathematical entities. Ernst Mach has reiterated that kind of criticism in the last century. Similarly, philosophical theories, according to Nāgārjuna have an instrumental utility in ordinary thinking, but no further claims to be correct descriptions. Nāgārjuna’s claims are part of a strand of Buddhist ideas at the basis of which is the notion that the world is not as it appears to be. The world does not consist of stable objects. Theories can never capture the truth. This is so because the world is merely appearance. All one ever has access to is the world of appearance. To say that there is more than appearance is unfounded. This raises the problem that if it is true that all objects, i.e., the physical world, and all statements/concepts/theories are empty then this statement itself must be empty as well. Nāgārjuna tries to get out of this by saying that the statement that all statements are empty, though it is a part of

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the Buddha’s teaching, is strictly speaking empty but that it is still useful in that it can demonstrate that objects and theories are empty. He wants to rescue the Buddha’s teaching and show in what way objects (the world) as well as statements (concepts, theories) can be empty without falling into the trap of nihilism. The prasaṅga technique occupies a central part of Nāgārjuna’s argument for emptiness. As mentioned above, the technique works along the following lines: take a hypothesis p and show that it leads to an unacceptable consequence. Then take the opposite hypothesis :p and show that it too leads to an unacceptable consequence. As a consequence, one can then assume that p and :p are empty. The Sanskrit term prasaṅga has different meanings. In this context, it is perhaps best translated as “(unacceptable) consequence.” The technique is supposed to show that all objects and all concepts/statements are empty by going through all the possible alternatives or cases and rejecting each of them. It will become clear shortly that simply rejecting a hypothesis and its negation is not enough for Nāgārjuna. His prasaṅga technique consists in setting out an exhaustive and exclusive list of the logically possible positions concerning some topic, and then showing that each in turn is untenable and therefore to be rejected.

What Are Nāgārjuna’s Four Alternatives? The law of the excluded middle states that, for any given proposition p, either p or :p. For example, it is either the case that dogs are mammals or it is not the case that dogs are mammals: there is no third option. The law of noncontradiction states that, for any given proposition p, it is not the case that both p and :p. If the proposition “dogs are mammals” is true then the proposition “it is not the case that dogs are mammals” must be false and vice versa. In the case of the law of the excluded middle the two options are exhaustive, which means that it is not possible for both of them to be false. In the case of the law of noncontradiction the two options are exclusive, which means that it is not possible for both to be true. At first sight, Nāgārjuna seems to reject these classical laws of the excluded middle and noncontradiction. He repeatedly claims that, if one wishes to have an exhaustive list, one must consider four, not two, exclusive possibilities. This is his doctrine of the “four alternatives” or catuṣkoṭi, usually translated as the “tetralemma.” Here are some examples from the Mūlamadhyamakakārikā: Neither from itself nor from another, Nor from both, Nor without a cause, Does anything whatever, anywhere arise. (MMK 1.1) Everything is real and is not real, Both real and not real, Neither real nor not real. That is Lord Buddha’s teaching. (MMK 18.8)

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“Empty” should not be asserted. “Nonempty” should not be asserted. Neither both nor neither should be asserted. They are only used nominally. (MMK 22.11)

There have been different interpretations of Nāgārjuna’s claims. In what follows a few of them will be discussed in brief.

The “Superficial” Interpretation It looks at first sight as if Nāgārjuna’s tetralemma takes the following form: p v :p v (p &:p) v (:p &::p) There are, however, two problems with this interpretation: first, the third clause (p & :p) is a contradiction and hence always false. It is not a genuine alternative because Nāgārjuna is seen as endorsing the principle of noncontradiction: An existent and non-existent agent Does not perform an existent and non-existent action. Existence and non-existence cannot pertain to the same thing. For how could they exist together? (MMK 8.7)

So, this interpretation allows for an alternative that violates the law of noncontradiction while Nāgārjuna, at least at times, seems to support this law. The second problem with this interpretation is that the fourth clause is equivalent to the third (rule of double negations) which means that if one is true then so is the other. This means that the alternatives are not exclusive.

Robinson’s Interpretation Robinson (1957) replaces the propositional analysis with one that takes each clause to have a quantified subject. This is motivated by examples such as the one about origination/causation (“everything is self-caused,” “everything is other-caused,” etc.). He reads the four clauses in the following way: 8x Fx _ 8x :Fx _ (∃x Fx & ∃x :Fx) _ (:∃x Fx & :∃x :Fx) Robinson tries to avoid a contradiction by interpreting the third clause, not as claiming that everything is both F and :F, but by claiming that some things are F and other things are :F. But what about his fourth clause? This is now logically equivalent to “everything is both F and :F” because “:∃x Fx” is equivalent to “8x :Fx” and “:∃x :Fx” is equivalent to “8x Fx.” So, by way of example, “there does not exist a man who is wise” is equivalent to “all men are unwise” and “there does not exist a man who is unwise” is equivalent to “all men are wise.” Therefore, the fourth clause can be read as “‘all things are F’ and ‘all things are not F’” which is a contradiction. To say that whatever exists has contradictory properties is to say that

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nothing exists, so the fourth alternative means, on Robinson’s interpretation, that the domain of quantification is empty. However, Jayatilleke (1967, p. 75) points out that one must take the universal quantifiers in the first and second clauses to have existential commitment, because otherwise the four alternatives are not exclusive. For example, if nothing exists then both the first and the second alternatives would be true. If there are no men then it is true that “all men are wise” and it is also true that “all men are unwise.” This is due to the fact that the statement “all men are wise” can be analyzed as “for all x, if x is a man x is wise.” This is a conditional and a conditional is true if its antecedent is false. So, Robinson’s reading of MMK 1.1 is: either things exist and are self-caused, or things exist and are all other caused, or things exist, some being self-caused and others other caused, or nothing exists. These alternatives are exhaustive and exclusive. The problem with this reading is that it does not fit the text very well. The text seems to postulate the existence of a single subject of predication in the third and fourth alternatives (“Everything is real and is not real, both real and not real, neither real nor not real. That is Lord Buddha’s teaching.” (MMK 18.8)). Nor can this reading deal with cases which have a nonquantified subject. For example, either Mount Everest is over 8000 m high or it is not. There are no third and fourth alternatives.

Jayatilleke’s Interpretation Jayatilleke’s (1967) interpretation seems to avoid many of the problems of the other interpretations. Consider the following example: 1. 2. 3. 4.

A is east of B, or A is west of B, or A is both east and west of B, or A is neither east nor west of B.

Alternative (3) is only apparently contradictory, for suppose that A is a line running east-west through B. Then it is both east and west of B, in the sense that part of it is east of B and part of it is west of B. Alternative (4) on this account does not entail that neither A nor B exist, for suppose that A is a line lying above B. Jayatilleke’s reading is: x is wholly F v x is wholly :F v (x is partly F & partly :F) v (x is neither F nor :F) It is important that one takes F and :F to be contraries rather than contradictories (as “west” is not “anything other than east” but “the direction opposite to east”) for otherwise it would be impossible to read the fourth alternative as saying that x is a thing to which neither F nor :F applies. Another example will clarify the point: Either x is wholly white (e.g., chalk) or x is wholly black (e.g., charcoal) or x is partly white and partly black (e.g., a zebra) or x is neither white nor black (i.e., it is something to which other color predicates apply, e.g., the blue sky).

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MMK 1.1 now reads: every object is either wholly self-caused, wholly othercaused, partly self- and partly other-caused, or neither self- nor other-caused (=noncausal/abstract, e.g., a universal or purely accidental object (yādṛcchika), occurring without any cause.). This interpretation seems to make most sense of the tetralemma. It is important to point out that that if the adverbs “wholly” and “partially” are read as involving quantification over parts, this proposal has the same logical form as Robinson’s. The improvement however lies in what is being quantified over (parts rather than objects). Note that this interpretation of the four alternatives entails no revision of classical two-valued logic. “Two-valued” means having the truth-values true and false only. So, according to this interpretation, Nāgārjuna has not rejected the law of the excluded middle. He has only extended it to four alternatives instead of two.

How Is It Possible to Reject All Four Alternatives? Nāgārjuna formulates the exclusive and exhaustive list of logical possibilities only in order to reject each and all of them. That is the essential point of a prasaṅga, a dialectical destruction of a theoretical position. However, it seems paradoxical, for if “1 v 2 v 3 v 4” is a logical truth then the rejection of all four, i.e., “:1 & :2 & :3 & :4” is equivalent to “:(1 v 2 v 3 v 4)” and hence a contradiction. So, the rejection involved cannot be a straightforward propositional negation. Nāgārjuna has to explain how he “rejects” philosophical theories without having a theory himself. If he thought of himself as proposing a theory, that theory would have to be empty as well. This means that Nāgārjuna’s rejection of theories cannot be the same as the assertion of the negation. Here are two ways in which one might try to explain what sense Nāgārjuna gives to the notion of rejection: 1. Matilal (2005) draws attention to the well-entrenched Indian distinction between “nominally bound” (paryudāsa) and “verbally bound” (prasajya-pratis´edha) negation. The first is like the “in-” as in “snails are invertebrates” – it turns one predicate or noun into another, namely its complement. The second kind of negation is a type of illocutionary negation like the “not” in “I do not say I will go.” The important difference is that whereas the use of a nominally bound negation still involves an act of assertion (“it is a vertebrate” and “it is an invertebrate” both make an assertion), the use of a verbally bound or illocutionary negation is a simple rejection without assertion (“I do not say I will go” does not entail “I say I will not go”). If one interprets Nāgārjuna as invoking illocutionary negation, then failing to assert some proposition p is not at all the same as asserting :p. So, it is possible for Nāgārjuna to withhold assent from all of the four alternatives “1 v 2 v 3 v 4” without asserting “:1 & :2 & :3 & :4” Nāgārjuna is portrayed as someone who refuses to be drawn when confronted with any of the alternatives. Withholding assent individually from 1 and 2 and 3 and 4 is consistent with assenting to the disjunction “1 v 2 v 3 v 4.” Nāgārjuna might just be incapable of making up his

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mind. However, this does not capture the sense in which rejecting all the alternatives leads to the claim that all theories are “empty.” For on this interpretation, one alternative might not be empty but Nāgārjuna does not decide which one. 2. The second way of making sense of Nāgārjuna’s term of rejection is the following: some philosophers (e.g., Peter Strawson) argue that sentences containing nonreferring singular terms, such as “The present king of France is bald,” are neither true nor false, because in order for a predication to be significant, the object has to have been singled out for further description. Perhaps Nāgārjuna’s rejection of the four alternatives with regard to some concept F indicates that the presupposition behind the four alternatives, namely that F refers to something, is false. Consider again MMK 1.1. The disjunction of the first three alternatives amounts to the claim that objects have some sort of cause, while the fourth alternative denies that this is so. In saying that the sentence “Objects have causes” is neither true nor false, Nāgārjuna is highlighting the fact that the presupposition behind the sentence, namely that “cause” refers to something, is false. Hence by rejecting each of the four alternatives, he demonstrates that the concept of a cause is “empty.” It is only at this point, when each of the four alternatives is rejected, that Nāgārjuna abandons the law of the excluded middle. In fact, this interpretation takes Nāgārjuna’s “rejection” to be like the “internal negation” in a three-valued logic. This means that a statement is negated, not because it is false but because the truth-value “undetermined” applies to it. It is negated because it cannot be asserted to be true, not because it is false. So the tetralemma and the prasaṅga technique are supposed to show that concepts and objects are empty. The question is, do they really show this? On the one hand, they seem to be successful in establishing that all objects and concepts are empty because they provide four exhaustive alternatives and show that none of them apply to these objects and concepts. However, there is one general problem with this idea which applies to all views that test the limit of thought in this way: the problem is that as long as a theory is used to convey a point, one has to assume some common ground between the speaker and the listener or reader. In the case of the prasaṅga technique, the common ground is that if one can reject all four alternatives, a theory is empty. So the question is why should one be concerned with logical alternatives if concepts, including the concepts of logic, are empty? If logical concepts are themselves empty then rejecting all four alternatives does not show anything. This is a general problem for any view that challenges the concept of theories on the basis of argument. In order to talk about the theories one wants to reject, one has to employ some form of argument and hence one has to have some agreement about what constitutes a good argument. In Nāgārjuna’s case, the prasaṅga technique is supposed to provide a reason for accepting that all theories/concepts/statements are empty. The technique itself relies on the notions of logically exclusive and exhaustive alternatives and the idea that a rejection of all of these alternatives is supposed to show a problem with any of these views. In order to do so, however, Nāgārjuna has

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to rely on certain assumptions which are expressed as statements and theories. It is possible for Nāgārjuna to argue that these are empty as well but he then has to provide some reason for why this constitutes an important insight. Nāgārjuna is trying to show that rational discourse does not yield the results one thinks it does, namely nonempty theories/concepts/statements. However, in order to show this rationally he is relying precisely on the audience accepting theories of rational argumentation as nonempty. In the end, what he can hope to gesture toward is an internal contradiction within the claim that objects/theories/statements are nonempty.

Din˙nāga on Inference This section will discuss Diṅnāga’s (fifth–sixthcentury CE) theory of inference, which is one of the main contributions to logic by any Buddhist thinker. It is worth noting, however, that his rules of inference were a criticism of the Nyāya syllogism which can be traced back to the Nyāyasūtras. In this text the nature of a good inference of the form “p has s because it has h” (or, “p is s because it is h”) is discussed. Subsequent writers of this and other philosophical schools tried to improve on this syllogism in their commentaries and criticisms of the Nyāyasūtras. According to the Nyāyasūtras (1.1.32) a good inference consists of five steps: (i) the statement of the thesis (pratijn˜ ā); (ii) the reason for the thesis (hetu); (iii) a general rule plus an example (udāharaṇa); (iv) the application of the rule to the case in question (upanaya); (v) the restatement of the thesis or conclusion (nigamana). This is best illustrated by citing an inference. One standard example in Indian logic is “Wherever there is smoke, there is fire.” The inference goes as follows: (i) (ii) (iii) (iv) (v)

There is fire on the hill. For there is smoke. Wherever there is smoke, there is fire, as in the kitchen. This is such a case (smoke on the hill). Therefore, there is fire on the hill.

The hypothesis to be tested is (i). The data on which the inference is partly based is (ii). Step (iii) is a universal generalization. Step (iv) is not mentioned in western accounts of inferences. It serves to show that one is observing the proper inferential sign. Step (v) is the conclusion which in a good inference is the hypothesis of step (i). The Nyāyasūtras (1.2.4–8) mention several cases in which this inference might go wrong. Among those, one is called “deviating” (savyabhicāra). This is the case where the inferential sign does not warrant the hypothesis. If for example somebody wanted to infer the presence of smoke from the presence of fire the inference would deviate because there are cases where there is fire without smoke. A sign would be called “contradictory” (viruddha) if it proves the opposite of the conclusion. Thus a pool of water is normally a sign for the absence of fire. Another bad

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inference is the case where the universal generalization is not established. A nonestablished generalization cannot count as evidence for the hypothesis. An important requirement is that the generalization is not “counter-acted” (prakaraṇasama). This means that it is not supposed to be countered by another generalization showing exactly the opposite. The fallacy of petitio principii is called “unproved” (sādhyasama). In Chap. 2 of the Pramāṇasamucaya the Buddhist philosopher Diṅnāga claimed that the five step syllogism of the Naiyāyikas could be stated in the form of three conditions. Thus the form of logical inference is replaced by general criteria which have to be fulfilled for an inferential sign to be counted as evidence. In order to state these conditions, Diṅnāga introduces a few key concepts. He takes the concepts of hetu (inferential sign) and sādhya (thesis-to-be-inferred) from the Naiyāyikas. He adds the concepts of pakṣa (locus of inference), sapakṣa (homologue or place which is known to possess the sādhya already), and vipakṣa (heterologue or place which is known not to possess the sādhya). With the help of these concepts, he formulates three conditions for a good inferential sign. The three conditions are: 1. The inferential sign should be present in the case under consideration. 2. It should be present in at least one homologue. 3. It should not be present in any heterologue. In the case of the inference concerning smoke and fire, the first condition states that smoke has to be present on the mountain. The second condition states that it has to be present in a similar case, such as the kitchen. The third condition says that smoke has to be absent in dissimilar cases, such as the lake. One might object to this account of reasoning that the second condition is redundant. It says that among all the places which have fire, there has to be at least one that also possesses smoke. This means that within the class of fiery objects it is true that “Wherever there is smoke there is fire.” It is, however, possible that there is a place where there exists smoke without fire. This is why the third condition, which says “Wherever there is no fire, there is no smoke,” is needed. This can be rendered as: a) (8x)(:Fx ! :Sx) Given this condition, the case of a smoky place without fire is ruled out. If this condition holds then not only is it true among the class of fiery objects that “Wherever there is smoke there is fire” but it becomes generally true. Thus the second condition can be formalized as: b) (8x)(Sx ! Fx) Clearly, b) is the contrapositive of a) which means that they are logically equivalent. It would therefore seem that condition (1) and (3) are sufficient to account for a good inference. It is however possible that Diṅnāga introduced the

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second condition in order to account for a logical problem. He divides the universe into pakṣa, sapakṣa, and vipakṣa. It is important that a sapakṣa is a case different from the pakṣa. The second condition ensures that a universal generalization of the form (8x)(Fx ! Gx) is not made true by there being no Fs apart from the one under investigation. If there were no Fs apart from the pakṣa then the generalization (8x) (Fx ! :Gx) would be true as well which is to be avoided. The second condition also ensures that there are examples outside the pakṣa which support the inference. Diṅnāga might have been motivated by the same idea as Mill (1843) who thought that the inference, Fa (8x)(Fx ! Gx) Therefore, Ga begs the question. The general rule already implies that the pakṣa is an instance of the rule, which is to be established. However, this also means that the real inference is not strictly deductive. For if the general rule says that every smoky place other than the mountain has fire, then the mountain might be the exception although that is unlikely. One important question which is raised by Diṅnāga’s discussion and which has applications for discussions of explanation in western philosophy is whether his method of inference implies a deductive or inductive argument pattern. At first sight, it seems that the pattern is deductive. One can use the membership and classinclusion relations of set-theory in order to illustrate this point. The inference “p has S, because it has H” can be rendered as: a) p  H b) H  S Therefore, p  S Diṅnāga, however, is not just concerned with the validity of an argument but also with the soundness of its premises. His three conditions are supposed to indicate under what circumstances it is rational to make certain inferences. On his account, not every H has to be included in S. It suffices if every known H is included in every known S. If K(h) stands for the class of objects which are known to be h then premise b) can be rendered as follows: b*) K(h)  K(s) This is clearly an inductive element in his account. It seems to be Diṅnāga’s intended meaning of the second and third conditions because the homologues and heterologues in his scheme are really the known cases which possess sādhya and which do not possess sādhya, respectively. The idea is that if one never observes an exception to a rule, then one is allowed to infer that the rule holds generally.

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But this creates a discrepancy in Diṅnāga’s account because although a) p  H b) H  S Therefore, p  S is a valid form of argument, a) p  H b*) K(h)  K(s) Therefore, p  S is not. In his work Hetucakraḍamaru (Hetucakranirṇaya), Diṅnāga argues that there are nine possible inferences to be drawn from his three conditions, of which only two are good inferences. He argues that the inferential sign might be present in all, some, or none of the class of homologues. Likewise it might be present in all, some, or none of the class of heterologues. The two good inferences are (i) when the inferential sign is present in all of the homologues and none of the heterologues and (ii) when the inferential sign is present in some of the homologues and none of the heterologues. Diṅnāga’s doctrine was criticized by the Naiyāyika Uddyotakara who points in his text Nyāyavārttika that either the class of homologues or the class of heterologues might be empty. One example, which is mentioned by Matilal (1990), would be “cows are distinct from noncows because they have dewlap.” There are no objects which possess “cowhood” apart from cows. Thus the class of homologues is empty. Yet, the inference indicates that the property “having dewlap” distinguishes cows from noncows. Uddyotakara’s extension of the wheel of reason allows therefore for three additional inferences: (i) the class of homologues is empty and there are no heterologues, (ii) the inferential sign is present in some homologues and the class of heterologues is empty, (iii) the inferential sign is present in every homologue and the class of heterologues is empty.

Dharmakīrti on Din˙nāga on Inference The Buddhist logician Dharmakīrti (seventh century CE) tried to clarify and improve Diṅnāga’s account of inference. In his Hetubindu, he pointed out that Diṅnāga’s conditions are not clearly stated. The problem with Diṅnāga’s formulation of the issue is that in the case of the pakṣa being a class, it is not clear whether the inferential sign has to be present in every member of this class or only in some. The same problem arises for the class of homologues. Does the inferential sign have to be present in every member of the class or only in some? Dharmakīrti resolves this problem by introducing the particle eva, meaning “only, exactly, really.” Hence the three conditions are restated as follows:

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1. The inferential sign should be really (eva) present in the case under consideration (which means it should be present in the whole, not just in a part). 2. Among the remaining cases, it should be present only (eva) in homologues. 3. It should not at all (eva) be present in any heterologue. With this formulation the inferential sign has to be present in every member of the pakṣa class. This was not clearly stated by Diṅnāga. Moreover, from the remaining cases, it should be present only in members of the class of homologues and never in any member of the class of the heterologues. This permits the existence of the sign in some homologues and not all. At the same time, it excludes the possibility of the existence of the sign in any heterologue whatsoever. The restatement of the second and the third conditions was implicit in Diṅnāga’s hetucakra (which, as mentioned above, gives a ninefold classification of inferences) because for him both inferences, one in which the sign exists in a part of the sapakṣa and one in which it exists in the whole of sapakṣa, are good inferences. Dharmakīrti, however, makes this explicit by his reformulation of the three conditions. In addition, Dharmakīrti inserted the word nis´citam (determinately, necessarily) in his formulation as applicable to all the three conditions (5.12). By this he indicated that the existence of the sign in homologues and its nonexistence in the heterologues is not accidental but rather a lawlike generalization. The next question was, under what conditions the generalization in question could be deemed lawlike. Dharmakīrti specified three such conditions, which provide three types of hetu. The second condition is that of natural reason (svabhāva-hetu). This comprises what is often called metaphysical necessity. An example of an inference would be “this is a mammal because it is a tiger.” The relation between being a tiger and being a mammal is not causal but it is not accidental either. The first condition is called reason qua effect (kārya-hetu). This condition applies if the sign is an effect of the property which is to be inferred. The inference “the mountain has fire because it has smoke” is an example of this kind of logical reason. The idea is that if the causal relation holds between the reason and the inferred property, then the smoke would not occur unless the cause occurred. The cause is necessary for its effect. The final condition is reason based on nonobservation (anupalabdhi-hetu). This is comparable to counterfactual statements. An inference of this type would be “there is no pot here because it is not observed.” The idea is that if there were a pot here it would be observed. Naturally, nonobservation of an object can imply its nonexistence only if the object is perceptible by nature. In other words, it is an object about which it can be asserted that “if it were there, it would have been perceived.” (For example, nonexistence of air in the room cannot be inferred on the basis of its nonobservation.) Dharmakīrti expressed this condition by the word upalabdhilakṣaṅaprāpta (observable, that which fulfills the condition of observability). Hence nonobservation as hetu was qualified as “nonobservation of the observable.” Dharmakīrti noticed that observation of a contrary object has the same epistemic role as nonobservation of the required object. For example, experience of excessive cold in a room also implies that there is absence of fire. Hence

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Dharmakīrti treated “observation of the contrary” (viruddhopalabdhi) as a kind of nonobservation. By combining this wider concept of nonobservation with other conditions he prepared a list of eleven types of nonobservation in his work Nyāyabindu.

Meaning as Exclusion: apoha Diṅnāga’s theory of the meaning of terms as presented in his Pramāṇasamuccaya draws on the hierarchical account of concept application. This means that concepts are organized by the relations of exclusion and containment, for example, “tree excludes pot” or “tree contains beech.” Historically, philosophers have provided different answers to the question of what this says about the meaning of words like “tree,” “pot,” or “beech.” The most obvious seeming answer is that the conceptual hierarchy corresponds to a real hierarchy in nature. The world itself instantiates a certain natural organization and conceptual classifications are not “invented” but stand for natural categories. This view is called “realism” in Western philosophy. According to this view, words such as “tree” denote things on the basis of being members of a real category of things, namely those which belong to the natural kind “tree-hood.” As a Buddhist, Diṅnāga rejects this realist view and develops a theory of meaning entitled apoha (exclusion). The idea behind this theory of meaning is that the conceptual hierarchy holds itself together functionally, without resting on the world. Diṅnāga’s idea is that concepts and words have content, not because they correspond to external objects, but because of their location in the hierarchy of concepts. Consider a familiar example: people often form themselves into groups, not on the basis of something they have in common, but rather in terms of what they reject. For example, a clique can be united only by its not allowing in anyone who is poor. Likewise, it is a familiar point that thinking of oneself as, for example, American is more a matter of who is excluded than of any unifying common feature. In this way, a concept can gather together a collection of people on the basis of who is excluded rather than by any common property. Now consider a word/concept like chair. Again, one would not want to say that “being-a-chair” is a natural property of the world. Nor could one find a property or set of properties that all chairs have in common. What would such a property be? Having four legs? Having a back? None of these are necessary or sufficient to account for a chair. Diṅnāga says that what makes something a chair is rather its ability to fulfill a certain function (“being-sittable-on”) which can be fulfilled by objects of vastly different shapes and materials. This means that a chair is defined by its relations with other things (human bodies). The idea is that all concepts are defined by their place in the hierarchy, by their relations with other concepts, of which exclusion is the most important. What is the relation between the term “lotus” and the term “blue lotus”? Diṅnāga’s answer is that the second term excludes more things than the first. The first excludes only nonlotuses while the second excludes both nonlotuses and nonblue things. So, as one moves down the hierarchy one

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arrives at concepts which exclude more and more things. But Diṅnāga claims that one never arrives at a concept so finely grained that it excludes everything except a single particular. Particulars are inexpressible. This means that words do not refer to objects. Instead, they exclude sets of objects.

The Case for apoha Why does Diṅnāga think that this is the right way to analyze meaning? He argues that the meaning of a word conveys knowledge to the hearer or reader. So the question is what does knowledge consist in and how is it conveyed? In order to answer this question, one has to know what the meaning of a word consists in. Diṅnāga goes through an exhaustive list of candidates for the meaning of a term and rejects each of them. These candidates are the following: 1. An individual or particular. The meaning of tree could be a particular tree that the speaker refers to. 2. A universal. The meaning of tree is the universal tree-hood. 3. A relation between the universal and a particular. The meaning could be this particular instance of a relation between the particular and the universal. 4. A thing possessing a universal. This is the idea that a particular instantiates a universal and that this instantiated universal is the meaning of a term. Diṅnāga provides the following reasons for rejecting these candidates: 1. The meaning of a word is a particular. The idea is that there are sets of particulars like the set of all trees and that the word “tree” picks out a particular tree. However, Diṅnāga objects to this by arguing that the word “tree” does not just apply to one tree. It applies to all trees, i.e., to all members of the set. So the problem is that as long as one says that even though knowledge is gained about a subset of trees when one hears the word “tree,” nevertheless the word applies to all members of the set of trees. So what about saying that when one hears the word tree one gains knowledge of the whole set of trees? The problem with this is that it is clearly not the case. When one hears the word “tree” one does not thereby gain knowledge of all existent trees. So if the meaning of a word is a particular, then it is either excluding certain particulars or it is including too many. Hence Diṅnāga concludes that particulars cannot be the meaning of a word. 2. The meaning of a word is a universal. Take the example to the standard inference “The mountain has fire, because it has smoke.” The meaning of the word “fire” in this example is not a universal. When one hears this sentence the word “fire” does not provide knowledge of a universal but of a particular fire. So the universal cannot be the meaning of a word either. 3. The meaning of a word is the relationship between a universal and a particular or between universals. This interpretation raises problems with regard to the meaning of sentences. When you hear the sentence “The tiger has stripes,” you might

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think that this sentence expresses the relationship between the universal’s tigerhood and stripeness. The universal tigerhood falls within the universal stripeness but not vice versa because all tigers are striped but not all striped things are tigers. However, the meaning of this relationship is not expressed by the words “tiger” and “stripes” in itself. The string of words “tiger, stripes” does not express the relationship between the two universals. Likewise, if you had a particular tiger called Bob and you said “Bob has stripes” the two words “Bob” and “stripes” do not express the relationship between the particular and the universal per se. The relationship is expressed by grammatical rules but not by the individual words. 4. The meaning of a word is an object possessing a universal or the idea that particulars instantiate universals. This apparently solves the problem of the meaning of sentences like “The tiger has stripes.” According to this view, the meaning of the terms “tiger” and “stripes” is the particular locus, i.e., the particular animal where tigerhood and stripeness occur. So the animal in question is the intersection of tigerhood and stripeness. Diṅnāga rejects this view on the following grounds: he argues that there are two ways in which one can interpret the idea that the meaning of a word is a particular instantiating a universal. First, one can think of it as a particular. The idea is that the animal is the particular place of the universals tigerhood and stripeness. But this raises the same problem as in the previous case when the meaning of a word could not be a particular because it picks out every particular under its extension. Secondly, one can interpret this hypothesis by way of saying that a word does not express a particular place of a universal but the general fact of a universal being present. This fact is common to all the places where a universal occurs. But this means that the word either expresses a universal itself, namely the universal of “universal-presence,” or it expresses the relation between a universal and its place of occurrence. However, as was demonstrated before, there are problems with the view that a word expresses a universal or a relation. The issue is that if one says that a particular instantiates a universal then one is talking about a certain place. This leads to the problem that this place is either a particular or a universal. So since all of these candidates fail to account for the meaning of a word, Diṅnāga concludes that the meaning of a word is not some “positive” entity. Instead, it is the absence of the exclusion class from a certain place. Matilal (1986) gives the example of the term “cow.” Diṅnāga argues that in order to find out what the meaning of the term cow is one would have to define the exclusion class. This goes back to his theory of inference where he divides the universe into homologues and heterologues. The exclusion class is all those things that are noncows. Matilal calls the class of noncows “woc.” One can then say that “no cow is woc.” The problem is whether or not this is a positive property that is attached to cows. Matilal suggests that one has to treat the property “woc” as simply saying what a cow is not without thereby attributing any positive property to a cow. It is helpful to refer back the concept of illocutionary negation which was mentioned in connection with Nāgārjuna’s prasaṅga technique. One interpretation

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of this technique was that all four possibilities are negated in the sense of saying “I did not say that x.” In saying so, the speaker does not make a statement. The same might be said here. All that is expressed by the term “woc” might be that it is not a cow. It does not express anything positive. Just like referring to somebody as “non-American” does not say anything specific about that person’s nationality. All it says about a person is that the person does not belong to the class of American people. Matilal (1986, p. 400) mentions the following example: suppose the universe was made up of only six things: three cows, two horses, and one cat. For each individual animal there is one corresponding mental representation. Each mental representation has an exclusion class. One can say that the three mental representations of a cow and the two mental representations of a horse are only tokens of the same type. Diṅnāga wants to say that the reason that the three mental representations of a cow are tokens of the same type is not that they share certain features but the fact that they have the same exclusion class which is “horse, horse, cat.” The general criticism of the apoha theory has focussed on two questions: (1) In what way do sentences about members of the exclusion class refer to members of the original class? (2) How can one identify the exclusion class without first identifying the original class? Discussing these questions would go beyond the scope of this chapter. In general, the motivation for arguing that the meaning of a word is its exclusion class is that it does not require the existence of universals. One common answer that Diṅnāga had mentioned is that the meaning of a term such as cow might be the reference to its universal cowhood. But this means that for every general term there has to be a universal. Even realists about universals, such as the Naiyāyikas, do not believe that every general term refers to a universal. One can also find this in Plato. Plato believed that there are universals and he was quite happy to accept that there was a universal of beauty or gold. But he did not want to admit that there was a universal for dirt, for example. His explanation is that dirt is simply the absence of cleanliness. So it is the absence of the universal for cleanliness. Diṅnāga on the other hand argues that if not every general term refers to a universal why should one believe that some do? What is special about them? His answer is that there are no universals and that the meaning of a term is its exclusion class. So given that there are problems with universals and particulars as the meaning for words the apoha doctrine seems to present an interesting alternative, in spite of its own problems.

Header Summary • Introduction: Buddhist logic. • Nāgārjuna’s prasaṅga technique: this part deals with Nāgārjuna’s prasaṅga technique and his claim that reality is empty (s´ūnya). • Diṅnāga on Inference: the second part provides an overview of Diṅnāga’s view on inference and his theory of the three conditions of an inferential sign (hetu).

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• Dharmakīrti on Diṅnāga on Inference: the third part explores Dharmakīrti’s developments of Diṅnāga’s view on inference, particularly his introduction of the particle eva and the three types of hetu. • Meaning as Exclusion: apoha: the final part of this chapter presents an outline of Diṅnāga’s apoha (exclusion) theory of the meaning and function of concepts and words.

References Primary Sources Hetubindu by Dharmakīrti (texts and English translations): Gokhale, P.P. 1997. Hetubindu of Dharmakī rti (a point on probans). Delhi: Sri Satguru Press. Steinkellner, E. 2016. Dharmakī rti’s Hetubindu, critically edited by Ernst Steinkellner on the basis of preparatory work by Helmut Krasser with a translation of the Gilgit fragment by Klaus Wille. Vienna: Verlag der Österreichischen Akademie der Wissensschaften. Hetucakraḍamaru (Hetucakranirṇaya) by Diṅnāga (text and English translation): Chi, R.S.Y. 1969. Buddhist formal logic: A study of Dignāga’s Hetucakra and K’uei-chi’s great commentary on the Nyāyapraves´a. London: The Royal Asiatic Society of Great Britain. Mūlamadhyamakakārikā (MMK) by Nāgarjuna (texts and English translations): Garfield, J.L. 1995. The fundamental wisdom of the middle way. Oxford: Oxford University Press. Inada, K. 1970. Nāgārjuna: A translation of his Mūlamadhyamakakārikā with an introductory essay. Tokyo: The Hokuseido Press. Kalupahana, D. 1986. Mūlamadhyamakakārikā of Nāgārjuna: The philosophy of the middle way. Delhi: Motilal Banarsidass, repr. 2006. Siderits, M., and S. Katsura. 2013. Nāgārjuna’s middle way. Somerville: Wisdom Publications. Streng, F. 1967. Emptiness: A study in religious meaning. Nashville: Abingdon Press. Nyāyabindu by Dharmakīrti (texts and English translations): Gangopadhyay, M. 1971. Vinī tadeva’s Nyāyabinduṭī kā. Calcutta: Indian Studies Past & Present. Stcherbatsky, Th. 1930. Buddhist logic. Vol. 2. Delhi: Motilal Banarsidass, repr. 2004. Nyāyasūtra (NS) (texts and English translations): Gangopadhyay, M. 1982. Gautama’s Nyāya-Sūtra with Vātsyāyana’s Bhāṣya. Calcutta: Indian Studies Past & Present. Jha, G. 1912–19. The Nyāya-Sūtras of Gautama, 4 Vols. Delhi: Motlal Banarsidass, repr. 1984 (includes a translation of the Nyāyavārtikka by Uddyotakara). Pramāṇasamuccaya by Diṅnāga (texts and English translations): Hayes, R. 1988. Diṅnāga on the interpretation of signs. Vol. 9, Studies of classical India. Dordrecht: Kluwer. Steinkellner, E. 2005. Dignāga’s Pramāṇasamuccaya, Chapter 1: A hypothetical reconstruction of the Sanskrit text with the help of the two Tibetan translations on the basis of the hitherto known Sanskrit fragments and the linguistic materials gained from Jinendrabuddhi’s Ṭī kā. http://www. ikga.oeaw.ac.at/Mat/dignaga_PS_1.pdf, with revisions of 2014: http://www.ikga.oeaw.ac.at/ mediawiki/images/f/f3/Dignaga_PS_1_revision.pdf

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Vigrahavyāvartanī (VV) by Nāgārjuna (text and English translation): Bhattacharya, K., E.H. Johnston, and A. Kunst. 1978. The dialectical method of Nāgārjuna: Vigrahavyāvartanī . Delhi: Motilal Banarsidass, repr. 2002.

Secondary Literature Jayatilleke, K.N. 1967. The logic of four alternatives. Philosophy East and West 17: 69–83. Matilal, B.K. 1986. Perception: An essay on classical Indian theories of knowledge. Oxford: Clarendon Press. Matilal, B.K. 1990. The word and the world. Delhi: Oxford University Press. Matilal, B.K. 2005. Epistemology, logic and grammar in Indian philosophical analysis. Edited by J. Ganeri. New Delhi: Oxford University Press. First published in 1971 by Mouton, The Hague. Mill, J.S. 1843. A system of logic, in his Collected works. Vols. VII and VIII. Edited by J.M. Robson. London: Routledge, 1973, repr. 1996. Robinson, R.H. 1957. Some logical aspects of Nāgārjuna’s system. Philosophy East and West 6: 291–308.

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Logic of Syād-Vāda Anne Clavel

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Development of the Sapta-Bhaṅgī . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Denominations: Syād-Vāda and Sapta-Bhaṅgī . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . External Influences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Plurivocal Doctrine . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Structure of the Sevenfold Predication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Number of Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Combinations of the Predicates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Necessity of Seven Predications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Beyond Inadequate Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Syād-Vāda and Logic Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Meaning of Syāt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contradiction and Tautology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The syād-vāda (“doctrine of syāt”), also called “sevenfold predication” (saptabhaṅgī ), constitutes a cornerstone of the Jaina doctrine of multilateralism, since it prevents a predicate from being attributed absolutely to a subject. According to Jainism, in order to consider a single state of affair as exhaustively as possible, one has to submit every predicative relation between a subject and a predicate to a structural rule made up of seven propositions, which are not conceived of as alternative truths but are all endowed with the same truth-value. A thorough under-

A. Clavel (*) Department of Philosophy, University of Lyon, Lyon, France © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_16

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standing of the syād-vāda raises tricky issues, such as the meaning of the adverb syāt in the context of this logical pattern, the compatibility of Jaina logic with basic logical principles, or the oddness of the number of propositions – why seven?

Introduction It is well known that one of the most emblematic features of Jaina philosophy is to give a faithful account of reality without excluding any possible perspective. By regarding the teachings of each of its philosophical adversaries as constituting a single, possible viewpoint among several others, Jainism proposes a conceptual synthesis that encompasses all other doctrines, the materialistic thesis as well as the monistic theory developed by Advaita-vedānta or the Buddhist theory that everything is momentary. Ontological tenets are consistent with such a purpose, inasmuch as Jaina philosophers defend the doctrine of the multiplex nature of reality (anekānta-vāda): Jaina philosophy aims at taking into account two complementary components of being, permanence and transitoriness, in other words the capacity of a substance to endure changes. Such a realistic ontology goes beyond the contradiction embodied in the Indian philosophical framework, by the opposition between the Buddhist theory of momentariness and the doctrine of immutability professed by Advaita-vedānta. Jainas recognize that all substances (dravya) in universe share a common structure combining consubstantial characteristics called properties (guṇa) and transitory modes (paryāya). Thus, each real thing is associated with two complementary aspects, permanence and transitoriness, without contradiction. Since each being is spoken of in terms of origination (utpatti), stability or permanence (dhrauvya, sthiti), and cessation (vyaya), a permanent substance can endure constant changes. Since no being can be reduced to only one aspect, reality must always be apprehended as multiplex (anekānta) and endowed with an infinite number of characteristics. This is why Jaina ontology is called the “doctrine of multiplexity of reality,” in contradistinction to all other systems which consider reality in a partial and unilateral way. Jaina thinkers see any other view as a “unilateral doctrine” (ekānta-vāda). Since matter and mind are not two realities absolutely different one from another, human faculties are not radically antagonist with reality but are rather adequate tools for its analysis. As a corollary of ontological theories, Jaina epistemology examines how things can be known through different procedures: on the one hand through the means of knowledge (pramāṇa), and on the other hand through two methods which are specific to Jainism, that aim at overcoming unilateral perspectives, namely viewpoints (naya) and sevenfold predication (sapta-bhaṅgī ). According to Jaina epistemology, the means of knowledge are intuitive or discursive operations that constitute the instrumental causes of knowledge. In its pristine purity, the soul knows everything, but its capacity is highly reduced by the hindering power of karman. Thus, when a soul endeavors to reach the final goal, namely liberation (mokṣa), its ascension necessarily implies reaching omniscience:

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the various stages of intellectual and spiritual progression, through which a soul is supposed to traverse in order to escape the cycle of rebirth, lead to a state where it regains its pristine purity back, especially its capacity to know everything perfectly. However, as long as a soul transmigrates, it must overcome the necessary limitations of its cognitive faculties by using the method of viewpoints and the sevenfold predication. The doctrine of viewpoints (naya-vāda) takes for granted that every judgment expresses only a part of reality; it depends on certain conditions, especially on the speaker’s intention, which makes it incomplete and partial. To choose to adopt a particular viewpoint which represents the most pertinent or efficient approach in a given circumstance does not imply an absolute rejection of all other possible viewpoints: the truth of an utterance does not imply that another utterance may be false. The different viewpoints share the same cognitive value: they provide a partial account of reality. Even though any object or situation can be accounted for from an infinite number of viewpoints, seven nayas are traditionally enumerated in Jainism, following an order that emphasizes a decreasing extension of their respective point of reference: they are namely the comprehensive viewpoint (naigama), the collective viewpoint (saṅgraha), the empirical viewpoint (vyavahāra), the instantaneous viewpoint (ṛjusūtra), the synonymous viewpoint (s´abda), the etymological viewpoint (samabhirūḍha), and the factual viewpoint (evaṃbhūta or itthaṃbhūta). The multilateral approach that features the naya-vāda is especially underlined by the structural distinction between substantial viewpoints (dravya-naya) – corresponding to the first three nayas – and modal viewpoints (paryāya-naya) – the last four nayas – a distinction which relies on the necessary coexistence of permanence and change in every existent thing. Whereas the naya-vāda can be seen as the analytical aspect of the doctrine of multiplexity of reality, its synthetical aspect is embodied by the sevenfold predication (sapta-bhaṅgī ). For, complete knowledge can be obtained if and only if the predicative content of a statement is subjected to the Jaina method called saptabhaṅgī . Without being false, a statement like “soul is existent” cannot claim an absolute truth because it is only a unilateral and partial judgment. However, the most complete and adequate description is reached when one considers the whole set of seven propositions that exhaust all possible predicative relations between the subject “soul” and the predicate “existent.” Following most of Jaina thinkers, this set of seven statements runs as follows: (S1) In some respect (syāt), soul is only (eva) existent. (S2) In some respect, soul is only nonexistent. (S3) In some respect, soul is existent and nonexistent. (S4) In some respect, soul is only inexpressible. (S5) In some respect, soul is existent and inexpressible. (S6) In some respect, soul is nonexistent and inexpressible. (S7) In some respect, soul is existent, nonexistent, and inexpressible.

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Historical Development of the Sapta-Bhan˙gī Two Denominations: Syād-Vāda and Sapta-Bhan˙gī Two denominations are concurrently applied to the set of seven statements: either syād-vāda (the “doctrine of syāt”) or sapta-bhaṅgī (the “sevenfold predication”). These two compounds share neither the same denotation nor the same connotation. Each term lays stress on a different feature of this logical pattern. The term saptabhaṅgī points out its structure by giving the number of propositions, i.e., seven (sapta-), and the status of each proposition: each bhaṅga is nothing but a part of the whole discourse, constituted by the set of seven statements, that can be held concerning the relation between a subject and a predicate; this is the meaning of the noun bhaṅga (literally “a portion” of a larger whole), derived from the root BHAJ- “to divide, to distribute.” The feminine ending –ī , that gives a collective value to this singular noun, underlies the fact that the seven statements are not alternative judgments but that each of them must be taken as an indefectible part of the whole. On the other hand, the compound syād-vāda draws attention to the fundamental presence of the particle syāt in each and every statement. This word is namely a cornerstone of the sevenfold predication, inasmuch as its presence is sufficient to make the coexistence of seemingly contradictory statements possible and consistent. Thus the terms syād-vāda and sapta-bhaṅgī are complementary, inasmuch as they highlight two fundamental aspects of this logical framework: the inclusion of the term syāt in each statement serves as a reminder that every bhaṅga is only one member of a larger whole, the seven constitutive parts of which throw light upon the multiplex nature of reality without bearing any contradiction. The contexts in which the words syād-vāda and sapta-bhaṅgī are used can also differ significantly. Whereas the term sapta-bhaṅgī is restricted to the logical framework constituted of seven propositions, the word syād-vāda can also be used in a broader sense (Nyāyavijayajī 2000, p. 328), as a synonym for the compound anekānta-vāda (the “doctrine of the multiplexity of reality”). In this wider meaning, it is then endowed with an ontological significance. This synonymity is expressed by several Jaina thinkers, e.g., Haribhadra, and it explains the fact that Samantabhadra calls anekānta-vāda this method of associating a pair of contrary features by predicating them from a real thing. In his ĀM Samantabhadra often repeats this point. The word sapta-bhaṅgī appeared as such quite late in the Jaina tradition, after the logical pattern of seven statements began to be elaborated. Balcerowicz (2015, p. 207) has shown that the existence of the word sapta-bhaṅgī was “not so well established or not so universally widespread among Jaina theoreticians even still around 500 CE.” Before the noun sapta-bhaṅgī itself appears for the first time, seeds could already be found in the form of the bahuvrīhi adjective compound saptabhaṅga (Pkt.: sattabhaṁgam, “to which the seven predications apply”), attested in Kundakunda’s Pan˜ cāstikāyasāra (PAS 14) and in Samantabhadra’s Āptamī māṃsā (ĀM 104), or in a periphrasis used by Siddhasena Divākara in his Saṃmatitarkaprakaraṇa (STP 1.41): sapta-vikalpaḥ vacana-panthaḥ (Pkt.: sattaviyyapo vayaṇa-paho), “a verbal procedure that consists of seven options.”

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Historical Process The earlier strata of the so-called Canon do not indicate a set of seven predications as they have been later formalized by Jaina logicians. The sapta-bhaṅgī in the form in which it is now dealt with by contemporary scholars who address its logical consistency results from a slow elaborative process that took place over many centuries. Various elements that were originally dispersed were progressively integrated in a single framework so as to form the logical pattern constituted of seven propositions that has now become one of the most stimulating issues in the field of Indian logic. A predication of a pair of two mutually contrary predicates to the same subject, sometimes associated with the adverb syāt, resulting in three propositions is attested in the Bhagavatī-sūtra (Pkt.: Viyāha-pannatti), the fifth aṅga of the Canon: x is A, x is non-A (henceforth :A), x is A and :A. Dixit (1971, pp. 25–27) adds that this very text already aims at escaping contradiction by mentioning the distinction between the viewpoint of the substance’s own properties, according to which the predicate is affirmed, and the viewpoint of alien properties, according to which the predicate is negated. No explicit mention of the syād-vāda can be found by Umāsvāti, neither in the sūtras of the TS – that provide a compendium written in Sanskrit of the Jaina tenets in accordance with the scriptures – nor in the commentary recognized as being genuine by the Śvetāmbara tradition only. However, the sūtra TS V.32 (“The non-emphasized (anarpita) aspect of an object is attested by the emphasized (arpita) one”) has sometimes been understood as an allusion to syād-vāda; this distinction between emphasized and nonemphasized aspects has then laid the ground to a widespread interpretation elaborated by later commentators in order to solve the problem of a seeming incompatibility of contradictory judgments. Kundakunda goes one step further in the systematization of the doctrine, inasmuch as he associates many features to it: he explicitly enounces the first four predications, mentioning once the particle syāt, alludes to possible combinations of the three predicates previously mentioned (existent, nonexistent, inexpressible), and associates this kind of reasoning to the bahuvrīhi compound saptabhaṅga (cf. PAS I.14 and PS II.23). Moreover, while they tried to explain and interpret the logical structure of the syād-vāda, especially in glosses and commentaries, many philosophers integrated an ancient set of four nikṣepas (extensively dealt with by Bhatt 1991) as the four parameters that have to be considered to determine the viewpoint according to which a property is or is not predicated of a subject, namely substance (dravya), place (kṣetra), time (kāla), and condition (bhāva). The introduction of the restrictive particle eva by the Digamabara Samantabhadra (c. 580–640?) is the last consequent innovation in the logical pattern. Thus, as Balcerowicz (2015, p. 206) brings it convincingly to the fore, “a mature theory of the sevenfold modal description took final shape by approximately the turn of the fifth and sixth centuries at the earliest.”

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External Influences The most important influence goes back to the sect of the Ājīvikas: as far as it can be deduced from the very few texts that have been preserved through indirect sources, various seeds of the Jaina sapta-bhaṅgī were already present in the Ājīvika doctrine (Basham 1981, pp. 274–275): the existence of three fundamental predicates, the fact that the third one results from the simultaneous association of the first two, the number of figures – seven – that is obtained from combining the three initial predicates. On the other hand, the introduction of the restrictive particle eva (“only”) attested for the first time by Samantabhadra may bear the influence of the Buddhist logician Dharmakīrti (Balcerowicz 2015, pp. 206–207).

A Plurivocal Doctrine Significant variations can be found in the formulation of the seven predications. Most Jaina philosophers use the particle syāt only once in each statement, be it a simple statement or a complex one. In contradistinction, later authors like Malliṣeṇa (thirteenth century) adopt a distributive construction and repeat the word syāt in complex statements (SVM 23.104–112, p. 143.3–11), twice when there are two predicates (S3, S5, S6) and three times in the proposition involving the three different predicates (S7). For instance, the propositions S3 and S7 are formulated as follows: (S3) In a certain sense, x is only existent and in a certain sense it is only inexistent (syād asty eva syān n^ a sty eva). (S7) In a certain sense, x is only existent, in a certain sense it is only inexistent and in a certain sense it is only inexpressible (syād asty eva syān n^ a sty eva syād avaktavyam eva). The presence of the particle eva may also be subjected to variations. Many Jaina authors (e.g., Akalaṅka or Prabhācandra) use this word only in propositions containing – linguistically speaking – one predicate, in other words in propositions S1 (“in a certain sense, x is only (eva) A”), S2 (“in a certain sense, x is only (eva) :A”), and S4 (“in a certain sense, x is only (eva) inexpressible”). In this perspective, the restrictive value of the adverb eva lays stress on the possibility in one of these three bhaṅga to predicate only one property to the concerned subject, excluding any other predicate. In contradistinction to this general trend, other authors like Vādirāja (cf. NViV ad NVi III.66, II p. 350.17–19), Malliṣeṇa (SVM 23.104–112, p. 143.3–11), and Abhayacandra (SVBh ad LT 62, p. 85.4–14) integrate eva in each predication, without distinguishing between those which are made up of one predicate only (S1, S2, and S4) from those which contain two (S3, S5, and S6) or three predicates (S7). In such a case, two alternative cases are attested, depending on the number of occurrences of the particle eva in complex propositions: only once in Vādirāja or Abhayacandra’s version but as many times as there are predicates by Malliṣeṇa.

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The third variation pertains to the order of propositions S3 and S4 that has wavered during ancient times (Dixit 1971, pp. 25–26). Such a hesitation is especially noteworthy when it occurs even by one and the same author, e.g., Akalaṅka. In his commentary on Umāsvāti’s Tattvārthādhigamastra, the predicate avaktavya (“inexpressible”) appears sometimes in the third predication (RVār ad TS IV.42, I p. 253.4–6 ) and sometimes in the fourth one (RVār ad TS I.6, I 33.17–18). Such an inversion is nevertheless devoid of real philosophical consequences inasmuch as the difference between propositions S3 and S4 does not concern the nature of predicates but only the way in which they are combined.

The Structure of the Sevenfold Predication A common structure can be brought out beyond the minor variations attested in the tradition. Fundamentally, the syād-vāda or sapta-bhaṅgī is a structural rule permitting a combination of two contrary predicates {A, :A} with one another in all possible ways, so as to consider one and the same state of affairs as exhaustively as possible. The whole set, which expresses all the possible ways to combine two contrary predicates, A and :A, is built on the following pattern (Model 1): Model 1 (S1) In some respect (syāt), x is only (eva) A. (S2) In some respect, x is only :A. (S3) In some respect, x is A and :A. (S4) In some respect, x is only inexpressible. (S5) In some respect, x is A and inexpressible. (S6) In some respect, x is :A and inexpressible. (S7) In some respect, x is A, :A, and inexpressible. Thus, every predicative relation between a subject (x) and a predicate (A) – even predicative relations obtained through the viewpoints (naya) or the means of knowledge (pramāṇa) – has to be submitted to this set of seven predications. Applied to the assertion “soul is existent,” the sevenfold predication runs as follows (Model 2): Model 2 (S1) In some respect (syāt), soul is only (eva) existent (syād asty eva jī vaḥ). (S2) In some respect, soul is only nonexistent (syān n^ a sty eva jī vaḥ). (S3) In some respect, soul is existent and nonexistent (syād asti ca n^ a sti ca). (S4) In some respect, soul is only inexpressible (syād avaktavya eva jīvaḥ). (S5) In some respect, soul is existent and inexpressible (syād asti c^ a vaktavyas´ ca). (S6) In some respect, soul is nonexistent and inexpressible (syād n^ a sti c^ a vaktavyas´ ca). (S7) In some respect, soul is existent, nonexistent, and inexpressible (syād asti ca n^ a sti c^ a vaktavyas´ ca).

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In such a case, the verb asti is not to be understood as a mere copula, but it is endowed with an existential value, since it also contains the predicate “existent.”

The Number of Predicates While it is obvious that the subject (x in Model 1; “soul” in Model 2) is identical in the seven predications, the number of predicates involved in a single set of seven propositions is a more tricky issue. Two different trends are represented among Jaina philosophers as well as contemporary scholars. Many scholars, such as Matilal (1981, 2000, 2008), Padmarajiah (1986), and Uno (2000), following an interpretation set forth by Vādideva Sūri, say that the sapta-bhaṅgī is based on three fundamental predicates: A, :A, and “inexpressible” (avaktavya). In this perspective, three judgments are considered as simple (S1, S2, and S4) inasmuch as they contain only one predicate, whereas four judgments are complex (S3, S5, S6, and S7) because they involve two or three predicates (Padmarajiah 1986). However, another interpretation, going beyond the seeming oddness of the predicate avaktavya, reduces the number of fundamental predicates to two (A and :A). This thesis is supported by most Jaina philosophers, for instance, the Digambara Akalaṅka (eighth century) who explains that the third predicate, “inexpressible,” is nothing but a peculiar association of the predicates A and :A. Thus, according to this logician and his successors, the sapta-bhaṅgī is based on two fundamental judgments only (S1 and S2), one based on assertion (vidhi), S1, and the other on negation (pratiṣedha), S2 (cf. Akalaṅka, RVār ad TS I.6, I p. 33.15). Both are then combined with one another in order to produce the five remaining ones. It can thus be understood that the predicate “inexpressible” never changes, whatever the content is expressed by the predicate A.

Combinations of the Predicates If only two fundamental predicates are involved in the sapta-bhaṅgī , it should be explained how seven different propositions can be obtained through their combination.

Propositions S3 and S4 The propositions S3 and S4 involve a first level of combination of the two fundamental predicates A and :A, because the predicate “inexpressible” (avaktavya) is a special association of A and :A: (S3) In some respect, x is A and :A. (S4) In some respect, x is only inexpressible (avaktavya). In proposition S3, the predicates A and :A are uttered explicitly and separately while their combination can no longer be detected in the enigmatic predicate

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“inexpressible” (avaktavya) of the fourth proposition. Jaina logicians like Akalaṅka (RVār ad TS IV.42, I p. 261.6–10) express this difference by distinguishing a mere association, expressed by the participles samuccita or pracita (“gathered, collected”) from a complete fusion, denoted by the adjective avibhakta (“undivided, unseparated”): on the one hand, the predicates that are associated (samuccita, pracita) in the predication S3 can still be identified and separated; on the other hand, the components that have merged (vibhakta) in the adjective avaktavya of the predication S4 have lost their genuine individuality to become an indissoluble unity. Later commentators have deepened this distinction between two kinds of association of the fundamental predicates A and :A by taking into account the criterion of temporality. Thus Abhayacandra (SVBh ad LT 62, p. 85.6–8) explains the third and the fourth predications as follows: (S3) In some respect, soul is only existent and nonexistent by expressing successively (krama-vivakṣā) its own substance, its own time, its own place, its own condition and another substance, another time, another place, another condition. (S4) In some respect, soul is only inexpressible by expressing simultaneously (yugapat-vivakṣā) its own substance, its own time, its own place, its own condition and another substance, another time, another place, another condition. Thus the predicate avaktavya (“inexpressible”) constitutes a new way to coordinate the predicates A and :A: the subject is said to be inexpressible because language has such limitations that it cannot express the simultaneous possession of two different properties (cf. for instance Jayatilleke 1998, p. 348 and Matilal 2000 pp. 3–4). In every act of speech, separating two elements can be realized only if two different words are uttered at two different moments, in other words successively. Two words can never be pronounced in the same breath: no language, whatever powers it may have, is able to escape temporality, neither Sanskrit nor any modern language. Even nominal compounds do not solve the problem because a copulative compound (dvandva) is nothing but a shortened coordination of two distinct terms that still can be separated from each other (Balcerowicz 2015). As a consequence, uttering simultaneously contrary predicates or laying a simultaneous emphasis on both of them (cf. Balcerowicz 2015, pp. 214–221) results in losing the content expressed by these predicates. Such is the price one has to pay in order to recognize their contradiction – as it appears in the predicate inexpressible – as well as language limitations. Thus, when two attributes are simultaneously predicated of a subject, there are two ways to express the discrepancy between being and language: either the periphrasis “x is A and :A simultaneously” where the adverb counterbalances the impossibility to express explicitly A and :A exactly at the same time – such is the solution adopted by commentators who aim at glossing the predicate avaktavya; or the peculiar meaning of A and :A is erased in order to have both united in a single attribute that gives preeminence to neither of them: then only a predicate such as “inexpressible” can fulfill this function; there is then no need to mention the criterion of temporality. Both solutions reveal that there is no perfect coincidence between being (astitva) and expressibility (abhidheyatva).

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This parallelism between the constitution of assertions S3 and S4, both made up of a peculiar combination of the predicates A and :A explains their possible inversion in ancient times, when the pattern was not definitely set.

Propositions S5 to S7 Once it is established that the coexistence of two contrary predicates in a single proposition can be expressed through two different kinds of combination, based either on succession (krameṇa) or on simultaneity (yugapat), one can wonder why the Jaina model goes beyond a fourfold predication, by adding three more propositions. In other words, why seven, neither more nor less? Does the number seven entail any necessity? This number seems rather amazing; no wonder that it has aroused mockery from opponents. This tricky problem has been noticed by many Jaina logicians who basically found two different ways of explanation. Several Jaina thinkers, for instance, Vādirāja (NViV ad NVi III.66, 350.14–16) and Prabhācandra (PKM ad PMS VI.73, p. 682.18–23), followed by contemporary scholars such as Nyāyavijayajī (2000, p. 348), Uno (2000, p. 50), and Venkatachalam (2000, p. 71), have tried to justify the number seven, in spite of its seeming oddness. A thorough version of this reasoning has been luminously presented by Mookerjee (1978, pp. 117–118) who grounds his development on the Saptabhaṅgī taraṅgiṇī by Vimaladāsa: But why should the number be seven, neither more nor less? The answer is that each proposition is an answer to a question, possible or actual. And only sevenfold query is possible with regard to a thing. The questions are seven because our desire of knowledge with regard to any subject assumes seven forms in answer to our doubts, which are also seven. Doubts are seven because the attributes, which are the objects of doubt, are only of seven kinds. So the sevenfold assertion is not the result of a mere subjective necessity, which has nothing to do with the mere objective status of attributes. All assertions are in the last resort traceable to an objective situation, which actually possesses seven modes or attributes as an ontological truth.

However, this reasoning entails significant drawbacks. First of all, grounding the number of bhaṅga on the number of characteristics of an object implies that these characteristics are reduced to a definite number, namely seven, which challenges one of the main ontological tenets of Jainism, i.e., the fact that every being is endowed with an infinite number of features (anekānta-vāda). Moreover, far from being really satisfactory and from setting forth a real necessity, this explanation of the internal structure of the sapta-bhaṅgī by extrinsic reasons and by a reasoning based on an infinite regress may appear as purely contingent. To solve this difficulty, one has to adopt a purely linguistic perspective and deal with the predicate “inexpressible” (avaktavya) as if it were a primary predicate – as well as A and :A – so that it can be combined, on the successive mode, respectively, with A (S5), :A (S6), or with A and :A (S7). Thus, the most elaborate proposition, i.e., the seventh (“in some respect, x is A, :A and inexpressible”) predicates A and :A twice, applying the modes used respectively in the third and the fourth propositions, i.e., firstly a successive association and then a simultaneous association

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expressed through the adjective avaktavya (“inexpressible”). If a successive predication is symbolized by ^ and a simultaneous one by a dot (.), the sapta-bhaṅgī can be formalized as follows (brackets are used from the fifth proposition for the sake of convenience, to let the predicate avaktavya appear obviously): S1 : A S2 : :A S3 : A ^ :A S4 : A.:A S5 : A ^ (A.:A) S6 : :A ^ (A.:A) S7 : A ^ :A ^ (A.:A) The order of the various propositions is not fortuitous: it matches a logic of increasing complexity, the most emblematic evidence of which is the increasing number of predicates. But then, what serves as a warrant that no other predication can be uttered so as to shed a new light on a state of affairs? Since there are only two combinatory modes, succession and simultaneity, but no third one, as it is acknowledged by Jaina logicians themselves (cf. Akalaṅka, RVār ad TS IV.42, p. 252.19–20), one has to try other combinations between the three predicates – A, :A, and avaktavya –, and see whether a new predication can be produced that could not be reduced to one of the seven original bhaṅga. Such a test has been experimented by Vādirāja (NViV ad NVi III.66, II p. 350.32–351.4) who concludes that the seven predications exhaust all possible combinations. For instance, associating successively propositions S1 and S3 leads to a useless repetition of the predicate A and is thus tantamount to proposition S3. Being only linguistic, repetition adds no new ontological fact and leads to a tautology. Vādirāja does not even try new combinations based on simultaneity: uttering two, three, four, or more predicates simultaneously does not change anything since only one word, namely “inexpressible” (avaktavya), can express the simultaneous association of many predicates. Thus, once the basic principles that grounds the sapta-bhaṅgī have been recognized – two predicates A and :A, that can be associated in two different ways, successively (krameṇa) or simultaneously (yugapat) – one can understand that seven propositions, neither more nor less, can be produced in order to express the complex relations between a subject and a predicate.

Necessity of Seven Predications Why the seventh predication, which is the most complete one since it contains the three different predicates (A, :A, and avaktavya), is not sufficient to give a faithful account of the multiplex nature of reality? In other words, what would be the need for the predications S1 to S6? The seven propositions of syād-vāda are nothing else than the seven possible attempts, complementary as well as necessary, that a speaker has at his disposal in

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order to give the most complete account for the complex relation between a subject and a feature that is predicated of it. Taken all together or separately, they reveal the incapacity of language to achieve its task; language is meant to fail when it has to face the multiplexity of reality: separately because each of the seven predications only gives a partial insight on reality; all together, because the fact that seven propositions are necessary to provide the most complete description of a single state of affairs puts to the fore the inadequacy of language to really express the quintessence of being. Even if the seventh and last bhaṅga seems to be the most complete one, it cannot be considered to be a better description than any other bhaṅga: the fact that this proposition must repeat twice each predicate (A and :A), combining their successive and their simultaneous associations (which corresponds to propositions S3 and S4, respectively), in order to shed light on their complex relation, is the best evidence of this incapacity of language to express the multiplexity of reality. No proposition is endowed with a higher adequacy than any other, and each of the seven propositions can be held as a complete description of reality (sakalādes´a), inasmuch as the statement is qualified by the adverb syāt: “the function of the term ‘syāt’ is to imply all possible standpoints and widen the scope of the discourse” (Uno 2000, p. 47). This is not contradicted by the presence of the restrictive particle eva. When used in simple propositions, the restriction expressed by the adverb eva aims at excluding unrelatedness (ayoga-vyavaccheda-bodhaka), in other words at excluding properties that are contrary to the very property that is predicated of the subject. In the proposition “the jar is only red,” the adverb “only” aims at excluding any other color adjective other than red. At the same time, using the word syāt counterbalances the restriction of eva: it acts as a reminder of the unilateral and partial description contained in a single proposition; each proposition must be completed by the others. The sevenfold predication is thus a method meant to overcome limitations inherent to language and to any kind of empirical knowledge. That is why it wrecks neither the possibility of omniscience nor the fact that absolute truths can be professed by omniscient beings. The objection raised by Naiyāyika thinkers (cf. Joshi 2000) does not really question the validity of the sevenfold predication: according to them even a statement like “a Tīrthaṅkara’s teaching is true” should be submitted to the sevenfold predication, with the absurd consequence that Jaina would have to acknowledge that in some respect the teaching professed by a Tīrthaṅkara is not true! But the Jaina method avoids such an objection because some statements – those that are expressed from the transcendental viewpoint – escape the syād-vāda and hold true absolutely (Clavel 2012).

Beyond Inadequate Comparisons Jainism and Advaita-Vedānta: Avaktavya and Anirvacanīya Because of their analogous formation and their usual synonymy in everyday language, the adjectives avaktavya and anirvacanī ya, used respectively by Jainas and Advaita-vedāntins, have sometimes been considered to be philosophically

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tantamount (Venkatachalam 2000). However, their conceptions of inexpressibility are totally opposite: in the Jaina sevenfold predication, the subject is said to be inexpressible (avaktavya) because language is limited and cannot express simultaneously the fact that a being is endowed with contrary predicates; on the other hand, inexpressibility in the Advaita-vedāntin doctrine implies exclusion of contrary predicates: being eternal and devoid of any conditionality, the nature of the Brahman cannot be characterized by predicates, since they are illusory.

Jainism and Buddhism: Sapta-Bhan˙gī and Catus-Koti ˙ implicit – between the ˙ or A second comparison has often been drawn – be it explicit Jaina and the Buddhist logical frameworks, because the first two propositions of the Buddhist catuṣ-koṭi are obviously similar to the first two propositions of the Jaina sapta-bhaṅgī . Many scholars have tried to compare both logical patterns in order to put to the fore a similarity between them or to explain their differences. Some of them assumed either that the Buddhist catuṣ-koṭi matches the first four propositions of the Jaina sapta-bhaṅgī (e.g., Raju 1954) or that the fourth proposition of the catuṣ-koṭi is tantamount to the last four propositions of the sapta-bhaṅgī (e.g., Miyamoto 1960). But such comparisons are grounded on a misunderstanding of the Jaina logical framework, or at least of the meaning of the predicate “inexpressible” (avaktavya) as used in the Jaina perspective: according to Jainism, saying that something is inexpressible does not mean that it is neither existent nor inexistent, but actually both of them simultaneously. Thus both patterns are fundamentally different from one another because they do not exactly interpret the logical principles of contradiction and of excluded-middle in the same way (Clavel 2010).

Syād-Vāda and Logic Principles Since neither the subject nor the fundamental predicates changes throughout the sapta-bhaṅgī , the compatibility between the sapta-bhaṅgī and logical principles, especially the principles of contradiction and of excluded-middle, raises serious problems. The conformity between Jaina sapta-bhaṅgī and such logical principles constitutes one of the main issues addressed by contemporary scholars.

The Meaning of Syāt The word syāt is a cornerstone of the sevenfold predication because this very word appears in each and every proposition. This is why this logical pattern is called sapta-bhaṅgī (“sevenfold predication”) when one aims at highlighting the fact that this logical structure consists of seven propositions, neither more nor less, as well as syād-vāda when stress is laid on the invariable presence of the adverb syāt in every proposition. No wonder that a misunderstanding of this fundamental word has often led scholars to a general misunderstanding of the sapta-bhaṅgī. Several models entail such a flaw. According to Pandey (1984) who claims the model of

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many-valued logic, statements S1 and S2 “challenge the law of contradiction” because Jainas “give some truth-value (other than falsehood) to contradictory statements” (Gokhale 2000, p. 76). Pandey does not take into account the presence of the term syāt, which is however sufficient to remove contradiction, because syāt does not refer to the same respect in S1 and in S2. The model of modal logic as presented by Barlingay (1965) and Marathe (1984) provides a wrong description of the syādvāda, because by characterizing a proposition as an incomplete truth one cannot conclude the indeterminateness of each proposition. The model of conditionality by Sagarmal Jain and B.K. Matilal is not satisfactory either because it aims at specifying the exact respect according to which the property A can be predicated from the subject x, whereas the originality of the Jaina logical framework precisely lies in the mention of some respect – without any other precision – and in the repetition of one and the same word throughout the syād-vāda. Lastly, Gokhale’s model of existential quantifier fails to conceive what sapta-bhaṅgī really is, in claiming that a “syātstatement” is actually “metalinguistic.”

Syāt = “Perhaps, Probably”? Among all models mentioned above there is a strong temptation to rely on the interpretation of the adverb syāt as “perhaps, maybe.” In Sanskrit, the third person of singular of the optative of the root AS- (“be”), taken adverbially, is usually translated as “maybe.” However, translating this word in such a way in the context of the syādvāda has significant consequences, because it leads to a general misunderstanding of the logic involved in the syād-vāda, and then to a misunderstanding of the whole Jaina philosophy. Scholars who acknowledge in the adverb syāt the usual meaning of a probability (“maybe”) are invariably led to a probabilistic interpretation of the syād-vāda. That is the thesis adopted by Barlingay (1965) who qualifies the sapta-bhaṅgī as a “logic of possibilities.” Doing so, he opened the way to the model of “modal logic” set forth by Marathe (1984). An analysis, which leads to subjective indetermination and uncertainty, can be expressed in two different models, depending on the number of occurrences of the adverb syāt: in the first case, syāt is repeated each time a new predicate appears (Model 3.1), whereas in the second case, it is expressed only once in each proposition (Model 3.2). Such an alternative is not totally artificial inasmuch as Jaina logicians themselves have wavered between both formulations: for instance, Malliṣeṇa uses the adverb syāt in a distributive way, repeating it as many times as there are predicates in a single proposition, while Prabhācandra never uses it more than once in a single proposition (cf. supra, § 1.4.). Model 3.1 (S1) x is maybe (syāt) A. (S2) x is maybe :A. (S3) x is maybe A and maybe :A. (S4) x is maybe inexpressible. (S5) x is maybe A and maybe inexpressible. (S6) x is maybe :A and maybe inexpressible.

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(S7) x is maybe A, maybe :A, and maybe inexpressible. Model 3.2 (S1) x is maybe (syāt) A. (S2) x is maybe :A. (S3) x is maybe A and :A. (S4) x is maybe inexpressible. (S5) x is maybe A and inexpressible. (S6) x is maybe :A and inexpressible. (S7) x is maybe A, :A, and inexpressible.

The probabilistic interpretation, especially the model 3.1, seems to bring a convenient solution to the problem of compatibility between the syād-vāda and the principle of contradiction. This logical principle is wrecked neither by the coexistence of the various propositions nor by the formulation of a single one. For, modalization does not concern the whole proposition but only the predicative relation between the subject and each attribute taken apart (A, :A, inexpressible). Thus propositions S3, S5, S6, and S7 are no longer complex and are nothing but associations of two or three fundamental predications (as found in propositions S1, S2, and S4). Since the seventh proposition includes the three predicates and shows them as the three possible alternatives, it is not necessary to express the six other propositions. Consequently, if seven propositions (bhaṅga) are meant to build up a set, it has to be acknowledged that there is only one predicative relation per proposition, even though a predicative relation can consist in more than one attribute. Thus, the adverb syāt should not have a distributive value in complex propositions (S3, S5, S6, and S7). According to the probabilistic interpretation of the word syāt, the syād-vāda would only express several possible judgments without being able to assert anything as being indubitable. It would bring to the fore the absolutely unknowable character of the world, since it would always be possible to utter contradictory judgments about everything without choosing between them. Therefore, translating syāt by “maybe” or by any other modalization aiming at expressing a probability or a modality is a wrong translation which implies interpreting Jainism as a form of ontological indetermination or skepticism (cf. Soni 1996). Such a consequence has been perfectly set forth by Nyāyavijayajī (2000, p. 345): Wrong interpretation of the term ‘syāt’ as ‘may be’ imparts a sceptical form to syād-vāda. But in fact syād-vāda is not scepticism. It is not the uncertainty of judgement, but its conditional or relative character, that is expressed by the qualifying particule ‘syāt’. Subject to the conditions under which any judgement is made, the judgement is valid beyond doubt. So there is no room for scepticism. All that is implied is that every assertion which is true, is true only under certain conditions. Syād-vāda is not of the nature of doubt arising from the difficulty or inability of ascertaining the exact nature of a thing in regard to existence and non-existence, permanence and impermanence, etc. It is not the doctrine of uncertainty. It is not scepticism.

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More generally, every attempt to explain the syād-vāda in the light of modern logical tools that were unknown to ancient and medieval times fails to provide a consistent conceptual framework. Only textual evidences and interpretations that are drawn from Jaina texts can be taken for granted. Moreover, syāt cannot be seen as a finite verb because with the finite word asti from the same root it would make a statement with two finite verbs meaningless.

Syāt = Kathan˜cit A precious clue provided by Jaina logicians invalidates the probabilistic and modal analyses of the syād-vāda. Many texts reveal that the adverbs syāt and kathan˜ cit are considered as synonymous. Their equivalence goes beyond the mere grammatical gloss and has also a semantic value. In this way, it is clear that the adverb syāt aims at expressing the nonunilateral or multiplex (anekānta) nature of reality, and also of every predication. To reject a unilateral thesis does not lead to the impossibility of asserting propositions with certainty. Thanks to the equivalence between syāt and kathan˜ cit, it is now taken for granted that the adverb syāt restricts the validity of a judgment by having it depend on a peculiar viewpoint. Using the word syāt (or kathan˜ cit) in a judgment acts as a remainder of its status: the judgment should not be considered as the only valid one; it is true only from a certain point of view, in some respect. Thus it is now taken for granted that the word syāt is to be translated by expressions like “from a certain point of view,” “in some respect” (cf. for instance Pardmarajiah 1986, p. 338, Kulkarni 2000, Nyāyavijayajī 2000, pp. 341–349 and Uno 2000). The syād-vāda is really a cornerstone of the doctrine the multiplex character of reality (anekānta-vāda) that avoids taking into consideration only a single aspect of a state of affairs or a univocal relation between a subject and a predicate. Nevertheless, the presence of this modalization does not obviate certainty to judgments. The predicative relation contained in the sentence “in some respect (syāt) x is A” is really asserted and is devoid of doubt.

Contradiction and Tautology As it has been expressed in Metaphysics Γ by Aristotle (1005b19–20), the principle of contradiction runs as follows: “It is impossible that the same thing at the same time both belongs and does not belong to the same object in the same respect” (τὸ γὰρ αὐτὸ ἅμα ὑπάρχειν τε καὶ μὴ ὑπάρχειν α᾿ δύνατoν τῷ αὐτῷ καὶ κατὰ τὸ αὐτὸ). The principle of contradiction seems to be wrecked by the sevenfold predication. This is why contemporary scholars have set forth many-valued logic models in order to solve the problems of compatibility between judgments that seem contradictory (for instance Barlingay 1965, Bharucha and Kamat 1984, Matilal 1981, Ganeri 2001 and 2002). However, such models are not consistent to give a right account of the syād-vāda, because Jaina thinkers themselves claim the principle of contradiction, such as the great logician Akalaṅka who makes it clear that the sapta-bhaṅgī really takes into account this principle: “Sevenfold predication is the set of judgments based on affirmation and negation, that are expressed without any contradiction

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(avirodhena) on account of a question with respect to one and the same real thing” (RVār ad TS I.6, I p. 33.15). Before explaining how the whole set of seven predications is consistent with that logical principle, one has to understand how the first two propositions can be uttered without contradiction, in other words, how two contrary attributes can be predicated of a single subject. One must not conclude from the recurrence of an identical word, i.e., syāt, in every proposition and especially in S1 and S2, that syāt always refers to the same viewpoint or perspective. Later commentators have explicated the viewpoints according to which a subject can be respectively said to be A and :A: are thus distinguished identity and alterity factors. In spite of the seeming equivocity of the word syāt, the subject is considered under two different sets of conditions. Soul is said to be existent from the viewpoint (vivakṣā) of its own substance, its own time, its own place, and its own condition, whereas it is said inexistent from the viewpoint of another substance, another time, another place, and another condition. Thus, the coexistence of S1 and S2 is consistent with the principle of contradiction inasmuch as the same subject is not qualified by two contrary predicates under the same set of circumstances (for an extensively developed instance, cf. Malliṣeṇa, SVM 23.113–120, p. 143.12–18). However, solving this difficulty makes another one emerge: if the coexistence of S1 and S2 is not inconsistent with the principle of contradiction, uttering successively the propositions S1 and S2 could appear as nothing but a verbal game, or in other words as a tautology. Saying on the one hand that a real thing is existent from the viewpoint of its own substance, place, time, and condition (S1) and on the other hand that the same thing is inexistent from the viewpoint of another substance, place, time, and condition (S2) may appear as a tautology, since S2 is nothing but the negative counterpart of S1. As such, the second assertion does not bring any new element that would not have been provided by S1. Several interpretations have been set forth to explain how the sapta-bhaṅgī can escape the objection of a tautology. Balcerowicz (2015) thus tries to explain the Jaina logical system in the light of the Buddhist theory of apoha (“exclusion”). According to Madhyamaka Buddhism, especially to Dharmakīrti (seventh century), one of its greatest representatives, if two things can be characterized as red, their redness is fundamentally the consequence of an exclusion process (x is red because x is not non-red, in other words x is A because x is : :A) and it only secondarily results in a similarity. From this perspective, the proposition S1 would predicate a property from a subject in the only way that is linguistically convenient, i.e., by asserting, while the proposition S2 would take into account thanks to negation the exclusion of what x is not. However, the Buddhist theory does not perfectly match the Jaina conception because it restricts the opposition between affirmation and negation to the linguistic field and it makes their ontological opposition disappear, or at least it reduces ontology to a mere effect of language. Jaina logicians invalidate this interpretation since they recognize that two contrary predicates (for instance, jarness and non-jarness) are not separated from one another (cf. Akalaṅka, RVār ad TS I.6, I, p. 35.1: ghaṭatvam aghaṭatvaṃ ca parasparato na bhinnam). Stating the inexistence of :A in the subject x does not

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only aim at drawing a strict delimitation between x and everything which is not x. Rather than rejecting negativity outside of the being, one has to understand, according to the Leibnizian theory of inter-expression, that negations are inscribed in the core of beings themselves. Such a conception matches another fundamental tenet of Jainism, namely the multiplex nature of reality (anekānta-vāda), according to which each and every substance has an infinite number of characteristics (cf. Uno 2000, p. 42). This complexity of beings is perfectly expressed by Akalaṅka: “Thought which consists in false and true, which consists in visible and invisible, in difference and identity, establishes by itself that one and the same reality consists in existence and non-existence” (LT 9cd-10ab). This inter-expression of substances sheds a new light on the whole Jaina doctrine: considering that each substance is a viewpoint on the world, one can understand that knowing perfectly a single thing requires knowing everything (cf. Kundakunda, NiS 158), since in each substance the determinations characterizing every other substances are inscribed, as it has been convincingly expressed by Balcerowicz (2015, p. 183): “the world is a complete network within which all the existents are related with all the remaining ones and that their essential character and nature is not only determined by what is in things themselves but also by all the relations in which they enter vis-à-vis all other existents.” The syād-vāda can thus be considered as a method that plays the part of an ersatz of omniscience for those who are not able to obtain this perfect and absolute knowledge of reality.

Definitions of Key Terms anekānta-vāda avaktavya

bhaṅga

eva kathan˜ cit

naya

pramāṇa sapta-bhaṅgī

“doctrine of multiplexity of reality.” third linguistic predicate used in the sapta-bhaṅgī “inexpressible,” generally considered as the only means to express two contrary predicates simultaneously. one member, i.e., one proposition among the set of seven that constitutes the sapta-bhaṅgī . The way each proposition is made up allows one to reconstitute the other six in order to give a full account of a single state of affair. restrictive particle “only.” “somehow,” adverb often used by commentators as a paraphrase of the term “syāt” in the context of the sevenfold predication. “viewpoint.” The Jaina tradition counts seven viewpoints: comprehensive (naigama), collective (saṅgraha), empirical (vyavahāra), instantaneous (ṛjusūtra), synonymous (s´abda), etymological (samabhirūḍha), and factual (evaṃbhūta or itthaṃbhūta). “means of knowledge.” “sevenfold predication,” set of seven predications that can express all possible relations between a subject and a predicate.

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syād-vāda

syāt

vastu

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“doctrine of ‘in some respect,” synonym of sapta-bhaṅgī when used in its most restrictive meaning, and of anekānta-vāda when used in its broader meaning. third person of singular of the optative of the root AS- (“be”). Taken adverbially, it is usually translated by “maybe” in Sanskrit, but in the context of the sapta-bhaṅgī , it is glossed by kathan˜ cit and rather means “in some respect.” a real thing.

Summary Points The syād-vāda (“doctrine of syāt”) or sapta-bhaṅgī (“sevenfold predication”) is an epistemological method of Jainism which is consistent with its ontology, i.e., the doctrine of multiplexity of reality (anekānta-vāda). The word sapta-bhaṅgī insists on the number of propositions in the logical framework, whereas the term syād-vāda lays stress on the fundamental adverb syāt used in each and every proposition. The sevenfold predication is the structural rule permitting a combination of two contrary predicates (A and non-A), with one another in all possible ways, so as to consider one and the same state of affairs as exhaustively as possible. The syād-vāda is based on two predicates only; the third one, namely “inexpressible” (avaktavya) is obtained when the first two are associated simultaneously. Two kinds of combination are used throughout the sapta-bhaṅgī, namely succession and simultaneity. All predications share the same truth-value: each can be considered as a complete description of reality inasmuch as the adverb syāt alludes to the six other. One judgment cannot be sufficient to give a faithful account of a single state of affairs because of the limited power of language. The word syāt has to be translated as “in some respect” but not as “perhaps.” Syād-vāda is not a doctrine of ontological indetermination or a skepticism. Nor does it challenge the logical principles, especially the principle of contradiction.

References ĀM = Samantabhadra. 1999. Āptamīmāṃsā. In Samantabhadra’s Āptamī māṁsā, critique of an authority, along with English translation, introduction, notes and Akalaṅka’s Sanskrit commentary Aṣṭas´atī , ed. & trans. Nagin J. Shah. Ahmedabad: Dr. Jagruti Dilip Sheth. Balcerowicz, Piotr. 2015. Do attempts to formalize the Syād-vāda make sense? In Jaina scriptures and philosophy, ed. Peter Flügel and Olle Qvarnström, 181–248. London: Routledge. Barlingay, Surendra S. 1965. A modern introduction to Indian logic. Delhi: National Publishing House.

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Logic in Tamil Tradition

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Nirmal Selvamony

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic as Rhetoric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tarukkam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tarukkam in Literature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tarukkam in Research Methodology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tarukkam in Political Oration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vākai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The End of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Logic, in Tamil tradition, is inseparably connected with philosophy and rhetoric. It is a reasoning praxis, which includes knowing the truth and communicating it. If the former is the epistemic aspect of logic, the latter is its rhetorical. The rhetorical branch has two main divisions, the monological (tarukkam) and the dialogical (vākai). Examples are provided from classical literature, folk songs, folk drama, religious polemics, political oration, screen play, research methodology, forum for disputation, and debating forum. The purpose of logic is to affirm the normative mode of truth of each type of society. If the truth of the Tamil primal society is differentiated continuity, that of the Tamil state society is foregone conclusion, and that of the Tamil industrial society is anarchic indeterminacy.

N. Selvamony (*) Madras Christian College, Chennai, Tamil Nadu, India Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_5

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Introduction The phrase otta kāṭci in tolkāppiyam (hereafter, tol.) (III. 9. 112: 1) means literally “proper or befitting vision,” “seeing how something fits with truth.” Though the term kāṭci often denotes philosophy in early Tamil sources (Selvamony 1996), it particularly refers to philosophy as logic in marapiyal (of tol. III. 9. 112: 1) as the term is used in the context of debating techniques (uttikaḷ). Some other terms for logic are tarukkam (tarukku [tol. III. 1. 53: 4], to exalt [often] oneself verbally + am, a nominal suffix ¼ tarukkam, verbal exaltation, logic), vākai (tol. III. 2. 18, 19; vātu/vātam, verbal duel; “vāti” [verbal dueler], malaipaṭukaṭām 112; logic), aḷavai (literally, measure; cf. tol. III. 8. 186: 3; validative criterion; validation of knowledge; logic; see the present author’s “Logic in tolkāppiyam” in the present Handbook; civañāṉa cittiyār), and niyāyam (“niyāyam pēcu,” vāṉamāmalai 558–559). Though validation (aḷavai) is an important part of logic, the latter is not wholly a matter of validation to the ancient Tamil people because the ultimate end was the vision of truth, not validation of knowledge. Logic, then, is basically finding truth and communicating it to others effectively. If the former addresses the epistemic aspect of logic, the latter does the rhetorical. In the non-Tamil context, logic could focus on the “theory of pramāṇas or accredited means of knowing in general, perhaps with particular emphasis upon the specific theory of aṉumāṉa, inference considered as means of knowing” (Matilal 1) and thus lay emphasis on the epistemic aspect. The epistemic branch has contributed the criteria for validation of knowledge (aḷavai), the method of concluding an assertion (kāṇṭikai, Selvamony 1990, 2000), and the techniques of reasoning (uttikaḷ), all of these resulting in the development of a separate domain of study, aḷavaiyiyal (epistemology). The epistemic logical branch has also yielded what we today call “research methodology.” However, it must be pointed out that this branch of logic has overemphasized what David Abram calls “a calculative logic” (267) or “the exclusively human logic,” a derivative of the alphabetized intellect of the humans which stifles “a kind of logic in tangible qualities” (Levi-Strauss 1). The latter consists in direct perceptual contact with the world (Abram 268). Our epistemology has a biological basis in direct perceptual contact with the world, and, as Abram shows, our sophistication of epistemic logic is disproportionate with our basic organismic need for adequate perceptual contact with the world. In deed perceptual contact facilitates our ontically differentiated continuity with the world. But difference and continuity have to be negotiated with the help of epistemic and rhetorical aspects of logic.

Logic as Rhetoric Though the epistemic and the rhetorical aspects of Tamil logic may be studied separately, this essay will focus on the rhetorical as the latter incorporates the former almost entirely. In other words, to communicate the “proper vision” to others, we use the criteria of knowledge, the argumentative text, and techniques. We may also say

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that rhetoric is applied logic. Aristotle defined rhetoric as “the faculty [power] of discovering in the particular case what the available means of persuasion are” (Cooper 7). Put differently, speech can be properly persuasive only when it is logical or when it states the fitting, the reasonable, and hence the adjective, “otta” (befitting, proper) in “otta kāṭci.”

tarukkam The rhetorical branch of traditional Tamil logic has two main subdivisions: the monological and the dialogical. If monologically rhetorical logic is known as “tarukkam,” dialogically rhetorical logic is “vākai.” In later times, vākai came to be known as “vātu” or “vātam.” As tarukkam and vākai or vātu include rhetoric in their semantic range, it would be absurd to conclude that there is no Tamil equivalent for rhetoric (Bate 48). We find monological rhetoric in both the Inner (akam) and Outer (puṟam) domains of life. In the former it is traceable to courtship, the burden of the tiṇai known as kaikkiḷai in which a lad tries to woo a non-reciprocating girl (tol. III. 1. 53), and in the latter, it is known as “pāṭāṇ” (singing the ruler) in which skilled artists (who are also rhetors or orators) seek gifts from a benevolent donor (tol. III. 2. 25– 36). If the courting rhetor (the persuasive speaker) exalts himself in kaikkiḷai using hyperbole, the gift-seeking rhetor underplays himself/herself. Such praxic equivalence justifies the pairing of the Inner and Outer tiṇaikaḷ. Even as the courting lad exaggerates the charm of the girl, the gift-seeker overstates the virtues of the donor. Both rhetors (of the Inner and the Outer domains) employ knowledge-validating criteria such as example and reason; in fact, predominantly, the former to vindicate their claim. If the lad piles up similes and metaphors to express the girl’s beauty, the gift-seeker does so in order to underscore the donor’s generosity. Both lavish analogical language only to stress that the implorer is the legitimate (reasonable) beneficiary. If the boy reasons his rightful claim to the girl’s beauty, the gift-seeker does his to the donor’s generosity. When one’s claim is not conceded, argument ensues. As contention is its necessary condition, argument may be considered the verbal equivalent of combat. It creates maximal positive mental distance between the two parties. But if disagreement crosses the threshold point, the mental distance between the parties becomes negative and they could even become enemies.

tarukkam in Literature Even as the argument during heterosexual courtship precedes premarital love relationship, tiff (another form of verbal duel) does sexual union in marital relationship (tirukkuṟaḷ 1330). If the purpose of argument in courtship is to separate the partners in order to strengthen their unity, the end of tiff is to effect greater passionate sexual union (continuity). Both courtship argument and tiff between married couple should

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not exceed proper limits. This is evident in a courtship monologue (from the primal society/tiṇai; kalittokai 56) of a boy who pursues a girl (walking on a street) and makes wild allegations against her. His charge is that she is the source of great distress to the young males when she chooses to walk on the street without any escort. Instead of stating the charge, he chooses to express it in the form of questions. The reasons adduced as evidence are in the form of figures of speech, which express the girl’s beauty. When the same reason is given in different forms, the repetitiveness creates a sort of hypnotic effect on the listener. However, his allegations are provocative and awaken her rationality though she does not respond in any way. In other words, his rhetoric balances both rational sally (which ensures difference) and arational hypnosis (which promotes continuity). Finally, he acquits her by squarely shifting the blame onto the chieftain and wonders how the latter, who forewarned the people when an elephant in musth was at large chose not to alert the people when a pretty girl walked on the street without proper escort (kalittokai 56). Evidently, the final acquittal prevents the argument from taking negative effect. The kalittokai song we discussed belongs to the tiṇai called kaikkiḷai (an akattiṇai we discussed earlier), which means “minor relationship.” Though it is not one of the major five tiṇaikaḷ based on love (aṉpoṭu puṇarnta aintiṇai, tol. III. 3. 1), it is an important early stage in the development of sexual life. It deals with unrequited love during courtship, a kind of primordial behavior, not only of humans but also of several other creatures. In order to understand courtship, we may do well to remember that male animals have “stronger passions than the females. Hence it is the males that fight together and sedulously display their charms before the female.” In most animals, the male seeks the female (Darwin 578–79) and gets extremely pugnacious when pairing (Darwin 698, 703). Courtship behavior in humans (in tiṇai society) includes such acts as making sexual advances despite lack of any response from the female, constructing a provocative, hyperbolic, and yet logical monologue consisting of sensitive remarks about him and her, often challenging the ethicality of female behavior (tol. III. 1. 50). In humans, especially, males, unlike in the other organisms, the word is the most effective instrument of persuasion. In other species, the equivalents of the word are vocalization and “instrumental music” (Darwin 704–714), courtship antics (Darwin 714–716), the bright coloring, and the “various ornaments, which often become more brilliant during the breeding-season. . .” (Darwin 704). Compelled to use the word (logos>logic) in the most rhetorical manner possible for basic biological purposes, namely, mating and perpetuation of the species in tiṇai societies, the alpha human male asserts his dominance (like the males of most other species) to win the female. Humans may also resort to combat with other males or subdue animals, or go through ordeals to win the female not during tarukkam but in a prior competitive stage called vākai. Though vākai particularly refers to verbal duel, it may involve other forms of contest and trial too. But vākai does not involve the prized female. Even when the contest is physical, vākai is incomplete without verbal boast. After subduing the competitors successfully, the alpha male enters the tarukkam stage. His rhetoric has a double effect on the female. On the one hand,

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his argument addresses her rational mind; on the other, his words have a hypnotic effect on her. The female partner has to be rationally active enough in order to witness male self-assertion through eloquence. At the same time, she has to suspend her disbelief in his truth-claims. Though self-assertion is isolative in nature, ironically, it is the prelude to sexual union. On the one hand, he desires union with her, which should involve merging his self with the girl’s. On the other, he seeks to indulge in logical verbal display, which activates the rationality of both maximally. But when he loses his self and unites with her, he will not need logic anymore. Just as courtship is a desirable prelude and justifiable if it culminates in sexual union, divisive logic is justifiable if it leads to befitting vision or otta kāṭci which will facilitate integration (or continuity) of the members of tiṇai (humans and nonhumans). If logic served the integrative purposes of the rhetors of the primal (tiṇai) society, it set different faith-communities in opposition to each other in the state society. However, both societies employed monological rhetoric. Examples of monological rhetoric may be found in folk songs also. Consider the song titled, “araṇmaṉaikkāri” (literally, palace girl/woman) in which the male addresser tries to get the attention of the female addressee not unlike in kaikkiḷait tiṇai (cakannātaṉ 51–53): pallākku mēlē ēṟip pavaṉi pōṟavaḷē –rācātti pavaṉi pōṟavaḷē (O girl, my queen, who rides a palanquin, leading a procession, leading a procession). (Trans. Nirmal Selvamony)

Another type of monological rhetoric in injunctive mood can be traced to a practice (of the Outer domain) in the primal (tiṇai) society called “ceviyaṟivuṟū,” meaning, “counsel” (tol. III. 2. 34: 8). Here is an example: O you great chieftain, you belong to the forest country where elephants wander like cattle among the buffalo-like black rocks. Matchless you are; so, will I tell you this: Avoiding the company of those that have no love and grace and head towards hell, may you nurture your land like how a mother does her baby; proper it is; rare too. (puṟanāṉūṟu 5; trans. Nirmal Selvamony)

This rhetorical genre was effectively employed in the public assembly (avai) also. An ancient Tamil song tells us that a bard admonished the elders in an assembly probably because they were not discharging their duties properly (puṟanāṉūṟu 195; cf. Bate 48–52). We may note in the song quoted above how the analogy and the

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bard’s own status in the society lend authority to the admonition. The analogy appeals to a context that can validate the relation between the ruler and the ruled when both enjoy necessary continuity. The injunctive monological rhetoric of the tiṇai assembly anticipates the sermons and exhortations of the religious leaders of the state society. A classic example of this type occurs in the canto of cilappatikāram where the heroine kaṇṇaki charges the king of pāṇṭiyar with the crime of the murder of her husband. Through different rhetorical modes the rhetor seeks to persuade the addressee.

tarukkam in Research Methodology Persuasion is certainly a goal of the modern researcher also. The dissertation has to be written in a persuasive manner, and in this regard, the role of the argumentative techniques gains importance. The argumentative tradition of the tiṇai assembly may be regarded as the ultimate source of what we today call “research methodology.” The Tamil scholars tamiḻaṇṇal and ilakkumaṇaṉ (1977) maintain that in Tamil, the grammar texts, commentaries on grammar, and philosophical texts have always adopted their own research methods (161). They point out that classification and synoptic account are methods the grammarians adopt in their texts (163). If the commentator parimēlaḻakar brings out the significance of the arrangement of the ten couplets in tirukkuṟaḷ (164), the theologians of caiva cittāntam employ the research methods in their texts (161). The research methods adopted by the grammarians, commentators, and theologians are either what tolkāppiyar called argumentative techniques (uttikaḷ) or similar ones. tamiḻaṇṇal and ilakkumaṇaṉ (1977) show how many of the argumentative techniques discussed in naṉṉūl and tol. are applicable to the modern dissertation (161–164). What we call synopsis (or abstract) is a kind of tokuttuk kūṟal (tol. III. 9. 112: 3), and the argument is based on the technique of maṟutalaic citaittal (tol. III. 9. 112: 18). Stating of the hypothesis is tolkāppiyar’s “taṉkōḷ kūṟal” (tol. III. 9. 112: 10). Citing others involves either agreement with earlier scholars (piṟaṉ uṭaṉpaṭṭatu tāṉ uṭaṉpaṭal, tol. III. 9. 112: 11) or disagreement (maṟutalaic citaittal, tol. III. 9. 112: 18). When one disagrees, one has to say why, and this involves stating one’s view on the matter (taṉ tuṇipuraittal, tol. III. 9. 112: 18). When one engages another scholar’s view, either to agree or disagree, one has to cite relevant facts or principles (muṭintatu kāṭṭal, ñāpakam kūṟal, tol. III. 9. 112: 15, 22). In all this, consistency is taken care of by iṟantatu kāttal (supra) and etiratu pōṟṟal (infra; III. 9. 112: 12). By employing the argumentative techniques, the dissertation, an example of monological type of tarukkam, appeals to the “befitting vision” of the examiner by arguing a thesis. When truth is more a matter of form than content, rational evidences to justify the assertion are adequate as in a court of law. The reduction of truth to interpretation gives rise to multiple truths and subsequent epistemic anarchism which typically represents the zeitgeist of the industrialist society.

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tarukkam in Political Oration Another example of monological rhetoric is the modern political oration in Tamil. It is believed that ci. eṉ. aṇṇāturai (1909–1969) and karuṇāniti (1924) “were the first general practitioners of the new mode within mass politics in the early 1950s” (Bate 35). The phrase “new mode” here means, “mēṭait tamiḷ” (oratorical platform Tamil), a kind of literary Tamil used in political orations. In fact, such Tamil was in evidence earlier too. ma. po. civañāṉam discusses the mēṭait tamiḻ of a few Tamil orators during the Independence Struggle (211–223). But one would not want to ignore such orators as na. mu. vēṅkaṭacāmi nāṭṭār (1884–1944), and cōmacuntara pāratiyār (1879–1959) both of whom earned the title “nāvalar” because of their oratorical skills. The latter’s political speeches against the British Raj (from 1905), and the imposition of Hindi in the Tamil state in the 1930s were literary and argumentative at once as he was a Tamil scholar, and, by profession, an advocate. ci. eṉ. aṇṇāturai was quite familiar with nāvalar’s speeches as he had argued with him even in the debating forum on the question, “Should kamparāmāyaṇam Be Burnt?” (iḷañcēraṉ 227–37). It is highly likely that the popular Dravidian politicians are indebted to eminent public speakers like na. mu. vēṅkaṭacāmi nāṭṭār and cōmacuntara pāratiyār. However, ultimately, modern Tamil political rhetoric could be traced to the monological rhetoric of the tiṇai society. We may illustrate the use of logical devices such as the proposition, reason, and illustration in modern political oration with the help of an excerpt from ci. eṉ. aṇṇāturai’s speech on Islam on 7 October 1957, at the event of mīlāp, at Madras Beach. Proposition: Why is Islam a great way of life? Reasons: 1. 2. 3. 4. 5. 6.

Because it has several doctrines which can dispel the doubts of humans Because one of its doctrines forbids idolism Because another Islamic doctrine is about making humans whole Because it removes caste identity Because Bernard Shaw has stated that Islam will be the only best life-way Because it can be used for great purposes Illustration: 6.1 Like diamond which can be put to good use such as making jewels, and also to bad use, such as gambling.

In the abovementioned example, the speaker adopts the strategy of multiplication of reasons. In fact, this is one of the significant features of popular Dravidian rhetoric in the movies and in politics. If the lad who was courting a non-reciprocating girl (in kaikkiḷait tiṇai) was multiplying analogies (civaliṅkaṉār 418; kalittokai 30), the modern popular Dravidian rhetor multiplied reasons for the same reason, namely, poetico-dramatic effect. It must be noted that the multiplication of reasons is a rhetorical technique, which ultimately derives from the folk song. Again, all oral texts (the folk song, the kaikkiḷai song, and political speech) use repetition in order to create symmetry (Selvamony 2017), which has the power to cast a spell on the

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listener. In this regard, the rhetoric of the pursuant passionate lad and the popular Dravidian politician tend to get alogical rather than logical. Such alogicality helps the listener overlook the truth of the argument. By analogizing religion (of the state society) to the diamond’s ambivalent usage, the political rhetor of the industrialist society anarchizes truth. The rhetoric of a Tamil scholar like cōmacuntara pāratiyār employed chaste Tamil, classical literary style in a strictly argumentative form. If māyūram vētanāyakam piḷḷai (1826–1889), who was in the legal profession, attempted to Tamilize legal language and used ordinary Tamil in public platforms, cōmacuntara pāratiyār, who was also in the same profession, introduced the rigor of legal argument in literary Tamil in his public orations (nāvalar cōmacuntara pāratiyiṉ nūlkaḷ; while teaching tolkāppiyam, pāratiyār had also imparted oratorical skills to his disciple, V. P. K. Sundaram [1915–2003], who, in turn, adopted a similar pedagogical method while he taught tolkāppiyam to the present writer).

va¯kai So far we dwelt on the logical forms that are monologically rhetorical. Now we may turn to the dialogical ones. Dialogical rhetoric or vākai is verbal duel. In tol. it occurs as one of the stages of combat (III. 2. 18, 19). Apparently, verbal duel is found in other primal cultures too. Though it figures in the Tamil tradition, in the public (Outer) domain of tiṇai life, its origin may be traced to the Inner domain (akam). Like the male birds that try to attract the female with their song, the human males try to excel each other rhetorically when one of them attempts to capture the other. In vākai, the prize of the contest is not a third entity, such as a female, but the subdued opponent himself. If so, the origin of both logical forms – monological and dialogical—could be traced to courtship.

va¯kai in Folk Songs Dialogico-rhetorical logic informs a genre known as “tarkkam” in folk song and drama. Evidently, the older form “tarukkam” is changed in popular usage to “tarkkam.” But it must be noted that tarukkam, which figures in kaikkiḷai is a monological form, whereas “tarkkam” in folk song and folk drama is a dialogical one. In this genre we usually find a young man who pursues a non-reciprocating woman as in the tiṇai called kaikkiḷai (literally, “small relationship”). But unlike her counterpart in the tiṇai, this woman is quite vocal about her feelings. Consider the following example: Female: Why did you come over here you beggar, paṟaiyaṉ brat? You don’t want any hassles, quickly you may leave, believe me. Argue not with me. Go your way, go you may. (8th stanza)

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Before the brothers who left for the river return to raise a ruckus quietly you go away, roundly will you be thrashed. Argue not with me. Go your way, go you may. (27th stanza) Male: If the brothers come and ruckus raise I’ll whisk you away in a jiffy and fly in the sky. O my swan, poṉṉammā, I can’t keep my eyes open anymore. O my swan. (28th stanza) Female: If you fly in the sky carrying me, into the earth will I go and come up as grass. Argue not with me. Go your way, oh go you may. (29th stanza; cakannātaṉ 59–65) (Trans. Nirmal Selvamony)

A closer look at the song will show that statements (propositions) are couched in the question form and each statement is validated with reason(s). For example, “Why did you come over here. . .” restates the form, “Do not follow me,” and this injunctive utterance is followed by the reason, “Because, you do not want any hassle.” In another song, a woman’s male cousin tries to persuade her to go and watch the temple car festival. Male: O beauty with swan-like gait, O adorning jeweled beauty, O plaited beauty, desirable apple of my eye, Get ready to watch the temple car, my loving apple of the eye. (1st stanza) Female: O mataṉ*-like handsome man, O handsome man with entrancing word, O handsome man with lovely colour and form, My forehead mark-like maccāṉ,** To watch the temple car I will not go, Oh maccāṉ of our clan. (2nd stanza) (*male love god; **male cousin) Male: If you say you will not go O thou with round forehead mark, Even a few reasons, O desirable apple of my eye, Tell me without hesitation, my love, the apple of my eye. (3rd stanza) Pearl necklace will I buy you, The best earring will I give, I’ll give you, my virgin girl, O desirable apple of my eye, Marble-like coral I’ll give you, my love, the apple of my eye. (4th stanza) Female: Oil of campaṅki I need, China silk I need,

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Bombay soap I need, my forehead-mark-like maccāṉ, Ventilated blouse I need, Oh maccāṉ of our clan. (5th stanza; cakannātaṉ 68–69) (Trans. Nirmal Selvamony) (campaṅki: Magnolia champaca)

The first two stanzas state only the proposition. In fact, the predicate (pakkam) of the proposition occurs only in the fourth line of the first stanza and the fifth and sixth lines of the second. The remaining lines in those two stanzas state the subject (poruḷ) of the proposition in different poetic ways. Since the reason for not going to the festival with him is not given by the woman, the man seeks the reason in the third stanza and gives three reasons himself in the fourth as to why she should accompany him. The multiplication of reasons is a rhetorical technique the folk song contributes to the argumentative tradition in Tamil. We may contrast the rhetorical genre like cousin mutumoḻi (which does not explicitly state the reason) with the folksong, the argument in the court, the screenplay, and the political speech, which multiply reasons. In the fifth stanza, the woman states three conditions, in fact, three veiled reasons, for going with him. Analogy, as in the fifth stanza, also garnishes such songs. In yet another song, a woman goes looking for a lost bull in pouring rain out of mortal fear of having to face the wrath of her step mother. It is then does she chance upon a young man she calls, “poṉṉumāmā” (literally, “golden uncle”) with whom she engages in a verbal duel. The man asks her to forget her bull and take the one he would give her, but she insists on having her own bull. It is then does he ask, Why do you try to outsmart me? I will not be deceived. A thousand people like you have I seen – Oh golden ruby, I have seen all over the world, golden ruby. (stanza 7; cakannātaṉ 75) (Trans. Nirmal Selvamony)

It may be recalled that outsmarting is a characteristic function of the type of verbal duel called “vākai” described in tol. (III.2. 18, 19).

va¯kai in Folk Drama The rhetorical genre “tarkkam” occurs in Tamil folk drama too. In vaḷḷi tirumaṇam, we find this genre employed six times. The following is the first instance: Song: Hunter: O hill-dweller-deer that escaped the net, Did you see a deer come this way? Escaping the bow, the arrow of the bow, Running, distraught, straight towards you today – O hill-dweller-deer Confidanté: Who could see? You are a bowed hunter Looking for a deer among people. Does the deer you saw have feet and arms? Horns on the bosom, facing you directly? – Who could see? Prose: Well, smart guy! Will you look for deer among girls? You do look like someone looking for a deer! What an intrigue!

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Song: Hunter: No deceit or trickery! Great trouble entering this forest. The deer I saw in the forest Has name, domicile, and the company of girls – O hill-dweller-deer Confidanté: Thinking the girls would be unprotected Did you come over here by design? Don’t you know in this world irāvaṇaṉ because of cītai had to die? – Who will see? Hunter: O jesting beauty schooled in sciences! A flirting deer got out of control And hereabouts for red honey in this forest, a deer Straight towards the millet came over here – O hill-dweller-deer. (kuṟṟālam piḷḷai 34–35; trans. Nirmal Selvamony)

It may be noted that the confidanté phrases in question form her reasons for the impossibility of finding a deer among girls. The stanza in which she cites the authority of the canonical text (in this case, The Ramayana, which is regarded as scripture also) she implies a proposition and a reason: You shall not pursue vaḷḷi, Because, it will be fatal for you

A little later, there is tarkkam between the hunter and the central persona, vaḷḷi herself: Hunter: O divine damsel shooing birds crying ālōlam To my joyous desire yield – O divine. . . vaḷḷi: O hunter who came mysteriously, Disappear without losing dignity, O stupid fool – O hunter . . .. (kuṟṟālam piḷḷai 38; trans. Nirmal Selvamony)

If tarkkam in vaḷḷi tirumaṇam occurs only in kaikkiḷai situations between an unrelenting female and a pursuing male partner, in arccuṉaṉ tapacu also it figures in kaikkiḷai context, but between a pursuing female and an unyielding male partner. Here are the first couple of stanzas from the latter play: arccuṉaṉ: Woman, stay away from me. How can I embrace you? Without a doubt you say so much, Without fear you quarrel with me In this forest, you’re coming in my way. My intense penance you disturb Trying to enchant me saying you have a crush on me In vain you are babbling, yāk, yāk – Woman . . . pēraṇṭi: O sage in penance come near me, Embrace me give me a kiss. My soul-lord praised by the world,

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Speak to me with love, prakkiyātā, Uncontrolled is my mind, I melt in pain. Let us in the world, fondle a little, prattling the teaching Praising the feet of civaṉ, let us play – O sage . . . (arccuṉaṉ tapacu 18). (Trans. Nirmal Selvamony)

It may be noted that the genre tarkkam is used only once in arccuṉaṉ tapacu. In folk drama, tarkkam figures in kaikkiḷai situations as well as in warlike situations, and for this reason tarkkam is referred to as fight (caṇṭai, 15) in the play tiruppati veṅkaṭēcap perumāḷ ṇāṭakam. However, there seems to be a difference between tarkkam and vākai. If the former does not appeal to the audience to resolve an issue or settle a dispute, the latter does.

va¯kai in Religious Polemics If tarukkam in kaikkiḷai, vākai in puṟattiṇai, and tarkkam of folk song and drama employed several rhetorical or argumentative techniques and also knowledgevalidating criteria, mostly unconsciously, these techniques and criteria were identified and consciously deployed in the formal public assembly called avai of tiṇai society. When the latter transformed into scholarly academies (caṅkam), these techniques and criteria were employed by poets and scholars. For example, the preface of tol. composed by paṉampāraṉār invokes the previous texts (muntu nūl) in accordance with the technique known as “quoting” (III.9. 112: 22). When institutional religions like Saivism and Vaishnavism emerged in the state society displacing the primal (tiṇai) society, the earlier rhetorical tradition was deployed for religious purposes. The argumentative techniques helped establish the final truth (cittāntam) of a religious sect when the assembly became a forum for religious debate in the state society. cittāntam is, literally, conclusion of knowledge: cittu, will, mind, knowledge +antam, end, conclusion or cittāntam. Significantly, the word, antam, is used here as a synonym of the fifth member of kāṇṭikai (conclusion). In short, cittāntam is conclusive truth. When you know conclusive truth, you do not argue and deploy the argumentative techniques and genres to find the truth in a dialogical and heuristic manner, but defend the conclusive truth or refute the opponent who brings up something that counters or menaces your conclusive truth. Embroiled in forensic arguments, each sect tries to charge the opponent with falsehood. One’s religious position is true because it is sacralized. This is clearly evident in the strategies of “poruḷ vātam” (material argument), which consisted in the successful performance of a miracle such as healing an incurable disease, and “aṉal vātam” (fire argument) and “puṉal vātam” (water argument) of the bhakti period. Contenders of two religious sects, who performed the latter two types of argument, put their “truth-texts” to test by fire or water. If the text was not consumed by fire, the sect of the unburnt text won, and when the text was immersed in a flowing river, if it was not washed away, it was considered the text that asserted the truth (veḷḷaivāraṇaṉ 102–19). It might be noted that the message in the text needed to have power enough to survive fire and water. Only the Truth can invest the text with such power.

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Of the 14 canonical texts known as meykaṇṭa cāttiram (literally, Treatises That Saw Truth) that embody the doctrinal truths (conclusions) of Saivism, civañāṉa cittiyār is of special interest to us, particularly because it is rooted in the local logical tradition. The text has two sections, parapakkam and cupakkam. If the former is the doctrine (cittāntam) of the opponents, the latter is that of the author. The author states the views of 13 opponents, refutes them, and establishes his own Saivite position. In the latter part of the text (cupakkam), we find 14 verses under the heading of “aḷavai iyal” (logic). These verses are about the following topics: three criteria for validating knowledge, four types of perception, inference, proposition, reason, kāṇṭikai (which is here named “col”; syllogism-like text), other inferences, authoritative text, and inferential fallacies. These might have been regarded, in the medieval Tamil country, as the fundamentals of logic. We may consider just two stanzas from this text, the 9th and the 11th, as they are of significance to our deliberations. The ninth states that “pakkam tuṇiporuḷuk kiṭamām,” which means that the proposition (pakkam) is the locus (iṭam) of the predicate (tuṇiporuḷ, that which is concluded). For example, in the proposition (pakkam), “This hill is on fire,” the predicate is “on fire.” The predicate is what remains to be concluded in the argument. The 11th stanza tells us that kāṇṭikai was known as “col” (saying) in the medieval times and that it had two types — the affirmative and the negative: Affirmative saying (col) avers the presence of fire from the presence of smoke, as in the case of the fireplace, and negative saying avers the absence of fire from the absence of smoke, as in the case of the pond with lotus flowers and buds. There are those who will decisively state that these two forms of saying have five limbs each. (meykaṇṭa cāttiram, volume 1, civañāṉa cittiyār, 19; murukavēḷ 30; trans. Nirmal Selvamony)

Evidently, kāṇṭikai enjoyed wide application in the debates among scholars from different religions. The Saivites rallied their forces to intimidate the well-established scholarship and popularity of the Jains and Buddhists invoking the ideology of Tamil identity. It is not hard to find evidences for logical disputes among rival religious groups in the devotional literature of the period (tēvāram. campantar. 312.5; veḷḷaivāraṇaṉ 119–121; 285–288). In these disputes kāṇṭikai played a major role. Probably kāṇṭikai was alternatively known as “col” even during the pre-state period, but it is hard to find any evidence for such usage. But this medieval equivalent resembles the earlier one in terms of its five-part structure, and also in respect of all names of parts but one. The earlier name for the fourth part was not ēṟṟal (murukavēḷ 30) but naṭai (tol. III. 9. 103: 3).

va¯kai in the Modern Court of Law The purpose of the use of kāṇṭikai in religious polemics was to defend or refute a foregone conclusion. If the religious forum hierarchized truth as the orthodox and the heterodox, abandoning the open-endedness that characterized the tiṇai forum of argument, the modern court of law of the industrialist society desacralized (or secularized) truth and expressed it in the form of dictates of the state which

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went by different names – statutes, enactments, orders, directives, and so on (Jayaraman, Personal interview 2016). Now truth was neither the opposite of falsehood as it was in the state society until then, nor doctrine, but rule. Apart from its disengagement from the ultimate, the notion of truth as rule was anarchic as it lent itself to be interpreted to one’s advantage. In the forensic legal game of ingenious interpretations of statutes, the role of the foregone conclusion was significant. The ratio decidendi of a court is useful not only to those in the legal profession but also to the common people who are likely to use the courts of law (Jayaraman, Personal interview 2016). Understanding the close similarity between the ruling and the conclusion in the argumentative text, māyūram vētanāyakam piḷḷai (1826–1889) translated ruling, “cittāntam” (adopting a medieval term for a modern entity) and published in 1863 a compendium of the rulings of the court (Sudder Udalut) between 1850 and 1861. When there was no publication in Tamil on modern law, for the first time, vētanāyakam piḷḷai brought out a modern legal document in Tamil. This publication helped ensure transparency in matters of law, empowered those who did not know legal language and English (vētanāyakam piḷḷai iii) to participate in modern democracy better (Sen), and, most of all, ushered in Tamil modernity (Though modernity in South India is attributed to Serfoji II, a Marathi king who ruled tañcāvūr principality between 1798 and 1832, Tamil modernity, in particular, commences with vētanāyakam piḷḷai (Peterson; Selvamony 2011: 5).

va¯kai in Screenplay One of the important agents of Tamil modernity was cinema as it helped propagate the ideal of democracy. Further, the argumentative tradition and institutions, which nurtured it, including the court of law, played a significant role in the democratic processes in the society (Sen). We have seen how the argumentative tradition that commenced in the public assembly of the tiṇai society, despite historical changes from time to time, shaped the Tamil society. In the early twentieth century, the rhetorical strategies were deployed for the purpose of producing the screenplay also. Here is an example from the screenplay karuṇāniti wrote for a movie called, parācakti (literally, “transcendent power,” here, goddess who is “cakti” [power], 1952). Consider the following excerpt of the forensic argument of kuṇacēkaraṉ in the court of law defending his alleged antisocial actions. I created trouble in the temple (proposition 1) Not because I am against temples (reason expressed negatively, or negative reason) Because the temple should not be the place of the wicked (affirmative reason) I attacked the priest (proposition 1) Not because he is a devotee (of God) (negative reason) To reprimand hypocritical devotion (affirmative reason) Why are you so concerned about it? (question 1) Why do you show concern nobody else in the world does? (question 2) (Part of each question stated separately for formal effect): You will ask, I myself was the victim (answer 1) Direct victim was I (answer 2)

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You will say I am selfish (anticipation of counterargument) My selfishness is not without a tinge of common good (reply to counterargument) Like a fish that cleans the pond by eating the dirt in it (illustration) How do they say I am a criminal? (question 1) If you retrace a few steps in the life-path of this criminal, you could count the number of flash floods he has crossed (answer 1) There are no koels in my path in the jungle, Snakes with raised hoods abound there (answer 2) I have not known breeze I have crossed fire (answer 3). (Trans. Nirmal Selvamony)

In this speech, the question is in fact an alternative form of the proposition, and the answer, that of the reason. For example, “Why are you so concerned about it?” is a dramatic and rhetorical alternative of the proposition, “I am greatly concerned about temples and priests” and its reason: “I am a victim.” The technique of multiplication of reasons found in the folk song is adopted in the Tamil screenplay too. The following dialogue, from the play vīrapāṇṭiya kaṭṭappommaṉ (1959), is very popular among the Tamil-speaking people all over the world. Written by a Tamil scholar-politician, ma.po.civañāṉam (1906–1995), the argument in this screenplay adopts the technique of multiplication of reasons. Here is an excerpt: vari, vaṭṭi, kisthi. . . tax, interest, land tax. . . yāraik kēṭkiṟāi vari? Who are you asking to pay the tax? etaṟkuk kēṭkiṟāi vari? Why do you ask for tax? vāṉam poḻikiṟatu, pūmi viḷaikiṟatu It rains, the earth yields uṉakkēṉ keṭṭavēṇṭum vari? Why pay tax to you? eṅkaḷōṭu vayalukku vantāyā? Did you go with us to the fields? nāṟṟu naṭṭāyā? Did you plant the seedlings? ēṟṟam iṟaittāyā? Did you draw water? allatu, koñci viḷaiyāṭum eṉkulap peṇkaḷukku mañcaḷ araittāyā? Or, did you grind turmeric paste for my prattling playful kin women? māmaṉā? maccāṉā? Are you my uncle? Son-in-law? māṉam keṭṭavaṉē Shameless you! (Trans. Nirmal Selvamony)

The argument may be analyzed in the following manner: Proposition 1: We do not have to pay you tax Reason: 1. Because we have not taken our food from you 2. Because we have not taken your labor for

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2.1 planting 2.2 drawing water 2.3 grinding turmeric paste for our women Proposition 2: We do not owe you any money Reason 1: Because you are a stranger to us 1.1 you are no uncle to us 1.2 you are no son-in-law to us 2. Because you are a shameless person, you demand money from us who do not owe you anything In emotionally charged dialogues, the argument employs mostly reasons (ētu) rather than analogy and does not develop to a conclusion through application in order to qualify as a five-member argumentative text. Like the modern court of law, courtroom in a screenplay too approaches truth in an anarchic manner, leaving the validity of truth to the force of evidences.

va¯kai in valakka¯tu manram ¯ ¯ ¯ other cultural practices, which nurture the argumenta˙there are Besides the movie, tive tradition. One of these known as vaḻakkāṭu maṉṟam (forum for disputation) is a forum for formal examination of evidence to decide guilt. The plaintiff levels the charges with evidences, and the defendant refutes each one of them with evidences, and finally the arbiter sums up the argument, stating the ratio decidendi and the judgment. In between, the plaintiff or the defendant may press the arbiter to give evidence for the latter’s claim (vēṅkaṭakirucṇaṉ 11). As the judgment for each charge is given at the end of the debate of each charge, the final judgment is no suspense as in the case of the paṭṭimaṉṟam, but an opportunity for the judge to advance more reasons by way of justification (11). This forum also employs the rhetorical strategy of multiplication of reasons. Though vaḻakkāṭu maṉṟam also may be traced to the public assembly of the tiṇai society, modern version of this practice is evident from the nineteenth century, when the colonial court of law influenced the native argumentative traditions. A typical example is the literary court held (sometime before 1876) in an open hall in the temple of cīrkāḻi to try the worth of the new text, cīrkāḻik kōvai authored by makā vittuvāṉ mīṉāṭci cuntaram piḷḷai. The arbiter of this forum was māyūram vētanāyakam piḷḷai (naṭarācaṉ 60–63). va¯kai in pattimanram ¯¯ ˙˙ maṉṟam, Like vaḻakkāṭu paṭṭimaṉṟam also owes its origin to the Avai of tiṇai societies. Though the term, paṭṭimaṇṭapam, which refers to a “hall for the meeting of scholars” (Tamil Lexicon IV, 2425), is attested in the epics (cilappatikāram 5: 102; maṇimēkalai 1: 61), there is no evidence of the word paṭṭimaṉṟam (an assembly for literary debate) in ancient Tamil literature. In the twentieth century, it became a popular feature of Tamil festival celebrations, particularly, poṅkal (spring festival).

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Several media and academic institutions patronize it as it has tens of thousands of Tamil enthusiasts all over the world. In a typical paṭṭimaṉṟam, the arbiter (naṭuvar, literally, mediator) opens the debate by briefly introducing the topic of argument saying that (s)he will give the judgment at the end after duly considering the points made by each side. After each speaker resolves the problem posed for the debate by asserting any one of the sides of the argument, the arbiter pronounces the judgment by choosing the winning side and stating the ratio decidendi. In the example given by vēṅkaṭakirucṇaṉ, the problem for the debate is this: “Which predominates in the songs of cuppiramaṇiya pārati – love of god or love of land or love of language?” After listening to the case made by the speaker from each side, the arbiter concludes the argument in favor of love of land as the dominant theme of the songs of cuppiramaṇiya pārati. How are the two debating fora – vaḻakkāṭu maṉṟam and paṭṭimaṉṟam – related? The former is said to be an offshoot of the latter (iḷañcēraṉ 240–241). However, if we consider the modern-day practice of these, it is possible to distinguish them. The debaters in both fora are learned persons (pulavar, iḷañcēraṉ 1, 39). This is so because both require knowledge of texts as the authority and validity of the arguments finally rest with the texts cited. Some of these texts are cilappatikāram, Mahabharata, Ramayana, and tirukkuṟaḷ. Of all these texts, tirukkuṟaḷ is quite unique because it is regarded as a compendium of statutes. Further, if the objective of paṭṭimaṉṟam is maximal persuasiveness, that of the vaḻakkāṭu maṉṟam is finding and establishing the truth. In other words, the former is a rhetorical forum, whereas the latter is a court that settles disputes. For example, the legendary dispute kaṇṇaki had with the king of pāṇṭiyar about the culpability of her husband sought to answer the question, not unlike the modern crime thriller, “Whodunit?” Here, the authority did not lie with texts but with objective evidence for the right anklet of the queen, namely, the pearl pellets in the anklet. Since the contention (vaḻakku) of the debate is resolved by finding the truth, this forum is called vaḻakkāṭu maṉṟam and the canto (in cilappatikāram), which deals with the contention, “vaḻakkurai kātai” (kātai, canto; vaḻakku, contention; urai, state). If the truth envisaged by the vaḻakkāṭu maṉṟam in the epic is unassailable fact, the truth presupposed by the modern literary court is that which is plausible.

The End of Logic So far we have shown how logic is both finding the reasonable and communicating it. The reasonable is that which befits truth. Though reason tends to separate the elements it analyses, the ultimate end of the pursuit of the reasonable (or logic) is envisioning the truth. This is why tol. calls both logic and philosophy, “otta kāṭci.” The adjective “otta,” meaning “fitting,” describes philosophy and logic as vision that fits reality because the basic mode of knowing the world has been analogy, which helps us know the unknown by seeing how it fits with what we already know. Analogy developed into an enthymemic text called “mutumoḻi” and “niyāyam.” An ancient Tamil verse traces the origin of niyāyam to Tamil sources: “From

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tolkāppiyam, and tiruvaḷḷuvar’s ancient text originated Sanskrit nyaya” (“tolkāppiyam tiruvaḷḷuvar ātinūl/vaṭamoḻi niyāyam vaṉtaṉa . . .” murukavēḷ 117). If so, the source of niyāyam is mutumoḻi defined in the former and exemplified in the latter Tamil texts. If non-Tamil “nyaya” is an equivalent of kāṇṭikai, then mutumoḻi as an early form of the partial kāṇṭikai must be the source of non-Tamil nyaya. The latter came into being in the first century of the present era with the introduction of kāṇṭikai in the science of debate (Vidyabhusana xvi). In fact, the Sanskrit word, “nyaya” meant the syllogism-like text, and since this type of textual evidence is the basis of logic (and not just perception), “nyaya” itself came to mean logic in the 1st c. ACE in the non-Tamil circles (Vidyabhusana 41–42). Apparently, the first non-Tamil logical text is Aksapada Gautama’s Nyaya-sutra (Vidyabhusana xiii), and its ultimate objective was (like the other six non-Tamil systems of philosophy) “liberation, which means the absolute cessation of all pain and suffering” (Chatterjee and Datta 187). However, it must be pointed out that mutumoḻi, the precursor of nyaya, originated in the debates of the Tamil primal society, whereas Nyaya, one of the six systems of non-Tamil philosophy, emerged in the state society. The Tamil practice of niyāyam closely relates to the concept of nayam. Both Tamil terms mean justice and logic. The term niyāyam is used in the sense of justice even today in rural places (which continue some of the primal practices) as evident in a folk song from tarumapuri district of Tamil Nadu, which refers to the practice of the villagers convening a court of justice in the platform under the tree in the village: “puḷiyamarattiṉ kīḻē. . .poṇṇu niyāyam pēciṉāḷē” (the girl held forth under the tamarind tree on what was just in the case, vāṉamāmalai 558–559; naṭēca kavuṇṭar). niyāyam derives from nayam (Ta. nayam>niyāyam), which is neither of the two sides of an issue but its middle (naṭu) alternative. If nayam meant niyāyam (katiravēl piḷḷai 1347), niyāyam did naṭu, justice (Winslow 672). We might do well to remember that the ultimate value of tiṇai society is not justice, but love (aṉpoṭu puṇarnta aintiṇai, tol. III. 3. 1: 2). But it is hard to rationalize the means to love. If justice is impartiality (nayam/niyāyam) in social dealings, it may be regarded as a rationalized form of love, a necessary condition for living together amicably. For just and peaceful coexistence, the epistemic criteria helped validate people’s statements, claims, and assertions. Singly or collectively, these criteria were used at different points of time especially in social discourse. For a very long time, the statement of what one observed, the reason for affirming the statement, and analogy to establish the validity of the observation would have remained separate logical entities without getting consolidated into a single logical text as such (like mutumoḻi or kāṇṭikai). Yet these separate logical entities were necessary for socially acceptable communication, and justice with peace. Indeed, justice was necessary for the harmonious integration of the members of tiṇai. Theoretically, tiṇai is a harmonious community as the name suggests (from iṇai, joining, fifth relation in music, spouse; t + iṇai ¼ tiṇai, joining, a type of community). The members of this community, namely, humans, nonhuman beings such as trees and animals, and the spirit beings, form a family in which each is bound to the other by kinship. Each member is ontically continuous with the other. Like the tonic and the fifth notes (musicologically, iṇai pitches) in an “octave,” which are

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distinguishable as distinct notes but yet blend so well as to sound as if they were a single pitch, the limbs of the body are distinguishable and yet remain organically identical with the rest of the body. Such a state of being is neither oneness nor discreteness but ontic differentiated continuity (Selvamony 2005, 2007, 2008, 2010, 2013, 2016–17). The implications of the state of ontic continuity for logic are significant. For one thing, definition of entities becomes problematized. If we say that x is continuous with y ontically, the conventional definition of x as “not y” or y as “not x” is no more valid. In the tiṇai world, x is x, as well as y, which is a middle alternative (naṭu), the perspective of nayam. This third middle combines both x and y yielding an ontology of differentiated continuity, which is crisply stated in tol.: oṉṟē vēṟē eṉṟiru pālvayiṉ oṉṟi uyarnta pālatu āṇaiyiṉ. (III. 3. 2: 1–2)

(By the decree of the higher part, the two parts, one and the other, united; trans. Nirmal Selvamony). The third is the higher part, which unites the two parts in a particular way. Let us explain this with the help of a couplet from tirukkuṟaḷ: Even falsehood is considered truth provided the former yields good. (292)

Truth (the first), and falsehood (the second) are differentiable in a context only in relation to the ultimate value (which is the higher third). It is the ultimate good that will decide (by virtue of its decreeing power or āṇai) whether falsehood would count as truth in a particular context. The need for context-specificity shows that truth and falsehood are continuous, and yet differentiable. On the one hand, they become continuous when one becomes the other, and on the other, they are differentiable as truth and falsehood. As tirukkuṟaḷ emerged in the state society where good and evil had already become oppositional, its author had to point to occasions where they were not merely oppositional but ontically continuous. Knowledge of the ontology of primal community called tiṇai is necessary to understand the nature of logic that prevailed in such communities. The logic of this era is markedly different from that of the state society. If the logic of the latter is based on the ontology of discontinuity (as evident in its law of non-contradiction), that of the former is based on the ontology of differentiated continuity. The ontology of discontinuity informs the truth claims of the different religious sects of the state society wherein truth is not that which fits the ultimate in a given context, but a foregone conclusion, a doctrine of a religious institution. Truth and falsehood have become opposites. In the industrialist modern era, when the British helped establish the modern court of law in India, truth was further redefined. Now truth is not the opposite of falsehood (as in the state society), but a statute, a form of desacralized and mechanical truth. Apart from its disengagement from the ultimate, the notion of truth as rule in the industrialist society is also anarchic by virtue of its interpretability.

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Definitions of Key Terms aḷavai Avai cittāntam kāṇṭikai

kaikkiḷai mutumoḻi nayam ontic continuity otta kāṭci paṭṭimaṉṟam tarukkam

vākai

vaḻakkāṭu maṉṟam

also aḷavu, criteria for validating knowledge; epistemic logic; logic. public assembly in tiṇai society. foregone conclusion. five-member argumentative text. The members are cūttiram (pakkam; proposition), ētu (reason), eṭuttukkāṭṭu (illustration), naṭai (application), and muṭipu (conclusion). a tiṇai of unilateral relationship in which a boy tries to court a non-reciprocating girl. an enthymeme-like argumentative text which gave rise to the five-member kāṇṭikai. Its major component is example. justice, logic. nayam>niyāyam, justice, logic. The argumentative text that used nayam/niyāyam was called mutumoḻi. a condition of being in which the members of the community are continuous and yet differentiated, as in primal societies. the term in tol. for philosophy which includes logic. a debating forum. the rhetoric of a boy who attempts to court a non-reciprocating girl. By extension, it referred to monologically rhetorical logic and later, logic itself. tarkkam, a corruption of tarukkam; dialogical rhetoric or verbal duel, usually employed in Tamil folk song and theatre. an archaic form of vāṭam or vātu, both of which mean, specifically, verbal duel and generically, dialogically rhetorical logic and logic itself. a forum for disputation.

Summary Points • Logic is denoted by such terms as “otta kāṭci,” tarukkam, vākai, aḷavai, and nayam. • There are two major strands in Tamil logic: knowing what befits (truth, the reasonable), which is the epistemic, and communicating it, which is the rhetorical. • Epistemic logic is aḷavai, which deals with validating knowledge with various criteria such as perception, and inference. • Rhetorical logic has two major forms: the monological (tarukkam) and the dialogical (vākai). • Examples of monologically rhetorical logic could be found in classical literature, folk song, dissertation, and political oration.

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• Examples of dialogically rhetorical logic could be found in folk song, folk theatre, religious polemics, court of law, screenplay, and contemporary argumentative practices such as vaḻakkāṭu maṉṟam and paṭṭimaṉṟam. • The purpose of logic is to affirm the normative mode of truth of each type of society. If the truth of the primal society is ultimate-oriented differentiated continuity, that of the state society is foregone conclusion and that of the industrialist society is the unstable rule or statute.

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Causal Reasoning in the Trika Philosophy of Abhinavagupta

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Historical Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philosophical Rationalization of Tantric Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Encountering Buddhists . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Seed and the Sprout . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meditator’s Desire as Cause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Complex Causality of Totality Factor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Śiva as the Perfect Agent and His Autonomous Agency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Abhinavagupta ( fl.c. 975–1025 CE) is a tantric philosopher whose rigorous epistemological discussions are deeply rooted in his Śaiva metaphysics. In order to strongly withhold the Trika doctrinal principle of non-duality, Abhinavagupta, like his predecessor Utpaladeva ( fl.c. 925–975 CE), is struggling to interpret the philosophical question of causality that rests in the analysis of cause and effect or subject and object duality. In this chapter, a short example from his magnum opus tantric manual, the Tantrāloka (9.1–44), and its elaborate commentary titled -viveka by Jayaratha ( fl.c. 1225–1275 CE) is discussed while also contextualizing the process of philosophical rationalization in the history of Trika Śaivism. The champions of the theory of causality (kāryakāraṇabhāva), the Buddhists, are precisely targeted, and following rational enquiry, Abhinavagupta and Jayaratha want to prove that Śiva alone is the supreme agent (kartā) or cause (kāraṇa) and He indeed is also the effect (kārya) since both cause and effect are the manifestation of M. Kaul (*) Department of Humanities and Social Sciences (HSS), Indian Institute of Technology-Bombay (IITB), Mumbai, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_31

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and in a single consciousness. Even though the Tantrāloka is based on the revealed knowledge from early scriptures like the Mālinī vijayottaratantra, yet at every step, compelling efforts are being made to justify the revealed (āgama) knowledge with reason (yukti). Keywords

Causality · Abhinavagupta · Tantrāloka · Pratyabhijñā · Kāryakāraṇabhāva · Trika · Śaivism · Reason · Tantra · Buddhism Abbreviation

conj. em. ĪPK ĪPKvivṛti ĪPKvṛtti ĪPV ĪPVV Ked. KSTS MMK MŚV MVUT PHṛ PTv ŚD SK ŚS ŚSV TĀ TĀV TS

conjecture emendation Īśvarapratyabhijñākārikā – TORELLA 2002 Īśvarapratyabhijñā-vivṛti Īśvarapratyabhijñā-vṛtti Īśvarapratyabhijñā-vimarśinī Īśvarapratyabjijñā-vivṛti-vimarśinī Kashmir edition – KSTS Kashmir Series of Texts and Studies Mūlamadhyamakakākirā -> Madhyamakaśāstra of Nāgārjuna Mālinīślokavārttika Mālinīvijayottaratantra Pratyabhijñāhṛdaya Parātriśikāvivaraṇa Śivadṛṣṭi Spanda-kārikā Śivasūtra Śivasūtra-vārttika Tantrāloka Tantrālokaviveka Tantrasāra

Introduction Logic and reason are the hallmarks of rational thinking in any philosophical tradition. Following the Eurocentric bias, many modern commentators have often interpreted the South Asian philosophical traditions either lacking in rational thinking or being highly inclined toward spirituality/religion and mysticism and thus deemed irrational. This gives rise to some very crucial questions: first, what does it mean to be “logical” and “rational” and should only “logic” and “rationality” qualify as philosophy? In other words, should philosophical investigations be limited to “rationality” alone? If that is so, what kind of structure do we expect rationality to have? This rationality that was inbuilt into the logical philosophical traditions of

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pre-modern South Asia like the Nyāya-sūtra was formulated in terms of 16 categories (padārthas) or even in the process of 5-membered syllogism (pan˜ cāvayava). The Nyāya-sūtra, while putting forth the theory of debate, distinguishes between the good debate that is based on reasoning (vāda) and the bad form of debate that is based on “wrangling” (vitaṇḍā) (see Matilal 1998: 44–54). Even the pre-modern medical treatises like the Charaka Saṃhitā has theorized debating on the principles of logic, reason, and argument (see Preisendanz 2009; Matilal 1998: 38–43). No doubt that the art of philosophizing is synonymous with the idea of rationalizing, yet a rational argument for the sake of argument alone may neither do justice to the idea of rationalization itself, nor can it logically justify the attempt of rationalizing that concept because the rationality of something should certainly lie in what one can do with it, rather than just formulating an argument. Second, should philosophy as one of the fundamental disciplines of humanities ignore the “irrational” and “illogical” patterns of thinking those have survived in the past or continue to survive in the present as a part of any domain that involves thinking? Just as ontological questions have been raised in case of logic and rationality, is it possible to raise same concerns for “irrational” and “illogical.” Is all philosophical query synonymous with rationality alone? If not, then what is the rationale for rationalizing what is otherwise deemed to be “irrational” and how and why do we want to do it? This is all the more reason to evaluate the concepts of “rational” and “irrational” simultaneously. What is really meant by “reason?” Is it converting irrational into rational, and if something is already rational enough, then perhaps reason is not required at all. In classical Indian thought, we witness compelling examples where revealed, or scriptural (āgama) knowledge has been justified with reason (yukti), or rather I should say that the former itself is converted into a proper means of valid knowledge (pramāṇa) using the philosophical tool of reason (yukti). “[I]n India,” mentions Isabelle Ratié (2017b, paragraph 1), “the philosophical field has remained essentially scholastic, notably in the sense that most Indian philosophical systems never ceased considering scripture (āgama) as a proper means of valid knowledge (pramāṇa). And despite great differences of opinion as to the way in which scriptural authority should be defined, these movements have most often asserted the supremacy of revealed speech over reason and experience – including those, such as the Nyāya, that were concerned with logic and epistemology rather than with scriptural exegesis.” Another question to ask is: is “reason” and “logic” always successful in mapping reality? In other words how do we justify real and unreal in terms of rationality? A deeper metaphysical concern that has bothered all philosophers at all times is the question related to the “real-ness” and “unreal-ness” of the phenomenal world of our empirical experiences. And we cannot afford to claim that everything that may seem “illogical” or “irrational” to us is always unreal. Can we fully explain, using logic and reason, the world of our experiences even though it may sound completely irrational from an absolutely common point of view? Is there a way we can make sense of “illogical” or “irrational” since we experience it to be so and not choose to ignore it at the cost of not being able to explain it in terms of logic and rationality? Can experience be rationalized? Whatever is rational and logical, can it always be real? Can experience still be real even if it cannot be rationally explained? Is

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everything rational also real at the same time? This constant tension that permeated “revealed” and “rational” and its juxtaposition with real and unreal nature of experience gave rise to revolutionary philosophical strategies in classical India. For instance, to counter as one form of “irrational,” the early Buddhist logician Dignāga (c. 400–480 CE) devised the theory of exclusions (apoha) (see Matilal 1998: 98–107), the new school of Naiyāyikas emphasized the concept of negation or absence (abhāva), or someone like Śaṅkara subtly developed his whole non-dual epistemological apparatus around the concept of exclusion. The question was if the experiential knowledge cannot be verbalized, then how do we know it and how can we communicate it (see Gupta 2009 for more on such questions and their answers from Śaṅkara’s Advaita Vedānta point of view). Such questions with respect to “rational” and “experiential” were also being asked by the tenth to eleventh centuries Kashmirian non-dual Trika Śaiva thinkers like Somānanda ( fl.c. 900–950 CE), Utpaladeva ( fl.c. 925–975 CE), Abhinavagupta ( fl.c. 975–1025 CE), and Kṣemarāja ( fl.c. 1000–1050 CE). With them began the age of reasoning for Trika Śaivism. It was the time when the tension between the scriptural or revealed knowledge (āgama) and the logical reasoning (yukti) intensified and culminated into the emergence of the “Philosophy of Recognition” (Pratyabhijn˜ ā-s´āstra). In fact in their epistemological investigations, the āgama itself was brought to the fold of the valid means of knowledge (pramāṇa) (see the chapter by Navjivan Rastogi in this section of the present volume. Also see Ratié 2013, 2017b; Torella 2017). In case of the works like the Tantrāloka (TĀ) of Abhinavagupta that itself was a ritual manual (paddhati) of Trika Śaivism based on the scriptural teachings of the Mālinī vijayottaratantra (MVUT), this tension was very subtle as opposed to the Pratyabhijn˜ ā-s´āstra that, in fact, was meant to be a pure epistemological exercise. But before probing into this subtle tension, a short historical review would be strongly desirable here to understand the context.

Historical Context Kashmir possessed vibrant Tantric Śaiva traditions up through at least the thirteenth century of the common era (see Sanderson 2007a, 2009). Alongside the other major branches of this tradition, viz., Siddhānta, Krama, and others, Trika Śaivism emerged in the middle of the ninth century as a major post-scriptural ritual system of the Mantramārga (for Mantramārga see Sanderson 1988: 667. For a detailed history of the terms Atimārga and Mantramārga Śaivism, see Sanderson 2014: 8ff). It was at the same period that Mantramārga traditions were gradually evolving from their scriptural anonymity into an extensive body of Kashmirian exegesis (see Sanderson 1988: 690ff), in which the two major traditions competing with each other were the dualists (dvaita) and the non-dualists (advaita). The non-dualist authors adhered to the Trika and the Krama cultic systems, following the doctrine of the “left” (vāma) (see Sanderson 1995: 43ff). The dualistic Śaiva Siddhānta adhered to a ritual system functioning within the boundaries of Brahmanical purity, while on the other hand, Trika ritual involved transgressive contact with impure substances, persons, etc. (see

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Sanderson 1985, 1995: 17; Torella 2015). The Siddhānta ritual system centered on Śiva alone, who was to be worshipped without his consort, but the two non-dualistic currents were predominantly s´ākta, i.e., centered upon the worship of various female s´aktis, in addition to the worship of Śiva, particularly in the form of Bhairava. The most defining feature of the scriptures of the Trika was their pantheon of three goddesses, Parā, Parāparā, and Aparā, while the Krama or Kālīkrama was devoted to worship of a sequence (krama) of Kālīs (see Sanderson 1988: 673. See also Sanderson 2007a: 370–371; Dwivedi 2001: 539ff). Krama was a Kālīkula tradition that taught the esoteric worship of many forms of the goddess Kālī or Kālasaṃkarṣaṇī (see Sanderson 2007a: 250). The basis of this non-dualistic tradition mainly represented by Abhinavagupta was rooted in the teachings of the Śivasūtra (ŚS) (see Singh 1979; Dyczkowski 1992b; Sanderson 2007a: 402ff) that was further elaborated in the works belonging to the Philosophy of Vibration (Spandas´āstra) (see Dyczkowski 1987, 1992a; Singh 1980; Gurtu 1981). Based on the doctrine of the Mālinī vijayottaratantra (see Sanderson 1992; Vasudeva 2005), the great masters of this tradition Vasugupta (c. 825–850) and Bhaṭṭaśrī Kallaṭa (c. 875) taught that “ignorance” (ajn˜ āna) is the cause of bondage (bandha) (see ŚSV, 2.2:11–12). This “ignorance” was not defined as the absence of knowledge but as the incompleteness of knowledge or limited knowledge (apūrṇa-jn˜ āna) (see TĀ 1.25) (For Theory of Error in Pratyabhijñā, see Rastogi 1986; Nemec 2012). For unlike in the case of the sentient (cetana) objects, complete absence of knowledge is only found in insentient ( jaḍa) entities. Insentient entities are never subject to bondage and liberation. Thus the concept of mala that does not allow the complete manifestation of knowledge essentially refers to the incompleteness of knowledge and not not-knowledge (on the Śaiva concept of mala or impurity, see Acharya 2014). Abhinavagupta, in the first chapter of his Tantrāloka (TĀ), is clearly articulating the definitions of jn˜ āna and ajn˜ āna on the basis of the Śivasūtras (see TĀ 1.26–30). The Śaiva Siddhānta, on the other hand, was a tradition placing considerable importance on ritual, both doctrinally and in praxis, believing that emancipation (mokṣa) essentially transpires through the salvific power of Śaiva initiation (dī kṣā). But as opposed to this, Utpaladeva and Abhinavagupta endeavored to establish a system emphasizing the paramount significance of knowledge ( jn˜ āna). Thus they attacked the perceived ritualism of their Śaiva Siddhānta contemporaries, who adhered to doctrinal dualism (see Sanderson 1988: 692; Dwivedi 2000: 310ff). In Trika, the absence of knowledge ( jn˜ āna) was taught to be the true cause of the impurity (mala) which obscures the soul’s inherent Śiva-nature (s´ivatva) (see Sanderson 2007a: 372). This impurity (mala) is nothing but ignorance (ajn˜ āna) that gives rise to bondage (bandha), and it is the removal of this impurity (mala) that leads one to the state of liberation (mokṣa) (see MVUT 1.23cd: malam ajn˜ ānam icchanti saṃsārāṅkurakāraṇam|). This emphasis on gnosis did not of course mean that there was no ritual practice prescribed in the Trika itself, but Abhinavagupta understood the path of following ritual as inferior, positing superior means, such as meditation and imaginative visualization through which liberation was possible (see Sanderson 2007b: 114–115. Also see Sanderson 1990). It is also important to

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mention here that both the dual and the non-dual Śaiva traditions understood the descent of the power (of grace) (Śaktipāta) (see Wallis 2007) as the only means of getting rid of the ajn˜ āna or mala. However, while an individual (aṇu) is expected to make an effort toward attaining the grace of the Great Lord (Parames´vara), yet this grace descends on an individual only at Śiva’s own autonomous “power of willing” (Icchās´akti). In the earlier Tantric scriptures, the means of doing away with the mala was usually said to be knowledge ( jn˜ āna), Yogic practices (yoga), ritual (kriyā), and observances (caryā) (see Brunner 1992). Thus an arrangement of dividing the teaching of an āgama into four sections (pādas) corresponding to the four means was common (see Torella 2017). However, moving away from this tradition, Abhinavagupta categorizes the Trika paddhatis after the idea of means (upāyas) (see Rastogi 2012: 94). This non-dual metaphysics based on Śaiva scriptural sources was in need of concrete rational foundation that demanded deeper analysis and clear and precise explanations and arguments to let it stand on firm grounds. Thus, the Philosophy of Recognition (Pratyabhijn˜ ā) was not only a further elaboration of the teachings expounded in the Śivasūtra and the Philosophy of Vibration (Spanda) but also a way of strengthening the non-dualistic tradition on the strong and firm pillars of logical and analytical arguments. Thus, while the teachings of the Śivasūtra and the Spandakārikā were elucidating the spiritual path of this tradition, at the same time, the Śivadṛṣṭi of Somānanda ( fl.c. 900– 950 CE) (see Nemec 2011) was beginning to formulate an analytical structure for such teachings (see Torella 2002: XIII). Following the teachings of his venerable master Somānanda, Utpaladeva further laid down the concrete foundational structure for the Pratyabhijñā-śāstra in his I¯s´varapratyabhijn˜ ākārikā (ĪPK) (see Torella 2002) and the two auto-commentaries thereupon titled the -vṛtti (I¯PKvṛtti) and the -vivṛti (I¯PKvivṛti). It should be mentioned here that the latter was supposed to have been lost, but the fragments thereof have been recovered recently (see Torella 2014; Ratié 2017c). Abhinavagupta went on to expand upon this by writing two elaborate commentaries on the ĪPK titled the -vimars´inī (ĪPV) and -vivṛtivimars´inī (ĪPVV). Together, along with the immediate master of Abhinavagupta, Lakṣmaṇagupta (none of whose works have come down to us), all of them are named as, as Jayaratha puts it, the makers and commentators of logical reasoning (TĀV 2.10: tarkasya kartāraḥ vyākhyātāras´ ca|) (also mentioned by Rastogi 2012: 150–151). It may be important to mention here that while Utpaladeva and Abhinavagupta spoke for the non-dualistic Śaiva tradition, their contemporaries Bhaṭṭa Nārāyaṇakaṇṭha (eleventh CE) and his son Bhaṭṭa Rāmakaṇṭha (II) (c. eleventh CE) were performing exactly the same role translating the dualistic teachings of the Śaiva Siddhānta into the philosophical-analytical realm. Thus Bhaṭṭa Rāmakaṇṭha wrote commentaries on Sadyojyotiḥ’s Paramokṣanirāsakārikā, Mokṣakārikā, and Nares´ varaparīkṣā and also on Mataṅgapārames´varatantra, Kiraṇatantra, and Sārdhatris´ atikālottaratantra (for more details on the works of Bhaṭṭa Rāmakaṇṭha, see Goodall 1998: xviii ff). The strategies of philosophical rationalization were not missing in Bhaṭṭa Rāmakaṇṭha. Like Abhinavagupta, he too was a master of creative reuse (see Watson 2006: 388).

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Philosophical Rationalization of Tantric Sources Both Utpaladeva and Abhinavagupta are seen presenting their teachings following clear logical arguments (sattarka). Abhinavagupta took up the earlier form of Trika represented in the MVUT, which he profoundly reshaped through a combination of additional scriptural sources, oral teachings, and his own creative brilliance (Abhinava emphasizes svaparāmars´a. Cf. TĀ 4.41: gurutaḥ s´āstrataḥ svataḥ. Also see Pandey 1963: 292; Singh (PTv) 1988: 83). He ascribes primary importance to his own spiritual experience (svasaṃvit) (Jayaratha glosses svasaṃvit with svānubhava. See TĀV 1.106) followed by clear logical arguments (sattarka) and the scriptural authority of Siddhānta and Trika (TĀ 1.106: iti yajjn˜ eyasatattvaṃ dars´yate tacchivājn˜ ayā|mayā svasaṃvitsattarkapatis´āstratrikakramāt||). Though Abhinavagupta claims that he bases the TĀ on the MVUT, he evidently draws on a wide range of other scriptural texts of the Śaiva Mantramārga, from the wider pool of Trika scriptures, including the Siddhayoges´varī mata, Tris´irobhairava, Devyāyāmala, Tantrasadbhāva, and Trikasadbhāva (see Sanderson 2007a: 374) to Krama scriptures such as the Kākī kula, as well as the archaic Brahmayāmala of the Vidyāpī ṭha and various Saiddhāntika scriptures (see Sanderson 2007a: 374). Particularly Abhinavagupta is engaged in providing critical philosophical structure to the teachings of the Trika scriptures (see Lawrence 2000: 17–18). Like his master Utpala, he is a master of his exegetical craft and mines and even manipulates his sources, making use of their teachings to counter the arguments of his opponents. Their main philosophical opponents are the Buddhists, particularly the Vijñānavādins but sometimes typically Mādhyamakas. A central purpose in this philosophical rationalization is to counter antagonistic contemporaries, such as the staunch followers of Śaiva Siddhānta ritualism, who embrace philosophical dualism, and on another hand to build their own Śaiva system of thought by challenging the philosophical positions of the Buddhists who would deny the existence of the absolute-self. Abhinavagupta’s “higher non-dualism” (paramādvayadṛṣṭi) is itself as much made of the doctrinal principles of the Siddhānta and Buddhists as it seeks to transcend it. Even though from doctrinal point of view, the Siddhāntins and Buddhists are his opponents, in the former case, he affirms and justifies the claims of their dualistic scriptural sources and appropriates their theological principles to fit them in his non-dualistic Śaivism, and in case of the latter, he literary wears the philosophical armor of Buddhists and counters their principles by their own arguments while presenting them as his own in many cases, as if he was wearing a Śaiva garb (see Ratié 2010a, 2011). And he does all this masterfully and comes up with a critical philosophical structure and uses it for explaining scriptural claims and for developing philosophical arguments in debates with real or imagined opponents (see Pandey 1963: 294). By applying this strategic method, he is laying a solid foundation for a rational system that he is developing. An example of Abhinavagupta’s philosophical rationalization of Trika rituals and scriptural doctrines is his treatment of the theory of causality (kāryakāraṇabhāva) before dealing with the description of the ontic realities (tattvas) and their mutual relationship in book nine of the TĀ (9.1–44) (for more on this, see Allen 2003).

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To explain, in the hierarchy of the tattvas, the causal sequence cannot be denied in view of the cause and effect relationship (kāryakāraṇabhāva) between them. Thus each tattva is related to the next by a cause and effect relationship: each higher tattva permeates and pervades the succeeding ones, with the highest and most subtle pervading and permeating all tattvas. This makes it clear that each successive lower tattva exists in and draws its sustenance from the successive higher tattvas which are also its material cause (for a discussion on the tattvas in the TĀ, see Kaul 2018). Hence in book nine of the TĀ, Abhinavagupta ensures that he first establishes a robust model of the theory of causality prior to discussing the nature of tattvas according to the Śaiva doctrinal principles and their ontological hierarchy. Here he also feels compelled to challenge the Buddhist theory of causality since they are the ones who champion its cause. By doing this Abhinavagupta is achieving twofold aim – he is demolishing the Buddhist theory of causality while simultaneously paving a way ahead for developing his own Śaiva theory of causality. This is what will be demonstrated in the rest of this paper. In the same fashion, he also seeks to establish the Śaiva theory of knowability (vedyatā) (TĀ 10.19–97) at the beginning of the book ten of the TĀ, prior to discussing the nature of and relationships between the seven knowing subjects (saptapramātṛs) and seven objects of knowledge (saptaprameyas). Abhinava bases his discussion on the refutation of the position of the Mīmāṃsakas, establishing that “knowability” (vedyatā) is an essential nature of an object (see Allen 2011 whose thesis is precisely based on the study of TĀ 10.19–97). Another example is a short Sāṃkhya debate (TĀ 13.32–41) that opens up the Chap. 13 in the context of Śaktipāta (Grace). Following exactly the lines of argument discussed above, Abhinavagupta offers an analytical account of the theory of reflection (pratibimbavāda) (TĀ 3.1–65) in book three of the TĀ before discussing the doctrine of phonemic emanation (parāmars´odayakrama) (see Kaul 2019). Furthermore, as a part of a similar philosophical strategy, Abhinava, basing his idea of “six-limbed yoga” (ṣaḍāṅga-yoga) on his source text the MVUT, he opposes the eight-limbed model (aṣṭāṅga-yoga) of Patañjali. The uniqueness of this six-limbed yoga is that “reason” (Tarka) is accepted as the best among all the limbs of Yoga. Even if eight-limbed Yoga model is to be accepted, it would only be accepted if all of them function toward attaining the Tarka (see Muller-Ortega 2005; Rastogi 1992). By employing such critical methods and strategic models, Abhinavagupta is using a well-thought-out mechanism reusing the older scriptures: he reshapes their content and using dialectic methods offering them the efficacy of a convincing argument, thus defending and presenting his own new system. In other words Abhinava developed a Trika Śaiva system that, on one hand, adhered to key features of earlier Śaiva scriptures and, on the other, emerges as a distinct tradition with unique features of its own, much like an architect’s creative reuse of older structures and building materials. This process of creative reuse is visible not only in Abhinavagupta’s use of Āgamic scriptures but also at the polemic level in Abhinavagupta’s Philosophy of Recognition (Pratyabhijn˜ ā). In fact this process of creative reuse was not confined to Abhinavagupta alone. His celebrated predecessors

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like Utpaladeva himself, as also suggested by Rastogi, were influenced by Bhartṛhari’s dynamism of consciousness, Sāṃkhya’s ontology, the epistemology of Nyāya-Vaiśeṣika (particularly the ideas related to inferential cognition), the pramātṛ notions from the Siddhānta, and pramāṇa ideas of the Vijñānavāda (see Rastogi 2013: 4–5). Both Torella (2008) and Rastogi (2009) have put forth their assessments of how Bhartṛhari has strongly influenced the Śaiva Advaita system. As indicated earlier, Ratié has further given us detailed and critical accounts of how both Utpaladeva and Abhinavagupta skillfully, subtly, and purposefully appropriate the themes and arguments of the Vijñānavāda Buddhism to fit their own Śaiva theological framework (see Ratié 2007, 2009, 2010a, b, 2011). This is particularly the case with the Buddhist logical-epistemological school’s profound influence on the Pratyabhijñā system (see Torella 1992). One can also take an example of how Abhinavagupta appropriates the idea of Inherited Cognition (prasiddhi) of Mīmāṃsā in developing one of the fundamental pramāṇas of Śaivas viz. agama pramāṇa (see Rastogi 2013: 141–196; Ratié 2013; Torella 2013). As already mentioned earlier, the Buddhists, who were anātmavādins, were the staunch philosophical rivals of the Śaiva non-dualists, so much so that sometimes the non-dualistic Śaiva masters were seen siding with Siddhānta Śaivism to mark their animosity for the Buddhists (see Torella 2002: XXII). To challenge the position of Buddhists and Siddhāntins, a stream of thought gradually emerged adhering to a strongly non-dualistic position. The emergence of non-dualistic Śaiva traditions may be understood in part as a reaction to Siddhānta Śaivism and the Buddhist Vijñānavāda in the same way as, in the earlier philosophical realm, the advent of Buddhist logic was basically a reaction to the Naiyāyikas. This non-dualistic position, as already mentioned previously, manifested into such branches as the Philosophy of Recognition (Pratyabhijn˜ ā), which was purely a dialectic method to encounter the Buddhists and to restructure dualistic Siddhānta positions as non-dualist. This philosophical rationalization, however, subtly follows the syllogistic scheme of the traditional logicians (Naiyāyikas), so much so that the formal structure of the ĪPK itself is presented in terms of the inferential cognition for the sake of others (parārthānumāna). The inferential method itself is seen as a recognitive structure (Rastogi 2012: 158; Lawrence 2000: 50–51). Following the traditional five-membered syllogistic method of the Naiyāyikas, Abhinavagupta indicates the first verse of the ĪPK as the thesis (pratijn˜ ā), the text in-between indicates reason (hetu) and example (udāharaṇa) and application (upanaya), and the last verse 4.1.16 is the conclusion (nigamana). What is important to keep in mind here is that Śakti or the reflective awareness (vimars´a) that is nothing different from Śiva or the Light (prakās´a) is herself employed as “reason” (hetu) in the inferential cognition. It is also important to mention here that even though the importance of Sāṃkhya theory of causality in the context of both dual and non-dual Śaiva philosophical evolution cannot be ignored, yet a choice of not focusing on the Śaiva-Sāṃkhya debate has been made in this paper (for more on Śaiva-Sāṃkhya debate see Ratié 2014a, 2015; Moriyama 2016). Since the focus of this paper is a specific section of the TĀ where the Sāṃkhya position is not dealt with exhaustively, and since this topic in itself deserves

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separate platform for discussion, the author has planned to do this in a separate paper altogether. We are already aware of the strong connections between the authors of Siddhānta Śaiva and Sāṃkhya (see Torella 1999; Watson 2006). Much of Sāṃkhya has permeated into non-dual Trika Śaivism through Siddhānta Śaiva sources. For non-dual Śaivas, the cause and effect relationship is not like that of the Prakṛti and its evolutes in Sāṃkhya where Prakṛti has to be the material cause of the effect it generates. As K. C. Pandey (1963: 435–436) notes in the context of the Śaiva theory of causality: “Both the theories of causality, namely, the Satkāryavāda of the Sāṅkhya and the Asatkāryavāda of the Nyāya and others, therefore, cannot stand. For, how can the insentient, which is devoid of the capacity of placing itself in a conscious relation with that object, to which its productive activity is related, produce an effect? The two, the seed and the sprout are separate from each other, and, being insentient, are self-confined, i.e. there is no conscious relation between them similar to that which exists between the potter and the jar, that is to be created. Therefore, if such things be supposed to be related to each other as cause and effect, there is no reason why any two things should not be supposed to be related. Moreover, if the essential nature of the effect before it comes into being is non-existence, as the Nyāya holds, it can become existent in any way. For, the essential does not change. But if it be existence, as the Sāṅkhya maintains, what is then to be effected by the cause? It cannot be said that the cause effects manifestation. For, the same question can be raised with regard to manifestation also i.e. does the manifestation exist before manifestation or not? If it does, the activity to bring it about ceases to have any meaning. But, if it does not, how can it then be brought about? For, according to the Satkāryavāda, nothing that does not already exist can be brought about. The Pratyabhijñā, therefore, puts forth its own theory of causality.” The problem is that how can one bring into existence something which is not there at all, and how can one bring into being something what is already existing. Manifestation cannot be the cause of manifestation, and manifestation cannot be the effect of manifestation either. Reading the emphasis of Pandey on the concept of “manifestation” (ābhāsa) above, it must be remembered that it is not the case that there is pre-absence or pre-presence and/or postabsence or post-presence of manifestation in the process of manifestation. Can nonexistence generate existence or can insentient generate sentient? In line with Pandey’s argument above, Isabelle Ratié (2014a) has convincingly shown how Utpaladeva is completely transforming the Sāṃkhya interpretation of the theory of satkāryavāda by manipulating the ideas of abhivyakti and s´akti to fit his Śaiva model. The problem in hand is what is the ontological status of effect, i.e., the potentiality of the potent and the agency of the agent. How can the insentient entity bring about the power of effect? To mention Abhinavagupta’s foundational idea of the satkāryavāda in his ĪPV 1.4.3: An effect can be assumed [to be] either existing or nonexistent [before the operation of its cause]. As for [this thesis:] “[the effect] is both [existing and nonexistent], neither [existing and nonexistent], inexplicable (anirvācya),” it is contradicted by its own formulation, so what is the point [of considering it]? [Now,] if the pot is nonexistent [before the operation of its cause], then, since this [pot]’s ultimate reality is nothing but its having a nonexistent nature, how could it obtain [through the operation of the cause] an existence that is contradictory with its nature? For even innumerable prostrations at its feet cannot make the blue accommodate yellowness! (I have used the translation of Isabelle Ratié 2014a: 131)

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Also the Siddhānta point of view significantly differs from Sāṅkhya on the following point: as opposed to the Sāṅkhya, Siddhānta does not admit puruṣa to be originally pure because the Self has beginning-less impurities. From the Siddhānta point of view, the cognition of something arises when two reflections are united by ahaṃkāra, the reflection of the subject comes from inside, and buddhi receives the reflection of the object from without. According to them the buddhi is insentient and cannot be an agent, while puruṣa is an agent since it is sentient (see Pandey 1986: 67; Watson 2006: 92–103 for more details). On the other hand, in the Pratyabhijñā system, Abhinava makes the Sāṅkhyavādins his target because there he has to establish that the buddhi is sentient as opposed to the Sāṃkhya view that it is insentient. Even though Śaiva perspective attributes purity or luminosity to the buddhi, Torella (2002: 93, fn12) reiterates on the basis of the ĪPV and the ĪPVV, the purity of buddhi is anyway no clearer than puruṣa. This again drives us to the conclusion that ultimately consciousness alone is pure according to Abhinavagupta. In addition to this, in the present context, it must also be mentioned that David Lawrence has talked about the “Śaiva Syntax of Causation” where he brings our attention to the subsuming cause-effect relation of Śaivas with their “idealistic agential syntax.” Like all the grammatical cases (kārakas) in an action (kriyā)oriented sentence have their substratum in a conscious agent; in the same way, it is the recognitive synthesis (anusaṃdhāna) of Śiva itself that serves as the substratum of the cause and effect relationship and all the components involved in there (see Lawrence 1998, 2000: 147–149, 2008, 2014). Keeping the above discussion in mind, one may now focus on a specific example from the Tantrāloka and see an illustration of how Abhinavagupta carries out the process of philosophical rationalization. The ninth chapter of the TĀ is titled the Tattvaprakās´āhnika, and it describes the hierarchy of ontic-levels (tattvakrama) as accepted by the Trika Śaivism. However, it is important to keep in mind that Abhinavagupta begins this chapter with a philosophical enquiry into the nature of causality because he is intending to explain the inconsistencies in the earlier Śaiva accounts of the tattvakrama and gradually moving himself toward a position where he is offering a standard Śaiva model of the tattvakrama (see Kaul 2018). Besides this, it is obvious that he is standardizing the theory of causality in the context of the tattvakrama as taught in the MVUT since tattvas are connected by causal relations (TĀ 9.48cd–49ab: kāryakāraṇabhāvī ye tattve ithaṃ vyavasthite|s´rī pūrvas´āstre kathitāṃ vacmaḥ kāraṇakalpanām||).

Encountering Buddhists As mentioned previously, even though the Vijñānavādins, who are the main philosophical rivals of the non-dual Śaivas, also argue (like their Śaiva counterparts) that the entire universe is nothing but consciousness alone (vijn˜ aptimātra), but their definitions of consciousness radically differ from each other. For a Buddhist idealist (Vijñānavādin), the knowable objects cannot exist independently of mind, and they are nothing but a series of momentary interconnected events that alone exists. Things do not exist in reality, but a kind of collective consciousness arising out of a store

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house consciousness (ālaya-vijn˜ āna) does. However the difference between the two idealistic positions is that a Vijñānavādin believes that the objects apparently external to consciousness are not a part of consciousness at all. They merely appear to be there because of our ignorance. For a Vijñānavādin there is no evidence that external objects exist. On the other hand, Abhinavagupta’s thesis is diametrically opposed to that of a Vijñānavādin. In his idea of consciousness, there is nothing external to it. Even what are called external objects by a Buddhist idealist are very much a part of consciousness according to the non-dual Śaivas. For Abhinavagupta, consciousness is nothing but supreme reality itself, and since, according to him, whatever is in fact said to be existing outside the domain of consciousness is also cognized within the domain of consciousness alone, there is no question of external reality at all (Ratié 2007, 2010a, b, 2011, 2014b). Criticizing the Buddhist position, Abhinavagupta himself says: “And it has been said by the Buddhists that even in the presence of the external reality, the cognition, perceiving one and many, assumes the form of many, but (in reality) it is one” (TĀ 3.55: uktaṃ ca sati bāhye’ api dhī rekānekavedanāt|anekasadṛs´ākārā na tvaneketi saugataiḥ||). So even if Buddhist also maintains that the cognition of many external objects is one, where does the conflict with non-dual Śaivas lie? Śaivas maintain that except Śiva who is the only cause and is also the effect of himself, there is no other sentient (cetana) reality that has any potent agency of its own. Agency lies with Śiva alone because he is the only sentient agent (cetana pramātā), and his agency is nothing different from his act (kriyā-s´akti). Another question that arises is that if the cause is one, how can effect be many? For Abhinavagupta there is no fundamental contradiction between oneness and manyness. Just as, he explains, a mirror is able to manifest singularly the manyness (differentiation), in the same way, consciousness is also able to do it (see Ratié 2017a; Kaul 2019). For Vijñānavādins, however, the universe is dynamic albeit this dynamism does not belong to any potential agent. They reject the existence of a permanent self (ātman) and only admit the existence of mind. For them, the mind is able to perform all roles of self, and so there is no need to admit to the idea of self. There are actions but no permanent agent to those actions. Even if the Vijñānavādins are the main philosophical rivals of the non-dual Śaivas, yet when it comes to establishing the Śaiva theory of causality in the TĀ, the challenge is how to target the champions of the Buddhist theory of causation, the Mādhyamika school that is mainly represented by Nāgārjuna (c. 150–250 CE). Nāgārjuna has shown in his Mūlamādhyamikakārikā (MMK) how causation is not possible on an absolute basis and how according to his doctrine the first cause does not exist in any form since whole phenomenon is based on the principle of momentariness where phenomenal entities come into being and die out in time and space. The existential phenomenon is interdependent, i.e., there is a chain of cause and effect relationship which gives rise to the universal phenomena. This is what, in brief, the Buddhist theory of dependent origination (pratī tyasamutpāda) stands for. In the short section of the TĀ what is being focused in this chapter, Abhinavagupta is clearly not content with the Mādhyamika Buddhist’s theory of causation. But what do Śaivas want to achieve by not agreeing with the causality principle of the Buddhists. As David Lawrence (2000: 147) notes: “The Śaivas

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elaborate some of the same basic syntactic considerations to produce an interesting refutation of the Buddhist logicians’ understanding of causation as “dependent origination.” According to it, causality is a mere regularity of succession between evanescent entities, without any continuous or substantial “connection” between them. The Śaivas interpret the regular priority and posteriority expressed with the locative construction as a sort of expectation (apekṣā) between the moments. They contend that such an expectation could not exist between discreet entities that in themselves lack recognitive synthesis (anusaṃdhāna).” Thus, for Buddhists, it is essentially the consciousness alone that is the cause of all existential phenomena. If one should follow the interpretation of Navjivan Rastogi (2006: VII), then the concept of cause manifests fully in the Śaiva notion of agent on the one hand, and, on the other hand, the effect is understood as nothing but the agency or functioning of the potential agent. If seen from the point of view of cause, the relationship of cause and effect is reduced down to the relationship of agent and action, and if assessed from the point of view of effect, the expedition from cause to effect is called “manifestation” or “reflection.” The universe is not (Rastogi 2006: XVII) the effect of Śiva, but Śiva is the sentient agent who creates the universe out of his autonomous Power of Will. According to Rastogi it will not be wrong to say that the absolute freedom is the cause, and the manifestation (ābhāsa) or reflection (pratibimba) is the effect. In other words, the absolute freedom and the manifestation of it are not two different things but are simply, like cause and effect, two ways of looking at the same reality. Like, for instance, light and luminosity are not two separate things at all, but we still call them as “light” and “luminosity” where the latter is understood as the essence or essential property of being the former. Utpaladeva and Abhinavagupta in the ĪPK 2.4.14 and the TĀ 9.11ab, respectively, attack the Buddhist theory of dependent origination (pratī tyasamutpāda) that believes “that the phenomena are happening in a series and we see that there being certain phenomena there becomes some others” (Dasgupta 2004: 84ff). The causal formula of this system is; “This being, that arises” or “Depending on the cause, the effect arises” (asmin sati, idaṃ bhavati). Thus every object of thought is necessarily dependent, and because it is dependent, it is neither absolutely real nor absolutely unreal. In TĀ 9.11ab, Abhinavagupta sets up an argument saying “The essence of the situation in which some thing comes into existence upon the existence of something else is nothing but dependence” (tasmin sati hi tadbhāva ity apekṣaikajī vitam|). He further challenges this idea of Buddhists and asks “How can this theory be true in case of the things that are independent of each other in as much as they are selfcontained” (TĀ 9.11cd: nirapekṣeṣu bhāveṣu svātmaniṣṭhatayā katham|). Jayaratha writes a detailed commentary on this stanza, putting forward various arguments to refute the Buddhist position, and says that their only evidence of perception is that one sees only smoke and not-smoke after fire and before smoke only fire and not non-fire. However, it is unreasonable to decide this from the nature of perception alone, even without considering what actually are these perceived entities. Thus, Buddhists are bound to accept that there are some features within the specific natures of these two things which can limit them in this way. If this was not the case, it would be impossible to invert the conclusion that the smoke and nothing else, which

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follows fire, is a feature inherent in the nature of fire and that fire and fire alone which perceives the smoke is an intrinsic feature of smoke. But this is impossible because one inert thing cannot over determine the nature of another. Jayaratha goes on to suggest that no one, however many times and however much assisted by fate, sees a cloth after a pot even though they are thus internally unrelated still because of necessity the one is seen after the other which is not of course causally correct. He takes an example of Kṛttikā and Rohiṇī – the argument of Pleiades and Taurus is usually used to criticize Buddhists by their opponents, (see also Torella 2002: 183, fn.25) that would be related as cause and effect if the only criterion for this were to follow up the other, for Kṛttikā having arisen Rohiṇī must ascend. But Kṛttikā and Rohiṇī do not bear any causal relationship. So if the Buddhist’s only criterion is the regular succession of the effect after its cause, why do they not consider Kṛttikā’s relationship to Rohiṇī to be so as in the case with fire and smoke? There is a relation of dependence between them, and this dependence is of two types; the first consists in mutual association (anyonyānuṣaṅgitātmikā) and the second in the intentional (abhiprāyātmikā). Neither of these can apply to inert entities which Buddhists want to call cause and effect: in the first case because fire and smoke appear separately unlike fire present in a hot object which is mutually associated and in the second case because the cause and effect are produced by their unconscious nature from concealing each other’s specific identity, unlike intentional dependency such as that between an eater and its food (TĀV 9.11). (It should be mentioned here as a comment that Jayaratha in TĀV 9.11 cites references from important Buddhist texts like the Dharmālaṅkāra of Śaṅkarānanda or Śaṅkaranandana (ninth to tenth CE) (see Krasser 2001) and the Traikālyaparī kṣā of Dignāga). Cause and effect must exist simultaneously, during some time at least, in order that the action of the one upon the other should take place. According to the realist, the potter and the pot exist simultaneously. But for the Buddhist, the potter is only a series of point-instants. One of them is followed by the first moment of the series called the pot. Therefore, the cause can exist no more when the effect is produced by it. It springs up, so to speak, out of nothing, because a simultaneous existence of cause and effect is impossible (Stcherbatsky 1962: 120). Abhinavagupta says “If one were to say that A comes first and B follows, then if the prior and the posterior do not extend beyond their nature because they are totally self contained phenomena then there is no dependence between them and what they (Buddhists) are left with is only A and B” (TĀ 9.12: sa pūrvam atha pas´cāt sa iti cet pūrvapas´cimau|svabhāve ’natiriktau  cet sa sa em. Sanderson] cet sama Ked. ity avas´iṣyate||This emendation suggested by Alexis Sanderson is also supported by ĪPV 2.4.14: tatra svarūpādanadhikā cet pūrvatā paratā ca tat bhāvadvayamātraṃ, sa ca sa ca iti cārtho’pi vā na kas´cit tasyāpapekṣārūpatvāt sa sa ity eva hi syāt|). Jayaratha asks his opponents whether this priority and this posteriority are different from the actual prior and posterior entities or not. The Buddhists cannot accept the first possibility, for there is no such real thing as priority and the rest in their doctrine which could come into existence separately in the things themselves, and if they take the alternative position that they are not different or more than the things themselves, then they are left with nothing but two things which it is true they call cause and

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effect but between which they fail to indicate the nature of real affinity. Nor in the Buddhist doctrine does the causal relation exist as a specific feature over and above the fireness of fire and the smokeness of smoke. For this reason they introduce the function of supernumerary judgment (abhyuccayabuddhi). A supernumerary judgment according to them is a subsequent analysis of what was already contained in the initial perception. Perhaps the Buddhist will argue that these two features are established like momentariness as the intrinsic nature of the perceived forms and that is for this reason that a supernumerary judgment is necessary because nothing can be void of form nor can there be any sensation unfelt. So the form that a capacity to cause smoke takes is that of fire, and it is of this form that supernumerary judgment is required (TĀV 9.11–12).

The Seed and the Sprout Does cause continue to exist once effect is produced? For instance, does a seed still continue to exist once a sprout has come out of it? But a sprout can only emerge once the seed ceases to be. For a Buddhist, cause and effect belong to two different moments of time. The effect will always remain different from the cause (see Murti 2016: 173ff). Emphasizing the non-dual Śaiva doctrine of Abhinavagupta, Jayaratha says that if the nature of cause and effect is first seed and then sprout, then why should not there be a relationship between first pot and then cloth. It is possible that there could be no relation between cause and effect or between inert things. But in the world of our experience, we do find such perceptions as the sprout is born out of the seed. So we are forced to accept that there must be some subjectivity whose essential form is consciousness in which these two entities might be grounded and therefore enabled to be spoken of as cause and effect. To support his argument, Jayaratha quotes a verse from the ĪPK saying “The inert could have no power to bring the non-existent into being, therefore the relation of cause and effect is really that of agent and agency” (ĪPV 2.4.2). Therefore, in the Śaiva doctrine, the relation of cause and effect is really that of the agent and its agency, as already mentioned earlier. Abhinavagupta also points out that mere succession does not constitute causality saying: “So if the cause and effect is seed and sprout then why should not there be a causal relation between a pot and cloth. Mere succession, in other words, does not constitute causality” (bī jamaṅkura ity asmin satattve hetutadvatoḥ| ghaṭaḥ paṭas´ceti bhavet kāryakāraṇatā na kim||TĀ 9.13) (Cf. MMK 20.19–20 and also ĪPV 2.4.8–9). The opponent argues saying surely the seed and sprout are still distinct, but how can Śaivas make them into cause and effect simply by grounding them into a single subject. If the Śaivas could do this and the phenomena of a pot and cloth could be equally so since for them grounding in subjectivity is all phenomena (Kṣemarāja answers this objection very clearly in his Pratyabhijn˜ āhṛdaya (KSTS), pp. 3–4), then they must conclude that cause and effect are identical so that they can say that it is the cause that effects in various forms. We are aware that in the Buddhist model, there is no dynamic connection whatsoever between cause and effect (Stcherbatsky

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1962: 126). On the other hand, from the satkāryavāda point of view that believes in whatever evolves is by definition unconscious, and whatever changes is not conscious concluding that there can be no change in the conscious. According to satkāryavāda the universe is a self-evolving totality of primal matter. And this is what Abhinava attacks saying if the seeds were to be transformed as the sprout, the leaf, etc., then something which has one nature cannot reasonably have another nature. The opponent is asked that if he says that a seed has its nature as these, i.e. (the subsequent transformations like the seed are of the nature of sprout or leaf and so forth), then the seed and the sprout would not be contained within their own spheres (TĀ 9.14–15: bī jamaṅkurapatrāditayā pariṇameta cet|atatsvabhāvavapuṣaḥ sa svabhāvo na yujyate||sa tatsvabhāva iti cet tarhi bī jāṅkurā nije|tāvatyeva na vis´ rāntau tadanyātyantasaṃbhavāt||). Abhinavagupta is suggesting the constant transformation of degrees so that it would follow that the seed and the sprout would not be coincident with the separate spheres of their existence. This is because in that case the last stage of the one would be the first stage of the other and we would end up either having a Buddhist model in which things actually do not come into contact or we are going to have transformation in the model itself. In that case the Buddhists do not have the need to have boundaries. There has to be one thing which is manifest as a complex, which consists of seed or sprout, etc., because of the consequences of the beginning and the end of the other. Therefore it follows inevitably that this is some being taking on different forms co-extensive with the whole. Jayaratha cites the possible objection from the opponent saying: if you say so, let it be. Then we reply; some object cannot be both red and not-red. Let it be so, you may say, but we cannot allow this since this single entity is for you without consciousness, and it is not possible for something non-conscious to have a mutually contradicted form (TĀV 9.15–16). And therefore Abhinavagupta introduces the concept of succession saying “Let the insentient have a variegated form successively (through the sequence of time) but then what is contradiction therein?” (TĀ 9.17ab: krameṇa citrākāro ’stu jaḍaḥ kiṃ nu viruddhyate|). There is not something in addition to the nature of the things. They are just forms of our perception. So if the thing is successive or non-successive, there is nothing added on to the thing in itself. Those two things, succession and non-succession, are just the perceiving of those things in that way. Further Jayaratha asks that surely succession and simultaneity are not characteristics in the nature of the things themselves; they are attributes of perception. But it is consciousness which establishes the succession and non-succession when it perceives that a cloth is after a pot. The things themselves are not endowed with succession or non-succession as something super added to their natures. So the Buddhists cannot avoid the fact that they are superimposing contradictory attributes upon a single entity even though they are trying to get away by introducing succession (TĀV 9.18). The Buddhists believe in the principle of the uniformity of nature in two relations, viz., tādātmya (essential identity) and tadutpatti (succession in a relation of cause and effect). The relation of tadutpatti is that of uniformity of succession of cause and effect (Cf. Dasgupta 2004: 345).

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Finally Abhinavagupta comes down to a simple affirmation saying “If you decide that this succession is none other than the subject’s intrinsic nature and is spontaneously manifest in accordance with the intimate Lord, what more could we say. It is in this way we define that the agent is Śiva and Śiva alone” (TĀ 9.20cd–21cd: svarūpān adhikasyāpi kramasya svasvabhāvataḥ||svātantryād bhāsanaṃ syāt cet kimanyad brūmahe vayam|ithaṃ s´rī s´iva evaikaḥ karteti paribhāṣyate||). This one being that manifests succession and non-succession out of its very nature is consciousness, and that is the one agent. And this agency of this being (namely of Śiva) is simply the fact that he appears in this way, i.e., in the modalities of subjectivity, and his manifesting in this way exists in the relation of cause and effect (TĀ 9.22ab– 22cd: kartṛtvaṃ caitad etasya tathāmātrāvabhāsanam|tathāvabhāsanaṃ cāsti kāryakāraṇabhāvagam||). In his gloss on the verse 22, Jayaratha interprets tathā in the tathāmātrāvabhāsanam as vicitreṇa rūpeṇa which he further glosses as pramātṛprameyātmanāṃ mātrāṇām aṃs´ānām avabhāsanam atiriktatayeva em. Sanderson] iva Ked (TĀ vol. 6, p. 29). Jayaratha has resorted to rather unconvincing explanation here. He understands mātrā as parts, and he interprets these parts as Pramātā, Prameya, and Pramāṇa. tathāmātrāvabhāsanam means making manifest of his parts. It indeed is odd when Jayaratha mentions subject and object as the parts of Śiva. Abhinavagupta further maintains that “His (Śiva) appearing in this way exists also in the relation of cause and effect” (TĀ 9. 22cd: tathāvabhāsanaṃ cāsti kāryakāraṇabhāvagam|). So the manifestation of the relation of cause and effect (sprout and seed) is just an instance of his power of manifestation. In fact his appearing in this way, moreover, exists in causality. Just as a pot can also be manifest concomitant with the cloth or one can experience a pot after a cloth, in the same way, a cloth can follow immediately after the pot. But that is bereft of law (TĀ 9.23). Here Abhinava has tried to narrow down the definition of causality. It is just not the matter of Buddhists saying evam sati, idaṃ bhavati. There has to be a regular succession. There has to be a substantial connection between the things in addition to the mere perceived sequence. Jayaratha says in his commentary that according to the Buddhist theory, Kṛttikā should be the cause of Rohiṇī because one is always preceded by the other. But actually there is no causal relation between the two of them. Hence Abhinavagupta maintains “Therefore, we define that as the cause of that (something else) if something else appears immediately after it, but only as a result of a causal necessity (not just a matter of chance)” (TĀ 9.24) (According to Alexis Sanderson, here Abhinavagupta seems to be setting up three levels: mere coincidental succession, rule bound succession, and rule bound succession where there is a continuity of form). There is no causal connection between the two. The perception presented by consciousness does not include the perception of any causal connection. This is essentially a phenomenological analysis. It looks at what is given in consciousness. When consciousness manifests causal connection, it manifests immediate succession, but that is not all. That does not amount a perception of causality, it also amounts perception in accordance with a causal necessity. Here the additional condition is satisfied in the last pāda of the verse which reads sati rūpānvaye’

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dhike (connection of material form). One might say that we can have a cause where two things appear in succession as a result of causal necessity but that does not mean that one is the cause of another, e.g., Kṛttikā and Rohiṇī might be such a case. Manifestation of the world is such that whenever Kṛttikā appears, Rohiṇī appears. But in that case also, one is not the cause of another.

Meditator’s Desire as Cause Since the non-dual Śaivas have to be able to establish both cause and effect into the domain of a single subject adhering to their doctrinal position, they put forth the concept of Yogic creation wherein sheer Power of Will (Icchā-s´akti) of a Yogī acts as the cause of the effect. Yogis do not require a seed to create a sprout, and this stands as an example that sometimes an effect can come into being without its direct cause (see ĪPV 2.4.10– 11; ŚD 3.34–36). Yogis can create a sprout ex nihilo. They do not need a seed. This is what Abhinavagupta means when he says that “And the rule here has its essence consisting of nothing but manifestation of that form, for the manifestation of a sprout from a seed does not always occur” (TĀ 9.25: niyamas´ca tathārūpabhāsanāmātrasārakaḥ|bī jādaṅkura ityevaṃ bhāsanaṃ nahi sarvadā||). And because we come across such cases and there are sprouts of that kind which arise immediately without the pre-existence of the seed, hence in such cases there is also a causal necessity of the meditator’s desire (TĀ 9.26: yogī cchānantarodbhūtatathābhūtāṅkuro yataḥ|iṣṭe tathāvidhākāre niyamo bhāsate yataḥ||). This is a proof that in consciousness, there is also manifestation of causal necessity. In fact, it is not mere a matter of succession alone like Buddhists maintain, but there is also a causal connection between the two. It is not just that first Yogī wills and then the sprout comes into being. There is a manifestation of immediate succession (ānantarya). It should not be understood as something like random collocation of the events, but also mere succession in accordance is necessary. Also in dreams and within the territory of dreams, the pot and cloth can indeed appear to be cause and effect, and that appearance is an appearance of a connection between them in the course of cause and effect until it is contradicted (TĀ 9.27: svapne ghaṭapaṭādī nāṃ hetutadvatsvabhāvatā|bhāsate niyamenaiva bādhās´ūnyena tāvati||). Here Abhinava has given two examples of the supremacy of cause and effect: succession and necessity. It is also important to keep in mind here that as far as Abhinavagupta is concerned, the sprout born out of a seed is of the same nature as that of the sprout born out of a Yogi’s will. In the invariable association of cause and effect, since one is supposed to be prior to the other, it seems to produce an impression for limited consciousness that the posterior entity is the effect and the prior entity is the cause. Even if a Yogi’s will is the cause of an effect, it is still ruled by the idea of prior and posterior causation. However, the non-dual Śaivas claim that in case of both the cause and the effect, the real cause is consciousness. That is to say that it is the consciousness alone that is operational in the cause-ness of cause and the effect-ness of effect. So it does not really matter if we have a seed as the cause of a sprout or a Yogi’s will as the cause of any effect. In both the cases, it is consciousness alone that is being manifested, both as cause and as effect.

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The problem here is that if Śiva’s will alone is the ultimate cause of all the effects, then why do we need to bother about seeds and clay as the potential causes for their effects like trees and pots (see Kaw 1967: 213). How do we know if something that is an effect of a certain cause (kārya) or it is an essential nature (svabhāva) of the effect? It will depend on the causal law in both the cases we are told. Jayaratha states that the basis of causality is nothing but uncontradicted law (TĀV vol. VI, p. 34: evaṃ nirbādho niyama eva kāryakāraṇatāyā nibandhanam|). But what is this uncontradicted law and if causality is not mere succession, then what is it? For Abhinavagupta, the firm relationship between cause and effect, i.e., the state of being cause and effect, is where and in what form the necessary relationship and succession is manifest (TĀ 9.28: tato yāvati yādrūpyānniyamo bādhavarjitaḥ|bhāti tāvati tādrūpyād dṛḍhahetuphalātmatā||). What does this mean? Abhinavagupta uses the example of a scorpion to illustrate this point (see Kaul 2018: 257). He says: “. . .[I]n the world a scorpion can come out of cow-faeces, from another scorpion, from imagination, from memory, from the desire of a Yogi, from such factors as the power of certain substances and mantras” (TĀ 9.42cd–43ab: loke ca gomayāt kī ṭāt saṃkalpāt svapnataḥ smṛteḥ||yogī cchāto dravyamantraprabhāvādes´ ca vṛs´cikaḥ|). Elsewhere, in the Tantrasāra (Chap. 8, p. 73), Abhinavagupta reflects upon the same idea saying even if a scorpion as an effect is generated out of various causes like cow feces, insect, Yogi’s will, a mantra, and a medicine, in each of these cases, the effect (i.e., a scorpion) is the same each time despite each cause being different. Abhinava has also used this example as a maxim supporting his theory of causation in the case of a Yogi. A thing produced by a Yogi is accepted to be similar to something that is produced naturally. To clarify, he further says that this is unlike the case of a scorpion that is produced out of natural birth as opposed to the one produced from cow-faeces (see also ĪPV 2.4.11: yogī cchāpi sarvathā tādṛs´ameva na tu vṛs´ cikagomayādisaṃbhūtavṛs´cikādinyāyena kathaṃcit rasavī ryādinā bhinnaṃ kāryaṃ janayati|). What Abhinava is arguing here is that even if a scorpion produced out of a natural birth is not similar to the scorpion produced from cow-faeces, yet the idea that it is a scorpion is the same. Jayaratha elaborates the same argument of Abhinava, answering the hypothetical opponent, saying that if they consider a separate scorpion to be actually there because of some specific quality as a result of some specific cause, then surely there is some individual scorpions indexed to place, time, form, etc., that have come forth from a scorpion or faeces, etc. Each one of them has some specific characteristics pertaining to themselves, as a result of which they are different from each other. In spite of that, they always remain scorpions (the idea of being a scorpion is the same in all of them). That each is “a scorpion” is always constructed as “scorpion” through parāmars´a. So it is not wrong to teach variety in consistency of effects, both with respect to what they are, their definition, and then the order of their appearance (TĀ 9.43cd–44ab: kāmaṃ kutas´cit svavis´eṣataḥ||sa tu sarvatra tulyas tatparāmārs´aikyam asti tu|). This is what he means when he further says: “Within this law such as has been described which brings about the relation of cause and effect, causality is in reality nothing but the manifestation of consciousness and that is all pervasive” (TĀ 9.29: tathābhūte ca niyame hetutadvattvakāriṇi| vastutas´cinmayasyaiva hetutā taddhi sarvagam||).

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Complex Causality of Totality Factor The concept of complex causality (Sāmagrī ) is used by both Naiyāyikas and Buddhists in their discussions about causality. Referring to Jayanta Bhaṭṭa’s Nyāyaman˜ jarī, Dasgupta says “Nyāya Vaiśeṣika regarded all effects as being due to the assemblage of certain collocations which unconditionally, invariably, and immediately preceded these effects. That collocation (sāmagrī ) which produced knowledge involved certain non-intelligent as well as intelligent elements and through their conjoint action un-contradicted and determinate knowledge was produced, and this collocation is thus called pramāṇa or the determining cause of the origin of knowledge” (Dasgupta 2004: 330). Jayanta proposes a unique concept of pramāṇa emphasizing complex causality which is also seen as an exceptional attempt of interpreting the notion of karaṇa in Pāṇinian model of language (see Jha 2008; also see Matilal 1986: 376). While discussing about Sāmagrī in his Mūlamadhyamakakākirā (MMK), Nāgārjuna seems to criticize the idea of selfcausation (see MMK 20.1). However, Abhinavagupta typically seems to accept the totality factor of Buddhists but admitting that ultimately it is Śiva who is the supreme agent. He proposes that “in the production of a pot we say that the cause is the totality of the factors” (TĀ 9.30ab: ata eva ghaṭodbhūtau sāmagrī hetur ucyate|). In such a necessity which brings about the status of cause and effect, ultimately it is that whose nature is consciousness that is the cause and that is all pervasive. “For if one does not accept that the aggregate of causal factors forms a unity, the difference between the causes would give rise to a number of different effects” (TĀ 9.30cd: sāmagrī ca samagrāṇāṃ yady ekaṃ neṣyate vapuḥ||hetubhedān na bhedaḥ syāt phale taccāsaman˜ jasam|). Jayaratha tells us that here the word na used by Abhinavagupta in his text should be construed with an ironical sense (neti kākvā yojyam). Then if all these causal factors were distinct as the result of the multiplicity of the causes (each of these entities which refer to samagrāṇāṃ), then there ought to be a difference in the effect. It is impossible to bring about a single effect if there is not a unity in the causal factors. The grounding of all these factors in the single agent of cognition is the unifying factor. Jayaratha further maintains that resting in a single essence is the cause of all these causal factors. For if all these factors such as the clay, the potter’s stick, the wheel, etc., were causes of pot as isolated phenomena, then numerous pots would come into being because of non-disagreement that there is a plurality of effects from plurality of causes, and then the Buddhists would accept this principle (TĀV vol. VI, p. 35: kāraṇabhedāt kāryabhedasya avivādāt|). If there is plurality of causes, then there must be plurality of effects. If their causal factors do not constitute of singular causal complex, then the Buddhists would not be able to count for singularity of effects. Jayaratha adds that this Buddhist attitude is in accordance with the principle that they do not debate on the problem, namely, “if there is a plurality of causes, there would have to be plurality of effects.” Jayaratha therefore understands that these causal factors cohere in a single causal complex. Therefore because of grounding in a single agent of cognition, one must seek a single object of these things which is what they mean by the word sāmagrī (TĀV vol. VI, p. 35: ity ekapramātṛvis´rāntyā

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sāmagrī s´abdavācyamevaiṣāṃ ekaṃ conj. Sanderson] sāmagrī s´abdavācyamevaiṣāṃ Ked. vapur avas´yaiṣaṇī yaṃ|). But since the Buddhists accept the plurality of causes which is contradicted by sense perception therefore Abhinavagupta uses the expression ‘taccāsaman˜ jasam’ (and that is clearly false). We have a sense perception that guarantees plurality of causal factors. There must be something unifying those causal factors to guarantee the unity of the effect. So, Abhinava questions, would there be no plurality in the effect, and if that is so, then that will clearly be wrong because it is contracted by sense perception. Both Buddhists and Śaivas can agree to what is given to sense perception. This is where Mādhyamika Buddhists leave the discussion because they did not accept that there has to be any shared positions. Jayaratha is now defending the idea of complex causality (sāmagrī ) saying whatever is the precondition of something coming about that is its cause. He spells it out in the preceding introduction of the verse 31 saying it has been taught that that which conforms to the positive or negative of something is the cause of that. He defines the cause of something that which accords with both positive concomitance and negative concomitance. And that condition does not pertain to the clay, the potter’s stick, and the rest when taken separately, and it does exist when they are taken together. Here probably there is an embedded reference to the “triple character of inferential sign” (traikālya) of Dignāga (see Matilal 1998: 90–94). So something is the cause of something if that thing comes about when it is present and does not come about when it is absent. And Abhinavagupta tells Buddhists that if they say that then this blunt reasoning holds no appeal to them (TĀ 9.31cd: yadyasyānuvidhatte tāmanvayavyatirekitām||tattasya hetu cet so ’yaṃ kuṇṭhatarko na naḥ priyaḥ|). The Buddhist argument is that there are no real wholes and all wholes are imaginary. They reduce everything down into the minutest causal factors. According to them, it is the parts which illusorily appear as the whole, the individual atoms arise into being and die the next moment, and thus there is no such thing as “whole” (see Dasgupta 2004: 165). Keeping this in mind the Buddhists say that there is no purpose of this objective reality which Śaivas denote with the word sāmagrī. If the plurality of things is not together, then they do not produce the effect. Therefore there is no need to invent this additional thing called “collectivity” which is just an abstract entity. Jayaratha introduces the next verse by saying that if the “were not” of these things is some form of unity in addition to their merely being taken together, then these would be the undesirable consequences that even things far separated in time and place might be the cause. As Abhinavagupta goes on to say: “We believe just as the potter’s stick, cord, wheel, hand etc. are causes, so likewise distant and future entities; so we consider that these too would be causes as a necessary consequence of that position” (TĀ 9.32cd: samagrās´ ca yathā daṇḍasūtracakrakarādayaḥ||dūrās´ ca bhāvinas´cetthaṃ hetutveneti manmahe|). To this an opponent argues maintaining that in this case the positive and negative concomitance of the distant and the future has not been established so what proof is there that the Meru and Karki have some bearing on the production of the pot. “If Meru or some future being were there, surely a pot would not arise (without those obstacles). Just as potter’s wheel becomes a cause at a specific place and time, likewise the Karki and

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Sumeru will remain in their own limit” (TĀ 9.33cd: yadi tatra bhaven merur bhaviṣyan vāpi kas´cana||na jāyeta ghaṭo nūnaṃ tatpratyūhavyapohitaḥ|). The opponent poses the problem asking if what is meant is that the wheel, etc., has proximity in place and time or proximity in form. If that is the case, then it would be possible for cloth, pot or mat, etc., to be proximate. To illustrate the problem of proximity further a counter question is asked: if the potter in sitting on the mat which is present in right time and at a right place, is that also the part of the causal process? And in that case, there would be the undesirable consequences that they too would be bringing about the same single effect, i.e., pot. But the Buddhists do not accept the proximity of a substance, as it would imply that proximity is being in a fixed place and time. They too ought to be causes, i.e., Meru, etc. Therefore they must accept that there is a single form whose nature is the fact that they are rooted within a single perceiver by means of which this universe can become manifest, there being the Lord who is one reality as the agent of conjoining and disjoining of these multiple factors (TĀV 9.34). Abhinava is showing here an undesirable consequence. There has to be some additional factor which accounts for the fact that Meru and Karki are not actually causal factors in the production of the pot. It is very hard for a Buddhist not to accept that the totality thing in the universe is the collective cause of each individual event. There is no dynamic force in the blind Buddhist universe of causality which privileges some things over others. So the argument is that we have to accept that these various factors which we recognize as the cause in the production of the pot are grounded in a single substance which is the agency of the Absolute Agent, i.e., Śiva, and that provides the autonomy in the act of combining and disjoining the infinitely diverse factors of the universe, so that only some are brought together in the production of the pot and not Meru and Karki, even though they exist, especially from Sarvāstivādin point of view.

Śiva as the Perfect Agent and His Autonomous Agency According to the non-dual Śaivas, Śiva alone being autonomous can be the agent in the specific conjunctions and disjunctions of these various causes (TĀ 9.35cd: tathā ca teṣāṃ hetūnāṃ saṃyojanaviyojane||niyate s´iva evaikaḥ svatantraḥ kartṛtām iyāt|). The argument here is that the autonomous consciousness sees to it that only certain factors are relevant in the production of the pot and not others. But the opponent is still not satisfied and asks that how can Śaivas claim that Śiva is the only cause in every production, for in such cases as the pot, it is the potter that we see engaged in pot-making. The awareness of the potter in the conjoining of potter’s wheel, stick, etc., is nothing else but Śiva. There is no other specificity of consciousness. Consciousness “is” consciousness (TĀ 9.36cd: kumbhakārasya yā saṃvit cakradaṇḍādiyojane||s´iva eva hi sā yasmāt saṃvidaḥ kā vis´iṣṭatā|). If the awareness of the potter is as it were because of the manifestation of circumscription, then one must count it among the causal factors like potter’s stick, wheel, etc. (TĀ 9.37cd: kaumbhakārī tu saṃvittir avacchedāvabhāsanāt||bhinnakalpā yadi kṣepyā

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daṇḍacakrādimadhyataḥ|). The awareness of the potter is limited, but fully expanded consciousness is not, because it is not circumscribed by time, place, and form and is of the nature of absolute non-duality. Jayaratha provides a gist of the whole discussion saying that like a potter’s stick, etc., this is just a concomitant cause. The consciousness of the potter when seen from the point of view of contracted nature becomes just as it were another factor in the causal process. We have to have a conscious potter in order to produce a pot. But the fact that the consciousness is Śiva is the reason that the potter is not the immediate agent. The direct agent is Śiva. Just as consciousness that is Śiva projects its contraction itself and that contraction becomes a concomitant cause in the production of a pot, so it has been established that it is Śiva that is the agent in every cause. In the fashion of every single entity, Śiva is embodied entirely as the universal agent. Even the individual’s false notion that an individual itself is an agent is the action of the Lord (TĀ 9.38cd: tasmād ekaikanirmāṇe s´ivo vis´vaikavigrahaḥ||karteti puṃsaḥ kartṛtvābhimāno’pi vibhoḥ kṛtiḥ|). Here Jayaratha asks his opponent if actually Śiva alone is totally impersonal dynamic agent and its essential property is being identical with consciousness, then how is it that the potter thinks, even if he is in the same category as the pot and other factors of the production of the pot, that “I made this” (yan mayedaṃ kṛtam). And Jayaratha‘s answer on behalf of Śaivas follows that the notion that a potter is the agent has been established thus by the power of the causal necessity which itself pertains to Śiva. Indeed there is no plurality of selves in the non-dual Śaivism except that it is consciousness alone behaving spontaneously. Abhinava justifies that Śiva is the most competent agent of his creation (TĀ 9.8ab: vastutaḥ sarvabhāvānām kartes´ānaḥ paraḥ s´ivaḥ|) and maintains that the principle of causal relations which he is going to explain is projected forth by the will of Śiva (TĀ 9.7cd: kāryakāraṇabhāvo yaḥ s´ivecchāparikalpitaḥ|). The autonomy of Śiva is that it modifies individual consciousness on many different levels. It is always grounded in self-awareness, and that self-awareness is perennially present. There is no differentiation in that, and it is always the same dynamic force which is perceived as just conscious, and it freely manifests its own contraction.

Conclusion Either through his vast commentaries on the Pratyabhijñā literature or through his original treatises like the Tantrāloka, Abhinavagupta reiterates the crucial point having demonstrated the autonomy of Śiva as the true agency of consciousness. He maintains that self-perception is indeed a process of self-recognition which leads to the elimination of impurity (mala) or incompleteness of knowledge (apūrṇajn˜ āna) and thus to liberation. Abhinava’s metaphysics is indeed rooted in the Krama system in the kind of inchoate form and he is simply rationalising this and trying to direct the same kind of the path to expand the awareness through intellectual inquiry which brings one to suddenly see to intuit something about the consciousness. That intuition is the recognition of Śiva, and it is the same fundamental intuition that is also being pointed out in the philosophical principle of Pratyabhijñā. In fact, he is

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pursuing his own intellectual pursuit in redefining the theory of causality in the light of non-dual Śaiva metaphysics. And this he achieves by making the Buddhist’s theory of causality a target. While the principle of causality is inherently dependent on the binary of cause and effect, the major philosophical project of Abhinavagupta is to eliminate the duality of binary and present it as a principle that is invariably concomitant to the unitary cognition of the binary, i.e., the manifestation of the binary is singular. As far as the unique contribution of Abhinavagupta to the theory of causality is concerned, it must be said that he is mostly basing his premise on the philosophical principles laid down by his grand teach Utpaladeva primarily in his ĪPK, ĪPvṛtti, and ĪPvivṛti and also in his other shorter but seminal works like the I¯s´varasiddhi (see Ratié 2015) and the Ajaḍapramātṛsiddhi (see Lawrence 2009). What, however, remains a desideratum is how in the TĀ, Abhinavagupta executes and appropriates the theory of causality in illustrating the causal relationship between the 36 ontic realities (tattvas). While it might sound illogical or unreasonable to make statements like “everything is consciousness” or “both cause and effect are rooted in consciousness,” yet as this paper attempts to illustrates, Abhinavagupta makes no judgements or claims without offering any concrete arguments in favor of his philosophical position. He very carefully crafts his own philosophical stance while targeting other dominant philosophical positions on causality, mainly Buddhists, while at the same time critically evaluating their view-points and indirectly also using their philosophical tools to strengthen his own doctrinal standpoint. This paper should by no means be considered as a rigorous study of causality in Abhinavagupta. However, the context of causality has only been used to establish the point how Abhinavagupta successfully created a set of canonical texts like the TĀ that even though were meant to serve as a Trika paddhati or a ritual manual, yet they did also serve as an excellent example of how he practiced the art of philosophical rationalization by employing reason (yukti) to the revealed scripture (āgama). Note: All translations of the TĀ and TĀV in this chapter are my own, unless indicated otherwise. Readers may also consult the translations by Gnoli (1999) into Italian and both Miśra (1994) and Chaturvedi (2002) into Hindi.

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Part III Particularities

Some Issues in Buddhist Logic

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Difference Between Nyāya and Buddhist Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . History of the Dialectics Between the Two Systems of Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Approaches to Ontology and Epistemology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Major Points of Difference Between the Theories of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . How/How Far Dharmakīrti’s Logic Differs from Diṅnāga’s Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . The General Issue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nature of Avinābhāva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nature of Parārthānumāna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anvaya, Vyatireka, and Their Interrelation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Is the Statement of “Instance” Required? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Fallacy Called Viruddhāvyabhicārī . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Questions About Pervasion in Dharmakīrti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identity and the Question of Antarvyāpti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Causal Necessity and Inductive Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Issues Involved in Prasaṅga Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Prasaṅgānumāna According to Dharmakīrti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nāgārjuna’s Use of Prasaṅga Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Issue Concerning the Doctrine of Apoha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Possible Ways . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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P. P. Gokhale (*) Department of Philosophy, Savitribai Phule Pune University, Pune, India e-mail: [email protected] K. Bhattacharya Department of Philosophy, Rabindra Bharati University, Kolkata, West Bengal, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_3

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Abstract

The chapter deals with some important issues related to the Buddhist logic. Since Nyāya logic is treated as the mainstream Indian logic by many scholars, the true nature and the real contribution of Buddhist logic are many a time overshadowed by the studies in Nyāya logic. The first issue discussed in the chapter, therefore, is the differences between the Buddhist and Nyāya logic. Within the Buddhist logic, the contributions of Diṅnāga and Dharmakīrti are many times studied in a combined (and confused) way. How one can distinguish between the two is the second issue. Dharmakīrti is known for his view that probans-probandum relation should be a necessary relation, and this necessity can be derived from identity or causality. Some questions are raised such as whether identity leads to “internal pervasion” and how causal necessity can be ascertained. They are discussed in the third section. The fourth issue is concerned with the logical consistency in prasaṅga method which was used as the core method of Mādhyamika Buddhism. The last issue is concerned with the doctrine of apoha, which, though primarily concerned with word-world relation, is also concerned with the theory of inference in an indirect way.

Introduction This chapter will deal with some issues in the logical theory or theories developed in Buddhist tradition. They can be called issues in the sense that they involve a major or minor controversy either with a non-Buddhist logical theory or within the Buddhist tradition. The chapter will deal with the following issues: 1. 2. 3. 4. 5.

How Buddhist logic differs from Nyāya logic How/how far Dharmakīrti’s logic differs from Diṅnāga’s logic Some questions about pervasion in Dharmakīrti Theoretical issues involved in prasaṅga method An issue concerning the doctrine of apoha

Difference Between Nyāya and Buddhist Logic History of the Dialectics Between the Two Systems of Logic Indian logic has developed through inter-systemic and intra-systemic dialectics, and the main agents of this dialectical movement are Nyāya and Buddhist epistemologists-cum-logicians. Though Naiyāyikas were the pioneers of the theory of inference and argumentation in India, Buddhists tried to give a different turn to it sometimes by questioning it and sometimes by systematization of it. Nyāya logicians while

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responding to the Buddhist critiques and systematizations revised and reformed their own theory, and again there was a critical response from the Buddhist side. Hence the difference between the two logics has to be understood in terms of historical dialectics between the two. It is a generally accepted view that Akṣapāda Gautama, the author of Nyāya Sūtra, the earliest Nyāya text, was not the earliest Nyāya philosopher. Nyāya ideas were already prevalent, and Gautama did the task of collecting and organizing them so that a comprehensive treatise can be formulated. The date of Nyāya Sūtra can be traced back to the second century CE. The first critical reaction from Buddhist side to Akṣapāda Gautama’s categories came from Ācārya Nāgārjuna, who belonged to the second century CE. Nāgārjuna not only repudiated the categories of Nyāya logic but any possible theory of pramāṇas. From Nyāya side, the first commentator of Nyāya Sūtra, Vātsyāyana (fourth to fifth century CE), tried to establish the Nyāya categories and also address the Buddhist criticism. Around this period Asaṅga (fifth century CE) and his brother Vasubandhu (fifth century CE), the celebrated Yogācāra philosophers, are seen as taking up the study of Nyāya logic in a constructive way and trying to adapt it to strengthen the foundation of their own philosophical systems. But though Pre-Diṅnāga Buddhist texts on logic may contain seeds of certain ideas, a radical reconstruction of the theory of inference was carried out by Diṅnāga (fifth century CE) who was aptly regarded as the founder of the school of logic in Buddhism. From the Nyāya side Uddyotakara (sixth to seventh century CE), the author of Nyāyavārtika vehemently criticized Diṅnāga’s theorization and through this criticism also brought out some reformation in the Nyāya logic. Dharmakīrtī (seventh century CE) from Buddhist side not only criticized Uddyotakara, but his main task was to give Diṅnāga’s epistemology and logic a more sound and sophisticated form and to shape the version of logic which later Buddhist scholars accepted as standard. Dharmakīrti’s sophisticated theory of inference had a long lasting influence on Indian logic. Hence later Nyāya logicians such as Vācaspati Miśra (tenth century CE), Udayana (tenth to eleventh century CE), and Jayanta Bhaṭṭa (tenth century CE) appropriated and also criticized Dharmakīrti’s innovations. From the Buddhist side, the commentators of Dharmakīrti such as Dharmottara (ninth century), Arcaṭa (ninth to tenth century), and Prajñākaragupta (tenth century) played their role in this debate. Hence Ratnakīrti (tenth to eleventh century) is seen to have criticized the Naiyāyikas like Trilocana and Vācaspati Miśra in defense of Dharmakīrti. After eleventh century, however, there was a decline in Buddhist logic for various historical reasons. The Nyāya logic which developed after that at the hands of Gaṅgeśa Upādhyāya (fourteenth century CE) and various Navya-nyāya Ācāryas did not have Buddhism as their main rival system. Hence the development of the Nyāya logic and that of the Buddhist logic cannot be studied in isolation but in terms of their dialectical relationship. The contrast

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between them is not superficial or just a matter of detail. It goes much deeper than that. It was also not uniform in nature because it had a historical dimension.

Approaches to Ontology and Epistemology Before proceeding to point out the differences between Nyāya logic and Buddhist logic, it is necessary to mention that unlike other Indian philosophical schools where epistemology precedes ontology, epistemological theories of Diṅnāga and Dharmakīrti were constructed on the foundation of their ontological claims. In Pramāṇa-samuccaya, Diṅnāga says that since the knowables (prameya) are of two kinds, namely, svalakṣaṇa or the particular and sāmānyalakṣaṇa or the universal, the instruments through which they are known are also of two kinds, namely, perception and inference. Therefore, while for the Nyāya school, the object of perception can as well be an object of inference, for Buddhists like Diṅnāga and Dharmakīrtī, object of inference can be sāmānyalakṣaṇa alone. Sāmānyalakṣaṇa for Diṅnāga and Dharmakīrtī is not a real object but a mental construct. The indirect knowledge of a sādhya based on the knowledge of a hetu has certainty and indubitability within this limit. On the other hand, svalakṣaṇas, which are the objects of perception, are described as unique particulars, distinct from both similar and dissimilar entities; and they alone are said to be real in the strict sense of the term. The ontology of the Nyāya, which it borrows from and develops over Vaiśeṣika, is much more populated. It contains particulars as well as universals and substances in atomic as well as composite form, qualities, movements, universals, particularities, inherence, and absences. In fact whatever is knowable and expressible in language is real according to Nyāya-Vaiśeṣika thinkers. Naturally the use of reasoning has a very wide scope aimed at proving a large variety of “reals.”

Major Points of Difference Between the Theories of Inference The comparison between Nyāya and Buddhist theories of inference can be made at various levels. Here comparison on the following points will be presented: (i) (ii) (iii) (iv)

The distinction between inference for oneself and inference for others The nature of inference for oneself and the conditions of good hetu The nature of hetu-sādhya relation The structure of inference for others

The Distinction Between Inference for Oneself and Inference for Others Though Gautama and Vātsyāyana’s account of anumāna and stating avayava as a separate padartha suggest that they were aware of the distinction between inference as a cognition occurring to oneself and inference as a communicative act generating cognition in another person, it was not explicitly spelled out by them. Diṅnāga was the first Indian logician to clearly distinguish between these two kinds of inferences

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and apply the terms svārthānumāna and parārthānumāna to denote them, respectively. This distinction was later accepted by the Nyāya philosophers, but there is a difference between the Buddhist and the Nyāya understanding of the nature of svārthānumāna and parārthānumāna. Gautama’s conception of avayavas suggested a specific structure of a sound argument. The steps in it follow a specific order. Naiyāyikas maintained that the inference for oneself also has an analogous structure. A person first sees a line of smoke on the mountain (hetujn˜ āna), she then recollects the relation between smoke and fire (sambandha-smaraṇa), then she applies that relation to the present case (parāmars´a), and at the next moment, she has the knowledge of the fire on the mountain (anumiti). Naiyāyikas in this way envisaged isomorphic relation between the process of knowing and that of justification. Buddhist logicians did not think of such an isomorphic relation in mechanical way. However, they did talk about the logical link between the two. The logical link is brought about by the concept of triple character of hetu. Inference for oneself for them is the knowledge of sādhya brought about by the hetu having triple character, and inference for others is the presentation of the hetu as having the triple character. The point is that the inference for oneself according to Buddhist logicians does not follow a rigid temporal order as it does according to the Nyāya logicians.

The Nature of Inference for Oneself and the Conditions of Good Hetu One technical difference between the Nyāya and Buddhist conceptions of svārthānumāna needs to be taken into account at the very outset. The two conceptions differ because in Nyāya anumāna is substantially different from anumiti, whereas the Buddhist logicians do not make such a substantial distinction. If we consider the inferential process from the knowledge of hetu to the knowledge of sādhya, then, Naiyāyikas hold, the intermediate step called “recollection of hetusādhya relation” is called anumāna, and the resultant knowledge of sādhya is called anumiti. Buddhist logicians on the other hand regard the resultant knowledge of sādhya itself as anumāna. The other important difference between the Nyāya and Buddhist conceptions consists in the condition of good hetu. Diṅnāga presented the theory of the triple character of hetu. Though it is true that in some pre-Diṅnāga Buddhist texts on logic these three characteristics are mentioned, Diṅnāga seems to be the first logician to work out the theory of reasoning based on it. The three characteristics of a hetu, according to Diṅnāga, are as follows: 1. Pakṣasattva (the hetu must be present in the pakṣa) 2. Sapakṣasattva (the hetu must be present in the sapakṣa) 3. Vipakṣāsattva (the hetu must be absent in the vipakṣa) Uddyotakara raised important objections against Diṅnāga’s formulation of the triple characteristics of a hetu. According to Diṅnāga’s theory, every sound hetu should be related (by its being present or absent in) to the pakṣa, sapakṣa, and vipakṣa. Uddyotakara pointed out that a hetu can be sound even if it is related to the pakṣa and either of the two cases, namely, sapakṣa and vipakṣa. An anvayī hetu or

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universal positive mark, he says, is a sound hetu which has the pakṣa and the sapakṣa as its loci. It is nowhere found absent. A vyatirekī hetu, on the other hand, is a hetu which is absent in the vipakṣa, but does not reside in the sapakṣa because a sapakṣa, in this case, does not exist. Anvayī and vyatirekī hetus were later named kevalānvayī hetu and kevalavyatirekī hetu, respectively, and a sound hetu which involves the pakṣa, sapakṣa, and vipakṣa is named anvaya-vyatirekī . Almost all Nyāya philosophers accepted this division of hetu into three kinds, and it is acknowledged as a very important contribution toward formulation of inferential thought process. Nyāya philosophers after Uddyotakara suggested another improvement on Diṅnāga’s formulation of the triple characteristics. These Naiyāyikas advanced criticism from the opposite direction. They claimed that the three characteristics of hetu are not sufficient. The reasoning in which the hetu possesses all the three characteristics could lead to a false conclusion, under two conditions: 1. The conclusion can be shown to be false on the basis of another pramāṇa. 2. Another reasoning may be available which is at least as much strong as the proposed reasoning but proves the contrary of the conclusion. Of these three, the first condition falsifies the conclusion and the second one raises doubt about the truth of the conclusion. The first condition designates the fallacy called bādha and the second condition designates satpratipakṣatva. In order to exclude these fallacies, Naiyāyikas after Uddyotakara augmented the list of three characteristics by two more characteristics: 4. Abādhitaviṣayatva (the hetu must not be about the sādhya which is sublated) 5. Asatpratipakṣatva (the hetu must not be countered by an opposite hetu) While explicating inference for oneself, Diṅnāga specifies the conditions that yield certainty whenever one infers the sādhya on the basis of a hetu in a particular pakṣa. A hetu, to be adequate must, therefore possesses three characteristics. Dharmakīrti’s discussion of the conditions of good hetu shows that he was aware of both these objections against the theory of triple character of hetu. He answered these objections and defended the doctrine of the triple character of hetu. He, however, interpreted the triple character of hetu as implying the necessary relation between hetu and sādhya such that the former cannot exist without the latter.

The Nature of Hetu-Sādhya Relation Naiyāyikas claim that the Nyāya theory of inference had the concept of universal vyāpti relation between hetu and sādhya since the stage of Nyāya Sūtra. Such claims are made because generally the commentators claim that the interpretation they are providing exhibits the original view of the source book even if they are introducing new ideas. In this way Sūtra-Bhāṣya tradition hides the changes that take place in a system. This is true of the concept of vyāpti as well. The concept of vyāpti has evolved through the interaction between Nyāya and Buddhist logicians.

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Gautama while defining hetu and explaining the elements (avayava) of the inference repeatedly refers to the concepts of sādharmya or similarity and vaidharmya or dissimilarity. Hetu is similarity or dissimilarity of the properties of the udāharaṇa and the sādhyadharmī; udāharaṇa is the case similar to sādhya, either positively or negatively. Upanaya refers to the collection (upasaṁhāra) of the sādhya and the udāharaṇa together by demonstrating their similarity or dissimilarity (by the expressions “tathā” and “na tathā,” respectively). The term “tathā” means “is positively analogous to” and “na tathā” means “is negatively analogous to.” This shows that Gautama relied only on the relation of analogy (expressed as “being analogous to”) between sādhya and udāharaṇa for formulating a good inferential form. Obviously, analogy can never sufficiently account for the universal relation between hetu and sādhya which is needed to make a logically sound inference possible. Vātsyāyana, the commentator on Gautama’s Nyāya Sūtra, while elaborating on Gautama’s notion of inference, emphasizes on the relation between hetu and sādhya rather than only on the similarity or dissimilarity between the sādhya and the udāharaṇa. But a close examination of the way he explains hetu-sādhya relation suggests that he was talking about the relation obtained in the “observed” world and not a universal relation which pervades both observed and unobserved world. Secondly, though he talked about the observed coexistence and coabsence of hetu and sādhya, he was not clear about the specific direction of the implicative relation between them. This becomes clear when he regards “hetu existing without sādhya” and “sādhya existing without hetu” both as the cases of inconclusive hetu. Diṅnāga’s doctrine of the triple character of hetu specifies the exact direction of hetu-sādhya relation. He allowed partial existence of a good hetu in sapakṣa but insisted on its absolute nonexistence in vipakṣa. This implies that in the domain consisting of similar and dissimilar cases, if hetu exists, sādhya too exists, and if sādhya is nonexistent, hetu too is nonexistent. Accordingly, sādhya can exist without hetu, but hetu cannot exist without sādhya. Hetu-sādhya relation is properly directed in Diṅnāga’s scheme in this way. But it is limited to the world of similar and dissimilar cases. It is neither universal nor necessary. Dharmakīrti removed this deficiency and presented hetu-sādhya relation in the form of universal and necessary relation of pervasion. Naiyāyikas, who came after Dharmakīrti, included universal and unconditional (anupādhika) vyāpti in their scheme of inference. Still, there remain some points of difference between the conceptions of vyāpti according to Dharmakīrti and Naiyāyikas: (a) For Naiyāyikas vyāpti relation is universal and also categorical. Because of categorical nature, it has existential import and has an instantiation. For Dharmakīrti, on the contrary, though vyāpti is generally stated in categorical form, it is to be understood as a necessary conditional relation. Vyāpti relation in this form can stand without instantiation. (b) As a result of (a) above, Naiyāyikas hold, though generally we have both positive and negative vyāpti because both have instantiations, sometimes we

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have only positive vyāpti without its negative counterpart because the latter has no instantiation. Similarly sometimes we have only positive vyāpti without its negative counterpart. For Dharmakīrti, on the contrary, every positive vyāpti has its negative counterpart and vice versa. Hence there is no purely positive (kevalānvayin) or purely negative (kevalavyatirekin) hetu. (c) When Dharmakīrti held that vyāpti is a necessary relation, he was concerned with the question as to how this necessity is possible. He held that it is possible in either of the two ways: Either hetu is identical with sādhya or it is caused by sādhya. Naiyāyikas were not ready to restrict the scope of hetu-sādhya relation to these two relations. It sufficed for them if hetu is observed with sādhya, it is not observed without sādhya and no obstructive condition (upādhi) is observed.

The Structure of “Inference for Others” According to the Nyāya Sūtra, there are five avayavas in a structured reasoning, and these can be expressed through five statements. The five statements are used to publicly demonstrate a reasoning process in which the sādhya or an unperceived object is established as a property of the pakṣa or subject with the help of a hetu or signifying mark. If the debater, for example, tries to prove that “the hill has fire,” the five statements will assume the following form: 1. 2. 3. 4. 5.

Pratijn˜ ā: The hill is fiery. Hetu: Because it is smoky. Udāharaṇa: Like the kitchen (or, unlike the lake). Upanaya: The hill is like that (or, unlike that). Nigamana: Therefore the hill is fiery.

The first statement asserts the pakṣa qualified by the property to be proved sādhyadharma. It, therefore, states the thesis (pratijn˜ ā). The second statement expresses the reason (hetu). The third statement expresses an example (Udāharaṇa), which is defined as the case which has similar or dissimilar characteristics as the pakṣa. The fourth statement is called upanaya. Upanaya which literally means “that which leads to the conclusion” (“upasaṁhriyate anena”) is to be understood as the statement which asserts the pakṣa as characterized by the presence or absence of the features present or absent in the udāharaṇa. The fifth statement, namely, nigamana, is the assertion of the thesis again as proved. One can notice that (1) this structure of argument involves a psychological order or a rhetorical sequence (2). It does not involve a universal statement of vyāpti (3). It is based on positive or negative analogy as it refers to similarity or dissimilarity between pakṣa and udāharaṇa. In response to this initial Nyāya formulation of argument, Diṅnāga presents a more systematic formula with three statements representing three elements of the inference: 1. Pakṣa: The hill is fiery. 2. Hetu: Because it is smoky.

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3. Dṛṣṭānta: Smoky things are seen to be fiery like the kitchen and things lacking fire are seen to be lacking smoke like a lake. Diṅnāga’s formula is oriented more by logical considerations than by psychological ones as it tries to get rid of unnecessary repetitions as one may find in the Nyāya formulation. Diṅnāga’s formula is still governed by similarity and dissimilarity. But here these considerations are systematically and logically arranged as a part of the doctrine of the triple character of hetu. Hence after stating the thesis in the first statement, Diṅnāga’s formula presents the first character of hetu (“existence in the property bearer”) in the second statement and the other two characteristics (“existence only in similar cases” and “definite nonexistence in dissimilar cases”) in the third step. Dharmakīrtī’s formulation of the inference for others follows the consideration of validity rigorously. His form of argument is no more based on positive or negative analogy, but is based on universal statement of vyāpti. The statement stating the example is redundant because its implication is revealed by understanding the nature of a hetu. The presence of the hetu in sapakṣa implies availability of positive instances. Similarly the absence of the hetu in the vipakṣa implies the availability of negative instances. (The point to be noted here: Naiyāyikas insist on availability of a real instance, whether positive or negative whereas in the Buddhist scheme even vacuous instantiation is permissible.) Now for Dharmakīrti, both the characteristics (existence in sapakṣa and nonexistence in vipakṣa) need not be stated separately because they are implied by a single statement of vyāpti whether it is stated in positive form (anvaya) or negative form (vyatireka). Dharmakīrti’s model of argument in this way consists of stating the two premises – statements of pakṣadharmatā and vyāpti.

How/How Far Dharmakīrti’s Logic Differs from Din˙nāga’s Logic The General Issue Diṅnāga and Dharmakīrti are the two pioneering figures of the Buddhist logic. According to historians like Satish Chandra Vidyabhushan, Diṅnāga belongs to the latter half of the fifth century and the first quarter of the sixth century AD. Dharmakīrti’s period is said to be about 635 AD. Dharmakīrti is said to belong to the teacher-disciple tradition of Diṅnāga. The latter had disciples such as Śaṅkarasvāmin, who taught Diṅnāga’s logic to Dharmapāla, who was the teacher of Dharmakīrti. Dharmakīrti is also said to have learned Diṅnāga’s logic thoroughly from another Diṅnāga expert, namely, Īśvarasena. Like Diṅnāga, Dharmakīrti also wrote many independent treatises on logic and epistemology. But more importantly he wrote an elaborate commentary on Diṅnāga’s work Pramāṇa-samuccaya. The commentary is entitled Pramāṇavārtika. A question can arise, as to whether Dharmakīrti simply continued the tradition of Diṅnāga or made some alterations in it. The question is important because there is no doubt about the fact that Indian

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philosophy has developed through sūtra-bhāṣya tradition. But there is no clarity on how this development has taken place. In fact, there are different ways in which such developments have taken place: (i) On the same source literature, different commentators give radically different interpretations which give rise to different schools of a system. Different schools of Vedānta have come into existence in this way. (ii) A system develops through commentaries and subcommentaries. This development is sometimes linear. A commentator or a subcommentator clarifies or elaborates the points which are available in unclear or abridged form in the source literature. But it may also have a dialectical aspect. A commentator has the task of defending the system against the criticism coming from the defenders of rival systems of his time, and he introduces newer ideas and theories in order to strengthen the system. But this is generally done as a part of elaboration and clarification of the source text and not by criticizing it. This has happened to a large extent in the Nyāya tradition. One finds there a chain of commentaries and subcommentaries where the latter commentary does not explicitly deviate from the earlier commentary, though one may find if one goes in minute details that it actually deviates in many respects. (iii) Sometimes the development brought about through commentaries may not be linear but explicitly dialectical. A commentator may sometimes radically deviate from the source material. At least sometimes he may do so explicitly, though sometimes in a disguised way. In the present context the question is: How is one to understand the development of logic from Diṅnāga to Dharmakīrti? The following alternatives come to the foreground: (i) It was a linear development. Dharmakīrti introduced new ideas or doctrines only as a part of elaboration and clarification of Diṅnāga’s theory and not in opposition to it. (ii) Dharmakīrti did differ from Diṅnāga on some details. But there was agreement at fundamental level. (iii) Dharmakīrti in his theory of inference deviated from Diṅnāga. Differences in detail were the manifestations of this fundamental deviation. However, Dharmakīrti generally avoided expressing the radical differences explicitly. Modern scholars are found taking one or the other position from above. B. K. Matilal seems to be taking the first kind of position. He claims that the idea of invariable concomitance was already introduced by Diṅnāga. Dharmakīrti made an advancement to the extent that he described it as based on own nature or causality. (Matilal 1998: 11–12). Similarly Diṅnāga had introduced twofold classification of inference into inference for oneself and inference for others. Dharmakīrti introduced another threefold classification: inference based on own nature, causal relation, and nonperception (Matilal 1998: 108–109). Satish Chandra Vidyabhushan seems to have taken the second type of position. He brings out the points on which

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Dharmakīrti criticizes Diṅnāga. The latter had introduced the fallacy of hetu called implied contradiction (iṣṭavighātakṛt, a kind of viruddha); Dharmakīrti denied its independent status by including it in the first type of contradiction. Diṅnāga had introduced the fallacy called viruddhāvyabhicārī as a variety of uncertainty. Dharmakīrti rejected this fallacy. Thirdly in opposition to Diṅnāga, Dharmakīrti maintained that example is not a part of a syllogism as it is included in the middle term (Vidyabhushan 2005: 315–318). Here it seems to be necessary to go to the root of the matter. If one goes deeper, one may find that the third possibility suggested above is stronger than the other two possibilities. Dharmakīrti was not simply clarifying or elaborating Diṅnāga’s frame of inference but was deviating from it and doing so at fundamental level. The points on which Dharmakīrti expressed explicit difference are like the manifest portion of an iceberg with the major portion hidden below it.

The Nature of Avinābhāva Although Diṅnāga talked about the invariable concomitance (avinābhāva) type of relation between hetu and sādhya, the relation of concomitance as formulated by Diṅnāga was a part of the triple character of hetu. It was limited to the domain of similar and dissimilar cases (sapakṣa and vipakṣa). The passage of inferential knowledge in Diṅnāga’s framework was from “observed” to “unobserved.” That is why Diṅnāga, while giving examples of anvaya and vyatireka, uses the word “dṛṣṭam” (“it is observed that”). In contrast with this, the concomitance relation understood by Dharmakīrti was universal and necessary. The following statements given by Dharmakīrti denoting concomitance may be considered: 1. “Yad upalabdhilakṣaṇaprāptaṁ san nopalabhyate so’sadvyavahāraviṣayaḥ siddhaḥ” (Nyāyabindu 3/8) (If an object is not perceived even if all the conditions for its being perceived are present, then it is proved to be the object of a linguistic usage like “this is not present.”) 2. “Yat sat tatsarvamanityam” (Nyāyabindu 3/9) (Everything real is impermanent). 3. “Yatsadupalabdhilakṣaṇaprāptaṁ tad upalabhyateeva” (Nyāyabindu 3/23) (That which is existent and has all the conditions for its being perceived present will certainly be perceived.) 4. “Anityatvābhāvekṛtakatvāsambhavaḥ” (Nyāyabindu 3/122) (If something is not impermanent, then it is impossible that it will be a product.) 5. “Yadutpattimattadanityam” (Nyāyabindu 3/10) (That which has origination is impermanent.) These statements denting concomitance show that for Dharmakīrti, the relation between the probans and the probandum should be already proved (siddha), universal (sarvam), and certain (eva) and its negation should be impossible (asambhava) (Gokhale 1992, 88–89).

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This shows that while for Diṅnāga the relation of concomitance was restricted to observed similar and dissimilar instances, for Dharmakīrti the relation should be pervasive enough to include all cases, observed and unobserved. Dharmakīrti uses the term “sarvopasaṁhāra” to express such universality. Universality, however, is here obtained by showing that the relation is necessary. In order to ensure a necessary relation, Dharmakīrti introduces the notion of existential dependence (svabhāvapratibandha) which subsumes under it the notions of essential identity (tādātmya) and causality (tadutpatti). A probans necessarily implies a probandum if it depends on the probandum for its own existence or if it is identical or causally connected with it. The relation between the probans and the probandum will therefore be necessary if any of the following conditions obtain: (1) when the probans is essentially identical with the probandum, that is, when the inferential relation is a relation of class inclusion, as in, “This is a tree, because this is s´iṁs´apā (a kind of tree),” and (2) when there is a causal connection between the probans and the probandum, as in, “The hill is on fire as it has smoke.” In this way the difference between the approaches of Diṅnāga and Dharmakīrti to inference is not peripheral but at the core. Different points of difference between the two theories follow from this basic difference. A few points are discussed below.

The Nature of Parārthānumāna For Diṅnāga inference has three elements, namely, the inferential subject, probans, and the example. The structure of a sound inference for others should be, accordingly, as follows: Thesis (pratijn˜ ā): Sound is impermanent. Reason (hetu): Because sound is a product. Example (dṛṣṭānta): (a) Statement of similarity (anvaya), anything other than sound which is a product is impermanent, for example, pot, and (b) statement of dissimilarity (vyatireka), anything other than sound which is not impermanent is not a product, for example, sky. Dharmakīrti did not accept the above structure of three limbs of inference. He says that an inference made for others has only two constituents: the statement of the existence of hetu in the inferential subject (pakṣadharmatā) and the statement of concomitance (vyāptivākya). Hence according to him, the structure would be: Reason (hetu): (Sound is impermanent) because sound is a product. Concomitance (vyāpti): Whatever is a product is impermanent. Or: Whatever is not impermanent is not a product. This had implications to their approaches to fallacies. Since inference for others had three elements according to Diṅnāga, fallacies of an inference accordingly are of three kinds – that of the inferential subject (pakṣābhāsa), probans (hetvābhāsa), and

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the example (dṛṣṭāntābhāsa). Of these, Dharmakīrti was more concerned with the fallacies of probans (because as mentioned above, he did not give much importance to the thesis and the example) and discussed the different kinds of hetvābhāsas with greater details. In Diṅnāga’s framework of “inference for others,” instance covers “invariable concomitance.” The latter is limited to similar and dissimilar cases, and the subject of the thesis is kept outside the domain of the concomitance. Hence the thesis is not entailed by the premises. In Dharmakīrti’s framework of the inference for others, the invariable concomitance is universal, and hence it covers the subject of the thesis as well. Hence the thesis is entailed by the premises. Therefore Dharmakīrti claims that only two steps are sufficient. The thesis need not be stated because it is entailed by the two premises which state vyāpti and pakṣadharmatā.

Anvaya, Vyatireka, and Their Interrelation One of the issues in Buddhist logic concerns with the relation between the second and the third characteristic of hetu. Dharmottara, the commentator of Dharmakīrtī’s Nyāyabindu, argues that the second and the third characteristics of the hetu imply each other and, therefore, either of them is rendered redundant. This is one of the issues which become complicated if one identifies Dharmakīrti’s theory of inference with that of Diṅnāga. In order to tackle the issue, therefore, one has to consider the two theories separately. The second and the third characteristics according to Diṅnāga’s formulation can be stated as follows: 2) Hetu exists only in sapakṣa. 3) Hetu is definitely nonexistent in vipakṣa. The universe of discourse of an inference in Diṅnāga’s theory is exhausted by pakṣa, sapakṣa, and vipakṣa. In the first characteristic, Diṅnāga talks about the status of hetu in pakṣa. Naturally in the second and third characteristics, he is concerned with the world outside pakṣa, which consists of sapakṣa and vipakṣa. In this world (outside pakṣa), if hetu exists “only in sapakṣa,” it follows that it is nonexistent in vipakṣa (provided that vipakṣa exists). This follows by the force of the word “only” which excludes the other possibility. In this way the second characteristic implies the third characteristic. Similarly if hetu exists somewhere outside pakṣa but is definitely nonexistent in vipakṣa, then it follows that it exists in sapakṣa (provided that sapakṣa exists). In this way the third characteristic implies the second characteristic. If in this way the second and third characteristics imply each other, the question arises, why Diṅnāga states both? This questions the very core of Diṅnāga’s triple character theory of hetu. A possible answer within Diṅnāga’s framework is that the second and third characteristics in this framework are contingent characteristics having existential implications. The second characteristic implies that there is sapakṣa and hetu exists in it. Similarly the third characteristic implies that vipakṣa

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exists and hetu is nonexistent in it. (In exceptional cases vipakṣa may be nonexistent, for which Diṅnāga had to make a special provision.) If the existence of similar and dissimilar cases are presupposed or implied by the two characteristics, then the second and third characteristics cannot be reduced to each other. That is because the existence of similar cases is implied by the second characteristic, but the existence of dissimilar cases is not. Similarly the existence of dissimilar cases is implied by the third characteristic, but the existence of similar cases is not. But the way Dharmakīrti explains the two characteristics is different from the way Diṅnāga does. Dharmakīrti de-emphasizes the existential contexts of the two characteristics and expresses them by universal and necessary statements. The two characteristics then stand for anvaya-vyāpti and vyatireka-vyāpti, respectively, which can be expressed by the universal statements of the form: Anvaya-vyāpti: For all x, if x has hetu property, then x has sādhya property. Vyatireka-vyāpti: For all x, if x does not have sādhya property, then x does not have hetu property. It is obvious that the above two statements entail each other by the rule of transposition.

Is the Statement of “Instance” Required? In Diṅnāga’s framework of inference for others, just as both positive and negative concomitance need to be stated, each of them needs to be accompanied by an instantiation. That is because both the statements of concomitance have existential import. Dharmakīrti categorically states that separate mention of example, as an element of inference for others, is redundant. That is because the statement of concomitance for Dharmakīrti is a necessary statement. Though instances might have played some role in the formation of the statements of invariable concomitance, once the statements are formed, they stand not as contingent statements, but necessary ones. Now they can stand without examples. Accordingly, statements of concomitance in Dharmakīrti’s writings can be cited where the example is not stated (e.g., Nyāyabindu 3/24 and 3/25).This difference between Diṅnāga and Dharmakīrti is reflected in their formulations of fallacies of example (dṛṣṭāntābhāsa). If a statement of concomitance lacks an example, Diṅnāga would consider it as a fallacy of insufficient reasoning (nyūna). Since according to Dharmakīrti, example has no function over and above giving a hint for knowledge of concomitance, the above for him would not be a case of insufficient reasoning.

The Fallacy Called Viruddhāvyabhicārī Diṅnāga in his theory of inference had introduced a fallacy called viruddhāvyabhicāri. Though it is included within anaikāntika by Diṅnāga,

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viruddhāvyabhicārī is different from all other varieties of pseudo-probans. The other varieties are based on nonfulfillment of one of the three conditions of a probans. Viruddhāvyabhicārī meets all the three conditions and is yet fallacious because of the availability of a counter-probans which equally fulfills all the three conditions. Thus viruddhāvyabhicārī is defined as that which is concomitant (i.e., nondeviant) with the contradictory of that which is proved by another probans (viruddhaṁ na vyabhicaratī ti viruddhāvyabhicārī ). This kind of pseudo-probans leads the hearer to doubt as to which of the two alternatives he would accept, as both, being supported by “good” probans, are cogent. The fallacy occurs when with reference to a given sādhya, there are two hetus available: one hetu which “proves” the sādhya and the other hetu (which is called “prati-hetu” or counter-probans) which “disproves” the sādhya (i.e., “proves” the absence of sādhya). Dharmakīrti is vehemently against acknowledging this fallacy. This is quite natural because given the basic difference between the two frameworks of inference, the fallacy is possible in Diṅnāga’s framework, but it is not possible in Dharmakīrti’s framework. The point can be explained with the help of an example. Diṅnāga’s example of the fallacy is like this: Argument A(Di) – Mīmāṁsaka argues: Sound is eternal. Because it is audible. Audible things (other than sound) are seen to be eternal, for example, soundness. Non-eternal things (other than sound) are seen to be inaudible, for example, a pot. Argument B(Di) – Naiyāyika argues: Sound is non-eternal. Because it is a product. Products (other than sound) are seen to be non-eternal, for example, earthen pot. Eternal things (other than sound) are seen to be non-products, for example, ether. Here Diṅnāga points out that both the hetus are concomitant (avyabhicāri) with their respective sādhyas. But what they prove are the two incompatible properties of sound, namely, eternality and non-eternality. Hence the two hetus together become inconclusive (anaikāntika) with respect to eternality or non-eternality of sound. Here, both the hetus are good hetus insofar as the fulfillment of the three conditions is concerned. But the point to be noted here is that fulfillment of the three conditions does not imply universal relation between hetu and sādhya. The two hetus therefore are two strong analogies leading to opposite conclusions. The conclusions of the two arguments are contradictory, but the premises of them do not contradict with each other. Hence the odd situation of inconclusiveness occurs. But this is not possible in Dharmakīrti’s frame of inference. Here concomitance relation is understood as universal in nature. In that case the two arguments will be stated as follows:

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Argument A(Dh) – Mīmāṁsaka argues: Sound is eternal. Because it is audible. Whatever is audible is eternal (like sound). Argument B(Dh) – Naiyāyika argues: Sound is non-eternal. Because it is a product. Whatever is a product is non-eternal (like a pot). Here the premises of each argument entail their respective sādhya. If the premises of the argument A(Dh) are true, then sound must be eternal, and if the premises of the argument B(Dh) are true, then sound must be non-eternal. Hence if the premises of both the arguments are true then sound must be both eternal and non-eternal. But this is a contradiction. Hence the premises of both the arguments cannot be true. This implies that there must be the fallacy such as asiddha, viruddha, or anaikāntika in at least one of the two hetus. Hence Viruddhāvyabhicārī is not an independent fallacy, but is reducible to the other fallacies of hetu. In other words, there can be two good hetus in Diṅnāga’s sense leading to two mutually incompatible sādhyas. But there cannot be two good hetus in Dharmakīrti’s sense leading to mutually incompatible sādhyas. This difference is due to the difference between the concepts of good hetu according to them. It may be noted here that a pseudo-probans of this kind definitely conforms to the notion of pseudo-probans classified by the Nyāya as “the one accompanied by a counter-probans” (satpratipakṣa). This kind of fallacy, according to Nyāya, follows when a probans has the property of being contradicted by a counter-probans (asatpratipakṣatva). Dharmakīrti, consequently, does not consider this property a necessary prerequisite for the probans’ being legitimate.

Some Questions About Pervasion in Dharmakīrti Identity and the Question of Antarvyāpti It has been maintained above that the relation of pervasion according to Dharmakīrti should be universal and this universality should be based on some kind of necessity. But how this universal and necessary relation is to be obtained was an issue. For Dharmakīrti this relation was twofold: identity and causality. When probans is an essential aspect (svabhāva-hetu) of probandum, the relation between the two was identity (tādātmya) which can be understood as class inclusion. In the inference “this is a tree because this is s´iṁs´apā,” the pervasion relation of the form, “whatever is s´iṁs´apā is a tree,” simply means that s´iṁs´apā class is a subclass of the tree class. This relation of universal concomitance

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between s´iṁs´apā and tree is known by understanding the essential nature of some other s´iṁs´apā tree which was other than the subject of the inference (i.e., which was outside pakṣa). But Dharmakīrti was also concerned with the inferences based on identity, where the subject of inference was itself universal in nature. For example, in the inference of the form “All that is produced is momentary” (Sarvaṁ saṅskṛtam kṣaṇikam), the whole class of products is the subject. The question is, how is the relation of the form “Whatever is a product is momentary” to be known outside the subject? According to the Buddhist logicians such as Ratnākaraśānti, the relation is known in such cases in the inferential subject itself, and this would be a case of “internal pervasion” (antarvyāpti). Like some Buddhist logicians, some Jaina logicians such as Vādidevasūri also accepted internal pervasion. Internal pervasion was defined as the pervasion which exists and is known on the pakṣa (inferential subject) itself. And this has become a problem area in Buddhist (and Jaina) logic. It is difficult to explain how pervasion can be known on pakṣa itself. It seems that such a pervasion is possible only if it is understood as conceptual or analytical in nature. Pervasion which is an analytical or conceptual relation will be a priori, and hence it will be acceptable independently of instantiation. If on the other hand it is an empirical relation, then it will be “external pervasion,” that is, the pervasion which can be substantiated through instantiation.

Causal Necessity and Inductive Reasoning According to Dharmakīrti, even the empirical pervasion cannot be established only on the basis of observation and non-observation. Such a relation will be binding if it is based on causality. For example, there is invariable concomitance between smoke and fire because smoke cannot come into being without fire. However, how the so-called causal relation between smoke and fire can be ascertained is a problem. Dharmakīrti suggests that causal relation has an invariable character which a simple coexistence does not have. “When a type of effect is seen once as arising from a type of a cause, the former does not deviate from the latter.” (“Tad yādṛs´aṃ kāryaṃ yādṛs´āt kāraṇād dṛṣṭam ekadā tat tanna vyabhicarati” – Hetubindu 3.2). This necessity for him is due to the power or capacity (s´akti) of the cause to produce the effect. (“Kāraṇas´aktipratiniyame hi kin˜ cid eva kasyacit sādhanāyopadī yeta, nāparam, . . .” –Hetubindu 3.3). This suggests that Dharmakīrti rejected random causality and strongly stood for a regular or necessary relation between cause and effect. While explaining the ascertainment of causal necessity, Dharmakīrti and some post-Dharmakīrti Buddhist logicians refer to observation and non-observation such as non-observation of the effect before its production, observation of the cause before the effect, and observation of effect after the cause. The question here is that if the necessity involved in a causal relation is derived from the regular sequence observed between cause and effect, then the necessity advocated by Dharmakīrti will not be logical, but empirical.

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Prajñākaragupta, while commenting on Pramāṇavārtika, says that causal necessity is known through direct transcendental perception. Dharmakīrti does not seem to subscribe to this view. As Nagin J. Shah (Shah 1967) points out, the general trend of Dharmakīrti’s philosophy suggests that for him necessity expressed in terms of identity and causality is apprehended through a mental construction (vikalpabuddhi). Perception knows unique particulars (svalakṣaṇa) only. Mind constructs the concept of necessity and imposes it on the real particulars to derive an ordered world. Of course this only gives a tentative explanation of how inductive reasoning takes place. It does not solve the logical problem of induction. Dharmakīrti can perhaps claim only that inductive generalization on the basis of co-occurrence (or rather, sequential occurrence) of two types of phenomena cannot yield invariable concomitance unless the regular sequential occurrence is indicative of cause-effect relationship. In other words, all synthetic necessity is causal necessity. There is another issue about Dharmakīrti’s conception of causal necessity. Dharmakīrti generally understands cause as necessary condition of the effect so that one can infer cause from the effect and that gives the type of inference called “kārya-liṅgaka anumāna” (the inference which has effect as probans). However, while discussing the varieties of anupalabdhi-hetu (non-apprehension as probans), he also acknowledges cause as sufficient condition. For example, while talking about “non-apprehension of effect” (kāryānupalabdhi) as probans, he talks of “causes whose power (to generate the effect) is not obstructed” (apratibaddha-sāmarthyāni kāraṇāni). If such a cause is accepted, then it should be possible to infer effect from cause as well. But Dharmakīrti does not include kāraṇa-hetu (cause as probans) in his classification of hetus. This can perhaps be called a deficiency in Dharmakīrti’s theory of inference.

Theoretical Issues Involved in Prasan˙ga Method Prasan˙gānumāna According to Dharmakīrti An account of inference for others in Buddhism remains incomplete without discussion on prasaṅgānumāna. Dharmakīrti in Pramāṇavārtika (“. . .parakalpitaiḥ prasaṅgo dvayasambandhād ekāpāye’nyahānaye” – Parārthānumāna-pariccheda, kārikā 12) argues that an argument based on prasaṅga does not have the status of anumāna-pramāṇa because it does not operate on an established fact but on the opponent’s view which the proponent accepts hypothetically and by applying a general rule of vyāpti to it derives an absurd or unacceptable consequence from it. In this way he refutes the opponent’s view by an indirect method. For example, if a Buddhist wants to refute the Nyāya conception of sāmānya, then he may argue: Sāmānya according to you is “one that resides in many.” For instance, potness is one, but it resides in many pots. But if potness resides in many pots, then potness will be many (i.e., the potness of each pot will be different). But potness is not many according to you. Therefore sāmānya, which is one among many according to you, does not exist. This argument is not “inference for

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others” proper, because what is argued here is not a categorical or factual statement but an unacceptable consequence derived from a counterfactual conditional. Buddhist logicians after Dharmakīrti tried to give a more constructive role to prasaṅga type of reasoning. In Tarkabhāṣā, Mokṣākaragupta defines prasaṅga as the “reasoning for bringing out an absurd conclusion which is undesirable to the opponent by means of statement based on a vyāpti established by proof” (“pramāṇaprasiddhavyāptikena vākyena parasyāniṣṭatvāpādanāya prasan˜ janaṁ prasaṅgaḥ”). He also refers to the view of some logicians that prasaṅga and prasaṅga-viparyaya are useful for establishing vyāpti in dṛṣṭānta. How can be prasaṅga applied to grasp vyāpti? A formulation of prasaṅga establishing vyāpti of a probans as essential identity with its probandum can be given as follows: [A general statement to be established]: All produced things are momentary. [Contradictory statement]: Something, for example, a jar, is not momentary. [General statement already proved]: Whatever is capable of producing a certain effect at a certain time does produce that effect at that time. [Absurdity]: A jar, if not momentary, will be capable of producing past and future effects at the present time. [Conclusion]: It is true that all produced things are momentary. It may be noted here that although prasaṅga type of argument is useful in this way even to grasp vyāpti, it is not given the status of pramāṇa, but it is said to operate only by reminding one of the pramāṇa which grasps vyāpti (“sādhyasādhana-vyāpti-grāhaka-pramāṇa-smārakastu prasaṅge prayogaḥ,” Manorathanandin’s commentary on Pramāṇavārtika, 4.12). Reality can be known according to these Buddhist logicians only through pramāṇa and not through prasaṅga. [This view can be compared with the Nyāya view about tarka, which was defined as aniṣṭaprasaṅga (deriving an undesirable consequence), the role of which was accepted in the knowledge of vyāpti, without giving it the status of pramāṇa.]

Nāgārjuna’s Use of Prasan˙ga Method Prasaṅgānumāna was employed in a very different way by Nāgārjuna. Nāgārjuna was critical about pramāṇas in general. But he was not critical about prasaṅga method as other logicians were. That is because the goal of Nāgārjuna’s arguments was not to establish any ontological truths in their essentialist form. It was rather to “establish” which meant “to develop an insight into” the nonessential nature (niḥsvabhāvatā), that is, emptiness (s´ūnyatā) nature of reality. All nonMādhyamikas held some essentialist view (svabhāvavāda) in some form or the other, and Nāgārjuna tried to establish s´ūnyatā by refuting all types of the essentialist views of the opponents (particularly those of ābhidharmikas). For doing this, he elaborately used the prasaṅga method in his magnum opus called Mūlamadhyamakakārikā or Madhyamakas´āstra.

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Nāgārjuna’s prasaṅga method can be described as the method of refutation without assertion. It differs from the method of Reductio ad Absurdum in this sense. While reductio involves accepting a view as valid by showing that absurdity can be deduced from its negation, prasaṅga has recourse just to the deduction of absurdity without implying its positive counterpart. Hence when existence of an entity is denied, that does not imply for Nāgārjuna affirming nonexistence of the same (Matilal 1985, 164). Rather, he aims at denying all the four possible views, existence, nonexistence, existence, and nonexistence, and also neither existence nor nonexistence (“Sarvaṁ tathyaṁ na vā’tathyaṁ tathyaṁ cātathyameva ca/ Naivātathyaṁ naiva tathyam etad buddhānus´āsanam// ” – Mūlamadhyamakārikā 18/8). Nāgārjuna applied prasaṅga method for refuting all types of views, and this gives rise to some logical issues. One important issue is whether his universal application of the method violates the laws of classical two-valued logic. While applying prasaṅga method, Nāgārjuna considers different possible combinations concerning the concept under scrutiny which generally assume the form of four alternative categories (Catuṣkoṭi). Then he refutes each category by deriving an unacceptable consequence from it. By refuting all such combinations, he tries to show the emptiness of the concept under scrutiny. The four alternatives are of the form: “yes,” “no,” “both yes and no,” and “neither yes nor no.” According to the classical two-valued logic, if the alternative, namely, “yes” is negated, then the alternative, namely, “no,” should be accepted and vice versa. Denying both will amount to violation of the law of noncontradiction. Nāgārjuna not only does that; he even considers the alternative “yes and no” and “neither yes nor no” as genuine alternatives which are obvious violations of the laws of noncontradiction and excluded middle. Hence Nāgārjuna’s use of the prasaṅga method does not seem to be prima facie defensible within the framework of classical two-valued logic. The scholars like Galloway have tried to answer this problem by pointing out that the apparently contradictory alternatives are not contradictory, but contrary (i.e., they can be false together though not true together). While refuting them, Nāgārjuna is trying to show that they are based on certain common false presuppositions. The claim of this kind, however, cannot be defended beyond a limit. It can be suggested here that the application of prasaṅga type of argumentation may not lead to transgression of the laws of logic if the application is restricted to some specified views. But the global application or unrestricted application does pose a problem. To illustrate, Nāgārjuna in the second chapter of Madhyamakas´āstra (called “Gatāgata-parī kṣā”) presents a trilemma with reference to the act of going, gata, agata, and gamyamāna (“gone to,” “not gone to,” and “being gone to”), and shows that there cannot be “going” in relation to any of them. Following the position of Galloway, one can say that all the three alternatives can be denied because they are based on a common presupposition that there is the act of going. However, Nāgārjuna’s argument does not stop here. Because denial of the act of going leads one to think that since there is no going, there is staying. Nāgārjuna does accept this implication. He comes out with another trilemma and by applying

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prasaṅga method shows that what is applicable to gati (going) is equally applicable to sthiti (staying). Now is it possible to deny both “going” and “staying” without contradiction? One might argue that Nāgārjuna can deny both because both are based on another common presupposition which he wants to deny. The common presupposition is that things exist. But in the fifteenth chapter, viz., Svabhāvaparī kṣā, Nāgārjuna denies svabhāva (self-nature), parabhāva (other nature), as well as bhāva (existence). Now denial of the presupposition that things exist would lead one to accept its negation that things are nonexistent. But Nāgārjuna does not accept this logical consequence either. In the fifteenth chapter, he not only denies bhāva (existence) but also abhāva (nonexistence). So this leads to the next dilemma according to which things are neither existent nor nonexistent. It seems obvious that one cannot deny both the existence and nonexistence without transgressing the laws of logic such as noncontradiction, excluded middle, and double negation. Hence there are logical difficulties in the global application of prasaṅga method which Nāgārjuna exercises in Mūlamadhyamakakārikā. Buddhist thinkers and scholars have attempted to overcome the difficulties in different ways. Three major ways can be considered here: 1. Svātantrika approach: There was schism of the Mādhyamika school in the sixth century CE into two branches – the Prāsaṅgika-Mādhyamika School founded by Buddhapālita and the Svātantrika-Mādhyamika School founded by Bhavya. The Prāsaṅgika-Mādhyamika School tried to demonstrate the emptiness or essencelessness of all dharmas recognized in the Abhidharma by prasaṅga method while the latter school thought it necessary to supplement Nāgārjuna’s prasaṅga arguments by independent (svatantra) arguments formulated in accordance with the rules of logic. Both these alternatives were not without difficulties. If according to Svātantrika methodology independent inferences are presented for proving s´ūnyatā, then s´ūnyatā will itself be a kind of view. But s´ūnyatā amounts to rejection of all views (sarvadṛṣṭiprahāṇa) according to Nāgārjuna. Hence s´ū nyatā too will have to be rejected. If, on the other hand, prāsaṅgika method is accepted for rejecting all views, then, as suggested above, Nāgārjuna cannot do this without self-inconsistency because negation of a view and negation of this negation (which is also a view) cannot go together. 2. Ineffability approach: According to the second alternative, Nāgārjuna’s position does not amount to breaking the rules of logic, but it amounts to critique of logic. This critique of logic is rooted in the critique of language. According to this interpretation, reality is s´ūnya which implies that it is ineffable, and because it cannot be expressed in language, it cannot be captured by the rules of logic. Nāgārjuna’s position on this interpretation amounts to a kind of mysticism. 3. Negation as non-truth-functional operator: Nāgārjuna’s position appears to violate the laws of classical two-valued logic, because the “negation” in it is interpreted as truth-functional negation. If on the other hand negation is understood as a non-truth-functional operator, “p and negation-p” will not be a contradiction, and “negation of negation-p” will not be equivalent to “p.” Matilal

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(Matilal 1985: 17), for example, distinguishes between refutation and propositional negation. For him refutation of a position does not commit one to assertion of its negation. Similarly Seyfor Ruegg (Seyfort Ruegg 1986) regards Nāgārjuna’s negation as performative negation. A performative negation is different from a truth-functional negation in that by performatively negating a statement, one is not making another statement, but one is only expressing a negative attitude toward the statement.

An Issue Concerning the Doctrine of Apoha The Problem The doctrine of apoha (exclusion) introduced by Diṅnāga and developed by Dharmakīrti and some other Buddhist logicians was primarily concerned with the question as to how the words in language are related to the objects. But the doctrine has application with respect to the Buddhist pramāṇa theory in general. The unique particular, which is the object of perception, is unique in that it is excluded from everything else – both similar and dissimilar things. It is sajātī ya-vijātī ya-vyāvṛtta. Through an inference, one cognizes a universal object. For example, through inference from smoke, one cognizes the existence of “fire in general” on the hill. The “fire in general” simply means “something excluded from non-fire.” Through inference for others, one proves fire to be there on the hill. Here too the word “fire” means “something excluded from non-fire.” The Buddhist theory of exclusion (apoha) has become a target of criticism at the hands of non-Buddhists. Two points of criticism are frequently advanced against the apoha theory of meaning. (a) The problem of tautology: According to the apoha theory of meaning, “fire” means “something excluded from non-fire” or “not non-fire.” But this is a tautology which yields no information about the meaning of the word “fire.” (b) The problem of circularity: If the double negation involved in the notion of anyāpoha (exclusion of the other) is regarded as informative, then it will be a case of vicious circularity. Suppose there are three types of objects in the world: fire, tree, and dog. Then the word “fire” will mean “what is other than tree and dog,” “tree” will mean “what is other than fire and dog,” and “dog” will mean “what is other than fire and tree.” Hence the meanings of “fire,” “tree,” and “dog” will depend upon each other, and none of them will be understood independently (Siderits Mark et al. 2011: 27).

Two Possible Ways One way by which the charge can be answered is to hold, as Matilal does, that in “not non-cow” (a-go-vyāvṛtta), the first negation represented by “not” (“vyāvṛtta”) is external negation (prasajya-pratiṣedha), whereas the second negation represented by “non” (“a” in “a-go”) is internal negation (paryudāsa). To use Nyāya

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terminology, one is talking here of “absolute absence (atyantābhāva) of mutual absence (anyonyābhāva).” And since the two absences are different, it is not a case of “double negation” or “tautology.” The answer does not seem to be satisfactory because though Naiyāyikas regard absolute absence and mutual absence as different from metaphysical point of view, for logical purpose they can be treated as equivalent. “Pot has mutual absence with respect to cloth” and “there is absolute absence of clothness in pot” are equivalent expressions. The other way by which the charges of tautology and circularity can be answered is by treating the doctrine of apoha as not about the meaning of a word, but about the function of a word. The distinction is crucial to the theory of apoha, because the meaning of a word is something which is expressed or referred to by a word, whereas the function of a word is something which is done or performed by a word. The apoha theory seems to imply that exclusion of non-cows is done or performed by the use of the word “cow” and not that the negative object “not non-cow” is expressed or referred to by the word “cow.” It can be said that if apoha is regarded as the meaning of a word, then difficulties like tautology and circularity are insuperable. They will not arise if exclusion is treated as a function.

Summary The five sections of the chapter deal with some significant issues concerning five themes in Buddhist logic. The first section highlights how the Nyāya and Buddhist logic have developed through mutual criticism and how the contributions of the latter need to be understood by comparing and contrasting with those of the former. The second section brings out how the differences between the logical positions of Diṅnāga and Dharmakīrti are not superficial or peripheral, but they are deep rooted. The third section deals with two questions concerning Dharmakīrti’s concept of vyāpti as necessary relation. It suggests that the so-called internal pervasion can better be understood as a conceptual or analytical relation. It also tries to bring out some limitations of Dharmakīrti’s conception of causal necessity. The fourth section refers to the charge that Nāgārjuna through the use of prasaṅga method violates the laws of logic and considers different ways of answering the charge. The last section suggests that the charge of tautology and circularity against the doctrine of apoha can perhaps be lifted by treating the doctrine as about function of a word rather than meaning of a word.

Definitions of Key Terms Anvaya-vyāpti (positive pervasion or positive concomitance) Hetu (probans)

Positive invariable relation between hetu and sādhya of the form: “Wherever there is hetu, there is sādhya.” A property which provides the reason or ground for the existence of sādhya in pakṣa.

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Sādhya (probandum) Sapakṣa (similar cases) Vipakṣa (dissimilar cases) Vyāpti Vyatireka-vyāpti (negative pervasion or negative concomitance)

P. P. Gokhale and K. Bhattacharya

Locus or property bearer in which a property is inferentially known to exist. The method of refutation involving derivation of an unacceptable consequence of a thesis proposed by the opponent. The property which is inferentially known to be existent in pakṣa. A set of property bearers similar to pakṣa in that they possess sādhya. A set of property bearers dissimilar to pakṣa in that they do not possess sādhya. Pervasion, invariable concomitance. Negative invariable relation between hetu and sādhya of the form: “Wherever there is absence of sādhya, there is absence of hetu.”

References Dharmakīrti. 1975. Nyāyabindu, included in Shriniwasa Shastri (Ed.), Nyāyabinduṭīkā of Dharmottara, Sahitya Bhandar, Merath. Galloway, Brian. 1989. Some logical issues in Madhyamaka thought. Journal of Indian Philosophy 17: 1. Gokhale, Pradeep P., ed. and trans. 1997. Hetubindu of Dharmakī rti (a point on probans), Sri Satguru Publications Delhi. Gokhale, Pradeep P. 1992. Inference and fallacies discussed in ancient Indian logic. Delhi: Sri Satguru Publications. Matilal, B.K. 1971. Epistemology, logic and grammar, in Indian philosophical analysis. The Hague: Mouton. [Chap. 5: “Nagation and the Madhyamika Dialectic”.] Matilal, Bimal Krishna. 1985. Logic, language and reality: An introduction to Indian philosophical studies. Delhi: Motilal Banarsidass. Matilal, Bimal Krishna. 1998. In The character of logic in India, ed. Jonardon Ganeri and Heeraman Tiwari. Albany: State University of New York Press. Seyfort Ruegg, D. 1986. Does the Madhyamika have a thesis and philosophical position ? In Buddhist logic and epistemology, ed. B.K. Matilal and Robert D. Evans, 229–238. Holland: D. Reidel Publishing Company. Shah, Nagin J. 1967. Akalaṅka’s criticism of Dharmakī rti’s philosophy. Ahmedabad: L. D. Institute of Indology. Siderits, Mark, Tom Tillemans, and Arindam Chakrabarti, eds. 2011. Apoha: Buddhist nominalism and human cognition. New York: Columbia University Press. Vidyabhushan, Satish Chandra. 2005. A history of Indian logic, ancient, mediaeval and modern schools. Delhi: Motilal Banarsidass.

The Nya¯ya on Logical Thought

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J. L. Shaw

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 1: Some Technical Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 2: Part I: Definition of a Cause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operation (vyāpāra) and Special Instrumental Cause (karaṇa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II: Belief . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 3: Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sources of Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nyāya on Knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Analogy or Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Verbal Cognition or Testimony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 4: Negation and Its Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 5: Universal Quantifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Common Assertion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 6: The Principle of Contradiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 7: Gadādhara’s Theory of Definite Descriptions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Section 8: Six Pairs of the Nyāya Philosophers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In this chapter, I shall emphasize the following features of the Nyāya logical thought: (1) some of the uses of Occam’s razor or the principle of simplicity; (2) how to avoid the postulation of tertiary entities, such as propositions or images; (3) how the concept of relevance has been used in the context of an inference; (4) the Nyāya view about understanding the meanings of contrary or J. L. Shaw (*) Department of Philosophy, Victoria University of Wellington, Wellington, New Zealand © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_36

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contradictory expressions; (5) the Nyāya conception of negation, as it cannot be equated with the term negation or the proposition negation in Western philosophy; (6) the definition of the quantifier “all” as well as the pervader-pervaded relation; and (7) the reconstruction of Gadādhara’s theory of definite descriptions. I would also like to discuss the distinction between the pairs of terms anuyogī-pratiyogī (first term-second term), ādhāra-ādheya (substratum-superstratum), viśeṣya-viśeṣaṇa (qualificand-qualifier), viśeṣya-prakāra (qualificand-relational qualifier), uddeśya-vidheya (subject-predicate), and pakṣa-sādhya (the locus of inference-probandum), which are analogous to the subject-predicate distinction in Western philosophy. These terms are used for the explanation of the distinction between perceptual, inferential, and verbal cognitions, as well as for the distinction in meaning between transformationally equivalent sentences and for suggesting a solution to Frege’s problem, why predicate alone is to be considered as unsaturated or to use the term of Russell “incomplete.”

Introduction In this chapter, I would like to focus on the following aspects of the logical thought of the Nyāya system of philosophy. Since the Nyāya philosophers emphasize the logical aspects of thinking, it is called tarkavidyā or ānvīksikī (Tarkavāgīśa 1982, pp. 3–7). The latter expression is derived from īksā, which means substantiation of a view or truth through argument or critical thinking. Logical thought or critical thinking implies consistency, relevance, justification, valid inferences, fallacies, and discussion on logical locutions such as “not” and “all.” It also involves the application of Occam’s razor or the principle of simplicity, delineating sharp distinctions for transparency or the removal of vagueness. According to Occam’s razor, “Entities should not be multiplied without necessity.” Hence one should select the solution with the fewest assumptions. Since this principle has been used in several places by the Nyāya philosophers, I shall begin with their conception or definition of causation as well as focus on their explanation of false beliefs, among many others. The conception of relevance has also been emphasized so widely that it relates almost everything to mokṣa, which is the summum bonum of life. I shall focus on their discussion on sources of knowledge and the relation between premises and conclusion, as well as the relation between premises of an inference. In their discussion of meaning, the Nyāya philosophers have drawn the distinction among a sentence, the meaning of a sentence, and understanding the meaning of a sentence. They have emphasized the absence of contradiction not only at the level of reality but also at the level of thought. Hence the thought of an impossible object is also impossible. From this, it does not follow that we cannot assign the value falsehood to a sentence or expression which is inconsistent or contradictory. In this context, I shall point out that the Nyāya view is a mean between the extreme views of contemporary Western philosophers.

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In this chapter, I would also like to mention how the Nyāya philosophers have opposed certain types of reductionism. If certain distinctions are obvious to us, then they are to be retained, not to be dispensed with for the sake of a theory. In this context, I will refer to my formulation or the explication of Gadādhara’s theory of definite descriptions, which retains the distinction between the expressions “The author of Waverley” and “The author of Waverly exists.” A comparison between Bertrand Russell’s theory of definite descriptions with that of Gadādhara’s will also reveal how certain shortcomings of Russell’s theory can be overcome. This is due to the fact that Gadādhara’s theory does not postulate supposed objects, in addition to things in the world. I shall also mention how the Nyāya philosophers have used certain categories or pairs of terms, analogous to subject-predicate distinction in Western philosophy, for explaining the phenomenological distinction between perceptual, inferential, and verbal cognitions, which is lacking in Western philosophy. The first section of this chapter will explain certain technical terms, which are useful for understanding the Nyāya conception of cognition, as well as for drawing the distinction between true and false beliefs and the fine-grained distinction in the meanings of expressions. In this context, I shall also introduce the concept of relevance and its types. The second section will deal with the application of the principle of simplicity, especially in the context of causation and the explanation of false beliefs. Since the Nyāya philosophers have explained false beliefs without postulating tertiary entities, they have falsified Russell’s claim that no one has succeeded in explaining a false belief without postulating the existence of the nonexistent. The third section will deal with the source knowledge, with special reference to inference. In this context, I shall discuss the Nyāya conception of fallacy and the classification of fallacies as well as the application of tarka (a type of counterfactual conditional), which is an auxiliary to an inference or gives rise to an inference. In this section, I shall also discuss how the Nyāya philosophers would explain the distinction in meaning between transformationally equivalent sentences, such as “Brutus killed Caesar” and “Caesar was killed by Brutus,” as well as the nature of atomistic and holistic understanding. The fourth section will deal with the Nyāya conception of negation as well as the types of negation in the classical Nyāya. The fifth section will refer to some of the definitions of the quantifier “all,” including the words “if” and “then,” as well as a definition of pervaded-pervader relation. The sixth section will focus on the Nyāya conception of the principle of contradiction as well as their view about understanding the meanings of contrary or contradictory expressions or sentences and the assignment of truth values to them. The seventh section will deal with the formulation of Gadādhara’s theory of definite descriptions, which will be compared with Russell’s theory of definite descriptions. The eighth section will focus on the six pairs of terms of the Nyāya philosophers for the phenomenological distinction between perceptual, inferential, and verbal

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cognitions. In this context, I shall also discuss how the Nyāya would explain the saturated nature of the subject and the unsaturated nature of the predicate, discussed in contemporary philosophy beginning with Frege.

Section 1: Some Technical Terms With the above aims in view, I would like to mention the following points, as I shall be focusing on the above topics from the Nyāya Perspective: (A) The Nyāya has drawn a distinction between qualificative (svavikalpaka) and non-qualificative (nirvikalpaka) cognitions. The Nyāya concept of qualificative cognition can be expressed by the form “aRb.” A qualificative cognition involves at least three elements, namely, a qualificand (viśeṣya), a qualifier (viśeṣaṇa), and a qualification relation (viśeṣya- viśeṣaṇa-sambandha), which relates the latter to the former. According to the Nyāya, the possibility of qualificative perception cannot be explained without postulating non-qualificative perception. Let us consider the qualificative perception of a flower, which is atomic in nature. This cognition has three elements, namely, a particular flower which is a substance (dravya), flowerness which is a class character ( jāti), and the relation of inherence (samavāya sambandha) which relates the latter to the former in the ontology of the Nyāya. Since the perceptual cognition of a relation presupposes the cognition of its relata, the cognition of the inherence relation in this case presupposes the cognition of both the particular flower and the flowerness. These relata are cognized in a non-qualificative perceptual cognition. Now the following points are to be noted in this context: (i) Since only the qualificand and the qualifier of an atomic qualificative perceptual cognition are cognized in a non-qualificative perceptual cognition, they are not cognized as qualificand or qualifier. They are cognized as such without any mode of presentation. (ii) The objects of a non-qualificative cognition cannot be cognized by expressions. Hence a non-qualificative cannot be generated by an expression. For example, the expression “a flower” will not generate a cognition of a flower which is not qualified by a property. (iii) Regarding the truth of a non-qualificative cognition, the Navya-Nyāya philosophers claim it to be neither true nor false. This is due to the fact that both true and false cognitions presuppose qualificand-qualifier relations. Since there is no qualificand-qualifier relation in a non-qualificative cognition, it is outside the scope of true and false. (iv) Since it is a causal condition of perceptual qualificative cognition, it is always immediately prior to it. When a sense organ is related to its objects, the initial cognition which is due to this contact is non-qualificative. Thereafter, a qualificative cognition is generated. (v) Since it has been postulated to give an account of qualificative perceptual cognition, it is also considered as perceptual in character. The objects of

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non-qualificative cognition, such as the particular flower and flowerness in the above example, are cognized as being related in a qualificative cognition, such as the flower qualified by flowerness in the relation of inherence. Hence the objects of non-qualificative cognition are public, not private sense data. Therefore, the Nyāya view does not lead to relativism, phenomenalism, or solipsism. For this reason, the problems of the supporters of sense-data theory in contemporary philosophy do not arise in the Nyāya philosophy. From the above discussion, it follows that a qualificative cognition has the form aRb. If we consider a nonatomic qualificative cognition, such as a flower is red, then it will be described in the following way: The cognition in which the property of being the qualificand (viśeṣyatā) residing in a flower is limited by (avacchinna) flowerness but determined by (nirūpita) the property of being the qualifier (viśeṣaṇatā) residing in the red color, which is limited by redness and the relation of inherence (samavāya).

Now let us explain the Nyāya conception of relation, as anything can play the role of a relation, and the distinction between the relation limited by and the relation determined by. R is a relation if and only if (Ǝx) (Ǝy) (it is due to R that x appears as the qualificand and y as the qualifier in the cognition xRy) and (Ǝx) (Ǝy) (it is due to R that there is a unified or qualified object or fact xRy), where x and y range over entities of the Nyāya system. It is to be noted that in this definition, the x and the y of a cognition need not be the same as the x and the y of a fact. If the cognition is true, then the x and the y of it would be the same as the x and the y of the fact xRy. The limitor-limited (avacchedaka-avacchinna) relation is usually defined in the following way, as there are exceptions: x is limited by y if and only if (i) both x and y are properties, (ii) x is a relational property, and (iii) the property y is a mode of presentation of the object where the relational property x resides.

The determiner-determined relation (nirūpya-nirūpaka-sambandha) may be stated in the following way: x is determined by y if and only if x and y are relational properties of correlatives.

As mentioned before, a qualificative cognition has the form aRb, where a is the qualificand (viśeṣya), b is the qualifier (viśeṣaṇa), and R is the qualification relation (viśeṣya-viśeṣaṇa-sambandha). The qualificand-qualifier category is used to differentiate the qualificand from other objects in terms of the qualifier. If R is a mode of presentation of b, which happens in almost all cases, then b is called prakāra (“relational qualifier”).

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The Nyāya has postulated several relational properties which signify the roles of objects, especially in epistemic contexts. The Nyāya also emphasizes the direction of the relation for the explanation of the meaning of a sentence as well as for semantical analysis. The relation of cognition ( jñāna) to the qualificand (viśeṣya) is called viśeṣyatā (“the property of being the qualificand”), the relation of cognition to the qualifier (viśeṣaṇa) is called viśeṣaṇata (“the property of being the qualifier”), and the relation of cognition to the relation (saṃsarga) is called saṃsargatā (“the property of being the relation”). The relation of cognition to the relational qualifier (prakāra) is called prakāratā (“the property of being the relational qualifier”). As regards the ontological nature of these properties, there is no unanimity among the Nyāya philosophers. Barring the question of their ontological status, they are very useful for drawing epistemic distinctions, including the distinction between true and false cognitions. Let us consider the cognition of the brown table or the table is brown. The relation of cognition to the table which is the qualificand is viśeṣyatā, the relation of cognition to the brown color is viśeṣaṇatā, and the relation of cognition to the relation of inherence (samavāya) is saṃsargatā. But the relation of cognition to the brown color presented under the mode of the relation of inherence is prakaratā. It is to be noted that both the table and the brown color are presented under the modes of tableness and brownness, respectively. So we have altogether two objects, namely, the table, the particular brown color, and the relation of inherence, two propertylimitors (avacchedaka dharma), and three relational properties of being the objects of this cognition (viṣayatās). The Nyāya claims that they are related in the following ways: 1. The property of being the qualificand residing in the table is limited by tableness. 2. The property of being the qualifier residing in the brown color is limited by brownness. 3. The property of being the qualifier residing in the brown color is also limited by the relation of inherence. 4. The property of being the qualificand residing in the table is determined by the property of being the qualifier residing in the brown color. 5. The property of being the qualifier residing in the brown color is determined by the property of being the qualificand residing in the table. 6. The property of being the qualification relation (saṃsargatā) residing in the inherence relation is determined by the property of being the qualifier residing in the brown color. 7. The property of being the qualifier residing in the brown color is determined by the property of being the qualification relation residing in the brown color. The first three relations are called “limitor-limited” (avacchedakaavacchinna), but the remaining four are called “determiner-determined relations” (nirūpya-nirūpaka-sambandhas). It is to be noted that the above seven relations are present both in a true and in a false cognition. In other words, if the cognition

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of the table is brown is false, then also these relations are present. But when the cognition is true, it is related to the fact or the qualified object (viśiṣta-viṣayatā). Hence it is related to the fact the table being brown. This relation of the cognition to the qualified object is called viśiṣta-viṣayatā (“the property of being the qualified object”). It is to be noted that the viśiṣta-viṣayatā resides in the whole which includes the qualificand, the qualifier, and the relation. Hence it is not something over and above these three entities. If a qualificative cognition is represented by aRb, then the cognition of R which relates b to a, in addition to cognizing a as the qualificand and b as the qualifier, will amount to the cognition of the qualified object (viśiṣtaviṣayatā). As mentioned before, the function of a relation at epistemic level is to make one object as qualifier of another. Hence, in this case, b is cognized as the qualifier of a. Another function is to make a fact or a qualified object. In the case of a false cognition, the former function is present, but not the latter with respect to the same a and b, although it relates two other objects elsewhere or elsewhen. But in the case of a true cognition, both the functions are present with respect to the same items. When we put a book on the table, a new fact occurs, and the novelty of this fact is explained in terms of the novelty of the conjunction relation of the book to the table. But in the case of a false cognition, this novelty is missing, as a previously cognized relation makes one the qualifier of another. So far we have explained the relation of a cognition to its objects and the relation among the objects. Now let us point out the relation of objects to the cognition. In our above example, the relation of the table to the cognition is called viśeṣyitā. This relation is the converse of the property of being the qualificand (viśeṣyatā). The relation of the brown color to the cognition may be called viśeṣaṇitā, which is the converse of viśeṣaṇatā, although this term has not been used by the Nyāya philosophers. The relation of the inherence relation to the cognition is saṃsargitā, which is the converse of the property of being the qualification relation (saṃsargatā). And the relation of the brown color under the mode of the relation of inherence to the cognition is prakāritā, which is the converse of prakāratā. The relation of the table being brown, which is a qualified object, to the cognition is viśiṣta-viṣayitā, which is the converse of viśiṣta-viṣayatā. As the properties residing in the objects of cognition are related to each other by the determiner-determined relation (nirūpya-nirūpaka-sambandha), so are the properties of a cognition which are due to relations of the objects to the cognition. Hence viśeṣyitā is determined by viśeṣaṇitā and the latter by the former. Again, saṃsargitā is determined by visesanitā and the latter by the former. Similarly, prakāritā is determined by viśeṣyitā and the latter by the former. As in a true cognition the relation of cognition to its qualified object is viśiṣtaviṣayatā, similarly the relation of the qualified object to the cognition is viśiṣtaviṣayitā. As the nirūpya-nirūpaka-bhāvāpanna-viṣayatās (the objects related to each other by determiner-determined relation) explain the unity of the objects of a cognition, similarly nirūpya-nirūpaka-bhāvāpanna-viṣayitās (the elements of a cognition related to each other by determiner-determined relation) explain the unity of the elements

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of a cognition. As viṣayatās are related to each other by the determiner-determined relation and the viṣayitās are also related to each other by determiner-determined relation, so are the relations between viṣayatās and their respective viṣayitās. That is to say, the relation of viśeṣyatā to viśeṣyitā, and its converse, the relation of viśeṣaṇatā to viśeṣaṇitā and its converse, the relation of saṃsargatā to saṃsargitā and its converse, the relation of prakāratā to prakāritā and its converse, as well as the relation of viśiṣta-viṣayatā to viśiṣta-viṣayitā and its converse are all determiner-determined relations. In a true cognition, all of them will hold good, but in a false cognition, the last one will not hold good, as the cognition is not related to the qualified object. In our above example, the cognition would be related to the table being brown by the relation of inherence if it is true, but not otherwise. By introducing the determiner-determined relation (nirūpya-nirūpakasambandha) at different levels, the Nyāya not only emphasizes the unity of the cognitive situation but also explains the difference between a true and a false cognition. It is to be noted that in the definition of truth, the Nyāya philosophers have used the terms viśeṣyakatva (the property having the converse of the qualificand at cognitive level) and prakārakatva (the property having the converse of the relational qualifier at cognitive level). I think this is due to the fact that these terms emphasize reference to the objects of cognition, although they are due to relations of objects to the cognition. The following diagrams will represent the points mentioned in the above discussion: 1. Objects of qualificative cognition

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2. Relation of cognition to objects

3. Relation of objects to cognition

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4. Relation among the objects of cognition

5. Relation among the elements of cognition

(B) In this context, I would like to explain the technical term saṅgati (“relevance”), as I shall be referring to it especially in my discussion of inference. Relevance is a relation which holds between the contents of expressions or sentences. According to some Indian logicians such as Gaṅgeśa, if P is relevant to Q, then Q is an answer to a question, say S, and S is due to a cognition, say T, and the content or the object of this cognition, say R, is the relation of relevance. Hence the property of being the content of this type of cognition would be the

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defining property of the relation of relevance. Let us explain with an example. Suppose John would like to make an apple pie. We gave him an apple. Now he asked: What are the other ingredients for an apple pie? The answer is white flour, sugar, and butter. Here P is the expression “apple,” while Q is the expression “white flour, sugar, and butter.” S is the question what are the other ingredients for an apple pie? T is the cognition of other ingredients, and R is the content of this cognition. Since the content of P is related to that of Q by the relation R, the former is the second term, and the latter is the first term of this relation, and the latter is characterized by this relational property. The causal condition for Q is S which, in turn, is causally related to T. Since R is the object of T, it (R) is related to T by the relation called “the property of being the object-possessor.” Hence the relation of relevance may be defined in the following way: P is relevant to Q Df The content of P is related to that of Q by the relation R which is the object of the cognition T, and T is causally related to the question S which is causally related to Q. According to Gaṅgeśa, there are six types of relation of relevance: 1. Memory Context (prasaṅga) Let us call this type of relation R1 and define it in the following way: P is related to Q by R1 Df (i) R1 is a memory object, (ii) it is revived by the cognition of P, (iii) it is related to the content of P, (iv) it is not something which can be ignored, and (v) the cognition of it (i.e., R1) will give rise to a question to which Q is an answer. Let us consider the relation between valid and invalid inferences. Consider P as the statement or the definition of a valid inference and Q as the definition of an invalid inference. The statement or the definition of validity of an inference will give rise to a cognition of validity. But the cognition of validity may give rise to the memory cognition of invalidity by the relation of opposition. Since invalid or fallacious inferences cannot be ignored from our discussion of inference, the cognition of invalidity will give rise to a question about invalidity, which is answered by defining “invalidity.” Hence the content of the definition of “validity” is related to the content of the definition of “invalidity.” The statement or the definition of “invalidity” would be an answer to the question: What are the inferences which are different from or opposed to valid inferences? Here the object of the cognition of inferences which are different from or opposed to valid inferences would be the relation Rl, and the invalid inferences are characterized by this relation. According to a broader interpretation, this type of relevance incudes not only memory context but also other contexts, where things are observed together or related to each other. 2. Justification (upodghāta) This type of relevance deals with the justification of what the speaker intends to say. It is also concerned with the truth or the falsity of a cognition, for example, the justification for the truth of the cognition that Judy is crossing the street.

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We may justify its truth by reference to perception, or inference, or verbal testimony. Here P is the expression “Judy is crossing the Street,” and Q is “I saw Judy crossing the street” if the justification is perceptual. Similarly, the definition of a term is justified in terms of its scope. For example, the definition of the term “human being” is justified in terms of “rationality and animality.” It may be defined in the following way: P is related to Q by R2 Df P is related to Q by the relation of justification.

3. Cause (hetu) This type of relevance refers to the causal conditions of the object of our statement. For example, perception is defined in terms of sense-object contact. Now we may ask about other causal conditions of perception. So the enumeration of other causal conditions, such as manas as well as contact with the self and negative causal conditions, such as not being too far, would answer this question. Hence P would be the definition of perception, and Q would the expression for other causal conditions. The content of P would be related to that of Q by being other causal conditions. 4. Cessation of Objectionable Questions (avasara) This type of relevance emphasizes the sequence of statements, chapters, or topics. Let us consider the sequence of statements, such as P, Q, and R. Since we have stated Q before R, it deals with the question why Q is to be stated before R. In some cases, the questions about R can be answered easily if Q is stated before it and after P. So Q paves the way for establishing R or answers some questions about R or prevents us from asking some questions about R. Let us consider the relation between perception, inference, and comparison which are sources of valid cognition or knowledge in Nyāya epistemology. There is a relation of relevance between perception and inference as well as between inference and comparison. Since the questions about inference are presupposed by questions about comparison, inference is to be discussed prior to comparison. If we know what an inference is, then we can answer the question why comparison is not reducible to an inference. Similarly, a set of topics or chapters are related to each other by this relation of relevance. 5. Having the Same Cause (nirvāhaka-aikya or nirvāhaka-ekatva) After determining or stating the effect of some cause, the question would be: What are the other effects of the same cause? If there are other effects, then the former is related to the latter by having the same cause. Cooking is an obvious example of this type of relation of relevance. If both the softening of the food and the change of color of the food are due to the action of cooking, then they are related to each other by the relation of having the same cause, and the statements about them are related to each other by this type of relation of relevance.

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6. Having the Same Effect (kārya-aikya or kārya-ekatva) If a particular effect is due to several causal conditions, then they are related to each other by the relation of having the same effect. For example, the inferential cognition is due to the cognition of invariable concomitance or pervasion (vyāpti) and the cognition of the property of being the locus of inference (pakṣatā). Hence the relation between these two causal conditions is one of having the same effect. Similarly, since both water and air are causal conditions for sapling, they are related to each other by the relation of having the same effect, and the statements about them exhibit this type of relation of relevance.

Section 2: Part I: Definition of a Cause (A) Now let us discuss the nature of the causal conditions for the substantiation of the application of the principle of simplicity which is one of the criteria for logical thinking, among many others (Bhāṣāparicchedaḥ, pp. 5–6, pp. 102–118, pp. 287–288; Tarkasaṃgrahaḥ, pp. 30–31, pp. 227–245; PadārthatattvaNirūpaṇam, pp. 60–64; Bhāratīya Darśana Koṣa, Vol. 1, pp. 56–57; Sibajiban Bhattacharya, Gadādhara’s Theory of Objectivity, Part 1, pp. 111–138). The Nyāya philosophers have defined causal conditions in terms of the following three properties: • The property of being related to the locus of the effect immediately prior to the effect (avyavahita pūrvavartitva) • The property of being always present (niyatatva) • The property of being simpler than other competing conditions (ananyathāsiddhatva) From the first condition, it follows that if x is a causal condition for the effect E, then x is present immediately prior to E. From the second condition, it follows that x is always present whenever E occurs. The third condition specifies the principle for selecting the conditions which have satisfied the first two conditions. Let us illustrate with an example of the Nyāya system. When an earthen jar is produced, there are innumerable conditions which are present immediately prior to this effect. Moreover, all of them are, directly or indirectly, related to the locus of the effect. These conditions can be divided into three types. Some of the conditions are such that they are present whenever an effect is produced. Positive causal conditions such as space and time are always present whenever an effect is produced. Hence they are called “common causal conditions” (sādhāraṇa kāraṇa). But there are certain conditions which are present whenever a type of effect such as a jar is produced. The conditions such as the jar-maker, the parts of the jar, the conjunction between the parts of a jar, the wheel, the stick, and the thread are present whenever an earthen jar is produced. This type of condition would

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come under uncommon causal conditions (asādhāraṇa kāraṇa). The third type of causal conditions may be called “unique conditions” (ananya kāraṇa). These causal conditions would explain the particularity of the effect as distinct from the effects of the same type. In this example, the particularity of a jar is to be explained in terms of the particularities of its parts. All the parts and the relations of conjunction between the parts are positive unique causal conditions. The not-yet type of absence of the jar would be the negative unique causal condition. Hence the distinction between different types of effect would be drawn in terms of the uncommon causal conditions and the distinction between the effects of the same type in terms of the unique causal conditions. Now the question is whether the conditions, such as the color of the stick, stickness, etc., which satisfy the first two conditions in the case of a jar, are to be considered causes of a jar. Similarly, in the case of a particular jar, the conditions, such as the donkey which has brought the clay or the father of the pot-maker, are to be considered causes of it. The Nyāya philosophers have introduced the third condition to eliminate these conditions which satisfy the first two criteria of a cause. The third condition emphasizes the simplicity of a causal condition in relation to other competing conditions. Regarding the criteria of simplicity, the Nyāya claims that an entity is simpler than another in respect of quantity, or knowledge, or relation. Let us state these criteria: (i) x is simpler than y in respect of quantity iff the limitor of x has less elements than that of y. For example, in the case of perception, both the magnitude of the object (mahatva) and being present in its several parts by the relation of inherence (anekadravya samaveta) equally satisfy the first two conditions. It is to be noted that in the case of perceptual objects only there is mutual pervader-pervaded relation between them. The acceptance of any one of them would explain the occurrence of our perceptual cognitions. Now we have to consider whether one of them is simpler than another in quantity. The object which is present in several parts of a substance by the relation of inherence is qualified by properties such as manyness, substancehood, and the property of being inherence, but the magnitude is qualified by the universal magnitudeness (mahatvatva) only in the ontology of the Nyāya. For this reason, the latter is simpler than the former. Hence, the latter, not the former, is to be considered as a causal condition. (ii) x is simpler than y in respect of knowledge iff the knowledge of x presupposes less than the knowledge of y. Let us consider the causal conditions of the smell of a flower. According to the Nyāya, both the not-yet type of absence of the smell and the not-yet type of absence of the color of the flower satisfy the first two criteria of a causal condition. It is to be noted that in the ontology of the Nyāya smell resides in the earth only. Hence the not-yet type of absence of smell is a cause of smell. The relation between the absence of smell and the absence of color is pervader-pervaded. The absence of smell is the pervaded (vyāpya), and the absence of color is the pervader (vyāpaka). Since both of them satisfy the first two criteria of a cause and do not differ in quantity, the question is whether both of them are to be considered causes of smell. On this

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point, the Nyāya claims that the knowledge of the not-yet type of absence of the smell is simpler than that of the not-yet type of absence of the color. Since we are determining the causal conditions of the smell of a flower, we already know its smell or we know what a smell is. But in order to know the not-yet type of absence of the color, we require the cognition of a color which we may not have. Again, the knowledge of a color alone is not sufficient as we are determining the causal conditions of the smell of a flower. Hence we require the knowledge of the smell in addition to the knowledge of the color. This is how the Nyāya claims that the knowledge of the not-yet type of absence of the smell is simpler than that of the not-yet type of absence of the color. (iii) x is simple than y in respect of relation iff the relation of x to the locus of the effect involves fewer relations than the relation of y to the locus of the same effect. For example, the relation of the stick to the parts of a jar, which is the locus of the effect, involves fewer relations than the relation of the color of the stick or the universal stickness to the parts of the same jar. The stick is related to the parts of the jar by the relations S and T, where S is the relation of the stick to the movement of the wheel and T is the relation of the movement of the wheel to the parts of the jar. But the color of the stick or stickness is related to the parts of the jar by the relations R, S, and T, where R is the relation of the color or stickness to the stick. Here R is the relation of inherence in the ontology of the Nyāya. Hence the stick, not its color or the universal stickness, is considered a causal condition of a jar. Similarly, the father of the jar-maker and the donkey which has brought the clay are not considered as causal conditions of any jar or a particular jar even if they satisfy the first two criteria of a cause in the case of a particular jar. Since the jar-maker is a simpler condition than his father, the former is to be considered as a causal condition. Similarly, the lump of clay is simpler than the donkey which has brought it. Hence the lump of clay is a causal condition, not the donkey which has brought the clay. From the above discussion, it follows that the Nyāya philosophers have applied the principle of simplicity in defining causal conditions in three ways, either by knowledge, quantity, or relation. In modern science also, the principle simplicity is used in selecting a theory provided other conditions remain the same.

Operation (vya¯pa¯ra) and Special Instrumental Cause (karana) ˙ Now let us explain the distinction between the terms vyāpāra (“operation”) and karaṇa (“special instrumental cause”), which are technical terms of the Nyāya (Tarkasaṃgrahaḥ, pp. 227–229; Bhāṣāparicchedaḥ, pp. 287–289). An operation (vyāpāra) is defined in terms of the relation of one causal condition to another. An operation is itself a causal condition, but it is due to another causal condition (tajjanyatve sati tajjanyajanakatvam). Hence it may be defined in the following way:

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(a) x is an operation of the effect E Df (∃y) (y is a cause or a set of causes of E, and x is a cause of E, but x is due to y). In our above example, the movement of the wheel is due to the stick, and the jar is due to the movement of the wheel. For this reason, the movement of the wheel is considered an operation. Since the movement is due to the stick, the stick becomes the operation possessor (vyāpāravat). Other intermediary conditions such as the conjunction relation between the stick and the wheel are to be eliminated by applying the third criteria of a causal condition. Since the stick is related to the parts of the jar through this operation and becomes a cause by virtue of this relation, it is called karaṇa (vyāpāravat kāraṇaṃ karaṇam). Hence karaṇa may be defined in the following way: (b) x is a karaṇa of the effect E Df x is a causal condition, x is related to the locus of E through an operation, and it is considered as a cause due to this relation only. With reference to our example of the jar, two more points are to be noted. Since there are several movements of the wheel, which are due to the stick, there are several operations. Moreover, the wheel is also related to the parts of the pot through the movements which are due to the wheel, and the wheel becomes a cause due to this relation. Hence the wheel is also regarded as a special instrumental cause (karaṇa). Therefore, in this case, there are at least two special instrumental causes and several operations. The special instrumental causes are related to the parts of the jar through these operations only. So this is an example of many-many relation between operations and special instrumental causes. According to the Nyāya, all the four types of relation, viz., (1) many-many, (2) one-one, (3) many-one, and (4) one-many, hold good between operation and special instrumental cause depending on the examples of causation. The example of jar illustrates the manymany type of relation. The following examples would illustrate the remaining types of relation. In the case of felling the tree by striking an axe with certain velocity, the operation is the contact between the axe and the tree, and the special instrumental cause is the axe. Hence it is an example of one-one relation between an operation and a special instrumental cause. The woodcutter or the agent is not a special instrumental cause as it is determined by the agent. Hence in determining a special instrumental cause, we have to exclude the agent. The agent is simply an instrumental cause (nimittakāraṇa). When a piece of cloth is made by conjoining several threads together, the conjunctions between the threads would be the operations, and the loom (vemā) of the weaver would be the special instrumental cause. So it would be an example of many-one relation between the operations and the special instrumental cause. The act of cooking is an example of one-many relation between the operation and the special instrumental causes. The fire or the heat would be the operation and the logs of wood, or the pieces of coal used in generating this fire would be the special instrumental causes of cooking.

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From the above discussion, it follows that there is at least one operation and at least one special instrumental cause. It is to be noted that even in the case of destruction (dhvaṃsa), which is a negative effect, there is an operation and a special instrumental cause. Consider the destruction of a jar with a stick. The stick is the instrumental cause, and the contact between the jar and the stick with certain velocity is the operation. Hence, according to the Nyāya, every effect, positive or negative, has an operation and a special instrumental cause. The operation is defined in terms of the relation between causal conditions, but the special instrumental cause is defined in terms of the operation and its relation to the locus of the effect. In other words, it is related to the locus through the operation only. Now it may be asked whether the Nyāya philosophers would accept the operation of an operation. Since they have accepted the cause of a cause in determining the causal conditions of an effect, they might accept the operation of an operation as well. It may also be asked: Why do we need karaṇa at all? Is not operation adequate for determining the cause of an effect? In reply, the Nyāya philosophers have put forward two types of argument. As regards the operation of an operation, it is claimed that it would lead to a regress. Hence higher-order operations and thereby higher-order special instrumental causes of these operations are to be excluded by applying the third property of a causal condition (ananyathāsiddhatva). As regards other questions, the Nyāya makes an appeal to ordinary usage. In the case of felling the tree, if we consider the contact between the axe with a certain velocity and the tree as the only cause, then we are going against the ordinary usage. In our ordinary language (parlance), we consider the axe also as a cause. Since the Nyāya philosophers try to retain our ordinary usage as far as possible, they consider axe also as a causal condition. Since it is related to the locus of the effect through the contact which is the operation, it is considered the special instrumental cause (karaṇa) of the effect.

Part II: Belief In this context, I shall discuss the Nyāya conception of belief, as the Nyāya philosophers have explained false beliefs without tertiary entities, such as propositions or images. Since there is no proposition in the Nyāya as distinct from a sentence, beliefs are considered true or false. It is to be noted that belief is a doubt-free cognition. The Nyāya concept of doubt does not lead to skepticism, as a dubious cognition rests on certainty. Moreover, the Nyāya discussion of belief suggests solutions to some problems of belief in the Western philosophy. I shall also mention the view of Russell, as he claims that no one has succeeded in explaining a false belief without postulating the existence of the nonexistent. Russell, in “The Philosophy of Logical Atomism,” claimed that in a belief sentence, such as “Othello believes that Desdemona loves Cassio,” there are at least two verbs. Here the verbs “believes” and “loves” have occurred as genuine verbs, and the verb in the subordinate clause seems to relate Desdemona to Cassio; but in fact, it does not do so. He says:

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This is what constitutes the puzzle about the nature of belief. You will notice that wherever one gets to really close quarters with the theory of error one has the puzzle of how to deal with error without assuming the existence of the non-existent (Russell 1977).

Now the question is how to explain the nature of this belief without postulating nonexistent love as an entity, which will relate Desdemona to Cassio. Moreover, Russell claimed that “loves” should be treated as a verb. This requirement leads to the rejection of his earlier view proposed in The Problems of Philosophy (Russell 1980), where this sentence has been analyzed as a four-place relation between Othello, Desdemona, loves, and Cassio. Hence it takes the form: B (Othello, Desdemona, loves, Cassio).

Since the verb “loves” in this sentence is on a par with the terms “Desdemona” and “Cassio,” this analysis does not fulfil one of the above requirements of Russell. In spite of these shortcomings, I think, Russell’s great contribution lies in the view that what occurs in a belief sentence is not a proposition, but the constituents of a proposition, and in his suggestions that a satisfactory theory of belief should not postulate nonexistent objects and should not reduce the verb in the subordinate clause to a term. In the context of our discussion of the Nyāya, we shall see how the Nyāya philosophers have avoided the shortcomings of Russell’s theory and at the same time followed the suggestions of a satisfactory theory of belief. Let us begin with the Nyāya analysis of this sentence. It is to be noted that in this case, we are not talking about Desdemona or Cassio, but about the belief state of Othello, which is related to the self by the relation of inherence in the ontology of the Nyāya. In the content of this belief, there are three major elements, namely, Desdemona, Cassio, and the relation of love (loving relation). Desdemona is the qualificand, Cassio is the qualifier, and love is the qualification relation. The relation of the mental state of Othello to Desdemona is the property of being the qualificand (viśeṣyatā) residing in Desdemona, to Cassio is the property of being the qualifier (viśeṣaṇatā) residing in Cassio, and to the relation of love is the property of being the qualification relation (saṃsargatā) residing in love. As a belief mental state is related to its objects, so are objects are related to the belief state. Hence the relation of Desdemona to this belief is the converse of viśeṣyatā, i.e., visesyitā; the relation of Cassio to this belief is the converse of viśeṣaṇatā, i.e., visesanitā, to introduce a technical term; and the relation of love to this belief is the converse of saṃsargatā, i.e., samsargitā. In addition to these elements, there are a few more elements in the content of this belief. Since both Desdemona and Cassio have occurred in the content of this belief, according to the Nyāya, they must be presented under some mode of presentation. In other words, the property of being the qualificand residing in Desdemona is limited by the property of being Desdemona, and the property of being the qualifier residing in Cassio is limited by the property of being Cassio (For a discussion on this property, see Shaw’s 1985, pp. 327–72.). Since the relation of love has been mentioned, it is presented under the mode loveness. So we have, broadly speaking,

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two terms called “qualificand” and “qualifier,” the relation of love, three propertylimitors or modes of presentation, and three relational properties of being the content. Now the question is how to explain the relation among these elements. The Nyāya claims that they are related in the following way: 1. The property of being the qualificand residing in Desdemona is limited by the property of being Desdemona. 2. The property of being the qualifier residing in Cassio is limited by the property of being Cassio. 3. The property of being the qualification relation residing in love is limited by loveness. 4. The property of being the qualifier residing in Cassio is also limited by the relation of love. 5. The property of being the qualificand residing in Desdemona is determined by the property of being the qualifier residing in Cassio. 6. The property of being the qualifier residing in Cassio is determined by the property of being the qualificand residing in Desdemona. 7. The property of being the qualification relation residing in love is determined by the property of being the qualifier residing in Cassio. 8. The property of being the qualifier residing in Cassio is determined by the property of being the qualification relation residing in love. In this context, it is to be noted that the above three types of relations which relate a belief to its content, viz., the property of being the qualificand, the property of being the qualifier, and the property of being the qualification relation, are present in any belief, true or false. But in a true belief, there is another type of relation which relates the belief to the unified content or the fact by virtue of which a sentence is considered as true. Now let us discuss the nature of the belief state. As a belief is related to its contents, so are the contents related to the belief. If the above three relations, viz., the property of being the qualificand, the property of being the qualifier, and the property of being the qualification relation are called “R,” “S,” and “T,” respectively, then the relation of Desdemona to this belief is the converse of R, the relation of Cassio to this belief is the converse of S, and the relation of love to this belief is the converse of T. These are all properties of Othello’s belief, and they are related to each other in the following way: (a) (b) (c) (d)

The converse of R is determined by the converse of S. The converse of S is determined by the converse of R. The converse of S is determined by the converse of T. The converse of T is determined by the converse of S.

In other words, by introducing the relations (a–d), the Nyāya emphasizes the unity of the belief state. Moreover, the truth of the sentence “Othello believes that Desdemona loves Cassio” does not depend on the truth of the sentence “Desdemona loves

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Cassio.” Now the question is: How can the belief state of Othello be related to the relation love which does not exist between them? If there is no such relation, then the converse of the property of being the qualification relation cannot characterize the belief state of Othello. In reply, the Nyāya claims that the belief state of Othello is related to a real relation of love, for example, between John and Janet, which is real elsewhere or elsewhen. Since this relation is real elsewhere and the belief state is related to this relation, it is characterized by the converse of this qualification relation. It is to be noted that here also the relation performs both the functions. It relates John to Janet, as John loves Janet, and makes Desdemona the qualificand and Cassio the qualifier. For this reason, the relation has not been reduced to a term. This is how the Nyāya has avoided the postulation of nonexistent entities in their explanation of false beliefs or cognitions. Moreover, this explanation does not reduce the verb in the subordinate clause to a term. This explanation of belief can be represented by the following diagram: Othello

Desdemona

Love

Cassio

John loves Janet

Section 3: Knowledge In this section, I shall focus on the Nyāya conception of knowledge and the classification of the causal conditions of each of the sources of knowledge into four types, so that the Nyāya concept of justification can be demonstrated. In the context of inference, I shall show how the premises and the conclusion are related by the relation of relevance. The Nyāya conception of fallacy is broader than its Western counterpart. In this context, I shall also demonstrate how tarka (a type of counterfactual conditional) is related to an inference. The Nyāya conception of meaning will demonstrate why transformationally equivalent sentences do not have the same meaning. The logical distinction between the atomistic and holistic understanding will also be mentioned.

Sources of Knowledge As regards sources of valid cognition or knowledge, all the systems of Indian philosophy have emphasized perception. In this context, it is to be noted that there

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is a substantial difference of opinion among the different schools of Indian philosophy regarding the sources of knowledge. For the Cārvāka (a type of materialist) philosophers, perception is regarded as the only source of valid cognition. The Bauddha and the Vaiśeṣika philosophers accept both perception and inference as sources of knowledge. The Sāṃkhya, Rāmānuja, and Bhāsarvajña accept perception, inference, and verbal testimony. The Nyāya accepts perception, inference, comparison, and verbal testimony. But the followers of the Prabhākara school of Mīmāṃsā accept presumption in addition to the four sources accepted by the Nyāya. The followers of the Kumārila Bhaṭṭa school of Mīmāṃsā and the Advaita Vedānta accept non-apprehension (anupalabdhi) in addition to the previous five sources of knowledge. The followers of the Purāṇas accept two more, namely, entailment (sambhava) and tradition (aitihya). The followers of the Tantra accept gesture and posture (ceṣtā) in addition to the eight other sources of knowledge. The Jaina philosophers have accepted two more sources of valid cognition, namely, the use of a type of counterfactual conditional (tarka) and memory (smṛti). Since the Nyāya philosophers do not accept presumption as a source of valid cognition, it is reduced to agreement in absence type of inference (vyatirekī-anumāna). Similarly, non-apprehension is reduced to perception, entailment to inference, tradition to verbal testimony, and gesture (or posture) to inference. But tarka is not reduced to an inference. It gives rise to an inference and thereby becomes auxiliary to an inference. Similarly, memory is not reduced to some other source of valid cognition. But the truth of a memory cognition depends upon the truth of a previous apprehension which is derived from perception, inference, comparison, or verbal testimony. In this context, the Nyāya philosophers have also applied the principle of simplicity, as there is no need to accept more than four sources of knowledge.

The Nya¯ya on Knowledge The Nyāya philosophers have discussed the conditions or causal conditions of cognition, conditions of a true cognition, conditions of a false cognition, and conditions which justify the truth of a cognition. The causal conditions involved in the process are not exclusively internal. Hence some conditions are external.

Perception The Nyāya claims that there are both a set of positive and a set of negative causal conditions of perception. The perceiver (the self), the internal sense organ (manas), the external sense organs (such as eyes), the objects of perception, the sense-object contact, etc. are positive causal conditions. In addition to these causal conditions, there are certain negative causal conditions. In this context, it is to be noted that the Sāṃkhya philosophers have mentioned the following negative causal conditions of

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perception, some of which have been accepted by the Nyāya (Purnachandra Vedāntacuñcu, Sāṃkhyakārikā of Īśvarakṛṣṇa, and Tattva-Kaumudī of Vācaspati Miśra, with Bengali translation and commentary, first published in 1901, West Bengal Book Board, 1983, pp. 66–68): (a) Not being too far (atidūratābhāva) (b) Not being too close (atisāmīpyābhāva) (c) Absence of loss of sense organs, such as deafness, blindness, etc. (indriyanāśābhāva) (d) Not being inattentive (mano’navasthānābhāva) (e) Not being too subtle (sūkṣmābhāva) (f) Not having intervening objects such as wall, screen, etc. (vyavadhānābhāva) (g) Not being overshadowed (or covered) by a more powerful object (abhibhavābhāva), e.g., during the day, stars are not visible as they are overshadowed by the rays of the sun (h) Not being mixed up with similar objects (samānābhihārābhāva), e.g., rainwater cannot be perceived in a lake or a river separately as it is mixed up with similar objects But the Nyāya philosophers have not treated all of them as negative causal conditions. They would consider only (a), (b), (g), and (h) as negative causal conditions. The remaining four will be considered positive. Therefore the third one will be normal sense organs instead of absence of loss of sense organs. The fourth one will be attentive instead of not being inattentive and the fifth one having some magnitude (mahatva) instead of not being too subtle. The sixth one is to be rejected as negative on the ground that the sense-object contact is a positive causal condition. Hence the Nyāya philosophers would consider only (a), (b), (g), and (h) in the above list as negative. In the case of an ordinary perceptual cognition, sense organs are special instrumental causes (karaṇas), and the sense-object contact is the operation (vyāpāra). Let us consider the following example of the Nyāya philosophers: The floor has a pot. In this case, our visual sense organ is the special instrumental cause, and the contact between the visual sense organ and the floor is the operation. Since our sense organ is related to the floor, it is also related to the pot which is on the floor. Since the cognition that the floor has a pot is due to a sense organ, it is considered as perceptual. In this case, the objects of cognition such as the floor, the pot, and the relation of conjunction are related to the cognition. Hence the cognition is also related to all these items. The cognition will be related to these items even if it is false. Hence in terms of the relation between these items and the cognition alone, we cannot draw the distinction between a true and a false cognition. When a perceptual cognition is true, our sense organ is related to the qualified object. In the above example, our visual sense organ is related not only to the floor but also to the floor that is qualified by a pot on it.

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Hence the cognition generated by this process will be related to the qualified object or the fact. The relation of the cognition to the fact is called viśiṣṭa viṣayatā, which is a relational property of the object of cognition. The cognition is characterized by the converse of this relational property, which is called viśiṣṭa viṣayitā. Thus a true perceptual cognition presupposes certain additional conditions. A false perceptual cognition could be due to a defect (doṣa) or an inappropriate causal condition (kāraṇavaiguṇya). A defect (doṣa) is the negatum of a negative causal condition of a true perceptual cognition, but an inappropriate causal condition (kāraṇavaiguṇya) is the weakness of a positive causal condition of a true perceptual cognition such as a defective visual sense organ or the absence of a positive causal condition of a true cognition such as blindness or loss of a visual sense organ. So a visual perception could be false due to distance (dūratva), which is the negatum of a negative causal condition of a true cognition. Similarly, it could be false due to weakness of the visual sense organ or due to the absence of the visual sense organ. In our above example, if the cognition is true, then it is related to the floor, the pot, the relation of conjunction, and the qualified object, i.e., the floor qualified by a pot on it. The causal conditions of this perceptual cognition would include the relation of the visual sense organ to these items. But in addition to these relations of the cognition to its objects, the Nyāya philosophers have accepted the relation of the cognition to universal floorness and the relation of the cognition to universal potness. Now the question is: What is the need for these additional relations? In this context, it is to be noted that some contemporary epistemologists claim that identification and discrimination are necessary for knowledge. On Goldman’s theory, if S knows that p, then S can discriminate the truth of p from relevant alternatives. In his system, these alternatives are counterfactual. But his theory cannot explain why a person, say Smith, is able to discriminate the truth of p from relevant alternatives, but another person, say Jones, is not able to discriminate the truth of p from relevant alternatives. The Nyāya can explain this phenomenon in terms of the relation of Smith’s cognition to the universal floorness and the universal potness which are limitors of a floor and a pot, respectively. Since Smith’s sense organ is related not only to the floor and the pot but also to their limitors, his cognition is related to these limitors as well. Since the cognition of limitors can explain our ability to discriminate, there is a need for these limitors in epistemic contexts. This is how the Nyāya solves the problem of Goldman. From this discussion, it follows that the Nyāya philosophers have mentioned a set of conditions for perception, a set of conditions for its truth, a set of conditions for its falsity, and another set for the justification of its truth. 1. Inference Similarly, in the case of an inference, the Nyāya philosophers have discussed the causal conditions of an inferential cognition (anumiti), the causal conditions of its truth or falsehood, and the causal conditions which justify the truth of an inferential cognition or the ability to discriminate. An inference, according to the Nyāya, has three terms, namely, sādhya (probandum), pakṣa (locus of inference), and hetu

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(probans or reason). The term sādhya refers to what is to be inferred. The term pakṣa refers to the locus of inference where there is some doubt about the presence of sādhya. The term hetu refers to the reason by means of which the sādhya is inferred in the pakṣa. In this context, it is to be noted that an inference for others, according to the Nyāya, involves five members, which are related to each other by the relation of relevance. Relevance is a relation between the contents of expressions or sentences via some questions. Let us consider the following inference for others: Thesis (pratijñā): The mountain has a fire. Reason (hetu): Because of smoke. Example (udāharaṇa): Wherever there is smoke, there is fire, as in a kitchen, etc. Application (upanaya): The mountain has smoke which is pervaded by fire. Conclusion (nigamana): Therefore, the mountain has a fire. This inference has the following form: Thesis (pratijñā): a is G. Reason (hetu): Because of F. Example (udāharaṇa): Wherever there is F, there is G, as in b, etc. Application (upanaya): a has F which is pervaded by G. Conclusion (nigamana): Hence, a is G, or G is present in a, where a is the locus of the inference (pakṣa), F is the probans, G is the probandum, and b is the locus where G is known to be present (sapakṣa). An inferential cognition, according to the Nyāya, has certain instrumental causal conditions (nimitta-kāraṇas) such as parāmarśa (operation), vyāpti jñāna (cognition of invariable concomitance between the probans and the probandum), and pakṣatā (a special relational property of the locus). An inferential cognition (anumiti) is usually defined in terms of parāmarśa (operation). Parāmarśa (operation) is the cognition of the property of being the pervaded which appears as the qualifier of the probans which is present in the locus (vyāpti-prakāraka-pakṣadharmatā-jñāna). In other words, an inferential cognition of the form “a is G” is derivable from the cognition of the form “a is F which is pervaded by G,” where a is the locus, F is the probans, and G is the probandum. The latter is a causal condition of the former. But the truth of the inferential cognition does not depend on this causal condition. Hence the truth of the cognition a is G does not depend on the cognition of a is F which is pervaded by G. The truth depends on the fact that the locus which is cognized in the operation is characterized by the probandum. Now the question is whether a true inferential cognition would assume the status of knowledge. In this context, it is to be noted that a false operation such as “the mountain has fog which is pervaded by fire” might lead to the true inferential cognition “the mountain has fire.” Since the occurrence of a false cognition can be prevented by a true one, the occurrence of the above false operation can be prevented

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by the true cognition that fog is not pervaded by fire. If the occurrence of the operation is prevented, then the occurrence of the inferential cognition which is due to this operation would also be prevented. In other words, if a person knows that fog is not pervaded by fire, then he would not use this operation to infer that the mountain has fire. For this reason, the Nyāya would claim that the above true inferential cognition does not have the status of knowledge. In other words, if the inferential process which leads to a true cognition contains a false cognition, then the true inferential cognition does not have the status of knowledge, i.e., a justified true cognition. In an inference for others, all the five sentences are needed, because each of them is an answer to a different question and gives some new information. But in an inference for oneself, all of them are not required and there is no need to use a sentence. Hence a deaf and a mute person can also have an inferential cognition. What is required is the operation (parāmarśa), which corresponds to the application in our above example and the cognitions that will give rise to this operation. In our above example, the thesis (pratiñnā-vākya) is an answer to the question what is to be established in a (paksa). a is usually considered as something where there is doubt about the presence of the probandum. The reason (hetu-vakya) is an answer to the question what signifies the probandum. Hence it states that the probans signifies the probandum. The signifier-significate ( jñāpya-jñāpaka) relation holds between the objects of two cognitions. The cognition of the signifier ( jñāpaka) gives rise to the cognition of the significate ( jñāpya). The reason does not state that the locus a (pakṣa) is characterized by the probans. If it is stated as a is F, then one of the premises would be superfluous. Now it may be asked: Why should we consider the probans as the signifier? The answer is given by stating a rule (vyāpti) along with some examples which give rise to the cognition of the invariable concomitance of the probans with the probandum (vyāpti-jñāna). For this reason, the third step is called “example.” Both the examples of agreement in presence and agreement in absence are to be stated in support of this rule of invariable concomitance. The observation of the presence of the probans and the probandum in some loci and the nonobservance of the presence of the probans and the absence of the probandum in some other loci are required for the cognition of the rule of invariable concomitance between the probans and the probandum. The rule takes the form of a universal sentence that can be stated as: ðxÞ ðIf F x, then G xÞ: The application (upanaya-vakya) is an answer to the question whether a (i.e., paksa) is characterized by this type of F. Since the reason does not state that a is characterized by F, the application gives us some new information about a. The reason simply states that F is the signifier of G. Hence the application gives us some new information, which is not already contained in the previous sentences. The conclusion (nigamanavakya) is an answer to the question whether the probandum which is the significate of that type of probans is in a. Hence it is an answer to the question whether G which is the significate of F which is pervaded by G is present in a.

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The difference between the thesis and the conclusion lies in the fact that the thesis simply states what is to be established in the locus but the conclusion states how it is to be established in the locus. The word “hence” or its synonym in the conclusion means “the significate of the cognition of the probans.” Hence the conclusion (nigamana-vakya) means that G which is the significate of F, which is pervaded by G and is in a, is present in a. Here the new information lies in the fact that G is the significate of that type of F. This is how the Nyāya philosophers would explain the difference between the thesis and the conclusion. Since each of the members in an inference is related by the relation of relevance, an inference, valid or invalid, is considered a large sentence (mahavakya). The relation of the thesis to reason is upodghāta saṅgati, as the latter gives epistemic justification for the former. Similarly, the relation of reason to example is also upodghāta saṅgati, as it suggests an ontological ground for the relation of F to G. But the relation of example to application is kāryatva saṅgati, which is the converse of hetutā saṅgati. In other words, we cannot have the cognition of the latter without the cognition of the former. Hence they are related by cause-effect relation. Similarly, the relation of application to conclusion is kāryatva saṅgati, as they are related by the cause-effect relation. So we cannot have the cognition of the conclusion without the cognition of the application. From the above discussion, it follows that in the case of inference also, there are four sets of causal conditions. The guarantee for the truth of an inferential cognition depends on the truth of parāmarśa, although a false parāmarśa such as the mountain has fog which is pervaded by fire may lead to a true cognition: the mountain has fire. Again, a false parāmarśa, such as the lake has smoke which is pervaded by fire, will lead to a false inferential cognition the lake has fire. Hence the truth depends on the fact that the locus which is cognized in the parāmarśa is characterized by the sādhya. But the guarantee for its truth or its justification depends on the truth of the parāmarśa.

Valid and Invalid Inferences Now I would like to discuss the Nyāya distinction between valid and invalid inferences. Each of the sentences in an inference for others is an answer to a question, and each of them except the last one will give rise to a question. Moreover, each of them is used to generate cognition in the hearer. Since a self-contradictory sentence such as “a is both G and not G” cannot generate a cognition, it cannot be used either as a premise or conclusion of an inference. If the inference (not the inferential cognition) is valid (nyāya), then all the sentences must be true, and the conclusion will follow from the premise or the premises. In other words, there is no true preventer cognition. Hence the application (upanaya-vākya), which represents the operation (vyāpāra) of an inferential cognition (anumiti), will entail the conclusion. Invalid inferences (nyāyābhāsas) are divided into two types. One type of invalid inference contains a false premise or premises, but the other type does not contain any false premise. Hence the former may be called “logically invalid” and the latter “epistemically invalid.” Hence the truth of the premises and the conclusion is not sufficient to define the validity of an inference. But any inference, valid or invalid,

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must satisfy the relevance condition. This point also is very important for understanding the difference between the Western and the Indian concept of inference. Now let us discuss the nature of the probans in a valid inference. If the valid inference is of the agreement in presence and agreement in absence type, then its probans has the following five characteristics: (a) It is present in the locus of the inference (pakṣa). Hence it has the property of being present in the locus (pakṣa-sattva). (b) It is also present in some of the loci which are known to be characterized by the probandum. Hence it has the property of being present in similar loci (sapakṣasattva). (c) It is not present in those loci which are known to be characterized by the absence of the probandum. Hence it has the property of being absent from dissimilar loci (vipaksāsattva). (d) It has no counter-probans (prati-hetu) which will demonstrate the absence of the probandum in the locus of the inference. A counter-probans is different from the probans in question, and it is pervaded by the absence of the probandum. Hence it has the property of not having a counter-probans (asatpratipakṣattva). (e) It is different from the probans which can be used to establish the probandum in the locus which is characterized by the absence of the probandum. Hence it has the property of being different from this type of probans (abādhitattva).

Fallacies An inference, according to the Nyāya, will be fallacious if the probans lacks one of these characteristics. In other words, if the probantia of the inferences of the agreement in presence and absence type do not have all the five characteristics and the probantia of the other types of inferences (agreement in presence only or agreement in absence only) do not have the remaining four characteristics, then they are fallacious. Since a probans is used to infer the probandum, the fallacy of an inference has been ascribed to the probans. Hence a fallacious inference is called hetvābhāsa (“defective probans”). A fallacy or hetvābhāsa has been defined in the following way: x is a hetvābhāsa if the true cognition of x prevents the occurrence of an inferential cognition (anumiti) or the operation (parāmarśa) which is the vyāpāra of an inferential cognition, where x is a qualified object of cognition.

Let us consider a fallacious inference, for example, this lake has fire because of smoke. In this case, the inferential cognition this lake has fire is false. From the above definition of fallacy, it follows that if the person would have known that this lake has no fire, then the inferential cognition would have been prevented. The absence of fire in the lake which is the object of cognition is the defect of the probans. Since smoke is the probans in this inference, it is infected with this defect. Now the question is: How can smoke be qualified by this defect? The Nyāya philosophers explain the relation between them in terms of the relation of a cognition to its object, which is called viṣayatā, and the limitor of the property of

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being the probans (hetutāvacchedaka). In other words, it is explained in terms of a conjunctive cognition such that one of them is the defect and the other one is the probans. In our above example, one of the objects of this conjunctive cognition would be the lake qualified by the absence of fire and the other one would be smoke. If we would have known this property of smoke, then we would not have inferred the presence of fire in the lake. Since smoke was used to make this inference and since this function of smoke will be restricted by our cognition of smoke qualified by the absence of fire in the lake, smoke as a probans is considered defective. In other words, it will fail to perform its function as probans for the above inference. There are five types of fallacies, viz., (1) asiddha (unestablished), (2) vyabhicāra (deviation), (3) viruddha (opposed), (4) satpratipakṣa (existence of a counterthesis), and (5) bādha (absence of the probandum in the locus). asiddha (Unestablished) If the probans cannot be established, it is called asiddha. This type of fallacy can occur in five ways: 1. The locus of the inference (pakṣa) is not real. For example, the golden mountain has fire, because of smoke. Here the golden mountain is the locus (pakṣa), smoke is the probans, and fire is the probandum. Since the locus is unreal or unexemplified (aprasiddha), the probans cannot reside in it. Since the locus cannot be established, this fallacy is called āśrayāsiddha (“unestablished locus”). Here the defect is the absence of gold in the mountain. The cognition of this defect is opposed to the cognition of the presence of the probans in the locus (pakṣadharmatā-jñāna) and the inferential cognition (anumiti). Here the probans lacks the property of being present in the locus (pakṣa-sattva). 2. The probans does not reside in the locus of the inference, although the locus is real and the probans is real. For example, sound is non-eternal, because of visibility. Here both sound and visibility are real entities, but visibility does not qualify sound. Since the probans cannot qualify the locus of the inference, this type of fallacy is called svarūpāsiddha (“unestablished in the locus”). This type of fallacy is opposed to the cognition of the presence of the probans in the locus (pakṣadharmatā-jñāna). Here also the probans lacks the property of being present in the locus (pakṣasattva). The defect (doṣa) is the absence of visibility in the sound. 3. The probans is unreal or unexemplified, although the locus is real. For example, the mountain has fire, because of golden smoke. In this case, the golden smoke which is the probans is itself unreal. Since the probans is unreal, this type of fallacy is called hetvasiddha (“unestablished probans”). This type of fallacy is opposed to the cognition of the presence of the probans in the locus of inference and the cognition of the rule of invariable concomitance between the probans and the probandum. Since the probans is unexemplified, it cannot have any of the properties of a genuine probans. 4. Another type of asiddha (unestablished) fallacy will occur if the probans of an unexemplified probandum is not present in the locus of an inference. For

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example, the mountain has golden fire, because of smoke. In this case, smoke is present on the mountain, but not as the probans of the golden fire. Hence this type of fallacy is called sādhyāsiddha (“unestablished probandum”). Here the probans lacks both sapakṣasattva (the property of being present in similar cases) and vipakṣāsattva (the property of being absent from dissimilar cases). Here the defect is the absence of gold in fire. The cognition of this defect is opposed to both the operation and the inferential cognition. 5. There is another type of asiddha fallacy known as vyāpyatvāsiddha (unestablished property of being the pervaded). In this case, the locus is real, the probans is real, and the probans is present in the locus, but the probans is not qualified by the property of being the pervaded which is limited by a property. The mode under which the probans has been cognized becomes the limitor of the property of being the pervaded (vyāpyatāvacchedaka). This type of fallacy will occur when the mode under which the probans has been cognized does not limit the property of being the pervaded which resides in the probans. For example, the mountain has fire, because of blue smoke. If blue smoke is the probans, then the rule of invariable concomitance would be between blue smoke and fire. The property of being the pervaded residing in blue smoke will be limited by blue smokeness. But this rule of invariable concomitance cannot substantiate the rule of invariable concomitance between smoke and fire. Since there is no property of being the pervaded which is limited by blue smokeness and resides in blue smoke, the type of fallacy present in the above inference is called vyāpyatvāsiddha. Here the defect will be the absence of the property of being the pervaded which is limited by blue smokeness and which resides in blue smoke. The cognition of this defect would prevent the cognition of the invariable concomitance between blue smoke and fire and thereby the cognition of the operation. vyabhica¯ra (Deviation) There are three types of fallacy of deviation. In all the three cases, the cognition of the defect would prevent the cognition of the rule of invariable concomitance between the probans and the probandum. (a) sādhāraṇa-vyabhicāra (common deviation) If the probans is present in pakṣa (locus of the inference), sapakṣa (locus known to be characterized by the probandum) and vipakṣa (locus known to be characterized by the absence of the probandum), then this type of fallacy would occur, and the probans is called sādhāraṇa-vyabhicārī-hetu (“common deviating probans”). For example, the mountain has fire, because of knowability. Since the probans is present in the locus of the absence of the probandum, the cognition of deviation (vyabhicāra) is opposed to the cognition of the invariable concomitance between the probans and the probandum. If we take a lake as vipakṣa, then fire is absent from it, but knowability is present in it. Hence there cannot be a cognition of the invariable concomitance between knowability and fire. Moreover,

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since there is deviation, the rule of invariable concomitance will not hold good between the probans and the probandum. In this case, the defect (doṣa) is the absence of fire in a lake which has knowability. Hence the cognition of this defect will prevent the cognition of the invariable concomitance between knowability and fire and thereby the cognition of the operation. In this fallacy, the probans lacks the property of not being present in vipakṣa. (b) asādhāraṇa-vyabhicāra (uncommon deviation) If the probans is present in the locus of the inference (pakṣa) only, then it is called asādharaṇa-vyabhicārī-hetu (“uncommon deviating probans”). In other words, the probans is not present in sapakṣa (the locus of the probandum) and in vipakṣa (the locus of the absence of the probandum), but is present in pakṣa (the locus of the inference). For example, sound is non-eternal, because of soundness. In this case, sound is pakṣa, a non-eternal object such as a pot is sapakṣa and an eternal object such as space is vipakṣa. Since soundness is not present in a pot, it lacks the property of being present in sapakṣa. Since soundness cannot be perceived in non-eternal objects, there cannot be cognition of the agreement in presence type of invariable concomitance between the probans and the probandum. But the probans is absent from the eternal objects. Since the agreement in absence between the probans and the probandum can be observed, the agreement in absence type of invariable concomitance (vyatireka-vyāpti) can be cognized. In this example, the defect is the absence of soundness in a non-eternal object such as a pot, and the probans lacks the property of being present in sapakṣa. The cognition of this defect would prevent the cognition of the agreement in presence type of invariable concomitance (anvaya-vyāpti). But it will not prevent the cognition of the agreement in absence type of invariable concomitance (vyatireka-vyāpti). Since there are two types of invariable concomitance, there would be two types of operation. The agreement in presence type of operation will be prevented by this type of defect. Hence the cognition of this type of defect does not prevent the cognition of all types of invariable concomitance or operation. For this reason, it may be treated as an epistemic fallacy as opposed to a logical one (where some of the sentences or cognitions are false). In the example above, the sentences would not be false, but we fail to cognize the agreement in presence type of invariable concomitance and thereby the agreement in presence type of operation. Moreover, this type of epistemic defect can also be removed. In our example, this defect can be removed if there is certainty about the presence of the probandum in some sounds such as the sound of a music. If it were so, then the locus would not be sound in general as it is in the above example, but some specific sounds such as the one which follows lightning. Since this defect can be removed, it is called anitya (“impermanent”). (c) anupasaṁhārī-vyabhicāra (unsupported deviation) If everything becomes pakṣa and thereby the probans does not have either sapakṣa or vipakṣa, then the fallacy of anupasaṁhārī-vyabhicāra will occur. Let us consider the following two examples:

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(i) Everything is non-eternal, because of knowability. (ii) Everything is nameable, because of knowability. In both (i) and (ii), everything is the locus of inference. There is doubt about the presence of non-eternality in (i) and nameability in (ii). Since everything is pakṣa, there is no sapakṣa or vipakṣa. Since the copresence of the probans and the probandum cannot be observed, the agreement in presence type of invariable concomitance cannot be cognized. Similarly, since the co-absence of the probans and the probandum cannot be observed, the agreement in absence type of invariable concomitance cannot be cognized. Since neither type of invariable concomitance is cognized, neither type of operation will occur. Since there is neither sapkṣa nor vipakṣa, the probans lacks both the property of being present in sapakṣa and the property of being absent from vipakṣa. As regards the nature of this fallacy, it is not logical, but epistemological. If a person does not have doubt about the presence of the probandum in everything, then this epistemic defect can be removed. viruddha (Opposed) If the probans is pervaded by the absence of the probandum, the probans is called viruddha-hetu (“opposed probans”). Hence the invariable concomitance would be between the probans and the absence of the probandum, not between the probans and the probandum. In other words, wherever the probans is present, the probandum is absent. For example, sound is eternal, because of the property of being an effect. Since an effect is non-eternal, the probans, far from establishing the probandum, establishes the absence of the probandum. In the case of viruddha fallacy, the probans lacks the property of being present in sapakṣa and the property of being absent from vipakṣa. Hence the agreement in presence (anvaya-sahacāra) and agreement in absence (vyatireka-sahacāra) cannot be observed. From this it follows that neither the invariable concomitance in presence nor the invariable concomitance in absence can be cognized. Moreover, since both the types of invariable concomitance are false, the defect would be the falsity of the invariable concomitances. Hence the cognition of this defect will be opposed to the cognition of both the types of invariable concomitance and thereby both the types of operation. It is also opposed to the inferential cognition. Since it is a permanent defect, it may be called “logical fallacy.” satpratipaksa (Existence of Counter-thesis) ˙ The word satpratipakṣa has two meanings. It may mean either the thesis of the opponent or a type of defect (doṣa) which will prevent an inferential cognition. Hence as a fallacy, it refers to a defect. Consider the following operations: (a) The lake has smoke which is pervaded by fire. (b) The lake has water which is pervaded by the absence of fire. The operation (b) will prevent the occurrence of the inferential cognition “The lake has fire” which is due to the operation (a).

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The object of the operation (b) is the defect, and the probans of the operation (a) is infected with this defect. Since there is a counter-probans which is pervaded by the absence of fire, the probans of (a) lacks the property asatpratipakṣattva (the property of not having a counter-probans which is pervaded by the absence of the probandum). ba¯dha (Absence of the Probandum Characterizing the Locus) The fallacy of bādha occurs when a probans is used to establish a probandum in a locus which is characterized by the absence of the probandum. For example, fire is cold, because of substancehood, as in water. In the case of a bādha fallacy, the inferential cognition is directly prevented by the cognition of the absence of the probandum in the locus. In the above example, the operation is the cognition “Fire has substancehood which is pervaded by coldness.” This operation will yield the cognition “Fire is cold.” But the cognition “Fire has absence of coldness” will prevent the occurrence of the inferential cognition. Since the preventer cognition is true, its object is the defect (doṣa). Hence the cognition of bādha fallacy is directly opposed to the inferential cognition. It is to be noted that there is a difference between satpratipakṣa and bādha fallacy, although both of them are directly opposed to the inferential cognition. The difference may be explained in the following way. Let us consider the following satpratipakṣa: The lake has water which is pervaded by the absence of fire. This satpratipakṣa would prevent the occurrence of the inferential cognition the lake has fire, which is derivable from the operation the lake has smoke which is pervaded by fire. Moreover, this operation is directly opposed to the cognition the lake has fire as it yields the cognition the lake has absence of fire. Therefore, it is directly as well as indirectly the preventer of the cognition the lake has fire. But the cognition of the bādha fallacy is directly opposed to the inferential cognition the lake has fire. From the Nyāya discussion of different types of fallacies, it follows that the Nyāya philosophers are dealing not only with the falsity of the premise(s) or the conclusion of a fallacious inference but also with the different ways the operation or the inferential cognition of an inference can be prevented. From the above discussion, it follows that the Nyāya philosophers have emphasized the relevance condition for any inference, valid or invalid. Hence the inference does not have the form: P, therefore, Q, where “P” and “Q” range over sentences or cognitions

Hence the following valid inference of Western logic is not treated as an inference in the Nyāya logic: P and not P. Therefore, Q.

This is due to the fact that it violates the relevance condition as well as certain epistemic conditions for understanding the meaning of a sentence. As a result, we

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cannot derive 2 + 2 ¼ 4 from it is raining and not raining, which is valid in classical symbolic logic. Since the Nyāya logic has emphasized the relevance condition, it might throw some light on contemporary discussion on relevant logic. Since it deals with the preventer-prevented relation at epistemic level and the ways a cognition can be prevented, it will throw some light on epistemic logic as well. The Nyāya philosophers have also discussed our ability to discriminate in the case of inferential cognition. Consider the following inference: (a) Wherever there is blue smoke, there is fire. (b) The mountain has blue smoke. (c) Therefore, the mountain has fire. In this inference, the conclusion follows from the premises, and both the conclusion and the premises are true. Now the Nyāya raises the question whether the cognition expressed by the sentence “wherever there is blue smoke, there is fire” is such that the property of being the pervaded residing in blue smoke which is signified by the expression “wherever” is limited by blue smokeness or by smokeness only. In other words, the question is whether the property of being the pervaded is presented under the mode of blue smokeness (i.e., blueness and smokeness) or under the mode of smokeness. If it is presented under the mode of blue smokeness, then the person, who has inferred the mountain has fire from the above two premises, would not be able to infer the same conclusion from the cognition of “the mountain has black smoke.” On the contrary, if he/she would have inferred “the mountain has fire” from “wherever there is smoke, there is fire, and the mountain has smoke,” then he/she would be able to infer “the mountain has fire” from the observation of black smoke as well. This is due to the fact that the mode of presentation of the property of being the pervaded signified by the expression “wherever” smokeness, not blue smokeness, is. Since the property of being the pervaded residing in any smoke, blue or black, is limited by smokeness, the cognitions expressed by sentences such as “wherever there is blue smoke, there is fire” and “wherever there is black smoke, there is fire” would be true. In other words, if the property of being the pervaded is cognized under the mode of smokeness, then it reveals an ontological property of smoke, blue or black. Hence the cognition of smoke as qualified by smokeness, not as qualified by blue smokeness, gives us a guarantee for making similar inferences. Therefore, a person is able to infer fire from any smoke, blue or black, if he/she has cognized the property of being the pervaded under the mode of smokeness. Hence the Nyāya not only emphasizes our ability to discriminate in the case of inference but also explains this ability in terms of the cognition of certain properties. In this case, it is the cognition of the limitor of the property of being the pervaded which resides in the probans.

Tarka and Inference In this context, it is to be noted that tarka (the use of a type counterfactual conditional) cannot be identified with an inference. Let us consider the sentence “If there would have been smoke, then there would have been fire.” This

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counterfactual sentence is not used to support the vyāpti sentence “Wherever there is smoke, there is fire.” On the contrary, it presupposes the vyāpti (invariable concomitance) relation of smoke to fire. Moreover, it is used to generate an inference which dispels doubt about the presence of smoke or which demonstrates the falsity of the cognition of smoke. Suppose the opponent knows that there is absence of fire and the pervade-pervaded relation between smoke and fire but has some doubt about the absence of smoke or thinks that there is smoke. Now the proponent utters the above counterfactual sentence. The use of this sentence will give rise to the following inference: Pratijñā (thesis): There is absence of smoke. Hetu (reason): Because of absence of fire. Udāharaṇa (example): Wherever there is absence of fire, there is absence of smoke, such as a lake. Upanaya (application): There is absence of fire which is pervaded by absence of smoke. Nigamana (conclusion): Therefore, there is absence of smoke. Hence tarka, understood in the above sense, is an auxiliary to an inference, and its purpose is to remove certain doubts or to show the falsity of certain cognitions. There is another type of tarka sentence which helps in establishing the vyāpti relation between smoke and fire by giving rise to another form of inference. But this type of tarka presupposes some other vyāpti relation, and the inference it gives rise to is different from the previous one. Here the aim is to remove doubt about smoke being pervaded by fire or to show the invalidity of the cognition that smoke is not pervaded by fire. Here the tarka sentence is “If smoke were not the pervaded of fire, then it would lack the property of being caused by fire.” The vyāpti which it presupposes is the following: Wherever there is absence of the property of being pervaded by fire, there is absence of the property of being caused by fire. In a nontechnical way, it may be said that this tarka presupposes that smoke is caused by fire. In this case, the vyāpti relation which is presupposed by the tarka is not the same as the vyāpti relation which it establishes. Here also the tarka sentence gives rise to an inference which is much more complex than the previous one. Since this inference dispels doubt about the vyāpti relation between smoke and fire, the tarka is considered as an auxiliary to this inference. If the presupposed vyāpti sentence is “P” and the vyāpti sentence to be established by using this tarka sentence is “Q,” then there is a vyāpti relation of P to Q. In other words, the truth of “P” entails the truth of “Q.” If the opponent questions the truth of “Q” but accepts the truth of “P” and the pervader-pervaded relation of P to Q, then by using the above tarka sentence which gives rise to an inference, it can be shown that the acceptance of “P” would lead to the acceptance of “Q.” If the cognition of P is valid (prama), then the cognition of Q is also valid. Since the opponent has accepted the validity of P, he/she cannot doubt the validity of Q. But this type of tarka cannot remove the doubt about P, if there is any. It can only dispel doubt about Q if the opponent accepts P or

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considers the cognition of P as valid. Thus it establishes “Q” which is a vyāpti sentence. This is how the Nyāya philosophers would demonstrate the relation between a counterfactual sentence and an inference.

Analogy or Comparison Now let us discuss the causal conditions of analogical cognition (upamiti). In an analogical cognition (upamiti), we cognize the property of being the referent of an expression (vācyatva) in its referent. Hence it takes the following form: (A) y is the referent of “x” under the mode y-ness, where “x” is the expression and y is its referent. An analogical cognition presupposes the cognition of the following sentence: (B) That which is similar to z is the referent of “x,” where z is the referent of the term “z” which is different from “x,” and the cognizer already knows the referent of “z,” but not the referent of “x.” Moreover, an analogical cognition presupposes a perceptual cognition, which is described by the following sentence: (C) This is similar to z. Let us illustrate with an example of the Nyāya philosophers: (a) That which is similar to a cow is the referent of the word gavaya. (b) This is similar to a cow. (c) Gavaya is the referent of the word gavaya. In this example, (c) is the analogical cognition (upamiti). It presupposes the understanding of the meaning of the sentence (a), which the cognizer might have heard from someone else or read in a book. Here (b) represents the perceptual cognition. The inherent and the similar-to-inherent causes would be the same as other types of cognition. In our above example, the special instrumental cause (karaṇa) would be the cognition of similarity with a cow in the animal, which is being perceived. This cognition would give rise to the memory cognition of (a). Hence the memory cognition of (a) would be the operation (vyāpāra) of the analogical cognition (upamiti). In this analogical cognition, gavaya which is the referent of the word gavaya is cognized under the mode gavayaness (gavayatva), not under the mode of thisness or being similar to a cow. For this reason, according to the Nyāya philosophers, it cannot be reduced to an inferential cognition (anumiti). Here also there are four sets of conditions for the analogical cognition. One set of conditions will define the analogical cognition. In our example, the perceptual

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cognition of similarity with a cow is the special instrumental cause (karaṇa), and the memory cognition of that which is similar to a cow is the referent of the word gavaya is the operation (vyāpāra). The analogical cognition would be true if we have cognized similar to a cow in the referent of the word gavaya. But if we have not cognized similarity with a cow, then the analogical cognition would be false. In addition to the causal conditions for the truth of the analogical cognition, the Nyāya postulates gavayatva as the mode of presentation of gavaya, which gives us guarantee for its truth. Hence, it gives us the ability to discriminate in other cases. The property of being the qualificand residing in the perceptual cognition and the property of being the qualificand residing in the analogical cognition are limited by gavayatva, although they are determined by different properties of being the relational qualifier (prakāratā).

Verbal Cognition or Testimony With respect to a verbal cognition (testimony) also, the Nyāya philosophers have discussed its causal conditions, the causal conditions of its truth or falsehood, and the causal conditions which justify its truth. The chief instrumental cause (karaṇa) of the cognition of the meaning of a sentence is the cognition of the words contained in it, and the operation of this cognition is the memory cognition of the referents of the words. According to the Nyāya, the cognition of the meaning of a sentence, as distinct from the cognition of the meanings of its parts, lies in cognizing the relation of the referent of its second term to that of its first term. Hence the cognition of the meaning of the sentence “a flower is red” lies in cognizing the relation of a red color to a flower. If the sentence is true, then it would generate a true cognition, and the cognizer would apprehend the relation which holds between a red color and a flower. If the sentence is false, then it would generate a false cognition, and the cognizer would apprehend a relation which does not hold between a red color and a flower, but which holds between some other objects such as between a red color and a table. Now the question is whether a true cognition generated by a true sentence has the status of knowledge. On this point, the Nyāya claims that it would be a case of knowledge if the true sentence is uttered or inscribed by an āpta (a trustworthy person). This is due to the fact that a true cognition generated by the utterance of an āpta has justification. Therefore, it has the status of knowledge. From our above discussion, it follows that knowledge is a justified true cognition or belief, provided justification is a qualifier of true cognition or belief. A true cognition is justified by certain perceptual causal conditions, or by certain inferential causal conditions, or by an analogical causal condition, or by certain verbal causal conditions. Hence the Nyāya technique for justifying a true cognition may be used for interpreting or explicating the meaning of the word “knowledge.” In Western philosophy, justification, belief, and truth are not related in the way they are related in the Nyāya philosophy. It is similar to saying that “there is a person with a red iron mask in this room” is true by virtue of the fact that there is a person in this room and there is an iron mask in this room and there is a red object in this room. This is due to the fact that they

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are not related to each other as qualifier-qualificand. But in the Nyāya philosophy, truth is the qualifier of cognition, and justification is the qualifier of truth. From the above discussion, it also follows that the Nyāya philosophers have treated justification as a qualifier of a true belief and have emphasized the sources of valid cognition, which will explain why certain true beliefs have justification. Moreover, the Nyāya explains the ability to discriminate an object or a set of objects in terms of the cognition of limitor(s). This explanation will allow us to solve some problems of contemporary Western philosophy, including the Gettier and the postGettier problems or counterexamples, and explain the ability to discriminate. The Nyāya theory of meaning may be used to explain the difference in meaning between the members of the following pairs of sentences: (1) Brutus killed Caesar. (10 ) Caesar was killed by Brutus. (2) John gave a book to Tom. (20 ) Tom received a book from John. (3) John sprayed paint on the wall. (30 ) John sprayed the wall with paint. (4) Bees are swarming the garden. (40 ) The garden is swarming with bees. (5) The speeches preceded a buffet luncheon. (50 ) A buffet luncheon followed the speeches. (6) The audience liked the overture. (60 ) The overture pleased the audience (Fodor 1977, pp. 90–92).

Contemporary philosophers of language are concerned with the problem whether transformation preserves the meaning of a sentence. Western philosophers such as Chomsky, Katz, Fodor, Fillmore, Postal, and Jackendoff are concerned with this problem, but there is no unanimity among their views as they have not yet developed a comprehensive theory to deal with this problem. According to some linguists such as Fillmore, (6) and (60 ) are synonymous, but not according to others. Hence Western philosophers are either guided by intuitions or by a theory which has limited application. On the contrary, the Nyāya theory of relation and meaning can explain why the members of the above pairs of sentences do not have the same meaning. Since the direction of the relation is part of the meaning of a sentence, the meaning of (1) cannot be identified with that of (10 ). Similar will be the case with the remaining pairs of sentences in the above list. Hence the Nyāya theory of meaning will throw some light on the contemporary discussion of synonymity and meaning. This is how I would like to demonstrate the relevance of the Nyāya philosophy. In this context, I would also like to mention the atomistic as well as the holistic nature of understanding the meaning of a molecular or complex sentence. In the case of atomistic understanding, first we understand the meaning of embedded sentence (s) or complex expressions which have occurred in a sentence. Then we understand the meaning of the entire sentence. But in the case of holistic understanding, we apprehend simultaneously the meanings of the embedded complex expression(s) and the molecular sentence. Let us consider the nature of the cognition generated by the sentence “The table has a red book.” The question is whether we first apprehend the relation of a

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particular red color to a book and then apprehend the relation of the red book to the table. For the sake of simplicity, consider “the table” as one term. According to the atomistic understanding, the cognition of the relation of the red book to the table follows the cognition of the relation of the red color to the book. But according to holistic understanding, we cognize both the relations simultaneously.

Holistic Understanding

Let us explain the nature of the cognition corresponding to this sentence. In this cognition the table is the qualificand, the book is the qualifier, and the red color is the qualifier of the book. For the sake of simplicity, we are not considering the modes of presentation of the qualificand and the qualifiers of this cognition. In this cognition, the book is the relational qualifier in relation to the table, and it is also a qualificand in relation to the red color. Since the table is the qualificand, it has the property of being the qualificand (viśeṣyatā). The red color has the property of being the relational qualifier (prakāratā). But the book has both the property of being the qualificand (viśeṣyatā) and the property of being the relational qualifier (prakāratā). It is to be noted that these relational properties specify the ways objects are related to this cognition. Now the question is how these relational properties are related to each other. In this cognition, the property of being the qualificand (viśeṣyatā) residing in the table is determined by (nirūpita) the property of being the relational qualifier residing in the book, and vice versa. Similarly, the property of being the qualificand residing in the book is determined by (nirūpita) the property of being the relational qualifier residing in the red color, and vice versa. Now the question is whether the property of being the qualificand (viśeṣyatā) and the property of being the relational qualifier (prakāratā) residing in the book are related to each other. If they are independent properties, then we cannot draw the distinction between (a) and (b). (a) The table has a red book. (b) The table has a book, and that book is red. In order to draw the distinction between them, the Nyāya claims that in (a), the property of being the relational qualifier (prakāratā) and the property of being the qualificand (viśeṣyatā) residing in the same book are related to each other by the relation of limitor-limited (avacchedya-avacchedaka-sambandha). In other words, the property of being the qualificand is the mode of the presentation of the property of being the relational qualifier, and vice versa. But this is not the case with (b). Hence in (b), they are

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not related to each other by the relation of limitor-limited. In a holistic understanding, we cognize the relation of the red color to the book and the relation of the book which is red to the table simultaneously. But in an atomistic understanding, first we cognize the former relation, and then we cognize the latter relation. It is to be noted that the Nyāya has drawn a fine-grained distinction between (a) and (b). This distinction has been explained in terms of higher-order properties residing in the properties of being the qualificand and the relational qualifier. Since the classical symbolic logic cannot draw this type of fine-grained distinction, the Nyāya technique will add a new dimension to Western philosophy.

Section 4: Negation and Its Classification The aim of this section is to discuss the Nyāya theory of negation, as it cannot be identified with the term or proposition negation in Western philosophy, as well as its classification into four types. Since the Nyāya has discussed negation at the linguistic, epistemic, and ontological levels, the proper understanding of the Nyāya view would presuppose the Nyāya conception of cognition, relation, and meaning. According to the Nyāya, what is negated is the second term of a dyadic relation as the second member of this relation. Let us consider the cognition of a table expressed by the expression “a table.” In this cognition, a particular table is the qualificand, the universal tableness is the qualifier, and the relation of tableness to a particular table is the qualification relation, which in this context is inherence. Since a table is the qualificand, it has the property of being the qualificand. This property of being the qualificand simply specifies the role of this object at the epistemic level. Similarly, the universal tableness which is the qualifier has the property of being the qualifier. The relation of inherence neither is a part of the qualificand nor is it a part of the qualifier. It is a mode of presentation of the qualifier. That is to say, the universal tableness is cognized as the second member of the relation of inherence. In the technical language of the Nyāya, it is described as “the limiting relation of the property of being the qualifier.” In a more complex cognition expressed by the expression, say “a table is brown” or “a brown table,” the qualificand is a table, and the qualifier is a particular brown color. The property of being the qualificand residing in a table which is the qualificand is limited by the universal tableness, and the property of being the qualifier residing in a brown color, which is the qualifier, is limited by the universal brownness. The relation of inherence which relates a brown color to a table is also a mode of presentation of the brown color. Hence this relation becomes the limiting relation of the property of being the qualifier. The property of being the qualificand is limited by a property alone, while the property of being the qualifier is limited by both a property and a relation. This feature of the Nyāya can be compared to some extent with Frege’s distinction between saturated and unsaturated parts of a thought. From the above discussion, it follows that any qualificative cognition can be described by the form aRb, where a stands for the qualificand, b for the qualifier, and

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R for the qualification relation. When this description is expanded in the technical language of the Nyāya, it takes the following form: The cognition in which the property of being the qualificand residing in a is limited by aness and determined by the property of being the qualifier residing in b, which is limited by both b-ness and R (In this context, I have not introduced the Nyāya distinction between a sentence which gives rise to a cognition and a sentence which describes this cognition. The latter is essentially richer than the former.).

In this context, it is to be noted that the expression “mode of presentation” is used in such a way that it determines the referent(s) of a term. Moreover, the Nyāya use of the term “property” is much broader than the ordinary use of it. A property, according to the Nyāya, has been defined in the following way: x is a property iff (Ǝy) (y is a locus of x) In this context, it is to be noted that the determined by relation is symmetrical. That is to say, if x is determined by y, then y is also determined by x. But the limited by relation is not symmetrical. According to the Nyāya, all relations are dyadic, and all higher-order relations are reduced to a set of dyadic relations. All relations can be divided into two types depending upon whether the second term occurs in the first term or not. The relation in which the second term occurs in the first term is called “occurrence-exacting (vṛttianiyāmaka) relation.” The linguistic form “y is in x” or “y occurs in x” represents this type of relation. If the second term does not occur in the first term, then the relation is called “not occurrence-exacting” (vṛtti-aniyāmaka). Relations like conjunction and inherence are occurrence-exacting (There are a few conjunctions which are not occurrence-exacting relations.). But relations like identity, pervasion, the property of being the content, and the converse of the property of being the content are not occurrence-exacting. In this context, another important aspect of the Nyāya concept of relation should be mentioned. In some context, a term itself plays the role of a relation. This type of relation is a self-linking relation (svarūpa-sambandha). Relations like the relation of the property of being Socrates to its possessor and relation of the property of being the present President of India to its possessor are self-linking relations. In addition to these types of self-linking relations, there are spatial and temporal self-linking relations. The self-linking relation plays an important role in the context of a negation. When we say “x has the absence of y,” what we mean or understand is that the absence of y, which is a negative entity, is related to its locus x by an absential self-linking relation which is a special type of self-linking relation. That is to say, the relation of the absence of y to x is not a separate ontological entity. It is to be identified with at least one of the terms of a relation. According to most of the Nyāya philosophers, it is to be identified with the first term of a relation (anuyogin). Now let us formulate the criteria for forming a significant negative expression. If the following conditions are satisfied, expressions of the form “not-t” or “absence of t” or “non-t” would be considered significant:

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(i) If t is a meaningful expression or refers to an entity, then “not-t” would be significant provided t does not refer to an absolutely universal property such that nothing lacks this property. According to the Nyāya, properties like nameability, knowability, and existence are considered universal properties in this sense. Hence, expressions like “nonexistence,” “nonnameability,” and “nonknowability” are not considered significant negative expressions. (ii) If “not-t” is significant, then t is not an empty term. Since terms like “a hare’s horn,” “Pegasus,” and “unicorn” are considered as empty, their negations would not be significant negative expressions. From this condition, one should not conclude that any expression which contains an empty term is nonsignificant (For a discussion on empty terms, see Shaw (1980).). Instead of the sentence “A hare’s horn does not exist,” the Nyāya prefers the sentence, “There is an absence of a horn in a hare.” The expression “not-t” or “negation of t” will be meaningful if we know what it is for t to be present somewhere. If we know what it is for t to be present somewhere, then we know the manner of presentation of t. Since t is the counterpositive (negatum) of the negation of t, t has the property of being the counterpositive. This property simply specifies the role of t in the context of a negation. The manner of presentation of t in the cognition negation of t is the limitor of the property of being the counterpositive residing in t. If the manner of presentation of t is a property, then the limitor is called a “property-limitor,” and if the manner of presentation is a relation in which t is cognized, then the relation is called a “relation-limitor.” The relation in which t is present somewhere is called “The limiting relation of the property of being the counterpositive residing in t.” The property of being the counterpositive is limited by a property (simple or complex) and a relation (simple or complex). (i) At the epistemic level, the cognition of not-t presupposes or depends upon the cognition of t. If a person has not cognized t, then he cannot cognize not-t. The cognition of t such that t is presented under some mode of presentation is considered as one of the causal conditions for the cognition of not-t. But the relation between the cognition of not-t in the locus l and the cognition of t in the same locus 1 is preventer-prevented, which is the analogue of the contradictory relation between two contradictory propositions. According to the Nyāya, both “t” and “not-t” are non-empty terms. If t does not occur pervasively in its locus, then the negation of t is also present in the same locus, and conversely. But this does not lead to a contradiction because t and not-t do not characterize the same portion of the locus at the same time. Here also both “t” and “not-t” are non-empty terms. Broadly speaking, there are two types of negation, viz., relational absence and mutual absence or difference. The distinction between them can be drawn in terms of the limiting relation of the property of being the counterpositive which resides in

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the negatum. At the linguistic level, these negations can be represented by the following forms: 1. x is not in y or x does not occur in y, or not-x is in y. 2. x is not y, or x is different from y. (1) represents relational absence and (2) represents mutual absence. In (1), not-x occurs in the locus y, and x is the counterpositive of not-x. The property of being the counterpositive residing in x (i.e., the role of x) is limited by both x-ness and an occurrence-exacting relation. In other words, both x-ness and an occurrence-exacting relation are modes of presentation of x. Here x-ness is the property-limitor, and an occurrence-exacting relation is the relation-limitor. In (2), y is the counterpositive, i.e., the negatum, and the property of being the counterpositive residing in y is limited by both y-ness and the relation of identity. So the relation of identity is the limiting relation of the property of being the counterpositive. Most of the Nyāya philosophers have accepted three types of relational absence: The relational absence of an object before its production is the not-yet type of absence (prāgabhāva). The absence of a jar before its production is present in its parts. The cognition of this absence can be expressed by the sentence “A jar will be produced in these parts.” When the jar is produced, the not-yet type of absence does not exist in its part. Since it cannot exist anywhere else, it ceases to exist. This type of absence has no beginning, but has an end. Since we are not asserting the absence of all jars, but the absence of the jar which will be produced, the property of being the counterpositive is limited not by a generic property like jarness only, but by a specific property like a particular blue color and jarness. As regards the limiting relation of the property of being the counterpositive, there is some difference of opinion among the Nyāya philosophers. It is claimed that since the jar has not yet been produced, the property of being the counterpositive is not limited by any relation. But the old Nyāya has accepted a temporal relation as the limiting relation of the property of being the counterpositive. If the absence of the jar is in its parts at tn and the jar is produced in the parts at tn + 1, then obviously the jar is related to its parts by the relation of posterior existence. This temporal relation of posterior existence is considered as the limiting relation of the property of being the counterpositive. But the followers of the Navya-Nyāya do not subscribe to this view. The relational absence of an object after its destruction is the no-more type of absence (dhvaṃsa). The absence of a particular jar when it is destroyed is present in its parts. Since the destruction of a particular jar does not imply the destruction of all jars, the property of being the counterpositive is limited, not by a generic property, but by a specific property of the jar which has been destroyed. As regards the limiting relation, here also there is difference of opinion among the Nyāya philosophers. The followers of the Navya-Nyāya do not accept any limiting relation, while the followers of the old Nyāya have accepted a temporal relation as the limiting relation. If the destruction of a particular jar is the separation of its parts, then the whole jar ceases to exist at time, say tn, when it is destroyed. If “ceases to exist at time tn” is explained as “existent at time tn-1,” then the parts are related to the jar by the relation of previous existence

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(Jagadīśa says “Prāgabhāva-dhvaṃsayorapi uttarapūrvakālāveva,” quoted in Madhusūdana Nyāyācārya (1976).). For this reason, it is claimed that the property of being the counterpositive is limited by the temporal relation of previous existence. Apart from this temporal relation, the property of being the counterpositive is not limited by any other relation. A no-more type of absence has a beginning, but no end. (3) The third type of relational absence is the never type of absence (atyantābhāva), for example, the absence of color in air or the absence of a jar on the ground. Some of the followers of the old Nyāya do not consider the absence of a jar on the ground as a case of never type of absence. Since a never type of absence has neither a beginning nor an end and since the absence of a jar on the ground has both a beginning and an end, these philosophers think that there is a need to accept a fourth type of relational absence. But the followers of the Navya-Nyāya as well as some of the followers of the old Nyāya think that the acceptance of the fourth type of relational absence would lead to the postulation of innumerable absences of a jar on the same ground. Each time the jar is removed, a new absence is created, and each time the jar is brought back, the previous absence is destroyed. In order to avoid this consequence, it is claimed that what ceases to exist when the jar is brought back is not the absence of it, but the relation of this absence to the ground. An absence is related to its locus by a self-linking relation which is to be identified ontologically with its locus. Now the followers of the Navya-Nyāya are of the opinion that this self-linking relation in the case of the absence of a jar on the ground is to be identified not with the ground as such, but with the ground when a jar is not present. Since this self-linking relation ceases to exist when a jar is brought on the ground which had an absence of a jar, we cannot perceive this absence when a jar is present on the same ground. So on the ground of parsimony, these philosophers have included such examples under the third type of relational absence. The property of being the counterpositive of a never type of absence is limited by both a property-limitor and a relation-limitor. But the limiting relation is an occurrenceexacting one (But a section of the Nyāya philosophers do not subscribe to the thesis that a not occurrence-exacting relation other than identity cannot be the limiting relation of the property of being the counterpositive. Gadādhara in his Vyutpattivāda says, “Vṛttianiyāmaka-sambandhasya-abhāva-pratiyogitā-avacchedakatve-ko-doṣaḥ,” quoted in Kalipada Tarkāchārya (1973). For a discussion, see Shaw (1980).). The Nyāya philosophers have also discussed the law of double negation in this context. Since there are four types of negation according to the classical view of the Nyāya, there would be 16 types of double negation. In my paper entitled “The Nyāya on Double Negation,” I have discussed (i) whether each of the 16 double negations is identical with something or not; (ii) if it is identical with something, whether it is a positive or a negative entity; and (iii) if it is identical with a positive entity, whether it is the same as the negatum of the first negation (See my paper “The Nyāya on Double Negation,” Notre Dame Journal of Formal Logic, 1988.). The discussion on negation also throws light on the logical thinking of the Nyāya philosophers, as they have classified negations into several types and discussed it at the linguistic, the epistemic, and the ontological levels.

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Section 5: Universal Quantifier The Nyāya use of the word “all”: It is claimed by the Nyāya philosophers that the word “all” refers to a collection of objects. Hence in a sentence such as “All S is P,” the word “all” refers to the collection (or set) of subjects S without any remainder (aśeṣatvaviśiṣṭa), and the limitor of the property of being the referent is the property of being without remainder (Pandit Visvabandhu Tarkatīrtha, “The Nyāya on the Meaning of Some Words,” translated with explanatory notes by J.L. Shaw. Journal of Indian Philosophy, vol. 20, 1992a; Ghosh 1982, pp. 532–534). Now the question is: What is the nature of the property of being without remainder? Gadādhara, a Navya-Nyāya philosopher, explains this property in terms of the concept of number. He claims that the property of being without remainder is the same as the property of being a collection (or a class), which is nothing but a positive number. This number, in the sentence “All S is P,” is the pervader of the limitor of the property of being the subject and pervaded of the predicate. Hence what the sentence “All S is P” asserts is that wherever the limitor of the subject S is present, the number which qualifies the collection is present, and wherever the latter (i.e., the number) is present, the predicate is present. If the property of being the collection is F, the limitor of the property of being the referent of the term is G, and the predicate is H, then what the sentence “All S is P” asserts can be stated in the following way: ðxÞðGx  FxÞ and ðxÞðFx  HxÞ Hence, according to the Nyāya, what we mean when we say “All human beings are mortal” is that the individuals which are qualified by humanity are qualified by a number which belongs to the collection of human beings and the individuals which are qualified by this number are qualified by mortality. In this context, the Nyāya use of the word “number” requires some explanation. A particular use of a number such as “two” in the expression “two apples” refers to a collection, and the number two is a property of this collection. But this property should not be equated with the universal twoness. Hence there are as many particular number twos as there are pair classes. The relation of a particular number two to a collection is a self-linking relation which is called paryāpti sambandha in the technical language of the Nyāya. A self-linking relation in the Nyāya system does not have any separate ontological status, and it is to be identified with one of the terms of it. It has been introduced to solve some of the puzzles about relations or their ontological status. Moreover, according to the Nyava, the particular number two in “two apples” is a quality of each member of this collection, and it is related to each of them by the relation of inherence. Hence when we talk about the collection, we are talking about something to which the number two is related by one relation, and when we talk about the members of this collection, we are talking about the things to which the same number is related by another relation. These two uses of the word

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“two” explain the difference in our cognitions about the same object which is a member of different classes. In addition to particular number twos, the Nyāya has postulated the universal twoness which is present in each of these particular twos which are considered qualities in the Nyāya ontology (See my paper “Number: From the Nyāya to Frege to Russell,” Studia Logica, 1982.). So far we have discussed the use of the word “all” as a qualifier of the subject term; now let us discuss the use of it as a qualifier of the predicate term. Let us consider the sentence “Wellington has all the twenty-storied buildings.” According to the Nyāya, here also the word “all” refers to a collection which is qualified by a number depending on the number of twenty-storied buildings. This number which is the property of being the collection is the pervader of the limitor of the property of being the predicate and the pervaded of the relation of the subject to the predicate. In this example, Wellington is the subject, twenty-storied building is the predicate, and the property of being the twenty-storied building is the limitor of the property of being the predicate. If the relation of the predicate to the subject is belonging, then the relation of the subject to the predicate would be the converse of this relation of belonging. What this sentence asserts is that the things which are qualified by the property of being a twenty-storied building are qualified by a particular number which belongs to this collection and the things which are qualified by this number are qualified by the converse relation of belonging. If we consider F as the property of being a twenty-storied building, G as the number which belongs to the collection and H as the converse of the relation of belonging, then what the sentence asserts can be stated in the following way: ðxÞðFx  GxÞ and ðxÞðGx  HxÞ The Nvaya analysis not only explains the distinction between the use of “all” as the qualifier of the subject expression and the qualifier of the predicate expression but also explains how the sentence “Wellington has all the twenty-storied buildings” is transformationally related to the sentence “All twenty-storied buildings are in (belong to) Wellington.” The F in the former sentence would represent the limitor of the property of being the predicate, but in the latter sentence, it would represent the limitor of the property of being the subject. Since the subject of the former sentence has become the predicate in the latter sentence and the predicate of the former has become the subject in the latter, the relation of the predicate to the subject in the latter sentence would be the converse of the former. Since H represents the converse of the relation of belonging in the former sentence, it would represent the relation of belonging to Wellington in the latter sentence. Hence the Nyāya explains the equivalence between these two sentences and at the same time explains the difference in meaning between them. Hence a transformation of this sort does not preserve the meaning, although it preserves the truth value. Meaning and Equivalence of Universal Sentences Now let us discuss (i) whether the following sentences have the same meaning, (ii) whether they have the same truth value, (iii) whether there is some assertion common to all of them, and (iv), if there is some such assertion, whether it can be defined.

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All human beings are mortal. Whoever is a human being is mortal. Wherever there is humanity, there is mortality. If anyone is a human being, then he/she is mortal. If humanity is present somewhere, then mortality is also present there.

From the above discussion of the Nyāya, it follows that a universal sentence such as “Wellington has all twenty-storied buildings” has a reference to a collection and thereby to a number which characterizes the collection or the class. Similarly, the meaning of “All human beings are mortal” has a reference to a number which characterizes the class of human beings. Since there is no such reference to a class and thereby to a number in “Whoever is a human being is mortal,” the meaning of (1) cannot be identified with that of (2). According to the Nyāya, the subject in (2) is a human being, and the predicate is mortality. The relation of the predicate to the subject is self-linking (svarūpa). Hence this relation is to be identified with the first or the second term. In (2), a pervader-pervaded relation has been asserted between humanity and mortality. Humanity is the pervaded (vyāpya), and mortality is the pervader (vyāpaka). Hence, like (1), it does not refer to any collection or the pervader-pervaded relation between a number and humanity or between a number and mortality. Now the question is whether the meaning of “Wherever there is humanity, there is mortality” can be identified with that of (1) or (2). The Nyāya claims that the subject in (3) is humanity and mortality, and the predicate is the property of being the superstratum determined by the substratum, which is the meaning of “wherever there.” What this sentence asserts is that the property of being the superstratum resides in both humanity and mortality and this property is determined by the locus of humanity and mortality. If we take a person such as John as the locus (or substratum) of humanity and mortality, then the property of being the superstratum residing in humanity and mortality is determined by John. But if we take a table as a locus, then the property of being the superstratum determined by it does not reside in humanity and mortality. Here also the relation of the predicate to the subject is self-linking. Since the subject and the predicate in (3) are different from those in (1) or (2), the meaning of (3) cannot be identified with that of (1) or (2). Since they do not have the same meaning, the cognitions generated by them would not be the same. They would generate three different cognitions. As regards (4) and (5), the Nyāya claims that the meaning of (4) is the same as that of (2) and the meaning of (5) is the same as that of (3). Now let us discuss whether (1) to (5) are equivalent. The Nyāya claims that (2) to (5) have the same truth value and they are implied by (1). But neither (2) nor (3) implies (1) unless an omniscient being has already formed a collection which includes past, present, and future human beings. Hence neither (4) nor (5) would imply (1). From this discussion, it follows that (2) to (5) are equivalent but (1) is not equivalent to them.

Common Assertion According to the Nyāya, there is some assertion which is common to all the sentences in our above list, although all of them do not have the same meaning

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and are not equivalent to each other. This common assertion is the pervaderpervaded (or pervasion) relation between two entities. In (1), the pervader-pervaded relation has been asserted between humanity and the property of being the collection which is a particular number according to Gadādhara and between this number which resides in each of the members and mortality. In (2), the pervader-pervaded relation holds between humanity and mortality; but in (3), it holds between humanity and mortality on the one hand and the property of being the superstratum determined by the substratum on the other. Since (4) means the same as (2), it asserts the same pervader-pervaded relation as (2). Similar is the case with (5) which means the same as (3). Hence all of them assert the pervader-pervaded relation between two entities. Now let us discuss this pervader-pervaded relation which is common to all the above sentences. According to Ingalls, the Nyāya-Kośa has listed 34 definitions of pervasion, i.e., the pervader-pervaded relation (Ingalls 1951, p. 29). Most of them are associated with the names of particular logicians, and they were studied in groups such as vyāpti-pañcaka. Ingalls also makes the claim that hundreds of manuscripts of commentaries are still available on these single groups. Since the literature is vast and highly technical, I shall only mention Gaṅgeśa’s conclusive definition discussed in his siddhānta-lakṣaṇa. Let us consider the pervader-pervaded (or pervasion) relation in a sentence of the form “If there is F, then there is G,” where F is to be called the probans (hetu) and G is to be called “the probandum” (sādhya). Since the validity of an inference depends on the validity or the truth of the sentence which states the pervader-pervaded relation between the probans of the premise and the probandum of the conclusion, it is very important for any discussion of inference. In fact, it plays one of the central roles in the Nyāya discussion of inference. It is to be noted that Gaṅgeśa’s definition of pervasion overcomes the shortcomings of the definitions of his predecessors (For a discussion on the five definitions of Vyāpti-Pañcaka, see J.L. Shaw 1991b). His definition known as “the conclusive definition” may be stated thus: The probans has the property of being present (or occurrent) in the locus of the probandum which is not limited by the limitor of the property of being the counterpositive of the absence residing in the locus of the probans.

Now let us see how this definition applies to a valid sentence such as “If something is nameable (or has nameability), then it is knowable (or has knowability).” In this example, the probans is nameability, and the locus of the probans would be any object in the Nyāya ontology. Let us consider this table as a locus of this probans. If there is no cat on the table, then the absence of a cat resides in this locus of the probans. Since a cat is the counterpositive (negatum) of this absence, the property of being the counterpositive residing in this counterpositive is limited by the limitor catness. But the probandum is not limited by any limitor of this type. In this case knowability is not limited by catness. Since knowability is present in this table, it would be a locus of the probandum. Since nameability is present in the same table, it has the property of being present in the locus of the probandum.

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It has also been claimed that Gaṅgeśa’s definition applies to all other types of valid universal sentences. Hence it is free from the defect of undercoverage (avyāpti). For this reason, Gaṅgeśa’s definition of pervasion is superior to all other definitions. From the above discussion of the Nyāya, it follows that there are different types of universal sentences or propositions. If we consider our sentences (1–5), then it follows that the meaning of (4) can be identified with that of (2), and the meaning of (5) can be identified with that of (3), but the meanings of (1), (2), and (3) cannot be identified with each other. As regards their truth values, the Nyāya claims that all of them do not have the same truth value. (1) is not equivalent to (2) or (3), although (1) implies (2), and (2) implies and is implied by (3). Hence (2) and (3) are equivalent. Since (2) and (4) have the same meaning, they are equivalent. Similarly, (3) and (5) are equivalent. As regards the common assertion, the Nyāya claims that all of them assert the pervader-pervaded relation between two terms, and the Nyāya philosophers have tried to define this common assertion. Since Arthur Prior, in recent philosophy, has raised certain questions about universal sentences which have been extensively discussed in the Nyāya system, his questions might serve as a bridge between the Western and the Indian tradition (For the Nyāya discussion of the meaning of a universal sentence and Gaṅgeśa’s definition of pervasion, I am greatly indebted to Pandit Visvabandhu Tarkatīrtha.).

Section 6: The Principle of Contradiction Regarding contradiction, I would like to compare the thesis of the Nyāya philosophers with the views of Western philosophers: 1. Let us begin with Aristotle’s formulation of the principle of contradiction (Jan Lukasiewicz, “On the Principle of Contradiction in Aristotle,” translated by Vernon Wedin, Review of Metaphysics, no. 24, 1971.). It is claimed that Aristotle has formulated the ontological, the logical, and the psychological principle of contradiction in the following three ways, respectively: (a) Two no object can the same characteristic belong and not belong at the same time. (b) Two conflicting (contradictory) propositions cannot be true at the same time. (c) Two acts of believing which correspond to two contradictory propositions cannot obtain in the same consciousness. These formulations do not have the same meaning, although the logical formulation, according to Lukasiewicz, is logically equivalent with the ontological formulation. The equivalence is due to the “one-one correlation between assertions and objective facts,” as assertions indicate objective facts. Lukasiewicz also claims that Aristotle tries to prove the psychological principle of contradiction in terms of the logical principle. Since Aristotle could not demonstrate that acts of believing corresponding to contradictory propositions are incompatible, he fails to establish

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the psychological principle of contradiction. Lukasiewicz also claims that Aristotle in his investigation of acts of believing commits the fallacy of “logicism in psychology” which is the counterpart of the fallacy of “psychologism in logic.” As regards the Nyāya position with respect to the principle of contradiction, I would like to mention that the followers of the Nyāya would accept the following formulations of the ontological, logical, and psychological principle of contradiction: (a) Ontological formulations: (i) A cannot be both A and not A (ii) A cannot be both B and not B The relation between A and not A, or between B and not B, is called “opposeropposed” (bādhya-bādhaka), as one of the conjuncts excludes the other. Moreover, the Nyāya claims that everything has its own identity. Hence the principle that everything has the property of having identity, i.e., (λx) (x ¼ a), excludes the possibility of having contrary or contradictory property. For this reason, this table cannot be not a table, this red color cannot be not this red color, and so on. Since ontological entities B and not B are related to each other by the relation of opposeropposed, there is no contradiction at the factual or ontological level. It is to be noted that the Nyāya concept of identity is not a universal which is present in every object. The property of identity of John, i.e., (λx) (x ¼ John), is not the same as the property of identity of Tom, i.e., (λx) (x ¼ Tom). Since each object has its own identity which is unique, it cannot have its opposite. For this reason, John cannot have the property being Tom. Hence in the Nyāya system, we have an argument in favor of the ontological formulation of the principle of contradiction, which according to Lukasiewicz, is lacking in Aristotle. (b) Logical formulations: (i) It is not the case that p and not p, i.e., ~(p. ~ p). (ii) A sentence cannot be both true and false. The Nyāya philosophers would accept both (i) and (ii). This is due to fact that if a proposition is true, then it corresponds to a fact. Since the ontological principle excludes having contradictory properties or facts, there cannot be true contradictory sentences. Hence the logical principle of contradiction rests on the ontological principle of contradiction. Therefore, the Nyāya notion of reality and the concept of truth would substantiate the logical principle of contradiction. (c) Psychological (or epistemic) formulation: The thought (or cognition) of p prevents the occurrence of the thought of not p. It is to be noted that the relation between p and not p at epistemic level is called “preventer-prevented” (pratibadhya-pratibandhaka). Hence two contradictory beliefs cannot occur at conscious level of our mind. In order to explain the preventer-prevented relation between contrary beliefs, such as x is round and square, we require a cognition which is to be expressed by the sentence:

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(a) (x) (If x is round, then x is not square). (b) (x) (If x is square, then x is not round). It is to be noted that the Nyāya philosophers have not committed the fallacy of logicism in psychology, an objection raised against the formulation of Aristotle by Lukasiewicz, as the preventer-prevented relation is causal, not logical. Now the question is whether the content of thought can have contrary or contradictory objects or entities. In other words, the question is whether a person, say S, can think of p and not p at the same time, that is, whether the content of thought can have contrary or contradictory properties. On this point, the Nyāya philosophers would point out that the thought of a contradiction is itself a contradiction. In other words, the thought of p and not p would imply the thought of p and the thought of not p. Hence S thinks that p and not p would imply S thinks that p and S thinks that not p. Since the former prevents the occurrence of the latter, and vice versa, both S thinks that p and S thinks that not p cannot occur consciously at the same time. From the above discussion of preventer-prevented relation at epistemic level, it follows that p and not p cannot be thought consciously at the same time at the same conscious level of our mind. But the preventer-prevented relation between cognitions corresponding to contrary sentences depends on the apprehension of a pervasion relation such that if one of the contrary terms is present, then the other is absent (tad-abhāva-vyāpya-darśana-vidheyā-pratibandhaka). The sentence which expresses this type of pervasion relation corresponds to the meaning postulate of contemporary philosophers. Hence, according to the Nyāya, an act of thought (belief or cognition) cannot have contrary or contradictory content. It is also to be noted that according to the Nyāya, the principle of contradiction is a necessary presupposition of any inferential cognition, but not according to Aristotle. The following syllogism is valid according to Aristotle, although it contains a contradiction: B is A (and not also not-A). C, which is not-C, is B and not-B. Therefore, C is A (and not also not-A), where A is living creature, B is a man, and C is Callias.

It is to be noted that the Nyāya philosophers emphasize the relevance relation between the premises of an inference or between the premises and the conclusion. Since there is no unified thought corresponding to a contradiction or understanding the meaning of a contradiction, it cannot be used in an inference, syllogistic or non-syllogistic. From this thesis of the Nyāya philosophers, it does not follow that we cannot assign a truth value to a contradictory or contrary sentence. Let us consider the following sentences: (a) A table which is brown and not brown (b) A round-square

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As regards their truth values, the Nyāya claims that each of them is false. Now the question is: How do we know the truth value of a sentence if it does not generate a cognition? Since (a) and (b) do not generate any cognition, how can we know that they are false? The Nyāya claims that we come to know the falsity of (a) when we know the truth of the sentence “A brown table,” i.e., “A table which is characterized by a brown color,” or the truth of the sentence “A non-brown table,” i.e., “A table which is characterized by the absence of a brown color.” Similarly, we come to know the falsity of (b) when we know the truth of the sentence “A round object” and the truth of “(x) (If x is round, then x is not square, i.e., x has absence of square)” or when we know the truth of the sentence “A square object” and the truth of “(x) (If x is square, then x is not round).” 2. Now I would like to mention that the Nyāya has avoided some of the extreme positions present in contemporary philosophy. According to G. E. Moore, a contradictory sentence is meaningless. To quote Moore: “. . .if in the sentence ‘Some tame tigers don’t exist’ you are using ‘exist’ with the same meaning as in ‘Some tame tigers exist’, then the former sentence as a whole has no meaning at all-it is pure nonsense” (Moore 1959). Hence, according to Moore, a contradictory sentence is meaningless. Since an analytic sentence is a negation of a contradiction, it is also a meaningless expression according to the significance criterion of negation. But the Nyāya philosophers have avoided this extreme position, as contradiction is meaningful expression although it cannot generate a unified cognition. According to another extreme position, a contradictory expression is meaningful, and we can understand its meaning. David Armstrong claims that it is possible to believe contradictory propositions simultaneously. To quote Armstrong: The conjunction of Bap and Ba~p is a possible state of affairs.21 (D. Armstrong, Belief, Truth and Knowledge, p. 104)

But according to the Nyāya, one cannot consciously believe both p and not p. This is due to the fact that the thought of p prevents the occurrence of the thought of not p, and vice versa. The Nyāya has also avoided another extreme position. Regarding contradiction, Strawson says: “suppose a man sets out to walk to a certain place; but, when he gets half-way there, turns round and comes back again. This may not be pointless. He may, after all, have wanted only exercise. But, from the point of view of a change of a position, it is as if he had never set out. And so a man who contradicts himself may have succeeded in exercising his vocal chords. But from the point of view of imparting information, of communicating facts (or falsehoods) it is as if he had never open his mouth. He utters words, but does not say anything” (Strawson 1963).

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This is also an undesirable thesis, as the denial of a contradiction would be equally pointless according to the significance criterion of negation. Since the Nyāya assigns the value false to a contradiction, it is not a pointless expression. According to paraconsistent logicians, a contradiction is not only a meaningful proposition but sometimes true, although it does not imply every proposition. According to the classical logic, a contradiction, such as it is raining and not raining, implies any proposition, such as 7 + 5 ¼ 12. Paraconsistent logicians are motivated by inconsistent information coming from different reliable sources, or inconsistent evidences of witnesses, or paradoxes such as liar. Hence a proposition of the form p and not p may not be rejected outright (Priest 1998). Some paraconsistent logicians, such as Graham Priest, have even claimed that inconsistent theories may be true. Hence a proposition of the form “p and not p” has been claimed to be true. The most commonly cited examples are self-referential paradoxes. Hence the paradoxes discussed by Russell, including the liar paradoxes, are considered as true. Similarly, the inconsistent propositions about the objects on the borderline of vague predicates are also treated as true. Some paraconsistent logics reject the disjunctive syllogism: A V B and ~A entails B: Some paraconsistent logicians allow a nonclassical truth value, namely, both true and false. Some others give up the truth-functional nature of negation, so that if p is true, then not p may be either true or false. Regarding paraconsistent logic, the Nyāya philosophers would claim that this view is not only counterintuitive and leads to the rejection of certain logical principles but also violates the principle of relevance. Hence the Nyāya philosophers would agree with the criticisms raised by the followers of the classical logic. From the above discussion it, follows that the Nyāya philosophers have put forward a theory of contradiction, which avoids the extreme positions of contemporary philosophers or rejects counterintuitive or undesirable consequences. Moreover, the Nyāya avoids the shortcomings of Aristotle’s logic.

Section 7: Gada¯dhara’s Theory of Definite Descriptions In this section, I would like to reconstruct the Nyāya theory of definite descriptions, which is based on Gadādhara’s explication of one of the uses of the word “one,” and evaluate Russell’s theory of definite descriptions in light of this theory. Hence I shall mention (1) how the Nyāya would draw the distinction between a proper name and a definite description, (2) how the Nyāya would draw the distinction between a definite description such as “The author of Waverley” and a sentence such as “The author of Waverley exists” or “The author Waverley is the author of Waverley,” (3) how the Nyāya philosophers would avoid the postulation of concepts for the explanation of non-designating terms such as “unicorn,” and (4) how the Nyāya would avoid the identity between an actual and a postulated entity, as we come across this type of identity in Russell’s theory of definite descriptions.

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The first part of this chapter will deal with Russell’s theory of definite descriptions, which is the greatest contribution of Russell to logic or philosophy of logic. The second section will deal with my reconstruction of definite descriptions, following the suggestions of Gadādhara, a Navya-Nyāya philosopher of sixteenth century.

Part I It is claimed that in order to get rid of the Meinongian ontology or to solve one of the age-old problems of philosophy related to negative existential propositions, Russell has propounded his theory of definite descriptions. To quote Russell: This [Meinong’s] theory regards any grammatically correct denoting phrase as standing for an object. Thus “the present king of France’”, “the round square”, etc., are supposed to be genuine objects. It is admitted that such objects do not subsist, but nevertheless they are supposed to be objects. This is in itself a difficult view; but the chief objection is that such objects, admittedly, are apt to infringe the law contradiction. It is contended, for example, that the existent present king of France exists, and also does not exist; that the round square is round, and also not round; etc. But this is intolerable; and if any theory can be found to avoid this result, it is surely to be preferred. (B. Russell, “On Denoting,” in Contemporary Readings in Logical Theory, edited by Copi and Gould, pp. 96–97)

In the system of Meinong, we cannot assert “The round square does not exist” unless we presuppose the being of the round square. This is analogous to our presupposition of the existence of Napoleon when we assert “Napoleon was defeated at the battle of Waterloo.” Hence the universe of Meinong is populated not only with actual or existent objects but also with nonexistent possible objects such as the winged horse and impossible objects such as the round square. But Russell’s theory of definite descriptions avoids this type of ontology. Russell has drawn a sharp distinction between a definite description and a logically proper name. A logically proper name is a simple symbol, and its meaning is its denotatum, but this is not the case with a definite description. He has defined the meaning of a definite description in use. To quote Russell: A name is a simple symbol whose meaning is something that can only occur as subject, i.e. something of the kind that, . . ., we defined as an “individual’ or a “particular. And a “simple” symbol is one which has no parts that are symbols. Thus “Scott” is a simple symbol, because, though it has parts (namely, separate letters), these parts are not symbols. On the other hand, “the author of Waverley” is not a simple symbol, because the separate words that compose the phrase which are symbols. (B. Russell, Introduction to Mathematical Philosophy, p. 173)

Again he says: We have two things to compare: (1) a name, which is a simple symbol, directly designating an individual which is its meaning, and having this meaning in its own right, independently of the meanings of all other words; (2) a description, which consists of several words, whose meanings are already fixed, and from which results whatever is to be taken as the ‘meaning’ of the description. (Ibid, p. 174)

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Moreover, he says: “. . .true proper names can only be conferred on objects with which we are acquainted” (“On the Nature of Acquaintance,” in Logic and Knowledge, edited by R. C. Marsh, George Allen & Unwin, 1977). Regarding a definite description, he says it has no meaning in isolation. To quote: These things, like ‘the author of Waverley’ which I call incomplete symbols, are things that have absolutely no meaning whatsoever in isolation but merely acquire a meaning in a context. ‘Scott’ taken as a name has a meaning all by itself. It stands for a certain person, and there it is. But ‘the author of Waverley’ is not a name, and does not all by itself mean anything at all,. . .. (Ibid, p. 253)

His view may be summed up in the following way: 1. A proper name is a simple symbol, and it cannot be eliminated from a sentence in which it occurs. 2. It can only occur as a subject in a proposition. 3. It is a name for a particular or individual. 4. The meaning of a proper name is its referent or the particular it denotes. 5. Knowing the meaning of a proper name presupposes acquaintance with its referent. 6. Understanding the meaning of a proper name does not involve understanding a proposition in which it occurs. Now let us consider the following sentences: 1. Scott is bald. 2. The author of Waverley is bald. According to Russell, (1) is not analyzable into any other sentence if “Scott” is treated as a logically proper name, while (2) is analyzed out into the following sentences: (a) At least one person wrote Waverley. (b) At most one person wrote Waverley. (c) Whoever wrote Waverley is bald. In symbols, they can be stated in the following way: ða0 Þ ð∃xÞ ðAxwÞ: ðb0 Þ ðxÞ ðyÞ ðAxw:AywÞ  ðx ¼ yÞÞ: ðc0 Þ ðxÞ ðAxw  BxÞ:, where “Axw” stands for “x is an author of Waverley,” “Ayw” for “y is an author of Waverley,” and “Bx” for “x is bald.” The symbolic counterpart of (2) is (3): ð3Þ ð∃xÞ ððAxw:ðyÞðAyw  x ¼ yÞÞ:BxÞ:

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The sentence “The author of Waverley exists” is analyzed out into (a) and (b). Hence it is equated with the conjunction of (a0 ) and (b0 ). The symbolic counterpart of this conjunctive sentence is (4): 3:ð∃xÞ ðAxw:ðyÞðAyw  x ¼ yÞÞ: Russell’s analysis reveals the following features: Firstly, Russell has defined definite descriptions not in isolation, but in the context of sentences in which they occur. He has analyzed those sentences in such a way that all definite descriptions are eliminated. Secondly, Russell’s analysis also reveals the distinction between a grammatical subject and a logical subject. Since definite descriptions can be eliminated from the sentences in which they occur, they cannot be treated as logical subjects. Thirdly, since “the author of Waverley” has been eliminated, what appears in the analysans is “(an) author of Waverley” as a predicate expression. Fourthly, the above analysis also reveals that the sentence “the author of Waverley exists” is not a simple or atomic sentence. It is a conjunction of two sentences, and the sentence “the author of Waverley is bald” is a conjunction of three sentences. Fifthly, the analysans does not contain any atomic sentence. According to Russell, a definite description is an incomplete symbol, and the purported referent of it is a logical construction. Russell has also used the term “logical fiction” for the purported referent of it (For a more comprehensive discussion, see Shaw 1991a.). As regards terms which do not refer to real entities, Russell has postulated concepts for the explanation of their meanings. To quote: “I met a unicorn” or “I met a sea-serpent” is perfectly significant assertion, if we know what it would be to be a unicorn or a sea-serpent, i.e. what is the definition of these fabulous monsters. Thus it is only what we may call the concept that enters into the proposition. (Introduction to Mathematical Philosophy, p. 168.)

Part II Now let us discuss the Nyāya analysis of a definite description. The Nyāya, like the contemporary discussion on singular terms, has drawn a distinction between a proper name and a definite description. A proper name is called “an expression which refers to one and only one individual object” (eka-vyakti-vācaka), but a definite description is called “an expression which generates a cognition of one and only one individual object” (eka-vyakti-bodhaka). Regarding proper names, we come across several views in the Nyāya literature, and I think some ten views may be reconstructed from the suggestions of the Nyāya philosophers. However, all the Nyāya philosophers claim that a proper name is a

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non-empty term and it refers to a particular. Almost all the Nyāya philosophers also claim that the meaning of a proper name includes not only the referent but also a mode of presentation of the referent. Hence the meaning complex of a proper name, say Tom, may be expressed thus: ‘Tom’ means < Tom, R, F >, where the first term of the ordered triple is the referent, the third term F is a mode of presentation or an indicator of the referent, and the second term R relates third term to the first term. (For a comprehensive discussion, see, Shaw 1985)

Regarding acquaintance with the referent of a proper name, unlike Russell, the Nyāya claims that it is not required for the use of a proper name. Hence, unlike Russell, there is no distinction between ordinary names and logically proper names. Therefore, names like “Socrates” or “Plato” are to be treated as proper names, as they refer to particulars. The meaning complex of a non-empty definite description such as “the author of Waverley” may be expressed in the following way: (1) The expression “the author of Waverley” means

By using lambda-operator, (1) may be rewritten as . The meaning of a definite description, unlike the meaning of a proper name, is determined by the meanings of its parts and the syntactic relation between them. Moreover, the distinction between a definite and an indefinite description is to be explained in terms of the uniqueness condition. Hence the meaning of “an author of Waverley” is determined by the meanings of “an author,” “of,” and “Waverley” and the syntactic relation between them. But the meaning of “the author of Waverley” involves the uniqueness condition as well. Furthermore, according to the Nyāya, a description, definite or indefinite, is a sentence. Hence a description such as “the author of Waverley” or “the King of France” is true or false. A sentence, according to the Nyāya, is a set of morphemes which are related to each other by syntactic rules. Hence it is an ordered n-tuple such that n  2 morphemes. Now let us see how the meaning of a definite description can be explained in terms of “one and only one.” Since the use of the definite article “the” implies the uniqueness condition, “the so-and-so” means the same as “one and only one so-and-so.” By using Gadādhara’s technique, the meaning of “the author of Waverley” may be expressed in the following way: A person is an author of Waverley, and he/she has the absence of other persons qualified by the property of being an author of Waverley by the relation of having the same locus (sāmānadhikaraṇya-sambandha). This conjunctive sentence may be expressed in the following way:

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(a) (∃x) (x has the property of being an author of Waverley, and (y) (If y 6¼ x and y has the property of being an author of Waverley, then x has the absence of y by the relation of having the same locus)).

This sentence may be expressed by (b) (∃x) (x has Aw. (y)((y 6¼ x. y has Aw)  ((absence of y by the relation of R)x))), where R is having the same locus. Hence it is derivable from (i) (∃x) (x has Aw), i.e., at least one person wrote Waverley, and (ii) (x) (y) (((x has Aw). (y has Aw. y 6¼ x))  (absence of y by the relation of R)x), i.e., at most one person wrote Waverley. The form of (b) may be expressed by: ðcÞ ð∃xÞ ðFx:ðyÞððy 6¼ x:FyÞ  ððabsence of y by the relation RÞ xÞÞÞ: It is to be noted that the Nyāya explains the meaning of “only one” in terms of having a negative property. Let us take “Fx” as “x is an author of Waverley.” If there is another author of Waverley, then the second conjunct of (c) is false. Hence (c) as a whole is false. If y is not an author of Waverley, then the antecedent of the second conjunct is false. Hence the second conjunct is true. Since the first conjunct is true, (c) is also true. It is to be remembered that (c) is, in some respect, similar to Russell’s analysis of a definite description which has the following form: ðdÞ ð∃xÞ ðFx:ðyÞðFy  x ¼ yÞÞ: In spite of some similarities with Russell’s analysis, there are a few significant differences. The Nyāya philosophers, unlike Russell, do not assert an identity between an existent object which has the property F and an object which is supposed or imagined to have the property F. Hence a supposed or an imaginary object has been smuggled into the Russellian analysis. It is to be noted that there are four interpretations of the expression “x ¼ y,” three of them cannot be applied to Russell’s use of it in the context of a definite description. If we say that both “x” and “y” refer to the same thing and have only reference as Russellian names presuppose, then there is no difference between “x ¼ y” and “x ¼ x.” Hence nothing has been said thereby. If “x” and “y” refer to different objects, then obviously it is false. The fact that these two alternatives are not tenable is expressed by the following remark of Wittgenstein: . . .to say of two things that they are identical is nonsense, and to say of one thing that it is identical with itself is to say nothing at all. (L. Wittgenstein, Tractatus LogicoPhilosophicus, 5.5303)

The third alternative is to postulate the sense of Frege. According to this view “x” and “y” refer to the same thing, but they have different senses. Since the mode of

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presentation of the referent of “x” is not the same as the mode of presentation of the referent of “y,” they are not identical in every respect. This method is very useful for explaining the identity between definite descriptions, such as “the author of Waverley¼the author of Ivanhoe.” But this alternative does not apply to Russellian proper names, as they do not have sense. There is another alternative which is also not satisfactory. It may be said that in the sentence “x ¼ y,” we are assigning an identity between an actual and a supposed object. It is to be noted that in the analysans of Russell, we come across: If x has the property F and y has the property F, then x ¼ y. Since we rule out the first two alternatives and third one is not tenable in the analysans of Russell, as proper names do not have sense, we are left with the fourth alternative. The followers of the Nyāya claim that the fourth alternative postulates a supposed or an imagined object to give an account of “x ¼ y.” Since the Nyāya analysis does not postulate any supposed object, “x ¼ y” does not occur in the analysans. What occurs instead is a negative expression signifying a negative property characterizing an actual object if “the so-and-so” is true. Since the Nyāya analysis does not involve the postulation of this type of entity, it is better than the analysis of Russell (See also, Shaw 1998). It is also to be noted that the law of transposition, according to the Nyāya, is not universally valid, as there are restrictions on the use of negation. Moreover, the Nyāya has drawn the distinction in meaning between the sentences “the author of Waverley” and “the author of Waverley exists.” This is due to the fact that they do not generate the same cognition, but in Russellian analysis, “the author of Waverley” has no meaning in isolation. According to the Nyāya, the symbolic counterpart of the sentence “the author of Waverley exists” will be: ðeÞ ð∃xÞ ððFx:ðyÞððy 6¼ x:FyÞ  ððabsence of y by the relation RÞ xÞÞ:ExÞÞ, where “Ex” stands for “x exists.” Since (c) does not signify the existence of the so-and-so, the quantifier “∃x” is to be interpreted as “some” or “at least one,” not as “exists.” Since the Nyāya analysis can preserve the distinction between “the so-and-so” and “the so-and-so exists,” it is closer to our ordinary usage, and it can be used in a Meinongian ontology, although the Nyāya ontology does not include nonexistent or possible objects. Hence it is neutral to ontological commitment. Since the Nyāya theory avoids the postulation of supposed or imagined objects and retains the distinction in meaning between “the so-and-so” and “the so-and-so exists,” it is better than Russell’s theory of definite descriptions. Moreover, the Nyāya philosophers, unlike Russell, have explained the meaning or understanding the meaning of a non-designating or empty term without postulating a mental entity such as a concept. Since an empty term such as “unicorn” means “a horse with a horn,” the understanding of its meaning implies the relation of a cognition to the elements of its meaning. Hence the mental state is related to a horse and a horn as well as to an elsewhere or elsewhen relation of a horn to an animal which has a horn. This is due to the fact that a horn is not related to a horse. Hence

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the understanding of the meaning of an empty term does not imply the cognition of a concept. Since the Nyāya does not postulate concepts in addition to the things in the word, it is simpler than the thesis of Russell. This is how I would like to demonstrate the logical thinking of the Nyāya philosophers both in philosophy of language and logic. Moreover, here we come across something which is better than the best of Russell’s contribution to logic or philosophy of language.

Section 8: Six Pairs of the Nya¯ya Philosophers In the first part, I shall discuss the distinction between the six pairs of terms of the Nyāya philosophers and why all of them are required in the Nyāya conception of inference, but only four in perception and three in verbal cognition. The second part will deal with the concepts of saturated and unsaturated as found in contemporary Western philosophy. In this context, I shall deal with the Nyāya argument for considering the predicate alone as unsaturated or incomplete. Thus we claim to solve a problem of contemporary Western philosophers, beginning with Frege, from the Nyāya perspective.

Part I The Nyāya philosophers have discussed the distinction between the following pairs of terms: 1. 2. 3. 4.

anuyogī-pratiyogī (first term-second term) ādhāra-ādheya (substratum-superstratum) viśeṣya-viśeṣaṇa (qualificand-qualifier) viśeṣya-prakāra (qualificand-relational qualifier) (The Sanskrit term prakāra has been translated as relational qualifier so that it can be distinguished from the word viśeṣaṇa (“qualifier”).) 5. uddeśya-vidheya (subject-predicate) 6. pakṣa-sādhya (the locus of inference-probandum)

Now let us define these terms and explain their roles in different types of cognition, such as perceptual, inferential, and verbal. Suppose a and b are related by the relation R. (A) a is anuyogī (the first term) iff b is related to a by the relation R. At the level of cognition, the relation R is cognized in a, but not in b. The other term of the relation R, namely b, is pratiyogī (the second term). The relation of our cognition to this aspect of a is expressed by the term anuyogitā (“the property of being the first term”). Hence anuyogitā (the property of being the first term) is a relational property of a. Similarly, b has pratiyogitā (the property of being the second term). As regards the relation of a to b, the Nyāya claims that the former is related to the latter by the converse of R, i.e., Ř. Hence b can be cognized as anuyogī (the first

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term) and a as pratiyogī (the second term) of R converse, i.e., Ř. In this cognition, the relation of cognition to b is anuyogitā (the property of being the first term), and the relation of cognition to a is pratiyogitā (the property of being the second term). This is what happens when two objects are related by the relation of conjunction (or contact), but one is not in another, for example, the contact between the two palms in salutation or the contact between two animals in a herd. In such cases, one term is not cognized in another. The relation between anuyogitā (the property of being the first term) and pratiyogitā (the property of being the second term) is mutual determiner-determined (paraspara nirūpya-nirūpaka). In other words, they are correlative terms. Again, the cognition of anuyogī (the first term) presupposes the cognition of pratiyogī (the second term), and vice versa. Hence anuyogī (the first term) and pratiyogī (the second term) are also correlatives. But the entities referred to by these terms may not be correlatives. In this context, it is to be noted that in specifying the direction of a relation, the Nyāya philosophers use the terms anuyogitā (the property of being the first term) and pratiyogitā (the property of being the second term). If a is anuyogī (the first terms), b is pratiyogī (the second term), and the relation between them is conjunction, then the relation of b to a is the property of being the first term determined by the property of being the second term which is limited by the relation of conjunction (saṃyogasambandhāvacchinna-pratiyogitā-nirūpita-anuyogitā). The relation of a to b is the property of being the second term limited by the conjunction relation but determined by the property of being the first term (anuyogitā-nirūpita-saṃyoga-sambandhāvacchinna-pratiyogitā). The former and the latter may be represented by the following forms, respectively: (i) a (cNds) b (ii) b (dsNc) a where a stands for the first term, b for the second term, c for the property of being the first term, d for the property of being the second term, s for conjunction relation, ds for d is limited by s, and N for determined by relation. The Nyāya philosophers have used this technique for the explanation of the meaning of a sentence. According to them, the meaning of a sentence as distinct from the meanings of its parts lies in the relation along with its direction. Hence the meaning of “a is related to b by the relation R” cannot be equated with that of “b is related to a by the converse of R.” This technique of the Nyāya philosophers may be used to explain the difference in meaning between the pairs of sentences, such as “Brutus killed Caesar” and “Caesar was killed by Brutus.” (B) Now let us discuss the pair ādhāra-ādheya (substratum-superstratum). If an object resides in another object, then the latter is ādhāra (substratum), and the former is ādheya (superstratum). For example, a book is on the table. The table is ādhāra (substratum) and the book is ādheya (superstratum). If a pen is on the book, then the book becomes the ādhāra (substratum), and the pen becomes the ādheya (superstratum) of the book. Here the book is ādheya (superstratum) in relation to the table, but ādhāra (substratum) in relation to the pen. If everything resides in time and time

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does not reside in anything, then time becomes the ādhāra (substratum) but not the ādheya (superstratum). The relational properties ādhāratā (the property of being the substratum) and ādheyatā (the property of being the superstratum) signify the relation of a cognition to the ādhāra (substratum) and the relation of a cognition to the ādheya (superstratum), respectively. Hence they are viṣayatās (a type of relational properties of the objects). If b is related to a by an occurrence-exacting relation (vṛtti-aniyāmakasambandha) R, then the cognition of a as ādhāra (substratum) would imply the vṛttianiyāmaka cognition of b in a by the relation R, but the cognition of b would not imply the cognition of a in b by the relation R converse, i.e., Ř. When a is cognized as ādhāra (substratum), it is cognized not only as anuyogī (the first term) and b as pratiyogī (the second term) but also as something in which b resides. In other words, the cognition of a as ādhāra (substratum) is something more than the cognition of it as anuyogī (the first term). Here also we have to specify the direction of the relation in terms of ādhāratā (the property of being the substratum) and ādheyatā (the property of being the superstratum). If a is ādhāra (substratum), b is ādheya (superstratum), and R is the relation of conjunction (contact), then the relation of b to a is the property of being the substratum determined by the property of being the superstratum which is limited by the relation of conjunction (saṃyoga-sambandhāvacchinna-ādheyatā-nirūpitaādhāratā). The relation of a to b is the property of being the superstratum limited by the relation of conjunction and determined by the property of being the substratum (ādhāratā-nirūpita-saṃyoga-sambandhāvacchinna-ādheyatā). For this reason, the pairs of sentences such as “the floor possesses a pot” (ghaṭavad bhūtalam) and “a pot is on the floor” (bhūtale ghaṭaḥ) do not have the same meaning, although they are logically equivalent. (C) As regards viśeṣya-viśeṣaṇa (qualificand-qualifier) distinction, the Nyāya philosophers claim that it is applicable to every qualificative or relational cognition (savikalpaka-jñāna). Only non-qualificative cognition (nirvikalpaka-jñāna) is excluded from its scope. Let us consider the cognitions expressed by the following sentences: (a) A fire is on the mountain (parvate vahniḥ). (b) The mountain has a fire (vahnimān parvataḥ). In (a), a fire is the qualificand and the mountain is the qualifier, while in (b), the mountain is the qualificand and a fire is the qualifier. The qualifier plays the role of a distinguisher. Hence it distinguishes something from other things or a collection from other collections. That which is distinguished by it is the qualificand. Since in (a) a particular fire is distinguished from other fires in terms of the mountain, the mountain becomes the qualifier. Since a particular fire is distinguished from other fires, it is the qualificand of this cognition. The relation of this cognition to the fire is viśeṣyatā (the property of being the qualificand), and the relation of this cognition to the mountain is viśeṣaṇatā (the property of being the qualifier). These relational properties are used to characterize the roles of these objects in this cognition.

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In (b), a fire is the distinguisher. Hence it distinguishes a particular mountain from other mountains. Since it is distinguished from other mountains, it is the qualificand of this cognition. Since the terms “qualificand” and “qualifier” are correlatives, the cognition of one presupposes the cognition of the other. Here also the property of being the qualificand (viśeṣyatā) and the property of being the qualifier (viśeṣaṇatā) are related to each other by the determiner-determined relation (paraspara-nirūpyanirūpaka-sambandha). The category of qualificand-qualifier relation emphasizes the distinguisher and the distinguished aspects of the objects of a qualificative cognition. In this context, it is to be noted that according to the Nyāya philosophers, one of the causal conditions of a qualificative cognition is the cognition of its qualifier (For a discussion of this point, see author’s paper “Cognition of cognition,” Part II, note 17, Journal of Indian Philosophy, 1996.). Hence in (a), the cognition of the mountain is a causal condition of the qualificative cognition, but in (b), the cognition of a fire is a causal condition of the qualificative cognition. In the case of perception, the Nyāya philosophers have used the category of qualificand-qualifier (viśeṣya-viśeṣaṇa), not the category of subject-predicate (uddeśya-vidheya). Let us consider the above examples for the description of a perceptual cognition. If the cognition of a fire is present prior to the perception and the cognition of the mountain is absent, then the sentence “the mountain has a fire” will be used to describe this perceptual cognition. But if the cognition of the mountain is present prior to the perception and the cognition of a fire is absent, then the sentence “A fire is on the mountain” will be used to describe it. This is due to the fact that in the former case, the mountain is the qualificand, and a fire is the qualifier, while in the latter case, a fire is the qualificand and the mountain is the qualifier. But if the cognition of both of them are present prior to the perception, then it will take either of the two forms. Hence it can be described either by (a) or by (b). If none of them is present prior to the perception, then the perceptual cognition will not take place. (D) Now let us discuss the category of qualificand-relational qualifier (viśeṣyaprakāra). It is to be noted that in both (a) and (b), the relation of conjunction is the qualification relation. The Nyāya philosophers have tried to establish the thesis that the qualification relation is a mode of presentation of the qualifier, not the qualificand. (See also author’s article “‘Saturated’ and ‘Unsaturated’: Frege and the Nyāya,” Synthese, 1989.) Hence in (a), the conjunction relation is a mode of presentation of the mountain, but in (b), it is a mode of presentation of a fire. In general, according to the Nyāya, the qualifier is presented under the mode of the qualification relation. The qualifier which is presented under the mode of the qualification relation is called prakāra (“relational qualifier”), and the relation of the cognition to this type of qualifier is called prakāratā (“the property of being the relational qualifier”). Now the question is whether every qualifier is presented under the mode of its qualification relation. On this point, the Nyāya philosophers claim that certain qualifiers are not presented under the mode of their qualification relations. When one relation is a qualifier of another relation, the former is simply a qualifier (viśeṣaṇa), not a relational qualifier (prakāra). Consider the following example:

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A fire is in a container and the container is on the mountain.

Let us suppose the fire is related to the container by the relation R and the container is related to the mountain by the relation S. Hence the fire is related to the mountain by the indirect relation R and S. In the cognition of the fire being related to the mountain, the relations R and S are related by the qualificand-qualifier relation (viśeṣya-viśeṣaṇa-sambandha). Hence R is the qualifier of S which is its qualificand. The relation of R to S would be the property of being second term of the relation S such that the second term is the locus of the relation R (svāśrya-pratiyogīkatva). In other words, this relation is a property of S if a is related to b by R and b is related to c by S. According to the Nyāya, the relation of R to S is not considered a mode of presentation of R which is the qualifier of S. This is due to the fact that if the relation of R to S becomes the mode of presentation of R, then there will be an infinite regress at epistemic level. Suppose R1 is the relation of R to S. If R1 becomes the mode of presentation of R, then R1 becomes the qualifier of R. Again, the relation of R1 to R, say R2, becomes the mode of presentation of R1 and so on. In order to avoid this type of regress, the Nyāya claims that a relation which is a qualifier of another relation is simply a qualifier (viśeṣaṇa), not a relational qualifier (prakāra). In this context it is to be noted that a qualificative perceptual cognition involves all the four pairs discussed above if the relation is occurrence-exacting (vṛttianiyāmaka). But if the relation is non-occurrence-exacting (vṛtti-aniyāmaka), then it will involve only three pairs, namely, anuyogī-pratiyogī (the first term-the second term), viśeṣya-viśeṣaṇa (qualificand-qualifier), and viśeṣya-prakāra (qualificandrelational qualifier). These terms signify the ways our perceptual cognition is related to the same object. They also give us information about the roles of the objects of perception. (E) The pair uddeśya-vidheya (subject-predicate) is used in the context of some verbal cognition, but in every inference. In the verbal context, it occurs in a questionanswer or topic-comment situation. Suppose the question is: Where is fire? And the answer is: A fire is on the mountain. In this context, a fire is uddeśya (subject), and the mountain is vidheya (predicate). But if the question is what is on the mountain? and the answer is A fire is on the mountain, then the mountain is the subject, and the fire is the predicate. The subject, according to the Nyāya, is that which is already known to us, but the predicate is not already known to us. Here also the terms “subject” and “predicate” are correlatives. The relation of the cognition to the uddeśya (subject) is uddeśyatā (the property of being the subject), and the relation of the cognition to the vidheya (predicate) is vidheyatā (the property of being the predicate). In the context of verbal cognitions, the Nyāya uses only three pairs of terms, namely, viśeṣya-viśeṣaṇa (qualificand-qualifier), viśeṣya-prakāra (qualificandrelational qualifier), and uddeśya-vidheya (subject-predicate). This is due to the fact that the superstratum (ādheya) or the second term (pratiyogī) takes the role of a distinguisher in a verbal cognition. Hence in understanding the meaning of the sentence “a pot is on the floor,” a pot is cognized as the qualificand (viśeṣya), and the

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property of being the superstratum (ādheyatā) residing in it is cognized as its qualifier. This qualifier is limited by the relation of conjunction but determined by the property of being the substratum residing in the floor (bhūtala-niṣṭhaadhikaraṇatā-nirūpita-saṃyoga-sambandhāvacchinna-ādheyatā). Since a pot is to be distinguished in terms of this qualifier (viśeṣaṇa) or relational qualifier (prakāra), it takes the role of a distinguisher. F) Now let us discuss the pair of terms pakṣa (“the locus of inference”) and sādhya (“probandum”). Since this pair occurs only in an inference and since all the remaining pairs are also involved in an inference, I shall discuss it in the context of an inference for others. In this context, I shall also point out the significance of the distinction between the pairs anuyogī-pratiyogī (the first term-the second term) and ādhāra-ādheya (substratum-superstratum). In order to substantiate these claims, let us consider the following inference for others: 1. 2. 3. 4. 5.

Thesis (pratijñā): The mountain has a fire. Reason (hetu): Because of smoke. Example (udāharaṇa): Wherever there is smoke, there is fire, as in kitchen, etc. Application (upanaya): The mountain has smoke which is pervaded by fire. Conclusion (nigamana): Therefore, the mountain has a fire.

According to the Nyāya philosophers, these sentences will ultimately give rise to an inferential cognition in the hearer. Initially, the hearer will understand the meanings of these sentences, and thereafter he/she will have a mental cognition (mānasa-jñāna) of the operation (parāmarśa) which corresponds to the application or the fourth premise. The operation (parāmarśa) will yield the inferential cognition (anumiti). The parāmarśa (operation), which is a cognition, is not generated by external sense organs or by the causal conditions of indirect cognitions (parokṣajñāna). For this reason, it is considered a mental perception. In this inference, the locus (pakṣa) is the mountain, the probandum (sādhya) is fire, and probans or mark (hetu) is smoke. The old Nyāya defines pakṣa as something where there is doubt about the presence of the sādhya (sandigdha-sādhyavānpakṣaḥ). The word pakṣatā (“the property of being the locus of inference”) refers to this dubious cognition of the presence of the sādhya (probandum) in the pakṣa (the locus of inference). But this definition of pakṣatā is not acceptable to the followers of the Navya-Nyāya as the desire to infer leads to an inferential cognition (anumiti) even if there is no doubt about the presence of the probandum in the locus of an inference. For this reason, the followers of the Navya-Nyāya define pakṣatā as the absence of certainty about the probandum in the locus qualified by the absence of desire to infer (siṣādhayiṣā-viraha-viśiṣṭa-siddhyabhāva). Hence pakṣatā which is a type of dubious cognition or the absence of certainty or a desire to infer qualifies the self of the person who is making the inference, and it is related to the locus of inference by the relation of the property of being the qualificand (viśeṣyatā). In other words, the locus of inference is the qualificand of this mental state. For this reason, the qualificand of the mental state pakṣatā is called pakṣa. The above definition of pakṣatā is usually explained in the following way:

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There is absence of certainty about the probandum in the locus, or there is desire to infer the probandum in the locus. In this context, it is to be noted that this explanation cannot be identified with the definition of pakṣatā, although they are equivalent. If this explanation is taken as the definition of pakṣatā, then it would violate the law of parsimony as the former is simpler than the latter. This is due to the fact that the latter would refer to innumerable differences which would qualify the desire to infer and the absence of certainty. This is involved in the meaning of “one or the other” (anyatara) which will occur in the latter definition of pakṣatā. Since these innumerable differences (e.g., difference from a table, difference from a chair, etc.) are limitors of the property of being the cause (kāraṇatāvacchedaka), this definition would violate the law of parsimony. Moreover, it would be difficult to know so many differences residing in both the alternatives, viz., absence of certainty and the desire to infer. For this reason, the above explanation should not be taken as the definition of pakṣatā. From the above discussion of pakṣatā, it follows that a dubious mental state or a desire to infer or simply absence of certainty is related to pakṣa (the locus of inference). Therefore, when we say that pakṣa has pakṣatā, what we mean is that an attitude of doubt or a desire to infer is directed toward the pakṣa (the locus of inference). In our above example of inference, the mountain which has occurred in the cognition of the thesis has the relational property pakṣatā. Since a fire is to be inferred on the mountain and there is doubt about its presence or there is desire to infer it on the mountain, it is called sādhya (probandum). Hence it is something to be established or inferred in the pakṣa. The way this cognition is related to the sādhya is called sādhyatā. Moreover, sādhyatā is the correlative of pakṣatā. Hence they are dependent on each other. Now the question is whether sādhya (probandum) is a correlative of hetu (probans or mark). According to the Nyāya, the relation between them is signifier and significate ( jñāpya-jñāpaka-sambandha). Hence they are also correlatives. Again, the property of being the signifier ( jñāpakatva) is determined by (nirūpita) the property of being the significate ( jñāpyatva), and vice versa. Hence hetutā which is the same as jñāpakatva is the correlative of sādhyatā which is the same as jñāpyatva. Now let us contrast the cognition of the thesis (pratijñā) with the inferential cognition (anumiti). In an inferential cognition, which is the result of an inference, the locus (pakṣa) is no longer characterized by pakṣatā as there is no doubt about the presence of the probandum (sādhya) or desire to infer the probandum (sādhya). Hence the mountain which has occurred in the inferential cognition lacks pakṣatā, and thereby the fire lacks sādhyatā. Since the fire is the significate of smoke which is the hetu (probans), the fire would remain as sādhya or significate in the inferential cognition. Hence it has sādhyatā (the property of being the probandum) which is the correlative of hetutā (the property of being the probans), although it lacks another type of sādhyatā which is the correlative of pakṣata. From this discussion, it follows that there are two types of sādhyatā. The cognition of the thesis has both the types of sādhyatā, but the inferential cognition has only one type of sādhyatā, namely, the one which is the correlative of hetutā.

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As regards the subject (uddeśya), the mountain remains the subject both in the thesis and in the inferential cognition. This is due to the fact that the mountain is already known to us. Since the fire was not already known to us on the mountain, it remains the predicate (vidheya) in both the cognitions. As regards the difference between the predicate (vidheya) and the probandum (sādhya), it may be said that the probandum (sādhya) is the correlate of the probans (hetu) but the predicate (vidheya) is the correlate of the subject (uddeśya), although neither the predicate (vidheya) nor the probandum (sādhya) is already known to us if there is doubt about the presence the probandum in the locus of inference. The fact that the probandum (sādhya) is the correlate of the probans (hetu) is not conveyed by the use of the term “predicate” (vidheya). Hence the fire, in the above example, plays a dual role. As a correlate of the subject (uddeśya), it is the predicate (vidheya), but as a correlate of the probans (hetu), it is the probandum (sādhya). For this reason, the Nyāya philosophers have used the term vidheya (“predicate”), not the term sādhya (“probandum”), in the context of verbal cognition which cannot be identified with an inferential cognition. From the above discussion, it follows that both terms are necessary for the description of an inferential cognition (anumiti). Now let us demonstrate the relevance of the remaining pairs of terms in the context of an inference. It is to be noted that both the inferential cognition and the operation which are directly related to the inferential cognition may take different forms. The inferential cognition in the above inference is also expressed by the sentence “A fire is on the mountain.” In this cognition also, the pakṣa (the locus of inference) and the sādhya (probandum) would remain the same. Again, the uddeśya (subject) and the vidheya (predicate) would also remain the same. In other words, in this cognition also, the mountain would be both pakṣa (the locus of inference) and uddeśya (subject), and the fire would be both sādhya (probandum) and vidheya (predicate). But the qualificand, the qualifier, and the relational qualifier of this cognition would not be the same as those of the cognition of the mountain has a fire. In the latter cognition, the mountain is the qualificand (viśeṣya), the fire is the qualifier (viśeṣaṇa), and the fire presented under the mode of the qualification relation, which is conjunction in the ontology of the Nyāya, is the relational qualifier (prakāra). In the former cognition, the fire is the qualificand (viśeṣya), the mountain is the qualifier (viśeṣaṇa), and the mountain presented under the mode of the qualification relation is the relational qualifier (prakāra). This is how the Nyāya philosophers describe the fine-grained distinction between these two forms of the inferential cognition. Now I shall mention the importance of the difference between the pairs anuyogī-pratiyogī (the first term-the second term) and ādhāra-ādheya (substratumsuperstratum) with reference to the different interpretations of parāmarśa (operation) which is the vyāpāra of the inferential cognition. In our above example, the application (upanaya-vākya) expresses the vyāpāra of an inferential cognition. It is called parāmarśa in the context of an inference. The Nyāya philosophers define parāmarśa as vyāpti-viśiṣṭa-pakṣadharmatā-jñāna. One of the interpretations of this definition is as follows: The cognition of the property of being the superstratum (ādheyatā) residing in the probans qualified by the property of being the pervaded (vyāpyatva) in the locus of inference.

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If we take this interpretation, then there will be only one type of parāmarśa, viz., pervaded qualificand (vyāpya-viśeṣyaka), positive or negative, which can be stated in four different ways (For a comprehensive discussion, see Shaw 2010.). If we take this interpretation, then we cannot derive other formulations of parāmarśa, including the one mentioned in our above example, namely, The mountain has smoke which is pervaded by fire. Moreover, we get only one type of inferential cognition as there is only pervaded qualificand (vyāpya-viśeṣyaka) type of parāmarśa. Hence in our above example, the inferential cognition would be: A fire is on the mountain; instead of the mountain has a fire.

It is to be noted that there are four more formulations of parāmarśa which cannot be derived from the above interpretation which uses the category of ādhāra-ādheya. In order to derive all the formulations, the following interpretation has been suggested: The cognition of the relation which is the limitor of the property of being the probans or the pervaded whose first term (anuyogī) is the locus of inference (pakṣa), and the second term (pratiyogī) is the probans (or that) which is qualified by the property of being the pervaded. This is how the Nyāya philosophers have shown the importance of the category of anuyogī-pratiyogī in the context of an inference. Since all the six pairs of terms are involved in an inference, the Nyāya philosophers can express several types of epistemic differences, which cannot be expressed in terms of one or two pairs of terms such as “subject-predicate.” Hence the Nyāya distinction between the six pairs of terms would add a new dimension to contemporary epistemology. In terms of these pairs, we can describe the phenomenological properties of perceptual, inferential, or verbal cognitions. Moreover, since there are altogether 12 terms in the 6 pairs, our cognition may be related to the same thing in 12 ways in different contexts. Hence by using these terms we can specify the roles of the same object in different epistemic contexts. This is how we can show the relevance of our discussion to contemporary phenomenology.

Part II Now I would like to discuss how the Nyāya philosophers would interpret Frege’s use of the terms “saturated” and “unsaturated.” Frege claimed that the subject expression of a sentence is saturated (or complete), while the predicate expression is unsaturated (or incomplete). At the level of thought also, the subject is saturated, but the predicate is unsaturated. Following the suggestion of the Nyāya philosophers, I will discuss this problem of contemporary philosophers. Hence I will discuss why the predicate or the predicate expression alone is to be considered as unsaturated. In the context of the Nyāya philosophy, the question is: Why should we consider the qualification relation as the mode of presentation of the qualifier alone?

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The Nyāya philosophers claim that if the qualification relation is considered as the mode of presentation of the qualificand (or the subject), then in some cases, we cannot describe the cognitions which are prevented by a particular cognition. It is to be noted that the Nyāya conception of preventer-prevented relation between cognitions is the epistemic counterpart of the relation between contrary or contradictory sentences. Hence the Nyāya view may be stated in terms of sentences as well. Let us consider the cognition corresponding to the sentence “Everything has the absence of time by the relation of inherence” (sarvaṃ samavāyena mahākālābhāvavān). This cognition would be prevented by the following cognitions: (i) Everything is related to time by the relation of inherence (sarvaṃ samavāyena mahākālavān). (ii) There is at least one thing which is related to time by the relation of inherence (samavāyena mahākālavān). (iii) Rāma is related to time by the relation of inherence (Rāmaḥ samavāyena mahākālavān), Śyāma is related to time by the relation of inherence (Śyāmaḥ samavāyena mahākālavān), and so on. Since (iii) refers to each member of our domain, it is not possible to enumerate each of them if there are infinite objects. Now the question is whether the prevented cognitions can be described. The Nyāya philosophers have suggested the following descriptions: (a) The cognitions in which the property of being the qualifier is limited by the property of being time and the relation of inherence (samavāya-sambandhāvacchinna-mahākālatvāvacchinna-prakāratāśālī-buddhi). (b) The cognitions in which the property of being the qualificand is limited by the relation of inherence and determined by the property of being the qualifier which is limited by the property of being time (mahākālatvāvacchinna-prakāratā-nirūpita-samavāya-sambandhāvacchinna-viśeṣyatāśālī-buddhi). Now let us compare (a) with (b). Since (b) has mentioned the property of being the qualificand, it refers to all the qualificanda of the prevented cognitions. Hence it refers to (i), (ii), and (iii) mentioned above. Therefore, the previous problem of enumeration cannot be overcome. In order to avoid this problem, (a) has been suggested by the Nyāya philosophers. Since (a) does not mention the property of being the qualificand of the prevented cognitions, it does not refer to each of the qualificanda of the prevented cognitions. So the description (a) is simpler than that of (b). This is due to the fact that the relation is considered as the mode of presentation of the qualifier, not the qualificand. Hence the qualifier is dependent on the relation, not the qualificand. Therefore, in the Nyāya philosophy, we come across an argument for considering the predicate alone as unsaturated. This is how one of the problems of contemporary philosophy can be solved. From the Nyāya distinction between the six pairs of terms, we can demonstrate why transformationally equivalents sentences do not have the same meaning as well

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as explain the difference in perceptual, inferential, and verbal cognitions. Moreover, we explain in what sense a predicate is to be considered as incomplete and suggest an argument for being incomplete (For the Nyāya view, I am greatly indebted to the late Pandit Visvabandhu Tarkatīrtha, an authority on Navya-Nyāya.).

Concluding Remarks From the above discussion, it follows that the Nyāya philosophers have demonstrated their logical thought in the following ways: 1. They have applied Occam’s razor not only in defining causation but also in explaining false belief as well as in reducing 11 sources of knowledge to 4. The principle of simplicity has been used almost in every discussion and justified on logical grounds, such as the avoidance of infinite regress. 2. Regarding false belief, it has been demonstrated how to explain it without postulating the existence of the nonexistent and thus falsifying the claim of Russell. 3. The classification of causal conditions of perceptual, inferential, analogical, and verbal cognitions into four types not only has demonstrated the distinction between the true and the false cognitions but also introduces four types of justifications, one for each type. 4. Since justification is a property of a true belief or cognition, not a property of belief or cognition in isolation, the Gettier or the post-Gettier counterexamples to the JTB thesis are not applicable to the Nyāya view. 5. It can solve the problems of Goldman, as the cognizer has the ability to discriminate both in perceptual and inferential cognitions. 6. Regarding inference, it demonstrates the relevance between the premises, as well as the relevance between the premises and the conclusion. Since the relevance condition for any inference, valid or invalid, has been emphasized, the following valid inference of Western logic is not treated as an inference in the Nyāya logic: P and not P, Therefore, Q. This is due to the fact that it violates the relevance condition as well as certain epistemic conditions for understanding the meaning of a sentence. As a result, we cannot derive 2 + 2 ¼ 4 from it is raining and not raining, which is valid in classical symbolic logic. Since the Nyāya logic has emphasized the relevance condition, it might throw some light on contemporary discussion on relevant logic. The types of relevance have also been mentioned in this context. The classification of fallacies into different types has added a new dimension to the Nyāya philosophy. The distinction between tarka and inference has also been explained. 7. In the context of verbal cognitions, it has been shown why the transformationally equivalent sentences do not have the same meaning and how the logical distinction between atomistic and holistic understanding can be explained.

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8. The negation and its types have been discussed at linguistic, epistemic, and ontological levels. Since the mode of presentation is included in the negatum, it is doubtful whether it is to be identified with the term negation or the proposition negation of Western philosophy. 9. The Nyāya discussion on universal sentences focuses on the use of the word “all,” as well as the expression “If, then.” The discussion on the pervaderpervaded relation, which is common to all the types of universal sentences, is highly technical and very comprehensive. 10. The Nyāya view of the principle of contradiction has been compared with Aristotle’s principle of contradiction. It is also claimed that the objections raised by Lukasiewicz against Aristotle’s formulations do not apply to the formulations of the Nyāya. Regarding its status or role in epistemology, the Nyāya has avoided the extreme views of the contemporary Western philosophers. 11. The reconstruction of Gadādhara’s theory of definite descriptions has added a new dimension to Nyāya logic. Unlike Russell, it has avoided the identity between an actual and a supposed object. Moreover, it has retained the distinction in meaning between “The author of Waverley” and “The author of Waverley exists.” Hence this theory is applicable to the Meinongian ontology as well. 12. The Nyāya distinction between the six pairs of terms may be used to specify the distinction between the cognitions derived from perception, inference, and verbal testimony, as only four pairs are involved in perception, but only three in verbal testimony and all six in inference. In this context, the Nyāya philosophers have not only specified the meaning of the term “unsaturated” or “incomplete” but also put forward an argument for the predicate being unsaturated or incomplete and the subject being saturated or complete. Acknowledgments This is how I tried to demonstrate the logical thinking of the Nyāya philosophers.  I am very grateful to the Publisher Bloomsbury, London, for giving me permission to use some materials from my book The Collected Writings of Jaysankar Lal Shaw: Indian Analytic and Anglophone Philosophy, 2016.

References Fodor, J. 1977. Semantics: Theories of meaning in generative grammar. New York: Thomas Y Cromwell Company. Ghosh, Rajendranath. 1982. Vyāpti–Pañcaka. Calcutta: West Bengal State Book Board. Ingalls, D.H.H. 1951. Materials for the study of Navya-Nyāya logic. Cambridge, MA: Harvard University Press. Moore, G.E. 1959. Philosophical papers. London: George Allen & Unwin. p. 118. Nyāyācārya, Madhusudana. 1976. Padārthatattva–Nirūpaṇam of Raghunātha Śiromaṇi, with Bengali translation and commentary. Calcutta: Sanskrit College. Priest, Graham. 1998. Paraconsistent logic. In Routledge encyclopedia of philosophy, vol. 7, 208– 211. London/New York: Routledge. Russell, B. 1977. The philosophy of logical atomism. In Logic and knowledge, ed. R.C. Marsh, 225. London: Allen & Unwin.

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Russell, B. 1980. The problems of philosophy, 73. Oxford: Oxford University Press. Shaw, J.L. 1980. The Nyāya on cognition and negation. Journal of Indian Philosophy 8: 279–302. Shaw, J.L. 1985. Proper names: Contemporary philosophy and the Nyāya. In Analytical philosophy in comparative perspective, ed. J.L. Shaw and B.K. Matilal, 327–372. Dordrecht: D. Reidel Publishing. Shaw, J.L. 1991a. Descriptions: Contemporary philosophy and the Nyāya. Logique et Analyse 31: 153–187. Shaw, J.L. 1991b. Universal sentences: Russell, Wittgenstein, prior and the Nyāya. Journal of Indian Philosophy 18: 103–119. Shaw, J.L. 1992a. The Nyāya on the meaning of some words. Journal of Indian Philosophy 2: 41–88. Shaw, J.L. 1998. Cognition of cognition: A commentary on Pandit Visvabandhu, 69–70. Calcutta: The R. K. Mission Institute of Culture. Shaw, J.L. 2010. Subject-predicate and related pairs. Journal of Indian Philosophy 38: 625–642. Strawson, P.F. 1963. Introduction to logical theory, university paperbacks, 2. London: Methuen. Tarkavāgīśa, Phanibhusana. 1982. Nyāya–Darśana. Vol. I–V. Calcutta: West Bengal Book Board.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Philosophical Pragmatism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deduction, Induction, and Abduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: Three Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: Three Objections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: An Abductive Conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: Three Responses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: A Deductive Conception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Inference: A Pragmatic Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter sets out and explains the discussion of inference (anumāna) in the Nyāya Sūtra of Akṣapāda Gautama and the Nyāya Bhāṣya of Vātsyāyana Pakṣilasvāmin. Affinities with the pragmatic method of Charles Peirce and modern scientific method are presented. Specifically, the procedural methodology of reasoning to form beliefs and the scope for legitimate doubt are shown to have a pragmatic character. An initial definition of inference in NS 1.1.5 is followed by an objection in NS 2.1.37 and a response to the objection in NS 2.1.38. There is scope to read these either as a progressive refinement of a deductive schema or as a sequence of stages in a procedural logic of rational inquiry involving abductive and deductive J. P. Sahota (*) Inavya Ventures Ltd, London, UK e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_10

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elements. Accordingly, the definition, objection, and response are presented and analyzed according to a deductive syllogistic schema and according to an openended abductive-deductive schema. The extent to which either of these schemas conforms to the intentions of the early Nyāya authors is considered, with reference to Vātsyāyana’s remarks in the Nyāya Bhāṣya in particular. Although the deductive schema is found to capture an important aspect of how Vātsyāyana understands correct inference, nevertheless the abductive schema more clearly demonstrates how the inference processes described by the early Nyāya authors are driven by pragmatic considerations, whereby the scope for belief adoption, suspension, and revision are constrained by observable evidence. Keywords

Anumāna · Abduction · Deduction · Hetu · Monotonic logic · Non-monotonic logic · Nyāya Bhāṣya · Nyāya Sūtra · Oetke, Claus · Peirce, Charles Sanders · Pragmatism · Pūrvavat · Śeṣavat · Taber, John

Introduction The river banks in flood make my heart gay; where the moorhen cries, the snake lies sleeping on the cane tops, the gray geese call and herds of antelope gather in peaceful circles; where thick grass growing everywhere is bent beneath the swarms of ants. and the jungle fowl is mad with joy. – Yogeśvara, trans. Ingalls (2000, 101, verse 10.221)

Humans intuitively reason in ways which have a pragmatic or commonsense dimension. That is to say, human reasoning takes place in a context of certain background assumptions shaped in part by the prior beliefs and epistemic capacities of human reasoners, and the new beliefs formed as a result constitute hypotheses which are susceptible to revision or defeat based on further information. As such, they are defeasible and context-sensitive, making assumptions of certain frame conditions or default conditions. This type of reasoning is classed as non-monotonic logic, which indicates its defeasible, context-sensitive character. This reflects the situated nature of human reasoners and the empirical nature of facts about which we typically reason and motivates the hypothesis-based character of scientific method. By contrast, classical deductive, monotonic reasoning is context-free, having a formal structure of an inference in which the premises strictly entail the conclusion. As such, validity is preserved under all conditions. Deductive logic is easier to model formally, and the history of European formal logic since Aristotle has been largely focused on the analysis of formal structures of deductive logic. Ordinary commonsense human reasoning naturally has a pragmatic character. Human speech is also based on pragmatic reasoning, whereby the meaning of utterances is understood by speakers and listeners to be context-sensitive in the same fashion. By contrast, humans find deductive logic less intuitive and are prone to make systematic errors in its application, such as in the Wason selection task.

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As such, some artificial intelligence theorists who seek to replicate human intelligence in machines and to develop automated speech recognition and other natural language processing technologies aim to simulate the pragmatic character of human thought and reason through the formalization and technical implementation of non-monotonic logics. However, developing procedural and programmatic models of commonsense reasoning has proved to be a far more difficult task than developing formal models of deductive logic. Various strands of Indian philosophy, including works of early Nyāya logic, demonstrate pragmatic aspects and important affinities with philosophical pragmatism. For example, discussing the pragmatism of Charles Sanders Peirce, Sarukkai states that he has a “resemblance with Indian philosophers” (Sarukkai 2008, 164–165) and his views on doubt are “similar to the Nyāya view” (Sarukkai 2008, 165). As yet, research into affinities between Indian logic and pragmatism has just begun, and there is significant potential for future research into such affinities. Firstly, such research can provide greater insight into the character of Indian logic and especially to illuminate the relative significance of its abductive, inductive, and deductive dimensions. Secondly, the legacy of the classical Indian philosophers may constitute a resource to inform existing pragmatic approaches in diverse disciplines and theories. This chapter will discuss the extent to which early Nyāya logicians involved pragmatic aspects in their formal inference structures. The analysis will be based on the discussions of correct reasoning in the Nyāya Sūtra (NS) of Akṣapāda Gautama, a text that reached its completed form by about the second century, and the Nyāya Bhāṣya (NB) of Vātsyāyana Pakṣilasvāmin, a fifth century commentary on that text. Contemporary research into pragmatic reasoning and pragmatic conventions in linguistic communication originates from the work of Charles Sanders Peirce, who originally presented the conception of abductive logic. Accordingly, this chapter begins with a description of philosophical pragmatism, abduction, and the pragmatic maxim of Charles Sanders Peirce, which is followed by an overview on deduction, abduction, and induction in European formal logic. The NS classification of inference is then presented and explained, based on a preliminary definition in Chap. 1 of the NS and its commentary in the NB. It is shown how the examples of inference can be formalized using the schema of the deductive syllogism. A set of objections to this conception of inference canvassed in Chap. 2 of the NS and its commentary in the NB is then presented, and this motivates the presentation of an abductive conception of early Nyāya logic, based on an important paper by Claus Oetke (1996), in the following section. The possibilities for modelling the inference procedurally using a dynamic schema are demonstrated diagrammatically. A set of responses follows this set of objections and this is discussed next. Consideration of the manner of response motivates a critique of the abduction conception of early Nyāya logic, based on a response to Claus Oetke’s paper by John Taber (2004). Finally, the discussion returns to the procedural schema to illustrate how the examples of the NS and NB demonstrate a procedural approach congruous with the pragmatic maxim of Peirce and modern scientific methodology.

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Philosophical Pragmatism Pragmatism as a philosophical movement emerged in the intellectual climate of the post-civil war era in the United States, when there was a sudden disenchantment with previously held intellectual certainties and absolute principles. Thinkers from varied intellectual backgrounds, including Charles Sanders Peirce, William James, and John Dewey, developed classical pragmatism as an approach applicable to a broad range of intellectual disciplines. Pragmatism is an important tradition in American and British philosophy, which continues to have impact through the work of recent neo-pragmatist thinkers including Richard Rorty and Hilary Putnam. Philosophical pragmatism not only has important ramifications in fields of philosophy such as logic, epistemology, and ethics but has also influenced the development of scientific method and continues to made a broader impact by informing theories of scientific method, law, administration, urbanism, and more. Classical pragmatism has also informed the pragmatic theories of communicative acts of the ordinary language philosophers including John Searle, John Austin, and Paul Grice, and the work of artificial intelligence theorists seeking to model the pragmatic dimensions of human language and behavior in order to better simulate how humans think and reason. Pragmatism was presented most succinctly by Peirce in the form of the pragmatist maxim. Various formulations exist, with one canonical formulation as follows: [A] conception can have no logical effect or import differing from that of a second conception except so far as, taken in connection with other conceptions and intentions, it might conceivably modify our practical conduct differently from that second conception (Peirce 1997, 249)

For Peirce, pragmatism consists in a practical method to guide human inquiry including scientific discovery and philosophical inquiry. The essence of the method, given by the pragmatist maxim, consists in clarifying the meaning of concepts and hypotheses with reference to their conceivable practical effects alone. The formulation of a new hypothesis with some conceivable differentiation according to practical effect may initiate a process of inquiry, whereas hypotheses and concepts with no such conceivable differentiation may not. One major philosophical target of Peirce’s pragmatism is the Cartesian quest for infallible foundations for our beliefs and the challenge of radical skepticism addressed by it. Pragmatism does not deny the existence of a reality external to the process of inquiry but rather denies that radical philosophical doubts about the existence of such a reality are genuine, properly motivated doubts. Rather, according to Peirce, “the sole object of inquiry is the settlement of opinion” (Peirce 1877). This means that inquiry must be properly motivated by the psychological arising of doubt which stimulates the mind to struggle to reach a state of settled belief and the psychological abatement of doubt. The early pragmatists characterized pragmatism as a return to commonsense thinking, which is fallibilist rather than Cartesian. As William James explains:

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Common sense appears thus as a perfectly definite stage in our understanding of things, a stage that satisfies in an extraordinarily successful way the purposes for which we think. (James 1907a, 69)

In this way, philosophical pragmatism seeks to bring the empiricist and experimentalist character of scientific method to philosophy and other domains of inquiry. All our activities of inquiry must be based on of inquiry on the methodological principles within our ordinary psychological capacities as human inquirers. Philosophical activity is thus constrained by our ordinary processes of inquiry such as belief formation, belief suspension, and belief revision. These processes constitute a strategy to improve our epistemic standing between options that can be differentiated on the basis of attainable evidence. Philosophical concepts and claims should be evaluated with reference to real differences in practical consequences. In this way, the pragmatist approach promises to eliminate many of our most intractable philosophical problems from the scope of meaningful philosophical inquiry. As James explains: A pragmatist . . . turns away from abstraction and insufficiency, from verbal solutions, from bad a priori reasons, from fixed principles, closed systems, and pretended absolutes and origins. . . . That means the empiricist temper regnant . . . as against dogma, artificiality and the pretence of finality in truth. (James 1907b)

Peirce introduced the term “abduction” to describe a type of inference which is guided by the logic of the pragmatic maxim. Abduction serves as a way of generating a new hypothesis, forming that part of scientific methodology concerned with the context of discovery and the generation of new hypotheses. The pragmatism maxim provides a constraint on this creative act of hypothesis generation because it “gives a rule to abduction and so puts a limit upon admissible hypotheses” (Peirce 1997, 249). This rule is the “rule as to the admissibility of hypotheses to rank as hypotheses, that is to say, as explanations of phenomena held as hopeful suggestions” (Peirce 1997, 249). As Fann explains (Fann 1970), the term “uberty” is used by Peirce to denote the “value in productiveness” of adopting a hypothesis. That is, uberty refers to something like fruitfulness in generating new ideas or new content. Uberty contrasts with security, which is the “approach to certainty” made by the hypothesis. Fann explains that “from deduction to induction and to abduction the security decreases greatly, while the uberty increases greatly” (Fann 1970, 8). Building on an evolving set of ideas found in Peirce’s writings, abduction has subsequently come to be understood as Inference to the Best Explanation. It is the formal structure of the inference which reaches the conclusion which would best explain the evidence provided by the premises. Abduction applies within both the context of discovery and the context of justification, and it may also refer to a token act of following the process of abduction and to the particular conclusion reached by it. In this sense, abduction constitutes a form of defeasible reasoning, in which the conclusion reached is warranted as being the most probable explanation of the facts, yet may be defeated by future evidence, so is not deductively valid. At the

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same time, given that induction also seeks to provide a generic explanation of observed facts, there is also a certain propinquity between induction and abduction.

Deduction, Induction, and Abduction Deductive logic, which has been a dominant trend in the history of European formal logic, requires that the premises strictly entail the conclusion, such that the conclusion must be true if the premises are true. That is, they provide formulations of absolute infallibility. Deductive logic may be formally represented using the categorical syllogism such as in the following example of the Barbara type taken from William of Ockham (1974): Major premise: Omnis homo est. animal. Minor premise: Sortes est. homo. Conclusion: Igitur Sortes est. animal. It can be seen that, in its form, the deductive schema constitutes an explicative form of inference, which simply schematizes the formal relations among a closed set of propositions, such as the set of our existing beliefs. That is to say, if the premises are already known, such as “Every man is an animal” and “Socrates is an animal” in the above example, then the conclusion of the reasoning, “Socrates is an animal” should already be known by a rational epistemic agent. By contrast, abduction and induction are ampliative forms of reasoning, which are able to incorporate new hypothesis and new instances. As such, they are suited to represent the way in which the total content of an agent’s beliefs increases through the process of reasoning to a conclusion. Abduction however differs from induction in that it draws on our creative powers of reasoning to generate new ideas as hypothesized causes for observed phenomena, whereas induction merely extrapolates from statistical regularities and extends the reach of our existing ideas to new instances. As Fann explains, in abduction “we pass from the observation of certain facts to the supposition of a general principle to account for the facts . . . abduction is an inference from a body of data to an explaining hypothesis, or from effect to cause” (Fann 1970, 10). By contrast, induction is “an inference from a sample to a whole, or from particulars to a general law” (Fann 1970, 10). Thus abduction explains, whereas induction classifies, and all new ideas enter into the systematic process of inquiry and belief formation through abduction. Induction and abduction are non-monotonic logics, which are sensitive to new information in such a way that the validity of the inference may be vitiated under certain conditions. Generally, certain background assumptions are made as frame conditions. In an example given by Taber, “[w]hen I get in my car in the morning and turn the ignition key, I expect the car to start” (Taber 2004, 144). Here, the validity of

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the inference rests on the implicit assumptions that the engine is working, the battery is present, etc. As Kowalski explains, deduction is connected with forward reasoning, which “leads from known facts to inescapable conclusions,” whereas abduction is connected with backward reasoning, which is used to find explanations based on existing beliefs or new hypothetical beliefs.

Nyāya Inference: Three Classes The works of early Nyāya logic define and discuss the formal structures, characteristic features, and practical methods of correct and incorrect reasoning processes. The Sanskrit term “anumāna” is used to describe the formal structure or process of such reasoning schemas and as such is considered to be equivalent to the term “inference” in European formal logic. Anumāna is theorized in various early Indian texts, including the Nyāya Sūtra (NS) and its commentary, the Nyāya Bhāṣya (NB). In relation to these texts, Oetke conjectures that “the notion of inference as embodied in the term anumāna exhibited to a significant degree the quality of ‘topic neutrality’ which is often regarded as a characteristic feature of (formal) logical concepts” (Oetke 1996, 449). The text of the NS is sufficiently terse as to leave much room for ambiguity, and the NB provides further details to provide one plausible and philosophically interesting interpretation which will be discussed here. In the first chapter of the NS, correct inference is given a threefold classification by using three technical terms to name the three classes as follows: So, inference, which is preceded by [perception], is of three types, [viz.] 'pūrvavat', 'śeṣavat' [and] sāmānyato-dṛṣṭaṃ. (NS 1.1.5 at Tailanga 1984, 17; my translation)

The NB provides two alternative sets of explanations for each of these three classes of inference. The first NB reading is as follows, of which the second will be particularly important for the discussion here: A. The first type (pūrvavat) is described as “inferring from a cause to an effect” (Tailanga 1984, 18; my translation) and illustrated by the example, “when rainclouds are advancing there will be rain” (Tailanga 1984, 18; my translation). B. The second type (s´eṣavat) is described as “inferring from an effect to a cause” (Tailanga 1984, 18; my translation) and illustrated by the example, “when the current of a river is seen to be faster and fuller compared to the previous [level of] water, it is inferred that there has been rain” (Tailanga 1984, 18; my translation). C. The third type (sāmānyato-dṛṣṭa) is described as “seeing in one place what was in another place before it moved” (Tailanga 1984, 18; my translation) and illustrated by the example “there is movement of the sun even though [such movement] is imperceptible” (Tailanga 1984, 18; my translation).

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A second set of interpretations of the same three classes is also provided by Vātsyāyana in the NB. These include the well-known example of inferring fire from smoke which seems to have greater importance as a paradigm example in some other works of Indian logic. However, this alternative line of interpretation is not taken up when the topic is returned to in the objections and replies of Chap. 2; hence it will not be discussed here. The inference being made in cases A and B is likely to be of the approach of the Indian monsoon rains, which are of great significance in Indian literature and life, and the lead-up and aftermath of which involve rather more certain indicators than do light, scattered showers. As Taber explains, we may suppose that case A refers to “the massive buildup of clouds across the horizon that in India signals the onset of the monsoon. When you see that massive wall of black, you know it is going to rain” (Taber 2004, 152). Indeed, the monsoon season is known just as well for its naturalistic concomitants as for the actual rain, as depicted in the verse at the start of this chapter. Formally, it is possible to model these inference types by provisionally adopting the closed deductive schema of the Aristotelian syllogism, wherein the example states the major premise of the inference. For example B above, this would be as follows: Major premise: When the river current is faster and fuller than normal, there has previously been rain. Minor premise: The river current is faster and fuller than normal. Conclusion: Therefore, there has previously been rain.

Nyāya Inference: Three Objections In substantive terms, deductive logic requires a relation of strict entailment between premises and conclusion to preserve the validity of the inference under all conditions. That this may not be the case is suggested by a set of objections and responses provided in Chap. 2 of the NS and the NB commentary on those. The set of objections canvassed in the NS simply mentions three examples where an inference can lead to a wrong conclusion, in the following statement: Inference is a non-pramāṇa because there is deviation due to obstruction, damage or similarity. (NS 2.1.37 at Tailanga 1984, 86; my translation)

Again, the Nyāya Bhāṣya commentary explains these cases more fully. The three objections in the above line will now be presented before addressing the question of what type of inference is theorized in the NS and NB. The first objection, “obstruction,” relates to inference type B above, “inferring from effect to cause” (“śeṣavat”), and an alternative explanation for the fullness of the river is suggested. The Nyāya Bhāṣya states:

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a river is apprehended to be full [actually] due to an obstruction [downstream] and then [there is] an erroneous inference that it has rained. (Tailanga 1984, 86; my translation)

The second objection again relates to inference type B above, “inferring from effect to cause” (“śeṣavat”), but a different rain-related example is used. The Nyāya Bhāṣya states: movement of ants [with their] eggs is due to damage to their nest and then there will be an erroneous inference [that it will rain]. (Tailanga 1984, 86; my translation)

This example is based on the fact that ants abandon their nests and swarm to higher ground with their eggs in advance of seasonal rains, as described poetically in the verse at the start of this chapter (cf. also Chhaganbhai 1992). This engages with an alternative predictor of rains to that given in the first chapter, which was the approach of rain clouds. The third objection is that “a person imitates the cry of a peacock [and] there is an erroneous inference” (Tailanga 1984, 86; my translation). Although the presence of a peacock is one thing that will be inferred, the example again seems to relate to rain, based on the fact that peacocks cry out when rain is imminent. Indeed, the peacock’s cry is invoked as an indicator of rain in numerous works of Sanskrit kāvya poetry, such as the Meghadūta, where the rain cloud will be “met by the cries of peacocks as a form of welcome” (Kale 2002a, 44, verse 23; my translation), and the Ṛtusaṃhāra, where the rain cloud’s approach will cause “the pleasing cries of a muster of peacocks beginning to dance” (Kale 2002b, 17, verse 2.6; my translation). This set of objections is purely a negative one, with no attempt to explicitly formulate an alternative position on the question of inference. Nevertheless, the NB discussion is helpful in drawing out the meaning of the objection and indicates what the ramifications of the objection would be. The objection in each case depicts an alternative explanatory hypothesis of the observed evidence, and its implication is to provide an alternative conclusion in terms of the formal deductive schema that was provisionally adopted for the original example A inference above. The substance of the first objection above can also be provisionally represented in terms of deductive reasoning using the same syllogistic schema as follows: Major premise: When the river current is faster and fuller than normal, there is an obstruction downstream. Minor premise: The river current is faster and fuller than normal. Conclusion: Therefore, there is an obstruction downstream. In terms of deductive reasoning, the premises have been shown to additionally lead to an alternative conclusion, and the argument is thus strictly inconclusive. As such, the original inference is alleged to commit the logical fallacy of affirming the consequent as follows: Major premise: When there is an obstruction downstream, the river current is faster and fuller than normal.

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Minor premise: The river current is faster and fuller than normal. Conclusion (invalid): Therefore, there is an obstruction downstream. Rather than question the classification schema of NS 1.1.5 per se, then, the objection seems to have a more radical skeptical character in that it attempts to undermine the very possibility of undertaking a process of reasoning that could legitimately be termed “inference.” At the same time, the exact character and motivations of the objector remain somewhat shadowy. The next section will shift away from this syllogistic framework to present the proposed interpretation of Claus Oetke of the reasoning structure of these examples as a form of non-monotonic logic.

Nyāya Inference: An Abductive Conception In an important study of the development of Indian logic from the ancient period of the NS and NB through to the work of Dharmakīrti, Claus Oetke argues that there was a transition from non-monotonic to monotonic reasoning. In his thoughtful response to this paper, John Taber questions this thesis, proposing an alternative interpretation of the primary texts that Indian logic was guided by an ideal of deductive reasoning from the beginning. This section will present an abductive interpretation of the NS and NB discussion set out above in line with Oetke’s non-monotonic view of this early phase of Indian logic. A later section will present Taber’s critique and alternative reading. From the fact that the objections canvassed are such as to vitiate the putative inferences originally proposed, Oetke rightly surmises that “even quite early [in Indian history] the problem of fallibility seems to have been recognized as something which potentially endangers inference in general” (Oetke 1996, 450). More speculatively, Oetke ventures that this objection “suggests that at some earlier date inferences were accepted even if they did not fulfill the demands embodied in the specifications” (Oetke 1996, 451). That is, the examples presented in NS 1.1.5 and the NB commentary, which were discussed in the previous section, are considered to be genuine inferences despite not being deductively valid. However, Taber expresses some doubt about this claim (Taber 2004, 150–154). Taber highlights Vātsyāyana’s remark in the NB that “[i]f the arguer fails to make the reason specific enough, so that it strongly supports the conclusion, then the fault lies with him, not with the anumāna” (Taber 2004, 151) which would discourage this fallibilist reading of the nature of inference per se. Rather than the inference being fallible, it would be the reasoner who is fallible in applying the inference. Even if Oetke’s proposal is not fully endorsed, it is clear that there is scope to identify an ampliative dimension to the examples described above, whereby the content of our beliefs is increased by applying the reasoning to generate a new belief. Thus Oetke’s observation that example C above is among those that “can also be viewed as inferences aiming at establishing explanatory hypotheses of well known facts” indicates an abductive reading of the inference types, whereby a novel hypothesis is generated based on the empirical facts and the reasoning expands the

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contents of our beliefs. Indeed, referencing Oetke’s paper, Stephen Phillips explains that several of Vātsyāyana’s examples “are abductive in character, to use the term popularized by C. S. Peirce, informal reasoning to the best explanation . . . But instances of deductive, extrapolative, and sometimes properly inductive reasoning on topics of everyday life as well as philosophy are given by the two early Nyāya philosophers [Gautama and Vātsyāyana]” (Phillips 2012, 51). According to Oetke’s reading, then, this objection demonstrates an awareness of the context-sensitivity of inferences, whereby inference requires “the hypostatization of the fulfillment of normality conditions” (Oetke 1996, 451) such that “it is presupposed that things in the relevant case behave as they ‘normally’ do” (Oetke 1996, 451). We should thus “consider fulfillment of normality conditions as an essential ingredient of inferences in the earlier stages of so-called ‘Indian Logic’” (Oetke 1996, 451). As we have seen, context-sensitivity is a feature of non-monotonic logic, which proposes a defeasible hypothesis consistent with the evidence. As mentioned above, abduction involves backward reasoning, possibly to multiple conflicting hypotheses, each of which could sufficiently explain the available evidence. In an illuminating discussion of abduction (Kowalski 2011, 134–143), Robert Kowalski provides a schema to represent this backward reasoning, using a similarly rain-focused example to the NS and NB. Kowalski explains that “[t]he time-worn example of abduction in Artificial Intelligence is to explain the observation that the grass is wet when you get up one morning. Of course, there are many possible explanations, but in this part of the world the most likely alternatives are either that it rained or that the sprinkler was on. The easiest way to find these explanations is by reasoning backwards from the observation . . .” (Kowalski 2011, 136–137). Kowalski provides the following depiction (Kowalski 2011, 138):

Whereas the syllogism is better suited to the forward reasoning from a single explanatory hypothesis, the backward reasoning structure depicted by Kowalski provides a formal structure which represents the possibility of multiple rival explanatory hypotheses. The objection presented in the previous section portrayed precisely this situation, and as such, the schema of Kowalski seems to better capture the logic of the reasoning process presented here. Adopting this schema from Kowalski, then, the basis for the first objection canvassed by the NS can be depicted as follows:

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The second objection above was that “movement of ants [with their] eggs is due to damage to their nest and then there will be an erroneous inference [that it will rain]” (Tailanga 1984, 86; my translation). Using Kowalski’s system, the basis for this second objection can be depicted as follows:

The third objection above was that “a person imitates the cry of a peacock [and] there is an erroneous inference” (Tailanga 1984, 86; my translation). Using Kowalski’s system, the basis for this third objection can be depicted as follows:

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The strength of Kowalski’s diagrammatic method is that it allows multiple rival explanatory hypotheses and multiple deductive conclusions to be made explicit and thus better represents the procedural dimension to the process of inquiry, both in scientific method and in everyday reasoning. Notwithstanding this, in the next two sections, by examining the manner of response to the objections and the explanation of Vātsyāyana in the NB in particular, the possibility of a deductive reading of inference in the NS will be reinstated.

Nyāya Inference: Three Responses To recap, the three objections in the NS were as follows: Inference is a non-pramāṇa because there is deviation due to obstruction, damage or similarity. (NS 2.1.37 at Tailanga 1984, 86; my translation)

The first two of these objections related to inferring that it has rained as per the inference type B in the original NS classification, by providing two alternative hypotheses, and the last objection refer to inferring the presence of a peacock and hence possibly to the approach of rain. To these three objections, the NS now provides a set of three replies as follows: No, because it is a different case when there is [full flow in] a single part [of a river], fear [of the ants when their nest is damaged, or] similarity [of a human cry to that of a peacock]. (NS 2.1.38 at Tailanga 1984, 516; my translation)

This set of replies aims to make a distinction between the evidence on the basis of which the legitimate inference of rain is made and the superficially similar evidence which would properly lead to an alternative conclusion. So, in the first case, obstruction causes an observably different type of surge phenomenon in the river to rainwater. Specifically, Vātsyāyana explains, a person “infers that it has rained because of the fullness of the river, perceiving that the rainwater is indeed distinct from the earlier water, and that the speed of the current is carrying along much foam, fruits, leaves, bits of wood etc. [and] not merely due to the increase in water [level]” (Tailanga 1984, 516; my translation). We may also recall the description of the rain-swelled rivers in the Ṛtusaṃhāra, when, “downing the trees on the river banks around, with muddied waters of increased velocity, the rivers rush towards the sea” (Kale 1992, 17, verse 2.7; my translation). In the second case, the approach of the rains causes a different type of behavior in ants to that when their nest is disrupted. Specifically, Vātsyāyana explains, “it is inferred that it will rain when the majority of ants are moving with their eggs and not just a few” (Tailanga 1984, 516; my translation). Finally, the cry of a human imitator is held to be similar yet distinct from that of a peacock. Vātsyāyana supports this seemingly rather tenuous claim by raising the

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example of “snakes etc. which do apprehend the distinct cry of the peacock” (Tailanga 1984, 516; my translation). This supposition reflects the motif in Sanskrit literature of the peacock killing the snake, to which the Ṛtusaṃhāra alludes when it depicts the hot season as making predators too exhausted to chase their prey, so that even the snakes are able to shelter from the sun under the plumage of peacocks (see Kale 2002b, 8, verse 1.16). Consequently, Vātsyāyana explains, the situations described in the objections “are not [cases of] inference going wrong. Rather, they are not inference at all but the mere appearance of inference. What is not accurately distinguished cannot be a basis for inference” (Tailanga 1984, 516; my translation). Rather, “the mistake is [a feature] of the inferring subject who, based on perceiving an indistinct object, wants to apprehend an object [properly to be] inferred on the basis of accurately distinguishing [another] object [which is the basis for inference], [and is] not [a feature] of inference [as such]” (Tailanga 1984, 516; my translation). Vātsyāyana’s contention here would seem to recommend that, rather than rejecting the syllogistic schema, that schema be emended so that the inference of example A is stated as follows: Major premise: When the river current changes in a specific manner, it has rained. Minor premise: The river current has changed in that specific manner. Conclusion: Therefore, it has rained. Such a reading of early Nyāya logic is endorsed by John Taber in his critique of Oetke’s paper, and Taber’s discussion will inform the discussion of the next section.

Nyāya Inference: A Deductive Conception According to Vātsyāyana, it is the reasoner who is fallible in his ability to correctly apply an inference, and it is not the inference itself that is fallible. Specifically, the reasoner is prone to the error of not making one premise of his inference sufficiently specific, viz., the premise termed the hetu, or reason, in Indian logic corresponding to the major premise in the categorical syllogism which would represent a deductive schema. As mentioned above, Taber highlights this argument of Vātsyāyana to question Oetke’s reading of early Nyāya inference as fallible. As Taber explains, the NS requires that the hetu “must be formulated in an appropriately specific way: it is not just from ants carrying their eggs in general that we infer that it will rain, but from ants carrying their eggs in the peculiar way they do just before it rains” (Taber 2004, 160). Because deductive reasoning may also be liable to revision where the original premises can be shown to be flawed, Taber explains that sensitivity to new information per se cannot be taken to distinguish the non-monotonic logic of abductive reasoning from the monotonic logic of deductive reasoning. Rather, “monotonic reasoning is ‘sensitive to new information,’ too. Specifically, it is sensitive to new information that calls the truth of its premises into question” (Taber 2004, 146).

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As such, “[t]he criterion of being ‘sensitive to new information’ identifies a piece of reasoning as nonmonotonic only if it is vitiated by new information without any of its original premises being called into question” (Taber 2004, 148). Given that the type of sensitivity to new information involved here, according to Vātsyāyana, is indeed such as to remedy the hetu, Taber instead characterizes the logic of the NS and NB as “not strictly conclusive; but . . . strong arguments that approximate an ideal of reliability” (Taber 2004, 151). That is, “from the very beginning something like monotonic, that is, deductively valid, reasoning was the ideal or norm, but the conception of that ideal was continually refined” (Taber 2004, 144). Indeed, Taber posits that “[t]he history of Indian logic . . . represents a series of proposals regarding the criteria for such a hetu – the specific conditions that must be satisfied in order for a hetu to be conclusive” (Taber 2004, 160). Thus Taber’s first critique of Oetke’s reading of the early Nyāya works as involving non-monotonic logic is based on the sensitivity to new information of the premises of monotonic logic. Aside from this, Taber additionally critiques Oetke’s claim by noting that it is possible to assess inference according to its degree of conclusiveness, finding that, on this measure, the examples of reasoning in the early Nyāya works provide “quite strong arguments which beg to be distinguished from, not assimilated to, the provisional, rough-and-ready type of reasoning we typically think of as ‘nonmonotonic’” (Taber 2004, 151). Taber argues that the “normal conditions” or “background conditions” presupposed by the examples are “such that one has every reason to expect that they are fulfilled . . . it seems, overall, quite unlikely that the reason could be true and the conclusion false” (Taber 2004, 152). Although, being arguments about empirical facts, they cannot be strictly deductive arguments, nevertheless they represent “an ideal that comes close to that of deductive validity” (Taber 2004, 150).

Nyāya Inference: A Pragmatic Dimension The discussion so far has presented two alternative ways of understanding the character of early Nyāya logic, according to an abductive conception and a deductive conception, by drawing on the work of Claus Oetke and John Taber. Although the deductive conception seemed to cohere better with the explicit remarks of Vātsyāyana in the NB, the abduction schematization was able to schematize the procedural logic of rational inquiry, wherein rival hypotheses could be formulated and examined within the same process of inquiry. Although there is more to the debate between Oetke and Taber, both in terms of their understanding of the later phases of Indian logic and also due to subsequent remarks from Oetke in a later paper (Oetke 2009), it is beyond the scope of this chapter to discuss that debate further. Instead, setting aside the question of whether it is the reasoner or the inference that is fallible, this section will now examine the way in which the discussion of reasoning in the text extracts presented above reflects the pragmatic character of scientific inquiry. Indeed, this connection could indicate an alternative angle from which to investigate the significance of abductive, deductive, and

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inductive facets of early Nyāya logic and hence the issue of non-monotonicity in Indian logic. As described above, abductive reasoning is closely connected with the process of scientific inquiry, forming that part of scientific methodology concerned with the context of discovery and the generation of new hypotheses. For Peirce, scientific method consists in the coordinated use of abduction, deduction, and induction. Hypotheses are generated through a creative process of abduction, their implications are deduced, and experimentally tested in an inductive extrapolation to new cases (cf. Burch 2017). By iterating this process, the optimal hypothesis may be identified, and it is the process of Inference to the Best Explanation which has more recently come to be termed “abduction.” In either conception of abduction, it is the characteristic of being open to new hypotheses in virtue of which abduction supports scientific method, reflecting the way in which our beliefs are fallible and open to revision. Whereas the deductive syllogism constitutes a closed schema, reasoning forward from a fixed set of predetermined premises to a single conclusion, abduction involves an open schema, into which new explanatory hypotheses can be added through backward reasoning and new conclusions can be added through forward reasoning. As Kowalski explains, “[a]bduction is only possible for an agent who has an open mind and is willing to entertain alternative hypotheses . . . Generating abductive hypotheses and deciding between them includes the classic case in which Sherlock Holmes solves a crime by first identifying all the hypothetical suspects and then eliminating them one by one, until only one suspect remains” (Kowalski 2011, 135–136). Similarly, we can depict the above discussion of example A of the NS and NB through a process of forward and backward reasoning as follows:

The advantage of this schema is that it makes clear the procedural logic of inquiry, whereby change of belief must be motivated by both backward reasoning to explanatory hypotheses and forward reasoning to deduce the implications of each such hypothesis. In terms of artificial intelligence theory, this diagram provides a procedural representation of knowledge, in which the change in the truth-value assigned is

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explicitly represented in a way that was not possible within the deductive schema. Further, it represents abstractly the position of the situated reasoning agent in a concrete context of observation and engaged response, in which observable evidence constrains the agent’s options for belief adoption, belief suspension, and belief revision. This constraint on rational belief mirrors the pragmatism maxim of Peirce, which “gives a rule to abduction and so puts a limit upon admissible hypotheses” (Peirce 1997, 249). This rule is the “rule as to the admissibility of hypotheses to rank as hypotheses, that is to say, as explanations of phenomena held as hopeful suggestions” (Peirce 1997, 249). That is to say, as illustrated in this diagram, to be legitimate, explanatory hypotheses must be those motivated by observed evidence through backward reasoning, and their implications must be deduced and validated against further observations. Sundar Sarukkai similarly explains how a pragmatic notion of doubt is found in both Peirce and early Nyāya, whereby “the kind of universal doubt of Descartes is a psychological impossibility” (Sarukkai 2008, 32) due to an “intrinsic relation between action, conduct and doubt” (Sarukkai 2008, 32) such that “doubt is the beginning of inquiry. The purpose of inquiry is to resolve a doubt and reach a state of certainty” (Sarukkai 2008, 10). Further, both Peirce and the Nyāya logicians support “fallibilism, the view that no beliefs are known for certain and that all our beliefs are open to change, however much they may look like truth at a particular time” (Sarukkai 2008, 32).

Conclusion Human rationality is different in interesting and important ways from the reasoning processes formalized in deductive inference and implemented in many reasoning systems of artificial intelligence. By drawing on material from early Nyāya logic in comparative perspective, it may be possible to enrich our theoretical conceptions and models of human rationality. The relationship of the Indian theories to their counterparts in classical and modern European formal logic is a matter which has been explored in a range of ways and to a range of conclusions in the academic literature. The conceptions of abduction, induction, and deduction, both as formally theorized and as manifested in actual practice in a range of disciplines and contexts, have a role to play in explicating the nature of anumāna in the early Nyāya texts. In this chapter, the significance of the paradigm examples of inference has been explicated with reference to both abductive, non-monotonic logic and deductive, monotonic logic. As described above, Taber suggests that fallibility be ascribed to the agent in applying inference rather than the inference per se and accordingly posits that inference in the early Nyāya texts is guided by an ideal of deductive validity. Incorrect inference results from failure to make the premises of the inference sufficiently specific. Taber’s argument gains support from the explicit remarks of Vātsyāyana in the NB. By contrast, Oetke presents a non-monotonic reading of inference, whereby the inference itself is fallible, as its validity is dependent on

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certain frame conditions. Without attempting to adjudicate this debate, it was shown that the syllogistic schema of deductive, monotonic reasoning has limitations in presenting the procedural aspect of the reasoning process in these examples. An abductive-deductive schema which shows how explanatory hypotheses are abduced and their implications are deduced is better able to depict the logic of the inquiry procedure and thus to highlight the pragmatic dimension of inquiry in early Nyāya logic. Interestingly, such a reading demonstrates a continuity between the vāda-type arguments discussed in the previous chapter and the process of inquiry as represented in these passages from the NS and NB. Indian theories of proof and inference are the subject of complex and highly abstract discussions in the classical Indian literature. As such, no firm consensus has been reached, either in this chapter or in academic scholarship generally, and there is significant potential for future research. As Taber notes, “the data are extensive and complex and admit of a variety of interpretations . . . Perhaps in the end there will be no conclusive considerations that will allow us to decide between them” (Taber 2004, 162).

Definitions of Key Terms 1. Anumāna – one of the four means of knowing in Nyāya philosophy, generally equated with logical inference and having thematic connections with deductive, abductive, and inductive inference. Anumāna is classified into three types in the Nyāya Sūtra. 2. Abduction – a type of inference which involves the formulation of an explanatory hypothesis. Originally characterized by Peirce as a creative act of conjecturing a new hypothesis to explain new evidence, it has come to be characterized as Inference to the Best Explanation. Abduction is a form of non-monotonic reasoning because it is defeasible. That is, although the hypothesis formulated is consistent with available evidence and explanatorily powerful, it is not deductively valid and may be defeated by future evidence. Abduction plays an important role in scientific method. 3. Deduction – a type of inference which reasons to a conclusion strictly entailed by the premises, such that if the premises are true, the conclusion is necessarily true also. It is represented by the form of the Aristotelian syllogism. 4. Hetu – the reason as one of the five components in the process of inference defined in the Nyāya Sūtra. It may be compared with the major premise in the Aristotelian syllogism of deductive inference. 5. Monotonic logic – any formal logic possessing the following property known as monotony or dilution: “If X ‘ a, then X, Y ‘ a (for any sets of propositions, X and Y).” The truth value of the conclusion does not change with the addition of new premises. Deductive inference is monotonic. 6. Non-monotonic logic – any formal logic which represents defeasible inference, such as that of commonsense reasoning or abductive inference, where certain

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10. 11.

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background conditions are implicitly assumed. The conclusion is not logically entailed by the premises. Nyāya Bhāṣya – a fifth century commentary on the Nyāya Sūtra by Vātsyāyana Pakṣilasvāmin. Nyāya Sūtra – the foundational text of the Nyāya school, composed and redacted over a period of time up to second century and attributed to Akṣapāda Gautama. Pragmatism – a philosophical movement which began in the United States of America with thinkers such as Charles Sanders Peirce, William James, John Dewey, and Oliver Wendell Holmes. It is connected with logical and philosophical principles such as defeasibility, fallibility, and the role of abduction in theoretical and practical reasoning. More generally, pragmatism also refers to principles of reasoning congruous with the approach of these thinkers, applied in a variety of disciplines and contexts. Pragmatic reasoning is context-dependent and guided by the logic of abduction. Pūrvavat – one of the three types of anumāna (inference) theorized in the works of the Nyāya philosophers: inferring from cause to effect. Śeṣavat – one of the three types of anumāna (inference) theorized in the works of the Nyāya philosophers: inferring from effect to cause.

Summary Points • Inference in modern logic and scientific method comprises abduction, induction, and deduction, of which abduction, a form of non-monotonic reasoning, is of particular relevance in the philosophical pragmatism of Charles Peirce and American pragmatism. • Inference in the two texts of early Nyāya logic in India, the Nyāya Sūtra and the Nyāya Bhāṣya, is categorized into three forms, of which one form, pūrvavat, reasoning from effect to cause, is discussed more extensively in a set of objections and replies. • Vātsyāyana’s comments in the Nyāya Bhāṣya indicate that the theoretical definition of inference should aspire to reflect deductive validity, which should be achieved by making the premises of any putative token inference sufficiently specific in order to rule out all possibility of error. • At the same time, consideration of the examples of inference as presented in the Nyāya Sūtra and the Nyāya Bhāṣya indicates affinities also with the non-monotonic form of reasoning used in abductive inference, whereby new explanatory hypotheses can be generated through abduction, their implications in terms of observable evidence can be worked out through deduction, and compared against the evidence from further observation. • As such, the comparison of inference as theorized in modern logic and the classical Indian logic of the early Nyāya authors can contribute to enriching the paradigms of reasoning in use in artificial intelligence and other reasoning system disciplines.

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References Burch, R. 2017. Charles Sanders Peirce. In The Stanford encyclopedia of philosophy (Fall 2017 edition), ed. Edward N. Zalta. https://plato.stanford.edu/archives/fall2017/entries/peirce/. Chhaganbhai, C.K. 1992. Forecasting rain ants. Honey Bee 3: 16. Fann, K.T. 1970. Peirce’s theory of abduction. The Hague: Martinus Nijhoff. Ingalls, D.H.H. 2000. Sanskrit poetry from Vidyākara’s Treasury. Cambridge, MA: The Belknap Press of Harvard University Press. James, W. 1907a. Pragmatism, a new name for some old ways of thinking. New York: Longmans, Green and Co. James, W. 1907b. What pragmatism means. New York: Longmans, Green and Co. Kale, M.R. (1992) The Meghadūta of Kālidāsa. Delhi: Motilal Banarsidass. Kale, M.R. 2002a. The Meghadūta of Kālidāsa. Corrected edition. Delhi: Motilal Banarsidass. (Reprint). Kale, M.R. 2002b. The Ṛtusaṃhāra of Kālidāsa. 2nd ed. Delhi: Motilal Banarsidass. (Reprint). Kowalski, R. 2011. Computational logic and human thinking: How to be artificially intelligent. Cambridge, MA: Cambridge University Press. Ockham, W. 1974. Summa Logicae, ed. Boehner, Gal, and Brown. New York: St. Bonaventure. Oetke, C. 1996. Ancient Indian logic as a theory of non-monotonic reasoning. Journal of Indian Philosophy 24: 447–539. Oetke, C. 2009. Some issues of scholarly exegesis (in Indian philosophy). Journal of Indian Philosophy 37: 415–497. Peirce, C. 1877. The fixation of belief. Popular Science Monthly 12: 1–15. Peirce, C. 1997. Pragmatism as a principle and method of right thinking: The 1903 Harvard lectures on pragmatism. Albany: State University of New York Press. Phillips, S. (2012) Epistemology in Classical India: The Knowledge Sources of the Nyaya School, Routledge. Sarukkai, S. 2008. Indian philosophy and philosophy of science. 2nd ed. Centre for Studies in Civilizations, Delhi. Taber, J. 2004. Is Indian logic nonmonotonic? Philosophy East and West 54 (2): 143–170. Tailanga, G.S. 1984. The Nyāyasutras with Vātsyāyana's Bhāsya and extracts from the Nyāyavarttika and the Tātparyatika. 2nd ed. India: Sri Satguru Publications.

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Contents Context and Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Āgama Defined as Prasiddhi: [1] (prasiddhi-lakṣaṇa) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Āgama That Has Established Well-known Usage (Prasiddhi) as Its Characteristic: [2] . . . . . . The Unitary Syntactic Structure of All the Scriptures: The Authoritativeness of All the Scriptures (sarvāgamaprāmāṇya) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Twofold Aspects of Established Well-known Usage (prasiddhi): Formally Constructed (nibaddha) and Not-Formally Constructed (anibaddha) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Source of the Validity of Āgama: The Firm Rooting of Conviction (vimarśanirūḍhi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Āgama as Verbalizing (Śabdana) or Intuitive Reflecting (Pratibhāna) . . . . . . . . . . . . . . . . . . . . . . . . Threefold Contexts of Verbalizing (śabdana) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Āgama as āpti (Testimonial Authenticity) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Different Kinds of the Authentic Being (āptattva) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Testimonial Authenticity (āpti) Morphosized into Established Well-known Usage (prasiddhi) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Epistemological Structure of Āgama . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The epistemological structure of the non-dual Trika Śaiva philosophy mainly represented by Utpaladeva ( fl. c. 925–975 CE) and Abhinavagupta ( fl. c. 975–1025 CE) accepts three pramāṇas, viz., Pratyakṣa, Anumāna, and Āgama. In this tradition, the Āgama-pramāṇa is regarded as the most important for several This article is a chapter in the book titled Kāśmīra Śivādvayavāda meṃ Pramāṇa-Cintana, [in Hindi] by Navjivan Rastogi, L.D. Sanskrit Series: 156, L.D. Institute of Indology, Ahmedabad, 2013, pp. 141–196. The chapter is translated from Hindi into English by Mrinal Kaul. N. Rastogi (*) The Department of Sanskrit and Prakrit Languages, Abhinavagupta Institute of Aesthetics and Shaiva Philosophy, University of Lucknow, Lucknow, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_30

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reasons. This chapter critically maps the structure of Āgama as a pramāṇa that is not only characterized by established well-known usage (prasiddhi) but also has deeper implications embedded in the concepts of verbalizing (śabdana) and intuitive reflecting (pratibhāna). The scriptural validity of Āgama is firmly rooted in conviction (vimarśanirūḍhi) and the concept of testimonial authenticity (āpti) is morphosized into established well-known usage (prasiddhi). At the same time, this chapter brings forth the philosophical creativity with which Abhinavagupta argues for the supreme validity of the Āgama-pramāṇa and pleads that understanding the core of Āgama is the ultimate fruit of the entire epistemic enterprise and brings forth Abhinavagupta’s theory of knowledge as a ‘dynamic theory of knowledge.’ Keywords

Established well-known Usage · Prasiddhi · Abhinavagupta · Pratyabhijñā · Āgama · Pramāṇa · Āpti · Śabdana · Vimarśa Abbreviations

Bhā ĪPK ĪPV ĪPVV KSTS MVV NB NŚAB PV SCM ŚD SK SvT SvTU TĀ TĀV VP VPVṛtti YS YSB

Bhāskarī (Bhāskarakaṇṭha’s commentary on the ĪPV) Īśvarapratyabhijñākārikā Īśvarapratyabhijñāvimarśinī Īśvarapratyabjijñāvivṛttivimarśinī Kashmir Series of Texts and Studies Mālinīvijayavārttika Nyāyabindu Nāṭyaśāstra-abhinavabhāratī Pramāṇavārttika Stavacintāmaṇi Śivadṛṣṭi Spanda-kārikā Svacchandatantra Svacchandatantrodyota Tantrāloka Tantrālokaviveka Vākyapadīya Vākyapadīyavṛtti Yogasūtra Yogasūtrabhāṣya

Context and Background In the traditional hierarchy of pramāṇas, even if Āgama is enumerated as the third pramāṇa, it is nonetheless considered as one of the most important in the non-dual Śaiva tradition of Kashmir that is mainly represented by Utpaladeva ( fl. c. 925–975 CE) and Abhinavagupta ( fl. c. 975–1025 CE). According to Abhinavagupta,

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understanding the core of Āgama is the ultimate fruit of the entire epistemic enterprise (Īśvarapratyabjijñāvivṛttivimarśinī (ĪPVV), Vol. 3, p. 217: āgamatattvaveditvam eva vidyāsthānaphalam|). While making this comment he tells us that Utpaladeva is reminding us of the importance of the structure of pramāṇas. From this, it is concluded that it is Āgama itself that is the ultimate level of epistemic function. Āgama is discussed in three different contexts in the Śaiva tradition – viz., as speech (vāk), as means of valid cognition (pramāṇa) and as Tantra. One can witness a certain level of uniformity or even a certain overlapping in the discussions of all the three contexts mentioned above (For instance, Bhāskarakaṇṭha regards that the special characteristic of Āgama is rooted in the inner verbalization (śabdana) [Bhāskarī (Bhāskarakaṇṭha’s commentary on the ĪPV) (Bhā), vol. 2, p. 90]. This is the same definition that Abhinavagupta uses for Parāvāk that is of the nature of reflective awareness: pratyavamarśaś ca āntarābhilāpātmakaśabdanasvabhāvaḥ | [Īśvarapratyabhijñāvimarśinī (ĪPV) vol. 1 ‘Bhā’, p. 252]. In addition to this, when calling Parāvāk as pratyavamarśātmaka-citi, the emphasis is on the nature of Cit as activity [cetayati ityatra yā citiḥ citikriyā, p. 250]. The same emphasis is noticed in calling the Āgama as the inner activity (antaraṅga vyāpāra) of the consciousness-principle (cidrūpa) [āgamas tu. . ..citsvabhāvasya. . ..antaraṅga eva vyāpāraḥ, ĪPV vol. 2, p. 84]. The reason for this is indicated by the fundamental relationship that both the aspects share with the reflective awareness). The essential nature of the reflective awareness (pratyavamarśātmaka) of consciousness (Citi) is known as the supreme speech (parāvāk) [i. Īśvarapratyabhijñākārikā (ĪPK) 1.5.13ab: citiḥ pratyavamarśātmā parā vāk svarasoditā | ii. ĪPK 1.6.1ab: ahaṃpratyavamarśo yaḥ prakāśātmāpi vāgvapuḥ |]. The homologous usage of the terms like Āgama and Tantra is very common, yet there is a subtle distinction between the two. The Tantras are the main subject matter of Āgama. In other words, the Tantras constitute the content of the Āgama. This is clearly articulated by Abhinavagupta toward the end of the Tantrāloka (TĀ) [TĀ 37.83: sa tannibandhaṃ vidadhe mahārthaṃ yuktyāgamodīritatantratattvam |]. It would also be clear gradually that Tantra is the verbal or written form of Āgama and thus, Āgama and Tantra can be used synonymously only in the secondary sense. For now, our immediate concern is Āgama as a valid means of cognition (pramāṇa). This is how the meaning of the Āgama has been understood in the above-mentioned quotation from the Tantrāloka. It must be added here that the Āgama is better known as the Śabdapramāṇa (verbal testimony) in other schools of Indian philosophy. In the Vedic tradition, the verbal testimony is mainly represented by Śruti and/or the knowledge emanating from the authentic source (āptajñāna), while in the tradition of grammarians and Tantra, it manifests in the form of Āgama. This can also be said to be the historical or cultural nuance of the usage of the word Āgama. The famous thinker Govind Chandra Pandey believes in the two main fundamental engagements of Indian philosophy, viz., Āgamānusārī (that follows the Āgamas) and Nyāyānusārī (that follows the logic and analysis). From this point of view, the philosophical systems like Sāṁkhya, Vedānta, Buddhist, and Pratyabhijñā are simultaneously Āgamānusārī and Nyāyānusārī (See Rastogi 2012, p. 501). In this context, the meaning of Āgama is based on the living tradition that includes Śruti, Āgama, and Buddha-vacana.

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The Śaiva pramāṇavādins discuss three types of fundamental forms of the cognitional activity: ( jñāna-kriyā) or the cognitive function ( jñānātmaka-vyāpāra), i.e., immediation (sākṣātkaraṇa), determination (vikalpana), and verbalization (śabdana). These three can be directly linked to Perception (Pratyakṣa), Inference (Anumāna), and Verbal testimony (Āgama). Perception is direct knowledge. The inference is of the nature of determinate knowledge (vikalpanātmaka) and verbal testimony is of the nature of verbalizing (śabdanātmaka). The question here is that why Utpaladeva and Abhinavagupta do not make a separate mention of the functionality of verbalization (śabdana-vyāpāra) or the verbalization-power (śabdanaśakti) in the Jñānādhikāra of the ĪPKV like they do in the context of experience [anubhava (sākṣātkaraṇa)] and exclusion [apohana (vikalpa)]. Probably the reason for this could be that it is not possible to distinguish the direct experiencing (sākṣātkāra) from verbalizing (śabdana) because of the interpenetrating nature of the luminous consciousness (prakāśa) and reflective awareness (vimarśa). This becomes clear from the characterization of the Power of Knowledge ( jñānaśakti) in the section titled Jñānādhikāra of the ĪPK. This is the reason that both the perceptual valid knowledge (sākṣātkārī-jñāna) and the verbal valid knowledge from the Āgama or verbalizing reflexivity (śabdanātmaka-vimarśa) are understood as unmediated indeterminate knowledge [ĪPK: 1.5.19ab: sākṣātkārakṣaṇe’ apy asti vimarśaḥ katham anyathā | See also Vimarśinī on this verse]. Of the two, the former progresses through the medium of prakāśa and the latter actualizes through vimarśa. We will discuss this later in the appropriate context. The logic of subscribing to Āgama as pramāṇa by the Śaivas certainly resonates with that of the other traditions. The Āgama-pramāṇa is conceived in the cases where the Perception and Inference do not have access, or to comprehend such objects which may be imperceptible, or wherever it is not possible to comprehend the mark/reason etc., or whose expanse is unlimited [ĪPV vol. 2, p. 213]. The Āgama is considered functional in respect of what is beyond direct perception [ĪPVV vol. 3, p. 84: parokṣe ca arthe tasya prāmāṇyam |]. Even though we do observe that the inference also applies to the realm of the objects those are beyond perception (parokṣa), but here the difference is that the mark (hetu/liṅga) can be grasped by the senses. Therefore, what is regarded as absolutely imperceptible, that alone comes under the purview of Āgama. Thus, the following topics fall under the purview of the Āgama-pramāṇa: the decision about the merit (dharma) and demerit (adharma) [Mālinīvijayavārttika (MVV) 1.795cd-796ab: sa eva ca āgamo nāma vṛddhavyavahṛtikramaḥ | tataḥ samagra evāyam dharmādipariniścayaḥ ||], the ascertainment of categories (padārtha) [ĪPV vol. 2, p. 213] ontic-realities and the worlds (tattvas and the bhuvanas) [ĪPVV vol. 3, p. 85: tattvabhuvanādīnāṃ tu iyattāyāṃ nāsti anumānamiti āgama eva tatra śaraṇam |], the number of ontic realities (tattvasaṃkhyā) as well as worlds (bhuvana), heaven (svarga), the effect of mantras, the laying down of efficacy of the initiation (dīkṣā) [ĪPVV vol. 3, p. 179], the formulation of the nature of vidyā and avidyā [MVV 1.806: prasiddhir āgamo loke. . .. | vidyāyām apy avidyāyāṃ pramāṇam iti tatsthitam ||]. But Abhinavagupta does not exhaust the extent of Āgama at the absolute imperceptibility (atyanta-parokṣa) alone, he extends the scope of Āgama to the larger domain of worldly transactions.

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All worldly transaction is permeated through the idea of Āgama. Thus, while perception and inference fall under the category of individual means of knowing, Āgama emerges as the universal means of knowing: it can know and map the territory of the complete universe [punar āgamo viśvaṃ pramātuṃ samarthaḥ ĪPVV, 3, p. 255]. The etymological derivation of the word Āgama is ā þ gam þ ac and it can be divided into two: 1. based on understanding (avagatimūlaka) and 2. based on historical course of emergence (āyātimūlaka). In the first case, the prefix ā has been interpreted in two ways: i. from all sides (ā samantāt) or ii. thoroughly (samyak). What makes something understood from all sides (lit.) or in the complete sense that is named Āgama-pramāṇa (which consists in Śabdana the nature of which is firm conviction, that is, reflective awareness). This is the meaning of ‘samantāt’ (ā samantād arthaṃ gamayati) [MVV 1.806: prasiddhir āgamo loke. . .. | vidyāyām apy avidyāyāṃ pramāṇam iti tat sthitam | (a) ĪPV vol. 2, p. 87: ā samantāt arthaṃ gamayati iti āgamasaṃjñakaṃ pramāṇaṃ | (b) ĪPVV vol. 3, p. 85: dṛḍhavimarśarūpaṃ śabdanam āgamaḥ ā samantād arthaṃ gamayatīti |]. In the same category there is another analysis of compound: ā samyag arthaṃ gamayati meaning that which makes something thoroughly understandable is called Āgama [ĪPVV vol. 3, p. 85: dṛḍhavimarśātmakasamyagāgamarūpaśabdanopayoge prāmāṇyamāsādayan na pakṣapātādivācyatārhaḥ |]. ‘Thoroughness’ (samyaktva) is not explained here. But keeping the context in mind ‘thoroughness’ would mean rendering such a firm grasp of the deployment of the term śabdana in the form of an Āgama with such authenticity as is completely devoid of any bias or preference. The second category is ‘based on historical course of emergence’ (āyātimūlaka): ā gacchati iti āgamaḥ. The immediate denotation of this etymology is signified by Established well-known Usage (prasiddhi). This etymology conceptualises the Āgama in the form of a commonly held previous idea evolving over time (TĀ 35.10ab: prācyā ced āgatā seyaṃ prasiddhiḥ paurvakālikī |). This particular etymology has also been used for conveying the lineal tradition of teaching (upadeśa-pāramparya) [ĪPVV vol. 3, p. 105: mukhānmukhāgataṃ jñānaṃ karṇātkarṇam upāgatam | It is a statement cited by Abhinava conveying his approval]. In the former case, there is the idea of the unbroken continuity of prasiddhi and in the latter, there is the successive stream of traditions in the form of ear-to-ear transmission. Abhinava maintains that this meaning of Āgama is generally acceptable to all the āgamic texts [ĪPVV vol. 3, p. 105: yata evaṃ samasteṣu āgamagrantheṣu paṭhyate |]. Āgama has also been formulated as aiśvarī-vāk (the Speech of the Lord) by using the etymology of the same class – āgataḥ iti āgamaḥ, i.e., Āgama is that truthful speech (satyā vāk) which has originated from the Parameśvara or Īśvara [ĪPVV vol. 1, p. 36: nanu sa āgamaḥ kuta āyātaḥ | āha ‘aiśvarī’ īśvarāt āgatā sā ‘satyā vāk’ |]. Abhinava’s positing of Āgama as a way of knowing (pramāṇa) by fully exploiting the cues from the tradition is one of the extraordinarily original and matchless enterprises of the Indian intellectual tradition. He reduces down the essential definition of Āgama in three ways: prasiddhi (ĪPVV vol. 3, p. 217: avigītā ca prasiddhir āgama eva |), inner intuitive awareness (pratibhāna) (ĪPVV vol. 3, p. 93: pratibhānalakṣaṇā iyaṃ śabdabhāvanākhya āgama eva |), and instruction

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from the authentic source (āptopadeśa) (ĪPVV vol. 3, p. 103: sarva āgama āptopadeśaśabdena [. . ..] saṃgṛhītaḥ |). He then develops them into three types of Āgamas and explains each one of them individually. Following the same scheme, he also postulates another threefold/tri-layered categorization where the nature of Āgama is regarded as (i) of the nature of internal speech, (ii) of the nature of intensely firm reflective awareness, and (iii) of the intrinsic or characteristic functionality of the Lord who is essentially consciousness (or the intrinsic own-form of the consciousness-principle) (ĪPV vol. 2, p. 84: āgamas tu nāmāntaraśabdanarūpo dṛḍhīyas tamavimarśātmā citsvabhāvasya īśvarasya antaraṅga eva vyāpāraḥ |]. It is important to note here that it is the processive nature of Āgama that has been directly stated in words and has also been implied by the employment of the words like ‘vimarśana’ and ‘śabdana’ [In the ĪPVV, Abhinava emends ‘vimarśa’ into ‘vimarśana’: ĪPVV vol. 3, p. 84: śabdanarūpatvaṃ (read śabdanarūpaṃ) vimarśanaṃ yadāntaraṃ citsvabhāvasya antaraṅgaṃ rūpaṃ |]. In reality, it is the Āgama-epistemological application of the ontological notion of reflective awareness in the form of speech [ĪPK 1.5.13: citiḥ pratyavamarśātmā parā vāk svarasoditā |]. The uniqueness of Abhinava’s approach to the subject matter lies in the fact that he begins his discussion with the divine vāk, or the innermost activity of the Lord/ Consciousness and eventually halts at prasiddhi that culminates into day-to-day behavior or the practice of the wise people of yore (vṛddha-vyavahāra). Throughout this whole treatment, a linear uniformity informed by logical consistency is maintained. It is the same prasiddhi that is ultimately reduced into the Āgamapramāṇa. While explaining the word ‘prasiddheḥ’ in the Vivṛti on the ādivākya (opening sentence) of Utpala’s ĪPK, Abhinava leaves no room for doubt that it is prasiddhi itself that has been established as Āgama-pramāṇa at the time of enunciating the definition of pramāṇa (pramāṇa-lakṣaṇa) [ĪPVV vol. 1, pp. 35–36]. Abhinava also informs us that he has received the concept of Established wellknown Usage (prasiddhi) in the form of Āgama from his grandmaster. Utpala discusses this in his Vivṛti on the ĪPK, but he does not do so in the kārikās or the vṛti [ĪPVV vol. 1, p. 35: nahi mūlavṛttigranthayor etad bhaviṣyati |]. Apart from Utpala, another source within Abhinava’s tradition is his guru Śāṃbhunātha who puts forth Established well-known Usage (prasiddhi) as the basis for the mutual synthesis of the scriptures (śāstra-samanvaya or the ekavākyatā of the śāstras) (Abhinava uses the expression ‘śāstra-melana’, (lit.) (commingling of the scriptures) [TĀ 35.44]. Possibly the uniqueness of these two underlying streams is reflected by the respective methodologies of analysis in the two places. In the Pratyabhijñā texts, the nature of Āgama as pramāṇa forms the subject-matter of discussion (vivecya) and in the Tantrāloka the śāstric character of the Established well-known Usage (prasiddhi) does the same. It should be noted that both of them complement each other. Utpala and Abhinava had three types of challenges in front of them. In sequence, one can enumerate them under the headings of (i) internal (svagata), (ii) homogenous (sajātīya), and (iii) heterogenous (vijātīya). Under the internal (svagata), they had to establish that form of Āgama which was their

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philosophical target, i.e., to present one standard model of the Tantric corpus. Even this was threefold: the first was to envision the synthesis of the Tantric corpus or Tantric śāstric literature in the Trika philosophy (tantraikavākyatā or śāstraikyavākyatā) [TĀ 37.26: aśeṣatantrasāraṃ tu vāmadakṣiṇam āśritam | ekatra militaṃ kaulaṃ śrīṣaḍarthakaśāsane ||]. The second was to establish this unity of meaning of all the scriptures right from the scriptures of ordinary people to the Śaiva Āgamic scriptures. Abhinava calls this convergence of the scriptures (sāstra-melana) (also called (synthesis of the śāstras) śāstrasamanvaya) [TĀ 35.30 and TĀ 35.44]. Even this achieves its culmination in the scriptures of the Trika [TĀ 35.31: tasya yat tat paraṃ prāpyaṃ dhāma tat trikaśabditam |]. The third was to bring forth the overarching importance of the Śaivāgamas among all the Āgamas [TĀ 22.8cd-9ab: samastaśāstrakathitavastuvaiviktyadāyinaḥ || śivāgamasya sarvebhyo ‘py āgamebhyo viśiṣṭatā |]. All these three approaches are mutually integral aspects of the same logical process of thinking. This was a challenge because a Śaiva philosopher had to devise such an all-inclusive and all-comprehending Āgama-pramāṇa that was able to accommodate and evenly balance all philosophical viewpoints by reconciling their mutual opposition. It was also able to preserve their individual identity and allowed them to ward off a confrontation with Śaiva ideologies by allowing the Śaivas to attain their goal. To use a contemporary idiom, they had to establish a unitary democratic system that would have catered to the nourishment of all the divergent streams of thought. The second challenge was coming from one’s own camp. The original inspiration for the Āgama-pramāṇa of Śaivas came from Bhartṛhari and his lineage of grammarians. Almost all the formulations of Āgama proposed by the Śaivas are anticipated by Bhartṛhari in some form or the other. But the theories of Bhartṛhari had to face strong opposition at the hands of the Buddhists and the Mīmāṃsakas. The second aim of the Śaivas was to offer a new lease of life to the philosophy of Bhartṛhari by countering the opposite (virodha) arguments of the Buddhists and the Mīmāṃsakas and thereby to offer a creative re-interpretation of Bhartṛhari. It seems plausible to add one more source as an additional dimension to this second challenge. Even if it might not be labelled as something coming from one’s own camp, yet it cannot be called heterogeneous either. This intermediary role is played by Patañjali, the author of the Yogasūtras. The natural extension of the second aim of the Śaivas was the intensification and rigorous problematization of the new paradigm of the Āgama-pramāṇa based on re-interpreting Patañjali’s positing the verbal testimony as prasiddhi and āpti. The second challenge itself contains the seeds of the third challenge. Even if the Buddhists and the Mīmāṃsakas were the co-rivals of Bhartṛhari, yet the metaphysical and logical grounds of the arguments of the former were starkly in conflict with the latter. The successful examination of the challenges thrown to them by the opposite camp was the third challenge. And the third aim was to firmly establish their own view by refuting the Buddhists and the Mīmāṃsakas. Keeping the above in mind we will have a deeper look into the concept of Āgama of the Trika thinkers.

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A¯gama Defined as Prasiddhi: [1] (prasiddhi-laksana) ˙ ˙ We have just seen above that both Utpala and Abhinava reduce down the nature of Āgama into prasiddhi. The ĪPV begins the Āgama discourse with the words ‘āntaraśabdanarūpadṛḍhīyas tam vimarśātmā’ and ends in ‘prasiddhi.’ On the other hand, in the ĪPVV, the treatment of Āgama begins with the words ‘prasiddhi’ and the explanation concludes with the words ‘āntaraśabdanarūpatā.’ As far as the TĀ is concerned, it begins with the equation of ‘prasiddhi’ and ‘vyavahāra’ where one chapter (i.e., Chap. 35) ends with the śāstra-samanvaya (synthesis or unitary meaning of śāstras) and another (i.e., Chap. 37) with the tantra-tattva (essence of Tantra). One should pay careful attention to the fact that the tradition adheres to its own position constantly [Tantrālokaviveka (TĀV) vol. 2, p. 47; TĀV vol. 8, p. 3646; ĪPK vol. 2, p. 83 (Kashmir Series of Texts and Studies (KSTS)); ĪPK vol. 2, p. 83, fn 90; ĪPV vol. 2, p. 89; ĪPVV vol. 1, p. 36; ĪPVV vol. 3, pp. 84–85; MVV 1.805–806; See Torella 2013, p. 5, note 12] quoting the two verses beginning with ‘prasiddhir āgamo loke’ either completely or in part or by just mentioning it. In the TĀV 1.18, Jayaratha quotes both these verses as follows [TĀV vol. 2, p. 47; vol. 8, p. 3646]: prasiddhir āgamo loke yuktimān athavetaraḥ | vidyāyām apy avidyāyāṃ pramāṇam avigānataḥ || prasiddhir avagītā hi satyā vāgaiśvarī matā | tathā yatra yathā siddhaṃ tadgrāhyam aviśaṅkitaiḥ ||

Translation: “In the world, it is prasiddhi itself that is known as Āgama; this can be consistent with logical reasoning or may be different. When uncontradicted, it is pramāṇa toward ascertainment of vidyā and avidyā. This is because the uncontradicted prasiddhi is regarded as the true speech of the Lord himself. Whatever is established wherever [by this speech], it should be grasped exactly like that without being doubtful [about it].” [Utpaladeva in his -vivṛti reads tayā: ‘tayā’ prasiddharūpayā aiśvaryā vācā (ĪPVV vol. 1, p. 36; see also Torella 2013, p. 5). Following him, Bhaskara reads tayā yatra yadā instead of tathā yatra yathā (See Bhā. vol. 2, p. 89) which is also the reading adopted by Jayaratha.]. The nature of the Āgama as a source of knowing is brought into relief by the idea of this prasiddhi itself. Āgama is said to be of the ‘nature of pure prasiddhi’ [ĪPVV vol. 3, p. 89: āgama eva āgacchati svacchaprasiddhirūpaḥ |]. The meaning of pure has been taken from avigīta. Therefore, the defining feature of Āgama is ‘avigītaprasiddhi (one that consists of or is constituted by uncontradicted prasiddhi) [ĪPVV vol. 3, p. 92: āgamam avigītaprasiddhilakṣaṇaṃ |]. The literal meaning of vigāna is viruddha-gāna (incongruous/discordant song) and the contextual meaning is contradictory reflective awareness (viruddha-vimarśa). In other words, the contradictory reflective awareness does not take place [ĪPVV vol. 3, p. 101: ‘vigāne’ iti tadviruddhavimarśodaye |]. This is the crux of non-contradiction (abādhitattva). Despite the distinction as regards space, time, person, etc., the prasiddhi that remains unobstructed (nirbādha) constitutes the ground of the validity of inference in

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accordance with the Principle of Necessity (niyati). Such validly is dependent on the same Principle of Necessity (niyati) and not on relations (sambandha). It is on full ripening of one’s reflexive intelligence (pratibhā) that the uninterrupted functioning of pramāṇa, that consists in ascertainment of ‘thatness’ in respect of all entities [ĪPVV vol. 3, p. 91: sārvatrikavastutathātvaniścaye vyāpārḥ], takes place. This full ripening could be the result of careful examination of practice since birth or through countless previous births or from strength thereof or is based upon the fully matured intelligence of others flowing from the successful accomplishment of causal efficiency. The question of correspondence immediately succeeding causal efficiency has no role here except that it only nourishes the element of practice. It will arise only when there is absence of ascertainment in the nature of causal efficiency because of lack of practice. The chain of repetition (abhyāsa) gives birth to the residual impression alone whose existence is possible solely in identity or unity with the knower [See ĪPVV vol. 3, pp. 91–92.]. For this reason, the secret of non-rising of the contrary reflective awareness lies in prasiddhi’s acting in accord with niyati as part of its very nature [ĪPVV vol. 3, p. 91: prasiddhir abādhitā niyatisvarūpānuvartinī |]. And for this very reason while an object or an entity is known as ābhāsa (manifestation), the niyati is called mahābhāsa (universal manifestation) [ĪPVV vol. 3, p. 92: niyatiśaktis tu mahābhāsasāratayā tatra praveśanīyā |]. This is why niyati and prasiddhi are homologous. To properly understand the challenges from the same camp, we need to revisit Bhartṛhari. Subsisting in the form of residual impressions of the past lives, this Established well-known Usage (prasiddhi) is almost identical with the idea of what Bhartṛhari calls the intuitive awareness (pratibhā). Accompanied with or impregnated by the linguistic potency or verbal creative energy (śabdha-bhāvanā), the intuitive awareness is born out of Āgama. The division of Āgama depends on the proximity or the non-proximity of the intuitive awareness [Vākyapadīya (VP) 2.151: bhāvanānugatād etad āgamād eva jāyate | āsattiviprakarṣābhyām āgamas tu viśiṣyate || See also Torella 2013, p. 8]. This can only be the object of one’s own experience (svasaṃvedana) and it is not at all possible to make it understood to someone else by verbal communication. All behavior, including even the behavior of a new-born baby, what Bhartṛhari calls ‘itikartavyatā’ can be traced back to and explained in terms of linguistic potency (bhāvanā) gained from previous births or the words as such [i. VP 1.121: itikartavyatā loke sarvā śabdavyapāśrayā | yāṃ pūrvāhitasaṃskāro bālo ‘pi pratipadyate || ii. VP 2.146: sākṣāc chabdena janitāṃ bhāvanānugamena vā | itikartavyatāyāṃ tāṃ na kaścid ativartate ||]. So much so that even the wisdom of the sages is also dependent on Āgama [VP 1.30cd: ṛṣīṇām api yajjñānaṃ tad apy āgamapūrvakam ||]. Adding another dimension to this context, Bhartṛhari says that Āgama should be understood as the perennial tradition wherein the worded forms like Śruti and Smṛti are also embedded. Metaphorically speaking all the conduct of the wise and beliefs are also included therein [Vṛtti on the VP 1.41: tathā cāyaṃ śrutismṛtilakṣaṇaḥ sarvaiḥ śiṣṭaiḥ parigṛhīta āgamaḥ |]. Śruti is eternal and authorless, but it is not eternal in the sense Mīmāṃsā understands it. The Mīmāṃsā notion of cyclic creation is not acceptable here. Here the eternality is understood in the following two ways: one, it is eternal and uncreated in the sense unbroken-tradition [VP 1.144: anādim avyavacchinnāṃ

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śrutim āhur akartṛkām | śiṣṭair nibadhyamānā tu na vyavacchidyate smṛtiḥ ||]. The second is that the Vedic sentence occupies the same place as consciousness [Vṛtti on the VP 1.132: vedavākyāni tu caitanyavad apauruṣeyāṇi |], because it brings Āgama to the level of consciousness. The Āgamas other than the Veda are also regarded as composed by humans but in case they are lost or destroyed, the Veda always remain as their potential seed (VP 1.124).] The Āgama is eternal and uninterrupted like the spontaneous and beginningless consciousness. They are not refuted by logical polemic [VP 1.41: caitanyam iva yaś cāyam avicchedena vartate | āgamas tam upāsīno hetuvādair na bādhyate || In comparison to this, we will see that in the Śaiva tradition the words like ‘iva’, ‘vat’, ‘paryanta,’ etc. undergo the process of thorough transformation. What all remains of Āgama is the idea of reflective awareness]. The second point mentioned above is very important because its subtle implication is to raise the Āgama to the level of word-essence, i.e., the word in the form of reflecting awareness. This itself becomes very clear at the beginning of the Brahmakāṇḍa of the VP, alternately also known as the Āgamakāṇḍa. Āgama is the nature of the word. This is to say that the Veda is not only said to be the means toward attaining the Brahman but also that it represents (anukāra) Brahman. To say it is representational is metaphorical. The eternal speech that is devoid of internal hierarchy and is subtle cannot be presented in the same form to someone else. It is only possible to transmit or communicate its representational image. Therefore, it is the same represented form or symbol that constitutes the anukāra (representation or image) either of the Veda or the mantras [VP 1.5: prāptyupāyo ‘nukāraś ca tasya vedo maharṣibhiḥ | eko ‘py anekavartmeva samāmnātaḥ pṛthak pṛthak || VP Vṛtti, pp. 23–24: prāptyupāyo brahmarāśiḥ | [. . ..] anukāra iti | yāṃ sūkṣamāṃ nityām atīndriyāṃ vācamṛṣayaḥ sākṣātkṛtadharmāṇo mantradṛśaḥ paśyanti tām asākṣātkṛtadharmabhayo’ parebhyaḥ pravedayiṣyamāṇā bilmaṃ [¼praticchandakam, vedāṅgākhyam | na cāsau atīndriyā sūkṣamā vāk tathākhyātuṃ śakyata iti praticchandakathnam | Paddhati] samāmananti |]. From this it follows that there is another form that is made out of consciousness and is of the nature of speech and is beyond the Veda, the body of which is made out of mantras composed in words. This indeed is the body [or the form] of Āgama. For this very reason, even the spontaneity of the intuitive awareness of the seers gets manifested only after its refinement by the Āgama essence [VP 1.30: na cāgamād ṛte dharmas tarkeṇa vyavatiṣṭhate | ṛṣīṇām api yajjñānaṃ tad apy āgamapūrvakam || Vṛtti, p. 86: teṣv api tadarthajñānamārṣam ṛṣīṇām āgamikenaiva dharmeṇa saṃskṛtātmanām āvirbhavati ityākhyāyate |]. Abhinavagupta, without any hesitation, names this Āgama in the form of creative awareness [ĪPVV vol. 3, p. 103: ‘ārṣam’ iti pratibhārūpavedādiprasiddhirūpasadoditalokaprasiddhyātmakam |]. Therefore, Established well-known Usage (prasiddhi) itself is Āgama and it leads as a means of proof that surpasses all other proofs. In order to legitimize this, Abhinavagupta reproduces the words of Vyāsa (from the Gītā and the Mahābhārata) through the mouth of Bhartṛhari himself. The people who are not aware of this but who pray after having heard it from others, even they are emancipated. For determining the notion of merit and demerit, it is prasiddhi that is responsible for the functionality of all that any human would

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need. They do not require scripture for this. In the MVV, Abhinava clarifies it further by quoting Bhartṛhari [ĪPVV vol. 3, p. 102: ‘ata eva’ iti yataḥ prasiddhirūpa eva āgamas tato hetoḥ | tadāha “anye tvevam ajānantaḥ śrutvānyebhya upāsate | te’ api cātitarantyeva mṛtyuṃ śrutiparāyaṇāḥ ||” (Gītā 13.65) iti | idaṃ puṇyam . . .. | ityādimunivacanaṃ bhartṛhariṇā (VP 1.40) āgamaprāmāṇyadārḍhyāya upanyastamiti tanmukhena iha likhitam | According to Madhusudan Kaul the kārikā quoted here is from the Vākyapadīya (VP) 1.40: idaṃ puṇyam idaṃ pāpam ity etasmin padadvaye | ācaṇḍālaṃ manuṣyāṇām alpaṃ śāstraprayojanam ||] According to him even if the usefulness of the scripture in the context of merit and demerit be negligible, the relevance of Āgama can in no way be compromised. This is because it is not the collection of beginningless scriptures such as the Veda, etc. alone, but it is surely Established well-known Usage (prasiddhi) the essence of which is speech [MVV 1.799–801: labhante niścayaṃ samyag āgamākhyāt parīkṣakāḥ | tathā ca munir āhedaṃ puṇyaṃ pāpam iti dvaye || śāstraprayojanaṃ svalpaṃ nāgamasya prayojanam | āgamo hi na nāmaiṣa pustakagranthasaṃcayaḥ || kevalaṃ prathitābhikhyo ‘nādir vedādikaḥ kila | kiṃ tu prasiddhir evāsau sā ca śabdasvarūpiṇī || See also Torella 2013, pp. 8–9]. In his both books Abhinava is quoting this verse of Vyāsa because he wants to challenge the Mīmāṃsaka’s position of invalidating Āgama as a pramāṇa while quoting Bhartṛhari who was championing the cause of the prasiddhi. Quoting the same verse (Ślokavārtika, Autpattika Prakaraṇa, Vṛttikāragrantha 5.3. Quoted in Torella 2013, p. 9) by Vyāsa, Kumārila doubts the autonomous validity of the universal acceptance of popular convention (lokaprasiddhi) toward ascertaining the dharma and the adharma. The Established well-known Usage can claim validity only when it is sanctioned by the scriptures: nirmūlasaṃbhavād atra pramāṇaiḥ saiva mṛgyate [Ślokavārtika, Autpattika Prakaraṇa, Vṛttikāragrantha 5.4, See the Ṭīkā (Bhā vol. 2, p. 3) of Sucaritamiśra: ‘na svatantrāyāḥ prasiddhireva prāmāṇyam siddhyati |’ quoted there; Here itself it is shown how Bhāskarakaṇṭha is following Sucaritamiśra]. Established well-known Usage has no ontological roots of its own at all and therefore it must be subjected to logical scrutiny. Thus, it is not only in respect of popular usage, but even in respect of the Established well-known Usage of the seasoned people, i.e., mahājanaprasiddhi (Established well-known Usage of the cultural role models), scriptural validation is required. Realistically speaking, it is by raising the prima facie standpoint of the Mīmāṃsakas on prasiddhi that Abhinavagupta begins discussing it in his ĪPVV: nanu prasiddhir nāma na kiṃcana pramāṇam [ĪPVV vol. 1, p. 35, See Torella 2013, p. 10]. This is the theory of the Mīmāṃsakas [cf. Pramāṇavārtika 1, the svopajñā vṛtti of Dharmakīrti, p. 171: prasiddham aprāmānyataḥ | Karṇagomin connects it with Mīmāṃsā: mīmāṃsakasya | (p. 602), Quoted in Torella (2013, p. 10); Also see prasiddhiś ca nṛṇāṃ vādaḥ pramāṇaṃ sa ca neṣyate | Pramāṇavārrtika (PV) 3.322.]. The word prasiddhi is not clearly mentioned in Bhartṛhari even if he seems to be very well aware of the concept of prasiddhi [For instance see VP 1.31: dharmasya cāvyavacchinnāḥ panthāno ye vyavasthitāḥ | na tāṃllokaprasiddhatvāt kaścit tarkeṇa bādhate ||]. Therefore, I agree with Raffaele Torella when he says that

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Śaivas have primarily got the word prasiddhi from the Mīmāṃsakas. While explaining Mīmāmsaka’s reformulation of Bhartṛhari’s notion of pratibha and śabdatattva, Śaivas further elaborate and strengthen the notion of Āgama (that has its basis in prasiddhi) of the latter. This is what is regarded as the most intrinsic activity of the consciousness according to Śaivas, and aiśvarī vāk, for that matter. The germs of this idea galore in Bhartṛhari. Bhartṛhari has strong faith in the Veda, but for him, apart from being a collection of mantras, the Veda is also Brahma-rāśi (embodiment of Brahman) – thriving inside all living creatures in the form of reflective speech, intuitive awareness, and consciousness. As opposed to this, the Veda of the Mīmāṃsakas is caught up within in its own ideas of injunction and disjunction, is completely devoid of anything related to this world and the beings living in it. In the discussions of the Mīmāmsakas, it becomes clear that the Buddhists do not accept prasiddhi as a valid means of knowing. If worldly transactions are accounted for by prasiddhi, then it is not because it is a valid means of knowing (pramāṇa) but because it is a mental construction (kalpanā) and the mental construction itself becomes a cause of worldly transactions. Therefore, another league of opponents of Śaivas is the Buddhists who they target right after the Mīmāṃsakas: yadi paraṃ kalpitaḥ svabhāvahetuḥ, na ca kalpanayā paramārthavyavahārāḥ siddhyanti | [ĪPVV vol. 1, p. 35. Here the objection is raised from the point of view of the Parārthānumāna (inference for the sake of others) of Buddhists because Śaivas consider the Āgama as the independent and absolute authority source of knowing]. Here the Śaivas theorise that prasiddhi is svabhāvahetu because of its inherent strength (svasāmarthya) and not because of the strength of imagination. Therefore, there is no impediment if prasiddhi is employed both in paramārtha as well as in vyavahāra. This is because certain prasiddhi is so deeply intrinsic within the heart that it is not possible to dig it out unless the heart itself is totally uprooted. This remark is being made particularly keeping in mind the Buddhists [ĪPVV vol. 1, p. 37: kasyacit kācideva prasiddhiḥ . . .. hṛdayabhittau utpāṭanaśatair api hṛdayam anunmūlya nāpasarpatītyāśayena āha ‘bauddhasyāpi’ iti |]. So far as the specific objects of perception and inference are concerned, the Āgama is not required there. Perception and Inference are respectively applicable to the objects that are directly perceptible and imperceptible, but in the case of the absolutely or remotely imperceptible entities, the validity of the Āgama or Śāstra alone has to be resorted to (for Buddhists both the terms are synonymous) [ĪPVV vol. 1, p. 37: sa (bauddhaḥ) hi: tathā viśuddhe viṣayadvaye śāstraparigrahaṃ | cikīrṣoḥ sa hi kālaḥ syād yadā śāstreṇa bādhanaṃ || (PV 4.50) iti āha, tathā ‘tadvirodhena cintāyās tat siddhārtheṣv ayogataḥ | tṛtīyasthānasaṃkrāntau nyāyyaḥ śāstraparigrahaḥ ||’ (PV 4.51) iti |]. Buddhists themselves endorse it [ĪPVV vol. 1, p. 37: yenaiva pratyakṣānumānāvagamayogyametat viṣayadvayam avitathaṃ kathitam, tenaiva ayaṃ tadaviṣayas tṛtīyasthānātmā viṣayo bhāṣita iti |]. The difficulty is that the Buddhists do not hold Āgama as an autonomous means of knowledge, but they admit it as a part of the parārthānumāna. The problem Śaivas have with this is that they do not see anything in the parārthānumāna that might justify its labelling as Āgama or that might require some characteristics of Āgama [ĪPVV vol. 1,

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p. 38: parārthānumāne ca sarvameva āgamarūpatānapekṣam |]. In the cases where the instrument of knowing serves only the pragmatic purpose, it only caters to removing of delusion. There is no possibility of any type of new knowledge being born in such a case. According to Śaivas, the self is always self-luminous by virtue of its being Lord (Īśvara), and therefore for them, no problems arise in removing delusion by the epistemological process that accounts for the worldly practice. Therefore, wherever there lies the absence of proving either the probandum, reason, or the invariable concomitance, in that case either the speech would be unnecessary or it would only transpire to be the knowledge from Āgama-pramāṇa originating as it does from word, the inference will not be there (ĪPVV vol. 1, p. 38: dharmyasiddhau hetvasiddhau vyāptyasiddhau vā parasya vacanamātrasya anupayogāt vacanapratyaye vā āgamarūpataiva syāt, na anumānam iti |). If the sentence “Buddha has said this” is regarded as a valid means of knowing, then it is the same sentence that serves as prasiddhi for the followers of one’s own tradition and that of the other tradition as well. No one should entertain any doubt about this. Therefore, if Buddhists want to establish the understanding of the noble-truths (āryasatya), etc. by invoking the word of Buddha as a testimony, then the only way it is possible is by taking recourse to our path. (ĪPVV vol. 1, p. 37: atha buddhena idaṃ bhāṣitam iti svayūthyaparayūthyaprasiddhir anatiśaṅkanīyā ucyeta | tadasmaduktasanmārgārohaṇametat |). The Śaivas go a step ahead and challenge the Buddhist idea that human beings are devoid of theorizations and mental traces right from their birth, but these are to be earned in one’s own lifetime (ĪPVV vol. 1, pp. 36–37: tena ‘riktasya jantorjātasya’ (PV 4.54) iti riktā vāco yuktiḥ |). In other words, these prasiddhis are not a natural part of one’s innate personality. In fact, quite contrary to this, Śaivas strongly believe that such activities and reflective prasiddhis are the very part of the existence of human beings right from their birth and they help refine the evolution of the consciousness of an individual that is uninterrupted from series of all previous births.

A¯gama That Has Established Well-known Usage (Prasiddhi) as Its Characteristic: [2] The Unitary Syntactic Structure of All the Scriptures: The Authoritativeness of All the Scriptures (sarva¯gamapra¯ma¯nya) ˙ Surviving on the original ideas adopted from Bhartṛhari and having reflected upon the two main opponents, the Pratyabhijñā philosopher has become more emboldened in his confidence. Prasiddhi is such a tool based on which Abhinavagupta can present his two main principles and their importance lies in their mutual interdependence of far-reaching significance. These two important principles are the unitary syntactic meaning of all the scriptures (sarva-śāstraikavākyatā) and the ensuing validity of all the scriptures (tajjanmā-sarvāgamaprāmāṇya). All our dealings are pivoted on prasiddhi and it is this prasiddhi that is Āgama (TĀ 35.1cd-2ab: iha tāvat samasto ‘yaṃ vyavahāraḥ purātanaḥ || prasiddhim anusandhāya saiva

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cāgamā ucyate |). This is the first among the threefold formulations of Āgama and, in the words of Jayaratha, it is the general definition of Āgama (TĀV vol. 8, p. 3645: atra āgamasyaiva tāvat sādhāraṇyena lakṣaṇamāha |). Against the popular notion, Abhinavagupta believes that even the anvaya (positive concomitance) and vyatireka (negative concomitance) that lead to the determination of all worldly dealings resulting from an insight into the sādhyasādhanabhāva (relationship between the means of establishing and what is to be established), are the effects of prasiddhi. To use Abhinava’s own language – they can survive just because of prasiddhi (TĀV 35.2cd-3ab: anvayavyatirekau hi prasiddher upajīvakau || svāyatteva tayor vyaktipūge kiṃ syāt tayor gatiḥ |). If they were efficacious without recourse to prasiddhi, then for each person the ascertainment of concomitance would have taken place object-wise, but this is not what we observe in our daily practice. Even while there is yellowness, etc. present in the smoke invariably accompanied with fire by the relation of positive and negative concomitance, we do not have this synthetic experience (anusaṃdhāna) that wherever there is yellowness of the smoke, there is also fire present (yatra yatra dhūmapāṇḍimā tatra tatra agniḥ) [TĀV vol. 8, p. 3647: svātantryeṇa tāv eva yadi niścāyakau syātāṃ tat prativyaktibhāvitvād ekaikaviṣayāśrayas tābhyām avinābhāvāvasāyaḥ syāt | na ca evam iti tatrāpi prasiddhir eva mūlam |]. In the same manner even in the case of perceptual cognition, the cause of common worldly transaction is the prasiddhi marked by the reflective awareness of identity with the object of perception (TĀV vol. 8, p. 3647: tādrūpyāvamarśamayīṃ tāṃ sarvavyavahāranibandhanabhūtāṃ prasiddhim |). And it is the same prasiddhi that despite arising from the causal collocation consisting of senses, etc., acts as a driving force for the senses toward the perception of the respective object [TĀ 35.3cd-4ab: pratyakṣam api netrātmadīpārthādiviśeṣajam || apekṣate tatra mūle prasiddhiṃ tāṃ tathātmikām |]. To elucidate the idea of prasiddhi, Abhinavagupta picks up the example of a newly born infant. In a room full of innumerable provisions if a lonely and hungry infant does not have prasiddhi in the form of self-reflective awareness, how would he be able to take something he wants, what can he take, and from who should he take it [TĀ 35.4cd5ab: abhitaḥsaṃvṛte jāta ekākī kṣudhitaḥ śiśuḥ || kiṃ karotu kim ādattāṃ kena paśyatu kiṃ vrajet | The note by Jayaratha (TĀV vol. 8. p. 3648): vinā svāvamarśātmikāṃ prasiddhiṃ niyataviṣayahānādānavyavahāro bālasya na syād ity arthaḥ || This illustration is the reply to Dharmakīrti’s PV 4.54: riktasya jantorjātasya guṇadoṣamapaṣyataḥ | vilabdhā vata kenāmī siddhāntaviṣamagrahāḥ || As we have already seen above, Abhinava in his ĪPVV (vol. 1, pp. 36–37) has rejected this stance by saying ‘there is no substance in this argument’ (riktā yuktiḥ).]. Just above while discussing ‘perception’, we noted that the nature of the prasiddhi is that of awareness of identity with the respective object (tādrūpyāvamarśa) and in the case of a baby born the same day, it is in the very nature of self-reflexivity (ātmāvamarśana). [In this context while discussing Anumāna [ĪPV 2.4.11], it would be useful to remember the influence of Patañjali on Abhinava as the latter puts forth his own standpoint]. In reality, the prasiddhi has been abstracted in terms of – the reflective awareness that is of the nature of previous impressions (prāg-vāsanā-rūpa-vimarśa) [TĀ 35.9cd-10ab: prāgvāsanopajīvī ced vimarśaḥ sā ca vāsanā || prācyā ced āgatā seyaṃ prasiddhiḥ paurvakālikī

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|]. If it is the previous impression, it will naturally follow that this prasiddhi is coming down from the past – the prasiddhi that belongs to the previous times. Indeed, this is also the etymology of Āgama [TĀV vol. 8, p. 3651: Quoted by Jayaratha ‘vimarśa āgamaḥ sā sā prasiddhir avigītikā |’]. This foundational prasiddhi is the universal valid means and we have no other option but to accept this: mūlaṃ prasiddhis tanmānaṃ sarvatraiveti gṛhyatām | [TĀ 35.11ab]. Thus, the source of all acts of giving and taking (hānādāna) is the historically uniform homogenous Established well-known Usage (sajātīya prasiddhi) (Jayaratha coins this new word while exploiting another citation). The reflective awareness that consists of prior impressions and the age-old prasiddhi are two synonymous, interchangeable notions (vinimayanīya pratyaya) [TĀV vol. 8, p. 3651: Quoted by Jayaratha ‘sajātīyaprasiddhyaiva sarvo vyavahṛtikramaḥ | sarvasyādyo vāsanāpi prasiddhiḥ prāktanī sthitā ||’]. What is meant by the homogenous Established well-known Usage is it’s being enlivened by successively preceding prasiddhi, and also successively preceding vāsanā (latent impression). The question is that if the whole transactional activity derives its life-breath from its own successive prior activity of the elders and because that flow of transaction being beginning-less, this constant dependence on the behavior of the successive elders of yore does not even lead to regress ad infinitum destroying the very basis of this transaction, what does one gain from forcibly rendering prasiddhi as the source of human transactions? The solution offered by the Śaivas is that this race is not without a reason or is not ad infinitum. In reality, the sustenance of the present lies in the past and then the sustenance of that past in its own past and so on. This quest for continuous pre-causes eventually takes us to an omniscient subject. At that level, prasiddhi shines as enfolded selfawareness or parā-awareness since there is no expectation of any other in this primary level constituted by parā. Because of this, the unmanifested manifests into selfawareness or absolute awareness in the form of prasiddhi [TĀ 35.11cd-12ab: pūrvapūrvopajīvitvamārgaṇe sā kvacit svayam || sarvajñarūpe hy ekasminniḥśaṅkaṃ bhāsate purā |]. Abhinava comes to the conclusion that by merely subsisting on the continuously anteriorized previous prasiddhis, without taking support of the omniscient agent, the normal transaction (vyavahāra) cannot be established because the controller or the determiner of the ubiquity element (samastatā) cannot be non-omniscient (asarvajña) [TĀ 35.12cd-13ab: vyavahāro hi naikatra samastaḥ ko ‘pi mātari || tenāsarvajñapūrvatvamātreṇaiṣā na siddhyati |]. In the MVV, because the universal transaction is subservient to an omniscient agent, niyati is depicted to be the prompter of the universe [MVV 1.795: niyatiḥ saiva viśvasya pravartakatayā sthitā | sa eva cāgamo nāma vṛddhavyavahṛtikramaḥ || Abhinava ascribes this doctrine to some students of Bhūtaja: prāmāṇyaṃ niyateḥ śrīmadbhūtajāntanivāsinām MVV 1.807a. It is not very clear who this Bhūtaja is]. In this way, it is the Supreme Lord himself who is all-knowing and completely self-reflecting, and it is he who alone is the prima causa of the entire prasiddhi. That is why while explaining the nature of the Bhairava in this context, he is defined as being adorned by the hundreds of prasiddhis those bestow both enjoyment and liberation [TĀ 35.14: bhogāpavargataddhetuprasiddhiśataśobhitaḥ | tadvimarśasvabhāvo ‘sau bhairavaḥ parameśvaraḥ || This is the same situation with the world of day-to-day existence. It is also full of numberless beliefs. The only difference is that the Supreme Lord is the locus of all those reflective awareness in

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the form of established well-known beliefs, while in the ordinary life each knowing subject is related to his respective beliefs: prasiddhiśatapūrṇe jīvaloke kasyacit kācideva prasiddhiḥ ĪPVV vol. 1, p. 37]. The prasiddhi that has originated from the Supreme Lord is free of internal contradiction or is unobstructed by another thought (vitatā) accounts for the common-practice (loka-vyavahāra) in two ways [TĀ 35.15: tataś cāṃśāṃśikāyogāt sā prasiddhiḥ paramparām | śāstraṃ vāśritya vitatā lokānsaṃvyavahārayet ||]: 1. Soliciting support from the tradition – according to Jayaratha, this means oral tradition and 2. depending on the scriptures – by this Jayaratha means ‘written’, in other words, the scripture that is composed in words. The above two are the same ideas defined in the short commentary (ĪPV) and the long commentary (ĪPVV) as anibaddha-prasiddhi and nibaddha-prasiddhi. The divisions like time and space, classic or Vedic become instrumental for the variety of the worldly activity that takes place through the means of prasiddhi. The question is that if it is the primaeval prasiddhi that is driving itself one by one in the form of different Āgamic scriptures, what would be the rationale for adhering to a fixed (niyata) Āgama? Here the answer of Śaivas is influenced by their leaning toward their faith (even though taking reflective awareness as the basis they try to repeatedly rationalize it with arguments but it does not seem to be well articulated here). In case of opting for a fixed Āgama, the reason lies in differentiation of caused by fullness (pūrṇatā) and the lack thereof (apūrṇatā) as well as the differentiation of the respective fruits thereof. The discerning people who aspire for pūrṇatā that is liberation, take refuge [TĀ 35.16cd: santaḥ samupajīvanti śaivam evādyam āgamam || In this context, it will be useful to make the mention of a quotation referred to by Jayaratha: tasmāt sampūrṇasambodhaparādvaitapratiṣṭhitam | yaḥ kuryāt sarvatattvārthadarśī sa para āgamaḥ || TĀV vol. 8, p. 3655] in the Śaivāgamas because it propounds the complete meaning. On the other hand, the non-discerning people who do not desire the complete result in the form of liberation, take refuge in other limited (fixed) Āgamas. But this does not let the principle of “all-āgamas are valid” (sarvāgamaprāmāṇya) of the Śaivas get compromised because in proportion to their trust and devotion they are anyway the enjoyers of limited fruits [TĀ 35.17: upajīvanti yāvat tu tāvat tatphalabhāginaḥ ||]. In the above context, Abhinava is raising a radically original question having enormous epistemological implications. In the behavior of a newly born child, the role of prasiddhi might be admissible because he is not aware of agreement or disagreement (anvaya and vyatireka). But as regards an adult whose discerning acumen is mature enough, why should his behavior be expected to be guided and informed by prasiddhi. Why should the role of perception and inference not be considered in this case? In response Abhinava reminds us of his principle of viṣayatāpatti. The means of knowing only becomes operational when a thing attains objecthood. A subject who is now grown-up, who is not a child anymore, whose object of perception is food, but there is no causal element readily visible in that knowledge that can lead one to decide if that food is worth eating or not. [TĀV vol. 8, p. 3656: tatra na tāvat pratyakṣaṃ sambhavati tasya hy annaṃ viṣayaḥ, na tadbhojyatvaṃ tasya jñāne vikārakāritvābhāvāt, tat katham asya viṣayabhāvam

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aprāpte vastuni pravartakatvaṃ syāt |] In other words, the edibility of the food has not attained the state of the object-hood, therefore the perceptual cognition cannot prompt as regards its edibility. Even inference cannot be deemed as of any consequence because it banks on agreement and disagreement and both agreement and disagreement are in turn dependent on prasiddhi (this we discussed just a little while ago). Therefore, eventually, the behavior of someone who is not a baby is, in fact, based on prasiddhi [TĀ 35.18: bālyāpāye ‘pi yad bhoktum annam eṣa pravartate | tat prasiddhyaiva nādhyakṣān nānumānād asambhavāt ||)] This is the reason, in the worldly life – both a baby and a knowledgeable elder are said to behave in like manner. (TĀV vol. 8, p. 3656: yadabhiprāyeṇaiva ‘laukike vyavahāre hi sadṛśau bālapaṇḍitau’ ityādy uktam |]. Even here Kashmiri Śaivas seem to follow Bhartṛhari who in a different context seems to suggest that as far as the approach of scriptures and day to day life is concerned, neither of them makes a distinction in their approach toward the mundane affairs [Vṛtti (p. 64) on VP 1.24–26: śāstravyavahārasadṛśaṃ ca laukikaṃ bhedavyavahāram |]. Abhinava emphatically says that in this case, one should not even think of any other cause apart from prasiddhi. A hungry person runs after food and not after anything else because he cannot decide on his own if anything else other than food can satisfy his hunger [TĀ 35.19ab: na ca kāpy atra doṣāśāśaṅkāyāś ca nivṛttitaḥ |]. Even if this be the case, the question still remains that why is there no doubt arising only in the mind of the person who is being impelled by the prasiddhi. The answer is that prasiddhi is nothing but the nature of the cognizing subject in the form of verbal apprehension where the conflicting reflective awareness (viruddha-vimarśa) does not arise [TĀ 35.19cd-20a: prasiddhiś cāvigānotthā pratītiḥ śabdanātmikā || mātuḥ svabhāvo . . .. . . |]. The important thing to note is that the Supreme Lord is regarded as the agent of the reflective consciousness that is activity in the nature of reflective awareness (vimarśa/parāmarśa-kriyā) [TĀ 35.20bcd: . . .. . .. yat tasyāṃ śaṅkate naiṣa jātucit | svakṛtatvavaśād eva sarvavit sa hi śaṅkaraḥ ||]. Precisely this is the reason why prasiddhi in the form of reflective awareness is regarded as the valid means of knowledge. This is because himself being an agent of knowing, the pramātā does not entertain any doubt or uncertainty in respect of the Established well-known Usage which is essentially identical with self-reflectivity. In reality, the knowing agent referred to here is none other than the omniscient Śiva himself. Even the absence of such doubt can occur in the state of Parameśvara alone. But the problem is that in our ordinary life we are not Parameśvara and because we are not Paramēsvara, self-reflection devoid of doubt (śaṅkā) seems rather inconsistent. Abhinava does not find any inner contradiction with this premise since so long as Śivahood is not completely attained, until then the knower does not have any doubt about this (limited) prasiddhi that keeps resonating the self (svātmānusāriṇī). For, this indeed happens to be his self-authored reflective activity (parāmarśanakriyā). He does doubt the prasiddhi of others and attaches more importance to his self-reflective discernment over their prasiddhi [TĀ 35.21–22ab: yāvat tu śivatā nāsya tāvat svātmānusāriṇīm | tāvatīṃ tām eṣa prasiddhiṃ nābhiśaṅkate || anyasyām abhiśaṅkī syād bhūyas tāṃ bahu manyate |]. In such a situation, the theory of Śaivas that proclaims that wise humans take recourse to

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scripture (Āgama) alone – indeed signifies that they treat it to be self-reflexive (svāvamarśamayī) in the sense of its being their own (sva-) prasiddhi. This is because the very destiny of human beings is to attain Śiva-hood [TĀ 35.21–22cd: evaṃ bhāviśivatvo ‘mūṃ prasiddhiṃ manyate dhruvam ||]. By positing this argument Śaivas draw two mutually complimentary conclusions: one, there is only a single Āgama and the other, this Āgama is nothing but the Śaivāgama. Here Abhinava performs a remarkable task quite silently – he assimilates the spirit of the Vedic Āgama and adapts himself to the main cultural stream of the Indian philosophy. The omniscient Supreme Lord reveals only one Āgama that situates itself in both the paths of worldly pursuit (pravṛtti) as well as reversion therefrom (nivṛtti), termed by Jayaratha as the paths of action (karma) and knowledge ( jñāna), respectively [TĀ 35.23: eka evāgamaś cāyaṃ vibhunā sarvadarśinā | darśito yaḥ pravṛtte ca nivṛtte ca pathi sthitaḥ || cf. TĀ 35.35 tadeka evāgamo’ yam |]. If the Supreme Lord has revealed one Āgama alone, then the reason for the scripture-wise variety and division that is noticed vis-à-vis its nature and fruition as far as the four goals of human pursuit (puruṣārthacatuṣṭya) are concerned, its objective lay in their intrinsic gradual fulfilment and non-fulfilment of the four puruṣārthas. The only means of the variegated fruition that is born out of the division into complete (pūrṇa), complete-incomplete (pūrṇāpūrṇa), and incomplete (apūrṇa) is the Āgama of Śiva (Śāṃbhavāgama) [TĀ 35.24: dharmārthakāmamokṣeṣu pūrṇāpūrṇādibhedataḥ | vicitreṣu phaleṣv eka upāyaḥ śāmbhavāgamaḥ ||]. There is a self-contradiction in this style of argument – there is a contradiction between the agent being one but his/her teachings being many – these two contradict each other. The Śaiva philosophers solve this logical anomaly by reducing the notion of singular means (ekopāyatā) into that of the variegation of means (vicitropāyatā). [TĀ 35.25: tasminviṣayavaiviktyād vicitraphaladāyini | citropāyopadeśo ‘pi na virodhāvaho bhavet || This insight of the Śaivas can be described as variegated non-dualism (vicitrādvaitavāda) from the perspective of their non-dualism. Here Arindam Chakrabarti notices markable influence of Buddhist point of view up untill Jñānaśrīmitra. Among the masters before Abhinava, Bhaṭṭa Nārāyaṇa (SCM 9) and Somānanda seem to have anticipated the similar idea. Somānanda in his Śivadṛṣṭi mentions the Citrabrahmavāda propagated by a section of the Vedāntins, but he distinguishes his Citrādvaita from the Citrabrahmavāda. But we do not have adequate information about this theory.] Citropāya, the word used by Abhinava is pregnant with meaning. The word citra does not signify the variety alone, but a kind of picturesqueseness (citrarūpatā) is implicit in it, which is suffused by the forms of the pictorial object (ālekhyākāratā), and the canvas that is overlaid with the myriad beautiful hues of the form (asaṃkhyavarṇī-ākāravicchitti) – where there is variety but no contradiction. The singularity of Āgama and variety of forms are equally applicable in the case of prasiddhi. Therefore, prasiddhi too is unitary and boasts of a plethora of forms at the same time [TĀ 35.35: tad eka evāgamo ‘yaṃ citraś citre ‘dhikāriṇi | tathaiva sā prasiddhir hi svayūthyaparayūthyagā ||]. In this Āgama, all the Āgamas right from the ordinary scriptures up to the Vaiṣṇava,

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Buddhist, and Śaiva Āgamas are situated. Its absolute goal (upeya) itself is known as Trika or Kula. The Sāṃkhya, Buddhist, etc. Āgamas are just a slice of it and it is only under delusion that they are taken to be autonomous and plural in the world [TĀ 35.37: ekasmād āgamāc caite khaṇḍakhaṇḍā vyapoddhṛtāḥ | loke syur āgamās taiś ca jano bhrāmyati mohitaḥ ||]. The thematic division or the content-induced variety (viṣayabheda), based on which there is the possibility of contradiction-resolution in the various articulations of the same Āgama, is simply unnecessary from the point of view of a class of Āgamas. This is because if there is equal force among the various segments of a single Āgama, then they should be optionally adopted in place of each other, hypothesizing the variety as caused by subject-matter (viṣayabheda) is of little use [I have understood ‘tulyapramāṇaśiṣṭānāṃ’ used by Jayaratha in the introductory remark on TĀ 35.28, it in the sense of ‘tulyapramāṇena upadiṣṭānāṃ’]. But Abhinava does not agree with this. In the alternative, even after admitting the plurality of the Āgamas, one has to concede the thematic distinction by way of hierarchical progression among the Āgamas implying some among them are higher and some lower. Otherwise, no one will be able to legitimize the authorly of any Āgama due to mutual clash in between the options [TĀ 35.38–39ab: anekāgamapakṣe ‘pi vācyā viṣayabheditā | avaśyam ūrdhvādharatāsthityā prāmāṇyasiddhaye || anyathā naiva kasyāpi prāmāṇyaṃ siddhyati dhruvam |]. Therefore, based on the hierarchy of eligible seekers, Śāstra can only become a testimony in the form of an instructor of a definitive means [TĀV vol. 8, p. 3667: kasyacid evādhikāriṇo niyatopāyopadeśakaṃ śāstraṃ pramāṇam |]. Abhinava strongly emphasizes the difference in respect of the content, of which the hierarchy of a deserving seeker is the integral component. In doing this, he should be seen endorsing the general trend of Indian philosophy. As against the content-wise division, Abhinava even rejects the authoritativeness of eternality (nityatva) and non-contradiction (avisaṃvāda). Even if perception is not eternal, it is still called valid means of cognition. The gross elements like ether, etc. are considered to be eternal, but they are not counted as the valid means of cognition; ‘svargakāmo yajeta’, ‘agnihotraṃ juhuyāt svargakāmaḥ’ – even if the non-contradiction/agreement (avisaṃvāda/saṃvāda) is not seen at all in these sentences, yet they are regarded as the valid means. At times, the ordinary sentences of daily life like ‘there is water in the well’ fail to generate valid conviction even if one can see water in the well. Even though eternality and non-contradiction (avisaṃvāda) be considered as determinant of the validity of Āgama, yet one will have to accept the validity of that very Āgama in the injunctions or teachings like ‘these are the sources of validity’ [TĀ 35.40ab: asminn aṃśe ‘py amuṣyaiva prāmāṇyaṃ syāt tathoditeḥ | Here Jayaratha takes the meaning of ‘amuṣyaiva’ in the sense of ‘amuṣya śaivasyaiva.’ This also seems plausible from the traditional point of view. But I think if this is not specifically understood in terms of the Āgamas of the Śaiva tradition but only in terms of just Āgamas or Established well-known Usage (prasiddhi), then the argument put forth by Abhinava on the validity of the Āgamas gets larger grounding]. As far as Abhinavagupta’s personal priority is concerned, he believes in a single Āgama alone. One-āgama theory is discussed at three levels: absolute (ātyantika) and relative (sāpekṣa). At the absolute level, there is only one Āgama, i.e., Śaiva

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Āgama that shoots forth in several divisions and outflows owing to the variety in its content. Even at the relative level, Āgama is one for respective qualified seekers who have a dedicated Āgama as a center of their belief. It is in this sense alone that all the Āgamas lay their claim on validity in respect of their given contextual field (niyataviṣaya). The interpretation of the Āgama and their objective correspondence is also determined by this. It is for this very reason that even the arthavāda sentences in the context of the Vedic exegesis do not sound contradictory and meaningless. This is because their veracity as a subsidiary of an injunction or correspondence in the form of an injunction too gets firm footing. Here, it does seem like Abhinavagupta is reminded of the Śivadṛṣṭi (ŚD) of Somānanda in his exposition that has been motivated by his teacher Śaṃbhunātha [ŚD 3.67–68ab: so’ arodīditi vede’ asti nārthavādo nirarthakaḥ | vidhyaṅgatvena cet sattā nāsatyasyāṅgatā sthitā || arthavādād api phalaṃ rātrikratuṣu darśitam | Also see Utpala’s Padasaṃgati on this (pp. 123–125)]. Otherwise, if one Āgama is explained based on another Āgama, every Āgama will prove to be a refuter of the other Āgama because of the unwarranted overlap of the process. For this reason, this valid means of cognition known as Āgama must be respected [TĀ 35.41: atiprasaṅgaḥ sarvasyāpy āgamasyāpabādhakaḥ | avaśyopetya ity asminmāna āgamanāmani ||]. And there is only one way to show this respect and that consists in unflinching conviction in that scripture [TĀ 35.42ab: avaśyopetyam evaitac chāstraniṣṭhānirūpaṇam |]. This general principle of Abhinavagupta specifically rests on the faith in Śaiva Āgama because by serving the central purpose, the ancillary purposes are automatically taken care of [TĀ 35.42cd: pradhāne ‘ṅge kṛto yatnaḥ phalavānvastuto yataḥ ||]. It is possible to prove the validity of all the Āgamas only when together with resolving their mutual conflict, they are brought to the plain of mutual convergence. Abhinava presents the following resolution options as constituting three models of synthesis – the content-wise division, complete-incomplete division, and variegatedness of being (citrarūpatā). All the three among them are mutually independent, yet they can be said to complement each other and for the sake of convenience, they can be labelled as first, second, and third. This first model consists of progressively growing efficacy and subjugating the successively preceding each anterior. Abhinava explores this at length while dwelling upon the method of the descent of Grace (Śaktipāta) [TĀ Āhnika 13]. There also he derives inspiration from his preceptor Śaṃbhunātha [TĀ 13.102cd: śrīśambhuvadanodgīrṇāṃ vacmy āgamamahauṣadhīm || Also see TĀ 13.349ab: tasmān na gurubhūyastve viśaṅketa kadācana |]. One of the outcomes of Śaktipāta also is that an aspirant who is seeking to tread on subsequent higher paths is motivated by the desire to get rid of the successively inferior teachers and Āgamas. Here it is marked by the process of attainment of the progressively higher (uttarottara) and that of the rejection of each preceding (pūrva-pūrva) [TĀ 13.356: yas tūrdhvordhvapathaprepsur adharaṃ gurum āgamam | jihāsec chaktipātena sa dhanyaḥ pronmukhīkṛtaḥ || Here I emend the reading ‘jihāsec chaktipātena’ to ‘jihāsecchā śaktipātena.’ The Guru is the one who has earned entitlement in that discipline and the one lacks this entitlement is reckoned as ‘other teacher’ or ‘other āgama.’ For this reason, one should not be too doubtful about the plurality of either the Āgamas or the Gurus if

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the intention is directed toward progressive attainment of knowledge [See TĀ 13.349–355]. In the case of violation of or making an exception to the above, one has to take recourse to the sacraments like the liṅgoddhāra, etc. Just like in the Vedic āśrama-system one is allowed to migrate from one āśrama to another following a proper sacrament, in the same manner Āgamic rites such as liṅgoddhāra, etc. bestow eligibility on the aspirant. The argument is, just as even after emerging from Śiva himself, one is not able to reap the fruit of transition from Brahmacarya-āśrama to Gṛhastha-āśrama, in the same way one is not able to achieve the direct result in the nature of Śiva from the inferior Āgamas like the Pāñcarātra, etc. [TĀ 35.28–29: yathaikatrāpi vedādau tattadāśramagāminaḥ | saṃskārāntaram atrāpi tathā liṅgoddhṛtādikam || yathā ca tatra pūrvasminn āśrame nottarāśramāt | phalam eti tathā pāñcarātrādau na śivātmatām ||] In this way, one may understand the mutual synthesis of the scriptures right from the Vaiṣṇava Āgamas (in reality from the phenomenal world) up to the Trika Āgamas. This viewpoint may be taken to represent the traditional standpoint of the Śaivas [TĀ 13.348: prāg vaiṣṇavāḥ saugatāś ca siddhāntādividas tataḥ | kramāt trikārthavijñānacandrotsukitadṛṣṭayaḥ ||]. In the same sequence, the second model consists in the hierarchicality of the parts in the wake of placing them in the whole-part or principle-subsidiary setting. The original Āgama is the whole/principle (aṅgī). It infuses life into all parts of the body in the form of vital-breath (prāṇa) or soul. These parts exist in a reciprocal order of higher and lower. In this way, this can be depicted as a two-tier formation. The situation of the Trika and its coordinate Kula school is akin to Prāṇa, transpires to be its essence, and all the Āgamas emerging from the five sources reside as parts or limbs in the hierarchy of above and below (TĀ 35.32: yathordhvādharatābhāksu dehāṅgeṣu vibhediṣu | ekaṃ prāṇitam evaṃ syāt trikaṃ sarveṣu śāstrataḥ ||). Here the root Āgama is only one. The second option in the form of upper-lower positioning of Āgamas is feasible in the event of upholding the plurality of the Āgama. The situation can be likened to that of a ladder-like sequence (sopānakrama) though the word sopānakrama is not used here. Because of the difference in respect of the content, the Āgamas should be literally considered as constituting a staircase where the constituent Āgamas are situated from top to bottom and bottom to top [TĀ 35.38: anekāgamapakṣe ‘pi vācyā viṣayabheditā | avaśyam ūrdhvādharatāsthityā prāmāṇyasiddhaye ||]. The third model consists in accepting the infinite number of ābhāsas by imitative structural formation of a particular (entity) technically called svalakṣaṇa. [ĪPVV vol. 3, p. 101: kaścit punar āgamo maheśatāvibhāgalakṣaṇaparamanirvāṇaphalo bhavannanantasāmānyanikurumbasvīkārighaṭābhāsavadanantābhāsasvīkāreṇa vartamāno’ adharaśāsanābhihitabhogāpavargasamartho’ api bhavati, natu adhara ūrdhvaphaladānasamarthaḥ |]. For instance, as it is only after these universal manifestations are accepted/cognized or their being in the same locus is established, the manifestation of the particular entity in the form of a pot keeps on containing in itself infinite universal ābhāsas. In the same way, among all these Āgamas which propounds Lordliness (Maheśatā) and whose fruition lies in absolute liberation, assimilating within itself innumerable Āgamas like a gestalt of ābhāsas (ābhāsanikuruṃba), is capable of bestowing pleasure and liberation

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as taught by the lower Āgamas. But, the lower Āgamas are not capable of bestowing the fruit of the higher Āgamas. This process can be applied to all the Āgamas in their relative hierarchical contexts merely in proportion to unification of the universal ābhāsas. By analyzing the derivation of the word ‘prasiddhi’ – ‘pratīteḥ siddhiḥ prasiddhiḥ’ (pra þ siddhi), Abhinavagupta offers an original interpretation in the sense of firmness of or conviction in belief (pratīti) [‘prasiddha’ (read ‘prasiddhi’) iti | pratīter nirūḍhir dṛḍhatā | ĪPVV vol. 3, p. 84.]. Conviction (pratīti) is the reflective awareness in the form of verbalization (śabdana) and the firmness consists in thatness of that is being reflected upon, i.e., as-it-is-ness of the subject-matter of that belief. This reflective awareness is the innate inner nature of the consciousness [ĪPVV vol. 3, p. 84: śabdanarūpatvaṃ (read: ‘śabdanarūpaṃ’) vimarśanaṃ yadāntaraṃ citsvabhāvasya antaraṅgaṃ rūpaṃ pratyakṣāder api jīvitakalpaṃ tena yat vimṛṣṭaṃ, tat tathaiva bhavati |]. Abhinava sees synonymity between verbalization and reflective awareness and for the same reason, the verbalization which is firmly rooted in reflective awareness is Āgama [ĪPVV vol. 3, p. 85: dṛḍhavimarśarūpaṃ śabdanamāgamaḥ |]. Here, the definition of the process involved in the etymology of Āgama in the sense of ‘understanding from all sides’ (avagatimūlaka): the Āgama is that valid means of knowledge which is instrumental in the firm reflective awareness of the meaning that is being propounded [ĪPV vol. 2, p. 87: dṛḍhavimarśanarūpaṃ śabdanam āsamantāt arthaṃ gamayati iti āgamasaṃjñakaṃ pramāṇaṃ sarvasya tāvat |]. It is this reflective awareness that is the immediate denotation of Āgama. The body of words (śabdarāśi) that leads to this reflective awareness can be designated as Āgama only in the secondary sense owing to pragmatic necessity [ĪPVV vol. 3, p. 84: tataḥ sa eva vimarśa āgama iti ucyate mukhyatayā, tadupayogitayā tu upacāreṇa tajjanako’ api śabdarāśiḥ |]. Here Abhinava conceives of a new paradigm of a dialogical correspondence between knowledge and its object. While the valid means like experiential knowledge follows the dictates of the object, the Āgama is determined by the ontological essence of the object [ĪPVV vol. 3, p. 90: vastutattvānusāriṇa āgamāḥ |]. This means that the object of cognition is the same as the reflective awareness of Āgama perceives it to be, i.e., it does not follow the object of cognition but the ontological reality of the object of cognition [ĪPV vol. 2, pp. 85–86: tena yat yathāmṛṣṭaṃ tat tathaiva yathā naitat viṣaṃ māṃ mārayati garuḍa eva aham iti |] In other words, in place of the reflective awareness that follows the object, it is the object that follows the reflective awareness in the case of Āgama. In this way, the firmness of the reflective awareness is the central criteria of the reflective nature of Āgama. Firstly, it is responsible for Āgama’s being unbiased as source of knowledge and secondly, it also defines that whoever believes in that Āgama is also worthy of the praxis laid down by that Āgama. The latter reminds us of the earlier notion of faithfulness toward the given scripture. An Āgama is not a valid means for someone who does not have an unflinching faith in it. The impartiality of the Āgama is not subject-indifferent flat premise. Without reference to the knowing subject even the perceptual and inferential cognitions do not become valid means. The perception that Mohan has of the pot or the fire he is inferring is not the same as the perception of pot or inference of fire that Shyam is having. The perceptual or inferential cognitions of both are subject to respective expectancies

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of the knowing subject. Their validity lies in their commonality (ekajātīyatā) and the openness to all the percipients (sarvasādhāraṇya) and not in their indifference or neutrality toward the knowing subject. This commonality is also shared in equal measure by Āgama as a valid means [ĪPVV vol. 3, p. 84: atha tathājātīye pratyakṣānumāne maitrasya api kadācit bhavata iti apakṣapātitā, sā tarhi āgamasya api aviśiṣṭā |]. It is for this reason that the firmness of conviction (pratīti) in case of Buddhists and Cārvākas lies in their Āgama or in the word of the person they trust, and thus it becomes a valid means for them. To see two moons is not a valid cognition, but even in that case, it is the same light, eye, etc. that are used in normal perception. But this does not render neutrality of entire perception as a valid means [ĪPVV vol. 3, p. 85: na ca iyatā samyak pratyakṣasya pakṣapātitā prasajyate kācit |] The implicit meaning of the criteria of firm reflective awareness is that it is not Āgama but merely the pseudo-Āgama (āgamābhāsa) for the person who does not have a firm reflective awareness. ‘jyotiṣṭomena yajeta svargakāmaḥ’ is only a semblance of an Āgama for someone who is ineligible or of a lower cast and is devoid of firm reflective awareness, whereas for a dedicated twice-born, it is verily Āgama. For this reason, the meaning of the Āgama derived from the etymology ‘samyak āgama’ (‘ā samyak [ava] gamaḥ’) indeed culminates into the firm reflective awareness [ĪPVV vol. 3, p. 85: tathā jyotiṣṭomādivākye śūdrādīnāmanadhikāriṇāmadṛḍha-vimarśarūpa āgamābhāse upayogaṃ vrajannapramāṇabhūto’ api śraddhādaravati dvije dṛḍhavimarśātmakasamyagāgamarūpaśabdanopayoge prāmāṇyamāsādayan na pakṣapātādivācyatārhaḥ | See also ĪPV vol. 2, p. 88]. This leads Abhinavagupta to arrive at a conclusion that all Āgama can either injunct or prohibit only when it duly takes into account or is governed by eligible person, specific space, specific time, specific ancillaries, etc. [ĪPVV vol. 3, p. 85: sarva eva hi āgamo niyatādhikārideśakāladaśāsahakāriprabhṛtīnāmṛśya vidhiniṣedhādivimarśamayaḥ | See also ĪPV vol. 2, pp. 88–89. Rastogi 1977, p. 299]. This is precisely implied by the words ‘yatra’ and ‘yadā’ in the verse ‘prasiddhir āgamo loke’ [ĪPV vol. 2, p. 89: tena ‘prasiddhiḥ’ iti śloke ‘yatra yadā’ ity uktam |]. From this, it clearly follows that Lord creates someone by connecting him with the ascertainment of requisite praxis associated with certain deity, perfected being or venerable entity appropriate for it and someone by connecting him with some such other ascertainment. For this reason, Abhinavagupta once again refutes Dharmakīrti’s statement – ‘a living person is born blank’ [ĪPVV vol. 3, p. 85: tataśca kaścit puruṣaḥ kaṃcideva devasiddhādyanya-tamakaraṇīyocitavimarśaṃ svātmasaṃyojanena vimṛśan bhagavatā sṛṣṭaḥ, anyastu anyaṃ vimarśamiti ‘riktasya jantoḥ’ (PV 3.54) iti asadetat |]. In the same way, the criteria of the firm reflective awareness or firm conviction is applicable to the definition of an eligible person. Thus, the firm conviction constitutes as the exclusive characteristic of an eligible person [ĪPVV vol. 3, p. 85: dṛḍhanirūḍhireva ca tattadadhikārilakṣaṇaṃ mukhyam |]. Abhinavagupta suggests that this eligibility criterion does not apply to the adherents of the Pratyabhijñā philosophy alone, but the complete Vedic tradition too respectfully subscribes to this definition [ĪPVV vol. 3, p. 85: iti darśitaṃ śrutyaiva ‘yaścainam evaṃ veda’ iti ‘vidvān yajet’ iti | tadartham eva ca uktaṃ ‘śraddhāmayo’ ayaṃ puruṣaḥ’ ityādi |]. This prasiddhi could be endowed

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with logical reasoning (yuktimāna) or may go even without it (tad itara). Inference of God’s existence from an aggregate, i.e., effect, acting as reason exemplifies the Āgama which is endowed with logical reasoning (yuktimāna) and reality principles (tattvas) and regions (bhuvanas), etc. represent the one without it (ĪPVV vol. 3, p. 85: īśvarasadbhāve hi sanniveśakāryādiliṅgajam anumānamastyeva, tattvabhuvanādīnāṃ tu iyattāyāṃ nāsti anumānamiti āgama eva tatra śaraṇam |).

Twofold Aspects of Established Well-known Usage (prasiddhi): Formally Constructed (nibaddha) and Not-Formally Constructed (anibaddha) In general, there are two kinds of Established well-known Usage: (1) reduced to writing in the form of scriptures (2) that is reflected in the practices of people [ĪPVV vol. 3, p. 85: evaṃ śāstranibaddhāyām ubhayaṃ vyākhyāya lokaparasparāpratiṣṭhitāyāṃ vyācaṣṭe |]. The latter is also known as unwritten Established wellknown Usage. The two forms described above, viz., yuktimāna and tad-itara, are represented in the form of formally constructed scriptures. The composed Established well-known Usage again has two forms: (a) that is reduced to a script, i.e., which is written [ĪPVV vol. 3, p. 99: śrotramanogocarīkāryāṇi api lipau yataḥ upacaryante |] and (b) that is reduced to a specific type of syntactic structure, i.e., even if not written or scripted but still structured in a specific syntactic construct [ĪPVV vol. 3, p. 100: ‘nibaddhaḥ’ iti viśiṣṭavākyaracanābhiḥ anibaddhas tu yatra tathā nāsti |]. In other words, the formalizing of the scriptures has two types, viz., written and unwritten. Even the scripture-bound belief is basically of the nature of reflective awareness. Because of seeing them in practice over innumerable births, we call the specific forms of reflective awareness composed in a script as Sarvavīra, Bhargaśikhā, etc. And it is because they are associated and understood as one with the Āgamas, we come to address them by such and such names. Even the cause of the gratification and repentance, etc. as fruiting from scriptures consists in the unification of a person or knowing subject in the form of light with the phonematic reflective awareness that is identified with the letters of alphabet homologized with the former. This indeed is the satisfaction of success or pain of the reflecting subject [See ĪPVV vol. 3, pp. 99–100]. If formulated in epistemological terms, like an object is created as a reflection in that light having a specific objective configuration of form, etc., similarly reflective awareness also having been affected by the reflection of that specific form, etc. is reflectively grasped. The words of such category cannot be premised as untouched by meaning [ĪPVV vol. 3, p. 100: tatra īśvarecchayā sa bhāvo yathaiva prakāśe pratibimbātmanā sṛṣṭo viśiṣṭarūpādiprakāśamayaḥ, tathā viśiṣṭarūpoparāgavicitravimarśaviśeṣamaya iti katham evaṃprāyāṇām arthāsaṃsparśitvaṃ śabdānām |]. Like the words of the scripture that acquire functionality corresponding to their meaning, similarly the unwritten particular syntactic structures accessed through the genius of common people reflect their identity with the respective meanings and efficacy. Illustrated by an example this characteristic of the Āgama is universally applicable [ĪPVV vol. 3, p. 100: evaṃ śāstradiśā kāryakāri-

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tvena tasya śabdasya tadarthātmakatvaṃ pradarśya lokapratibhāmukhena api āha ‘tathā’ iti | [. . .. . .] udāharaṇasphuṭīkṛtamāgamalakṣaṇaṃ sarvatra saṃcārayati . . .. . .. . .. etac ca ‘ghaṭo’ ayam ityadhyavasā’| (1.5.20) iti sūtre vivecitam | In the above quotation, by quoting the words ‘udāharaṇasphuṭīkṛtam’ Abhinavagupta probably referring to the example given by Utpala in his Vṛtti and Vivṛti. It is not clear to us what this example was.). Even in our everyday life, in the cases of determinate comprehension (adhyavasāya) like “this is a pot,” where there is mutual superimposition (adhyāsa) of the word and its meaning, it is the power of the Supreme Lord himself that unboundedly manifests in the form of mental apprehension. Prasiddhi, based in the popular tradition or what is called ‘Established wellknown Usage lacking the formal construction’ (anibaddha), is also of two types – (1) one having the nature of popular knowledge (lokaprasiddhi) and (2) the other having the nature of the knowledge of eminent elders of society (mahājanaprasiddhi). As is clear from the word itself, what is popular in the tradition is the popular Established well-known Usage. Taking cue from Utpala, Abhinava is discussing a heresy saying: in the medieval times in the mid-eastern part of Kashmir, it was laundrymen who used to do all domestic work instead of washing clothes [ĪPVV vol. 3, p. 85: ‘kaśmīreṣu’ iti madhyapūrvakāladeśādau hi dhīvarā eva gṛhakaraṇīyaṃ bhūyasā vidadhate |]. Here, the valid means lies in the popular Established well-known Usage in embracing the domestic chores that is different from the professional activity of laundrymen. While defining ‘loka’ Abhinavagupta says – ‘one who follows the practice or conduct specific to a certain place, that is what is known as common world (loka) in that context’ [ĪPVV vol. 3, p. 98: yo yatra vyavahāre vyavahartā sa tatra lokaḥ | ‘deśe’ iti tattadvyavahārasthāne |]. In this context, he also raises a question of conflict between the ordinary world and the scripture. Somewhere, in opposition to some prescription by an Āgama, certain local practice is unanimously adopted and also acted upon by the majority of people of that place without any doubt in their minds in doing so. Such acquiescence takes the form of an Āgama as popular Established well-known Usage. Here the cause of this contradictory behavior lies in the speech or the word itself [ĪPVV vol. 3, p. 98: kenacidāgamaniścayena yadyapi nirmitam anyalokācāreṇa ca viruddhaṃ tathāvidham api caraṇaṃ ceṣṭitaṃ yat; tadapi pratipadyante kāmaṃ bahavo’ api ekavākyatayā aṅgīkurvanti, na tu tatra eṣāṃ vicikitsā bhavatītyarthaḥ | ‘tatra hi’ iti tathāvidhaviruddhavyavahāranimittabhūtā vāg ityarthaḥ |]. The Established well-known Usage of the eminent elders of society (mahājanaprasiddhi) is just another name of the Established well-known Usage that partakes of the nature of popular tradition (lokaparamparātmaka). The word ‘mahājana’ is a very well-known term in the cultural milieu of India. We are familiar with the famous line from the Mahābhārata – ‘mahājano yena gataḥ sa panthāḥ’ (that verily is the path which has been treaded by the eminent elders). Even here the mahājana is a culturally refined state of the lay world. A class of people that has made its mark in a certain field of activity, is hailed as mahājana, like for instance, the Chandas in respect of the performance of the Vedic ritual or like the Bhāgvatas as regards practicing the Vaiṣṇava code of conduct [ĪPVV vol. 3, p. 100: mahājano yatra

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kartavye yaḥ prasiddhiṃ yāto vargaḥ, sa eva mantavyaḥ, yathā vedānuṣṭhāne cchāndasaḥ, vaiṣṇavānuṣṭhāne bhāgavatākhyaḥ |]. In comparison to others, they can acquire the related fruits with much less effort and greater ease because of the spontaneous grasp of identification with the reflective awareness of one’s own code of practice [ĪPVV vol. 3, p. 101]. Here, this is how mahājana (eminent elders of the society) is precisely different from the jana (laymen). This Established well-known Usage also lacks in formal construction because it is gathered not from leaning on the scriptural sentences but from the ritual performance, action or the conduct of the eminent elders, the wise or the professionally competent in their respective fields [ĪPVV vol. 3, p. 101: anibaddhāpi iti mahājanānuṣṭhānaśeṣatayaiva sthitā |]. Its validity is confirmed due to the eventual visibility of the emergence of the final fruit because the hindering reflective awareness does not come into being in-between. If there is hinderance in the mid of this process, it will not be called a valid means, but this is highly unlikely. Like for instance, a single perception of the tree-ness leads to the ascertainment of the branch-ness etc. and that tree-ness holds good for all the three worlds and all the three times because of the absence of any hinderance in the intervening period. In the same way, the final reach of an unsuspected inherent cognition up to the external object is indeed proven as valid means by perception alone as regards a celestial body connected with a name or as regards a planet being poisonous [ĪPVV vol. 3, p. 101: evaṃ nirvicikitsaprasiddherbāhyārthaparyantatvaṃ nāmanakṣatrādau viṣabhūtagrahādau ceti pratyakṣeṇaiva prāmāṇyaniścayaḥ |]. The Established well-known Usage of the eminent elders of society (mahājanaprasiddhi) is also an Āgama in the form of Established well-known Usage. In order to substantiate this, Abhinavagupta quotes a verse from the Gītā (reference to Lakshman Joo’s edition, 1933): anye tvevam ajānantaḥ śrutvānyebhyaḥ upāsate | te’ pi cātitarantyeva mṛtyuṃ śrutiparāyaṇāḥ || (Gītā 13.26)

(Translation: Others, not so knowledgeable, worship on hearing from others. Such followers of the Śruti, do swim across the death.)

The Source of the Validity of A¯gama: The Firm Rooting of Conviction (vimarśanirūdhi) ˙ It has been repeatedly mentioned that it is because of the firm rooting in the conviction that Āgama is a valid means [ĪPVV vol. 3, p. 96: nirūḍhatayā āgamo mānam |]. Towards nurturing this concept, we have already been introduced to several arguments put forth by the Śaiva masters. Even then the Śaiva masters do not want to leave any stone unturned to establish this further. An effort has been made to interpret this conviction of Āgama in three ways [ĪPVV vol. 3, p. 96: atra tridhā vidhyanuvādayogo vyākhyātavya iti | Here the meaning of ‘atra’ is doubtful. ‘atra’ can be deemed as the predicate of the statement by Bhaṭṭa Nāyaka just quoted above in the ĪPVV text or alternately as referring to the topic under discussion. I have

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understood it in the latter sense]. The first explanation is to maintain the epistemic validity of all the Āgamas progressively in tune with the part of the last line ‘yat yadā yathā yatra’ from the verse ‘prasiddhir āgamo loke’ [ĪPVV vol. 3, p. 96: ‘yat yadā yathā yatra’ ity anena krameṇa sarvāgamānāṃ prāmāṇyam |]. Even though the Āgamas of the mlecchas and the yavanas etc. are also regarded as valid means by them, they turn out to be false Āgamas (āgamābhāsa, Āgama in appearance only) because of the possibility of their contamination or weakening by the non-aryan contact [ĪPVV vol. 3, p. 96: mlecchādyāgamo hi tāvati pramāṇaṃ bhavann api anāryasaṃparkasaṃbhāvyamānamlānirāgamābhāsaḥ |]. Owing to the temperamental leaning of the human mind with the rituals prescribed by the Vedic scriptures with the duties laid down by the Āgamas, Āgama will not be there if a person resorts to an action or practice following one’s desire, or intermingles with the premises of the other Āgama, or feels that his previous conduct was not appropriate. In such an eventuality, if one shuns his Āgama in places other than those of the Āgama emanating from the Highest Lord (Pārameśvarāgama), there is always a possibility of the infinite rise due to the descent of grace from Śaivāgama. But if in the case of the Āgama emanating from the Highest Lord (Pārameśvarāgama) there is a veil cast on one’s mind, then this possibility of rise is destroyed and only endless misfortune ensues as a result. In such cases, it is indeed for the removal of the defects arising out of the abandonment of one’s own Āgama, that the previously discussed liṅgoddhāra initiation has been prescribed. But no such difficulty arising for the one who has absolute faith in the Śaivāgamas because the very nature of Āgama is to reflect the intrinsic essence of the objective reality [ĪPVV vol. 3, pp. 96–97]. This explanation certainly cannot be called philosophical, but it aims to determine seriatim hierarchical limit of each Āgama in terms of the postulated validity of all the Āgamas. If Āgama is one then it is the Śaivāgama alone, in which all the Āgamic streams, Established well-known Usages in other words, ultimately realize self-fulfilment. From here, we will move on toward the second explanatory phase of conviction. The main problem of Abhinavagupta is – how to account for the logical consistency of homodensity of Āgama with the variegated multiplicity of the Established wellknown Usages. We just saw above that Established well-known Usages are countless and even their agential sources are not fixed. In such a situation if the speech that is without a fixed or identifiable source itself is regarded as divine speech, then we reach up to scripture named Nāda that emerges out of Śiva that is consciousness, and through that, we also reach further to the beginningless state of Āgama. In holding so, Abhinava is inspired by the Āgamas of Śaiva Siddhānta [ĪPVV vol. 3, p. 97: nanu evaṃ yadi aniyatakartṛkaiva vāk pārameśvarī, tarhi siddhāntaśrutyādeḥ prāmāṇyaṃ syāt – adṛṣṭavigrahācchāntācchivātparamakāraṇāt | nādarūpaṃ viniṣkrāntaṃ śāstraṃ paramadurlabham || iti | amūrtādgaganādyadvannirghāto jāyate mahān | śāntātsaṃvinmayāt tadvacchabdākhyaṃ śāstram————|| ity anena krameṇa anāditvāt parameśvare prakāśavimarśasvabhāve kālānullāsāt, buddhādipraṇītatvāt tu na bauddhāgamādīnāṃ bhavet | We are not completely sure wherefrom Abhinava is quoting these verses. The first two lines of the stanza closely resemble the two lines from the ST 8.27b-28a. Even Jayaratha quotes these lines in the TĀV 35.26–27

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supporting the stance of Abhinavagupta in the TĀ 25.27 based on the ST where it says that all the Āgamas like Siddhānta, Tantra, Śākta have originated from the five faces, i.e., Sadyojāta, etc.: adṛṣṭavigrahāyātaṃ śivātparamakāraṇāt | dhvanirūpaṃ susūkṣmaṃ tu suśuddhaṃ suprabhānvitam || TĀV, vol. 8, p. 3231. In his Udyota on the ST 8.27–31, Kṣemarāja quotes the parallel verses in the same context from the Śrīkaṇṭīyasaṃhitā. These verses are available as the opening verses of the Śrīkaṇṭīyasaṃhitā (See Hanneder 1998, Appendix 1). This verse is closer to the citation of Abhinava that reads: adṛṣṭavigrahācchāntācchivātparamakāraṇāt | jñānarūpaṃ viniṣkrāntamanavacchadanaṃ mahat || We are still in dark about the second verse. Torella has also mentioned this citation of Abhinavagupta (Torella 2013, p. 12), but our views are different]. Since there is no emergence of time in the Supreme Lord whose nature is light and reflective awareness, therefore even the coming into being of Nāda is also metaphorical in that case. Because of this, the Established well-known Usages or Āgamas that are said to originate from the historical figures like Buddha, Kapila, Mahāvīra, would cease to be regarded as valid means. As a result of this, multiplicity of Āgamas constituted by the variegated Established well-known Usages itself will be under crisis. From the way Abhinavagupta resolves this question, lends weight to one’s surmise that his theory of Āgama is as dynamic and flexible as his definition of pramāṇa. Abhinava maintains that the speech which cannot be traced to a definite speaker or agential source undoubtedly constitutes reflective awareness of the Supreme Lord. Even in the traditions that are considered to have origin in the historical personages, there too it does not allude to a historical figure tied to a definite place and time, but rather it refers to the one born in a tradition with continuous and uninterrupted flow and who is completely immersed in the firm awareness of momentariness etc. resulting from the power of meditative visualization (bhāvanā). The teaching as regards meditative visualization on momentariness, etc. is imparted to him by the previous Buddha who in turn receives it from earlier Buddha, who receives it from still anterior Buddha, and so on, thus eventually what seems to remain is the Supreme Lord’s reflective awareness alone whose agential authorship cannot be determined. The same can be said about Sage Kapila who was immersed in the meditative visualization of the 24 tattvas. This principle is universal [ĪPVV vol. 3, pp. 97–98: nahi buddho nāma niyataḥ kaścit, api tu bhāvanābalapratilabdhakṣaṇikādidṛḍhavimarśaḥ | tasya kṣaṇikādibhāvanopadeśī guruḥ pūrvabuddhaḥ, tasyāpi anyaḥ, iti krameṇa aniyatavaktṛkatvāt pārameśvaravimarśamayataiva vastutaḥ | evaṃ caturviṃśatitattvabhāvanābhāvitaḥ kapilo mantavyaḥ | ata eva sarvāgamā anādaya eva |]. Therefore, all the Āgamas are beginningless. Abhinava warns us that this does not mean that Buddha or Kapila are the authors of the works associated with their names. But their authorship is subject to prasiddhi. Prasiddhi alone is the valid means. It is due to the will of the Supreme Lord alone that Buddha and Kapila etc. also having entered into that prasiddhi, were able to earn the eligibility for bestowing grace, so that they could further favor those Buddhas, those Kapilas and others [ĪPVV vol. 3, p. 98: tatkṛtatvameva hi prasiddhimantareṇa kiṃpramāṇakam iti prasiddhir eva ekā pramāṇam | parameśvarechāvaśāc ca sugatakapilādayo’ api tatprasiddhyanupraviṣṭāḥ kṛtā anugrāhyāstān eva anyāṃś

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ca anugrahītum |]. In this way, the historicity specified by time and space can discover its intrinsic dynamism in the perenniality of tradition rendering the determining adjuncts meaningless. In the third formulation, the basis of conviction lies in the multiplicity of prasiddhi as opposed to its unitary nature. Although this is the third explanation it begins where the second interpretation ends. All prasiddhis are beginningless. Even that prasiddhi which occasions mutually contradictory behavior serves as a valid means for someone [ĪPVV vol. 3, p. 99: viruddhā prasiddhiḥ prāmāṇyalakṣaṇamaśnute iti saṃbhāvyate iti sambandhaḥ |] For instance, there is a Vedic injunction that beginning with Śukra onwards the consumption of liquor by the twice-born is permissible [ĪPVV vol. 3, p. 99: tathāca ‘śukrātprabhṛti madyasyāpeyatā dvijaiḥ’ ityādi śrūyate iti sāpi prasiddhireva | sāpica anādiḥ | Here, in ‘madyasyāpeyatā’ I am reading ‘āpeyatā’ instead of ‘apeyatā’]. This is also a prasiddhi and hence beginningless. In reality, all the prasiddhis are beginningless. Amongst them, a certain prasiddhi might emerge on surface at some point in time and a certain other may almost vanish having submerged [ĪPVV vol. 3, p. 99: anādikālabhāvinyo hi sarvāḥ prasiddhayaḥ | tatra tu kācit kadācidunmajjati, kācit nimajjati |]. This process keeps on going. The purport being that the validity of the prasiddhi that comes up is evidently noticed whereas in the latter case, it remains hidden from the sight. This can also be said that some prasiddhis are embraced and some are left behind because of eligibility-differentiation (pātratābheda) and level-differentiation (starabheda). Implicit also is the meaning that all the prasiddhis act as valid means but they are not valid means at the same level, i.e., synchronously in respect of the same person, place or time. Their validity is dynamic.

A¯gama as Verbalizing (Śabdana) or Intuitive Reflecting (Pratibha¯na) Next original conceptualization of Āgama or prasiddhi has crystalized into the form of pratibhā [ĪPVV vol. 3, p. 102: evaṃ pratibhārūpeṇa nibaddhānibaddhaprasiddhidvayātmanā ca trividhamāgamaṃ pradarśya . . .. . . | See Avasthi 1966, pp. 1–2]. This category is not mentioned in the ĪPVV. As the pre-text of his discussions on pratibha, Abhinava poses a crucial question that if all the mental and linguistic activity is a creation of the Supreme Lord, why does then this process turn out to be contradictory or fails (visaṃvādī) in the case of pure mental discursion or projection. Abhinava does provide an answer that even though being identical with the Supreme Lord when the empirical subject is not able to gauge that identification, its individuation or fettered subjectivity appears reasonable. In the same way, the linguistic and mental functioning of the empirical knowing subject is imprinted with differentiation arising from the formal diversity of the primordial divine dynamism that can occasionally be understood sometimes by reaching out to external objects and sometimes not. Even its unique identity left to itself is not perceptually grasped or is seen with great difficulty. As a result, popular conduct is also like that. There, this contradiction is also logically accounted for. [ĪPVV vol. 3, p. 93; See also the short note on this by Rameshvar Jha 1980, p. 29]. From this, it is clear that the reason for this contradiction in this activity

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consists in the will of the Supreme Lord. By obstructing this contradictory (visaṃvādinī) popular conduct, we all aspire to attain divine activity. After analyzing the cues from Utpala, Abhinava interjects the idea of Pratibhā here for deliberation. That divine activity is called Pratibhā which remains steady even after being repeatedly impelled by the same obstructing will [ĪPVV vol. 3, p. 93: niścalaiva iti nirodhecchayā cālyamānā api avicalā | pratibhāsaṃjñā iti |]. Right after this, reflecting upon the word ‘pratibhāsaṃjñā’ in Utpala’s Vivṛti, Abhinava exhaustively deals with the Pratibhā-centric nature of Āgama: pratibhāsaṃjñā iti prabhānalakṣaṇā iyaṃ śabdabhāvanākhya āgama eva | [ĪPVV vol. 3, p. 93]. Here three things invite one’s attention: one, Pratibhā has been defined as Pratibhāna. Second, Āgama is referred to by the word śabdabhāvanā and the third, by using emphatic particle ‘eva’ the uninterruptedness of the Āgama has been demonstrated by the reflective awareness that is the prime source of all the linguistic modifications according to Bhartṛhari. By using the word ‘pratibhāna’ Abhinava provides the dynamic functionality to Pratibhā. It is strange to note that except this one sentence about pratibhā or pratibhāna in this context, he chooses to say nothing else about it at all [See AB vol. 1, pp. 278–281; Also see my article titled “Pratibhāna aur Rasānubhava” in Rastogi 2012, pp. 399–420]. From what he says, the only thing that becomes clear is that in this context the pratibhā is nothing but śabdabhāvanā and because of its being essentially one with pratibhāna, it remains of the nature of verbalizing (śabdana). The term śabdana itself suggests dynamism of the word. The plausibility of my conclusion is reinforced by the fact that Abhinavagupta offers his long exegesis of Āgama that is of the nature of uncontradicted (avigīta) prasiddhi, in the context of manifold verbalizing (śabdana).

Threefold Contexts of Verbalizing (śabdana) The composition of prasiddhi in the long or short sentences is one kind of a speech. A meaning, which is made popular by its practice in a tradition that is repeatedly contemplated by the followers of their respective Āgamas, happens to be identical with the Supreme Lord, consisting of reflective awareness (vimarśa) characterized by uninterrupted luminosity (prakāśa), due to absence of any kind of delimiting factor, and hence is without a beginning [ĪPVV vol. 3, p. 92: The printed reading ‘parasparānuṣṭhānena’ seems incorrect. It should be ‘paramparānuṣṭhānena.’ Rameshvar Jha agrees with me in his explanatory reformulation: paramparayā anuṣṭhānena prasiddho yor’ thaḥ (Āgama-vimarśaḥ 1980, p. 29]. The delimitations take the form: ‘this popular meaning is spoken of in this Āgama alone’, ‘This indeed is the origin of this’, ‘It has come in vogue from this very moment’, etc. The popular practice of the sages like Kaṭha, Bhārgava, Mataṅga, and Nārada is put together in the form of texts/treatises as being verily beginningless because syntactical formations separately constructed become the instruments of verbalizing, i.e., expressing through the medium of words [ĪPVV vol. 3, p. 92: kaṭhādibhir bhārgavamataṅgādibhir nāradaprabhṛtimiś ca prasiddhānuṣṭhānamanādi eva nibadhyate yataḥ śabdanaṃ samāsavyāsopakalpitavākyayojanābhiḥ |]. One does not need to make

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a special effort to prove that these syntactic compositions are also that of the Supreme Lord or they are of the form of Divine Speech because it is self-proven. Utpala has been repeatedly propounding this and will do so in future also [ĪPVV vol. 3, pp. 92–93: naca asyā yatnasādhyaṃ pārameśvarītvam iti darśayati ‘sarvaiva’ iti | upapāditam iti ‘tatra jñānaṃ svataḥ siddham’ 1.1.5 ityādau | ‘upapādayiṣyate’ iti ‘itthaṃ tathā ghaṭapaṭādyābhāsa.’ 2.4.21 ityādisthāneṣu | Here the meaning of ‘asyāḥ’ is taken in the sense of ‘vākyayojanā’ by Rameshvar Jha (1980, p. 29): na cāsya vākyayojanāyā api pārameśvarītvam yatnasādhyam |. Though there is no incongruity in this meaning, I have some difficulty. This is because immediately before this ‘vākyayojanābhiḥ’ is in the plural. In my opinion, ‘asyāḥ’ is linked to prasiddhi (verbalizing that is without beginning). Abhinava has raised this question elsewhere also (this has been discussed in the previous pages): ĪPVV vol. 3, p. 97: nanu evaṃ yadi aniyatakartṛkaiva vāk pārameśvarī, tarhi siddhāntaśrutyādeḥ prāmāṇyaṃ syāt | Here the situation is a little different. Not only the ‘Established well-known Usage having uncertain source’ (aniyata-kartṛkā vāk/ prasiddhi) even the so-called Established well-known Usage claiming a definite source (niyata-kartṛkā vāk/prasiddhi) is Divine Speech (pāramēśvarī vāk) in reality, because such composition by the Āgamas like the Mataṅga, etc. also belongs to that Supreme Lord]. From this exposition, Abhinava seems to be of the view that this context of verbalizing relates to praxis. From here, Abhinava moves on to discuss verbalizing in its mental context [ĪPVV vol. 3, p. 94: iyatā mānasīṃ vṛttiṃ vicārya | Rastogi 1977, p. 302]. By exegeting the word ‘bhāvanā’ in the expression ‘śabdabhāvanā’, he lends a new dimension to the verbalizing essence (śabdarūpatā) of the intuitive awareness (pratibhāna). In the spiritual disciplines, particularly in Kashmir Śaivism, even generally speaking, ‘bhāvanā’, ‘bhāvana’, or ‘bhāvita’ have special significance. A little while ago we considered the issue of the undetermined authorship of the Buddha having the unflinching reflective awareness in the momentariness etc. or that of the Sage Kapila, having the firm reflective awareness in the 24 ontological realities (tattvas) realized through the force of meditative visualization (bhāvanā). Bhāvanā is simply verbalizing in the form of a specific instruction and bhāvita is that person whose inner sensory apparatus that is constituted by the triple psychoses of retrospection (saṃkalpa), I-attribution (abhimāna), and ascertainment (niścaya) is completely pervaded and enveloped by those verbalizations. Same is the case with the meditatively efficacious verbalizing of the mantras. The defining characteristic of the mantras is to completely identify with the a priori (ādisiddha) form of the Supreme Lord whose intrinsic essence is pure consciousness [ĪPVV vol. 3, p. 93: bhāvitam iti | . . .. . .. ityādyupadeśaviśeṣaiḥ śabdanarūpair otaprotīkṛtaṃ vyāptam antaḥkaraṇaṃ saṅkalpābhimānaniścayavṛttitrayamayaṃ yasya |]. Such mantras as permeate the reflective awareness of any knowing subject with their inner essence come to yield specific fruits accordingly by forcibly casting that person in the self-same mold [ĪPVV vol. 3, p. 93]. It should never be doubted how is I-attributive verbalizing (abhimānātmaka śabdana) of the fully permeated person (because the inner sense of the bhāvita is fully overtaken by that instruction or mantra) designated as Divine Speech (Pārameśvarī-vāk) because the ubiquitous Lordliness can never, nowhere, be

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restricted. It is only because of Her sheer will that divinity keeps variously and partly emerging and/or plunging [ĪPVV vol. 3, pp. 93–94: nanu paśupramāturyo’abhimānātmā śabdanaviśeṣaḥ, sā kathaṃ pārameśvarī vāgityāśaṅkya parameśvararūpatā sarvatraiva aniruddhā, kevalaṃ tadicchayaiva tasyāḥ kvāpi aṃśe vicitre pronmajjananimajjane bhavata | Here we may remind this that just a few lines back in the ĪPVV Abhinava has taken similar recourse to reason as regards the third interpretation of nirūḍhi]. When this ubiquitous divinity is not perceived, at that moment the prāṇa, buddhi, deha, etc. are made to carry out the transactional activity in the form of knowing subject (in other words prāṇa, buddhi, deha, etc. themselves are depicted as knowing subjects), but their knower-hood is like the lump of clay because of being devoid of freedom. It is not the true knower-hood [ĪPVV vol. 3, pp. 93–94. Also see ‘Āgama-vimarśaḥ’, p. 29]. To further clarify this point, Abhinava wants to catch our attention toward the tenet of the primacy of objective factor (bhāvāṃśakaprādhānya) which is hailed in the Āgamas. As is clear from the word itself, it is such a potent conviction emanating from the will of the Lord in the objective factor (bhāvāṃśa) that one cannot get rid of, even if one is inclined to do so because his individual consciousness has already attained oneness with that reflective awareness [ĪPVV vol. 3, p. 94: tata īśvarecchayaiva yo yatra vimarśāṃśe ekīkṛtaḥ samujjhitumicchur api na ujjhituṃ śaknoti, tata eva ‘rudrāṃśo rudrabhaktastu’ ityādinā bhāvāṃśakaprādhānyamāgameṣu darśitam |]. Taking support of the reflective awareness-based argument, Abhinavagupta says that the way the Supreme Lord makes this creation luminous, in the same way, he reflects on it. If light is unfragmented, so is reflective awareness. It is indeed here, the way the object of the senses, object of inner senses (antaḥkaraṇa), causal efficiency and the aggregate of ancillaries shine forth, in the same way they are reflectively apprehended [ĪPVV vol. 3, p. 94: etad uktaṃ bhavati yathaiva viśvaprakāśātmā parameśvaraḥ, tathā viśvavimarśātmā | tatra bāhyatvagrāhyatvārthakriyātatsahakārivargasya yathā prakāśaḥ, tathaiva vimarśaḥ |]. Then the obvious conclusion would be that depending on the will of the Lord anyone who is firmly rooted in a particular śabdana, or is reposed in a particular firm conviction, transpires to be the Āgama of that person. Then what is the stronger testimony – how can this relative superiority be gauged? Abhinava says one should not doubt so because who are those helpless reliable? It is the Lord himself who manifests in that way, who reflects and makes others reflect. Then one should just keep quite because if everything is owing to the will of the lord, then creation of the scriptures and performance of rituals is futile. Here Abhinava says that they would surely have been futile if the Lord would have willed them to be so. Here Abhinavagupta borrows both the doubt and its resolution from his great grandmaster Somānanda [Abhinavagupta refers to ŚD 3.75a in the ĪPVV vol. 3, p. 94]. The third context of the śabdana relates to the linguistic activity (vṛtti) [ĪPVV vol. 3, p. 94: vāṅmayīṃ vicārayati ‘vāgapi’ iti |]. By elaborating upon the significant cues in the Vivṛtti, Abhinavagupta develops his central thesis that the testimony of speech (vāk) in the nature of Established well-known Usage proceeds in two directions, i.e., valid for self and invalid for others. The criterion of validity is – firmness of conviction due to being reflectively aware (for self) and that of invalidity is – not being reflectively aware for want of firm growth (for others). The fundamental point

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is that in deciding the validity, the role of polemical argument (tarka), means of cognition (pramāṇa) or reasoning (yukti) is rendered completely superfluous [ĪPVV vol. 3, pp. 94–95: ‘tathā ca’ iti yataḥ prarūḍhā satī pramāṇam, tataḥ parayūthyeṣu prarohābhāvalakṣaṇena bādhanena avimarśīkṛtā naiva pramāṇam | nanu nirūḍhatvāt na pramāṇam, apitu pramāṇopapannatvāt | naivam ityāha yuktyaṃśas tu iti |]. There could be several reasons for this. One, there is no rule that reasoning (yukti) should always be a valid authority in every case. Second, even if it serves as a valid authority, it remains weak because there is no complete information available about the effect. It’s another aspect can be that a strong reason on one’s own turf can turn weaker when confronted with that of another proponent. Third, in this world, no one knows where the six kinds of logic and the arguments based on their innumerable types end [ĪPVV vol. 3, p. 95: Here the exact meaning of the tarkaṣaṭaka in tarkaṣaṭkatadbhedasahasrotthāpyamānānāṃ nyāyānāṃ | is not clear]. Bhartṛhari has already pointed out its futility: a man who is an adept in logical reasoning tries very hard to infer a certain meaning, but someone who is better at reasoning explains it differently offering more cogent arguments. This is not all. Everyone considers Āgama as a valid means in the case of the object that is its proper outcome. But it is equally possible that the same issues may be interpreted with opposite meanings in the case of the same Āgama [ĪPVV vol. 3, p. 95: yadāha “yatnenānumito’ apy arthaḥ [kuśalair anumātṛbhiḥ |abhiyuktatarair anyathaivopapādyate|| VP 1.34 iti, sarvaḥ phalocitānarthānāgamāt pratipadyate | viparītaṃ ca sarvatra śakyate vaktumāgame |” VP 1.142 iti ca | In the printed edition of the VP it reads ‘sarvo’ dṛṣṭaphalā’ instead of ‘sarvaṃ phalocita’]. Therefore, as far as reasoning is considered not completely holding on to solid grounds, Abhinava agrees with Bhartṛhari [ĪPVV vol. 3, p. 95: sarvathā tarko aprathiṣṭha eva | cf. VP 1.32: nāgamād ṛte dharmas tarkeṇa vyavatiṣṭhate |]. But Abhinava does not agree with Bhartṛhari on the point that reasoning does not jeopardise the Āgama because God has created reasoning in that very form [ĪPVV vol. 3, p. 95: tathaiva hi parameśvareṇa sa sṛṣṭaḥ |]. Here Abhinava assumes many worlds like the world of reasoning, the world of literary creativity and the world of experience in the created universe of the Supreme Lord. Owing to the respective identities of these worlds, their objective legitimacy and veridical authority lies in their own spheres. In the world of reasoning, the power of the Supreme Lord manifests this creation by these two-fold operations – one while gradually augmenting the powers and other by drawing the contracted reflective awareness within – creates this world of reasoning, being useful for worldly enjoyment (Bhoga) and liberation (Apavarga), which is altogether a separate world different from the world of Āgama just like that of literary creativity. Therefore, even if the reasoning is not established (in the world of Āgama), no harm is caused to the fundamental postulates of the Śaiva philosophy [ĪPVV vol. 3, pp. 95–96: tata eva iha śaktisaṃvardhanakrameṇa antaḥsaṅkucitavimarśaśaktisamākarṣaṇena tathaiva bhogāpavargopayoginā upayujyamānaḥ prameśvareṇa sṛṣṭastarkasaṃsāraḥ kāvyasaṃsāravad apara eva | vicitrā hi amī saṃsārāḥ | apratiṣṭhitatve’ api tu tarkasya na asmaddarśanasya khaṇḍanā kācit | ābhāsamānavastuvāde hi parameśvarecchayā ayamābhāsaniyamaḥ |]. Here what is manifesting alone is real. Therefore, reasoning stands established in the

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universe of objective reality. For this reason, the glory of reasoning lies in the fact that it is not able to even fathom the final level of depth [ĪPVV vol. 3, p. 96: tadayam alabdhagādha eva tarkaḥ śobhate | . . .. . . sarvathā anumāne na āśvasitavyam, apitu āgama eva | sa ca yo yasya hṛdaye nirūḍhimupagataḥ, sa eva | nanu evaṃ yasya na kiñcit nirūḍhaṃ, tasya kim | nanu evaṃ yasya cakṣuṣī na staḥ, tasya kim |]. If someone does not accept this, it would be something like owing to his/her arrogance or imitating a child he/she begins to imagine that there is the center of the whole earth in the mid of the compound of his/her house [ĪPVV vol. 3, p. 96: atra tu svabuddhigarveṇa pratiṣṭhāṃ paśyan svagṛhaprāṅgaṇam adhyakalpitaniḥśeṣabhūgolakamadhyadhruvakasthānavat mūḍho vā ḍimbaviḍambako veti āstām |]. Therefore, one should not completely rely upon sheer inferential reasoning but must believe in Āgama. And this is the very Āgama that has turned into whatever conviction in the heart of the person who adheres to it. The question – what about someone in whose heart nothing is firmly rooted – has no meaning. For the only answer to this would be what about someone who does not have both the eyes.

A¯gama as a¯pti (Testimonial Authenticity) In the foregoing discussion we have talked about the twofold division of Āgama consisting of Established well-known Usage (prasiddhi) and intuitive reflecting (pratibhāna) or verbalizing (śabdana), alternatively about the threefold division in case the two forms of Established well-known Usage, i.e., formally constructed (nibaddha-prasiddhi) and lacking in formal construction (anibaddha-prasiddhi) are considered separately. Apart from this there is a new category of Āgama, i.e., teaching of reliable authority (āptopadeśa) [ĪPVV vol. 3, p. 102: evaṃ pratibhārūpeṇa nibaddhānibaddhaprasiddhidvayātmanā ca trividhamāgamaṃ pradarśya rūpāntaram api asya darśayati anyo’ api iti |]. The basis of its being a separate category lies in its being rooted in another pramāṇa (authoritative testimony). [ĪPVV vol. 3, p. 102: etāsu tisṛṣu prasiddhiṣu pramāṇāntaramūlatvaṃ na anveṣyam, āptavāde tu anveṣyam eva |]. All the three kinds of prasiddhis are selfcontained while for ascertaining (niścaya) the ground of authenticity of a person (āptattva), we have to seek refuge in another pramāṇa (authoritative testimony). The original inspiration for the idea of a reliable authority (āpta) for Utpala and Abhinava comes from Patañjali. Patañjali includes the entire Āgama under the head ‘teaching of reliable authority’ (āptopadeśa) [ĪPVV vol. 3, p. 103: tata eva sarva āgama āptopadeśaśabdena bhagavatpatañjaliprabhṛtibhiḥ saṃgṛhītaḥ |. In this way, Śaivas distinguish themselves from Patañjali in the sense that the āptopadeśa is one among the three/four kinds of Āgama while in the Yoga, it embodies Āgama in its totality. Therefore, Abhinava devises a new model of prasiddhi in the form of āpti (testimonial authenticity). Testimonial authenticity consists in attaining, i.e., grasping, the reality that is being taught or spoken of. In other words, an āpta is one who is endowed with it: āptir vaktavye vastuni adhigatiḥ, tattaś ca vaktavyavastvadhigatiḥ, sā vidyate yasya sa āptaḥ | [ĪPVV vol. 3, p. 102]. If one leaves aside the parameter of

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being rooted in another means then all the specifications of Āgama in the form of prasiddhi, i.e., being subject to eligibility-criterion, firmness of conviction, non-contradiction of reflective awareness, and greater strength than inference also equally mark it. Here only that aspirant is eligible to receive this teaching who does not have even an iota of doubt about that knowledge of an āpta that its basis lies in some other authority (pramāṇa), nor is he uncertain that this knowledge is of the nature of doubt (saṃśaya), misapprehension (viparyaya), or ignorance (ajñāna) [ĪPVV vol. 3, p. 102: ya upadeśyo yasya āptasya saṃbandhi jñānaṃ pramāṇāntaramūlatayā utpannaṃ na abhiśaṅkate, naiva saṃśayaviparyayājñānarūpatayā abhimanyate . . .. . .|]. Since the imparting of such words of reliable authority culminates into performance-centric instruction no one in anyway can distract the recipient of that instruction from the knowledge of the object constituting crux of the said instruction [ĪPVV vol. 3, pp. 102–103: tamupadeśyaṃ tathābhūtāptavākyopadiṣṭavastusaṃvedanād anuṣṭhānaparopadeśaparyantāt nivartayituṃ na kenacit prakāreṇa anyo bhavati śaktaḥ |] This is the index of the firm conviction in that knowledge of the one who has been so instructed. Why does this happen? In order to explain this Abhinava looks into the knowledge of the person so instructed through the perspective of parameters set by Bhartṛhari [VP 1.39: yo yasya svam iva jñānaṃ darśanaṃ nātiśaṅkate | sthitaṃ pratyakṣapakṣe taṃ katham anyo nivartayet |]. According to him, in the knowledge of recipient of teaching (upadeśya) related to āpta, to use the phrase of Abhinava literally, he is ‘situated in the state of leaning toward the side of perception’ (pratyakṣa-pakṣāśrayatā). This means that the transmitted knowledge is reflected, as if perceptually, in the experience of the recipient. When the āpta says ‘I have directly known this’, at that time the recipient of teaching too believes that ‘I myself have known it directly’ [ĪPVV vol. 3, p. 103: yato’ asāv upadeśyas tatra āptasambandhini jñāne pratyakṣapakṣāśrayeṇa sthitaḥ | yadāptena uktaṃ mayā sākṣādetat jñātam iti, tatra asāvupadeśyo’ abhimanyate mayaiva etat sākṣātkṛtamiti |]. We know very well that it is not possible to manipulate even a child who has seen a thing directly, i.e., with his own eyes, even if one deploys several arguments to convince him/her. Abhinava says that this is what Dharmakīrti is trying to convey by saying ‘buddhipakṣapāta’ (lit. natural leaning of intellect). The buddhipakṣapāta evinces that wherever there is the absolutely firm and ineradicable reflective awareness, there would be testimonial validity (prāmāṇya). Therefore, because of its being unimpeded reflective awareness the Āgama, partaking of the words of āpta, also happens to be valid testimony [ĪPVV vol. 3, p. 103: yathāhuḥ pare’ api ‘nirupadravabhūtārthasvabhāvasya viparyayaiḥ | na bādhā yatnavattve ’api buddhes tatpakṣapātataḥ ||’ PV 1.223 iti | buddhipakṣapātena hi dṛḍhatamatayā anunmūlanīyavimarśatvameva prāmāṇyasya nibandhanaṃ vyāpakam . . .. . .. āgamaḥ pramāṇam abādhitavimarśatayā |].

Different Kinds of the Authentic Being (a¯ptattva) The question is how to determine the authenticity. The āpta has three forms: people (loka), scripture (śāstra), and an agent having a definite form (niyatākāra) that is a

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person (puruṣa). Therefore, a word of the people, a word of the scripture or a word of the person (puruṣa), any of these could constitute an āptavākya (testimonial sentence) [ĪPVV vol. 3, p. 102: lokavacanarūpo vā, śāstravacanarūpo vā niyatākāraṃ kartāraṃ ca apekṣya api vartamāno’ api |]. It is these different kinds that leads to the secondary classification of the āptavākya (teaching of reliable authority). According to Utpala the reliability is arrived at by the analysis of the particular kinds of actions of the āptas [ĪPVV vol. 3, p. 102: nanu āptatvamasya kathaṃ niścitam iti cet, āha ‘ceṣṭāviśeṣaiḥ’ iti | tatrāpi hi kāryasya suvivecanā kiṃ na sahyate |]. The thesis of an āpta (āptavāda) that is decided according to the action-based analysis has two types: the first type is dependent on the scripture that is also known as śāstrāptavāda or caraṇāptavāda. For instance, in grammar, the authority of Pāṇini and Vararuci is well recognized, but not that of Akṣapāda. The second is dependent on Puruṣa (person) or Loka (people) that is called puruṣāptavāda (the theory that holds a Puruṣa to be an āpta). This, in this sense, is comparatively a broader category since it even encompasses the lokāptavāda (the theory that holds people to be āpta) into itself [ĪPVV vol. 3, p. 102: ‘praticaraṇam’ iti śāstrāptavādaḥ | tathāhi pāṇinivararuciprabhṛtervyākaraṇe āptatā prasiddhā, na akṣapādādeḥ | puruṣāptavādena tu lokāptavādaparigrahaḥ |]. When the hermeneutic achievements of puruṣāptavāda gain complete endorsement among the people (loka), then it is the puruṣāptavāda itself that transforms into lokāptavāda [Rastogi 1977, p. 307]. Abhinava does not offer any example for this. It seems that as far as both these categories are concerned, Utpala and Abhinava are influenced by Bhartṛhari [cf. VP Vṛtti, 1, p. 97: santi praticaraṇaṃ pratipuruṣaṃ pāptāḥ |]. A few short comments on this topic would be useful on this point. Among the types of āpta, Abhinava also counts the people’s word (lokavacana) into it. But he does not mention it in the classification of āptavāda. In fact, Abhinava dwells on lokavacana under anibaddha-prasiddhi (lacking in formal construction), a division of prasiddhi. There are two sub-types of non-formally constructed Established wellknown Usage: lokaprasiddhi (popular convention) and mahājanaprasiddhi (Established well-known Usage of the eminent elders of the society). In-between the divisions of āpta and prasiddhi, the overlap or intralap can only be avoided when in all types of the theory of āpta (āptavāda), it is centered around an individual or puruṣa. This can be explained by the two illustrations offered by Abhinavagupta. In the illustration of śāstrāptavāda (where scripture is supposed to serve as a testimony), his example does not take into account the testimony ‘of’ śāstra, but the testimony (of someone) ‘in’ śāstra. Pāṇini and Vararuci are reliable authorities in the grammar, but not Akṣapāda. In a parallel fashion when he illustrates the mahājanaprasiddhi, he points out to a group of Chandasas in respect of the Vedic ritual or marks out the group of Bhāgavatas in that of Vaiṣṇavism. As regards lokaprasiddhi, he posits the popular tradition or lokapratibhāmukha (interfacing with peoples’ genius) and not an individual. Moreover, even after offering his sharp critical analysis, Abhinava is conscious of its tentativeness or limitation. Therefore, in his considered opinion this āptavāda (theory of āpta) includes within itself all the above stated varieties of Āgama by explaining them as per context and occasion. By saying this he also evinces his agreement with Patañjali’s thesis that all the Āgamas are

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subsumed under the teaching of āpta [ĪPVV vol. 3, p. 102: ayaṃ ca āptavādaḥ pūrvoktam āgamabhedaṃ yathāyogaṃ vyākhyānādidvāreṇa anugṛhṇāti | tata eva sarva āgama āptopadeśaśabdena bhagavatpatañjaliprabhṛtibhiḥ saṃgṛhītaḥ |].

Testimonial Authenticity (a¯pti) Morphosized into Established Wellknown Usage (prasiddhi) If we recall, Abhinava introduced the concept of the āptopadeśa (teaching of āpta), the root idea of which Śaivas inherit from Patañjali, as a qualitatively different kind of Āgama which seeks succor from another authority. The question is that if this can be parried away just by construing it as an inferior division of Āgama or it can be assimilated as its component in the mainstream of Āgama, as agreeable to Śaivas by logically rescrutinizing it. Abhinava obviously goes in for latter choice. As a result of this, he makes a historical attempt of reversing the interpretation of āpti in terms of prasiddhi by changing the āpti-based interpretation of prasiddhi which is how prasiddhi was understood up till his own time. [ĪPVV vol. 3, p. 104: nanu na sarvair āgamasya prāmāṇyam abhyupetaṃ, tat katham asya vimatipadapatitasya anapekṣatvaṃ bhavatā uktam | abhyupetamevetyāha ‘prasiddhiḥ’ iti | . . .. . . kimatra prasiddhyā | . . .. tatrāpi prasiddhiśaraṇatvaṃ na vighaṭate |]. He achieves this by bringing together Āgama with śabda. In the context of popular tradition or anibaddhaprasiddhi (Established wellknown Usage lacking in formal construction), Abhinava equates the body of Āgama with the reflection arising from śabdana (verbalizing). In order to buttress it he quotes Patañjali saying that śabda (word) is the only medium of teaching for transmitting one’s understanding to another [ĪPVV vol. 3, p. 89: āgamo hi nāma ayaṃ śabdanasaṃkrāntiśarīraḥ | yathāha bhagavān anantaḥ ‘paratra svabodhasaṃkrāntaye śabdena upadiśyate’ (quoted from the YSB 1.7). In this context, it is important to remember that the Pratyabhijñā tradition does not make a distinction between Patañjali and its commentator Vyāsa]. His analysis of Patañjali’s view has three steps: explanation, editing, and structural re-evaluation. In the citation from Patañjali, that is from Vyāsa to be more precise, he says that a reliable person takes recourse to the medium of word for making someone understand either what he has perceived or inferred about a certain object. That time the mental activity as regards the meaning that the word generates in the mind of the listener-recipient is termed as the Āgama-pramāṇa for that listener-recipient. Even though Abhinava does not refer to the whole statement of Vyāsa’s gloss, it would be fair to have a look at it as well [ĪPVV vol. 3, p. 89: āgamo hi nāma ayaṃ śabdanasaṃkrāntiśarīraḥ | yathāha bhagavān anantaḥ ‘paratra svabodhasaṃkrāntaye śabdena upadiśyate’ | YSB 1.7. The complete quotation from the YSB reads like: āptena dṛṣṭo’ numito vārthaḥ paratra svabodhasaṃkrāntaye śabdenopadiśyate, śabdena tadarthaviṣayā vṛttiḥ śrotur āgamaḥ | yasya aśraddheyārtho vaktā na dṛṣṭānumitārthaḥ syāt, sa āgamaḥ plavate, mūlavaktari tu dṛṣṭānumitārthe nirviplavaḥ syāt | Here, it may be important to remind the readers that the whole Pratyabhijñā tradition does not make a distinction between Patañjali

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and Vyāsa]. The preceptor of an Āgama who himself does not believe in its meaning (alternatively this may also mean an unreliable preceptor), and whose subject matter has not been grasped by that preceptor either by perception or inference, there Āgama would not be valid. The Āgama could be said to be valid or free from anomaly only in respect of its content which has been directly perceived or indirectly known through an inference by the original āpta preceptor. From this, it is very clear that in maintaining āpta as deriving sustenance for its being from some other valid source (pramāṇāntaropajīvī), Abhinava has been inspired by Patañjali’s ideas. Commenting on Patañjali, Abhinava says that here svabodha (one’s own knowledge) stands for the knowledge acquired through self-experiential perception (svasamvedanātmaka-pratyakṣa) and śabda (word) for the object of auditory perception acquired through the sense of hearing. A doubt might arise that if the validity of the perception turns out to be weak, its reflection in Āgama is likely to weaken it as well [Apparently the position of the Kashmiri Śaivas is somewhat different from Yoga philosophy. In the Yoga, the perception is the seed of verbal and inferential testimony, while in the Śaivas and Bhartṛhari the verbal (meaning Āgama here) is the seed of the perception and inference. ĪPV vol. 2, p. 85: pratyakṣāder api jīvitakalpaḥ. In his introductory remark on YS 1.43 Vyāsa says: tat paraṃ pratyakṣam, tac ca śrutānumānayor bījam, tataḥ śrutānumāne prabhavataḥ | na ca śrutānumānajñānasahabhūtaṃ taddarśanam, tasmāt saṃkīrnaṃ pramāṇāntareṇa yogino nirvitarkasamādhijaṃ darśanam iti |]. According to Abhinava, there is no room for this doubt because the establishment of an object directly grasped by a trustworthy person is only possible as a result of the uncontradicted reflective awareness [ĪPVV vol. 3, pp. 89–90: tatra svabodhaḥ pratyakṣeṇa svasaṃvedanātmanā, śabdaś ca śrotrendriyādhyakṣeṇa prameya iti pratyakṣasya cet prāmāṇyaṃ prati daurbalyaṃ syāt, tadā āgamasya api tadāpated ityāśaṅkāṃ . . .. abādhitavimarśena hi pratibhāsena tadavasthāpitaṃ sākṣāt, tat katham anyathā bhavet |]. Anyone can observe that in explaining Patañjali, Abhinava is embedding the language of Śaiva epistemological thought pattern. In addition, he is also re-editing Patañjali. Patañjali’s ‘śabda’ in the garb of ‘śabdana’ (śabdanasaṃkrāntiśarīraḥ) in Abhinavagupta’s presentation provides a potential new tool to the latter for a thorough re-evaluation. Now śabdana does not remain a word that is to be received by the sense of hearing rather it catapulates into the central axis of the word’s process of transmission characterized by verbalizing. We will talk later on the topic of śabdana, but by the immediate cultivation of the śabda into śabdana, Abhinava changes the entire parameter of āpta while enlarging and extending the semantic scope of āpta in Patañjali. Āpta is not only the one who has seen or inferred the object, but āpta is also the one whose object is apprehended through another āpta (āptāntaropajīvī) also [ĪPVV vol. 3, p. 104: na kevalaṃ dṛṣṭānumitārtha eva āptaḥ, yāvadāptāntaropajīvyapi ityāha upadeśa iti |]. What is important to note is that whereas an object is seen and inferred through the means of perception or inference, there is no other way out than the linguistic mode of transmission as regards the knowledge of the object grasped from another āpta. It is here that the project of Abhinava of transforming āpti into prasiddhi begins taking shape subtly and deftly. Abhinava questions: in the Āgamas the preceptors are said to

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be the āpta. The question is: should the ones who have not read the subject matter of the śāstra by themselves but are aware of it by knowing it through the tradition, be considered as the authority on that subject or not? He univocally answers saying they should be considered authoritative because only that person can be regarded as āptāgama (āpta identified as an Āgama) even if he has not read Āgamas by himself but has acquired teaching in the intact tradition. This interjection of the concept of āptāgamatā ultimately takes the shape of prasiddhi [ĪPVV vol. 3, p. 104: āgamagranthāvabhodhena vināpi upadeśaparamparāyāṃ kevalāyām api āptāgamasyaiva tatra prasaṅgādāptāgamatā prasajyate prasiddhirūpatā ca paryavasyati]. Abhinava notices three steps in this process. The primary step for this is that of the beginningless prasiddhi belonging to the Supreme Lord. The middle step is represented by that āpta – who has got the Āgama through teachings in the tradition and forms the soul of the succeeding āpta preceptor. The next step relates to that person who has sourced his teachings (upajīvī) to the middle āpta and is a recipient of the previous āpta and himself happens to be an āpta by virtue of being a preceptor of future student-preceptor [ĪPVV vol. 3, pp. 104–105: yena āptatayā madhya-vartinyā maulikyā ca pārameśvarānādiprasiddhirūpatayā tādṛśā api guravo’nyasya upadeṣṭāro bhavanti |]. In this way, prasiddhi ! āpta ! āpāntaropajīvī-āpta, this is the order. The same etymological semantic literality of Āgama indeed is that knowledge propounded by all the Āgamas, by common consensus, is transmitted from one mouth to another mouth, another mouth to the third mouth and from one ear to another ear, from another to a third ear so on. Abhinava inquires about śabdana in the context of the āpta knowledge. How can the knowledge in the form of śabdana that is connected with the āpta become a valid means in the form of inner awareness (saṃvedana) in a listener or a student, because the knowledge arising from a word spoken into the ear of a listener rests in the listener, then how can that be connected with the speaking preceptor? Resolving this Abhinava states that the sequential word of the speaker, propounder or the āpta which is grounded in the perception of the nature of trans-sequential awareness leads to the identical experience in the listener or the student (for whom the instruction is meant) as well, losing its distinction from the listener [ĪPVV vol. 3, p. 106: pratipādayitari hi asau sakramaḥ śabdo’kramavimarśātmakapratīti-mūla iti tathaiva pratipādye saṃkrāman pratītiparyavasāyī tato’nanyatvāt tasya | upayogaḥ iti āgamatveneti śeṣaḥ |]. This is what renders it into a pramāṇa (testimony) in the form of inner awareness. Because if this is not the case then so long as it remains the object of hearing or auditory perception, it would just remain an object like color or form, etc. (as in the case of visual perception), but would not become a pramāṇa [ĪPVV vol. 3, p. 106: śrotrajñānagamyatve hi asau rūpādivat viṣayamātraṃ, na pramāṇarūpaḥ |]. It is clear from the comment of Abhinava that here he is talking about śabda qua Āgama. One more question can also be asked about āpta. A word can become an Āgama only after having first turned into a denoter, i.e., expressive of meaning. The expressive word will first be in the speaker and then in the listener. But then time would be different in both the cases rendering their identification impossible. This question does not perturb the Śaivas. In their philosophy of language, it is not the spoken-word that has expressiveness (vācakatā), but it is the word located in

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mind or the mental form of the word that has expressiveness. Therefore, the expressiveness present in the inner ground of awareness remains untouched by time. The day-to-day usage of temporal distinction etc. is just a metaphorical usage due to connection with body etc. [ĪPVV vol. 3, p. 106: pramātṛtā iti | antaḥsaṃvedanabhūmau vācakatvaṃ, sā ca kālāspṛṣṭā, dehādyupacārāt tu kālabhedavyavahāra iti sarvam anavadyam |]. Therefore, there is no evident inconsistency as regards the homogeneity of reflecting awareness between the speaker and the listener. If the meaning of āpta is “all-knowing” (among the Śaivas the word ‘sarvañja’ is used both for Yogī and Īśvara) and if there is the tinge of time therein, so in this situation one has to accept the presence of time in all its prameyas because of the contracted knowing subjectivity. But then the nature of this time would amount to perennially active compresence (nityapravṛttavartamānatā) and it is only then one would be able to establish their omniscience. And by implication it would also follow that there is no incongruity between the non-dual apprehensions of the speaker and the listener [ĪPVV vol. 3, p. 107: sarvajñānām api kālasparśe tatpramātṛtāsaṅkocād aśeṣam api prameyaniṣṭham api jñānaṃ kālaspṛṣṭam eva, yadi paraṃ nityapravṛttavartamānatodrekeṇaiva teṣāṃ jñānaṃ vyavahriyate tadā sarvaṃ vidantīti | Here it may be noted that the word saṅkocāt in pramātṛtāsaṅkocāt has to be understood in the context of the time qualifying (kālasparśa) the prameya (object)]. Thus, in a sense, the description that the masters of Śaivism, specifically Abhinava, have offered about Āgama and its divisions and sub-divisions can be charted out in a tabular form below (Table 1):

The Epistemological Structure of A¯gama Abhinava deeply contemplates about the foundational structure of Āgama as a pramāṇa. Whereas both perception and inference are the valid means of knowledge dependent on artha or viṣaya (object), there the Āgama precisely depends on the knowing subject (pramātā). This aspect becomes all the more important because all the streams of the non-dual Śaivism are pivoted around the knowing subject in the center. Being (sattā) and subjectivity (pramātṛtā) transpire to be one in the ultimate analysis. This is the reason behind the overriding importance of the Āgama-pramāṇa. If the distinction between the three valid means of knowledge is to be understood from Abhinava’s point of view, then it should be said that perception comes into being from the efficacy (sāmarthya) of artha (object), inference comes into being from the efficacy/capability/strength of arthāntara (other object) and the Āgama comes into being from the efficacy of the pramātā that is the knowing subject [ĪPVV vol. 3, p. 82: pratyakṣam arthasāmarthyāt, anumānam arthāntarasāmarthyāt, āgamaḥ pramātṛsāmarthyāt tathā vitatāvitatapramātrantarasāmarthyād iti viśeṣaḥ |]. The efficacy of the pramātā is set to be operational from two points of view: as the efficacy of the (individual) pramātā or as that of the extended subjects or non-extended subjects (vitatāvitatapramātrantara), i.e., limited or unlimited

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Table 1 Āgama and its Divisions and Sub-divisions Āgama pratibhāna - / śabda-rūpa prasidhirūpa [pramā āntaramūlatvābhāva] [pramā āntaramūlatvābhāva]

nibaddhaprasiddhirūpa (śāstravacanarūpa) lipibaddha yuktimān (nyāyānug rhīta)

āptopadeśarūpa (lokavacanarūpa, śāstravacanarūpa and niyatākārakart vacanarūpa) [pramā āntaramūlatva]

anibaddhaprasiddhirūpa (lokavacanarūpa)a viśi avākyaracanābaddha

taditara yuktimān (āgamaikarūpa) (nyāyānug hīta)

taditara (āgamaikarūpa)

lokaprasiddhirūpab

āptavāda

śāstrāptavāda or cara āptavāda

mahājanaprasiddhirūpab

āptāntaropavīvitvavāda

ekagurutāvāda puru āptavāda or (upadeśapāramparya) lokāptavāda

samastāgamagurutāvāda (the conscious reading of all the Āgamas)

Discussed under the anibaddhaprasiddhi even after its inclusion in the āptopadeśa Principally these should also entail divisions into ‘yuktimān’ and ‘taditara’ but they are not charted in the table for want of any textual discussion on or allusion to these two a

b

subjects. We have seen that perception is a direct unmediated (sākṣātkārī) knowledge of the object (artha). The inference is the knowledge that arises due to the force of concomitance of reason (i.e., arthāntara: the other object ¼ different from the major term). On the contrary, the Āgama is the knowledge that arises (in fact, having essentially the same nature) due to the Supreme Lord’s autonomy which is called divine speech (pārameśvarī-vāk), verbalizing (śabdana) or ideating (vimarśana). This knowledge, through the chain of limited and unlimited subjects, takes also the nature of respective prasiddhis or tradition that ultimately rest in the beginningless prasiddhi by assimilating the historical and ahistorical pramātās. If seen from the point of view of Prakāśa and Vimarśa, the two constituent categories for logically analyzing the unitary essence, structurally the perception happens to be that knowledge which belongs to Prakāśa of a single cognizing subject while Āgama essentially consists in the reflective awareness (vimarśa) belonging to the Supreme Knower. If translated into the technical terminology of the Pratyabhijñā epistemology, the perception (knowledge) is the reflective awareness of an object, immersed in the limited luminosity of the individual subject. While the Āgama

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(knowledge) constitutes the uncontradicted reflective awareness of the entire mass of objects resting in the unbroken and uninterrupted luminous consciousness of all the knowing subjects [ĪPVV vol. 3, p. 92: ekapramātṛrūpasaṅkucitaprakāśāveśinīlādyavabhāsavimarśanarūpāt pratyakṣāt samastāvigītavimarśanarūpaparicchedaśūnyāsaṅkucitaprakāśaviśrāntabhāvatattvāvabhāsavimarśanasvabhāva āgamo balavān iti |]. (Therefore, Āgama is considered to be stronger Pramāṇa in comparison to perception). In simple words, whereas the individuality and the luminosity belonging to knowledge are the defining factors of perception, the universal totality of the knowing subject and the reflectivity of experience (pratīti) are that of Āgama. In this connection, the noteworthy contribution of Abhinava is that in the perception and Āgama, the definitional (pāribhāṣika) nature of the object of knowing (prameya) consists in the that of the ‘prakāśya’ (illuminable, i.e., the object of light) and ‘vimṛśya’ (reflectable, i.e., the object of reflection) respectively. In (illuminating) the ‘prakāśya’ (illuminable), the role of light (Prakāśa) belonging to the perception does not depend upon reflection (vimarśanirapekṣa). Exactly in the same way, in (reflecting) the ‘vimṛśya’ (reflectable), the reflective awareness belonging to Āgama does not require Prakāśa. In simple words, as regards perception, the grasping of object solely depends on Prakāśa and in the case of Āgama, it solely depends on reflection. Therefore, the light and reflective awareness are not mutually dependent [ĪPVV vol. 3, pp. 103–104: pratyakṣāgamayor hi yat prakāśyaṃ ca vimṛśyaṃ ca, tat yathākramaṃ prakāśavimarśamukhena anyāpekṣāśūnyam | See also Rastogi 1977, p. 300]. Because here, like the Buddhists, there is no distinction between the pramāṇa and pramā (the fruit of pramāna) where pramāṇa characterizes the element of prakāśa by way of knowing externally and pramā characterizes the element of reflective awareness by way of internalizing reflexively (vimarśana). Therefore, one can also say: in perception we reach the reflective awareness through the window of light and quite contrary to this, we access Āgama through the window of reflective awareness. In both the conditions there is a relation, immediate and devoid of expectation, between the Prameya (object of knowledge) and the Pramāṇa (means of knowing) – in the case of perception the object is related to light and in the case of Āgama, the object is related to reflective awareness. But in the case of inference, the situation is different. In inference, there is the expectation of both prakāśa (illumination) and vimarśa (reflection) with regard to the concomitant or invariably accompanying other object, i.e., hetu or sādhya (for hetu, anumeya/sādhya is the arthāntara, and for anumeya it is hetu that is arthāntara). For the same reason, both light and reflective awareness participate in anumeya [ĪPVV vol. 3, p. 104: pratyakṣe hi prakāśadvāreṇa vimarśo’nyatra tu viparyayaḥ | anumāne tu nāntarīyakavastvantaraprakāśavimarśāpekṣānumeye prakāśavimarśayoga iti sāpekṣatvāt dūrā iyaṃ pramitiḥ prameyāt |]. Hence both of them are mutually dependent. The basis of inference is the relation of invariable concomitance that subsists between the two entities (acting as arthāntara for one another). At the time of inference, the illumination (perception/grasp) of other object that is invariably accompanied by the awareness (i.e., reflection) of the law of concomitance, leads to ascertainment (vimarśana) of sādhya (i.e., concomitant other object). Because of the one depending upon the other – Prakāśa depending on Vimarśa, Vimarśa depending on Prakāśa – the resulting objective grasp ever remains

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mediated, hence interferential knowledge has always been treated as mediated cognition. From this, no one should doubt, Abhinava cautions, that there would be ‘dependence on the other’ even in the case of Āgama – meaning requiring the word – since apperception of meaning (arthavimarśana) presupposes the realization of word (śabdavimarśana) (as a result, knowledge arising out of Āgama too would be mediated). The Śaivas do not accept the independent existence of the reflective awareness of meaning as separate from the reflective awareness of the word [ĪPVV vol. 3, p. 104: na ca vācyamāgame’pi śabdavimarśanapūrvakam arthavimarśanam iti tatrāpi sāpekṣatādi bhaved iti | nahi śabdavimarśanād aparam arthavimarśanaṃ kiṃcit śabdasya svaparavimarśātmakatvāt |], because the very nature of the word is – to be simultaneously aware of self and other (sva-para-vimarśana). The word articulates the meaning while articulating itself first [ĪPVV vol. 2, p. 248: dīpaḥ svaparadīpanaḥ, śabdaḥ svaparaśabdanātmakaḥ, jñānaṃ svaparaprathārūpaṃ, vimarśas tu svaparavimarśarūpo na pṛthak gaṇyate |]. The question (that we are going to raise now) about the fundamental structure of Pramāṇa in the wake of Prakāśa and Vimarśa should have already been raised. Even if pramāṇa and pramāṇa-phala are considered to be one, yet in their conceptual analysis the element of pramāṇa is reduced to Prakāśa and the element of pramā to Vimarśa. As regards perception and inference it does not create a problem because in both the places the element of ābhāsa (i.e., Prakāśa) constitutes pramāṇa and the element of vimarśa constitutes pramā, its end-result. But then what will be the situation in the case of Āgama because the primary nature of Āgama itself consists of vimarśa (reflective awareness). In such an eventuality, what would be the nature of the element of ābhāsa, luminosity in other words, in Āgama-pramāṇa. It seems, both Utpala and Abhinava are aware of this difficulty and they almost seem to be raising this question when they say even in Āgama the element of ābhāsa (ābhāsāṃśa) remains a valid means like perception and the element of vimarśa (vimarśāṃśa) remains the fruit [ĪPVV vol. 3, p. 106: ‘pratyakṣasyeva ca’ iti ābhāsāṃśaḥ pramāṇaṃ, vimarśāṃśaḥ phalam iti |] but neither do they answer this question in this context nor do they take the discussion further. Nonetheless, this question must have an answer. One can make an effort in this direction based on what Abhinava has said at other places. They seem to hold that the positing of the reflective awareness as constitutive of Āgama is directed toward formulating Āgama’s core essence, and not toward establishing Āgama as a pramāṇa. Going strictly by the definition of pramāṇa, it is a logical necessity to have the element of ābhāsa and that of vimarśa in Aḡama by virtue of its being Pramāṇa and the fruit of Pramāṇa respectively. Taking recourse to the following equations we can explain these two thus (Table 2): The basis and meaning of the above formulation will become clear while exploring the structure of Āgama-pramāṇa as grounded in verbalizing (śabdanagatasaṃracanā). In the previous treatment, the verbalizing essence of Āgama has come up for discussion in various ways, but the examination of the different kinds of pramāṇa from the perspective of verbalizing (śabdana) remains yet to be attempted. The Śaivas are absolutely convinced that all the worldly transaction is not rooted in

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Table 2 Elements of Ābhāsa and Vimarśa in Āgama Pramāṇa ābhāsāṃśaḥ ¼ (śabda kā svarūpa þ śabda kī viṣayarūpatā) þ prayoktṛgataśabdanātmakaparāmarśa element of ābhāsa ¼ (the own form of word + the objective nature of word) þ reflective awareness in the form of verbalising (śabdana) as belonging to the speaker vimarśāṃśa ¼ (śabdasvarūpaparāmarśa þ śabdādyarthaparāmarśa) + prayoktṛgataśabdanātmakaparāmarśaparāmarśanaa element of vimarśa ¼ (reflective awareness of the own form of word + reflective awareness of the word as object) + reflecting awareness of the reflective awareness in the form of verbalising (śabdana) as belonging to the speaker Note: aFor the sake of clarity, slight repetition would be appropriate. In this compound-word, the first ‘parāmarśa’ stands for the element of Prakāśa, i.e., the element of ābhāsa or the element of pramāṇa and the subsequent ‘parāmarśana’ for the element of vimarśa or the element of pramā

convention, but instead in verbalizing. Though apparently, this phenomenon is related to the Pratyabhijñā philosophy of language (See Rastogi 2009, 2013a and the chapter titled ‘Vāk’ in Rastogi 2002), its epistemological implications nonetheless have direct bearing on the problem under discussion. In this context, Abhinava says that the śabdana that arises in perception belongs to oneself that is the knowing subject (pramātṛgata) or the user (prayoktṛgata), because it forms part of the selfawareness of the knowing subject [ĪPVV vol. 2, p. 262: śabdanaṃ svagataṃ svasaṃvedanāt |]. The inference here stands for the parārthānumāna (inference for the sake of others). The inference for the sake of others functions in two different ways: one is that when through the five-membered syllogism (in the form of five premises, that too is a mode of śabdana), one makes another person, treated as the inferer, infer. Even Buddhists accept this, however, the only difference is that the steps in syllogism reduce from five to three. [The definition of parārthānumāna in the Buddhist logic has been attempted in terms of ‘statement’ (ākhyāna) of triple mark: trirūpaliṅgākhyānaṃ parārthānumānaṃ | (Nyāyabindu) NB on which the commentary of Dharmottara reads: ākhānaṃ hi punas tat? vacanam | vacanena hi trirūpaṃ liṅgaṃ ākhāyate | Also see Dharmottara’s commentary on the NB 3.2: aupacārikaṃ vacanaṃ anumānam, na mukhyam ityarthaḥ | . . .. . . tasmālliṅgasya svarūpaṃ ca vyākhyeyam, tatpratipādakaś ca śabdaḥ | tatra svarāupaṃ svārthānumāne vyākhātaṃ | pratipādakaś ca śabdaḥ iha vyākhāyeyaḥ |]. And, two: when a word is inferred from a word. In both situations, the śabdana would relate to the other, i.e., it would belong to the inferer (anumātṛgata). With regard to the latter situation, i.e., the inference of the word from a word, Abhinava says that the śabdana taking place from inference belongs to the other. Realistically speaking, from gross point of view, the gross word (sthūlaśabda) is to be deemed as kāryahetu (effect as reason) and not as svabhāvahetu (essential property as reason), when inferring a word from a word. In this state of śabdana we can reach up to this point only. If viewed closely, Buddhists too subscribe to the inferentiality of the word [ĪPVV vol. 2, pp. 262–263: paragataṃ tu anumānāt, tadanumāne kartavye śabda eva sthūlaḥ sthūladṛṣṭyā kāryahetur vastuvṛtte, na tu svabhāvahetur eva | śabdanāvasthaiva hi tatparyantā | ānumānikatvaṃ tāvat parair api upagataṃ].

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In the Āgama-pramāṇa, there is unification between the verbalizing of the self and that of the other. Here the process apparently seems a little complicated. In the śabda-pramāṇa (word as a valid means), śabdana (verbalizing) consists in reflecting upon (comprehending) the reflective awareness (understanding) of the speaker (user) by the knowing subject (pramātā) or the knower (pratipattā) (śabdana ¼ prayoktṛparāmarśa-parāmarśana). As a consequence, in the śabda pramā, the pramā that is born out of Āgama, the reflective awareness of the speaker/user (prayoktṛgataparāmarśa) attains oneness with one’s own reflective awareness (svaparāmarśa). Here the prayoktā means the one who makes use of the word, i.e., speaker, preceptor, guru, scripture, trustworthy-authoritative person (āpta), Established well-known Usage (prasiddhi) or the tradition. The ‘sva’ is listener, knowing subject, perceiver, student. Here the interpretation of Abhinava is very complex, dense, coherent, and multi-layered. As far as I have been able to understand, this is what Abhinava means: the domain of word is the domain of mutual transaction – it is the domain of communication and transmission between the two where the understandings (parāmarśa) of both – one belonging to the speaker and the other to the listener – are fused together. The self-awareness of the knowing subject itself is the substratum of this uni-formity. The worldly transaction is nothing but consists in the mutual permeation in the form: “He/she understands (parāmṛśati) what I am thinking (madīyaṃ parāmarśam) and I understand (parāmṛśati) what he/she is thinking (etadīyaṃ parāmarśam)” [ĪPVV vol. 2, p. 263: madīyaṃ parāmarśamayaṃ parāmṛśati, etadīyaṃ ca aham ityeṣa eva parasparavyāptilakṣaṇo vyavahāraḥ |] The context is that how a śabda (word) enlivens the usage by evolving into the nature of śabdana (verbalizing). Abhinava says this usage of word functions the same way like it would have functioned in the form of śabdapramāṇa (verbal testimony). (This is the reason why we are overcoding this statement of Abhinava in the context of śabdapramāṇa). In the case of normal transaction (constituted by words), the word does in fact borrow the epistemic mechanism of the śabdapramāṇa and conducts itself like one for the simple reason that there is no contradiction because such obstacles as the remembering of the necessary concomitance (vyāptigata-smaraṇa) liable to disrupt the experience are totally absent therein. Just as the word, first grasps its own form, then grasps the object such as blue (denoted by the word) and then captures the user’s (the speaker’s) reflective awareness which essentially consists in verbalizing (prayoktṛgataṃ śabdanātmakaṃ parāmarśam) by necessarily reflectively homologising (abhedenaiva parāmṛśan) it with own reflective awareness. In the same manner, it also effects in the other (parasya: the listener) the realization of the reflective awareness taking place in that user (tatprayoktṛparāmarśaparāmarśanamayatām) by first capturing the reflective awareness of the word’s own form, then capturing the realization of the reflective awareness of the objects like blue etc. [ĪPVV vol. 2, p. 263: paramārthatas tu vyāptismaraṇādisaṃvedanavighnābhāvāt śabdapramāṇarūpatayaiva śabdaḥ svarūpam iva nīlādim iva ca prayoktṛgataṃ śabdanātmakaṃ parāmarśam api svaparāmarśābhedena iva parāmṛśan, parasya tatsvarūpaparāmarśamayatām iva tannīlādy arthaparāmarśamayatām iva ca tatprayoktṛparāmarśaparāmarśanamayatām api vidhatte |]. Thus we can see that the epistemic modality

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of the śabda as a pramāṇa exactly resonates in the epistemological functioning of the Āgama pramāṇa.

Conclusion From the forgoing analysis this is very clear that in the epistemological theorisation of the Trika philosophers, Āgama wields extraordinary importance and their approach has been remarkably profound. Whereas Pratyabhijñā constitutes the universalized modality of knowledge, Āgama embodies the foundational basis of the epistemological thinking. Even then all the three sources of knowledge have their individual roles to play and what is important to note is that all the Pramāṇas mutually support each other [ĪPVV vol. 3, p. 83: parasparānugrahatantratvaṃ pramāṇānām uktam |]. Without going into any further detail we want to conclude with some brief observations. Even though the Pramāṇa as such determines the nature of objective entity (vastu), yet what is manifested in the perception, i.e., the immediate experience, realistically speaking that is the object. In all modes of knowledge, it is the same objective entity that is variously explored. Even though in memory (smṛti), conceptual constructions (vikalpa), etc. the impregnation of objective entity is imperative yet because of being dependent on perceptual residuum (anubhavavāsanā) for their own coming into being, these knowledge-episodes do not dwell on the real nature of the objective entity because it does not figure there in its essential form. As far as the question of cognitions arising out of inference and Āgama, even in them, there is the manifestation of the objective entity as exclusively determined by experience, their inter se distinction is due only to their respective epistemic configuration (yojanāṃśa) [ĪPVV vol. 1, pp. 92–93: sākṣātkāre’nubhavātmani yat pratibhāsate, tadeva vāstv. . ..anubhavavāsanopajīvitvāt smṛter manorājyotprekṣādivikalpānāṃ ca na tadvicāra iti . . .. saṃbhāvanānumānāgamajaniteṣv api jñāneṣu tadeva bhāsate, kevalaṃ yojanāmātramadhikam | The saṃbhāvanānumāna can be interpreted in two ways: hypothesis and inference or hypothetical inference. I have taken it simply referring to inference]. The inference is that means of knowledge which is dependent both on perception and Āgama. As far as the question of arriving at the invariable concomitance is concerned, the crucial causal role of perception is acceptable to all. But the inference is also based on Āgama and, in this regard, the Śaivas find themselves in alignment with Patañjali. [This issue has been taken up in detail in my book: See Rastogi 2013b, pp. 113–140]. Towards determining the hetu (reason) our sole anchor happens to be lokaprasiddhi (popular convention), a form of Āgama, as another pramāṇa (pramāṇāntara), for deciding the absence of its having been created by a Yogi [ĪPV vol. 2, p. 176: yadi pramāṇāntareṇa lokaprasiddhyā yoginirmāṇanatvāsyābhāvo niścito bhavati |]. Realistically speaking, even among the Śaivas, it is not possible to ascertain invariable concomitance without ascertaining the operational extent of Niyati (Principle of Necessity) and in that case, prasiddhi is invariably necessitated [ĪPVV vol. 2, p. 172: nanu evaṃ dhūmābhāsārthī kimupādattām |

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uktamatra ‘yatraiva ābhāsāṃśe niyatiśaktyā kāryakāraṇabhāvo gṛhītaḥ, tatraiva upādānaṃ tadarthinaḥ | vakṣyate ca etat ‘yoginirmāṇatābhāve . . .. . .. . .. . .. . .. . . | 2.4.11’ ityatra |]. It is not the case that perception and inference always lean on Āgama. Sometimes even they lend support to Āgama as well. For example, if someone is looking for uniqueness and excellence in a painting and if the subject of painting relates to something that is a matter of sight, in such a situation, perception and inference, because of having perceptible object, strengthen Āgama, generally linked to an unseen object [ĪPVV vol. 3, p. 83: evam iyatā pratyakṣānumānayor dṛṣṭaviṣayatayā sātiśayānveṣaṇīyaṃ satyādau citrādau dṛṣṭaviṣayatāyām āgamam api anugrahītum asti sāmarthyam iti darśayatā parasparānugrahatantratvaṃ pramāṇānām uktam |]. In exact contrast to it, an object that is grasped through perception seeks shelter in Āgama alias prasiddhi that is hailed by the seasoned wise people (śiṣṭa). For instance, ‘This is that cow whose milk is healthy and sacred’, ‘This is that she-donkey whose milk burns like fire and is not sacred’ [ĪPVV vol. 3, p. 83: tathāhi pratyakṣadṛṣṭam api arthakriyāsu śiṣṭair yojyamānām āgamalakṣaṇāṃ prasiddhimapekṣate eva, iyaṃ sā gauryasyāḥ kṣīraṃ pathyaṃ pavitraṃ ca, iyaṃ sā gardabhī yasyāḥ payo’gnisādanam apavitraṃ ceti |]. For this reason, so far as the question of motivation for action of someone, who is aspiring to obtain or effect something, it is always due to a combination or group of several valid means of knowing. This is why all the sources of knowing become useful. [ĪPVV vol. 3, p. 83: tata eva pramāṇasamūhād eva pravṛttir iti vakṣyate ‘sā tu deśādikā———————————————————|’ 2.3.9 iti |].

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Śivadṛṣṭi of Somānanda with the vṛtti by Utpaladeva, edited by Paṇḍit Madhusūdan Kaul Shāstrī, KSTS 54, Srinagar, 1934. Spandakārikā with the Vivṛti of Kallaṭa Rāmakaṇṭha, edited by J.C. Chatterji with the cooperation of the Pandits of the Research Department of Kashmir State, KSTS 6, Śrīnagar, 1913. Stava-chintāmaṇi of Bhaṭṭa Nārāyaṇa with commentary by Kṣemarāja, edited by Paṇḍit Mukunda Rama Shāstrī, KSTS No. 10, Bombay, 1918. Tantrāloka of Abhinavagupta, edited by Paṇḍit Mukunda Rama Shāstrī and Paṇḍit Madhusūdan Kaul Shāstrī, 12 vols., Kashmir Series of Texts and Studies 23, 28, 29, 30, 35, 36, 41, 47, 52, 57, 58, 59, Srinagar, 1918–1938. Tantrāloka with Commentary by Rājānaka Jayaratha, edited by R.C.Dwivedi and Navajivan Rastogi, Vols. I–VIII, Delhi, 1987. The Vākyapadīya of Bhartṛhari with the Vṛtti, Chapter 1, Eng. tr. K.A.Subramania Iyer, Deccan College, Poona, 1965. The Vākyapadīya of Bhartṛhari, Kāṇḍa II, with the commentary of Puṇyarāja and the ancient Vṛtti, K.A. Subramania Iyer, Motilal Banarsidass, Delhi, 1983. Vākyapadīyam, Bhartṛhari, Svopajña vṛtti va harivṛṣabha kṛta paddhati ke sātha, Ed. K.A. Subramanya Ayyar, Part I, Deccan College, Poona, 1965.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Alphabetical List of Central Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Short Argumentation in the Satya-śāsana-parīkṣā . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logical Principles Used in the Argumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . prayoga and anumāna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . sādhana, avinābhāva, tarka . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . upalabdhi- or anupalabdhi-hetu? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . pakṣa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The Jainas used variations of the pan-Indian sets of dialectical, epistemological, and logical principles in order to justify their ontological and soteriological convictions. The article analyses the application of logical principles in an ontological argument of a Jaina Digambara author. After the explication of the ontological aspect of the argumentation, central logical principles are examined against the backdrop of their definitions in Digambara Sanskrit works. Keywords

avinābhāva · anyathānupapatti · pakṣa · hetu · prayoga · anumāna · sādhya · sādhana

H. Trikha (*) Centre national de la recherche scientifique, Paris, France e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_17

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Introduction Vidyānandin flourished in tenth-century Karnataka, where Jainism had prospered for centuries during the reign of the Western Gaṅga dynasty. In striving for religious supremacy, Jaina philosophers had refined Jainism’s ancient theories on the nature of reality and emancipation and defended them fiercely against the concepts of opposing traditions. Vidyānandin belongs to a group of authors who took part in this debate by frequently using dialectical, epistemological, and logical principles, which are often subsumed under the terms vāda (“scholarly debate”) and pramāṇa (“means of valid cognition”). These principles were widely used by scholars of various traditions of Indian thought and often constituted a common matrix for the justification and refutation of ontological or soteriological concepts. Today, the best-known variations of these principles are those defined in the Nyāyaśāstra and in the logical-epistemological school of Buddhism. The Jainas’ contributions to the literature on these principles are considerable but less well researched. Examples of Jaina Sanskrit works, where full sets of dialectical, epistemological, and logical principles were discussed and have been translated or closely paraphrased by modern researchers include Māṇikyanandin’s Parī kṣā-mukha-sūtra (Ghoshal 1940), Hemacandra’s Pramāṇa-mī māṃsā (Mookerjee and Tatia 1946), Vādidevasūri’s Pramāṇa-naya-tattvālokālaṅkāra (Bhattacharya 1967), Yaśovijaya’s Jaina-tarka-bhāṣā (Bhargava 1973), Siddhasena Mahāmati’s Nyāyāvatāra (Balcerowicz 2001), Akalaṅka’s Laghī yas-traya (Clavel 2008), or Prabhācandra’s Prameya-kamala-mārtaṇḍa (Balcerowicz 2013). The present article offers a very brief insight into how an acclaimed Jaina Digambara scholar used logical principles. Reflections on the consistency of the principles are beyond the scope of this introduction and the historical development of the various terms by which the principles are expressed is only mentioned in passing. After a list of terms discussed in more detail, a translation and the ontological background of a short argumentation from one of Vidyānandin’s ten works will be presented (part 1). Then some of the terms and concepts used in the argumentation will be examined with the help of definitions from other Jaina Digambara works (part 2). The article is a revised rendering of an extract from the author’s German study on the Satya-s´āsana-parī kṣā (2012: 253–283). The Sanskrit text of the argumentation is the one established in the German study. More details on the text, e.g., variants and passages corresponding to other Sanskrit works, and on Vidyānandin and his works in general have been stated in the original study. According to the wishes of the editors, Sanskrit lemmata in compounds have been separated in this article by hyphen (notable exceptions are names of persons and traditions). Some passages in this article were improved subsequent to the insightful comments of Ph. Maas, Vienna; G. Pellegrini, Turin; J. Soni, Innsbruck; and T. Watanabe, Vienna.

Alphabetical List of Central Terms Alternative meanings or translations are taken from Balcerowicz 2013 (B.), Ganeri 2001 (Ga.), Phillips 2012 (Ph.), or Steinkellner (S.) 2009.

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Logical Argument in Vidyānandin’s Satya-śāsana-parīksā ˙

Anumāna Anyathānupapatti Avinābhāva Tarka Pakṣa Prayoga Vyāpti Sādhana Sādhya Hetu

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“Inference” “otherwise not explicable” (B., p. 1), “impossible otherwise” (G: 145) “non-occurrence (of one) without (the other)” (S: 198) “suppositional reasoning” (Ph: 30), “suppositional knowledge” (B: 69) “inferential subject” (Ph: 54), “subject term” (B: 41), “logical subject” (ibid.) “formal demonstration” (Ph: 53), “formal expression” (B: 41) “pervasion” (Ph: 60), “invariable concomitance” (B: 42) “proving property” (B: 30), “prover” (Ph: 54) “inferable property” (B: 30), “target” (Ga: 36), “probandum” (Ph: 54) “logical reason” (B: 41), “evidence” (Ga: 21)

A Short Argumentation in the Satya-śāsana-parīksā ˙ In the Satya-s´āsana-parī kṣā (SŚP) Vidyānandin discusses central tenets of opposing philosophical traditions and refutes them from the Jaina point of view. The following short extract from the work is found in the section of the refutation of the Vaiśeṣika school of thought, where the notion of inherence (samavāya) plays a central role. The Vaiśeṣikas assert that a phenomenon of, say, a “cognizing soul” would presuppose two complete distinct things, a substance (dravya) “soul” and a quality (guṇa) “cognition” that are connected by inherence in order to become a unified whole. The Vaiśeṣikas’ opponents, among them Vidyānandin, argue that such an ontological concept would require a further form of connection: Each of the two elements, which is allegedly connected to the other by inherence, would also need to be somehow connected to inherence itself. If a “cognizing soul” is brought about by the connection of “soul” and “cognition” through inherence, then inherence must also be connected individually to both, “soul” and “cognition.” Inherence needs to be based (ās´rita) on them. The Vaiśeṣikas have several strategies to refute this objection. One is employed by Uddyotakara, who denies that inherence needs in fact to be based on the things it characterizes (samavāyin). In the following argumentation Vidyānandin attempts to show the fallacy of Uddyotakara’s position. atha “anās´ritaḥ samavāyaḥ” iti mataṃ [II] tadā na sambandhaḥ samavāyaḥ, sambandhibhyāṃ bhinnasyoˆbhayā^s´ritasy^ a iva saṃyogavat sambandhatva-vyavasthiteḥ. [III] tathā ca prayogaḥ: “samavāyo na sambandhaḥ, sarvathā^nās´ritatvāt. yo yaḥ sarvathā^nās´ritaḥ, sa sa na sambandhaḥ, yathā dig-ādiḥ. sarvathā^nās´ritas´ ca samavāyaḥ. tasmān na sambandhaḥ” iti. (SŚP 38,7–10 [II 33]) [I] If one now holds the opinion that inherence is not based (on the things it characterizes), [II] then (it follows that) inherence is not a connection, because a connection is established to be different from the two connected things (and) precisely based on both of them, such as, for example, contact (is different from the two things in contact but based on both of them). [III] And the formal demonstration (of this reads) as follows: “Inherence is not a connection, because it is in no way based. Whatever is in no way based, that, in fact, is not a connection, such as, for example, space, etc. And inherence is in no way based. Therefore it is not a connection.” [I]

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The argumentation can be divided into three sequences, which are indicated above by Roman numerals in superscript. Sequence [I] states the opponent’s opinion, sequence [II] articulates Vidyānandin’s objection, sequence [III] expresses the objection in a formal way. The two adduced examples point to an ontological distinction within the Vaiśeṣika’s system of categories: contact (saṃyoga) is a quality, space (dis´) a substance. As a quality, which always has to reside (ādheya, ās´rita) somewhere, contact is based on the two things it connects (sambandhin). Space, on the other hand, is one of the few eternal substances, which occurs “without a substrate (anās´rita)” (Halbfass 1992: 148), it is not based anywhere. Against this backdrop of ontological suppositions, the basic line of argumentation in sequence [2] holds against the opponent, that his concept of inherence lacks the basic requirement for any connection (sambandha) whatsoever, which can be conceived in the presently undisputed instance of contact. Without being firmly related to the things it allegedly unifies, inherence cannot connect them and the concept therefore fails, in turn, to explain how two distinct things become a single entity.

Logical Principles Used in the Argumentation Sequence [3] expresses Vidyānandin’s concern with Uddyotakara’s position according to a fixed scheme of argumentation. In summarizing what will be dealt with in more detail below, it may be said, that Vidyānandin understands this scheme as the “formal demonstration” (prayoga) of a cognitive process, namely, “inference” (anumāna). The potential of the linguistic device for conveying a valid cognition lies mainly, some Jaina authors say, in its components “proposition” (pratijn˜ ā) and “logical reason” (hetu). In these components it is claimed that a “subject term” (pakṣa) is characterized by two properties (dharma): an explicit “predicate” (sādhya) and “evidence” (hetu) for it. The claim is thought to be justified because of an underlying inference, in which these two properties appear as an “inferable property” (sādhya) and a “proving property” (sādhana). In a valid inferential cognition, these properties are invariably concomitant (! anyathānupapatti, avinābhāva, vyāpti). Due to this nexus, the cognition of the proving property entails the cognition of the inferable property and the knowledge, that it qualifies the “inferential subject” (pakṣa). The knowledge of the nexus is established on the basis of “suppositional reasoning” (tarka), which is according to Jaina authors an independent means of knowledge.

prayoga and anumāna A prayoga is the formulation of a fact known by inference (anumāna), in which the inference is formulated for another person (parā^rthā^numāna) and thus may be applied as a statement of proof (sādhaka-vākya) in an academic dispute (see TPS I: 95–99, lemma avayavaḥ). Statements of this kind differ from other forms of

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argumentative procedure in their standardized succession of individual components, the so-called members of proof (avayava, aṅga). In the SŚP two types of this proof formula occur: a trinominal pattern with the components proposition (pakṣa, pratijn˜ ā), reason (hetu), and example (udāharaṇa, dṛṣṭānta) as well as a fivefold pattern in which application (upanaya, upasaṃhāra) and conclusion (nigamana) are added as two further components. In the prayoga under discussion, the fivefold pattern is used: [A]

samavāyo na sambandhaḥ [B] sarvathā^nās´ritatvāt. [C] yo yaḥ sarvathā^nās´ritaḥ, sa sa na sambandho yathā dig-ādiḥ. [D] sarvathā^nās´ritas´ ca samavāyaḥ. [E] tasmān na sambandhaḥ. (SŚP II 33)

Component [A] expresses the proposition (pratijn˜ ā), i.e., it is claimed that the subject term (pakṣa) is qualified by a particular property, which is the property that needs to be established (sādhya). In Component [B] the logical reason (hetu) is stated, i.e., the property mentioned here is claimed to be invariably concomitant with the property that needs to be established and can thus function as the proving property (sādhana). In Component [C] the invariable concomitance of the two properties is explicated and exemplified (udāharaṇa). In Component [D] the proving property is explicitly applied (upasaṃhāra) to the subject term. In component [E] the conclusion (nigamana) is drawn by asserting the proposed property for the subject term. Being the “formal demonstration” (Phillips 2012: 53) of an inference, a prayoga conveys a cognition which is brought about by this particular means of valid cognition (pramāṇa). In Digambara circles of the tenth and eleventh centuries the following definition was popular: sādhanāt sādhya-vijn˜ ānam anumānam. (TAŚV 1.13.120ab = PrP 70,34 = PMS 3.14 = PKM 354,1) An inference is the cognition of the inferable property due to the proving property.

With this definition of inference the first two components of its formal demonstration gain more weight than the latter three. The two properties regarded here as central for the cognitive process are expressed in the first two components of the prayoga. The proposition has to express the inferable property (sādhya) and stating the logical reason has to consist of the proving property. Since the decisive factors are thus being stated immediately at the beginning, the further components of the formal demonstration are not regarded as compulsory for the successful communication of an inference. For some Jaina philosophers “ . . . just to mention the pakṣa (as subject term/logical subject) and the hetu is enough . . . ” (Balcerowicz 2013: 41, n. 1). Such a “minimal structure” (Dunne 2004: 26, n. 30) is advocated by, e.g., Māṇikyanandin, a Digambara author who was probably an older contemporary of Vidyānandin: bāla-vyutpatty-arthaṃ tat-trayoˆpagame s´āstra evā^sau. na vāde, anupayogāt. (PMS 3.46) When these three [viz., example, application and conclusion] are joined to them [viz., proposition and reason], happens this in the scientific treatise only in order that ignorant

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people understand. (These three are) not (applied) in the debate, because (they are) not conducive [to what needs to be established].

Only the logical reason (hetu), i.e., the proving property which occurs in the cognitive process or is stated in the formal demonstration of the cognition, is conducive: samarthanaṃ vā varaṃ hetu-rūpam anumānā^vayavo vā^stu, sādhye tad-upayogāt. (PMS 3.45) [Opposed to the conception that example, application, and conclusion are components of proof] it is better rather to have either a justification in the form of a logical reason or as the component of an inference, since (these alone are) conducive to what needs to be established.

In commenting on this statement, the Digambara Prabhācandra, who flourished at least one generation after Vidyānandin, adds: samarthanaṃ hi nāma hetoḥ . . . sva-sādhyena ’vinābhāva-sādhanam. sādhyaṃ prati hetor gamakatve ca tasy^ a ivoˆpayogo nā^nyasyêti. (PKM 376,16–18) Because . . . justification is precisely the proof that the logical reason does not occur without what needs to be established by it. And if the logical reason makes known what needs to be established, it alone is conducive, nothing else.

Against the backdrop of this conception, the main focus has to be on the initial thesis and the logical reason stated for it, in order to understand the prayoga under discussion. Therefore, which notions are attached in a Jaina context to the two respective components of the underlying inference and to their relation?

sādhana, avinābhāva, tarka sādhanāt sādhya-vijn˜ ānam anumānaṃ vidur budhāḥ | ... anyathānupapatty-eka-lakṣaṇaṃ tatra sādhanam | sādhyaṃ s´akyam abhipretam aprasiddham udāhṛtam || (TAŚV 1.13.120f. ad TA 1.13) The learned (persons) know inference to be the cognition of the inferable property due to the proving property. . . . Of these, the proving property’s single characteristic is to be impossible otherwise. The inferable property is explained to be apt, intended (and) not generally known.

Vidyānandin attributes a single characteristic to the proving property here: it is “impossible otherwise” (anyathānupapatti, s. Ganeri 2001: 145). In other contexts of Jaina philosophical literature, this term is to be interpreted in the sense of “inexplicability otherwise” (s. Balcerowicz 2003, Clavel 2008: 164 or Gorisse 2015: 14). Here, where not a linguistic explication but an element of the prior epistemic process is addressed, an interpretation in the sense of “the proving property does not occur in another manner” seems more suitable. In his own commentary to this passage, Vidyānandin explains anyathānupapattyeka-lakṣaṇa with sādhyā^bhāvā^sambhava-niyama-lakṣaṇa (TAŚVA 197,33): When the proving property is given, it is “necessarily impossible for the inferable

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property to not occur.” In other words, a particular property functions as a proving property only, when it complies with a “necessity” or “invariable rule” (niyama, s. TPS II: 132f.) which pertains to the impossibility (asambhava) of the absence (abhāva) of another particular propert, which is under this sole condition “inferable.” By insisting on the singularity of this characteristic (eka-lakṣaṇa), Vidyānandin confronts alternative definitions, notably those of Buddhists who assume “three conditions of a logical reason” (trairūpya-liṅga, s. Oetke 1994: 3). The Jainas’ claim for a single “mark of a good reason” (Ganeri 2001: 144) seems to have evolved in, or shortly before, the eighth century when Pātrasvāmin, Siddhasena Mahāmati and Akalaṅka attempted to show that the “three characteristics (of the logical reason) are impotent” (klībās tri-lakṣaṇāḥ; Tattva-saṅgraha 1364d, see Balcerowicz 2003: 343). Vidyānandin shares this concern, but unlike his predecessors “apparently equates vyāpti with anyathānupapannatva” (ibid., p. 354), i.e., he uses also the latter term to address the logical nexus of two properties. This becomes clear from the following two passages. In his Yukty-anus´āsana-ṭīkā, while reflecting on the nature of a “reasonable instruction” (yukty-anus´āsana) Vidyānandin states: anyathānupapannatva-niyama-nis´caya-lakṣaṇāt sādhanāt sādhyā^rtha-prarūpaṇaṃ yuktyanus´āsanam. (YAṬ 122,20f.) A reasonable instruction is the demonstration of a subject matter that needs to be established by a “logical reason the characteristic of which is the determination of the invariant rule ‘inexplicability otherwise’.” (Balcerowicz 2003: 354)

In the Pramāṇa-parī kṣā, Vidyānandin introduces his long examination of a Buddhist position as follows: tatra sādhanaṃ sādhyā^vinābhāvi-niyama-nis´cay^ a ika-lakṣaṇam, lakṣaṇā^ntarasya sādhanā^bhāse ’pi bhāvāt ... (PrP 70,35f.) Of these [i.e., of the inferable and the proving properties], the proving property’s single characteristic is the ascertainment of the rule that it does not occur without the inferable property because other characteristics pertain also to a fallacious proving property . . .

In the four passages referred to above, the following phrases are equivalent in the definition of the proving property (the additionally cited passages contain the respective similar definitions in Vidyānandin’s Āpta-parī kṣā and Patra-parī kṣā): anyathānupapatti-eka-lakṣaṇa (TAŚV 1.13.120f., cf. APṬ 64,7; PaP 3,18) sādhyā^bhāvā^sambhava-niyama-lakṣaṇa (TAŚVA 197,33) anyathānupapannatva-niyama-nis´caya-lakṣaṇa (YAṬ 122,20f., cf. APṬ 64,2; PaP 9,36f.) sādhyā^vinābhāvi-niyama-nis´caya-eka-lakṣaṇa (PrP 70,35f., cf. APṬ 47,26; PaP 2,28)

Due to this functional equivalence it can be safely assumed that Vidyānandin uses interchangeable terms here in order to address the invariable concomitance of the inferable and the proving properties. In the latter two definitions, the proving property is said to involve also a “determination” or an “ascertainment” (nis´caya, see Kellner 2004: 9), either the “ascertainment of the rule of its non-occurrence without the inferable

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property” or the “ascertainment that restricts the non-occurrence of the proving property to the inferable property.” These alternatives for analyzing the compounds reflect different positions of Mīmāṃsā and Buddhist philosophers and divergent opinions of modern scholars (see Steinkellner 2009). In order to pinpoint Vidyānandin’s exact position within this discourse, the additionally cited passages above are to be examined. Whatever Vidyānandin’s opinion here was in detail – according to the Jainas the inferential process is clearly limited to the ascertainment that the invariable concomitance applies to the properties in question and does not bring about the cognition, whether the concomitance exists in fact or is a mere figment of the imagination. The latter is decided upon by a separate means of knowledge, i.e., “suppositional reasoning” (tarka). sādhyā^vinābhāvitvena nis´cito hetuḥ. (PMS 3.15) The logical reason is ascertained to not occur without the inferable property. kuto ’sau . . . avinābhāvo nirṇī yata ity āha tarkāt tan-nirṇayaḥ, na punaḥ pratyakṣā^der iti. (PKM 369,11–13 with PMS 3.19) [To the question] “How is this ‘non-occurrence without an other’ determined?”, [Māṇikyanandin] replied “It is determined by suppositional reasoning, not however by perception or [one of the] other [means of knowledge].”

Rational processes addressed by the term tarka figure prominently in both the early Nyāya tradition (Kang 2010) and Navya-Nyāya (Guha 2012). For Ganeri (2012: 13) these kinds of processes represent one of two dominant patterns of rationality within Hinduism, namely, “hetu, ‘evidence-based-rationality’, and tarka, ‘hypothesis-based-rationality’” (ibid.). In Jaina Sanskrit works, the term tarka and its frequent synonym ūha gained significance from the fourth century onwards, since the terms are attested in the Tattvā^rthā^dhigama-bhāṣya. There they appear in a string of synonyms which are reminiscent of their use in the early Nyāya tradition (Balcerowicz 2001: 184f.). The mental process associated with these terms came to be considered as a separate means of knowledge (ibid. 2003: 354–359), because it was thought to cause a unique result which cannot be effected by another means of knowledge (s. TAŚV 1.13.84–119 or PP 70,26–33). Māṇikyanandin and his commentator Prabhācandra outline the process as follows: upalambhā^nupalambha-nimittaṃ vyāpti-jn˜ ānam ūhaḥ: idam asmin saty eva bhavati, asati na bhavaty eveti ca. yathā^gnāv eva dhūmas tad-abhāve na bhavaty eveti ca. (PMS 3.11–13) ‘Presumptive knowledge is the cognition of invariable concomitance by reason of comprehension or non-comprehension: x occurs only when y is there, and [y] does not occur only when [x] is not there’ (Balcerowicz 2003: 257) like, e.g., smoke occurs only when fire is there and [fire] does in fact not occur when [smoke] is not there. upalambhā^nupalambhau sādhya-sādhanayoḥ . . . sakṛt punaḥ punar vā dṛḍhataraṃ nis´cayā^nis´cayau, na bhūyodars´anā^dars´ane. (PKM 348,5f.) Comprehension and non-comprehension consist in the very firm determination or non-determination of the inferable and the proving properties (which) occur once or again and again; (they do) not (consist in) a repeated observation.

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The process of acquiring a firm knowledge about the invariable concomitance of two properties requires a variety of individual conditions. One of the auxiliary means to acquire this knowledge seems to be the information about different types of logical reasons, as many Jaina tracts on inference offer detailed discussions on the classification of types of the logical reason.

upalabdhi- or anupalabdhi-hetu? Ghoshal (1940: 120) sketches 22 types of logical reasons in Māṇikyanandin’s Parī kṣā-mukha-sūtra. This scheme is referred to here for the further discussion of the prayoga under consideration. There (see p. 5) it is claimed that the property “being in no way based” (! sarvathā^nās´ritatva) can function as the evidence for the property “being no connection” (! asambandhatva). The invariable concomitance of the two properties is expressed as follows: yo yaḥ sarvathā^nās´ritaḥ, sa sa na sambandhaḥ . . . (SŚP II 33) Whatever is in no way based, that, in fact, is not a connection . . .

Māṇikyanandin offers two basic distinctions for the logical reason. Firstly, he distinguishes those reasons which rely either on cognition or on non-cognition (sa hetur dvedhoˆpalabdhy-anupalabdhi-bhedāt; PMS 3.57). Secondly, both can be applied either to the affirmation or to the negation of the inferable property (upalabdhir vidhi-pratiṣedhayor anupalabdhis´ ca; PMS 3.58). The type of the logical reason in the prayoga examined in this article is clear with regard to the latter distinction: The inferable property expresses a negation (“not a connection”). But the attribution to one of the elements of the first distinction is not immediately evident: Does the logical reason rely on the cognition (upalabdhi) of the property “not being based on something” (anās´rita)? Or does it rely on the non-cognition (anupalabdhi) of the property “being based on something” (ās´rita)? In other words: Is something not a connection, because we perceive that it is in no way based? Or is it not a connection, because we do not perceive that it is in any way based? The question can be delimited with the help of further distinctions in Māṇikyanandin’s scheme. Below the negations supported by cognition (section “4.3.1”) and the negations supported by non-cognition (section “4.3.2”) are briefly touched upon, before possible candidates for the classification of the prayoga can be compared (section “4.3.3”).

4.3.1 If the logical reason supports a negation by cognition (upalabdhi-hetu), the proving property consists of a property which is contradictory (viruddha) to the inferable property (viruddha-tad-upalabdhiḥ pratiṣedhe . . . ; PMS 3.71). For example: nā^sty atra s´ī ta-spars´a auṣṇyāt. (PMS 3.72) There is no sensation of cold here, because it is hot.

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The properties “being hot” (auṣṇya) and “being cold” (! s´ī tatva) are contradictory. For a particular location, the cognition of “being hot” entails the absence and thus the negation of “being cold.” The example serves to illustrate the first of six subtypes for this form evidence. This subtype is classified as “cognition of the pervaded” (vyāpyoˆpalabdhi). Prabhācandra explains it as follows: auṣṇyaṃ hi vyāpyam agneḥ. sa ca viruddhaḥ s´ī ta-spars´ena pratiṣedhyena. (PKM 385,8f.) Because heat is pervaded by fire. And this is contradictory to the sensation of cold, which can be negated.

The other five subtypes of this category of logical reason seem to be of no relevance for classifying the evidence put forward in the prayoga in question. The further types pertain to the cognition of a contradictory effect (kāryoˆpalabdhi, PMS 3.73) or cause (kāraṇa, PMS 3.74) or to the cognition of contradictory forms of occurrence, namely, prior (pūrva-), posterior (uttara-), or synchronous occurrence (saha-cāra, PMS 3.75–77).

4.3.2 The types of logical reason that support a negation by non-cognition (anupalabdhihetu), are classified like their respective counterparts, except that the “non-cognition of the nature” (sva-bhāvā^nupalabdhi) is added as a seventh type (PMS 3.78) and that the “cognition of the pervaded” (vyāpyoˆpalabdhi) is paralleled by the “non-cognition of the pervasive” (vyāpakā^nupalabdhi). This latter subtype could match the type of evidence put forward in the prayoga examined in this article. Logical reasons pertaining to the category rely on the non-cognition of a property which is not contradictory (aviruddha) to the inferable property. Māṇikyanandin gives this example: nā^sty atra s´iṃs´apā vṛkṣā^nupalabdheḥ. (PMS 3.80) There is no redwood tree here, because a tree is not perceived.

The properties “being a redwood tree” (s´iṃs´apātā) and “being a tree” (vṛkṣatva) are not contradictory. Inversely, the proving property (! vṛkṣatva) pervades (vyāpaka) the inferable property (! s´iṃs´apātā): If we do not perceive a tree in a particular location, it can be precluded that we find a redwood there.

4.3.3 Against this backdrop, two alternatives for underlying presuppositions in the prayoga can be grasped more clearly. – First alternative upalabdhi-hetu: The property “being a connection” (sambandhatva) and the property “being not based on something” (anās´ritatva) are understood to be contradictory (viruddha). A negation of “being a connection” would presuppose a cognition of “being not based on something.” – Second alternative anupalabdhi-hetu: The property “being a connection” and the property “being based on something” (ās´ritatva) are understood to be not contradictory (aviruddha). A negation of “being a connection” would presuppose a non-cognition of “being based on something.”

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A final conclusion in favor of one of the two alternatives cannot be drawn here. This would require a more detailed study of Māṇikyanandin’s classifications scheme (see Balcerowicz 2013 and Gorisse 2015) and a comparison of the Jainas’ schemes with those of Nyāya (see, e.g., Preisendanz 1994: 528–540) and the Buddhist logical-epistemological traditions (see, e.g., Kellner 2007: 47–56). However, against the backdrop of the material presented so far, the second alternative seems to be more likely. According to the two alternatives, either “being not based on something” (anās´ritatva) or “being based on something” (ās´ritatva) has to be assumed as the proving property. The first alternative implies that a negative property would be an actual content of consciousness. But how could such a negative notion occur, if it is not conditioned by a positive notion prior to it? Thinking of Apohavāda, such a theory cannot be entirely excluded in Indian philosophical contexts, but here, in the context of Jaina epistemology, it is unlikely to apply. It appears therefore that “being based on something” is to be assumed as the proving property of our prayoga. The notion, that something is based on another, can be consolidated in various individual cognitions, so that its non-cognition in a particular instance can be deployed as an element of poof. With this, a preliminarily summary of the mental processes that lead to the notion of an invariable concomitance of the properties expressed in the prayoga reads as follows: On the basis of respective perceptions, suppositional reasoning (tarka) results in the knowledge, that “everything, which possesses the property ‘being a connection’ also possesses the property ‘being based on something’” (* yad yat sambandhatvamat, tat tad ās´ritatvamat). This in turn leads to the further reflection (ūha) that “something which does not possess the property ‘being based on something’ does not possess the property ‘being a connection’” (* yad yad anās´ritatvamat, tat tan na sambandhatvamat). It is this thought which appears in the SŚP in a simplified form: “Whatever is in no way based, that, in fact, is not a connection” (yo yaḥ sarvathā^nās´ritaḥ, sa sa na sambandhaḥ; SŚP II 33).

paksa ˙ When the invariable concomitance has been established by suppositional reasoning, the cognition of the proving property entails the cognition of the property invariably connected with it. In addition, as seen above, the non-cognition of the proving property entails the absence of the property invariably connected with it. The subject of these acts of inference is, in the strict sense, nothing but a property. In some cases the notion of the inferential subject (pakṣa) is extended: sādhyaṃ dharmaḥ kvacit tad-vis´iṣṭo vā dharmī – pakṣa iti yāvat. (PMS 3.25f.) At a particular instant, the inferable matter is a property, or the possessor of this (property), which is qualified by it. Insofar an inferential subject (can be spoken of).

Prabhācandra explains on what occasion the inferential subject is the possessor of the inferable property:

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kvacid vyāpti-kāle sādhyaṃ dharmo nityatvā^dis ten^ a iva hetor vyāpti-sambhāvāt. prayogakāle tu tena sādhya-dharmeṇa vis´iṣṭo dharmī sādhyam abhidhī yate, pratiniyata-sādhyadharma-vis´eṣaṇa-vis´iṣṭatayā hi dharmiṇaḥ sādhayitum iṣṭatvāt. (PKM 371,2–5) At a particular instant, (i.e.) during the time of (the cognition of) the pervasion, the inferable subject is a property like “being permanent”, etc., because only this (property) can pervade the logical reason. But during the time of the formal demonstration, the possessor of this inferable property, which is qualified by it, is expressed as the inferable subject, because one wants to establish the possessor as being precisely qualified by a qualification, which is the determined inferable property.

In the prayoga under consideration, the property “being no connection” (asambandhatva) is therefore understood to be a qualifying property for “inherence” (samavāya), its possessor (sādhya-dharma-vis´iṣṭa-dharmin). The articulation of such a subject in the proposition (pratijn˜ ā) of the prayoga serves to explicate the claim that the inferable property (sādhya) pertains to this particular subject: sādhya-dharmā^dhāra-sandehā^panodāya gamyamānasyā^pi pakṣasya vacanam. (PMS 3.34) The articulation of the inferential subject, although it is understood, serves to remove a doubt regarding the substrate of the inferable property.

Anantavīrya IV, another commentator of the Parī kṣā-mukha-sūtra, explains the phrase “although it is understood” (gamyamānasyā^pi): tad-arthaṃ gamyamānasyā^pi sādhya-sādhanayor vyāpya-vyāpaka-bhāva-pradars´ anā^nyathā^nupapattes tad-ādhārasya gamyamānasyā^pi pakṣasya vacanaṃ prayogaḥ. (PRM 32,15–17)

The articulation, i.e., the demonstration of the subject serves this (removal of a doubt), although it is understood, i.e., although the substrate of this (inferable property) is understood because the inferable and the proving properties do not occur [/ are not explicable] otherwise than in the exemplified manner, i.e., as being pervaded and pervading (properties).

Summary With this short glimpse into the conceptional background of some logical principles in Digambara works, their use in the argument SŚP II 33 can be summarized as follows: The argument is a “formal demonstration” (prayoga) of an “inference” (anumāna), i.e., claims to be the linguistic explication of a cognitive process, which is considered to be a “valid means of cognition” (pramāṇa). The argument is restructured below according to three decreasing degrees of the cognitive purport of its components. samavāyo na sambandhaḥ sarvathā^nās´ritatvāt.

Articulation of a “proposition” (pratijn˜ ā) and its justification (samarthana). In the proposition it is claimed that a particular predicate, i.e., “is not a connection” (. . . na

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sambandhaḥ), pertains to the subject “inherence” (samavāya). It is assumed that the predicate expresses a particular “property” (dharma), i.e., the “property that needs to be established” (sādhya1) and that the “subject term” (pakṣa1) expresses the “possessor of the property” (dharmin). The proposed property is thought to be an “inferable property” (sādhya2) and its possessor is understood to be a “subject of inference” (pakṣa2). The “logical reason” (hetu) for this assertion is another property, i.e., “being in no way based on something” (sarvathā^nās´ritatva). This property is stated in the justification as the “proving property” (sādhana), because it is assumed to be invariably concomitant with the inferable property. The invariable concomitance has notably three dimensions: The proving property would “not occur without” (avinābhāva) the proposed property, it would be “inexplicable otherwise” (anyathānupapatti) and would thus entail the cognition of the proposed property. This latter act of inference rests on the knowledge of the invariable concomitance, which is brought about by a prior act of “suppositional reasoning” (tarka). yo yaḥ sarvathā^nās´ritaḥ, sa sa na sambandho yathā dig-ādiḥ.

Explication of the invariable concomitance of the proving and the proposed properties and statement of an “example” (pradars´ana) for the concomitance. Like in the example of “space” (dis´), a non-cognition of the property “being based on something” entails the non-cognition and thus the negation of the property “being a connection.” sarvathā^nās´ritas´ ca samavāyaḥ. tasmān na sambandhaḥ.

Explication that the proving and the inferable properties are to be regarded as pertaining to the subject term. In the passages succeeding this argument, Vidyānandin examines in detail two of its underlying presuppositions. The first is that the proving property is in fact invariably connected to the inferable property and that its statement is not to be regarded as a “fallacious reason” (hetvābhāsa). The second examined presupposition is that the inferable property does in fact pertain to the subject term and that its statement is not to be regarded as a “fallacious subject term” (pakṣābhāsa). As shown with this short introduction, it is worthwhile to examine also Vidyānandin’s further application of logical principles and how he and other Jainas used and modified the methodical elements of a debate in which they aimed to justify their convictions to the best of their rational abilities.

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Balcerowicz, P. 2001. Jaina epistemology in historical and comparative perspective. Critical edition and English translation of logical-epistemological treatises (ANIS 53.1–2). Stuttgart. Balcerowicz, P. 2003. Is “Inexplicability Otherwise” (anyathānupapatti) Otherwise Inexplicable? Journal of Indian Philosophy 31 (1–3): 343–380. Balcerowicz, P. 2013. Summary of prabhācandra’s prameyakamalamārtaṇḍa. In: Potter/ Balcerowicz 2013: 85–211. [The references above refer to the page numbers of the philologically more exact version of Balcerowicz’ article, which is available at the author’s website, www.orient.uw.edu.pl/ balcerowicz, last accessed January 24, 2016.] Bhargava, D. 1973. Mahopādhyāya yas´ovijaya’s jaina tarka bhāṣā. With translation and critical notes. Delhi. Bhattacharya, H.S. 1967. Pramāṇa-naya-tattvālokālaṃkāra of Vādi Devasūri. Bombay. Clavel, A. 2008. La théorie de la connaissance dans le Laghī yastraya d’Akalaṅka. Vol. I-III. Thèse de doctorat. Lyon. Dunne, J.D. 2004. Foundations of dharmakī rti’s philosophy. Boston. Ganeri, J. 2001. Philosophy in classical india. The proper work of reason. London/NewYork. Ganeri, J. 2012. Identity as reasoned choice. New York. Ghoshal, S.C. 1940. see PMS. Gorisse, M.-H. 2015. The taste of the mango. A jaina-buddhist controversy on evidence. International Journal of Jaina Studies (Online) 11 (3): 1–19. Guha, N. 2012. Tarka as cognitive validator. Journal of Indian Philosophy 40: 47–66. Halbfass, W. 1992. On being and what there is. Classical vaiśeṣika and the history of Indian ontology. Journal of Indian Philosophy 39 (4-5): 553–569. Kang, S.Y. 2010. An inquiry into the definition of tarka in Nyāya-tradition and its connotation of negative speculation. Journal of Indian Philosophy 38: 1–23. Kellner, B. 2004. Why infer and not just look? Dharmakīrti on the psychology of inferential processes. In The role of the example (dṛṣṭānta) in classical Indian logic, ed. Shôryû Katsura and Ernst Steinkellner, 1–51. (WSTB 58). Wien. Kellner, B. 2007. Jn˜ ānas´rī mitra’s anupalabdhirahasya and sarvas´abdābhāvacarcā. A critical edition with a survey of his anupalabdhi-theory. (WSTB 67). Wien. Mookerjee, S., and N. Tatia. 1946. Pramāṇa-mī māṃsā. A critique of the organ of knowledge. Calcutta. Oberhammer, G., E. Prets, and J. Prandstetter. 1991, 1996, 2006. Terminologie der frühen philosophischen Scholastik in Indien. Band I-III. Ein Begriffswörterbuch zur altindischen Dialektik, Erkenntnislehre und Methodologie, ed. G. Oberhammer. Wien. Oetke, C. 1994. Studies on the Doctrine of trairūpya. (WSTB 33). Wien. PaP Patra-parīkṣā. Vidyānandin: See ĀPṬ. Phillips, S. 2012. Epistemology in classical India. In The knowledge sources of the Nyāya School. New York. PKM Prameya-kamala-mārtaṇḍa. 1941, 1990. Prabhācandra: Prameya-kamala-mārttaṇḍa by Prabhācandra. A commentary on Māṇikyanandin’s Parī kṣāmukhasūtra, ed. Mahendra Kumar Shastri. (Sri Garib Dass Oriental Series 94). Delhi. PMS Parīkṣā-mukha-sūtra. 1940. Māṇikyanandin: Parī kṣāmukham by Māṇikyanandī with prameya-ratnamālā by Anantavī rya. Edited and translated by S.C. Ghoshal. Lucknow. Potter, K.H., and P. Balcerowicz, eds. 2013. Jain philosophy (part II). (Encyclopedia of Indian philosophies 14). Delhi. Preisendanz, K. 1994. Studien zu nyāyasūtra iii.1 mit dem nyāyatattvāloka vācaspati mis´ras II. (ANIS 46.1–2). Stuttgart. PRM Prameya-ratna-mālā. Anantavīrya IV: Included in PMS. PrP Pramāṇa-parīkṣā. 1914. Vidyānandin: Samantabhadra-viracitā ... āpta-mīmāṃsā vidyānandasvāmiviracitā pramāṇa-parīkṣā ca. In Gajādharalāla-jaina-sampādite. (SJG 10). Kāśī. SJG Sanātana-jaina-granthamālā. Steinkellner, E. 2009. Further remarks on the compound avinābhāvaniyama in the early dharmakīrti. Wiener Zeitschrift für die Kunde Südasiens 51 (2007–2008[2009]) 193–2005.

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SŚP Satya-śāsana-parīkṣā. 2012. Vidyānandin: Vidyānandi-kṛta-Satya-śāsana-parīkṣā. sampādaka: Gokulchandra Jain. (Jñānapīṭha Mūrtidevī Jaina Granthamālā: Saṃskṛta Grantha 30). Calcutta 1964. [For references with Roman numerals see the edition in Trikha]. TA Tattvā^rtha(sūtra). Included in TAŚVA. TAŚV. Tattvā^rtha-śloka-vārttika, Vidyānandin: Included in TAŚVA. TAŚVA Tattvā^rtha-śloka-vārttikā^laṅkāra. 2002. Vidyānandin: Vidyānandisvāmi-viracitaṃ tattvā^rtha-śloka-vārtikaṃ Manoharlāl-nyāyaśāstriṇā sampāditaṃ saṃśodhitaṃ ca. (Saraswati Oriental Research Sanskrit Series 16). Bombay 1918, Ahmedabad reprint. TPS. Terminologie der frühen philosophischen Scholastik, see Oberhammer et al. Trikha, H. 2012. Perspektivismus und Kritik. (Publications of the De Nobili Research Library 36). Wien. WSTB. Wiener Studien zur Tibetologie und Buddhismuskunde. YAṬ Yukty-anus´āsana-ṭī kā. 1919. Vidyānandin: Samantabhadra-praṇītaṃ Yukty-anuśāsanam. Vidyānanda-viracitayā ṭīkayā samanvitam Indralālaiḥ Śrīlālaiś ca sampāditaṃ saṃśodhitaṃ ca. (Māṇikacandra Digambara Jaina Grantha-mālā 15). Bombay.

¯ NA” [Inference]: Jaina Theory of “ANUMA Some Aspects

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §1. Svārthānumāna and Parārthānumāna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §2. Antarvyapti and Bahirvyapti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §3. Analysis of “Antarvyāpti” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §4. Antarvyāpti: Its “Logical” and “Methodological” Aspects Dis-Entangled . . . . . . . . . . . . . . . . §5. Nature of Hetvābhāsa: An Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §6. “Prayojakatva” and “Aprayojakatva” of Hetvābhāsas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §7. The Three Hetvābhāsas Considered by the Jaina Logicians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §8. Minimum Number of Avayava-S Needed for an Anumāna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §9. Use of “Vikalpa” as “Existence Proof” in Jaina Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §10. Contradiction and Contextualization of LNC in Jaina Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §11. Jaina Logic Viewed in a Wider Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . §12. Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The chapter takes a thoroughly analytical, but somewhat nontraditional, look at the Jaina theory of anumāna in its various aspects. A critical analysis of the scheme of classification of anumāna-s into “svārtha” and “parārtha” is undertaken to show that a svārthānumāna is only a proto-anumāna, not a standard anumāna at all. So, it is wrong and misleading to consider both of them as “anumāna-s” in the same sense. Next, the features of bahirvyāpti and of antarvyāpti are compared, the respective roles played by each of them in universal generalizations and some severe limitations of bahirvyāpti in this respect, etc., are critically discussed one by one. After a detailed analysis of the definition of “antarvyāpti,” it is claimed that antarvyāpti signifies a “semantic-conceptual linkage,” and, in this respect, it has a thematic affinity to Kant’s T. K. Sarkar (*) Jadavpur University, Kolkata, West Bengal, India University of Waterloo, Windsor, ON, Canada © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_37

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notion of “synthetic a priori judgements.” “Logical” and “methodological” aspects of antarvyāpti are dis-entangled next. It is followed by a thorough discussion of the nature of Hetvābhāsas, the reasons why they are not to be viewed as purely formal fallacies, an enumeration of different types of hetvābhāsas, lessons to learn from some queer instances of “fallaciously valid” argument-patterns in Western logic, etc. Topics such as the Minimum Number of avayava-s required in an Inferential Unit, notion of “existence” and of Vikalpa as “existence proof,” notions of Contradiction, Contextualization of LNC in Jaina Logic, etc., are discussed after it. The need of balancing between the ontic and the epistemic conceptions of logic, replacing the tautology-centric notion of “deductive validity” by an information-theoretic-cumcontext-sensitive notion of “logical infer-ability” [Sanskrit, “anumeyatva”] are discussed next. In the section “Concluding Remarks,” attention is drawn to what the author considers to be an emerging trend of mutual convergence of the respective outlooks [viz., the respective “epistemic” and “ontic” outlooks] of Indian and Western logicians. A final such convergence may even need a radical “paradigmshift” in the patterns of logical thinking. It is not expected to be an easy task at all. Nevertheless, it feels better to keep dreaming about it as a realizable goal. Keywords

Antarvyāpti · Anumāna · Anumiti · Aprayojakatva · Avayava of an anumāna · Avinābhāva · Bahirvyāpti · Bhūyodarśana · Context-sensitivity · “Epistemic” and “ontic”outlooks · Explosion (principle of) · Factuality bias · Fallacious validity · HD method · Information-content (zero/nonzero) · Intuitive induction · Jñānātmaka · Parārthānumāna · Post-prediction confirmation · Prayojakatvaaprayojakatva · Projection rule · Proto-anumāna · Svārthānumāna · Upādhinirāsa · Ur-linkage · Vākyātmaka · Viṣayatā

Introduction This chapter discusses some of the salient features of Indian logic vis-à-vis Western logic, in a somewhat less general way, by focusing mainly on such characteristic features of Jaina theory of anumāna [i.e., inference], which distinguishes it from that of its counterpart in Western logic. However, it is clearly not possible to work out even such a less general approach without discussing a few more general features of the respective approaches to logic viz., the Indian and the Western. So, incidentally, I had to touch upon a few such general features as well. I consider the first three decades of the second half of the twentieth century the “transition era” – a period when logicians [both Indian and Western] began to reflect upon their own traditional systems and tried to relate it to those of the others. It was inevitable that it resulted in reports based on seeing the other side through one’s own looking glasses. Some Indian thinkers assumed that (i) the logical-epistemological significance of the Indian concept of “anumāna” and that of “inference” in Western logic are basically the same, (ii) the tradition-entrenched, five-membered pattern of

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“anumāna” of the Nyāya school is just an extended version of the traditional threemembered syllogistic form, (iii) as if, the five-membered pattern of “anumāna” of the Naiyāyikas can be considered [without having any conceptual downside] as the uniform, paradigmatic pattern of “anumāna” in Indian logic in general, (iv) hetvābhāsa-s are on par with “logical fallacies,” i.e., a hetvābhāsa-affected anumāna is simply an invalid argument by another name. To put it bluntly, this assumption is untenable for a number of reasons. In order to make my claim [viz., that hetvābhāsa-s cannot be forced into the purely formal “fallacy-invalidity” mold of Western logic] more perspicuous, and further, in order to capture and to express in language the basic conceptual incongruity that exists between a hetvābhāsa and a logical fallacy, I had to coin queer expressions like “fallaciously valid arguments” which, though it may sound like an oxymoron, has been fully explained in the chapter. (v) One misconceived but widespread assumption regarding the relationship between a svārthānumāna and a parārthānumāna is that both are “anumāna” in exactly the same sense, just as a “black cat” and a “white cat” are cats in exactly the same sense. As a matter of fact, however, a svārthānumāna is not all an anumāna [in the sense in which a parārthānumāna is] any more than pineapples are apples. Finally, (vi) another prevalent view found in the classical versions of Western logic is that the idea of “existence” and that of “logical contradiction” are nonnegotiable, i.e., absolute in nature in the sense that they hold universally without any restrictions. Hence, there is no way to tamper with or of doing any kind of conceptual fiddling with any of those two key concepts of logic. Most scholars of the “transition era” who wrote on the nature of Indian logic simply took it for granted that the nonnegotiability assumption regarding those two key concepts holds equally well in Indian logic as well. In this chapter, I challenged each one of the claims (i)–(vi) above and have either suitably reformulated or altogether rejected each one of them on the basis of clear arguments that are backed by relevant textual evidences. The entire exercise undertaken in this chapter is geared first, to highlight the characteristic features of Indian logic that can tell it apart from Western logic and secondly, to suggest a possible roadmap for convergence of these two distinct approaches [viz., an essentially epistemic one of Indian logic and hitherto, a predominantly ontic one of Western logic] in future.

§1. Sva¯rtha¯numa¯na and Para¯rtha¯numa¯na Indian logicians classify inference patterns or types by using different principles of classification. One such scheme of classification about which the Jainas, the Bauddhas, as well as the Naiyāyikas agree is that of classifying inferences into svārthānumāna (i.e., inferences where the subjective feeling of conviction of the inferer [anumātā] is the adequate criterion of justification) and parārthānumāna (i.e., inferences where the respective justifications need to be convincing to others, namely, the public]. Till now, the accepted and unchallenged practice has been to translate “svārthānumāna” and “parārthānumāna” as “inference for oneself” and

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“inference for others,” respectively. I consider these translations woe-fully inadequate and propose to use “individually acquired ground-level conviction about infer-ability [anumeyatva],” for “svārthānumāna”; and “expression/statement of inferential justification (meant for use) in a public-discourse,” for “parārthānumāna.” This deviant translation, as we shall see, would help avoiding a number of misunderstandings about the real purpose and true logicalmethodological significance of this scheme of classification of inferences. The way “svārthānumāna” and “parārthānumāna” are characterized in some texts (e.g., TS and TSD) tend to suggest misleadingly that, (i) Both “svārthānumāna” and “parārthānumāna” are inferences in the same sense except that “svārthānumāna” is only a less elaborate, covert, and compact version of “parārthānumāna.” It also suggests that, (ii) The role of “svārthānumāna” is to convince the inferer [anumātā] himself [Chattopadhyay, H. (1983), 25], (which suggests, as if, it is precisely for this reason that a svārthānumāna does not need to be as elaborate and expanded as a parārthānumāna) while, in contrast, the aim of “parārthānumāna” is to convince others; and as such, it needs to be more elaborately expanded. Nothing could be more obfuscating and wrong about the supposed reasons given by different writers, in order to explain why a parārthānumāna needs to have five components, while a svārthānumāna needs only three. “Svārtha” and “parārtha” are specially coined highly technical terms, meant for use in discussions about types of inference or anumāna. They must not be construed as simple adjectives [like, “red” and “white” in “red rose” and “white rose”] of “anumāna.” It is therefore very crucial to remember in this context that (a) svārthānumāna and parārthānumāna cannot be anumāna-s [inferences] in the same sense, because there is a crucial difference in their nature, viz., svārthānumāna is jnānātmaka [cognitively oriented] in nature, while parārthānumāna is vākyātmaka/śabdātmaka [linguistically/propositionally oriented] in nature. For our very limited purpose, “vākyātmakam” and “śabdātmakam” can be used interchangeably.] So, there cannot possibly be any general/common definition of “anumāna” which can be legitimately used to cover both svārthānumāna and parārthānumāna [Chattopadhyay, H. (1983), 16, 21, 25]. Moreover, (b) svārthānumāna is not at all geared to “convince” either the inferer [anumātā] himself or anyone else, of anything. To say that the role of svārthānumāna is to “convince the inferer [anumātā] himself” would be no more justified than saying that after waking up in the morning, I open my eyes in order to “convince myself” that I am awake. My awareness that “I am awake” is itself a conviction. It is not geared to generate any other conviction at all, in order to justify a claim, (like “I am awake now”), as the conclusion of a tacit inference. The real import of the distinction, however, is quite different. The role of svārthānumāna, according to our view, is to show how an associative bonding/ linkage, which Hume called a “custom bred habit of expectation,” between a “hetu” [link-concept] and its “sādhya” [probadum] gets spontaneously established as an

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internalized conditioned-cognition [somewhat analogous to Pavlovian “conditioned reflex,” except that instead of being a purely mechanical reflex-response it happens to be a reflexive cognitive awareness (a state of jnāna) according to the Indian logicians]. Once such a cognitive awareness of the linkage between a hetu and a sādhya is established, it automatically/spontaneously leads one [an inferer] to arrive, under appropriate initial conditions, at the relevant law-like generalization, e.g., “Wherever there is smoke, there is fire.” Such linkages/law-like generalizations [LLG’s] are technically called, “vyāpti.” Such law-like generalizations automatically/spontaneously crop up as soon as a “conditioned-reflex” type cognitive-arc gets established [just in the way a conditioned-reflex arc is established in the case of a Pavlovian dog] through repeated observation [bhūyodarśana]. However, unlike in the cases of Pavlovian dogs, the discerned vyāpti-relations always have a state of cognitive awareness accompanying it. I think that svārthānumāna-s constitute the pre-logical underpinnings of corresponding parārthānumāna-s. Therefore, it would be more plausible to view svārthānumāna as providing a psychological grounding for a corresponding parārthānumāna, the latter being an “inferential argumentpattern” formally dressed up for use in a public discursive context. [In view of the fact that (i) in case of a svārthānumāna the hetu-sadhya linkagae originates spontaneously (at the subliminal level) and (ii) thereby, enables a svārthānumāna to act as the pre-logical underpinning of a corresponding parārthānumāna, it seems more appropriate to designate/translate a svārthānumāna as “Ur-linkage-based anumāna,” instead of characterizing it as “inference for oneself.” In view of the automatic and spontaneous origination of the hetu-sādhya bonding that underlies a svārthānumāna, another alternative appropriate English paraphrase of “svārthānumāna” may be “sahaja vyāpti-samskāra-janya anumāna.” However, I consider “Ur-linkage-based anumāna” a better and compact paraphrasing/translation of “svārthānumāna.” Anyway, it cannot be “inference for oneself,” i.e., for the inferring agent himself.] In this context, the following points about the implications of my interpretation of svārthānumanā need to be seriously noted. First, after multiple observations of instances, [without coming across any counter-instances] of co-occurrence/concurrency of two things [e.g., hetu (smoke) and sādhya (fire)], the process of universally linking them through an associative bonding starts automatically and spontaneously (at the subliminal level) as a psychological activity. We may call such generalizations [as are made on the basis of a requisite kind of spontaneously arising associative bonding] “law-like associations” (¼ LLA’s,) in order to distinguish them from well-articulated, full-fledged law-like generalizations (LLG’s)]. As LLG’s have to be well-articulated formulations of a consciously entertained linkage between two entities, they cannot be regarded as svārthānumāna-based. Again, as the Jainas do not admit “sāmānyalaksana pratyaksa” (which is a sort of “alaukika pratyaksa” according to the Naiyāyikas), neither the question whether “sāmānyalaksana pratyaksa” is a legitimate tool for vyāptigraha nor the question whether a sāmānyalaksanā-based vyāptigraha would be an instance of “svārthānumāna” can be relevant in the Jaina context. Moreover, had it been a relevant question, still a sāmānyalaksanā-based vyāptigraha would not count as a case of svārthānumāna because, being a full-fledged vyāptigraha/

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vyāptijnāna, it would be a deliberate [i.e., non-spontaneous], articulated [i.e., not inarticulate], and conscious [i.e., not subliminal] one, in violation of all the required features of “svārthānumāna” in our sense. Secondly, from what has been said just now, it follows that (i) all such LLA’s have to be ultimately rooted in/trackable to one ground-level svārthānumāna or another and (ii) LLA’s only suggest un-articulated universal relationship, [we may call it “sahacāritā-mātra-janya prāthamika vyāpti-pratyaya/niścaya/jnāna”], which is distinct from a final, fully articulated, “vyāpti-jnāna”/“vyāpti-graha”[i.e., an LLG] that is supposed to truly and reliably (i.e., unfailingly), link a hetu with its sādhya. Finally, (iii) as it turns out, an LLA provides the pre-theoretic basis which is yet to be fully articulated as an avayava called “hetu-vākya” [which has to be a fully articulated LLG itself (and according to some logicians), is also an indispensable component of a pañcāvayavī [five-component] parārthānumāna.]. Consequently, as (α) there would possibly be no prāthamika vyāpti-pratyaya which is not rooted in, or grounded in, some corresponding svārthānumāna, (β) there would not possibly be any “vyāpti-jnāna” without a supporting LLA that occurs at the subliminal level, and thus, (γ) there would be no parārthānumāna either, without being rooted in a svārthānumāna. This is the basis of my very clear claim that the possibility of svārthānumāna is a prerequisite for there being any parārthānumāna at all. In this respect, the views of some modern writers lend support to my position when they point out [citing Jaina-Bauddha sources] that “inference for one’s self is really a case of conviction [svārthānumānasya bodharūpatvāt] on the part of the inferring agent [anumātā]” and a parārthānumāna (which is nothing but a verbal demonstration of what one is internally convinced of) may be based on bahirvyāpti, but still the point remains that “unless the avinābhāva between the hetu and the sādhya is already discerned by the inferring agent by taking recourse to antarvyāpti, bahirvyāpti, has no value in leading to a relevant inferential conclusion [anumiti].” [See: Bhattacharya, H. M. (1994, 252). Also Cp. Phaṇībhūṣaṇa (1989, 121) in §3 below.] In this sense, a svārthānumāna may be called a “proto-anumāna” – not at all a regular anumāna, as a parārthānumāna is. Just as the “grammaticality” of the sentence “I is an English personal pronoun” cannot be judged in the same way as that of “He is a man.” Similarly, it is important not to conflate the two senses of “anumāna” [inference] as it occurs in “svārthānumāna” and “parārthānumāna.” Not paying heed to this important point of caution can only generate needless confusion and avoidable controversies. The so-called “nigamana” of a svārthānumāna is not an inferential conclusion at all, any more than my acute awareness that “I am in pain now” is an inferential conclusion from the look at my swelled ankle. It implies, in its turn, that a svārthānumāna and a parārthānumāna are not “anumāna-s” in the same sense, any more than “Bull dogs” and “Hotdogs” are both “dogs,” or “Time Tables” and “Dining Tables” are both “tables” in the same sense. [Cp. Dharmakīrti in Nyāyabindu. Chattopadhyay, H. (1983), 20]. A vyāptijnāna, arrived at through svārthānumāna, functions like a “state-transition input in a machine” – an input that leads towards an anumiti. Therefore, svārthānumāna needs to be viewed as that which provides a psychological/epistemological grounding for a corresponding parārthānumāna, the latter being a language-based [i.e., vākyātmaka]

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inferential argument-pattern, formally dressed up for use in a public discursive context. In other words, were there no svārthānumāna to provide a required law-like associative bonding at the subliminal level (via LLA’s) as the very basis of law-like generalizations [LLG’s], there would not possibly be any parārthānumāna. [Cp. Dharmakīrti: “No parārtha-anumāna unless based on svārthānumāna.”] Hence, as a pre-logical ground/basis of all other types of anumāna [without being an “inference” itself], a svārthānumāna is better characterized as “proto-anumāna” to make it stand out from the other types. Use of this new terminology would also block the prospect of some avoidable confusions from arising. One example of what I consider to be an example of such avoidable confusions is this: svārthānumāna is an anumāna just as a parārthānumāna is, the two differs only in respect of their degrees of “elaborateness.” Moreover, any anumāna whatsoever is supposedly susceptible to some hetvābhāsa or other. If so, then the question that arises is this: what could a “hetvābhāsa” possibly mean in the case of a svārthānumāna? Or, what kind of defective probans [duṣta hetu] is it that may lead to a hetvābhāsa in a svārthānumāna? (a) It may be pointed out in this context that Heramba Chattopadhyay Shastri in his book [in Bengali] on the Nature of svārthānumāna in Buddhist Logic [Chattopadhyay, H. (1983), 26] points out that some Buddhist logicians were actually engaged in bitter controversy and hair-splitting analysis on this issue and some of them even maintained that all hetvābhāsas arise primarily in the context of svārthānumāna. However, so far as my interpretation is concerned, svārthānumānas are exempted, by definition, from the risk of “hetvābhāsa” [in the sense in which a parārthānumāna can be hetvābhāsa-afflicted], because they are only “proto-anumānas,” and not “anumānas” strictly speaking. Moreover, I think that the alleged controversy about “hetvābhāsa” in a svārthānumāna is a nonissue which looks like a real issue, due to a failure [on the part of the disputants] to detect a sort of category-mistake underlying the very formulation of the point at issue. An error, if any, in a svārthānumāna would be an error due to some psychological aberrations (a hardware problem, so to say) – not at all due to an “error of reasoning” or “hetvābhāsa” (which is an analog of a software problem or a programming error). (b) Secondly, if a svārthānumāna and a parārthānumāna are not “anumāna” in the same sense then, for any sensible person, it would be a futile exercise to try to find a common, nontrivial definition that equally applies to both. Actually, Dharmottara, in his commentary on Nyāyabindu, clearly states this point. “Parāthānumānam sabdātmakam, svārthānumānam tu jnānātmakam. Tayoratyantabhedāt naikam laksanam asti.” [Chattopadhyay, H. (1983), 16]

Hence, the two anumāna-s cannot be covered under one definition. (c) If it is maintained that a five-component, language-based, communicating-to-othersoriented inferential-pattern [vākyātmaka, parārtha, pancāvayavi nyāya] qua a five-

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membered set of sentences (of a special kind consisting of, e.g., pratijnā-vākya, hetu-vākya, etc.) alone are to be called an “anumāna” by definition, then a svārthānumāna fails to qualify as an anumāna on all three counts. Actually, some logicians [e.g., Gotama] propose to define an anumāna as a five-membered sentence-cluster [Phaṇībhūṣaṇa (1989), on Vātsāyana Bhāsya-49, 287289]. To view an anumāna just as a five-membered sentence-cluster is akin to the view that a syllogism is a three-membered cluster of propositions. [The Greek word for “syllogism” is “συλλoγισμóς” which is linked to “logos,” i.e., language/sentence, and thus predominantly highlights the vākyātmakatā aspect of an inference.] This, in its turn, delinks the cognitive [i.e., jnānātmaka] aspect of a syllogism and prepares the ground for a meaning-insensitive formulation of syllogistic inferences in terms of abstract formal schemas like, “All M are P, All S are M,/Therefore, All S are P.” This spirit of considering logic as a “purely formalist, abstract and sentence-oriented discipline” is captured and expressed by Kant in the Preface of his First Critique and was endeavored to be incorporated in the works on modern formal logic as developed by de Morgan, Boole, Venn, Peano, Frege, Russell, Whitehead, etc. Since substitution of any arbitrary term could be a permissible substituent in such purely formal schemas, it should end up [as it actually did] tending to authorize a “garbage-in, garbage-out” notion of deductive validity. [Respective structures and the contrasting logical features of Aristotle’s “syllogism” vis-à-vis that of “anumāna” in Indian logic are discussed in section §10] (d) Besides the above ones, viz., (a)–(c), two other questions arise here: (α) regarding the mutual relationship of svārthānumāna and parārthānumāna and (β) regarding their respective roles and contributions in an anumāna taken as a whole. We must not forget that (i) unlike a deductive inference which is basically abstract and language-centric, an anumāna is essentially linked to a knowledge-state [i.e., it has to have epistemic moorings]. Again, (ii) svārthānumāna being jnānātmaka is inalienably epistemology-centric, while in contrast, parārthānumāna is vākyātmaka [i.e., it is inalienably language-centric]. Thus, (iii) svārthānumāna directly embodies the requisite cognitive-content of an anumāna [due to (i) and (ii) above]. In this sense then, (iv) svārthānumāna is to be regarded as the principal and direct basis of an inferential knowledge [anumiti] in general. Therefore, (v) Buddhist logicians [e.g., Dharmottara] maintain that the logical link of parārthānumāna to inferential knowledge [anumiti] is to be taken in an indirect, nonliteral [lākṣanika] sense [Chattopadhyay, H (1983), 21]. Gangeśa, in the section on Anumāna in his Tattvacintāmani, does not seem averse to this Buddhist view]. Clearly, this is an interesting alternative theorization of “svārtha” and “parārtha” in the context of svārthānumāna-parārthānumāna discourse. Here, the table is turned, and instead of viewing svārthānumāna as a less important, condensed and truncated version of parārthānumāna, the former is allotted the rank of a primary contributor to anumiti. Viewing the nature of the distinction between the two inference-patterns in the way suggested above would also help one to tackle the following vexing questions, viz.,

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(i) How to understand and answer those who claim that, strictly speaking, justification of any inference cannot be public (i.e., there can be no “parārthānumāna” at all)? and secondly, (ii) How to reconcile the standard view that suggests as if the crucial point of difference between svārthānumāna and parārthānumanā is that the former only requires a lesser number of components or avayavas [viz., three] than the latter [which needs five] with the views of those logicians (who hold that even for a parārthānumāna there need to be only three components)? If the number of avayavas were the only crucial point of difference between the two kinds of anumāna, then the proponents of a three-component view would not have accepted the legitimacy of the classification of inferences into svārthānumāna and “parārthānumāna.” But they do. This clearly indicates that the rationale underlying the scheme of svārthānumāna-parārthānumāna classification needs a deeper look from a different angle. In order to do this, first, we must assign a “ground-level epistemic status” to svārthānumāna, instead of viewing svārthānumāna as a less elaborate condensed version of parārthānumāna; secondly, the respective status of these two types of anumānas need to be assessed by using two very different criteria – one epistemic (for svārthānumāna), the other linguistic-expressive (for parārthānumāna). The two criteria are noncomparable and hence cannot be used for prioritization of the types of anumāna on a common scale. On the basis of the interpretation proposed above, the second of the two questions get answered immediately. As “svārthānumāna” is not an “inference” at all (at par with “parārthānumāna”), the former cannot be regarded just as a condensed form of the latter. Consequently, the respective roles played by the avayava-s in a svārthānumāna [psychological-cum-epistemological] and in a parārthānumāna [linguisticexpressive] are completely different. They are so disparate in nature that counting and characterizing the two types of anumāna solely in terms of the number of their avayava-s would hardly make any better sense than counting and characterizing “one’s body temperature” and the “size of an angle” in terms “degrees,” say, 100 degrees and 90 degrees, respectively, and then, on the basis of that, seriously claim that there is only a difference of 10 degrees between “a body temperature” and “a right angle.” Comparing these two cases by simply counting their respective number of degrees would hardly be relevant in understanding the nature either of “fever” or of “angularity.” The same observation applies, mutatis mutandis, to comparing the number of avayava-s in svārthānumāna-s with that of parārthānumāna-s [even if we agree to call both of them “anumāna” or inference]. Once this is granted, question (ii) above, looks both “hollow” and philosophically/logically “unsubstantial.” Only genuine substantial questions deserve any serious answer, pseudo-questions do not. I consider question (ii) above nothing more than an unsubstantial pseudo-question. It follows that if svārthānumāna is viewed as providing the experiential/psychological grounding of parārthānumāna, as I indicated above, then there is a sense in

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which looking for an exclusively public justification of any inference (including parārthānumāna) becomes a nonstarter. All empirically justifiable inferential grounds have to depend, on ultimate analysis, on the establishment of an appropriate subjectively rooted bond of psychological association. It is analogous to the sense of “having a toothache.” In the intended sense, “I have a toothache” can have no “exclusively public” justification which is not ultimately based on some appropriate subjective experience, just as there can be no “exclusively public” justification of parārthānumāna without dependence on a jnānātmaka [i.e., subjective] svārthānumāna. Something similar in spirit to what has been said here just now is expressed by K.K. Dixit: “True, in a way the objective criterion .... too involves an appeal to self-evidence, .... Suppose, we ascertained through repeated experience a universal relationship between “drinking water” and “quenching of thirst.” “But how, (if not on the testimony of self-evidence) can you be certain that your thirst is actually quenched when you drink water?” [Dikshit, K.K. (1975): Indian Logic: Its Problems as Treated by its Schools, 34].

§2. Antarvyapti and Bahirvyapti All schools of Indian Philosophy [including the Cārvākas] agree on one point about the nature of inference, viz., it is the process of coming to know some as yet unobserved feature y [e.g., presence of fire on a hill] on the basis of the observed presence of some other feature x [e.g., smoke on that hill]. The reason given to justify this inferential passage from x to y is that there is an unfailing/invariable relation between a hetu x [smoke] and a sādhya y [fire]. Such relationships are technically called “vyāpti-sambandha” or “linga-lingī sambandha.” In short, unless there is a duly ascertained requisite kind of “vyāpti-sambandha,” there can be no inference [anumāna]. Even a Cārvāka’s refusal to accept anumāna as an acceptable/ accredited means of knowing, hinges on their argument that it is in principle impossible to ascertain any invariable relationship [vyāpti-sambandha] between a “linga” [a logical indicator, say, smoke] and a “lingī” [i.e., what is logically indicated by it viz., fire]. So, the entire controversy between the “pro-anumāna” schools and the “no-anumāna” group boils down to this: how is it possible, if at all, to ascertain an invariable universal relationship or vyāpti-sambandha between a hetu and a sādhya. This naturally leads to the question of classification of different kinds of vyāpti and their respective roles, in countering the challenge of the Cārvākas. Most schools of Indian logicians [e.g., Naiyāyikas, Vaiśeṣikas, Mimāmsakas, Vedāntins, etc.] claim that the “ascertainment [vyāptigraha] of such a universal invariable relationship or vyāpti-sambandha is a matter of instance-based empirical generalization based on repeated observations [bhūyodarśana].” Here crop up two problems: (a) the problems pertaining to making fully warranted empirical generalizations and (b) the question whether and how, if at all, those problems can be successfully tackled. [See: Gangopadhyay, M.K. (1975, in JIP): Ascertainment of Invariable Concomitance. Also, Goekoop (1967): The logic of Invariable Concomitance, Dordrecht: Reidel.]

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The Naiyāyikas and those who are committed to follow the standard instancebased model of universal generalization and consider vyāpti-relation only as a special case of such generalization, try to work out a kind of semi-algorithm-like procedure in the hope of successfully tackling the problems stated in (a) and (b) above. They maintain that vyāpti can be ascertained by repeated observation of constant conjunction, coupled with non-observation of any exceptions to it. The generally recommended and widely accepted procedural steps required for this purpose are the following: (i) Observation of a large number of corroborating instances [bhūyodarśana] where x (hetu) and y (sādhya) are found to occur together in a regular sequence [niyata sahacāradarśanaṁ] without encountering any exceptions [vyābhicāra adarśanam] and (ii) methodically eliminating all undetected perturbing factors [upādhi nirāsa]. This uniform model of instance-based empirical generalization, if it were practicable and tenable, would entail that observation of corroborating instances [dṛṣṭānta-s] is indispensable for vyāpti-ascertainment. In short, that would imply, “no dṛṣṭānta, no vyāpti-ascertainment.” This, coupled with another basic claim, viz., “No vyāpti-sambandha, no inference” [anumāna] entail that, “no dṛṣṭānta, no anumāna.” This means that in the standard instance-based models, citing actual examples [dṛṣṭānta] should be an indispensable requirement for an anumāna. [Some logicians make this claim, but others (e.g., the Jainas, who want to make an anumāna free of empirical constraints and fact-dependency), reject it.] The Jainas, as expected, challenged and rejected the rationale underlying all instance-based models and, as its corollary, also rejected the claim that citing actual examples is necessary at all. However, it is undeniable that no matter how many instances [without even a single exception] one may have observed, that cannot cover all the possible cases of past, present, and future, and hence, no exception-less bhūyodarśana can logically warrant any universal empirical generalization. Moreover, the number of potential undetected perturbing factors [upādhi-s] being endless, it is never practicable, or even logically possible, to eliminate all potential perturbing factors. [The wellknown “Tweety” example from Default Logic illustrates this point. (See: Sarkar1992, pp-212224).] Still it is a fact that this uniform model of vyāpti-ascertainment [technically called “bahirvyāpti”] is accepted without any qualms by most schools of Indian logic, except by the Bauddha and the Jaina logicians. Hemachandra in his Pramāṇamīṁāmsā [PM] categorically states that knowledge of vyāpti can be obtained neither by empirical perception nor by any ordinary inference. Hence, a genuine vyāptigraha, which is not amenable to any standard way of knowing, can be ascertained only by ūha, i.e., tarka [tarkāt tanniścaya]. Prabhācandra, in his Prameyakamalamārtaṇda [PKM], holds that the invariable concomitance between a hetu/sādhana and its sādhya is the subject matter of tarka pramāṇa. Consequently, the Jaina [and also the Bauddha] logicians needed to look at vyāpti from a very different angle. Breaking away from the standardly proposed instance-based model of empirical generalization, the Jaina logicians proposed to classify vyāpti into two types, viz., (i) antarvyāpti (i.e., intrinsic linkage/intrinsic vyāpti) and (ii) bahirvyāpti (i.e., extrinsic vyāpti or empirical instance-based linkage/concomitance) [See: Bhattacharya,

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H.M. (1994, 237254). Also see Mishra’s book: Antarvyāpti (2002) which contains the text of Ratnākaraśāntipāda’s Antarvyāptisamarthanam. On pages 109–111, Mishra maintains that “Jaina logicians .... advocated the theory of internal concomitance long before the Buddhist logician Ratnākaraśānti .... Mishra also draws attention to the fact that the ‘number of constituents of a syllogism, according to Jaina logicians, is relative to the intelligence of the people concerned’.” [Ibid, 109–110.] I propose to use “intrinsic semantic-conceptual linkage/concomitance” [or simply, “semantic-conceptual linkage”] as an English equivalent of “antarvyāpti.” The Nyāya logicians were acutely aware of these difficulties and prescribed a set of five requirements hoping that those, when fully complied with, would guard against the possibility of the instance-based model of vyāpti-ascertainment going astray. The five requirements [the Bauddhas proposed to trim it to the first three] prescribed are the following: (a) the hetu [¼ smoke] must be present at the required “pakṣa” (the hill). This requirement is called “pakṣavrttitva,” (b) the hetu should also be present in places similar to that of the pakṣa [¼ svapakṣavrttitva], (c) hetu must not be present in any dissimilar places, viz., places which are associated with absence of fire [¼ vipakṣāvrttitva], (d) the hetu must not have been counterbalanced/counter-mandated by any set of contrary evidence [¼ asatpratipakṣatva], and (e) the hetu must not be contradicted by or be incompatible with any well-established body of truth [¼ abādhitatva]. This set of five requirements is called determiner of the characteristic features of hetu [anumāpakatā-prayojakatva] or, ways of arriving at a proper hetu [gamakatā-aupāyikatva.] However, the Jainas showed with a number of convincing counterexamples that complying with these requirements does not guarantee that the right hetu has been picked up. The Naiyāyikas tried to tackle the problem of getting a fully warranted universal generalization based on empirical observations alone, by postulating a capacity of extraordinary perception [alaukika pratyakṣa] called, “Sāmānyalakṣaṇa pratyakṣa,” that would enable one, after observation of even a single instance, first to abstract the respective universals/class-properties of a “hetu-class,” [e.g., smoke-ness from the set of smoky things] and of the corresponding “sādhya-class” [e.g., fire-ness from the set of fiery things] and then to universally link smoke with fire in a causal sequence. Unfortunately, due to various reasons, sāmānyalakṣaṇa pratyakṣa fails to deliver: first, it is a totally ad hoc assumption which even some Naiyāyikas like Gangeśa [See: Matilal, B.K. (1968): Gangesa’s view on “Kevalānvayin,” in Philosophy E&W.], Jayantabhaṭṭa, etc., refused to admit, and secondly, it also fails, as the Jainas claim, to explain how bhūyodarśana can still be of any relevance at all, so far as ascertainment of a vyāpti-relation is concerned because, once sāmānyalakṣaṇā is postulated, observation of even a single relevant instance should be enough. Thirdly, the very idea of sāmānyalakṣaṇa pratyakṣa turns out to be a strangely hybrid creature. It is independent of the ordinary operations of the sense organs [i.e., alaukika] and yet, it is not fully a-priori because, the Naiyāyikas very categorically state that it is a kind of perception, except that it is “non-ordinary” in nature. Whether it is a hybrid creature or not, we must not overlook the fact that in the conceptual

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framework of Nyāya logic, postulating sāmānyalaksaṇā, plays the same methodological role as the postulation of Law of Uniformity of Nature [LUN] once played in the methodology of science in the West. Anyway, in view of the difficulties mentioned above (and some other similar ones), the critics of standard instance-based models [mainly, the Jainas and the Bauddhas] rejected the very idea of bahirvyāpti as a possible means of arriving at universal empirical generalizations. So, they proposed to replace bahirvyāpti by a kind of semantic-conceptual notion of vyāpti, technically called, “antarvyāpti.” [“Antarvyāpti” is discussed next in §3.]

§3. Analysis of “Antarvya¯pti” How do we go about characterizing antarvyāpti to make it stand clearly apart from bahirvyāpti? A Jaina logician Vādidevasūri says: If a given minor (pakṣa) is such that within it the concomitance between the hetu (probans) and the sādhya (probandum) holds/are co-located, then it is a case of antarvyāpti. Elsewhere, it is bahirvyāpti. [PNTL, Ch-3, Sutra-39.] We also find the following very similar characterization of antarvyāpti in Ratnākaraśāntipāda’s work, Ratnākarāvatārika: “pakṣīkṛta eva viṣaye sādhanasya sādhyena vyāptiḥ antarvyāptiḥ anyatra tu bahirvyāptiḥ.” [Part-2, Sutra-38.] Unfortunately, however, the notion of “concomitance holding within/inside a pakṣa” needs a lot of unpacking before it can make any clear sense. The traditional commentators, as we shall see, do not throw much light on it either. So, we discuss it more analytically in the following sections. (i) Meaning of “Concomitance holds within the pakṣa” Phaṇībhūṣaṇa also follows Vādidevasūri and says: in case of antarvyāpti, “concomitance of a probans and its probandum” holds internally. He explains it thus: “when it is a case where the pakṣa [i.e., the hill] to which the sādhya [i.e., the fire] is to be imputed by using anumāna, is such that the concomitance of the sādhana [i.e., the hetu, (viz., the smoke seen on that hill)] and its sādhya [i.e., fire on that hill] holds internally/within the pakṣa itself, that counts as an instance of antarvyāpti.” Similar other instances of “smoke-fire concomitance” observed elsewhere, e.g., in the kitchen, etc., are cases of bahirvyāpti. [Phaṇībhūṣaṇa (1989), Nyāyadarśana, Vol-1, 339; translation mine.] I propose to use “intrinsic semantic-conceptual linkage/concomitance” [or, simply, “semantic-conceptual linkage”] as an English equivalent of antarvyāpti. According to S.C. Vidyābhūṣaṇa, “Extrinsic inseparable connection (bahirvyāpti) occurs when an example from outside is introduced as the common abode of the middle term (hetu) and the major term (sādhya) to assure the inseparable connection between them. ... Here the reference to the kitchen is no essential part of the inference.” [Vidyābhūṣaṇa, S. C. (1978): History of Indian Logic, 177178]. I have a hunch that what Vidyābhūṣaṇa really meant to suggest, by implication at least, is that in cases of antarvyāpti the linkage between a hetu and what is to be inferred from that hetu [viz., its sādhya] is such that their concomitance can be “discerned” without depending on any external observation besides the pakṣa itself. If so, it becomes

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clear why in antarvyāpti the concomitance/unfailing linkage between a hetu and its sādhya is said to be holding internally/within the pakṣa. Still, the question remains: How, without observation of (some other) actual relevant instances, a universal and unfailing [and allegedly empirical] linkage between a hetu and its sādhya can be known to exist? It is unfortunate that all the writers [especially those who write on this topic in English] are very keen about faithfully restating the relevant textual expressions [of say, Siddhasena, Vādideva, etc.]. They seem to be even keener to avoid venturing into any kind of interpretation/explanation of what “holds internally/within the minor/pakṣa” is supposed to mean. I plan to offer an explanation later in this section. But before doing that, I want to make a critical, in-depth assessment of the views of some eminent scholars about the true purport/significance of postulating antarvyāpti as an explanatory parameter of vyāpti-jnāna. Let us now assess a few of their views one by one. Even, some eminent traditional scholars like Pt. S. C. Nyāyācārya (1975), 3940, fail to address with sufficient clarity the problem that I just mentioned. Instead of delving into the logic of antarvyāpti, Nyāyācārya focuses on an altogether different issue, viz., the intended scope of applicability of antarvyāpti. In this regard, he maintains that by antarvyāpti the Jainas simply meant the type of vyāpti used in justification of inferences that yield only pan-inclusive universal conclusions called, “kevalānvayī anumāna.” In cases of such anumānas, the hetu-class, being all-inclusive, excludes the possibility of any counter-instance. A kevalānvayī anumāna is non-counter-instantiable, ex hypothesi. [We may use the following often-cited example of kevalānvayī anumāna to illustrate this point: One stock example of kevalānvayī anumāna repeatedly used by the Naiyāyikas in this and similar other contexts is this: “x is nameable, because x is knowable” [e.g., “idam abhidheyam, jneyatvāt”]. Here, “knowability” is the “hetu” and “namability” is the “sādhya.” Whatever is a possible instance of “x” [here, the referent of the ostensive pronoun “idam,” i.e., the “pakṣa”] must be characterized by “knowability” due to the very fact that it is being ostensibly pointed out. There can be no conceivable exceptions or counterexamples to it. So, “knowability” turns out to be an all-inclusive hetu and it applies to any possible pakṣa, whatsoever. The rationale underlying a kevalānvayī anumāna looks very similar to that of Berkeley’s argument for his thesis “esse est percipi.” No one can cite an instance, say x, and claim, “here exists an x which is not perceived.” No wonder, a critic jibed saying, “Berkeley’s argument is irrefutable, but unconvincing.” It is to be noted here that “all-inclusive-ness” of the hetu entails the “impossibility of having any counter-instance.” [Cp. Chattopadhyay, H. (1983), 8788 (footnotes). Also see, Perrett, Roy W. (1999) where he remarks that “Naiyāyikas are fond of a slogan, which often appears as a kind of motto in their texts: Whatever exists is knowable and nameable.” He then asks, “What does this mean? Is it true?” The first part of his essay addresses the first question by giving a brief explication of this important Nyāya thesis; in the second part he argues that, given certain plausible assumptions, the thesis is demonstrably false.]

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Accordingly, both the Bauddhas and the Jainas had to invoke and rely upon antarvyāpti, in order to be able to justify their respective pet metaphysical theses like “whatever is real, is momentary,” (Bauddha kṣaṇabhangavāda), or “whatever is real, has an infinite number of facets” (Jaina anekāntavāda), etc. Such overarching metaphysical claims are not purely analytic ones because, unlike “a bachelor is an unmarried male,” they are supposed to have some factual content. Clearly, neither the idea of “having an infinite number of facets” nor that of “being momentary” are contained in the idea of “being real,” any more than the idea of 4 is contained [in the Kantian sense] in the idea of 2 + 2. Thus, in a way, such “metaphysical theses” are akin to “synthetic apriori judgements” in so far as they contain the elements of both “universatlity” and “necessity.” As the features of “universatlity” and “necessity” of a genuine vyāpti-relation cannot be ascertained by relying on bahirvyāpti alone, the Bauddhas and the Jainas argue that one needs to reject bahirvyāpti, and opt for antarvyāpti instead, for this purpose. I consider it both ill-advised and unfortunate that the Jainas as well as the Bauddhas did use some metaphysically loaded pan-inclusive theses/claims as typical examples to illustrate their moot point in favor of antarvyāpti and against the alleged methodological adequacy of bahirvyāpti. I think, it is because of this that the ultimate shape that the controversy about the rationale of and about the role played by antarvyāpti [in the process of ascertainment of vyapti] changed in such a way as could fool both the opponents as well as the proponents [of bahirvyāpti] into believing that the only point for postulating antarvyāpti [over and above bahirvyāpti] was to defend some pan-inclusive metaphysical claims or else, to clinch some such fairly uninteresting issues like whether a two-component or a three-component theory of inference is to be preferred. Consequently, the central point about the real logical significance of antarvyāpti was missed by most people. Rest of this subsection (a) is devoted to clarifying the often-missed “the real logical significance” of postulating antarvyāpti. It should be clear by now that if Nyāyācārya’s interpretation of antarvyāpti were correct, the question of looking for other instantiating examples outside the pakṣa-class woud not arise at all. Consequently, we should expect to find a “hetu-sādhya concomitance” only among the members of the pakṣa-class. This is, Nyāyācārya claims, what “concomitance holding within the pakṣa,” i.e., antarvyāpti, precisely means [1975, 40]. Unfortunately, this claim is as much an instance of a clear statement of Nyāyācārya’s view on antarvyāpti, as it is of a confused and misleading one about the nature of antarvyāpti. Moreover, Nyāyācārya does not tell us how his interpretation of antarvyāpti would fare with views like that of Jayanta Bhaṭṭa, who hold that “there is no kevalānvayī hetu” or, with the Vedāntins’ view, according to which neither kevalānvayī nor kevalavyātirekī anumāna-s are admissible [Chattopadhyay, H. (1983), 88. Also, Vedānta Paribhāṣā, p-152], though at the same time, the Vedāntins do make overarching metaphysical claims like “jagat mithyā, dṛśyatvāt,” which is practically indistinguishable from a kevalānvayī anumāna.

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Although, it is a fact that justification of such overarching metaphysical theses like, “everything is momentary” [“sarvam kṣaṇikam”] (of the Bauddhas) or, similarly, the claim, viz., “everything real, possesses innumerable features” [“anantadharmātmakam vastuḥ” (of the Jainas), etc., happen to depend, [qua anumiti] on antārvyāpti and yet, it need not mean that antarvyāpti can have no other logical role to play in a different context, nor does it show that anyone who makes such “overarching metaphysical claims” (like the ones mentioned here) has to admit “antarvyāpti” as a legitimate posit. It may be pointed out in this connection that according to the Nyāya theory, the Buddhists’ claim (viz., “sarvam kṣaṇikam”) is to be considered an instance of unacceptable, faulty inference that is vitiated by the hetvābhāsa known as “asādhāraṇa,” instead of being considered an acceptable case of anumāna proper which is legitimately based on a “kevalānvayī hetu.” However, in all fairness to Nyāyācārya, I must also point out that he is in good company. A number of reputed scholars of Indian philosophy seem inclined to share Nyāyācārya’s interpretation of antarvyāpti. For example, Phaṇībhūṣaṇa [(1989), Vol-1, 339], Vidyābhūṣaṇa [(1978), 177178], Bhattacharya [(1994), 247249], all of them simply recast Vadideva’s words in English and maintain that in cases of antarvyāpti, discernment of a vyāpti is possible “without depending on any external observation besides the pakṣa itself,” without really explaining the true significance of “holds internally/ within the pakṣa itself.” Besides having other difficulties, Nyāyācārya’s interpretation seems to suggest that antarvyāpti is only a limiting case of bahirvyāpti [in a somewhat kindred sense in which “invariance of mass” in Newton’s system is a limiting case of “mass” in Relativistic mechanics, though only under extremely low velocity conditions]. Had antarvyāpti been just a limiting case of bahirvyāpti, then the only philosophical utility of antarvyāpti would lie in providing a justification for kevalānvayī anumāna-s [Nyāyācārya (1975), 3940]. Naturally, outside of such a context, antarvyāpti would hardly have any other philosophical role to play. I plan to show, however, that antarvyāpti does have a different and deeper methodological significance, in so far as it can be reconstrued as an effort to address a problem somewhat similar to that of Kant’s problem of justifying “synthetic a-priori” judgements. The notion of antārvyāpti looks very similar in spirit to Kant’s notion of an “analytic judgement,” where “the subject-term contains the predicate-term within it” [das Prädikat B gehört zum Subjekt A als etwas.] All we need for such a re-construal is to substitute, “in an analytic judgement the subject-term contains the predicate-term within it,” in place of “in antarvyāpti, the concomitance of hetu and sādhya holds within the pakṣa.” [See (ii) below.] However, it is extremely important to remember here that “Holds internally/ within the minor/pakṣa” should not be taken to refer only to those cases where for the purpose of inferring “fire-on-the-hill” (on the basis of “smoke-on-thehill”), one concentrates on the cognitive awareness of concomitance between “this hill-smoke” and “this hill-fire” to the exclusion of similar other cases of smoke-fire relationship. If this claim is wrongly taken to mean that antarvyāpti

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must depend only/exclusively on the single observation of “this hill-smoke” and “this hill-fire” as empirical instances, then cases of antarvyāpti would turn out to be special cases of vāhirvyāpti in the sense that it now requires only a single empirical instance. This amounts to failing to appreciate the real methodological significance of antarvyāpti. Moreover, it also fails to square it up with the Jaina claim that anyathānupapannatvam yatra tatra trayeṇakim/nānyathānupapannatvam yatra tatra trayeṇa kim. //(Phaṇībhūṣaṇa (1989), 121)

“if antarvyāpti is competent enough to justify the inferential transition from a probans (hetu) to its probandum (sādhya), then vāhirvyapti (extrinsic/empirical concomitance with its three-point requirement) becomes redundant; on the other hand, if antarvyāpti is not capable of such a feat then taking recourse to vahirvyāpti becomes useless and infructuous.” [In the verse quoted, “anyathānupapannatva” stands for “antarvyāpti.” Translation mine.] Anyways, the above discussion shows that the view of Pt. S. C. Nyāyācārya fails to capture the insight that underlies the need for postulating antarvyāpti, and consequently, he fails to appreciate the true logical significance of “antarvyāpti” vis-à-vis “bahirvyāpti.” I claim further, that by letting the notion of antarvyāpti play the most pivotal role in their theory about ascertainment of the relation of invariable concomitance (vyāpti-relation between a hetu and its sādhya), the Jaina logicians ended up skillfully hitting the right cord, presumably without fully appreciating the associated acoustic dimensions of the musical note that it produced. Next, I am going to discuss the parallels between “antarvyāpti” on one hand, and Kant’s notion of “analytic judgements” and “synthetic a-priori judgements,” on the other. After that, I will elucidate with examples what I mean by “associated acoustic dimensions” of taking antarvyāpti seriously. [See (ii), (v) below.] (ii) “Antarvyāpti” and “Synthetic a-priori” When I claim that “semantic-conceptual linkage” has a thematic affinity to Kant’s idea of “subject-predicate relationship” in an “analytical judgement,” the qualifying phrase “thematic affinity” is studiedly used by me on purpose. However, unlike Kant’s analytical judgements, a “semantic-conceptual linkage” [i.e., antarvyāpti] is not informationally sterile. As Kant himself put it: Analytic judgements do not widen/enrich understanding [“... analytisches Urteil bringt den Verstand nich weiter”]. In contrast to it, my position about “semantic-conceptual linkage” in antarvyāpti vis-à-vis the subject-predicate relation in Kant’s “analytical judgements” may be stated thus: First, semantic-conceptual linkage must not be equated or conflated with purely analytical judgements in Kant’s sense. Johnson’s notion of “intuitive induction” [e.g., a generalization, “all Kangaroos have short forelegs,” made after seeing a Kangaroo only once] better captures the sense in which, granted antarvyāpti, even a single instance [sakṛt darśana] may yield “all smoky

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things are fiery” type of generalizations. Secondly, my idea of semanticconceptual linkage [unlike the Kantian idea of analytical judgements] would help to reconcile the Bauddha view that all vyāpti relations are ultimately reducible to tādātmya or tadutpatti with the Jaina view that antarvyāpti alone counts as genuine vyāpti relations. The reason is quite simple. It is obvious that “tādātmya” straightway entails “semantic-conceptual linkage,” whereas, the universality of pratītyasamutpādavāda ensures that there is an exception-less conceptual linkage between any two “caused” things that are subject to the law of pratītyasamutpannatva, (i.e., everything, without any exceptions). Thirdly, this way of looking at the issue of antarvyāpti, as a sort of “semantic-conceptual linkage” would very naturally dovetail into the general framework of Indian theory of anumāna [which should be able to yield an anumiti, which must have a “nonzero information content,” i.e., to have an element of novelty (¼ anadhigatatva)]. Satisfying the “novelty requirement” automatically makes an anumāna immune to the charge of “petitio” – a charge raised against purely deductive inferences. Fourthly, it may also be pointed out in this connection that without a sharp analytic-synthetic divide, the problem of “synthetic a-priori” does not arise at all, and honestly speaking, I did not find any such sharp line of analyticsynthetic-divide in Indian logic. The “thematic affinity” between the corresponding difficulties, [which are meant to be tackled by taking recourse to the notions of “antarvyāpti” and “synthetic a-priori judgements,” respectively] is more “notional” than “literal” in intent. [Strictly speaking, however, there are different shades of views in Indian philosophy regarding the “novelty-requirement.” That “novelty” is an indispensable requirement for an anumāna is not admitted by all schools of Indian philosophy. The Bhaṭṭa Mimāmsakas make this claim in an emphatic manner. For Buddhists like Dharmakirti, a pramāna must reveal some novel aspect of its object, since otherwise, it will not be able to lead to successful activity (arthanirṇaya). The Sāmkhya school also admits this view. But Jayantabhaṭṭa in his Nyāya- manjari, Udayana in his Nyāyakusumānjali and Sālikanātha Misra (a Prābhākara Mīmāmsaka) in his Prakaraṇa Pancikā have rejected this claim after a detailed discussion.] It is also a tricky question how, if at all, an analytic statement may acquire a “factual relevance.” This question is relevant here because if “antarvyāpti” (viewed as semantic-conceptual linkage) and “synthetic a-priori judgements” are kindred in spirit, then both should share the elements of “necessity” as well as of “novelty.” One example may make it clear. The first and the second law of Newton are analytic in nature in the sense of just being “definitions” of “force” and of “mass” [Mass ¼ df the constant of proportionality between “force” and “acceleration”] and yet, these two laws do have “factual relevance” in mechanics. This does not look so surprising, especially after Quine’s [1957] decisive onslaught on the very rationale that underlies a sharp analytic-synthetic divide. When I claim that there is a close “thematic affinity” between “antarvyāpti” and the “subject-predicate relation” that logically links the “subject” to the

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“predicate” in a synthetic a-priori judgement, that claim is naturally prone to mis-construal in several ways, which needs to be guarded against. First, the analogy drawn between antarvyāpti as a “semantic-conceptual linkage between a hetu and its sādhya that holds within the pakṣa” must not be taken to mean that the sādhya is contained in the hetu in the way in which the predicate is contained [in Kant’s sense] in the subject term of a purely analytic judgement. [A better analogy, (already given above), would be the Kangaroo example used by Johnson to illustrate his idea of “intuitive induction.”] Certainly, the idea of “having short forelegs” is not contained in the idea of “being a Kangaroo”; nevertheless, the concept of “having short forelegs” gets intuitively/naturally linked to the semantics of the word “Kangaroo.” Presumably, according to the Jainas, similar is the case/nature of the linkage that holds between “smoke” and “fire,” or between “having horns” [of course, not the kind of horns that are fitted on a Milk Van] and “being an animal.” “Non-analytic universalizability” is the precise point of analogy between the notion of antarvyāpti and Kant’s idea of “synthetic a-priori.” When viewed in this way, it becomes easier to reconcile the Bauddha’s claim about “reducibility of all vyāpti-relations either to tādātmya or to tadutpatti,” with the Jaina’s claim that “all genuine vyāpti-relations are ultimately based on a semantic-conceptual linkage between a hetu and its sādhya,” i.e., antarvyāpti. Although the Buddhists are “atomists,” whereas the Jainas are “panrelationalists,” yet I do claim that the Jaina view about antarvyāpti and the Bauddha tādātmya-tadutpatti thesis can be reconciled, without showing any fissure, in terms of a common model, viz., Leibniz’s model of “monadology.” [However, as a discussion on the topic would be more relevant to Jaina metaphysics, than to Jaina logic, I considered it advisable not to include it in this chapter. Moreover, it would be wrong to deny that the Jainas are “atomists” too, of course of a different kind. Theirs is, what I call, “non-insular” or, “panrelational atomism” which is more akin to Leibniz’s “monadic-atomism” or to Wittgenstein’s “Tractatus-type atomism.” [For details on this point, see, Sarkar (2020), forthcoming book; “Studies in Jaina Philosophy: Creating Dialogue with Western Philosophy,” Chap. 2.] There is another spin-off [viz., of looking at antarvyāpti as “semanticconceptual linkage between a hetu and its sādhya”] that has a deeper significance in the sense that it preserves in a very “natural” way the following two common salient features of the views held by all schools of Indian philosophers/logicians: (i) anumāna is a kind of “nontrivial-knowledge-generating process” [just like perception] and as such, (ii) all instances of genuine knowledge-claims [including an anumiti] must contain an element of new information or, novelty in it. Unless, of course, one tends to equate “a semantic-conceptual claim” with “a purely analytical claim” [the latter having only a “zero information-content,” i.e., no element of novelty at all in it]. However, our notion of antarvyāpti-based inferential linkage prevents Indian logic from lapsing into a system of un-informative pattern of purely formal deductive inference. As such, unlike a standard deductive inference, an

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anumāna is not susceptible to the charge of “petitio.” It is also very important to keep in mind that when I claimed that there is a “thematic affinity” between semantic-conceptual linkage of a hetu and its sādhya in antarvyāpti, and the kind of subject-predicate relation/linkage in a Kantian synthetic a-priori judgement (e.g., 7 + 5 ¼ 12), I did not mean that there is any formal isomorphism between them. A “thematic affinity” cannot be taken in the sense of an “exact point-by-point similarity,” just as despite having a “thematic affinity” between Democritus’s “atomism” and Bohr’s model of atoms, the two cannot be taken as “strictly isomorphic.” (iii) Jaina Concept of “Antarvyāpti” and Bauddha Concept of “Svabhāva hetu.” As already pointed out earlier, both the Bauddha and the Jaina logicians were advocates of “antarvyāpti.” Incidentally, in Buddhism, one comes across another technical term, viz., “svabhāva hetu,” which seems to play the same methodological role as antarvyāpti plays in Jaina philosophy. It is interesting, however, that the respective examples used (by the Jainas) for antarvyāpti and the ones used (by the Bauddhas) for what they call, “svabhāva hetu,” are uncannily similar. Actually, both parties use “It’s a tree, because it is an Oak” (or, some similar variants of it) as illustrative examples for their respective cases. This naturally prompts one to ask whether or not the two terms mean the same thing, except for being couched in different terminologies. The case gets a little complicated because Dharmottara defines “svabhāva hetu” differently in his two different works. In Hetubindu Tīkā [p-41] it says: “sa sādhanadharmaḥ bhāvaḥ svabhāvo yasya,” while in his Nyāyabindu [Chap. 2, Sūtra 22] it says: “sa sādhyo’rtha ātmā svabhāvo yasya yataḥ sādhanam.” [Information gathered from H. Chatterjee (1983).] Both of these definitions are compatible with Vādideva’s view, viz., “in antarvyāpti the linkage between a hetu and its sādhya is said to be holding internally/within the pakṣa” and I think both “svabhāva hetu” and “antarvyāpti” do share the spirit of, what I call, “semantic-conceptual linkage.” Moreover, both of them also have an unmistakable thematic affinity to Kant’s idea of “subject-predicate relationship” in an analytical judgement because, in a sense, “being a tree” can be viewed as an analytical consequence of “being an Oak.” (iv) Antarvyāpti and HD Method in Science: At this point, one may pertinently ask: if antarvyāpti [which is defined as a “semantic-conceptual relation” (and not an empirically derived one) between a sādhya and a hetu] is, by itself, an adequate tool for ascertaining requisite vyāpti-relations, then why even a sakṛt-darśana should ever be needed for, or be relevant at all to, the process of ascertainment of vyāpti? How to explain it? The answer is not far to seek. Strictly speaking, in applying HD method, scientists start by first, making an initial hypothesis in order to find out a general law-like regularity which can possibly account for some hitherto unexplained phenomenon. Clearly, such a hypothetically entertained general law-like regularity, say H, is not the result of any prior instance-based inductive generalization. For example, presence of gravitational pull exerted by some “not-yet-discovered planet X” was

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postulated first. It was initially postulated [by Alexis Bouvard] in order to explain the observed aberrations in the orbit of the planet Uranus. Subsequent sighting of the planet Neptune by Johann Galle, at a particular spot as mathematically deduced from H, had confirmed H. Similar was the case with Einstein’s 1916 prediction about “bending of the light-rays” in a strong gravitational field, and its later confirmation (in 1919) by Arthur Eddington. However, another very similar hypothesis, call it H1, failed to explain the perihelion of Mercury. The main point that emerges out of the above discussion is this: HD method has no need to depend on observation of any individual instance to discover a law-like regularity that can explain a given phenomenon. However, scientists who use HD method need to look for empirical observation only for postprediction confirmation of an initially postulated explanatory hypothesis, say H. Pre-observation of individual instances hardly has any role to play in the formulation of an initially postulated law-like hypothesis. Need for a “postprediction confirmation” is the reason why sakṛt-darśana [or, even of a few darśanas] still remains relevant, even after both bahirvyāpti and bhūyodarśana are jettisoned in favor of antarvyapti. [For a similar approach, see Srinivas, M.D. (1988).] (v) Associated Acoustic Dimensions of Hitting the Antarvyāpti Cord I may now explain what I meant when I claimed [a few paragraphs earlier] that the Jaina logicians failed to fully appreciate the related philosophical consequences that the innovative incorporation of antarvyāpti into their system] did produce. First, if antarvyāpti alone [as a conceptual-cum-semantic relation, supposedly ascertainable a-priori] is the real pivot on which the justification of the entire process of ascertainment of vyāpti-relationship depends, then instancebased empirical generalization becomes dispensable and the problems pertaining to making fully warranted empirical generalizations [by resorting to exceptionless bhūyodarśana] automatically drop to a secondary position and get skirted around as well. This is to take the first step towards to a non-inductivist approach to methodology of science. Secondly, given antarvyāpti, even seeing a single instance [i.e., sakṛt darśana] may be sufficient for ascertaining [as claimed by the Jainas as well as by the Prābhākara-s] a vyāpti-relation between an x and an y, by way of successfully latching on to avinābhāva [i.e., an invariable concomitance relationship between an x and an y]. Since, according to the Jainas avinābhāva is the necessary and sufficient condition for any vyāpti sambandha, it follows that wherever there is antarvyāpti, bhūyodrśana has no role to play; unless there is an antarvyāpti between an x and an y, bhūyodrśana must fail to ensure full warranty for any “invariable concomitance relation.” The Jainas show the infructuousness of bhūyodrśana as a fail-proof means of ascertaining vyāpti relation [even when all of the five-factor requirements (of the Naiyāyikas) or the three-factor requirement (of the Buddhists) is fulfilled]. The Jaina logicians cite a number of convincing counterexamples which show that co-location

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[sāmānādhikāraṇya] neither entails nor is entailed by vyāpti relation. There may be universal concomitance between an x and an y, even when the two are not co-located. [For details about the counterexamples used by the Jaina logicians, see: Sarkar (1997) in Saha (ed.)]. Similarly, bhūyodrśana of exception-less instances of x and y do not entail that there is vyāpti relation between them, although if there were a vyāpti relation between them, that would ensure exception-less bhūyodrśana of the two. Any attempt to capture the notion of invariable concomitance solely in terms of co-location [sāmānādhikāraṇya] is shown to be too restrictive. [In this context, the role of elimination of all possible vitiating accidental factors (upādhinirāsa) becomes crucial, and the possible ways of handling the problems pertaining to it which may require taking recourse to the machinery of default/nonmonotonic reasoning] also need to be discussed.] Some interesting methodological consequences of the above moves are the following: First, the theoretical need of bhūyodarśana is dispensed with, because bhūyodarśana, if it is good for anything at all, is at the most good for eliminating erroneous/accidental generalizations of “post hoc” type [kākatālīya nyāya nirāśa]. Even a Naiyāyika of Udayaṇa’s stature [in his Nyāyavārttika tātparya pariśuddhi] partly agrees. Secondly, as a consequence, it opens up the possibility that even a single observation [i.e., sakṛt-darśana] may be good enough for universal generalization. Thirdly, the need for ad hoc postulation of sāmānyalaksanā [resorted to by the Naiyāyikas as a damage-control measure] no longer remains. Fourthly, the requirement that dṛṣṭānta be retained as an indispensable avayava of an anumāna becomes dispensable. Fifthly, even the indispensability of pakṣadharmatā jñāna for an inference is categorically denied by some Jaina thinkers like Siddhasena, Akalanka, Vidyānanda, Vādibhasimha, etc. Finally, by dispensing with bahirvyāpti, switching over to antarvyāpti in its place by claiming the adequacy of antarvyāpti as the only means for ascertaining genuine vyāpti relations, and further, by refusing to meekly toe the line of “crash inductivism” [of Mill’s kind], the above chain of logical moves resulted in a sort of conceptual ground-clearing for what can be viewed as a precursor of hypothetico-deductive [HD] method in theory-construction. [See (iv) above.] To sum up: By rejecting bahirvyāpti, in favor of antarvyāpti, the Jainas brought a sea-change in the logic-scenario. As a result of their move, the general direction of thought about logical-linkage [vyāpti] changed (a) from being “empiricistinductivist” to becoming “conceptualist-non-inductivist” in nature. (b) This, in its turn, prepared the ground for making the theory of anumāna free of its traditional “factuality bias,” as well as made it more flexibly adaptive by opening up favorably to admitting (i) counterfactual patterns of argumentation in logic and (ii) by doing the necessary groundwork for using HD method for arriving at LLG’s. Finally, (c) these moves taken by the Jaina logicians exposed the artificiality-cum-untenability of arbitrary compartmentalization of different areas of logic. I think, it is unlikely that the Jaina logicians were in a position to be fully aware of, or able to fully appreciate all these subtler tones that emanated from their logical orchestration.

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§4. Antarvya¯pti: Its “Logical” and “Methodological” Aspects DisEntangled In all Jaina books that discuss the nature of vyāpti, one finds two characteristic features of it, viz., “avinābhāba” and “anyathānupapatti” repeatedly mentioned. Both traditional and more recent writers consider the use of these two terms very clear and unproblematic. Naturally, they keep using the terms interchangeably, as if, they were perfect substitutes of each other [e.g., “avinābhābo hi ananyathānupapannatvam” (Nyāyācārya (1975), 45)]. Sometimes, one of the two features, say “anyathānupapatti,” is selected, and highlighted as the sole criterion of vyāpti, to the exclusion of the other, viz., “avinābhāva.” Often, it is done so nonchalantly as if, using those two terms interchangeably does not matter at all [e.g., “Niścitānupapatti ekolakṣaṇo hetum” (PNT, 3/11), also Nyāyācārya (1975), 42, 46]. On the other extreme of it, we find this in Tātparya-tīkā, “tasmādantarvahir vā sarva upasamhāreṇa avinābhāvaḥ avagantavyaḥ.” [Quoted in Phaṇībhūṣaṇa (1989), Vol-1, 319]. Most of the people I talked to do not consider these differences as serious discrepancies. Instead, they regard the use of the two different technical terms [e.g., “avināvhāba” and “anyathānupapatti”/ “ananyathāsiddha,” etc.] interchangeably, simply as “minor stylistic variations” – something which is not worth scratching our heads about. Consequently, people belonging to such a group refuse to recognize a crucial problem that lurks behind the issue. Without entering into the rather pedantic issue, viz., whether “avināvhāba” and “anyathānupapatti” are synonymous, we can use the two “interchangeably.” [Cp. “avināvhābo hi ananyathānupapannatvam” (Nyāyācārya (1975), 45).] “Anyathānupapannatva” or “ananyathāsiddhatva” literally means “(as of now), not explainable/establishable otherwise.” Up to a point in time, some phenomena could not be explained without postulating “ether,” but it is no longer so. What can or cannot be explained in some “other” way (at a given point of time) is “epochal-cummethodological” in nature. This is the reason why I recommend adding “as of now” before “cannot be explained otherwise,” in translations of “anyathānupapanna” or, “ananyathāsiddha.” It should also be clear by now that practically, in all contexts of use of “anyathānupapanna” or, “ananyathāsiddha,” the hetu-component of a vyāpti concerned indicates causal ground of a sādhya. It has to be a “because-therefore” type of explanation, rather than being a purely logical “if-then” type explanation. Kant’s distinction between “antecedent-consequent” relation and “cause-and-effect” relation is very relevant here. It is very pertinent here to note that even in today’s formal logic, there is no notion of “implication” which can formally capture or reflect the distinction between “if-then” and “because-therefore.” According to some Naiyāyika-s avinābhāva relation obtains even between smoke and fire. Here, it should be noted that “avinābhāva” is used by them in a different sense to mean any empirically observed, exceptionless co-presence of two things. On the other hand, avinābhāva is a technical term in Jaina logic which signifies a sort of “semantic-cum-conceptual inseparability relation” as holds between say, “concavity-convexity,” “even-odd,” “two poles of a magnet,” [ignoring magnetic monopoles for now], etc. Being semantic-cum-conceptual in nature, avinābhāva of the

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Jainas cannot but be logical-cum-non-epochal in nature, which a “smoke-fire” type relation is not. In specific contexts, the hetu-component of the vyāpti relation [qua avinābhāva of the Jainas] stands for a purely logical “if-then” type implication between a hetu and its sādhya, whereas a vyāpti relation [qua “avinābhāva” as used by some Naiyāyika-s] signifies a factual “because-therefore” relation.] To conflate the senses of Jaina and Nyāya uses of “avinābhāva” would be a sort of “conceptual category mistake.” The truth of “tomorrow is a Monday, because today is a Sunday” is based on an empirical knowledge [viz., whether today is actually a Sunday or not]; whereas the truth of “if today is a Sunday, then tomorrow is a Monday” does not require one to know whether today is actually a Sunday or not. It can be shown by citing any number of instances that the Indian logicians lacked any keen awareness of the distinction between the “methodological” and the “conceptual-logical” aspects of a hetu that a vyāpti-relation may indicate. Consequently, they were prone to mix up the “methodological” and the “conceptual-logical” aspects of vyāpti, without realizing its implications. Yet, because of their instinctive and keen analytical acumen they had a hunch that something was amiss somewhere, so far as the question of how to prioritize the claims of “avinābhāva’ vis-à-vis ‘anyathānupapannatva” is concerned. This, I hope, explains the anomaly in the various formulations of the Jainas and thereby also removes a potential source of confusion. The Buddhists hold [as we saw in (b) above] that every vyāpti is either a case of identity relation [tādāatmya] or else, it is a case of cause-effect relationship [tadutpatti]. Of these, the identity-based cases would ensure that there has to be a conceptual link (which is ascertainable a-priori) between the hetu and the sādhya concerned. On the other hand, the cause-effect based vyāpti relations would account for pure empirical generalizations. Apparently, there is no reason why the Jainas should criticize the Tādātmyatadutpatti thesis, instead of incorporating it in their own system. As a matter of fact, however, the Jainas and the Naiyāyikas did criticize the tādātmya-tadutpatti vāda on the following grounds: (i) the Tādātmya-tadutpatti thesis of the Bauddhas depends on their doctrine of pratītyasamutpādavāda and the associated doctrines of asatkāraṇavāda and kṣaṇikavāda. But, as the Jainas point out, (ii) if everything exists only for one moment and is non-comparably unique [sva-lakṣaṇa], then comparison of items for discerning the points of similarity between them and for subsequently making necessary abstraction out of them [which is indispensable for discernment of universals] are not possible. Consequently, inside the Buddhist framework, the idea of vyāpti as an unfailing, invariable relation between a “hetuclass” and a “sādhya-class” loses the ground under its feet, and the Tādātmyatadutpatti thesis becomes untenable. However, the Bauddhas who deny the very admissibility of “universals” as something “real,” respond to the criticisms by maintaining that a universal is only a subjective projection, not something real that has an ontological status. Only, universals qua “subjective constructs” are logically needed for their Tādātmyatadutpatti vāda to stand on. So, the criticisms of the Jainas and of the Naiyāyikas [against the Buddhist position] fail to hit their target.

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It is interesting to note in this context that like the Bauddhas the Jainas also deny the admissibility of “universals as an ontological reals,” and yet being strong realists, the Jainas consider vyāpti-relations to be something objectively real. Hence, consistency requires that unlike the Naiyāyikas, the Jainas consider vyāpti relations as a sort of semantic-conceptual linkage [based on avinābhāva]. For them, universals are not full-fledged ontological reals [as “universals” are, according to the Naiyāyikas.]

§5. Nature of Hetva¯bha¯sa: An Analysis The Logicians of the Nyāya school, as we have seen, held that a legitimate probans must be characterized by a set of five characteristic features. A probans in its turn is considered illegitimate if it lacks any one or more of the five required characteristic features. An inference based on an illegitimate probans [duṣṭa hetu] is considered to be a case of hetvābhāsa. Accordingly, the standard view of the Naiyāyikas is that there are five types of hetvābhāsa or defective inference, each type corresponding to violation of a specific characteristic feature. Since, according to the Buddhists, any defect of inference is due to the fact that the probans used in an inference lacks one or more of the three legitimizing features mentioned by them, they admit of only three kinds of hetvābhāsa, viz., savyābhicāra, asiddha, and viruddha. [(Dingnāga, Dharmakīrti, Nyāyabindu)]. The Jainas, as we have seen, hold that neither the five nor the three of the characteristic features can guarantee the legitimacy of a probans and ensure its ability to logically lead to the targeted probandum [sādhya] as a conclusion [Yaśovijaya (1973), 12. (Ed. D. Bhargava)]. According to the Jainas, anyathānupapannatva (i.e., “as of now, it cannot be explained otherwise”) is the sole requirement (both necessary and sufficient) for being a legitimate hetu. So, there needs to be only one type of hetvābhāsa, [i.e., defect of inference due a faulty/ illegitimate hetu] which arises due to a violation of/failure to conform to that sole requirement, viz., anyathānupapannatva. Such a failure, according to the Jainas, is tantamount to jettisoning the semantic-conceptual inseparability [avinābhāva] relation between a hetu and its sādhya. That there is an inseparable semantic-conceptual relation [avinābhāva] between a hetu and a sādhya simply means that “it is impossible that the hetu exists but the sādhya does not, i.e.,  M (hetu &  sādhya).” [The use of “M” and other symbols in the formulation must not be taken as a suggestion that Lewis’s strict implication can capture or, even adequately represent the spirit of “avinābhāva” which, unlike strict implication, is both context-sensitive and epistemology-permeated. The symbols have been used here simply to enhance presentational perspicuity.] In contrast to “avinābhāva,” I propose to take “anyathānupapannatva” as indicative of a currently encountered factual, [not logical or, conceptual] methodological difficulty – only a temporary road-block to a plausible alternative theorization. In view of this crucial difference between “anyathānupapannatva” and “avinābhāvatva,” I decided (i) to add “as of now” as a prefix in my “translation of ‘anyathānupapanna’” and (ii) phrased my translation of it in the declarative mood [e.g., “cannot be explained otherwise”] instead of using a potential mood like, “not

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explain-able otherwise.” This would enable us to avoid using any modal operator, like “M,” in its formulation. In this context, it is extremely important to note that the Jaina logicians tend to use “anyathānupapanna” and “avinābhāva” interchangeably. This, I think, sowed the seed of confusion between the logical and the methodological aspects of a hetu, and it also resulted in conflating the senses of “anyathānupapannatva” and “avinābhāvatva.” [In my earlier writings on Jaina logic, I failed to notice this crucial difference between “anyathānupapannatva” and “avinābhāvatva” and uncritically toed the line of the dominant official position that glosses over their crucial difference and equates the two. I plan to explain, later in this section, the reasons behind my present position.] Apart from the above point, we also need to note that instead of using “hetvābhāsa” as a common tag for any and every possible defect of inference, the Jainas proposed to use a more inclusive term “anumānābhāsa” to mean “defects of inference in general.” Besides the inferential errors due to some defect in its hetu, the Jainas also mention different types of errors of inference [other than hetvābhāsas, taken in its strict literal sense] which include, pratijñābhāsa [defective initial formulation of the conclusion to be established], pakṣābhāsa [defect due to wrongly picking up the purported locus of an imputed feature], sādhyābhāsa [improper formulation of the characterizing feature to be inferred], drṣṭantābhāsa [improper selection of supposedly corroborative instances], etc. Some Jaina logicians like Vādirāja went on to further subclassify drṣṭantābhāsa-s into six different kinds. Pakṣābhāsa (or, pakṣadoṣa of the Bauddha’s) is reduced to a kind of hetvābhāsa by Jayantabhaṭṭa. Sometimes, pratijñābhāsas are subdivided into five types, asiddha hetvābhāsa being viewed as a particular type of pratijñābhāsa [Phaṇībhūṣaṇa (1989), Vol-1, 298]. The distinction between “hetvābhāsa” and “anumānābhāsa” was hardly ever strictly adhered to by the Jaina logicians. However, we may occasionally exploit this terminological point to our advantage for analytical clarity. Let us, for the time being, confine the use of “hetvābhāsa” to mean such defects of inference as are exclusively due to a defective hetu. Now, according the Jainas, the only defective hetu are the ones that fail to be anyathānupapanna/avinābhāvi. Hence, there should be only one type of hetvābhāsa. How come then that many Jaina writers [Yaśovijaya (1973), JTB, 18] speak and admit of three or four different types of hetvābhāsas then? [Yaśovijaya (1973), JTB, 18]. This apparent inconsistency in their position is explained by the Jaina thinkers, by taking recourse to a general strategy, namely, by maintaining that all three/four of the so-called varieties of hetvabhāsa are but different ways of failing to satisfy the anyathānupapannatva [i.e., “cannot be explained otherwise”] requirement. Akalamka goes even a step further and holds that all sorts of violation of anyathānupapannatva, (i.e., the “cannot-be-explained-otherwise” requirement) share a common feature, namely, akincitkaratva (i.e., lack of any inferential significance/relevance) [(Akalamka (1939), Nyāya Viniścaya, 2/120)]. Vādidevasūri (PNT 6/57) regards akincitkaratva as a defect that characterizes a pointless inference (e.g., an inference that tries to justify a tautology or something that is obviously/undeniably true.) [(Dharmabhūṣaṇa (1945), 102103).] It’s easy to see why Yaśovijaya refuses to accept akincitkara as a distinct type of logical defect, on top of the three mentioned by him [Yaśovijaya (1973), 18)].

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It is interesting to note that the nonorthodox Naiyāyika Jayantabhaṭṭa initially admitted akincitkara as the sixth type of hetvābhāsa (on top of the usual five admitted in Nyāya logic) but subsequently had succumbed to the pressure of tradition and instead of dropping it altogether, proposed to include akincitkara as one kind of asiddha hetvābhāsa. In the light of the very brief sketch given above, we may now take a deeper look at hetvābhāsas in the context of Indian logic and especially, of Jaina logic. I need to frankly admit here that I was not at all enthused by my first exposure to hetvābhāsa for a number of reasons. Initially, the treatment of the topic of hetvābhāsa looked needlessly circuitous, loaded with use of tons of archaic terminology and burdened with repetitious use of a fixed set of hackneyed illustrative examples. Above all, it looked cut off from all routes of relevance to contemporary logical thinking. However, as my ideas matured and I graduated, so to say, from “Junior KG grade” to “Senior KG” in Indian Philosophy, I realized that one’s attitude to “Indian Logic” need not be so harsh after all, because a “seeming irrelevance” may also be due to an “inability to see” the points of relevance. The relevance of a “positron track in a cloud chamber” failed to be noticed even by the brilliant physicists for a number of years. In the next few paragraphs of this section, my endeavor will be to take a fresh, unconventional look at the problem of hetvābhāsa in a way that should make the ideas hidden behind its traditionally presented uninviting façade, look both sensible and relevant. Our first step in that direction will be to recast the theory of hetvābhāsa in contemporary jargons. A proper hetu is supposed to be able to “invariably latch on” to its sādhya. If it succeeds, it is a legitimate hetu, otherwise it is a defective one [duṣṭa hetu]. Standard presentations of “theory of hetvābhāsa” are made to look like a recipe for identifying and sifting out inferences infected by a duṣṭa hetu, i.e., a hetu that fails to properly latch on to its sādhya. I propose to reconstrue the relation of “latching on to” as a mapping relation between the elements of a “hetu-set” and those of its corresponding “sādhya-set.” Before proceeding further in that direction, we may note some peculiarities of the examples used by the traditional writers on anumāna and hetvābhāsa. What kind of information is required in order to justify the inferential passage from “being smoky” to “being fiery”? After having observed that a good number of smoky things [woodburning fire places, kitchen hearths, wildfire spots, etc.] are invariably fiery things too (irrespective of other features in which they may differ), one arrives at a hetuvākya or a law-like generalization [LLG], viz., “whatever has smoke, also has fire.” All inferences similar to “smoke-on-this-hill,” therefore, “fire-on-this-hill” depend on some relevant hetu-vākya or other. Clearly, for formulation of such LLG’s, the instances observed need to be (i) analogous, i.e., relevantly similar in respect of being a location that is shared by both smoke [hetu] and fire [sādhya] and (ii) observed co-location-ness [sāmānādhikaraṇya] of hetu and sādhya be the empirical ground of a required inductive generalization. This explains why Keynes observed long time ago that there is a close affinity between an analogical argument and the process of induction. [Keynes (1957), Chapters XVIII, XIX, Harper Torch-book, NY. For Aristotle’s treatment of analogy,

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see Joseph, H.W.B. Introduction to Logic [OUP], Chap. XXIV. See, Stebbing, L.S. (1961) p. 250, Chap. XIV for a few other examples of analogical arguments similar to the “natural abode” example used by Aristotle.] A similar analogy-based projection [which is a more primitive basis of induction] had been used by Aristotle to arrive at an LLG in formulating his “natural abode” theory of falling bodies. However, it is not that all analogy-based projections do lead to a legitimate LLG’s. Clearly, in the case of Aristotle, it did not. What really then is the problem that the Indian logicians were concerned with? It is worth noting that it was neither the development of a formal theory of inference (following a syllogistic model), nor was it to find a solution of the problem of induction (as we understand it now). Their real concern was centered around the following issues: (a) formulating rules for distinguishing between projectable and non-projectable properties (e.g., the property of “being smoky” [¼ Ψ]) is a non-projectable property, i.e., it is not a target of projection [ToP] in respect of “being fiery” [¼ Φ]. Obviously, we cannot legitimately infer, “fiery, therefore, smoky.” Consequently, a projection rule that authorizes inference of or, projection to “being smoky” [¼ Ψ]) from “being fiery” [¼ Φ] would be an improper one. It would be a schema for any anaikāntika/savyābhicāri hetvābhāsa. However, it is important to emphasize here that there is nothing absolute either about a property being projectable or, its being non-projectable or about whether a projection-rule is legitimate and proper or, illegitimate and improper. Any answer to such questions has to be essentially context-relative. Since a law-like generalization [i.e., the projection rule “g (Ψ ⇝ Φ)” from “smoke ¼ Ψ” to “fire ¼ Φ”] is already incorporated as the required hetu in any anumāna [inference] and the hetu is an “indispensable avayava” [i.e., an un-eliminable component of an inference], therefore, the problem is not that of “reinventing the wheel” again [i.e., of re-discovering the LLG, over again] but instead, it is that of deciding whether the specific mapping-rule/projection-rule concerned, (here it is “g”) is a legitimate one. Hence, the whole issue boils down to the problem of identifying the right sort of projection base [PB], with a suitable set of items that are characterized by a set of required projectable properties [PP] and finally, to the problem of formulating an appropriate kind of mapping-rule/projection-rule [PR] that maps the elements of a projection base, to elements [i.e., the targets of projection (¼ ToP’s)] in the set of the projectable properties.

§6. “Prayojakatva” and “Aprayojakatva” of Hetva¯bha¯sas Traditionally speaking, different hetvābhāsas are ways of specifying how violation of a particular restriction imposed on a given projection-rule results in a specific kind of flawed/faulty anumāna. Keeping this in mind, we proceed to discuss the three types of hetvābhāsas, viz., anaikāntika, asiddha, and viruddha, which both the Buddhist and the Jaina logicians admit. [Logical status of the remaining two hetvābhāsas (viz., satpratipakṣa and bādhita) is quite different.] To take a computer analogy, anaikāntika, asiddha, and

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viruddha, hetvābhāsas prevent the inferential machinery from getting started, while satpratipakṣa and bādhita hetvābhāsas let the inference-machine run through the whole course leading to an anumiti. However, only when the anumiti-generated [i.e., the inferential output produced by a human inference-machine which is assumed to act like a deterministic knowledge-machine (DKM)] clashes with another legitimately generated/derived anumiti, [which is already stored there in the data-base of the DKM], that the human inference-machine [¼ the anumātā] stops entering/registering the recently generated anumiti in its existing data-base. So, satpratipakṣa and bādhita hetvābhāsas are not due to a violation of any restriction on a projection-rule which would prevent the machine from functioning. Instead, these two let the inference-machine run its full course, and also let it generate an inferential output [i.e., an anumiti], and then goes into the “Sleepmode,” until it [i.e., the anumātā, functioning like a DKM] receives further “instructions.” Looked at from this angle, the satpratipakṣa and bādhita hetvābhāsas are, rather, about how to handle a situation where two equally legitimate projection-rules end up yielding mutually incompatible anumitis (i.e., inferential conclusions). [For details, see: Sarkar in SRS (1997), 371; also, Bhattacharya, G.N. (1983), 244268.] Besides the abovementioned machine-analogy-based explanation of why some philosophers dropped satpratipakṣa and bādhita hetvābhāsas from the original Nyāya list of five hetvābhāsas, there is also another more traditional way of looking at the issue. Interestingly, this “more traditional way” is not altogether un-linkable to the machine-analogy-based explanation. According to the author of Nyāyakandalī [(p-15], the distinguishing feature shared by all five of the hetvābhāsas is aprayojakatva, i.e., inability to yield any reliable and definite outcome [“aprayojakatvam ca sarva hetvābhāsānām anugatam rūpam” (Ibid, p-15.)]. Such inability [i.e., [aprayojakatva] may be due to two different reasons: (i) The “outcome” may be wrong because inadmissible inputs are “fed into” the machine, and consequently, it fails either (a) to register the inputs altogether (as in cases of asiddha hetvābhāsas) or else (b) it ends up giving wrong outputs, i.e., wrong anumitis, (as in cases of anaikāntika and viruddha hetvābhāsas). Alternatively, (ii) such inability may be due to the fact that though the inferencemachine registers the input-data and is functioning perfectly well, except that its currently generated inferential outcome [i.e., the anumiti], clashes with an equally legitimate previously derived inferential outcome [this is what happens in the cases of satpratipakṣa and bādhita hetvābhāsas.] Naturally, due to presence of such a clash [occurring in the cases of satpratipakṣa and bādhita hetvābhāsas], the decision about choosing any one of two or more incompatible anumitis remains inconclusive, i.e., remains aprayojaka, in one sense. Thus, taken in the wider sense [of (i) and (ii) combined], aprayojakatva is the feature shared by all five [anugatam rūpam] of the hetvābhāsas. But, taken in the narrower/more restricted sense (ii) above, only satpratipakṣa and bādhita are aprayojaka. This is why the Jaina and the Bauddha logicians (who did use “aprayojakatva” in its more restricted sense (ii)) had good reasons for excluding satpratipakṣa and bādhita from the Naiyāyika’s list of five hetvābhāsas.

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§7. The Three Hetva¯bha¯sas Considered by the Jaina Logicians (a) Anaikāntika hetvābhāsa: Suppose, “Ψ” and “Φ” are the projection-base and the projection-target, respectively, of a relevant projection rule, say “f (Ψ ⇝ Φ).” [Here, “f (Ψ ⇝ Φ)” stands for “ ‘f’ maps the elements in the projection-base ‘Ψ’ (e.g., all smoky things) to elements in the projection-target ‘Φ’(e.g., all fiery things), and ‘⇝’ indicates ‘mapping relation’. ” In cases, where under “f,” the elements of “Ψ” stand in a “one-one” or in a “one-many” relation, [which can be either “into” or “onto”], to the elements of “Φ,” “f” is considered an admissible projection rule. How are we to figure out whether an admissible projection rule, say “f,” is also a legitimate projection-rule or not? Let us consider another mapping relation, say “g,” such that it says “g (Φ ⇝ Ψ).” In this case, it turns out to be non-legitimate. How are we to guard against such illegitimacies? On the above interpretation, “f” happens to be a legitimate projection rule. So, the anumāna [inference], viz., “it is fiery, because it is smoky,” is free of any hetvābhāsa. However, on the same interpretation the anumāna [inference], say “g (Φ ⇝ Ψ),” i.e., “it is smoky, because it is fiery,” involves anaikāntika/ savyābhicāra hetvābhāsa. The reason? Very simple. There are many cases of fire without presence of any smoke, e.g., a red-hot piece of iron, or a gas burner with blue flames, etc. A closer look shows that in “g (Φ ⇝ Ψ),” [the projectionbase “Ψ” and the projection-target (¼ToP) “Φ,” of the previous example, viz., “f (Ψ ⇝ Φ)”] switched their respective positions. So, Ψ is now a proper subset of Φ [because, now Φ contains all fiery things, both smoky and non-smoky, whereas Ψ contains only the smoky things.] Hence, an inferential passage from “fiery” to “smoky” would involve an over-generalization or illicit generalization. This shows that all anaikāntika hetvābhāsa-s are actually cases of such “illicit generalization.” [For details, see: Sarkar (1997), 376 ff.] (b) Asiddha hetvābhāsa: Majority of Indian logicians, excepting the Jainas, broadly agree on the point that there can be no legitimate anumāna in the absence of an actual exemplar. Hence, they claim, we cannot arrive at any legitimate anumiti jñāna [¼ inferential conclusion] about such purported entities as “sky-lotuses” or “dream-flowers” or “Pegasuses.” Consequently, even such arguments as “All sons are males,” therefore, “all sons of a childless woman are males,” etc., are considered flawed/defective in the context of Indian theories of anumāna; although they would be considered valid, without any question, in Western logic. Violation of this strong existence-requirement by the required examples results in a kind of inferential illegitimacy called “asiddha hetvābhāsa” [¼ un-grounded projection-base/inference-base]. The maxim to follow in such cases is this: “No Projection-rules are applicable, unless its projection-base is non-empty.” [For details, see: aprayojakatva. Sarkar (1997), 369 ff.] (c) Viruddha hetvābhāsa: Suppose, we are given a projection rule, viz., “g (Φ ⇝ Ψ),” where Φ is the projection-base (consisting of different input options) and Ψ is the projection-target (consisting of different output options) and “g” is the projection rule which is supposed to “map” a suitably chosen input item in Φ to a

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desired output-item in Ψ. Let the desired output-item be “producing a fertilized ovum through in-vitro fertilization,” in order to ensure that the baby born from that fertilized ovum will be a boy. Let us assume further that the technology for manipulating and combining individual sex-chromosomes in a gene is available. Suppose, the task is to be done in a modern fertility clinic, equipped with all necessary facilities. The job is assigned to a competent lab technician. On completion of the job, the technician concerned submits the job-completion report to his supervisor, stating that he had used an “XX” combination of chromosomes for the assigned purpose. Naturally, the report upsets the supervisor. Obviously, the technician had selected the wrong input-item from Φ because, “XX” is mapped to an inappropriate output-item, viz., “the baby born will be a girl.” To ensure the desired output, the lab technician needed to make a different selection of input-item. Selecting an “XX” combination as input turns out to be counterproductive. So, it is not a permissible input-item, just as in the textbook example, “kṛtakatva” is not a permissible input-item to be selected to justify the claim that a pot is eternal, because it is a product [“ghataḥ nitya, kṛtakatvāt.”]. A similar modern example would be like selecting the interference-pattern of a “double slit” experiment, in order to justify the claim that an electron is a particle. It is counterproductive. Selecting a wrong input in violation of the required restrictions results in a faulty inference called, “viruddha hetvābhāsa.” [For details, see: Sarkar (1997).] (d) Quasi-hetvābhāsas? Some queer instances of “fallacious validity” in Western Logic: After having explained a plausible alternative approach [in which hetvābhāsa-s in Jaina logic are viewed as “a mapping relation which has gone wrong somehow”], I will end this section by drawing attention to some other relevant points in this connection, without going into the details of any one of them, as they are simply some general observations on Indian theories of hetvābhāsa, having no specific relevance to the Jaina view. It is to be noted here that (i) the Hetvābhāsa-s are: (α) not “purely formal fallacies” at all. Western logicians are mixed-up in this respect, despite the fact that they seriously claim to have made a “neat classification of fallacies” into “deductive” ones and “inductive” ones. Let us take a closer look at the so-called “neat classification of fallacies” in Western logic. Even the supposedly “pure deductive” fallacy of ambiguous middle turns out not to be a purely deductive one, it is actually of a mixed sort – it is a “semantic-cum-logical” fallacy. This becomes obvious, if we remember that a computer program that relies on a purely abstract schema, based exclusively on rules of “formal syntax,” would fail to be sensitive to the two different contextual senses of “dates” in its two occurrences [viz., in “dates are edible” and in “12th and 13th of May are dates”]. Naturally, such a context-insensitive logic program would put “12th and 13th of May are edible” in the category of valid inferences. (β) Again, despite the fact that they maintain a very sharp line of demarcation between “inductive” and “deductive” logics, the Western logicians, unlike their Indian counterparts, are hardly concerned with the problem of

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formulating a general definition of “fallacies,” which applies both to “Inductive” as well as to “Deductive” fallacies with equal plausibility. (γ) Moreover, the theoretical position of Aristotelian logic [AL] (i.e., traditional logic) is not internally consistent at all, even when one takes into consideration only the purely formal deductive fallacies. Let us take just one such example: In traditional logic, “Most P” ¼ “Some P.” So, “Most S are P” ¼ “Some S are P,” it is an “I” proposition in which both the subject and the predicate terms are undistributed. Yet, from “Most teachers are graduates” and “Most graduates are reliable” we can validly infer that, “Some teachers are reliable.” Although, (a) the argument is a syllogism, (b) it violates the syllogistic requirement that the middle term must be distributed at least once in the premises, and (c) it is also valid in the sense that if its premises are true, so must be its conclusion. Although, this very same argument has to be counted as definitely invalid, as per the rules of Aristotelian logic. Such queer cases may be called, “fallaciously valid” arguments. Nothing can better highlight the difficulties of working out a totally unproblematic scheme of neat compartmentalization of logic into “deductive-inductive,” of fallacies into “formal-informal,” of arguments into “valid-invalid,” etc. In our college days, we grew up being constantly exposed to the claim that “Indian logic” blurs/lacks clear lines of “area-compartmentalization” vis-à-vis the “surgically clean dissection” of areas in Western logic.

§8. Minimum Number of Avayava-S Needed for an Anuma¯na Among the Indian logicians, there is a controversy about the minimum number of avayavas (components) needed for an anumāna considered as “an inferential unit.” [For example, a syllogism, as an inferential unit, has three components, whereas an immediate inference, as an inferential unit, has only two.] In other words, the point of controversy is this: How many avayavas in an anumāna should be considered sufficient for leading to an anumiti (conclusion of an anumāna /inference). Interestingly, the views in this regard range from “only a single avayava is really needed” [ekāvayava vāda, (proposed by some Buddhists and some Jainas), going all the way to “up to ten (or more) avayavas (daśāvayava vāda) may be needed” as and when required in a specific context.] Daśāvayava vāda was proposed by some older Naiyāyikas, some Buddhists and also by some Jainas. The two-component view [dvi-avayava vāda, (of some Buddhists)], the three-component view [tri-avayava vāda (of the Sāṁkhyas)], the four-component view [caturāvayava-vāda (of the Mīmāmsakas)], etc., coming in between “ekāvayava vāda” at one end and “daśāvayava vāda” at the other. Although, the typical and representative view of the Jaina-s in this regard is known as “two-component view” of inference [dvi-avayava vāda], this must not be taken to mean that all Jaina thinkers subscribe to this view. The “two component view” [¼dvi-avayavavāda] of inference is supported, among others, by Māṇikyanandī, in his Parīksāmukhasūtra [¼ PMS], by Hemacandra, in his

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Pramāṇamīmāmsā [¼ PM], Vādidevasuri in his Prāmaṇanayatattvālokālankāra [¼ PNTL], Prabhācandra in his Prameyakamalamārtaṇda [¼ PKM], etc. According to these thinkers, pratijñā and hetu are the only two constituents of anumāna that are really needed for leading to an inferential cognition. However, the supporters of two component view make it amply clear that any restriction on the number of required components [¼ avayavas] in an inference, say to 2 or 3 or more is not a logical, but only an extralogical requirement. One may go on adding more and more avayava-s if needed, by taking into consideration such contextual factors as the intellectual level, the background knowledge of the person for whose conviction the inference is being made, etc. For a dull-witted person [mandabuddhi], an exemplar may be needed, but for a really smart person, it may be dispensable, just as for a 5- or a 6-year old, doing an addition [of, say 5 and 9], counting on fingers is needed, but for a grown up, it may not. Use of geometrical figures in proving theorems is unavoidable for most of us but, in principle, theorems in geometry can be proved without taking recourse to actual geometrical figures. Hemacandra and Yaśovijaya are quite emphatic on the point, however, that an exemplar [¼ udāharaṇa] is not really necessary for arriving at an inferential conclusion. In the same way, for a person who cannot easily figure out how an exemplar relates to and supports a given hetu, upanaya may have to be added as a further component. Stripped to its barest minimum, an inferential unit takes the following form: “The hill is fiery” [¼ pratijñā] because, “it is smoky” [¼ hetu]. This resembles an enthymeme in Aristotle’s logic, in which some premises are withheld/kept implicit, on the assumption that a normal person would be able to fill-in the resulting information-gap by using his common-sense or, the store of his background knowledge [¼BGK]. Similarly, the Jainas’ two-component view [¼ dvi-avayavavāda] of inference, proposes/recommends to make good use of BGK or, background information that the person who is inferring possesses. This naturally fits in well with the Jaina assumption that in an inferential situation, our intellectual machinery functions like a deterministic knowledge machine [¼ DKM], following a fixed set of sequential steps. If inferences are considered simply as products of such a fixed, deterministic input-output-sequence-generating machine, then, depending on how rich the in-built data-base of a DKM is or, how it gradually improves/enlarges itself, the amount of stored information accessible to it, etc., the step-by-step “operational instructions” needed to be fed into such a machine may be proportionately decreased. Keeping this background in mind, we may try to understand why the Bauddha logicians proposed to go to the extreme and embraced a single component view of inferential components by claiming that hetu alone should be regarded as the only necessary component of an inference [Nyāyācārya (1975), 47; Sarkar (1997), 379]. Actually, the controversy regarding the minimum number of avayava-s needed for an inference is not at all a logical controversy in the sense in which the question about the independence of an axiom in a formal system is. It is actually a controversy about how much background information may be taken for granted for arriving at an inferential result/conclusion [anumiti]. But, interestingly, “the extent of what may or may not be taken for granted” depends also on who it is that takes a decision on what or how much may be taken for granted.

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The point that I am trying to make here, may be illustrated by using a popular story about the famous mathematician G.H. Hardy [without whose help the hidden genius of Indian mathematician Ramanujan would remain unrecognized]. The story runs thus: Hardy was lecturing on a difficult theorem in Number Theory to some advanced graduate students. While proving the theorem, he wrote a few steps on the board, and then wrote the final step, as he muttered “now, it is obvious that .... (putting emphasis on ‘obvious’).” He then turned around, looked at the class and asked, “Isn’t it obvious, boys?” There was no response. So, Hardy asked the brightest guy in the class, “What about you?” “No Sir. It is not obvious at all,” came the timid response. After a brief pause, Hardy incredulously asked, “Do you really mean, it’s not obvious?” “Yes Sir,” the student said. Hardy remained silent for a moment and then somberly said, “OK, then, let me think over it again,” and left the classroom. After a couple of minutes, he came back with a beaming smile on his face and said, “Well, boys! I thought over it again. IT IS obvious.” No one needs to be told now, how subjective a decision on, “what to take (or, not to take) for granted,” is. Isn’t it pretty obvious? Hemacandra maintains that the hetu, [which is indispensable for explaining something (viz., a sādhya)], and the pratijñā, these are the only two constituents that are really needed for an anumiti to be possible. Since pratijñā does not have any assertional commitment, it can act only as a signpost indicating the direction of the desired conclusion. The only one operative [¼ prayojaka] or, result-generating, [i.e., anumiti-generating] constituent of an inferential unit is, therefore, the hetu. The Bauddha and the Jaina views do not really differ much in this regard. The real and truly controversial point at issue comes to relief only when we seriously analyze the implications of the views of Jaina thinkers like Anantavīryācārya, (1941) [PRM], who in his note on Sūtra 36 of PMS rejects the three-component theory [pratijñā, hetu, udāharaṇa] of the Sāmkhya-s; and the four-constituent [pratijñā, hetu, udāharaṇa, upanaya] theory of the Mīmāṁsaka-s. As the Jaina-s deny any role to pakṣadharmatā and udāharaṇa in an inference, this third component, viz., udāharaṇa, is dropped by them from the list of indispensable components. Similarly, upanaya is dropped from the list as well, because, it is actually a meta-logical component, rather than a logical one. Upanaya only tells us that a proposed substitution instance, say “m,” of the universal variable “x” that occurs in the law-like generalization “(x) (Lxh ! Lxs)” [¼ whatever has (i.e., is the locus of the) hetu x, also has the sādhya, y] is a legitimate one. This is like the rule of “E.G.” which entitles us to infer “(∃x) Fx” from “Fa,” provided “a” is a non-empty name. The role that a upanaya vākya plays is that of explicitly stating that the said proviso, which is actually a meta-logical rule, has been fulfilled. [See: Sarkar (1997), 379380.] Now, if udāharaṇa and upanaya be dropped, we are left with only three [out of the five traditionally admitted components of an inferential unit], viz., pratijñā, hetu, and nigamana. Of these, pratijñā is the provisionally entertained but assertionally noncommittal, expression of the conclusion that a given inference unit is trying to arrive at. As such, it acquires the right to become an assertion, so to say, after it becomes a nigamana on the basis of the given hetu. Hence, once the hetu and the

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pratijñā are given/stated, the conclusion need not be stated separately, except for the very dull people [see Yaśovijaya (1973),16]. On the final count, therefore, only two components [i.e., avayava-s], viz., the hetu and the pratijñā, are retained. Why may we not go a step farther, make maximal use of one’s store of available background information and compress all inferences to single-membered inference-units consisting of hetu as the only required component [¼ avayava] of it? From the point of view of conceptual economy, that would be ideal, only if it did not have a snag. The snag is that from the very same hetu one may infer a number of different things, e.g., from the presence of the same hetu, viz., presence of “smoke” on the hill, one can infer, with equal logical plausibility, not only the presence of fire there but also that there is wet fuel [i.e., wet logs and sticks stoking the fire], that there is a source of heat, etc., What one actually infers on the basis of a given hetu depends on what is intended by him to be established [¼ abhīpsita]. A pratijñā, therefore, can be viewed as a signpost, the only function of which is to direct the application of the hetu along the right track so as to reach the desired inferential conclusion [¼ anumiti]. Clearly, with the very same hetu, but with a different pratijñā, an equally legitimate, but very different, conclusion could have been drawn. Hence, it follows that in addition to a hetu, a pratijñā also should be regarded as a non-dispensable component of an inferential unit. [Here, “hetu” has to be taken to mean the sign (e.g., smoke) itself, which is an indicator of a specific signified (e.g., fire). “Hetu,” must not be taken here to mean the hetu-expressing sentence (¼ hetuvākya) that states the universal concomitance between a hetu and its sādhya.] It is true that the Jaina philosophers never tried to justify their two-component theory of inference, as explicitly as we have done above. The Jaina logicians, however, always insisted that a sādhya [in our terminology, “the target of an inference”] must have the following features: (i) it must be inferentially intended [¼ abhīpsita] – it is one of the epistemic factors involved in an anumāna, (ii) it [¼ sādhya] must not have been already shown to be definitely un-entertainable [¼ a-nirākṛta], e.g., trying to infer how may digits are there in the largest prime or how sharp is a rabbit’s horn – it is the logical entertainability factor in an anumāna. Moreover, (iii) that any sādhya must also be “as-yet-un-established” [¼ a-pratīta], i.e., it must still have a new information-content – it is the novelty of informationcontent factor in an anumāna. It is clear that anyone who considers (a) abhīpsitatva of the sādhya, as an indispensable requirement and (b) also appreciates the fact that in the absence of a pratijñā, the same hetu can be used to draw very different conclusions, must also admit (c) that besides hetu, it is pratijñā which needs to be taken as an indispensable component of an inferential unit. The Jaina logicians were fully aware of (a), (b), and (c) above. The same awareness of the ineffectiveness of a hetu, standing alone and all by itself, in the absence of any pratijñā, was expressed by Jayantabhaṭṭa in his Nyāyamanjarī [NM] (p-144), “pratijnām vinā nirāśrayo heturbhavet iti sā purvam prayoktavyā.” With this discussion as our background, we may now proceed to take a quick look at the different versions of the ten-component view [¼ daśa-avayavavāda]. Both Bhadravāhu [PM, 2/1/10] and Yaśovijaya [(1973), 16] point out that depending on the specific nature of the context of an argumentation, e.g., the intellectual smartness,

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the depth and extent of background knowledge of an inferer [¼ anumātā, i.e., the person who infers], an inferential unit may be optionally gradually expanded to a full ten-constituent [or, even more] sequence of components [¼ avayava-s]. According to Bhadravāhu, the ten components of an inferential unit are the following: (i) pratijñā, (ii) pratijñā-vibhakti, (iii) hetu, (iv) hetu-vibhakti, (v) vipakṣa (vi) pratiṣedha, (vii) drṣṭānta, (viii) āśankā, (ix) āśankā-pratiṣedha, and (x) nigamana. Out of these ten, (iii), (vii), and (x) are the familiar ones. Of the rest, (ii) and (iv) have to do with semantic clarification of the meanings of respective avayava-vākyas. Notion of such clarification, in its turn, is intimately linked with the Jaina doctrine of vibhajya-vāda. [Sarkar (2020, forthcoming), Chap. 5.] The components (v), (vi), (viii), and (ix), in Bhadravāhu’s list, are concerned with alleviating our state of doubt. Their import is purely epistemological-cum-psychological – not at all logical. Hence, except (i), (iii), (vii), and (x), all the other components are nonlogical in nature and thus may be dropped from consideration, without any logical harm, just as instructions like “put the plug in the power socket,” “turn the switch on,” etc., can be dropped, without any real loss, from the manual of instructions for operating a computer. In Vātsāyanabhāṣya, the ten-components mentioned are the standard five of the Naiyāyika’s, plus the following five: (i) jijñāsā [an inquisitive urge], (ii) samśaya [being in a state of un-surity], (iii) śakyaprāpti [motive to achieve the desidiretum], (iv) prayojana [relevant need-perception], and (v) samśayavyudāsa [overcoming the state of un-surity]. It is clear that each one of these is a nonlogical component in an inferential unit. As such, they cannot be regarded as avayava-s of the same logical status with the other logical ones. Hence, those are dispensed with by the Jaina-s. The way they proposed to dispense with three of the other five components in order to support their “two component view” [¼ dvi-avayavavāda] of an “inferential unit” has already been discussed. It should be pointed out here that the Jaina’s refusal to consider “udāharaṇa” as an indispensable avayava of an anumāna may be viewed as a corollary of (i) taking avinābhāva as the sole requirement for invariable concomitance or, vyāpti-relation, and (ii) denial of any effective role to vahirvāpti entail that actual instances [“udāharaṇa-s”] can have no role to play in an inference. Hence, it [udāharaṇa] cannot be considered one of the “indispensable” components of an anumāna. Another not so obvious spin-off from the Jaina theory of vyāpti-ascertainment is this: viz., that on ultimate analysis, an invariable concomitance can be definitely ascertained only by taking recourse to tarka or hypothetical reasoning [tarkāt tanniścaya]. In order to methodologically legitimize this claim, the Jainas needed to admit tarka as a full-fledged pramāṇa. They did this by going against the Naiyāyikas and some other mainstream traditionalists. At least from our vantage point of view, I prefer to consider this bold and away-from-the-tradition approach of the Jainas as a primitive inkling of hypothetico-deductivism [Popper-Lakatos type] away from a simplistic Mill-type “Inductivism” of the Naiyāyikas, in so far as framing/formulation of empirical law-like generalizations [LLG’s] are concerned.

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§9. Use of “Vikalpa” as “Existence Proof” in Jaina Logic If logic, as a theory of inference, is always required to be tied down to the actual world and if “udāharaṇa” and bahirvyāpti-based empirical law-like generalizations were essential for logical inferences, then arguments based on counterfactual conditionals [CFC’s] would not be logically admissible. That would drastically restrict the scope of logic. The Jaina logicians were bent on easing such restrictions and with this aim in mind, (a) they denied that bahirvyāpti can have any role at all to play as a means of generating true law-like generalizations [LLG’s], (b) they claimed that avinābhāva [linkage due to semantic-conceptual inseparability] alone is the characteristic feature of any invariable concomitance [vyāpti], which can be ensured only by antarvyāpti. They also claimed that (c) neither “udāharaṇa” nor “pakṣadharmatā” is an indispensable procedural step in the direction of inference [¼ anumāna], and that (d) tarka [counterfactual conditionals] must be admitted as a pramāṇa in its own right on top of the standard ones like perception [pratyakṣa], inference [anumāna], etc. Furthermore, (e) the Jainas rejected “pure inductivism” for sound methodological reasons and were inclined to opt for a sort of non-inductivist methodology, similar in spirit to what may be called, “hypothetico-deductive method” [HD method] in modern terminology [See: §3, and also §7, for a discussion on it.] Finally, (f) by denying that “third linga parāmarśa” has any role to play as an essential step on the way to anumāna, the Jainas also tried to get rid of “psychologism in logic.” [G. Frege severely criticized Husserl for resorting to psychologism in mathematics.] Jaina logicians’ move along the above line [(a)–(f)] freed logic of its prevalent “factuality bias” and also of its “psychologism” [both of these elements are embedded in the logical systems of Nyāya, as well as of other Indian schools of logic] on the one hand, and opened up the possibility of developing a more abstract approach to logic, on the other. In order to keep the length of this chapter reasonable and also to avoid needless repetitions, I will not go into the details of “psychologism in logic” in this chapter. [Interested readers may see: Thomas Sheehan (2015), Making Sense of Heidegger: A Paradigm Shift, Rowman & Littlefield. NY.] However, the following points, relating to the notion of “existence” in Jaina logic, need to be mentioned here. Because of some of their metaphysical commitments, the Jainas needed to make inferences about things which cannot be known by any of the ordinary pramāṇas [accredited means of knowing]. So, they tried to solve the problem by formulating a sort of “existence proof” which they called, “Vikalpa.” They maintained that so far as the existence of an ordinary thing is in question, some ordinary pramāṇa, e.g., sense perception, is to be used. Their existence is not subject to proof by Vikalpa. Jaina logicians maintain that in a vikalpa-proof, it is legitimate to hypothetically entertain as a posit the existence of the entity that is sought to be proved. Vikalpaproof depends on the principle of absence of any definite impossibility/inconsistency in the idea of the thing sought to be proved “Vikalpasiddho dharmī yathā sarvajño’sti

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suniścita asambhavadvādhakapramāṇ atvādityastitve sādhye sarvajnaḥ.” [Yaśovijaya (1973),14.] For example, the existence of imaginary numbers would not be “provable”/ “logically entertainable” unless the square root of “minus-1” were free of any definite impossibility-proof. The Jaina logicians further maintained that in order to deny the existence of something without inconsistency, we need to change the negative existential statement to its proper logical form. For example, “there are no hare-horns” is to be transformed into “there are no hares with horns.” [Yaśovijaya (1973),15.] Lastly, the Jaina logicians are also very clear about it that one can use vikalpa only in order to prove the existence (or nonexistence) of a thing which is not knowable by any ordinary pramāṇa but is supposedly characterized by certain properties [i.e., it is a dharmī]. On the basis of the above remarks, it is possible to formulate some rules [by using the notion of a “sahaja dharma” (an “original”/“in-built” characteristic feature), the modal operator “M,” and some additional rules of “instantiation” (which are especially relevant to Vikalpa proofs)]. This logical outfit was formulated with a view to justifying the Jaina’s belief that “morally perfect, omniscient human beings are both conceivable and also do exist” [sarvajña asti]. It should be noted in this context that Vikalpa neither does, nor can it, prove the actual existence of any ordinary thing. From what has been said above, it should be clear that Jaina logic was clearly shaped, to a large extent, by their ontology, especially anekāntavāda, and also syādvāda [¼ doctrine of unavoidable conditionality of all propositionally expressed truth-claims.] These two, coupled with Jaina theory of language, made their joint contribution by developing an elaborate, nonstandard scheme of classification of propositional expressions. Keeping such logical ramifications in view, the Jainas classified all propositional expressions (i.e., any grammatically correct, meaningful sentence to which a truth-value can be assigned) by going beyond the artificial True/ False dichotomy of the logical positivists. Naturally, the resulting Jaina scheme of classification has some highly interesting features. As a consequence of breaking the barrier of True/False dichotomy, the Jaina logicians were able to include not only the purely truth-functional expressions but also the non-truth-functional ones in their scheme and classified all purported truth-claims into (α) satyāpanı ya (paryāpta) bhāṣā [i.e., potentially truth-value assignable expressions of a language [Prajaha Sūtra. Bhāṣāpada, 1519.] and (β) a-satyāpanı ya (a-paryāpta) bhāṣā, [i.e., nonalethic ones to which no truth-value (T or F) can be assigned [Ibid]. The potentially truth-value assignable expressions again are of three types, viz., T (true), F ( false), and imprecise ones [i.e., expressions to which only a non-sharp truth-value can be assigned (e.g., “current population of India is one billion.”)]. This shows that the Jaina-s are never happy with an “all-or-none” type scheme of bifurcation of truth-values. The non-alethic expressions, on the other hand, are sentences/expressions (e.g., “May God bless you,” “Listen to your parents,” “Wish you the best of luck,” etc.) which are not classifiable under any one of the three classes of potentially alethic [i.e., truth-value assignable] expressions listed above. In some Jaina texts [e.g., S.M. Jain (1986): Jaina Bhāṣādarśan [in Hindi], Chap. 7], non-alethic

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expressions of a language are classified into two subgroups, viz., quasi truthfunctional expressions (satyāmṛṣa bhāṣā) and pure non-truth-functional (asatyāmṛṣa bhāṣā). Nonetheless, according to the Jaina-s, such non-alethic expressions are logically as significant as are the potentially alethic ones. Accordingly, the non-alethic expressions are graded and classified by the Jaina-s into various subclasses of non-truth functional, yet informationally non-empty, expressions. This idea of a non-truth functional and yet information-wise non-empty sentential expression/proposition stands in sharp contrast to the logical positivists’ view, according to which a sentence which is neither T nor F, must not be counted as having any information-content whatsoever. [See Brahma (1998), Chap. 5, for further details of the Jaina scheme of classification of statements, also see, S.M. Jain (1986).]

§10. Contradiction and Contextualization of LNC in Jaina Logic In the last section on “existence proof,” which is called “Vikalpa,” it was pointed out that the said proof depends on the assumption that if the idea of an “x” is not definitely, [i.e., not provably/ logically “inconsistent”], then “x” can be provisionally entertained as being really true. The question, “how to define the notion of inconsistency itself,” particularly in the context of Jaina logic, naturally arises here. The present section is devoted to a discussion of that question. Before we can proceed further, we need to have a few basic background information. (i) The Jainas, like all other Indian logicians, define contradiction/inconsistency in terms of epistemic inconsistency and not in terms of propositional incompatibility, [i.e., being guided solely by the abstract form (say, “P & ~P”) of the propositions concerned]. Looking at logical contradiction in terms of “epistemic inconsistency” is quite unlike the way “contradiction” is treated in Western logic. (ii) Contradiction is said to occur only when one epistemic state ( jñāna) is blocked (pratibaddha) by some blocker (pratibandhaka). (iii) Any given negated proposition, say not-q, does not form a contradictory pair with any arbitrarily selected proposition p. A contradiction occurs only when both the assertion and the negation of the propositions conjoined, do refer to exactly the same referent. For example, an epistemic state J1 can form an incompatible pair with another epistemic state J2, only if J1 and J2 have exactly the same epistemic content (viṣayatā). (iv) Viṣayatā is a multicomponent composite notion consisting of a specific set of relata and a specific relation R that holds between the given set of relata. The viṣayatā of the knowledge, “a (specific) pot is on the floor,” consists of: (a) The characterizer (prakāra) [here, it is the referent of the word “pot”]. (b) The characterized (viśeṣya) [here, it is the referent of the word “floor”]. (c) The specific floor-pot-contact relation R viz., of “being on” [here, R is called the samyoga sambandha].

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(v) J1 and J2 cannot have the same epistemic content (viṣayatā) unless their respective (viṣayatās) have exactly the same componential-cum-relational specificity. [Notes: (α) Grammatical peculiarity of the Sanskrit sentence: “Bhūtale ghato’sti,” plays an important role in this discussion about the nature of “inconsistency.” In order to avoid needless complications, I decided to translate the said Sanskrit sentence simply as, “a pot is on the floor” (without attaching any definite article or any qualifier-word to “pot”). So, in the present context, “a pot is on the floor” is supposed to be taken as an exact equivalent of “Bhutale ghato’sti.” (β) See: Banerjee, H. (1972): “On the Mistranslation of the Terms “Viśeṣya” and “Prakāra.” Philosophy East and West no. 22 (i): 9396, for problems pertaining to proper translation of “prakāra,” “viśeṣya,” “viṣayatā,” etc.] We may now proceed to analyze the notion of inconsistency in Jaina logic, keeping the above points in mind. Let J1 ¼ a pot is on the floor, and J2 ¼ a pot is not on the floor. Are J1 and J2 mutually incompatible? It depends. Actually, the two need not be incompatible, because someone with knowledge J2, may be referring to the pot (a different one) which he saw here yesterday on the very same spot. Let us call the prakāra (related to the epistemic content of J2) P2 . For obvious reasons, a different pot which J1 sees on the floor now, is a different prakāra, say P1, (as related to the epistemic content of J1). Even a slight difference in any single component of a viṣayatā can change its identity. Clearly, simply because we are using the same propositional form “P” [once with and once without the negation sign “~”], that need not mean that the prakāras of J1 and J2 are the same. Hence, although form-wise “(J1 & ~J1)” is an inconsistent pair, yet in our present case it is not. According to the Jainas, the epistemic content of any jñāna has four dimensions of freedom (un-specificity), viz., dravya (substantiality), kṣetra (location), kāla (temporality), and bhāva (features). Each one of these admits of an infinite number of variations [Yaśovijaya (1973) JTB, 19]. It should be clear by now that any two epistemic claims like J1 and J2 can be shown to be non-incompatible (i.e., jointly entertainable) so long as even one dimension of freedom or a single degree of variations of their corresponding viṣayatā remains unfixed/unspecified. Exact specification of all dimensions and of all degrees of variability of a specific dimension is not possible (for obvious reasons) in normal cases. [See: Haribhadra (1986), 215. (Ed.) L. Suali, Ṣad-ḍarsana Samuccaya. (ṢDS).] I may mention here a few more recent similar approaches to contextualizing the notion of logical inconsistency and of the notion of contradiction, so that the Jaina approach may not look like an inane and pointless philosophical exercise having hardly any relevance to contemporary logic. In his book Putnam uses a mereological example to show how even such a simple question as “How many objects are there in the room,” happens to be so highly context-dependent that it may elicit apparently inconsistent responses like, “there are five objects in the room” [¼P] and “there are not five objects in the room” [¼ ~ P] without lapsing into any inconsistency [Putnam

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(1992), 110120; Also see, Sarkar (1998), LBC for a detailed analysis of the reasons why there cannot be any absolute and purely syntactic definition of logical inconsistency, i.e., a definition of logical inconsistency which is free of all semantic considerations.] Nancy Cartwright, a trained physicist and a philosopher of science, strongly argues against “absolutizing” the notion of “inconsistency.” She argues against what she calls, “intolerant realism,” and pleads for a kind of tolerant non-exclusivist pluralism in scientific methodology. According to Cartwright, both members of an incompatible pair [of theories] may be right, when each is viewed from its proper perspective. [Cartwright, N, (1994), The Metaphysics of the Disunified World, also, PAS, pp-357364.] It is also instructive to note in this connection that according to some competent physicists [e.g., D.S. Kothari (1985) “The Complementarity Principle and Eastern Philosophy,” (in Niels Bohr Centenary Volume. Edited by A.P. French and P.J. Kenney); P. Ghose (1991) “Relativity and Complementarity,” Occasional Paper No. 15, in PHISPC, New Delhi] tried to relate Syādvāda and Anekāntavāda [the so-called pillars on which the Jaina arguments for contextualizing LNC are based] to some interesting aspects of Quantum Physics. Even some of the brightest minds [e.g., von Neumann [See: Goswami (1995), for related discussions] proposed an epistemology-oriented approach to handling the so-called “measurement problem” in Quantum Physics. Wittgenstein also changed his earlier position on logical inconsistency (held, in TLP: 3.02, 3.03, 3.032, etc.) and switched over to his later position [LFM: Wittgenstein (1939)]. After this switch-over, he even insisted that his goal was “to change our entire attitude to contradiction and proof of consistency.” [LFM: Wittgenstein (1939), edited by C. Diamond. Also, in his RFM Wittgenstein (1956), [edited by G.E.M. Anscombe, Rush Rhees, G.H. von Wright; Oxford.] The foregoing discussion highlights three important points: (i) How crucial the epistemic moorings of Indian logic in general, and of Jaina logic in particular, actually are, (ii) How the very notion of inconsistency/contradiction can be contextualized by defining it and duly embedding it in an epistemic context, and (iii) How the Jaina view of contextualization of LNC fares with the views of some contemporary philosophers of science.

§11. Jaina Logic Viewed in a Wider Perspective In the hay-days of logical positivism, an academically fashionable slogan was, “logic without metaphysics.” It took people [including myself] some time to wake out of this dream. Until then, the idea of “types of logic” seemed no more sensible than talking about “types of English alphabet” – one British and one American. Today, the issue is looked at differently. Scholars like John Corcoran (1994) [“The Founding of Logic: Modern Interpretations of Aristotle’s Logic,” Ancient Philosophy, 14, 1994 pp. 924] do claim that there are at least two different conceptions of the nature of logic – an ontic conception and an epistemic conception.

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On the ontic conception, logic investigates certain general aspects of “reality,” of “being as such, in itself,” without regard to how (or even whether) it stands related to the knowing minds. In this sense, logic needs no epistemic moorings and, as such, it may be called, “formal ontology.” In contrast, on the epistemic conception, logic amounts to an investigation of various ways of reasoning without trying to deny or cut-off its umbilical relationship to epistemology. I consider this to be the most plausible way of looking at the issue, because, I believe that: (i) Philosophy is as deeply rooted in a culture, as literature or music is. Naturally, the task of getting philosophical ideas across, accurately from one culture to another is a daunting one. It is not simply a matter of “good translation” [Übersetzen] alone, but ideally, that of crossing over to the other side [Setzen ūber] of a cultural divide. (ii) It also holds when it comes to accurately getting the core concepts of “Indian logic” to readers steeped in a “different culture of doing logic.” By “different culture,” I mean the “Aristotelian-formalist” culture of doing logic in the West vis-à-vis “epistemology-steeped Indian culture of doing logic.” (iii) Russell is absolutely correct when he claims that, “logic is concerned with the real world just as truly as zoology, though with its more abstract and general features.” [Introduction to mathematical philosophy, 1919, 169]. Russell does not mean to deny thereby that there is always an epistemic dimension to logic as an ontic science. Every science, he says, in so far as it is science, has to have an epistemic dimension. (iv) The relation between the “ontic” and the “epistemic” dimensions of logic is of that kind which holds between the “shape” and the “size” of a solid object – distinguishable but not separable. (v) The underlying significance of the “epistemology-steeped Indian culture of doing logic” [especially, of Indian theories of anumāna] can be properly appreciated only when it is not forgotten that a theory of anumāna is not a theory of pure deduction, but a blend of “methodology of science,” [in the sense of being concerned with laying down the legitimacy-requirements for “law-like generalization,” i.e., LLG’s], and also of a theory of anumāna, i.e., sound inference. Obviously, formulation of proper LLG’s is the very crux of theory-construction in any science. Thus, an ability to draw proper conclusions [on the basis of the information currently available] is an indispensable part (but only a part) of the whole strategy. I should frankly admit here that I consider Indian logic to be primarily “epistemic” in orientation, while Aristotelian logic/Western logic is primarily “ontic” in its orientation. [In the very limited and specific context here, “Aristotelian logic” and “Western logic” will be used interchangeably, without any harm or confusion.] In this connection, I may also point out that besides Corcoran, there are other well-reputed interpreters of Aristotelian logic who would agree with my idea of “Two conceptions view” [e.g., Russell (1919, p-169), Boger, Gorge (2001), Leszl,

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Walter (2004), etc. Perreiah, Alan R. (1993) discusses some possible approaches to resolving the “dilemma between the axiomatic theory (i.e., an ontic orientation) of demonstrative science” and the “non-axiomatic practice (i.e., an epistemic orientation) of demonstrative science” (i.e., an epistemic orientation) in the physical treatises.] Similarly, on the other hand, there are some equally good scholars of Aristotelian/ Western logic [e.g., Łukasiewicz, J. (1957), Striker, Gisela. (1998)] who do not seem to agree with my understanding of Aristotelian logic in this regard. Keeping these points in mind, I now need to elaborate on my earlier “garbage-in, garbage-out” characterization of Aristotelian logic by pointing out some of the basic differences in the respective outlooks of Indian and Western logics. This is done with the aim of (A) Presenting a unifying approach to the diverse and apparently un-connected distinguishing features of Indian theories of anumāna, vis-à-vis the Western. (B) Of viewing the entire issue, especially the Jaina theory of anumāna, from a wider integrative perspective. (A) We may start by discussing the unifying approach above. Some features that really make “anumāna” stand apart from a syllogistic inference are (i) the thoroughly epistemology-centric orientation of the former, and (ii) the associated claim that anumāna is an accredited means of knowledge-acquisition and as such, (iii) must ensure that an anumiti must be capable of providing some hitherto unknown new information [a-jñāta-jñāpakatva/anadhigatatva, i.e., novelty requirement]. This view, though often formulated in different ways by different Indian logicians, is unanimously held by every school of Indian logic. Let us consider feature (i): “Anumāna” is defined as that logical process which definitively yields [is a karaṇa of] an anumiti, i.e., a state of cognition that results from an anumāna. This amounts to a reversal of the “reference point” for evaluation. In a deductive inference, the “truth of a conclusion” is “inherited” from the “truth of its premises,” whereas in an anumāna, the truth of an anumiti being given, the aim is to pick up its most plausible cause/hetu, i.e., some “justificatory reasons” and/or “explanatory grounds” (in the sense of being “ananyathāsiddha”) for the anumiti. Similarly, the definitions of “pakṣa” of an anumāna as “sandigdha sādhyavān pakṣaḥ” and of “sādhya” as that feature [dharma] “which is yet to be ascertained,” all these go to show the in-eliminable epistemic moorings of theories of anumāna. This also clearly shows that uncritically translating “pakṣa” and “sādhya” as “minor” and “major” terms and then to proceed to compare anumāna with a syllogistic inference [as is done by many writers, e.g., B. N. Singh (1986), Dikshit (1975), especially, in Appendix-A, etc.] would not only be highly inappropriate, it would be quite misleading as well. Frits Staal (1973) very clearly recommends ample caution to guard against possible confusions engendered by indiscrete translation of logical terminology of Western logic and its glib use in the context of discussing Indian logic. He draws attention to the fact that the customary assumption that the Indian concepts of “hetu,” “sādhya,” and “pakṣa” correspond to the

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Aristotelian middle, major, and minor terms, respectively, is incorrect. Moreover, the concept of “pakṣa” is used ambiguously in Indian logic, where it denotes either the term whose property is the “sādhya” or the relation between that term and the “sādhya.” Another ambiguity of the Sanskrit originals is due to fudging the line between “pakṣa as used” and “pakṣa as mentioned.” It is resolved in a “Chinese translation.” Hence, instead of trying to identify the purely abstract forms of deductive validity (e.g., the valid “Figures” and “Moods” of a syllogism), we need to view/locate the goal of anumāna in its proper epistemic context as a process of reasoning that is supposed to lead to an anumiti [i.e., an inferential knowledge]. Several interesting corollaries of far-reaching consequence follow from (i)–(iii) above. For example, the “novelty requirement” of anumāna immediately disqualifies: (α) any “entirely tautology-centric” criterion of validity. An argument like, “p/therefore, p” [although it satisfies the “relevance-requirement” in one sense] not only fails to be a valid one, it does not even count as an anumāna by the Indian logicians, and this, in its turn, suggests (by default) that (β) the tautology-centric notion of validity be rejected in favor of an information-theoretic notion of “logically infer-able” [“inferable” for short]. Again, (γ) the fact that in Indian logic “being inferable” is an information-theoretic relation, ensures that the requirements of context-sensitivity, relevance, and probability, are already built-into it. (δ) A strict adherence to context-sensitivity and relevance-requirements would block accepting the “principle of explosion” [(p& ~ p) ➔ q], “addition,” viz., [p➔(p v q)], etc., as valid theorems/axioms in any system of Indian logic. Interestingly, ability to block the “principle of explosion” is also a step towards an inconsistency-tolerant logic. Furthermore, (ε) I have shown earlier [in §3] that by using “ananyathāsiddhatva” or “anyathānupapannatva” and “avinābhāva” in an indiscrete and interchangeable way, the Jaina thinkers betrayed their own confusion about the “logical” and the “methodological” aspects of vyāpti-relation. Incidentally, it also shows (ζ) that the “confusion” is a very natural one because, the so-called “boundary-line” between “hetu-as-a-logical-ground” and “hetu-as-a-causal-ground” is itself an artificial construct for convenience. Granted this, any proposal for a sharp demarcation between “inductive logic” and “deductive logic” has to be somewhat artificial as well, and thus, would be liable to frequent inadvertent transgressions. The examples cited as cases of “fallaciously valid” argument-patterns, the hybrid character even of an allegedly “pure deductive fallacy” (e.g., “fallacy of ambiguous middle”) etc., [discussed in §6] are some clear instances of such transgressions. Moreover, (ή) the Jainas’ claim, viz., “No legitimate vyāpti-relation, no anumāna,” would need falling back upon some kind of default logic [DL] or non-monotonic reasoning [NMR] because, the number of upādhis to be eliminated [in order to ensure that the vyāpti-relation concerned is really a legitimate one] is endless. Similarly, when [in order to justify their claim about the highly context-relative nature of the number avayavas required in an anumāna] the Jainas invoke the notion of an anumātā’s potentially unending store of background knowledge [BGK] that shows the relevance of default logic [DL] and non-monotonic reasoning [NMR] in the theorization of Jaina logic.

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Finally, (θ) a look at the details of the Jaina scheme of classification of “propositions” also reveals that the Jaina logicians are not averse to incorporating “fuzzy” and/or “quasi-truth functional” propositions in their system of logic. The above discussion [(α)–(θ)] clearly suggests that “ideally speaking,” an adequate systematization of Jaina logic (theory of anumāna) would require (i) softening and suitably adjusting the currently dominant exclusively formalist-deductivist tautology-centric notion of “validity,” in favor of a more “intuitively natural” notion of “soundness” or “logical inferability.” The features that an ideal system of “logical inferability,” say S, needs to be able to incorporate, are mainly of three types, viz., (i) incorporating context-cum-relevance sensitivity, (ii) incorporating the machinery for handling “fuzziness” into the system S. [These two requirements constitute the so-called, “epistemic moorings” of S. [Clearly, “fuzziness,” when it is taken seriously, would be antagonistic to the spirit of “absolutizing” such dichotomies as, “deductiveinductive,” “valid-invalid,” “consistent-inconsistent,” etc.] finally, (iii) S would also need to be flexible enough to accommodate a way of de-linking the ideas of “logical rigor” and “deductive validity,” so that some conclusions (e.g., “All men are mortal,” “The Sun will rise in the East tomorrow,” etc.) may be inferred [on the basis of the data currently available] with “logical rigor,” without the supporting argument for it needing to be “deductively valid.” If such a logical system S were ever fully realizable, that would naturally amount to being flexible enough to accommodate elements of “fuzzy logic” and of “default-cum-non-monotonic modes of reasoning” as parts of its inferential machinery. However, such flexibility of an S would come only at a cost. At the “metalogical level,” the resulting system can be only “non-semi-decidable” [Sarkar (1992), 226 m]. A few points need to be noted regarding my observations above. First, it is important to recognize that all the far-reaching consequences (α)–(θ) can be viewed as logical spin-offs from only two basic underlying commitments listed above, viz., A (i) epistemology-centric orientation of Jaina theory of anumāna, and A (ii) indispensability of anadhigatatva [i.e., novelty-requirement] for all genuine knowledge-claims. Secondly, the consequences (α)–(θ) need not be taken to mean that the Jaina logicians were consciously pursuing those as a program. It is only my way of looking at what the possible logical features a fully worked out system of Jaina logic would have to be compliant with, were it possible to develop such a logical system incorporating all the tall claims listed above. (B) A part of what we mean by situating the Jaina theory of anumāna within a “wider integrative perspective” has been suggested in our discussion of the consequences (α)–(θ), under the unifying approach above. There the discussion always gyrates around the pivot called the “epistemic moorings” of Jaina logic. The idea of “epistemic moorings as the pivot” might wrongly suggest that the Jaina logicians were bent on indulging in a sort of “psychologism” in logic. [“Psychologism” in a pejorative sense – a sense in which, Frege accused, Husserl succumbed to doing “Psychologism” in Mathematics.] In the remaining part of our discussion, I plan to show how some other in-built features of Jaina logic [when viewed from a suitable “wider integrative perspective”]

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can be reasonably construed as working together to open up the possibility of developing it [i.e., Jaina logic] in the direction of an abstract, semi-formal but epistemology-rooted model of inferential reasoning. It has been pointed out earlier that the Jainas rejected the instance-based model of universal generalization; rejected bahirvyāpti and opted for antarvyāpti; replaced the actual co-locationality [sāmānādhikaraṇya] criterion of vyāpti-relation by a semantic-conceptual notion of inseparability [avinābhāva]. Incidentally, they also prepared the case for introducing HD method in scientific theory construction, by rejecting (i) the crude inductivist model for generating law-like generalizations [LLG’s] on the one hand, and by giving (ii) top priority to hypothetically entertained counterfactual reasoning as the only means of confirming of LLG’s [tarkāt tanniścaya]. On top of all these moves, the Jaina logicians also maintain that (a) pakṣadharmatā is not at all an indispensable requirement for an anumāna, (b) that, citing actual examples [udāharaṇa] as an avayava of an anumāna is not indispensable either, and finally, (c) that, linga parāmarśa’ has no logical role to play in an anumāna. All these, especially, (a)–(c) above, unmistakably indicate that the Jaina [and to some extent, also the Bauddha] logicians were keenly interested in freeing Indian logic of its tradition-imposed “factuality bias,” in preparation for some possible model of abstract/formal reasoning. When looked at in this way, we may claim that Jaina logic, as a theory of anumāna, has always contained two different (but non-divergent, intertwined, and mutually complementary) strands of thought, viz., “staying epistemology-centric,” while trying at the same time to “free logic of its factuality bias.” We may call it, “bi-potential root of Jaina logic.” This is quite in keeping with the all-pervasive “exclude none” [Sibjiban (1984)] attitude of Jaina anekāntavāda.

§12. Concluding Remarks During the last 3–4 decades, a number of excellent papers, written by competent scholars, on different aspects/schools of Indian logic, have been published. Many of them are “analytically penetrating,” but most of them are extremely narrowly focused. The respective authors make painstaking efforts to select a particular tree out of the forest [e.g., “anumāna,” to the exclusion of other associated logical topics], carefully count the number of leaves on each branch [e.g., focussing on the number of “avayavas” in an anumāna, while ignoring the significance of the controversy in the light of its epistemological moorings, Jaina scheme of classification of linguistic expressions, etc.]. All these overspecialized approaches to Jaina logic look so uncannily similar to the proverbial way of “failing to see the forest, for the trees.” Sometimes, ignoring the climatic region where the forest is located may result in complete wash-out of some crucial information. The sociocultural environs of the Indian systems of logic constitute such “climatic region.” The case of focussing solely on the “logic” of a school, without looking at the “metaphysics” that ensconces it, should not be considered any less futile. In this chapter, I have tried to show (i) how the Jaina logicians try to address both the “ontic” and the

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“epistemic” aspects of logic and (ii) how the complementarity between these two aspects can be linked to the overall framework of Jaina metaphysics. If we take a careful look at the growth pattern of recent thought about both Western and Indian logic, an interesting pattern begins to emerge. On the one hand, in the post-PM [“Pricipia Mathematica”] period, the growth of Western logic has been away from “strictly rigid formalism” to more “flexibly inclusive” diversified systems of logic which include, “Fuzzy Logic” [FL], “Relevance Logic” [RL], “Default Logic” [DL], “Para-consistent Logic” [PCL], “Epistemic Logic” [EL], etc. In short, Western logic is moving away from its initial, rigid, predominantly “ontic” view of logic to a more flexibly accommodative “epistemic” view of logic. On the other hand, during the last 50 years or so, approach of reputed scholars of Indian logic is moving away from the original nebulously formulated, non-deductivist, information-theoretic and predominantly “epistemic” view of logic, to a more wellregimented but semi-formal analog of “ontic” view of logic. [See: Bina Gupta (1980): Are “Hetvābhāsas” Formal Fallacies? – The first part of Bina Gupta’s paper examines the “Hetvābhāsas” of the Nyāya school. The second part analyzes the differences between Indian and Western conceptions of fallacy and deals with the question whether the Indian account of the “Hetvābhāsas” is totally devoid of the notion of formal fallacy as it is understood in the West. I have suggested that though the “completed” Nyāya inference includes the properties of formal validity, the notion of “Hetvābhāsa” presents only the necessary conditions for satisfactorily completing such an inferential process. Thus, while the Nyāya inference adequately accounts for the validity of the final “product” of inference, the Nyāya “Hetvābhāsas” account for the inferential process leading up to a sound product of inference. [Also see: Gokhale, P. (1992), Oetke, Claus (1996): “Ancient Indian Logic as a Theory of Non-Monotonic Reasoning.” Journal of Indian Philosophy no. 24:447539.] In his paper, Oetke presents a sweeping new interpretation of the early history of Indian logic. His main proposal is that Indian logic up until Dharmakirti was nonmonotonic in character – similar to some of the newer logics that have been explored in the field of Artificial Intelligence, such as default logic, which abandon deductive validity as a requirement for formally acceptable arguments. Dharmakirti, he suggests, was the first to consider that a good argument should be one for which it is not possible for the property identified as the “reason” (hetu) to occur without the property to be proved (sādhya) – a requirement akin to deductive validity. Oetke’s approach is challenged by John Taber (2004): Is Indian Logic Non-monotonic? He maintains that from the very beginning logic in India has been something like monotonic. In other words, deductively valid reasoning was the ideal or norm, although the conception of that ideal was continually refined and the criteria for determining when it [viz., the “ideal” or “norm”] is realized were progressively sharpened. [Bhattacharya, Kamaleswar (2001): “A note on Formalism in Indian Logic.” JIP no. 29:1723, etc., discusses certain other relevant points on “Hetvābhāsa.”] I strongly believe that a proper and balanced blending of “epistemic” and “ontic” views of logic is needed for balancing out their respective one-sidedness. As I see it, taking the first step in the direction of tackling this formidable task requires working

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out a plausible account of an information theoretic [not a tautology-centric] notion of implication. If successful, this itself would take care of both “context-sensitivity” and “relevance.” A number of excellent publications by others on Jaina logic has taken place in the past few decades [e.g., by Goekoop (1967), Schang, F. (2010), Priest (2008, 2012), Sibajiban (1984, 2009), T. K. Sarkar (2009, 928941), B.K. Matilal (1998), p. 4453; R.C. Pandey (1984), etc.]. However, hardly any of these papers did take a holistic-integrative approach to Jaina logic. Anyway, the enormity of the task is so overwhelming that we may have to wait until an appropriate paradigm-shift occurs. To be honest, such “paradigm-shifts” may ultimately turn out to be made of “dream-stuffs,” but still it is imperative that we do not give up dreaming. The bottomline is this: If we are not daring enough to dream, we forfeit our right to complain about our dreams having been shattered. Anyway, I think logicians can learn a lot from the more recent examples of paradigm-shifts in physics, some of which were real “dream-stuffs” until actualized.

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Pratyabhijñā Inference as a Transcendental Argument About a Nondual, Plenary God

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Philosophical Reenactment of Nondual Śaiva Myth and Ritual . . . . . . . . . . . . . . . . . . . . . . . . . . Philological Objections to Dialogical Engagements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Anticolonial Resistance to All Western Theorization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Applicability of Kantian Understandings of the Transcendental, Charles Hartshorne’s Logical Corollary of Divine Contingency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meanings of the Category Transcendental Beyond Kant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dialectics Between Nondual and Theistic Givenness (A Priori) and Śakti Interpreted Philosophically as Epistemological and Onto-Grammatical Dependence (A Posteriori) . . . . . . Translation of IPV 2.3.17 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Śakti Overcoded with the Pratyabhijñā Vocabulary of Recognition . . . . . . . . . . . . . . . . . . . . . . . . . . . Additional Subsumptions of Inference: Pleromatic Fragmentation and Inductive Noncommitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Niyati Śakti as Generating Substantive Inferential Concomitance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scriptural Traditions as Grounds of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identity of Cosmogony and Teleology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The two most well-known and important nondual Śaiva and tantric philosophers are Utpaladeva (c. 900–950 CE) and Abhinavagupta (c. 950–1020 CE). Utpaladeva furthered the initiatives of his teacher Somānanda (c. 900–950 CE) in creating the originary verses and commentaries of the Pratyabhijñā system of philosophical theology. Abhinavagupta is famous for brilliant and extensive D. P. Lawrence (*) Department of Philosophy and Religious Studies, University of North Dakota, Grand Forks, ND, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_29

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commentaries on Utpaladeva’s writings as well as for the creation of a vast corpus of philosophical hermeneutics, itself comprising both a poetics and an exegetics of tantric symbolism and ritual. This chapter clarifies and build upon the interpretations of the overarching Pratyabhijñā inference as a transcendental argument in the article, “Tantric Argument: The Transfiguration of Philosophical Discourse in The Pratyabhijñā System of Utpaladeva and Abhinavagupta” (1996) and the book Rediscovering God with Transcendental Argument (1999). It considers how the Pratyabhijñā inference may be characterized as transcendental, answering various objections to that characterization, provides translations of key passages for understanding the Pratyabhijñā method (especially, IPV 2.3.17 and a paragraph from IPV 1.17.1 on the identification of cosmogony and teleology), and summarizes some of the author’s later research that pertains to this subject. Abbreviations

BIPV IPK IPKV IPV IPVV VAP VAPV

Bhāskarakaṇṭha’s commentary Bhāskarī on the IPV I¯s´varapratyabhijn˜ ākārikā by Utpaladeva, for convenience cited in edition with BIPV and IPV, rather than better edition of Torella I¯s´varapratyabhijn˜ ākārikāvṛtti by Utpaladeva, commentary on IPK, cited in edition of Torella I¯s´varapratyabhijn˜ āvimars´inī by Abhinavagupta, commentary on IPK, cited in edition with BIPV I¯s´varapratyabhijn˜ āvivṛtivimars´inī by Abhinavagupta, commentary on Utpaladeva’s I¯s´varapratyabhijn˜ āvivṛti Virūpākṣapan˜ cās´ikā by Virūpākṣa, cited in edition of Kaviraj Virūpākṣapan˜ cās´ikāvṛtti by Vidyācakravartin, commentary on VAP, cited in edition of Kaviraj

In fact, one may – this simple proposition, which is often forgotten, should be placed at the beginning of every study which essays to deal with rationalism – rationalize life from fundamentally different basic points of view and in very different directions. Rationalism is an historical concept which covers a whole world of different things. Max Weber The Protestant Ethic and the Spirit of Capitalism

Introduction The two most well-known and important nondual Śaiva and tantric philosophers are Utpaladeva (c. 900–950 CE) and Abhinavagupta (c. 950–1020 CE). Utpaladeva furthered the initiatives of his teacher Somānanda (c. 900–950 CE) in creating the originary verses and commentaries of the Pratyabhijñā system of philosophical theology. Abhinavagupta is famous for brilliant and extensive commentaries on Utpaladeva’s writings (I will not endeavor to distinguish their views in this area)

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as well as for the creation of a vast corpus of philosophical hermeneutics, itself comprising both a poetics and an exegetics of tantric symbolism and ritual. The author has learned that it is challenging to write on the topic of inference in the Pratyabhijñā, about which one thought to have already said all that one could. After much thought, he has decided here to clarify and build upon the interpretations of the overarching Pratyabhijñā inference as a transcendental argument in his first article, “Tantric Argument: The Transfiguration of Philosophical Discourse in The Pratyabhijñā System of Utpaladeva and Abhinavagupta,” and his first book Rediscovering God with Transcendental Argument. The author will take the opportunity to consider how the Pratyabhijñā inference may be characterized as transcendental, answering various objections to that characterization, provide translations of two of the key passages for understanding the Pratyabhijñā method (i.e., IPV 2.3.17 and a paragraph from IPV 1.17.1 on the identification of cosmogony and teleology), and summarize some of his later research that pertains to this subject.

The Philosophical Reenactment of Nondual Śaiva Myth and Ritual First, a brief overview of the Pratyabhijñā philosophy. In current scholarship, one of the most definitive characteristics of what is called “tantra” is the pursuit of power, the theological essence of which is the Goddess, Śakti, and the manifestations of which vary from limited siddhis, through royal power, to the saint’s omnipotent agency in assuming the divine cosmic acts. (The five Acts of Śiva, along with Viṣṇu, Śakti, and most other deities conceived by Hindus to be Ultimate, are creation, preservation, destruction, delusion, and graceful liberation.) In “nondual Śaivism” Śakti is, as Alexis Sanderson would say, “overcoded” within the metaphysical essence of the God Śiva. Śiva is the s´aktimān, “possessor of Śakti,” encompassing her within his androgynous nature as his integral power and consort. According to the predominant myth, Śiva out of a kind of play divides himself from Śakti and then in sexual union emanates, embodies himself within, and controls the universe through her. The basic pattern of practice, which reflects the mythic-cum-historical appropriation of Śāktism by Śaivism, is the approach to Śiva through Śakti. One pursues identification with Śiva as the s´aktimān by assuming his narrative agency in emanating and controlling the universe through Śakti. In the manner described by Mircea Eliade, and Christian and non-Christian Platonism and Aristotelianism, the ritual of return recapitulates the cosmogony. (See the discussion of this subject below.) The same mythico-ritual process is articulated in a great number of reciprocally encompassing codes, and codes of codes (if A=B and B=C, then A=C, and so on) – in terms of the overemphasized sexual ritual and other ritually transgressive practices, theosophical and philosophical contemplations, mantras, maṇḍalas, and so on (see the summary in Lawrence 2008a, 5–18). The Pratyabhijñā system replicates this modus operandi in s´āstraic philosophical discourse as the programmatic disclosure that one has Śiva’s Śakti (s´aktyāviṣkaraṇa) (see IPK and IPV 1.1.2 in IPV 1:56–59; and IPK and IPV 2.3.17, ibid., 2:139–149).

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Śakti is conceived as the reason in an inference-for-the-sake-of-others. I am Śiva because I have Śakti (Lawrence 1999, 49–57). The disclosure of emanatory and cosmocratic Śakti is also described as the awakening of pure wisdom (s´uddhavidyā), in which all objective things or “thisness” (idam) are absorbed into the omnipotent I (aham) as “I am this” (aham idam) (IPK and IPV 3.1.3–7 in IPV 2:221–32; see Lawrence 1999, 61–65). To address a variety of problematics, the Pratyabhijñā thinkers interpret Śakti epistemologically as an idealistically constitutive self-recognition or recognitive synthesis (ahampratyavamars´a, pratyabhijn˜ ā, anusaṃdhāna), also identified with a principle of supreme speech (parāvāk) derived from the linguistic philosopher Bhartṛhari. The philosophers interpret Śakti ontologically as universal first-person agency (uttamapuruṣa, kartṛtva, svātantrya), of the action (kriyā) that constitutes existence (sattā, see Lawrence 1999, 2008b, 2014; Dupuche 2001; Baumer 2011). (Elaborating on similar assertions by Utpaladeva, Abhinavagupta explains, “Being [sattā] is the agency of the act of becoming [bhavanakartṛtā], that is, agential autonomy [svātantrya] regarding all actions.” sattā ca bhavanakartṛtā sarvakriyāsu svātantryam. IPV 1.5.14, [Abhinavagupta 1986, 1: 258–259].) Abhinava features the first-person index of agency especially in his tantric exegetics and aesthetics, although he mentions it in his Pratyabhijñā commentaries. The student learns to participate in Śiva’s enjoyment of Śakti as self-recognition/speech/agency/discursive agency, by contemplating her as the reality underlying all immanent experiences, interlocutors and objects of experience and discourse.

Philological Objections to Dialogical Engagements There have been mostly favorable responses to the interpretation of the Pratyabhijñā argumentation as “transcendental.” Some have also objected to this description. In this and the following three sections, I will address these objections in various ways to better clarify the idea. Firstly, some operating entirely within the philological and historical style of scholarship predominant in Indology and Sanskrit studies, particularly in Europe but also some in the USA, indicate opposition to more constructive, intercultural philosophical interpretations for the very agenda to interpret texts beyond the strictly empirical data. It seems undeniable that an understanding of the empirical contents of texts, the immediate circumstances of their production, and the relationships between them is essential to any more theoretical – philosophical or nonphilosophical – interpretation of them. John Nemec has observed that many early Indological translations of Sanskrit texts retain their value, even though the scholars who produced them had pejorative views about Indian culture. Nevertheless, Nemec explains, a great deal more translation work is needed to help emancipate scholarship from those scholars’ narrow understanding of the South Asian canons (Nemec 2009, 757–780). Of course, it is legitimate that scholars have their own individual interests and priorities, and it is admirable that anyone devotes himself or herself to serious

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philological work. Nonetheless, it is highly prejudicial to conceive South Asian philosophy reductionistically as nothing but the empirical stuff that comprises texts and their contexts and relationships. The most well-established leader of nondual Śaiva and tantra studies, Alexis Sanderson, in chiefly following this method, evidently does not assume it is the only proper one. While philology and descriptive history are invaluable, they should not be viewed as the final goal of scholarship on Hindu philosophies. Especially young careeroriented scholars following only this methodology use rhetorical strategies of ignoring more dialogical and constructive interpretations while reiterating some of the same points, dismissing them by citing the disparateness of the comparisons, making unsubstantiated criticisms of them, or refuting incidental comments or conjectures. At the same time, the authors claim to keep abreast of every work in the field. The author must certainly be included among those who have made mistakes in interpretation and translation. In the exclusivistic kind of philological research, as with more theoretically reductionistic studies, there is no consideration of the truth claims of South Asian intellectual traditions and the challenges they pose to the scholars’ own beliefs and values. Comparisons are at most an accessory to empirical description or causalinstrumental reduction in more theoretical studies. [One is reminded of the largely historicist program for comparative philosophy advocated in the early twentieth century by Paul Masson-Oursel (1926). On the overall historicist and reductive causal-functional explanations also see Heidegger on the scientific-technological understanding of the world as intellectually controllable and practically exploitable “standing reserve” (1977). Max Weber and subsequent modernization theory focus on the instrumental rationality of capitalist bureaucracies. However, Max Weber, as indicated in the epigraph also used before (Lawrence 1999, 1), was actually aware of the plurality of concepts of rationality. See Levine 1981 for a classification of types of rationality discussed by Weber. The Pratyabhijñā, like many philosophies, would be classified as a type of substantive rationality, although Weber does not specify or relate inductive and deductive logic.] Peter J. Park has analyzed anthropologists and related scholars, such as Christoph Meiners in the eighteenth century, excluding Asia and Africa from the history of philosophy on the basis of racial considerations, and their influence on Hume, Kant, Hegel, and other historians of philosophy (Park 2014). An often implicit rule of what Nigerian musician Fela Kuti calls the “colonial mentality,” followed by Colonialism, Neocolonialism, and their accepting subjects, is that the “subaltern” cannot speak (Spivak 1988). (The scholars in question may be experts at not listening, but they will analyze the subaltern’s sentences and trace their genealogies.) Amartya Sen characterizes such scholarship as a variety of colonialist Orientalism which is both “exoticist” and “curatorial” (Sen 2005). The assumption is that the scientific-instrumental rationality of the post-Enlightenment West provides a way of understanding other cultures better than they understand themselves. One thinks of Sanskrit traditions as skeletons or mummies in the museum. Robert Yelle has discussed how a Protestant instrumentalizing and disenchantment about language undergirds this kind of scholarship (Yelle 2012). Vishwa Adluri and Joydeep Bagchee similarly examine the Protestant and historical roots

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of deconstructive philological scholarship (Adluri and Bagchee 2014). Contemporary Western and South Asian scholars need to use imagination and critical thought to consider the challenges of alternative ways of thinking rather than dismissing all normative and dialogical interpretations. [Frazier (2009, 14) describes the historicist bias against comparative engagement as “a postcolonial xenophobia endemic to the idea of academic research as an accumulation of knowledge within set boundaries, like capital stored safely in a vault.” Cf. the revision of Quentin Skinner’s historicism in Ganeri (2008). The historical context of Indian philosophies may be identified in ongoing discussions of intellectual issues rather than only immediate social circumstances. See the discussion of philology in Lawrence 2011.] Listening to the views of the colonized cannot be postponed indefinitely, like delayed justice in allowing human rights for oppressed groups.

Anticolonial Resistance to All Western Theorization Some indigenous South Asian scholars and their followers have reacted negatively to comparisons with Western and especially Christian philosophy as a disguised form of imperialism. The author wishes sincerely to emphasize that that is not his intention and he strongly believes, like Jeffrey Kripal has represented, that what is often most distinctive about Abrahamic exclusivism is intolerance and colonialism (2014). This is evinced in the exclusion which condemns the already more pluralistic people, including Hindus, as “polytheists” and “idolators.” Likewise, it is the purport of more hierarchical “inclusivism” which views them as at best imperfect efforts of “anonymous Christians” to find the truth which Christians know more fully. Speaking bluntly, the idea too frequently is: Our God is the only true one, and he said that we can have your land. The archetypical narrative Exodus is often a story of religious imperialism as much as liberation. Bibles and guns have respectively been among the most persuasive expressions of Western religion and “Enlightenment” (see the discussion of philology in Lawrence 2011). Of course, there is also much authentic and laudable concern with equality and justice in all three Abrahamic religions, and many anti-oppressive teachings, including contemporary versions of liberal inclusivism and pluralism, in all three, and the author does not wish to condemn them. (On formulations of varieties of Abrahamic pluralism, see Levin 2016; Hick 1985, 1995; Diana Eck 2003; Farid Esack 1996.) Gandhi himself was at first suspicious of Christianity as a justification of empire but came to appreciate and be inspired by the ethics of Jesus (Gandhi 2013; Jordans 1987). Jesus’ evident deep mysticism and his beautiful and profound pacifism and radical opposition to privilege and hypocrisy were a variation of Jewish mysticism and ethics, also elaborated in expressions of Islamic mysticism and ethics. Jesus, Leo Tolstoy, and John Ruskin were among the Western influences on Gandhi complimenting the Indian influences, as Martin Luther King, Jr., understood. I do contend that the ideal of equitable intercultural philosophy is more coherent (on the role of coherence, see Lawrence 2012) than nativist cultural isolationism.

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In spite of the tragic history of imperialism, the author still maintains that there are Western thinkers such as Plato, Aristotle, and their followers, Augustine Thomas Aquinas and his followers, Immanuel Kant, Charles S. Peirce, Ludwig Wittgenstein, and Martin Heidegger; and more recently, Jesper Hoffmeyer, Terrence Deacon, and so on, who are brilliant and exciting thinkers, eminently worthy of dialogical engagement.

Applicability of Kantian Understandings of the Transcendental, Charles Hartshorne’s Logical Corollary of Divine Contingency The most poignant matter of disagreement is the contention that Indian philosophy, including the Pratyabhijñā system, does not have transcendental argument as defined by Immanuel Kant. The positive and negative examples (respectively, anvaya and vyatireka) in the classic inference often rely on inductions of probability (for a sophisticated examination of Navya-Nyāya on the probity of induction in Sanskrit philosophy, see Chakrabarti 2010). The inference deductively applies the generalizations of the usual inductions. Therefore, it is said, there is only pragmatic generalization in Indian thought rather than logical necessity. Kant conceived as transcendental argument only a priori analytic or synthetic inference. Mohanty (2001) himself has concurred on the paucity of the Kantian sort of pure logic in the overarching practical orientation of Indian philosophy. This notion of India as only practical rather than theoretical ignores Weber’s point about multiple rationalities as well, indeed, as Aristotle’s own foundation of Western traditions of pragmatism, and has itself sometimes been used as a colonialist stereotype and justified allegations of the inferior rationality of Indian culture. Purely a priori logic is perhaps not usually the basis of classic conceptions of inference (anumāna) itself in South Asia. It must also be acknowledged, however, that relationships of logical necessity are sometimes established in Indian examples, for example, according to Dharmakīrti’s logical classifications of pervasion and exclusion based on essential nature (svabhāva) rather than causal origination (utpatti). However, the former is supposed by the Śaivas actually to be contingent on idealistic creation in the latter (yoginirmāṇatābhāve pramāṇāntaracis´cite/kāryaṃ hetuḥsvabhāvo vāta evotpattimūlajaḥ, IPK, 2.4.11, 2:175; cf. IPV 2.4.11, 2:173–181; IPVV 2.4.111, 2:197–214). So, this does not prove the point. Indian conceptions of supportive reasoning (tarka) and other accessories to inference also invoke necessity. Such reasoning underlies the reductions in Nāgārjuna’s Mādhyamika dialectic. Necessity is also present in Indian theories of meaning (see Ganeri 2011). Theories of self-luminosity (svaprakās´atva, svasaṃvedana) may be taken as referring to a priori knowledge or cognition (as discussed below) but are not inferential. This chapter will first consider the classification on the assumption that only what Kant described as pure a priori knowledge of identities and logical entailments

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counts as transcendental argument. Later it will argue that the Kantian understanding of the transcendental is overly restrictive in both historical and normative senses. Ancient Neoplatonic metaphysics as well as the so-called ontological argument of Anselm will be treated as conforming to Kant’s standard, whether the latter is a sound argument or not. The scholastic Neoplatonism of someone such as Proclus would conform at least, in ostensible method, to Kant’s definition. The method employed in his Elements of Theology is almost entirely deductive, or what came to be known as a priori analytic or synthetic (Proclus 2004). However, that is not how the inference works in the Pratyabhijñā system. The main argument for the existence of God in the narrow sense of the a priori analytic or synthetic has been the widely discussed formulations of the so-called ontological argument by Anselm of Canterbury (c. 1993–1109) and later rearticulated by Rene Descartes (1596–1650). There have been several conflicting interpretations of the significance of this argument and whether some or other versions of it are cogent. At the outset, mention may be made of an alternative approach to the interpretation of Anselm by philosopher of religion, Eric Voegelin. Although his interpretation is complicated, Voegelin basically has contended that Anselm’s “ontological argument” is actually not an argument at all but is articulating facts that are known only by a priori faith or intuition (see the discussion of Anselms’ “ontological argument” in Voegelin, “The Beginning and the Beyond,” in 1990, 191–209). Whether this truth is communicated by dogmatic assertion or some kind of inference is a secondary consideration on which he does not focus. Leaving aside that alternative, most scholars unquestioningly accept the ostensible refutation of the “ontological argument” made by Kant, which is that existence is not a predicate. We allegedly cannot argue from logical entailments of a concept to whether or not its referent exists. Among the most famous contemporary defenses of an ontological argument in Anselm is Charles Hartshorne’s advocacy of it as a species of “modal argument.” Somewhat similar is the defense of the argument by Norman Malcom (Malcolm 1960). Hartshorne endeavors to refute Kant’s very claim that existence cannot be a predicate (see Hartshorne 1991a, 57–58 and the longer discussion of Kant in Hartshorne 1991b, 208–234). Focusing on Anselm’s second formulation of the argument, he interprets the argument as proving that an ostensible God is necessarily a sufficient reason. One decides on other grounds whether such a sufficient reason should be affirmed. So, the idea is that God must be understood as somehow perfect, entailing that he is either logically necessary or logically impossible (Hartshorne 1991b). Hartshorne also refutes “classical theism” on the nature of the perfection of God that he contends to be logically necessary or impossible. He claims that perfection could not be a static, unchanging condition. Rather it must include the process polarity of some contingent element (Hartshorne 1991a). The broader position is that God is both transcendent and immanent, necessary and contingent. “Panentheism,” a term originally invented by Karl Christian Friedrich Krause, certainly characterizes the nondual Śaiva version of God and his five cosmic Acts (see summary of the Acts above).

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In the Śaiva system the provision for contingency is established by the very conception of Śiva, which has also often been described as a form of panentheim (see Hartshorne 1991a, b, 2000 for the classic classification of panentheism versus classic theism and other doctrines; Hartshorne and Reese 2000; for a recent survey of panentheism which includes nondual Śaivism, see Biernacki 2013; on the classification as panentheism, see Lawrence 1999, 167–167; on the emphasis on simultaneous transcendence and immanence and a “vaguely panentheistic ultimate” in the nonaffiliated spirituality movement, see Forman 2004.) As Heinrich Zimmer interpreted the Natarāja symbol, Śiva is both smiling peacefully as quiet inside, eternal or transcendent, and dancing actively in controlling the cosmos (Zimmer, “The Cosmic Delight of Shiva,” in 1974, 123–188). Śiva’s activity is ascribed in the narratives and theologies precisely to his Śakti. We need to keep the notion of God’s polarity of immanence and contingency in mind, because in the interpretations of other kinds of metaphysical, transcendental argument, it is the very contingency of God which requires that argument about him includes an a posteriori element.

Meanings of the Category Transcendental Beyond Kant Moving beyond the narrow Kantian restrictions, in Rediscovering God it was actually stated that: My understanding of transcendental/metaphysical argument is indebted to that of David Tracy, and, somewhat less directly, to two of his sources, Emerich Coreth and Bernard Lonergan. This understanding unites the Kantian conception of the transcendental as the necessary with older Greek and Christian understandings of metaphysics as inquiry into ultimate reality. (Lawrence 1999, 171n.)

This should have already obviated objections, but the issue reaches deep into history. Actually, with ancient roots in Plato and Aristotle, and as developed especially in the Middle ages in thinkers such as Philip the Chancellor, Dun Scotus, and Thomas Aquinas, transcendental referred to metaphysically necessary affirmations of facts or qualities reality or Being (see Doyle 2012; Gracia 1992; Honnefelder 2003). However brilliant and epochally influential Kant may have been, he did not invent but rather limited the meaning of transcendental argument to something based on a priori (perhaps analytic or synthetic) judgments, just as he limited epistemology and metaphysics to a timid or perhaps incoherent representationalism (for a Thomist critique of Kant on metaphysics, see Clarke 2001, 11–14). His analysis of pre-linguistic categories paved the way for many sociohistorical and cultural-linguistic conceptions of the mediation of reality with which we are quite familiar in religious studies (see Forman 1999, “Non-linguistic Mediation,” 55–80). In any event, the contemporary Thomist W. Norris Clarke conforms to the medieval understandings that were backgrounds to the philosophy of Thomas in defining transcendental inquiry as follows:

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As applied to the idea of being, this means that the latter concept must be all-inclusive, both in its comprehension (i.e. the content included in its meaning) and its extension (the range of subjects to which it can be applied). Thus being signifies all that is, in everything that is, i.e., everything that is real in any way. Outside of this lies only “nothing” or nothingness, non-being. For this reason, the concept of being is called “transcendental” (from Latin transcendere = to climb over), that is, transcending or leaping over all divisions, categories, and distinctions between and within beings, pervading them all. It excludes only non-being. This is its purpose as a concept, to be the ultimate all-inclusive term, to express the ultimate horizon of reality itself and everything within it. It is through such an idea that we are able to embrace intellectually and express to ourselves the whole of reality. (Clarke 2001, 43)

The most influential contemporary Thomist, Bernard Lonergan himself, though prescribing various research operations that are clearly a posteriori, actually conforms to Kant in asserting that the transcendentals are a priori (Lonergan 1971, 11, 14n). Still, Emerich Coreth, another prominent Thomist writing at the same time, unites Thomist notions of metaphysics as transcendental with Kantian notions: The philosophical method which investigates the conditions of the possibility of our knowledge has been known, ever since Kant’s pioneering efforts, as the transcendental method. That is why the Thomism which we advocate might be called transcendental Thomism. The name may sound strange, since it combines the very new with the very old. (Coreth 1968, 10)

In Coreth’s understanding, the full dialectic comprises both the a priori and the a posteriori, which he describes as “reduction”: The transcendental method uses a double movement, consisting of what we may call reduction and deduction. Transcendental reduction uncovers thematically in the immediate data of consciousness the conditions and presuppositions implied in them. It is a return from that which is thematically known to what which is unthematically co-known in the act of consciousness, to that which is pre-known as a condition of the act. Transcendental deduction, on the other hand, is the movement of the mind which, from this previous datum, uncovered reductively, deduces a priori the empirical act of consciousness, its nature, its possibility, and its necessity. Whereas reduction proceeds from a particular experience to the conditions of the possibility, deduction goes from these conditions to the essential structures of the same experience. The two movements are in constant interaction, they influence each other; yet it is possible to emphasize one over the other (Coreth 1968, 37; cf. 42; see Coreth 1968, 10 for a quotation, possibly from Lonergan, that compares transcendental inquiry with Socrates’ “intellectual midwifery” that reveals what it “‘always already’ possesses.”).

[Another follower of Lonergan, David Tracy again recovers the medieval in his careful formulations: [One clear way of articulating the nature of the reflective discipline capable of such inquiry is to describe it as “transcendental” in its modern formulation or “metaphysical” in its more traditional expression. As transcendental, such reflection attempts the explicit mediation of the basic presuppositions (or “beliefs”) that are the conditions of the possibility of our existing or understanding at all. Metaphysical reflection means essentially the same thing: the philosophical validation of the concepts “religion” and “God” as necessarily affirmed or denied by all our basic beliefs and understanding. We seem to be unavoidably led to the conclusion that the task of fundamental theology can only be successfully resolved when the theologian fully and

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frankly develops an explicitly metaphysical study of the cognitive claim of religion and theism as an integral moment in his larger task. (Tracy 1975, 56; cf. 67–68, 159)

[Tracy’s preferred term for philosophical theology is “fundamental theology” (Tracy 1998, 89n) [Tracy explains that what Thomists know as transcendental, metaphysical reflection is both a specialist form of “fundamental theology” and critically structures symbolic exegesis, which he labels “systematic” theology and ritually and ethically oriented “practical” theology: [The major argument of this book will be to show that the hermeneutical character of systematic theologies, although not obviously “public” in the first sense, is nevertheless public in a distinct but related sense. In more traditional Aristotelian language, fundamental theology deals principally with “dialectics” and “metaphysics,” systematic theology with “rhetoric” and “poetics,” and practical theology with “ethics” and “politics.” Each of the three enterprises can achieve a public status distinct from but related to (in a word, analogous to) the other two. Each is concerned with both meaning and truth. In the alternative language of transcendental reflection: fundamental theology is concerned principally with the “true” in the sense of metaphysics, systematic theology with the beautiful (and, as we shall see, the beautiful as true) in the sense of poetics and rhetorics, practical theology with the good (and the good as transformatively true) in the sense of ethics and politics. The major role of fundamental theology is to explicate the transcendental in relation to the religious, the holy or the sacred. Hence Part I of this book is, in fact, an exercise in fundamental theology designed to show the truth status of the claims of systematic theologies. The major focus, therefore, will be on the art-religion relationship in the analyses of the classic and distinctively religious classics (chapters 4, 5, 6). Every discipline in theology must be concerned with the truth of its claims on the inner-theological grounds outlined in the chapter. But that concern, I propose, will operate “obviously” in fundamental theology (i.e., as dialectics or argument) and less obviously but no less really in systematic theology (i.e., like ethics and politic). The obviously ontological (as metaphysical or transcendental) character of these claims distinguishes this enterprise from Anders Nygren’s claim that transcendental reflection [following Kant] provides only purely logical sets of operations (Meaning and Method 209–27). (Tracy 1998, 85n; cf. 183)

[If this statement were interpreted in his terms, it seems that Abhinavagupta would fully endorse Tracy’s formulation. [In his comments, Lonergan suggests but does not elaborate some inadequacy of Coreth’s synthesis of analytic-deductive and synthetic-inductive (a.k.a. a posteriori) procedures in his remarks on Coreth (1968, 200–201, 218–219).] Coreth explains the ontological counterpart to his methodological synthesis of the a priori and a posteriori: Every being is both necessary and not necessary. It is not necessary, insofar as it is a finite being, which is not by itself, through its own essence, determined to the necessity of being. But if it is, it is as necessary as being; then, insofar as it is, it can no longer not be. Insofar as it is, it necessarily is. This is possible only if every being, over and above its own contingent essence, possesses something which determines it to the necessity of being. Else it would by itself be both necessary and not necessary, which is contradictory. Therefore, every contingent being which is posited in the necessity of being require a positive element, by which it is determined to the necessity of being: it requires a ground, a sufficient reason of its being. (Coreth 1968, 96–97)

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With regard to God, Coreth also supports Hartshorne’ claim that the God explicated by the ontological argument must include a moment of contingency. This seems, perhaps inadvertently, to qualify his orthodox theological doctrine of the static perfection of God: The principle of causality cannot be established through analytic deduction but only through synthetic reduction. It is not formally contained in the principle of identity and it cannot be deduced from it. The principle is one more example of mediated immediacy: it is immediately evident, but it can be critically established only through the mediation of the dynamism of our intellect, by which we are irresistibly urged to look beyond every being to the ground or cause of its being. (Coreth 1968, 97)

He again clarifies the admission of broadly a posteriori criteria on the basis of religious symbolism and freedom (Coreth 1968, 147; cf. 189–196). [Coreth, though a follower of the supposed empiricists Aristotle and Thomas, acknowledges the idealistic trajectory pointed out in the Pratyabhijñā prakās´a arguments and analogous moments of the divine mind in both Platonic and Aristotelian logos theologies (Lawrence 1999, 162–163): All in all, we stand nearer to idealism than to rationalism, to Hegel than to Wolff. We agree with the idealistic contention that we must start with the self-positing and self-mediating spirit, and that the spirit, even the finite spirit, manifests a real infinity in its self-actualization. (Coreth 1968, 43)

Coreth also asserts that what we would describe as the self-recognition of God is the ground of our knowledge in content and act (Coreth 1968, 70). In regard to the dialogical setting of transcendental inquiry, we see a little bit of the Śaiva attitude to the doubt of the pūrvapakṣin in the classic Thomist method of questioning the act of questioning. We recall that Abhinavagupta said: The nature of Ultimate Reality here [in this system] is explained through the consideration of the views of opponents as doubts and the refutation of them; it is thus very clearly manifested. (IPV 1.2 introduction, 1:82. Cf. IPV 4.1.16, 2:309–310)

Likewise, Coreth explains: Questioning the question would not make sense if there were not in the question more than what there seems to be at first. A question is an action and has a content (Coreth 1968, 39). The ultimate aim of our groping is neither totally determined, nor totally undetermined. Where shall we discover this positive aspect? In the pure pre-knowledge of the question as such, of the act of questioning in general. (Coreth 1968, 60–61).]

We may finally briefly consider the classic discussion between Gārgī Vācaknavī and Yajñavalkya in the Bṛhardāraṇyaka Upaniṣad 3.6 as pertaining to transcendental argument in the larger sense, and an ancient background to that of the Pratyabhijñā (Upaniṣatsaṇgraha 3.6, 104). In the translation of Robert Hume, this discussion begins:

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Gārgī Vācaknavī questioned him. “Yājñavalkya, said she, ‘since all this world is woven, warp and woof, on water, on what, pray, is the water woven, warp and woof.’”

Yājñavalkya then suggests a series of hypothetical substrata, and each time Gārgī asks “On what, pray, [is that or are they] woven, warp and woof?” He finally comes to suggest the worlds of Brahma, a.ka. Hiraṇyagarbha. She asks: “On what, pray are the worlds of Brahma woven, warp and woof.” Yājñavalkya answers: “Gārgī, do not question too much lest your head fall off. In truth you are questioning too much about a divinity about which further questions cannot be asked. Gārgī, do not overquestion.” Thereupon Gārgī Vācaknavī held her peace. (3.6, 113–114)

(As was discovered later, this is also discussed by Voegelin, “The Beginning and the Beyond,” in 1990, 173–232.) There is a deeper meaning in Yājñavalkya’s final answer, beyond the humor in his attempt to stop someone who is asking an irritating number of questions. It pertains to the transition of the dialectic from the a posteriori to the a priori. The two ancient thinkers have discussed a number of increasingly grand contingent a posteriori hypotheses. Then they have arrived at the sufficient reason. By a priori reasoning that reason is something “about which further questions cannot be asked” in the Kantian sense, without what Karl Otto Apel calls “performative contradiction” or Bernard Lonergan describes as a self-contradictory “counter-position.” On the contrast, Buddhist analyses of infinite regress and vicious circularity in dependent origination are trying to make “your head fall off.” They view the quest for a sufficient reason (svabhāva) as a form of attachment. Śāṅkarācārya largely concurs with this interpretation. Advaita Vedāntins often object to contemporary nondual Śaiva “straw-man” representations of them as denying all immanence. They do accept immanent, mundane cognition, but say that its real object is brahman. It seems that the transcendence from the world is always combined with immanence in representations of an ultimate, because otherwise it would not be both the Ultimate and in some way accessible. What differs is both the qualitative understanding of transcendence and immanence and the relation of one to the other (Lawrence 2001). The theological phenomenology of John Mbiti in Concepts of God in Africa makes the same point. Mbiti’s comments that summarize the views of many African ethnic groups and would also apply to other religions: The transcendence of God is a difficult attribute to grasp, and one which must be balanced with God’s immanence. The two attributes are paradoxically complementary: God is “far” (transcendent), and men cannot reach him; but God is also “near” immanent, and he comes close to men. (Mbiti 1979, 12)

Śāṅkarācārya accepts the (a posteriori) inferences of successive causes as in effect pertaining respectively to the realm of provisional truth, as leading to the (a priori) self-luminosity of the brahman/ātman (Śaṅkarācārya 1983, 422–425). To quote the translation of Swami Madhavananda:

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These worlds, arranged in an ascending order of subtlety, are each composed of the same five elements transformed so as to fit abodes for the enjoyment of beings. By what is the world of Hiraṇyagarbha [Brahma] pervaded? Yājñavalkya said, ‘Do not O Gārgī , push your inquiry too far – disregarding the proper method of inquiry into the nature of the deity: that is, do not try to know through inference about a deity that must be approached only through personal instruction (Āgama), lest by so doing your head should fall off.’ The nature of the deity is to be known from the scriptures alone, and Gārgī’s question, being inferential, disregarded this particular means of approach. (Śaṅkarācārya 2008, 344)

[The understanding of the religious Ultimate or high God as sufficient reason in its transcendent aspect is common in the religions of the world. It seems that this understanding is part of the reason why the famous Ṛg Veda 10.129 (Tad Ekam) says that originally That One “breathed, breathless” (Ṛg Veda 1986, 10.129.2, 633). See Mbiti 1979, 19–25, on how numerous African ethnic groups conceive the “Self-Existence of God.” For the Zulu, for example, “God is uncreated, without parents, without family, without any of the things that compose or sustain human life” (Mbiti 1979, 20).]

Dialectics Between Nondual and Theistic Givenness (A Priori) and Śakti Interpreted Philosophically as Epistemological and Onto-Grammatical Dependence (A Posteriori) As the author has shown elsewhere, the Pratyabhijñā project of demonstrating identity with God is at the outset situated in a context of “givenness” based on the metaphysical reality of that God as nondual and therefore self-luminous as well as a theistic “super-person” (Lawrence 1996, 168–169; Lawrence 199; 44–49, 2016). This may be interpreted as a form of a priori knowledge or cognition but is not inferential per se. Utpaladeva’s second IPK 1.1.1 (kartari jn˜ ātari svātmanyādisiddhe mahes´vare/ ajaḍātmā niṣedhaṃ va siddhiṃ vā vidadhī ta kaḥ; 1:48) and IPK 2.3.15–16 (vis´ vavaicitryacitrasya samabhittilopame/viruddhābhāvasaṃspars´e paramārthasatī s´ vare// pramātari purāṇe tu sarvadā bhātavigrahe./kiṃ pramāṇaṃ navābhāsaḥ sarvapramitibhāgini.//; 2:134–135), as well as Abhinavagupta’s commentary (respectively, 1:47–56, 2:134–139), negate ritual and argument concisely for the reasons given. The same understanding of the (a priori) nondual epistemic and theistic givenness underlies Abhinavagupta’s explanation of his description in his Tantrāloka and Tantrasāra of the “nonmeans” (anupāya) as the highest means-type (upāya) of spiritual practice. His accounts of this means also seem to owe much to Buddhist skepticism such as the catuṣkoti, and also appear to indicate Abhinavagupta’s employment of material that became the basis for Tibetan Dzogchen and Chinese sudden Ch’an (Lawrence 2016). It is such statements that underlie the frequent view, it seems mistaken, that the Pratyabhijñā should be classified within Abhinavagupta’s anupāya. [Bettina Baumer suggests that the Pratyabhijñā “is at the border between the s´āmbhava and anupāya” (Baumer 2011). A probable s´ākta upāya rather than anupāya classification has been supported by Alexis Sanderson, Hemendra Nath Chakravarty, and Navjivan Rastogi in

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personal conversations. See the discussion in Lawrence (1999, 57–65). Baumer also criticizes there the use of the word monism because of its ostensible denial of the reality of the world and uses the word nondualism, which has become popular more recently among those who wish to affirm its reality. While because of nondualism’s popularity to express that, this author also switched to that term, he does not think we can redefine older terms to mean what the authors did not intend. Monism, as he understands it, just means belief in unity, whether or not the world is part of that. Some people who have also maintained the reality of the world, including the author, have used the word popular earlier. The Monist is the name of reputable journal of philosophy and is not committed to any such belief. Indeed, materialist monism also accepts the reality of the world. Monism, nondualism, or panentheism all also have aspects of illusionism to explain the nonascertainment of whatever is nondual. That also varies in kinds and degrees. See the remarks on panentheism below.] Utpaladeva and Abhinavagupta, however, also elaborate positive accounts of inferential-ritual methodology within the Pratyabhijñā and Pratyabhijñā-derived exegetics, which are “dialectically” qualified by the negations based on givenness. See the parallel assertions of givenness already mentioned at IPK, IPKV, IPV, and IPVV 1.1.1 and 2.3.15–16; and positive formulations of the method at (kiṃtu mohavas´ādasmindṛṣṭe ‘pyanupalakṣite/s´aktyāviṣkaraṇeneyam pratyabjn˜ opadars´yate, IPK 1.1.2, 1:57), IPV 1.1.2 (1:56–59) and IPVV 1.1.2 and IPK 2.3.17 (apravartitapūrvo’tra kevalam mūḍhatāvas´āt/s´aktiprakās´enes´ādivyavaharaḥ pravartyate; 2:141); IPV 2.3.17 (2:139–149); and IPVV 2.3.17 (2:165–183). This methodology reconceives the ritual process as an overarching inference-forthe-sake-of-others (parārthānumāna) with Śakti as the reason (hetu). The interpretation of Śakti may be described, still at a more programmatic rather than technical philosophical level, as an epistemological-idealistic reduction of investigated objective experience, along with an ontological reduction of the same to agency. Abhinavagupta explains in his hermeneutics and aesthetics that that this features the first person indexical (Lawrence 2008b; for the philosophical psychology of this approach to the first-person narrative, see Lawrence 2008a). There are programmatic explanations of the approach at IPK and IPV 1.2-4. (The Bhāskarī edition actually numbers both the benediction as 1.1.1 and the subsequent verse as 1.1.1. These are Utpaladeva’s last two verses of the chapter: tathā hi jaḍabhūtānām pratiṣṭhā jī vadās´rayā/ jn˜ ānaṃ kriyā ca bhūtānāṃ jī vatāṃ jī vanam matam// IPK 1.1.3, 1:61. tatra jn˜ ānṃ svataḥ siddhaṃ kriyā kāyās´ritā satī / parairaṇyupalakṣyeta tayāyajn˜ ānamūhyate// IPK 1.1.4, 1:70.) The epistemology and ontology are respectively spelled out in the Jn˜ ānādhikāra and Kriyādhikāra (IPV 1.4–1.8, 1:147–425; IPV 2.1–4, 2:1–208; IPVV 1:1–2:439). The Pratyabhijñā system may be classified with much Yogācāra, whether or not we include particular Buddhist thinkers, as a species of metaphysical “absolute idealism” inasmuch there are both the generation of the world from consciousness and an interindividual coherence. This characterization is supported by recent works of Sebastian Rodl. Rodl interprets absolute idealism as the experience of coherent and relatively independent or nonindividual objects by a subject, which objects are constituted as integral epistemological and ontological features of that subject’s selfconsciousness (Rodl 2018).

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Translation of IPV 2.3.17 The passage translated in a still provisional way below is from IPV 2.3.17 (2:139–149) on IPK 2.3.17 (2:141), that as mentioned closely parallels and further interprets IPK 1.2, synthesizing the demonstration of Śakti as emanatory dependence with a previously asserted nondualistic and theistic givenness. As such, Śakti is clearly conceived in several examples as the “reason” (hetu) of the inference: VERSE 2.3.17 There may be this doubt: If a means of knowledge [pramāṇa] is neither useful nor possible regarding the Blessed One, what is the purpose of the academic discourse [s´ā stra] concerning Him? For that [academic discourse] is nothing but a means of knowledge [pramāṇa, other than that of sense perception]. Indeed, academic discourse [s´āstra] has the nature of an inference for the sake of others [parārthānumana]. That [academic discourse] ultimately consists entirely in [utilizing] the sixteen categories [substantive concerns and methodological principles of public philosophical discourse, as formulated by the Nyāya school of philosophy], such as the means of knowledge [pramāṇa, and so on]. However, the Buddhists have disputed that [the inference] has five parts, and so on. That [disputation] is nothing but intransigence. For when the sixteen categories [pardāṛṭha, of philosophical discourse] are articulated, the other is made to ascertain [pratipādyate] fully that which is to be made to be ascertained [pratipādyam]. This accords with the book where there are [statements] such as “the obtainment of that which is good and the relinquishment of that which is bad.” What is the purpose with regard to the other? Indeed, that [an academic discourse] is for ascertainment by the other. And there is that [ascertainment] from the inference for the sake-of-others [parārthānumana]. And in that [the inference for the sake of others] there is the use of the thesis, and so on [parts of the inference, five according to Nyāya]. The creator of the Nyāya corpus, Akṣapāda, has explained that every academic discourse except a scriptural tradition [āgama] is really an inference for the sake of others [parārthānumana] that brings about the complete ascertainment by the other. Thus, he explains here, in order to remind of what was stated previously [in the verse containing] “however, due to the force of delusion” [IPK 1.1.2, 1:57; “However, this recognition of him, who though experienced is not noticed due to the force of delusion, is made to be experienced through the revealing of (his) Śakti (s´aktyāviṣkaraṇa)”]: 2.3.17. Here, by means of the illumination [prakās´a] of Śakti, there is merely [kevalam] caused to be employed [pravartyate] the cognitive practice [vyavahāram, of the understandings with regard to oneself] of Lord [ī s´a], and so on, that were not employed previously [apravartitapūrva] due to the force of delusion [mūḍhatā]. [Just as IPK 2.3.15–16 paralleled the second IPK 1.1.1, so this verse parallels IPK 1.1.2 on the operation of the s´āstra as the disclosure of Śakti. The commentary spells out the inference in more detail.]

Here, the supreme agential autonomy [svātantrya] of the Supreme Lord [parames´vara] is indeed the accomplisher of the thing which has heretofore seemed to us [asmāaddṛk]

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impossible or extremely difficult to accomplish in that particular state of a bound creature [pas´u]. What could be more difficult than this, that there shines [prakās´amanaḥ] the manifestation of the negation of shining [prakās´ananisedhavabhāsa] in him who has the nature of awareness [prakās´a], and shines in unbroken identity [akhaṇḍitatādrūpye]? Therefore, the supreme agential autonomy [svātantrya] of the Supreme Lord [parames´vara] is such a nonmanifestation [tathānavabhāsnanam] called the manifestation of the condition of the bound creature and is the accomplisher of the aspect of the cognizer [grāhaka, that is as limited]. [As indicated in the edition’s note, the earlier Kashmir Series has tathāvabhāsanam. One could get approximately the same meaning by glossing that word as a manifestation or appearance of not shining.] And by means of that [limited cognizer] there is also the manifestation of the cognized [grāhya]. This is that [saiṣā] which is said to be the Māyā Śakti of the Blessed One. [Cf. tadidam iti (IPV 1.2.1–2, 1:88); eṣa saḥ (IPK 1.4.3, 1:160); ‘eṣa sa’ iti ācchāditasyeva pramātṛtattvasya sphuṭāvabhāsanaṃ kṛtam, idam iti vismayagarbhayānayā uktyā pratyabhijn˜ ā eva sūcitā (IPV 1.4.3, 1:165); tau nāmarūpalakṣaṇau s´abdārthau ekarūpatayā ‘so’yam’ ityevaṃrupatvena parāmṛs´antī adhyavasā yā sā parmes´varas´ aktiḥ vimars´rūpā ātmavadeva ahamityanavacchinnatvena bhāti. na tu kadācit idantayā vicchinnatvena bhāti (IPV 1.5.20, 1:295); vakti arthaṃ svādhyāsena so’yamityabhisaṃdhānena (IPV 1.6.1, 1:303).]. As it has been said: That very Māyā is called Delusive [vimohī nī ] [Vijn˜ ānabhairava 1979, 95, 87].

Due to agential autonomy [svātantrya] having such a form of the Māyā Śakti, there is delusion, which is the condition of destroyed perfect consciousness [vinaṣṭapūrṇacetanatā]. [That is] the egoistic conceptualization [abhimanana] as not shining [aprakās´amānatayā] of what is shining [prakās´amānasya] – that plenitude [pūrṇatva], which has the nature of bearing internally all things which have been clearly manifested through the arising of intention [icchā] and creative vibration [spanda]. [That plenitude also] has the nature of the agential autonomy [svātantrya] consisting of the Śaktis of memory, and so on, and comprises the qualities of omnipresence [vaibhāva] and eternity that are established without effort due to the lack of the contractions [samkoca] of place and time. From the force, that is the capacity [for that delusion] there is originally not employed what was explained in the antecedent two verses, about the cognitive usage of the Lordship, perfection [pūrṇatā], and so on regarding the Blessed One who is the knower. Let those people be caused to employ that cognitive practice [vyavahāram, of the understandings with regard to oneself] having the form “That is I who manifest as perfect [pūrṇa], omnipresent [vibhu], agentially autonomous [svatantro] and eternal,” and so on. By means of this academic system [s´āstra] which illumines [prakās´aka] the Intention [icchā], Cognition [jn˜ āna] and Action Śaktis, which has the form of recognition [pratyabhijn˜ ā], and which has the nature of an inference-for-the-sakeof-others [parārthanumāna] for the establishment [sādhana] of that cognitive practice [vyavahāram, of the understandings with regard to oneself], there is effected

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[kriyate] the employment [pravartayatām], which is the utilization of the capacity for that [tatsamarthācaraṇam]. There are two causatives in “caused to be employed” [pravartyate]. “Only” [kevalam] means: Nothing which there was not previously is made. Nor is what is really not shining [aprakās´amanam] illuminated [prakās´yate]. [Rather] the conceptualization pertaining [falsely] to the Self [abhimanana] regarding that that which is shining [prakās´amāna] is not shining [na prakās´ate ity] is removed. For liberation [mukti], which is the obtainment of the state of the Supreme Lord [parames´varatā] is nothing but the removal of that [selfish conceptualization just described]. Saṃsara is nothing but the nonremoval of that. For both of these are essentially nothing but conceptualization pertaining to the Self [abhimananamātra]. And both are entirely yawned forth [vijṛmbhitam; notions of having “yawned forth” (vijṛmbhita) are a common metaphor for the Lord’s creative and other activities.] by the Blessed One. This is what has been said: It is as the delusion [moha] of one who is possessed by a demon [bhauta] and, although his Self is indeed manifesting [bhasamāne evātmani], believes due to that delusion [mohān] “I have been taken away” – is removed [by questioning]: “Who indeed are you?” If [he answers] “One of whom the clothing is such and face is such,” then [one replies]: “See, that [such clothing and face] is yours.” Nothing new is accomplished for him by the one who states this again and again. Similarly, the delusion [moha] is removed of bound humanity [pas´uloka], who, though the Self [ātman] is manifesting [bhāsamāne], believes about the Self [abhimanvato] due to delusion [moha], “I am not Lord [ī s´vara].” [This follows the enumeration of the arguments in Pandey’s translation, in IPV 3:164:] [#1:] For just as it is well known in Siddhānta [Dualistic Śaiva theology and philosophy], the Purāṇas [popular texts containing narratives and theology of various gods], and so on, that the Lord [ī s´vara] is endowed with agential autonomy [svātantrya, or “absolute freedom”] of cognition [jn˜ āna] and action [kriyā], so also are you. [#2:] If one is depended upon over something [that person] is the Lord [ī s´vara] of that, like a King [is depended on] over his domain [maṇḍala]. So does the universe [depend upon] you. Thus the usage [vvyavahāra] of Lordship [īs´varatā] does not have another cause [nimitta]. Thus is the inductive concomitance [vyāpti]. [#3:] That integral to which something manifests [bhāti] is said to be full [pūrṇa] of that, like a treasure with jewels. The universe [appears] to you. [#4:] That in which something manifests [bhāti] is the pervader [vyāpaka] of so much, like a chest regarding jewels. The universe, beginning with the earth and ending with Sadāśiva [a lower emanation of the Supreme Śiva or Supreme Lord], has been explained by the method of the text [s´āstra] [to manifest] in you who have the nature of consciousness [saṃvid]. [#5:] Something that, when it already exists [sthite] something [else] arises and disappears, is the pervader of the prior and later parts of that, like when there is [already] the earth the sprout [arises and disappears]. So, when there is [already] you who have the nature of awareness [prakās´a], the universe [arises and disappears].

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Similarly, thousands of other qualities [dharma] established in the scriptural traditions [āgama], and so on, are to be attributed. Thus, even when delusion [vyāmoha] has been removed, while there is the body, and so on, due to nothing but mnemonic impressions [saṃskāra] of that [ignorance], there remains the delusion of understanding as the self [abhimāna] what is not the Self. When there is the belief as the self of the Self [ātmatā], the pot, and so on, still shine [prakās´amana]. Just as one seeing a magical illusion [indrajāla], who has known the truth of the magical illusion [indrajāla], does not really have a delusion [vyāmoha], so it is for one who has recognized the Self [pratyabhijn˜ ātma]. Therefore, when the body has been extinguished in the end obtained by death, there is Supreme Lordship [parames´varatā]. However, when through practice [abhyāsa] and contemplation [bhāvanā] taught in the Śaiva academic works [s´āsanā], and devotion to submersion [samāves´a] of [objective identifications such as] the body, [objects of experience such as] the pot, and so on in Supreme Lordship [parames´varatā], there is the origination [for the practitioner] of the qualities of the Supreme Lordship even while seeing here in the body. However, there is not really perfection [pūrṇatā]. For when there is the loss of embodiment enlivened by contraction [saṃkoca], there is the attainment of having the nature of the universe which is as has been established. However, he for whom, even though he has had the explanation of reasons [hetukalāpa] for the establishment [sādhana] of everyday life [vyavahāra], there is the self-conception [abhimāna] of nonrealization [asiddhatā] – his ignorance [vyāmoha] is to be removed only by [additional] justifications [sādhana] of everyday life [vyavahāra]. However, he whose [delusion] is not removed in any way has nothing but delusion [mūḍhatā], due to the force of the Śakti of the Lord [īs´vara]. Because of going to the path of the ears [of this teaching], with the ripening of the mnemonic impressions [saṃskāra], even that one will necessarily at some time attain his essential nature [svarūpa]. Thus, what has been stated with the two verses beginning “who is the agent [kartari] [IPK 1.1.2]” has been explained again with three verses beginning “diversity of the universe [vis´vavaicitryasya] [IPK 2.3.15–17].” Thus, how can what amounts to a means of knowledge [pramāṇa] be possible regarding such a Blessed One who is such? [Bhāskara here, BIPV 2.3.17, 2:149, includes nitye in his gloss of evaṃbhūte.] Therefore, what has been said is now stated in the text [s´āstra] in another way. Thus Śiva.

Śakti Overcoded with the Pratyabhijñā Vocabulary of Recognition To address a variety of problematics, the Pratyabhijñā thinkers interpret Śakti in the technical arena of epistemology, as an idealistically constitutive self-recognition or recognitive synthesis (ahampratyavamars´a, pratyabhijn˜ ā, anusaṃdhāna), also identified with a principle of Supreme Speech (parāvāk) derived from the linguistic philosopher Bhartṛhari. This is articulated in the well-known Pratyabhijñā epistemology of recognitive apprehension (vimars´a) as the essential nature of awareness (prakās´a) (Lawrence 1999, 107–122).

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The article reiterates what was stated originally in Rediscovering God. The Pratyabhijñā theory of recognition is not only articulated in derivatives form abhijn˜ ā or prati-abhi-jn˜ ā, which literally mean recognition, but in two other sets of terms (Lawrence 1999, 86–87, 208–209n). Derivatives from anu, prati, or abhi plus saṃ-dhā are usually translated as “recognitive synthesis.” Supporting this interpretation, Abhinavagupta defines pratyabhijn˜ ā in terms of anusaṃdhāna and pratisaṃdhāna at IPV 1.1 on IPK benedictory verse (1:36–38). Abhinava similarly defines pratyabhijn˜ ā in terms of anusaṃdhāna at IPV 1.4.8 (1:188–189). The later Pratyabhijñā text, Virūpākṣapan˜ cās´ikā, simply classifies the “This is that” experience as anusaṃhiti (VAP 3.38, 16). The commentator Vidyācakravartin again makes the equation: “Anusaṃhiti is pratyabhijn˜ ā” (VAPV 3.38, 16; also see VAPV 3.39, 17). The free alternation between the terms pratyabhijn˜ ā and anusaṃdhāna is seen in the discussions of action, for example, at IPV 2.1.5 (yadā tu gāḍhpratyabhijn˜ āprakās´abalāt tadeva idaṃ hastasvarūpam iti pratipattau mūrterna bhedaḥ, atha ca anyānyarūpatvam bhāti tadaikasmin svarūpe yadanyat anyat rūpam tadvirodhavas´āt asahabhavatkriyā ucyate, tasyā yat vaicitryam parimitāparimitātmakaṃ tadekānusaṃdhānena phalasiddhyādinibandhanavas´āt yathāruci carcitena nirbhāsayan kālarūpaṃ kramamevāvabhāsayati/2:17). It is also articulated in derivatives, with various prefixes, from mṛs´, which the author has translated as both “recognitive apprehension” and “recognitive judgment,” now usually the former because it has less dualistic connotations. (Alexis Sanderson stated in personal conversation that there are sometimes small differences in meaning between the terms. On the “mṛs´ terms,” see also the classic study of Alper 1987. For an effort to recover the historical associations of “touching” in the meaning of the terms, see Skora 2009.) Bhāskarakaṇṭha equates the terms parāmars´ a and pratyabhijn˜ ā in commenting on IPV 2.2.2, 2:39. (He glosses the expression parāmars´abalādeva with pratyabhijn˜ ābalādeva. BIPV 2.2.2, 2:39.) Bhāskarakaṇṭha on 1.5.20 (1:294) uses the word pratyabhijn˜ ā to describe the means (by using the instrumental case) by which parāmars´a unifies word and object. (For example, he explains paramṛs´antī – s´abdārthaikī karaṇarūpayā pratyabhijn˜ ayā parāmars´aviṣayatāṃ nayantī , ata eva tābhyāmatirekeṇa yukteti bhāvaḥ. BIPV 1.5.20, 1:294.). At 2.3.10–11 Utpaladeva and Abhinavagupta use the words vimars´a and pratyavamars´a identically to pratyabhijn˜ ā as invoked by the Naiyāyikas against the Vijñānavādins. They explain through it the knowledge “This is that thing” regarding objects which appear successively as far and near, inferred and directly perceived, external and internally imagined, and as seen in incorrect and correct cognitions. IPK 2.3.10–11 (2:117); IPV 2.3.10-11 (2:117–119); and BIPV 2.3.10–11 (2:117–119). (These are Utpaladeva’s verses: dūrāntikatayārthānām parokṣādhyakṣatātmanā/bāhyāntaratayā doṣairvyan˜ jakasyānyathāpi vā. IPK 2.3.10, 2:117. bhinnāvabhāsacchāyānāmapi mukhyāvabhāsataḥ/ ekapratyavamars´ākhyādekatvamanivāritam. IPK 2.3.11, 2:117.) Abhinava similarly uses the expression pratyavamṛs´yate to describe the recognition of the continuity of material cause and effect at IPV 2.4.18 (nanvevamapi bī jamaṅkurādivicitramavabhātaṃ dī rghadī rghaparāmars´as´ālibhiḥ srotovadavicchinnasvarūpameva nirbādham pratyavamṛs´yate/. . ..2:194). Parāmars´a and other mṛs´

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terms are used to describe the soteriological recognition at IPV 4.1.16 (yata evaṃ s´ā stragurusvapratyayasiddho ‘yamarthaḥ, taditi tasmādatra prameyapade parāmars´aṃ vis´ramayan vis´vakartṛtvalakṣaṇamais´varyamātmano vibhāvya dārḍhyena yadā parāmṛs´ati tarhi tatparāmars´amātrādeva tāvajjī vanmkto bhagavān˜ chiva eva, 2:310–311) and IPV 4.1.17 (2:314–315). Bhāskarakaṇṭha identifies anusaṃdhāna with pratyavamars´a in his commentary at BIPV 1.6.10 (yojanā cānusaṃdhānam, sa eva pratyavamars´aḥ; 1:340; also see below). Anusaṃdhāna, anusaṃdhi, etc., are employed in synonymous or intrinsic functional relationships with vimars´a, parāmars´a, etc., at IPK 1.5.19 (sākṣātkārakṣaṇa’pyasti vimars´aḥ kathamanyathā/dhāvanādyupapadyeta pratisaṃdhānavarjitam//; 1:284); IPV 1.5.19, 1:291–292). In commenting at BIPV 1.6.1, Bhāskarakaṇṭha identifies anusaṃdhāna as the effect (kārya) of pratyavamars´a (e.g., anusandhānasyāpi – pratyavamars´akāryasya yojanasyāpi; 1:301). The terms are used disjunctively in analyses of states with different degrees of contingent empirical, rather than transcendental, recognitive synthesis. In elaborating a typology of cognitive states, Abhinava thus describes a form of direct experience (anubhava) which lacks synthesis (abhisaṃdhi), despite his usual stress on the invariable concomitance of synthesis with consciousness. He is endeavoring to describe what seems to be the most discrete, uninterpreted sort of experience. However, Abhinavagupta emphasizes that even here there is (recognitive) apprehension (parāmars´a), which is necessary for awareness (IPV 1.4.8, 1:187–188). On the basis of differentiation of such an underlying transcendental apprehension (parāmars´a), he analyzes in this section a great variety of sorts of direct experience, memory, and recognition (pratyabhijn˜ ā; IPV 1.4.8, 1:187–189 and IPVV 1.4.8, 2:58).

Additional Subsumptions of Inference: Pleromatic Fragmentation and Inductive Noncommitance Now, in Sanskrit philosophies, the conceptually constructed cognitions (savikalpaka jn˜ āna) that make up all quotidian epistemic experience – in distinction from unconstructed states (nirvikalpaka jn˜ āna), comprise a variety of distinctions, between all the factors of cognitive processes. Likewise, the classic fivefold inference articulated in the Nyāya system has five steps with various constituent terms, invoking the probable or necessary regularities of things. To explain how there emerge various epistemic distinctions, constituting cognitions and invoked in inference, the article adverts to previous and unpublished analyses of the Pratyabhijñā accounts of the fragmentation of Śiva’s pūrṇatā, “perfection,” “pleroma,” “fullness,” or “completion.” [Pūrṇatā is another one of the basic conceptualizations of the Lord’s nature, which Abhinavagupta equates with the Lord’s creative Śakti Herself (TS, 4, 27–28; cf. Kṣemarāja in Vasugupta and Kṣemarāja, 1925, 3.13, 66; and Maheśvarānanda 1972, 122). Śiva is frequently described as comprising perfect or complete consciousness (pūrṇasaṃvid) (Lawrence 2013) that emanates into more-or-less erroneous and contingent, incomplete knowledge (apūrṇakhyāti); and as having perfect I-hood (pūrṇāhaṃtā), underlying the psychology of limited egoity (ahambhāva, abhimāna) (Dyczkowski 1992,

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Hulin 1978, Lawrence 2008a). This perfect I-hood is also explained as the highest level of mantra, encompassing within itself all of the Sanskrit letters.] Epistemologically, pūrṇatā defines Śiva’s idealistic comprehension of everything within, as identical with Himself. All things appear as one’s own perfect (pūrṇa) I in the Self (IPV 2.1.6–7, 2:22–23). Again the Lord also comprehends all limited subjects within his perfect omniscient and omnipotent nature (IPV 4.1.1, 2:281–282; also see IPV 1.1.4, 1:75–77; and Ajaḍapramātṛsiddhi, in Utpaladeva, 1921, 13, 5). [Abhinava is fond of the maxim “Everything has the nature of everything” (sarvaṃ sarvātmakam).] The Pratyabhijñā further overcodes this fragmentation by appropriating and subverting the theory of apoha, semantico-inferential exclusion (apoha) of the Yogācāra Buddhist logical-epistemological school of Dignāga (c. 480–540 CE), Dharmakīrti (c. 600–600 CE), and their followers. In his explanation, Abhinavagupta hearkens back to old conceptions of apoha as a kind of vyatireka limiting optionality: When a pot is seen, it is also possible that there could be, in the place of the pot, a non-pot, such as a cloth, etc. [Such a non-pot would like the pot also] naturally have a location believed to be suitable; it would produce cognition [of itself as object], and would be brought to its location by its own causes. [Abhinava is trying to show that the non-pot is a viable alternative.] Since the manifestations of both the pot and the non-pot are possible, there is an opportunity for a superimposition [in which one wrongly takes a pot to be a non-pot]. Since there is the [possibility of the] superimposition of a non-pot, there is the operation of exclusion, which is characterized by negation. Thus the ascertainment [nis´caya] “pot” has the nature of conceptual construction that is animated by that [exclusion]. [ghaṭe hi dṛṣte ghaṭasthāna evāghaṭo’pi yogyades´ābhimatasthānākramaṇas´ī lo vijn˜ ānajnakaḥ svakāraṇopnī taḥ sabhāvyate paṭādisvabhāvaḥ, ato ghaṭāghaṭayordvayoravabhāsasya sambhāvanāt samāropaḥ sāvakās´o bhavati, aghaṭasya satyārope niṣedhanalakṣaṇopohanvyāpāraḥ iti tadanuprāṇitā vikalparūpatā ghaṭa ityetasya nis´cayasya (IPV 1.6.2, 1:306)].

As cybernetic theorist Gregory Bateson states: “A ‘bit’ of information is definable as a difference which makes a difference” (Bateson 1972, 315). The basic, binary digits of information, by some sort of statistical derivation, define the differences of intelligible cognitions in practical life, including inferences with their diverse terms (see Lawrence 2018, for an interpretation of the Śaiva theory of apoha in relation to information theory, Saussurean structuralism, and Peircean pragmatics). The capacity for producing fragmentation-cum-exclusion that forms the basis of ordinary life is again further explained, from a nondualistic perspective, with theodicial notions of the Māyā Śakti, Power of Magic Illusion, that is one of Śiva’s five Acts (see above n. 1): In the condition without conceptual construction [avikalpa], the pot has the essential nature of consciousness [cit], and just like consciousness [cit], has the nature of everything [vis´vas´arī ra] and is perfect [pūrna]. However, there is no worldly activity with that [pot that has the nature of everything and is perfect]. Therefore, [the knower], manifesting the operation of Māyā, causes the thing, even though perfect [pūrna], to be fragmented. By means of that, is created semantic exclusion [apohana], having the form of negation of the non-pot, such as the self, cloth and so on. On the basis of that very exclusion [vyapohana], there is said to be the ascertainment [nis´caya] of the pot. The meaning of “only” [eva] in [the ascertainment] “only

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the pot” is the negation of other things that are supposed to be possible. Therefore, there is this complete distinction, by distinction all around, like cutting. [tadavikalpadas´āyāṃ citsvabhāvo’sau ghaṭaḥ cidvadeva vis´vas´arī raḥ pūrṇaḥ, na ca tena kas´cidvyavahāraḥ, tat māyāvyāpāramullāsayan pūrṇamapi khaṇḍayati bhāvaṃ, tenāghaṭasyātmanaḥ paṭādes´cāpohanaṃ kriyate niṣedhanarūpam, tadeva vyapohanamās´ritya tasya ghaṭasya nis´cayanamucyate ‘ghaṭa eva’ iti, evārthasya eṣa eva paritas´chedāntakṣaṇakalpāt sambhāvyamānāparavastuniṣedharūpatvāt, paricchedaḥ (IPV 1.6.3, 1:309–310).]

[For another description of the creation of the limited subject distinguished from objects, which invokes both the terms Māyā and Exclusion see IPK and IPV 1.6.4–5, 1:312–323. [The idea of fragmentations is also well articulated in the following passage: [That (object) which is manifested is differentiated from consciousness. Consciousness (is differentiated) from that. One consciousness (is differentiated) from another consciousness. One object of consciousness [is differentiated] from another object of consciousness. This differentiation is not really possible. Thus it is explained to be the mere manifestation of the differentiation. As such, that (differentiation) is not ultimately real. For this [manifestation] is the ultimate reality of everything which is created. Due to the differentiating all around (paritas´chedanāt), there is said to be all-around differentiation (pariccheda). The capacity for the manifestation of that [differentiation] is the Semantic Exclusion Śakti. [Tat ābhāsyate tat saṃvido vicchedyate, saṃvicca tataḥ, saṃvicca saṃvidantarāt, saṃvedyaṃ ca saṃvedyāntarāt, na ca vicchenaṃ vastutaḥ saṃbhavati iti vicchedanasya avabhāsmātram ucyate/ na ca tat iyatā pārmārthikaṃ nirmayamāṇakya sarvasya ayameva paramārthaḥ/ yataḥ eṣa eva paritas´chedanāt pariccheda ucyate, tadavabhāsanasāmarthyam apohanas´akti (IPV 1.3.7, 1:142–143).]

Niyati Śakti as Generating Substantive Inferential Concomitance According to the Pratyabhijñā thinkers, Śiva sets the limits in the construction of the plurality of cognitions by divine decree, as it were, through his Niyati, “Fixed Regularity,” Śakti. The Niyati Śakti thus determines that particular manifestations may not be combined in perception, as they are contradictory to each other. Some occur in regular relationships because of essential nature (svabhāva) or origin (utpatti). (On the combination of abhāsas, and contextual and personal factors causing their contingency, along with their epistemic and spiritual rectification, see Lawrence 2013). These form the basis of mundane cognitive regularities including those invoked in inference.

Scriptural Traditions as Grounds of Inference Abhinavagupta adduces a variety of considerations in order to demonstrate that scriptural traditions (āgama) are the final justificatory grounds of all cognition. Scriptural traditions for Abhinavagupta are expressions of Śiva’s Supreme Speech/ self-recognition. They are directly accessible as “innate” or “a priori” features of cognition. At the same time, they are manifest in their familiar cultural aspects such

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as oral and written canon, or public knowledge (prasiddhi), and the testimonies of authoritative people (āptavākya). These considerations may be understood as an extension of the Pratyabhijñā arguments about vimars´a, which disclose Śiva’s selfrecognition/Supreme Speech as the reality underlying all human experience. Abhinavagupta further supports the subsumption of inference by the Śaiva worldview by developing arguments of Bhartṛhari that make it depend on scriptural tradition (āgama). In this case, the scriptural traditions on which it depends are primarily the nondual Śaiva āgamas (see Lawrence 2000).

Identity of Cosmogony and Teleology “Tantric Argument” and Rediscovering God spoke of an “identification of cosmogony and teleology” in the nondual metaphysics of the Pratyabhijñā. The author has then and sometimes later remarked how this identification reminded one of Mircea Eliade on the relationship of myth and ritual. It also reflects a broadly Durkheimian understanding of the relation of world view and practice, as is evinced in Clifford Geertz on the relation of culture or religion and ethos. However, the author has come to understand that this identification in the West reflects the legacy of complexly intertwined, Christian and non-Christian, Platonism and Aristotelianism. Both Platonic and Aristotelian traditions view human actions as getting their genuine impetus and intentionality by somehow enacting or embodying respectively the One/Good/ Ideas or Being/Act of divine Intellect. Eric Voegelin has written an illuminating discussion of the issue (Voegelin, “The Beginning and the Beyond,” in 1990, 173–232) from his own mystical perspective. For example, he states: In fact, when the response becomes a questing movement, the directions are prescribed by the structure of the reality in which man finds himself situated as its part. In his search of the divine ground, man can do no more than move either in the time dimension of the cosmos or through the hierarchy of being from inorganic matter to his own questioning existence, in order to find it either in an event preceding the present state of things or in a place higher than the known hierarchy of things. Hence, when the response becomes reflectively conscious as a quest, the experience reveals a truth not only about divine reality but also about the structure of the cosmos in which it occurs. Beside the structure of appeal-response, the structures of cosmic lasting in time and of the hierarchy of being are a further area of reality that becomes visible through the movements toward the Beginning and the Beyond. (Voegelin “The Beginning and the Beyond,” in 1990, 17)

He also has the idea that the highest discovery is in some way self-luminous (and a priori). The Metaxy describes the middle ground of a linkage between worldly experience and the Ultimate, as realized in religious experience. He explains: The experience of the divine reality, it is true, occurs in the psyche of a man who is solidly rooted in his body in the external world, but the psyche itself exists in the Metaxy,

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in the tension toward the divine ground of being. It is the sensorium for divine reality and the site of its luminous presence. Even more, it is the site in which the comprehensive reality becomes luminous to itself and engenders the language in which we speak of a reality that comprehends both an external world and the mystery of its Beginning and Beyond, as well as the metaleptic psyche in which the experience occurs and engenders its language. In the experience, not only the truth of divine reality becomes luminous but, at the same time, the truth of the world in which the experience occurs. There is no “external” or “immanent” world as such by its relation to something that is “internal” or “transcendent.” Such terms as immanent and transcendent, external and internal, this world and the other world, and so forth, do not denote objects or their properties but are the language indices arising from the Metaxy in the event of its being luminous for the comprehensive reality, its structure and dynamics. (Voegelin, “The Beginning and the Beyond,” in 1990, 184–185) Here may be repeated the passage from IPV 1.5.17 quoted previously: That which is called recognitive apprehension [parāmars´a] is the absolutely final and true place of rest [vis´rāntisthānam]; and it only has the form “I.” In traveling to a village, the intermediate point of rest [madhyavis´rāntipadam] at the root of a tree, is explained to be created as expectant of that [final point of rest]. Therefore, what is the contradiction [between the divine self and various cognized objective representations of it and other things]? Thus also blue, etc., in the intermediate recognitive apprehension [parāmars´a] as “This is blue,” are established to consist of the Self. For they rest [vis´ranteḥ] upon the root recognitive judgement [parāmars´a] “I.” [parāmars´o nāma vis´rāntisthānam, tacca pāryantikameva pāramārthikaṃ, tacca ahamityevaṃrūpameva/ madhyavis´rāntipadaṃ tu yat vṛkṣamūlasthānīyaṃ grāmagamane tasya tadapekṣayā sṛṣṭatvam ucyate iti ko virodhaḥ/ anena nī lādeḥ api idaṃ nī lam iti madhyaparāmars´o’pi mūlaparāmars´o ahamityeva vis´rānteḥ ātmamayatvam upapāditameva. . . (IPV 1.5.17, 1:278–279).]

The identification in terms of absorption in a One, of cosmogony and teleology, is of a piece with the identifications of transcendent and immanent, Being and contingency, peaceful smiling and dance, means (upāya) and goal (upeya), reason (hetu) and conclusion (nigamana) (see Lawrence 1996, 186–187 and 1999, 100), and even a posteriori and a priori. As was said, the Śaiva form of transcendental argument is seen as a mode of return in a circular journey which does not go anywhere. Since that circle is also hermeneutic, in what we describe as a transcendental inference, something is still learned.

Definitions of Key Terms anusamdhāna pratyabhijn˜ ā vimars´a kriyā kāraka kartṛ svātantrya

Recognitive synthesis Recognition Recognitive apprehension Action Semantic-syntactic factor of action, according to Pāṇini Agent Agential autonomy

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Logic in Tamil Didactic Literature

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Thanga Jayaraman

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kāṇṭikai: Its Forms and Members . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Validity of the kāṇṭikai Form of Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Demonstrating that there had been an ancient logical tradition native to the Tamil society, Nirmal Selvamony has, in a paper entitled, “The Syllogistic Circle in tolkāppiyam” (The syllogistic circle in tolkāppiyam. 65th Session of the Indian Philosophical Congress. Madurai-Kamaraj University, Madurai, 27–29 Dec. Unpublished Paper, 1990; J Tamil Stud 57 & 58: 117–134, 2000), identified the five-member syllogism central to the tradition. He has also shown that kāṇṭikai had perfect application to Tamil didactic texts and was significantly different from the universally adopted Aristotelian three-member syllogism. The early tendency of the Tamils was to identify logic with philosophy and religion. Taking a departure from this tendency, he recovers the original definition of kāṇṭikai from iḷampūraṇār’s commentary on tolkāppiyam. This chapter tries to identify the structure of arguments in Tamil didactic verses employing the kāṇṭikai form. The application of kāṇṭikai to diverse Tamil texts as an analytic tool has no hermeneutic intent, but the tool gets itself defined in the process. The argument of this chapter is that the efficiency of the different members of kāṇṭikai can be better appreciated when their correspondence to those of the jurisprudential model of argument, described by Stephen Toulmin, is traced. Taking a clue from Toulmin, it is also argued that many Tamil didactic texts have what Collingwood calls T. Jayaraman (*) Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_8

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“absolute presuppositions” and that these presuppositions have correspondence to “warrants” in the jurisprudential structure of argument. The ubiquitous analogy and recursive analogical reasoning found in Tamil texts can also be accommodated in this model of argument when the features of an analogy are seen as the “datum” in the jurisprudential model. This helps us overcome the problem in claiming application of logic to didactic texts of preceptorial tone, the arguments of which, though often seen as assertions and affirmations on authority, do have warrants and also backing for their warrants. The arguments of Tamil didactic verses, when cast in the kāṇṭikai form of argument thus eclectically constructed, allow us to clearly see the unstated presupposition of every one of them. Keywords

Argument · Syllogism · Proposition · Claim · Presumption · Presupposition · Variant · Construction · Form · Warrant · Backing · Term · Premise · Assertion · Deductive · Plausibility · Establish · Issue · Hermeneutic · Structure

Introduction This chapter is an attempt at identifying the structure of the arguments in Tamil didactic texts using logical categories available in the ancient form of argument of Tamil society. Nirmal Selvamony, in his paper, “The Syllogistic Circle in tolkāppiyam” (1990, 2000), reconstructs the five-member Tamil syllogism adopted for the construction of arguments in the primal society. He shows how it persisted in the state society and was adopted by Tamil didactic verses and religious texts, especially by Buddhist and Jain composers. Though the kāṇṭikai form was adumbrated in tolkāppiyam, the later Buddhist text, maṇimēkalai, describes it in greater detail. Sanskrit texts dealing with logic, such as those of the Nyaya-Vaisesika system, came much later, and their Tamil provenance is marked by the presence of technical terms borrowed from the form of argument peculiar to kāṇṭikai (Selvamony 1990, 1996: 109, 113, 2000). In later periods, kāṇṭikai was used by commentators for interpreting texts although it was originally used for establishing doctrinal claims and formed part of religious and philosophical traditions (Selvamony 1996: 100, 101). With the doctrinal fervor abating in Tamil history, kāṇṭikai could have lost its salience but had not altogether disappeared from religious literature as it can still be seen in later didactic literature of successive periods and also in the devotional literature. Being exhortative and preceptive, didactic literature may not be credited with arguments designed to establish the truth of a claim on evidence, least of all, on empirical evidence. Didacticism has little to do with issues of fact and has even less of what we can call facts at issue. However, the presupposition they rely upon for the progression of their argument can be treated as the fact at issue, the acceptability of which claims a right to presumption. When verse 18 of the didactic text, naṉṉeṟi, says that harsh words do not please this world, only pleasant words do, one hardly suspects there being an impliedly present argument in the exhortation. This chapter is premised

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on the following assumptions: (1) that the Tamil didactic literary texts, from tirukkuṟaḷ to the latest in the line, naṉṉeṟi, have constructed their arguments with propositions that are exponents of members of kāṇṭikai and (2) that these arguments can be recovered by applying the kāṇṭikai form to them. Any assertion, persuasion, or justification as they are found in the didactic verses would necessarily involve arguments to render their claim plausible. One cannot expect empirical, inductive arguments there. Still, methods of argument are different in different fields. Didactic literature has its own logic, and the kāṇṭikai form of argument is its proven exponent. For centuries, the Tamil society has been using kāṇṭikai for constructing its arguments for audiences who ought to have had some acquaintance with and expectation of this logical device. The verses were addressed to them. That is of enough historical significance to merit a study. This chapter tries to make a contribution to recovering a nearly lost tradition in which religion, philosophy, and logic were found locked in an inextricable embrace. Critically studying the identity of kāṇṭikai established until now in one or two sources and moving toward a more precise definition of its members and their function are what this chapter would attempt. The issue of the validity of arguments in kāṇṭikai form has been dealt with in this chapter relying on the stand adopted by Stephen Toulmin’s The Uses of Argument (1964). Questioning the possibility of a universally applicable syllogism, he describes the comprehensive structure of jurisprudential argument. He states that the forms of arguments are field dependent and vary in different fields. The “absolute presuppositions” of R.G. Collingwood point to a useful line of investigation in didactic texts, especially when ethical arguments do not seek to base themselves on empirical data (Toulmin 1964: 255–258). It has already been said that the ethical arguments in the didactic texts, by their very nature, are assertive, authoritative, and exhortative, and hence they do not present anything as issues of fact. They, rather, rely on analogical reasoning. Logicians have extensively discussed analogical reasoning, and several Tamil didactic texts are built upon this kind of reasoning. Analogy cannot be left unaccommodated as mere rhetorical device in any form of argument, which we identify as properly belonging to Tamil didactic tradition. Let us briefly discuss the forms of kāṇṭikai summarized by Selvamony, notice the significant variations in different accounts of kāṇṭikai and their implications for the form, proceed to describe the jurisprudential form of argument, and find out how categories in one correspond to those in the other. Analogies and analogical reasoning of the verses will then be discussed to show that the character of kāṇṭikai is inductivedeductive. In the course of its argument, the chapter will try to assess the susceptibility of individual Tamil didactic verses of different periods to being adequately described in terms of the kāṇṭikai form of argument.

ka¯ntikai: Its Forms and Members ˙˙ Selvamony takes a departure from the practice of explaining Tamil philosophy in terms of Western Humanism and Marxism and attempts its reconstruction in terms of native tradition. In the introduction to his 1996 book, he states that the methods of

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argument and the grammar of syllogism used in presenting philosophy are first found in tolkāppiyam, a Tamil work that precedes texts on Indo-Aryan traditions of syllogism by more or less 600–1000 years. maṇimēkalai, a later Tamil Buddhist work, is the first to mention the members of syllogism with illustrations. Philosophy then comprised jurisprudence, hermeneutics, commentary, rhetoric, essay, and criticism (1). This being the situation, it should be no surprise that didactic texts in Tamil abound in analogy and constantly resort to analogical reasoning. Analogical reasoning gave rise not only to several figures of speech but also to the logical device known as kāṇṭikai. According to Selvamony, tolkāppiyam mentions kāṇṭikai as a method used to establish one’s claim (1996: 101). kāṇṭikai has five parts: claim (mēṟkōḷ or cūttiram), reason (ētu), example (eṭuttukkāṭṭu), naṭai, and conclusion (muṭipu). He explains mēṟkōḷ, which is synonymous with pakkam, as the claim, which has to be finally established in the argument (1996: 104). He mentions kāraṇam as an equivalent of ētu (1996: 106), which, according to the commentator of tolkāppiyam, falls into three classes: mutal or the primary, tuṇai or the secondary, and nimittam (1996: 107). nimittakāraṇam is the agent in whom inheres the intention or purpose. A pot made of clay has clay for its primary kāraṇam, the wheel on which the pot is made for its secondary kāraṇam, and the potter for its nimittakāraṇam. Selvamony cites the proposition “whichever has smoke has fire, as in the case of an oven” as a positive eṭuttukkāṭṭu (example) and the proposition “whichever has no fire has no smoke, as in the case of a lotus pond” as a negative one. This chapter will later argue that eṭuttukkāṭṭu corresponds to “warrant” in the jurisprudential model of argument. It has to be observed here that the “example” is but an illustration of the “warrant,” the truth of which is assumed in the argument. Hence, the oven having smoke or the lotus pond having no smoke is what in the jurisprudential model is known as “backing” for the warrant. naṭai is the step that leads to the conclusion through linking the reason with the claim. muṭipu (conclusion) repeats the mēṟkōḷ or cūttiram, and referring to ētu, eṭuttukkāṭṭu, and naṭai, it establishes the mēṟkōḷ (Selvamony 1996: 108). Here is the standard illustration: This hill has fire (claim). Since it has smoke (reason). Whichever has smoke has fire, as in the case of an oven (example). This hill also has smoke (naṭai). Therefore, this hill has fire (conclusion). An illustration given for negative kāṇṭikai is “This hill has no smoke (claim).” As this has no fire (reason). Whichever has no fire has no smoke, as in the case of a lotus pond (example). This hill also has no fire (naṭai). Therefore, this hill has no smoke (conclusion; Selvamony 1996: 108). For the sake of clarity, this chapter would recast this negative kāṇṭikai as follows: This hill has no fire (claim). This has no smoke (reason). Whichever has no fire has no smoke, as in the case of a lotus pond (example). This hill also has no smoke (naṭai). Therefore, this hill has no fire (conclusion). The author of maṇimēkalai identifies the five members of kāṇṭikai as claim or pakkam, reason or ētu, example or tiṭṭāṉtam, upanayam which corresponds to naṭai, and nikamanam, which seems to have been extricated from “example” and given an independent place replacing “conclusion.” The illustration runs as follows: This hill

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has fire (claim). Since this has smoke (reason). As in the case of an oven (example). This hill also has smoke (upanayam). That which has smoke has fire (nikamaṉam) (Selvamony 1996: 109). Here, the member forming the conclusion varies from the one found in forms adopted by texts preceding and following maṇimēkalai in time. Of this variance more will be said later. The first verse of civañāṉapōtam, a work that attests to the continuity of the kāṇṭikai tradition, lists the members of the kāṇṭikai as claim, reason, example, upanayam which corresponds to naṭai, and niyamaṉam, corresponding to conclusion. The illustration for this is as follows: This universe is made by an agent (claim). As it remains manifest in diverse forms (reason). As even the pot is made by the potter (example). This universe is a created one, like the pot (upanayam). Therefore, the universe is a created one (niyamaṉam) (Selvamony 1996: 115). Citing verse 70 in paḻamoḻi, Selvamony mentions a truncated form of kāṇṭikai where two members, reason and conclusion, are allowed to remain hidden yet inferable (1996: 111). The substance of the verse is that the noble, even when they decline into poverty, will not depart from their noble path, even as tiger, though hungry, will not have the lowly grass for its food. This is a form of argument that has been widely adopted by Tamil didactic texts and is inherent in analogical reasoning. Selvamony (1996: 149) explains analogical reasoning as one that points up the evidential link between the claim and the reason that is advanced for the claim. He quotes na. ra. murukavēḷ’s illustration of kāṇṭikai structure as he finds it in the first verse of tirukkuṟaḷ (1996: 112). The verse in translation is “Alphabets all have ‘A’ as their beginning/the world has the Primordial God as its beginning” (Trans. kō. vaṉmīkanātaṉ. 2007). This world has ātipakavaṉ for its beginning (claim). Since it is caused to move (reason). Whichever is caused to move, that has for its beginning something that moves it (example). This world is moved as even letters in the alphabet are caused to move (naṭai). Therefore, this world has ātipakavaṉ for its beginning (conclusion). Selvamony (1996: 112) also finds that there are verses in the canonical Tamil didactic collection, kīḻkkaṇakku, which do not articulate ētu or reason. Rational arguments have an element of commonality, though their structure may vary from field to field. It would now be necessary to summarize the jurisprudential structure of argument to see how kāṇṭikai compares with it. Stephen Toulmin describes the layout of jurisprudential argument as one of greater complexity than the three-member rational argument of the Aristotelian model. The structure has six members, three of which remain constant in any situation (96). The argument moves from “datum” to a “claim” enabled by a “warrant.” The claim may have a qualification expressed by words like “presumably,” “probably,” and an exception expressed by a word like “unless.” “Warrants” may need a “backing” as they may be of questionable authority (Toulmin 103). “Murugan is the son of Nallan” is the datum. “Murugan should take his father’s property” is the claim. “Father’s property goes to his son” is the warrant. “Sons in India have been taking their fathers’ property” is the backing for the warrant. “Murugan should ‘normally’ take his father’s property” is a qualification on the claim. “Murugan should take his father’s property ‘unless’ he incurs a disqualification” is the exception. If “claim” is put at the

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head of the list and modal qualifiers such as “probably,” “presumably,” or “normally” and also rebuttals such as the exceptions are excluded and finally the conclusion indicated is added to the list, a structure more or less comparable to kāṇṭikai is obtained. The illustration given above can be cast in the kāṇṭikai form of argument: Murugan takes his father’s property (claim). For, he is the son (reason). A son takes his father’s property (example). Murugan as son is similarly placed (naṭai). Therefore, Murugan takes his father’s property (conclusion). “Datum” in the jurisprudential form of argument corresponds to “reason” in the kāṇṭikai form as even “warrant” does correspond to “example” in the kāṇṭikai. Elucidating the jurisprudential form of argument, Toulmin says that warrants are implicit and general, while data are explicit (Toulmin 96). This observation about warrant and that it corresponds to “example” in the kāṇṭikai form are attested by the standard kāṇṭikai “example,” “whichever has smoke has fire, as in the case of an oven” Selvamony (1996: 114) quotes mātava civañāṉa yōkikaḷ on the first verse of civañāṉapōtam where he expands the standard “example” as “whatever exists has its maker, as in the case of a pot.” That “reason” in kāṇṭikai form corresponds to “datum” is obvious, as seen in the standard illustration for “reason,” “Since this hill has smoke.” It has earlier been seen that maṇimēkalai identifies the proposition “That which has smoke has fire” as “nikamaṉam” and assigns the function of conclusion to it. In the earlier kāṇṭikai forms, this is identified as “example” and “as in the case of an oven” is tagged to it. Put in terms of the jurisprudential model, the pre-maṇimēkalai kāṇṭikai clubs “warrant” with an illustration or embeds an unstated warrant in an illustration. It can even be said that illustration is treated as warrant which legitimizes the step to the conclusion. maṇimēkalai’s “nikamaṉam,” which has been seen to be corresponding to “warrant” in the jurisprudential form, takes the place of “conclusion.” This cannot be explained away as a variant of the kāṇṭikai form of argument. V. Kanakasabhai (2000:218), in his Tamils 1800 Years Ago, cites “This mountain has fire” as assertion and proceeds to say that “Whatever does not possess fire cannot be accompanied by smoke, as for instance, ‘a stream of water’ is an example of the Negative of the Assertion.” He summarizes the account of true knowledge given by the Buddhist monk, āticinēndra, to maṇimēkalai, the eponymous protagonist of the ancient Tamil epic, maṇimēkalai. To the Buddhists, the sources of true knowledge consist of perception and inference. Every inference or syllogistic text has five parts: assertion, reason, example, comparison, and deduction. “This mountain has fire” is assertion. “Because it has smoke” is reason. “Like the kitchen hearth” is example. “This mountain also has smoke” is comparison. “As it has smoke it has fire” is deduction (Kanakasabhai, 218). Arguing that the concomitance of smoke with fire alone does not prove the existence of fire, his summary points out that smoke perfects its probative value by the converse negative fact that “where there is no fire there is no smoke” (Kanakasabhai, 218). Illustrations in his summary argue that neither concomitance nor the converse negative proves the fact independently of each other. The argument is developed by a quote: “Both Comparison and Deduction are subordinate to Example.” Kanakasabhai renders eṭuttukkāṭṭu as “example” and naṭai as “comparison.” The subordination of both comparison and deduction to “example,” which is the translation equivalent of eṭuttukkāṭṭu, problematizes

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eṭuttukkāṭṭu as this chapter has earlier anticipated. It can fairly be claimed that eṭuttukkāṭṭu can be better understood if it is taken as corresponding to “warrant” in the jurisprudential model of argument. The illustration invariably suffixed to eṭuttukkāṭṭu is the “example” functioning as the “backing” for the warrant. Another quote by Kanakasabhai in his summary provides standards for testing the validity of arguments: Right Reason is of three kinds: it may be founded on the Assertion itself; or it may be from Analogy or from the Negative proposition. If it is from analogy, it should be from a perfect similitude, for instance, if it is intended to assert that sound is non-eternal (the Reason by Analogy would be) ‘even as non-eternal as a jar.’ If it is from a negative Proposition it is as follows: ‘whatever is eternal is not made as the sky.’ To be made and to appear during a certain action, constitute a proper reason for non-eternity, according to Assertion, Analogy and the Negative Proposition. (219)

maṇimēkalai, according to Kanakasabhai (219), finds two classes of “example” – positive example such as non-eternity being concomitant with jars and the like and negative example which shows that the effect (what is predicated) does not exist where the cause (reason) does not. M. Hiriyanna (2013: 256) gives the typical Indian syllogistic text as gathered from the commentaries on Gautama’s Nyaya Sutras: “1. Yonder mountain has fire. 2. For it has smoke. 3. Whatever has smoke has fire, e.g. an oven. 4. Yonder mountain has smoke such as is invariably accompanied by fire. 5. Therefore, yonder mountain has fire.” The earliest Sanskrit commentaries on the sutras, which treat of formal logic among others, are assigned by Hiriyanna (226) to the fifth century A.D., and it should be remembered that tolkāppiyam, the Tamil work that first mentions kāṇṭikai (syllogistic text), predates the Nyaya Sutras by 600–1000 years. Contrasting the Indian syllogistic text with that of Aristotle, Hiriyanna says that the third member is the major premise but, unlike in the Aristotelian syllogism, is supported by an example. It can be argued that this major premise occurs as “warrant” in the jurisprudential model and the “example” is the “backing” for the warrant. Hiriyanna (257) explains how, in the Indian syllogistic text, the major premise and the example came to be clubbed. He says, This step in inference seems to have consisted originally of only the example. . .. The general statement was introduced later. That is, according to early Indian logicians, reasoning. . . was taken to be from particulars to particulars. In its present form the statement implies that it was realized in course of time that reasoning proceeds from particulars to particulars through the universal. This innovation is now commonly ascribed to the Buddhist logician Dinnaga.

This fact helps Hiriyanna (257) to characterize the reasoning found in the apparently deductive syllogistic text, as “inductive-deductive.” The fourth step in the syllogistic text above, he argues, clubs the major and the minor premises and thus brings all the three terms together in the same proposition, making the formulation of the conclusion simple (Hiriyanna 257). In the context of the attribution to the Buddhist logician, Dinnaga, it ought to be mentioned here that Selvamony finds Jainism and

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Buddhism and such other philosophical traditions in Tamil alone giving the place that duly belongs to logic in their disquisitions. In support of this contention, he states that even though tolkāppiyam mentioned the members of kāṇṭikai, it dealt with it briefly and only the Buddhist work, maṇimēkalai, discussed it elaborately (Selvamony 1996: 20). Citing Vidyabhusana, Selvamony mentions that carakar, who wrote Charaka Samhita in 78 A.D., was the first Sanskrit scholar to speak of kāṇṭikai (1996: 42), and the others who did after him were Akshapada who called it avayava in 150 A.D. in his Nyaya Sutras (49), Nagarjuna who appeared around 150 years later and used the same designation for kāṇṭikai (258), and the logician Vasubandhu of the fifth century A.D. who referred to it as panchavayava and employed another form of kāṇṭikai with just two terms: cūttiram and ētu (Selvamony 1996: 269, 109). Ram Chandra Bose (1986: 213, 214) lists the members of the avayava or the syllogistic text as found in the Nyaya Sutras: proposition, reason, example, application, and conclusion. He finds the syllogistic text to be “more complex than that of Aristotle, being, as Monier Williams says, a compound of the enthymeme and the syllogism.” He quotes two examples that are commonly given as illustrations for the syllogistic text: (1) The hill is fiery (proposition). For it smokes (reason). Whatever smokes is fiery, as a culinary hearth (example). The hill is smoking (application). Therefore it is fiery (nikamaṉa or conclusion). (2) Sound is non-eternal (proposition). Because it is produced (reason). Whatever is produced is non-eternal, as pots (example). Sound is produced (application). Therefore it is non-eternal (conclusion). It should be noted that naṭai, the fourth term found in kāṇṭikai, is designated here as “application.” kāṇṭikai seeks to arrive at a conclusion through an appeal to an “example” that has the nature of a general truth. If “example” is taken to correspond to “warrant” in the jurisprudential model, the acceptability of the warrant, which is implicit or explicit in the “example,” as Toulmin (120) says, “is taken for granted.” Didactic texts in Tamil are mostly seen to be, to put it in the words of Toulmin (122), “warrant-using” arguments. They are not regarded as “warrant-establishing” arguments such as the ones found in the sciences. That is, they are not seen as inductive ones. The account of kāṇṭikai which Selvamony (1996: 109) finds in maṇimēkalai is a nuanced one, as it clearly disengages “example” from the “warrant” embedded in it. In the kāṇṭikai of maṇimēkalai, the premise “that which has smoke has fire” is not a step toward conclusion but is the conclusion itself. It would be legitimate to hold that maṇimēkalai speaks of a warrant-establishing form of argument, different from the warrant-using one. maṇimēkalai does not merely elaborate the kāṇṭikai structure by providing details, it presents the warrantestablishing kāṇṭikai as yet unknown in Tamil didactic texts. It would be useful to cite two verses from tirukkuṟaḷ in support of the contention that the didactic texts, in fact, employ both the warrant-using and warrantestablishing arguments. The first verse of tirukkuṟaḷ concludes that this world has ātipakavaṉ for its source or beginning. The warrant it uses to reach this conclusion is that whichever exists cannot but have a source. Thus it is a warrant-using verse. Verse 247 on charity is “The other world is not for those lacking in charity/Even as this world is not for those lacking in wealth” (Trans. vaṉmīkanātaṉ 2007). Using the

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terms in maṇimēkalai, the structure of this argument can be traced; thus, the pleasures of the other world are not for the uncharitable (claim or pakkam). Pleasures of either world are attainable but by proper means (reason or ētu). The pleasures of this world are not enjoyed by those wanting wealth (example). For the pleasures of the other world, charity proves to be the proper means (upanayam). The charitable alone attain the pleasures of the other world (nikamaṉam or warrant). Thus it is a warrant-establishing verse. This chapter will later have occasion to discuss the link between warrant and example as a pointer to an inductive argument of a kind in kāṇṭikai. The fourth member in the standard form of kāṇṭikai, naṭai or “application” as it is rendered in English, proves to be a problem area. The form in maṇimēkalai designates naṭai as upanayam. civañāṉapōtam also calls it upanayam. In the standard illustration, the proposition cited for naṭai is “This hill also has smoke.” This is followed by the conclusion, “Therefore this hill has fire” as the fifth member. In civañāṉapōtam the proposition for the third member, example, is “As even the pot is made by the potter,” and the proposition for naṭai/upanayam is “This universe is a created one, like the pot.” The third member speaks of a class of things, the created ones. The fourth member includes this world in that class of created things, and thus the passage to the final member, “This universe is a created one” becomes easy, and the conclusion in the final member has to follow necessarily. In contrast to this, “This hill also has smoke” as the proposition in the fourth member does not clearly get the smoking hill included in the class indicated by the third member, “Whichever has smoke has fire.” A rather crudely framed naṭai such as “This hill is also subject to that law,” the law being that whichever has smoke is bound to have fire, would bring the hill in question into the class of hills indicated in the third member and make the conclusion as a necessarily following one. Unless it is so done, the passage from the fourth member to the conclusion remains a problem. It would be difficult to argue, with the propositions as they are, that “This hill also has smoke” should necessarily lead to the conclusion, “This hill has fire.” The propositions in civañāṉapōtam for the third, fourth, and also the fifth (the conclusion) ensure a smooth passage from one to the other and help one get a better understanding of the logical function of naṭai/upanayam. Verse 247 of tirukkuṟaḷ discussed above can be referred to again to elucidate this observation. The substance of the kuṟaḷ in the kāṇṭikai form, when recast, would be: (1) Certain people cannot have the pleasures of the other world (claim). (2) They are not charitable (reason). (3) The charitable inherit the other world, as people with means have the pleasures of this world (eṭuttukkāṭṭu). (4) The uncharitable and the charitable alike are subject to this dispensation (naṭai). (5) Therefore, the uncharitable cannot have the other world (conclusion). What follows from X or Y being or not being charitable is sought to be established by the verse. The function of the fourth term, naṭai, is to have the uncharitable excluded from the class of people spoken of by the third proposition. Without this exclusion or as in the case of other verses, an inclusion, the conclusion would not necessarily follow. In the kāṇṭikai form, the fourth term has an importance equal to that of the third one. It is interesting to note that in contemporary Tamil, the word naṭai refers to the rather narrow passage in a traditional Tamil house which links the front part of the house with the hall inside.

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Validity of the ka¯ntikai Form of Argument ˙˙ Having discussed the nature of kāṇṭikai, let us now turn to its validity. About the validity of different forms of arguments, Toulmin states that it “can be judged by standards appropriate within that field.” This means, he argues further, that it will not be legitimate to expect necessary conclusions in some fields and only presumptive conclusions in others. In this context, he refers to R.G. Collingwood’s doctrine of absolute presuppositions. Even as the standards of validity are “field-dependent,” so are the structures of argument (Toulmin 256). The objective of inquiry should be to “study the ways of arguing which have established themselves in any sphere, accepting them as historical facts” (Toulmin 257). This stand would lead one, Toulmin says, into the history of ideas. “We can look with new sympathy on Collingwood’s vision of philosophy as a study of the methods of argument which at any historical moment have served as the ultimate Court of Appeal. . .” (Toulmin 258). Of what avail would an inquiry into the ancient forms of argument be if that is to yield only the absolute presuppositions? John E. Llewelyn (1961: 42) would argue that we are not to see whether absolute presuppositions are true or false but to find out “what absolute presuppositions have been made by this or that person or group of persons on this or that occasion in the course of this thinking or that thinking.” To focus on the absolute presuppositions of kāṇṭikai, we may see, successively, how the arguments of didactic verses get structured when they are cast in the kāṇṭikai and jurisprudential forms. The structures will have to show how one proceeds from the initial step to that which follows it and finally to the conclusion. Arguments from analogy involve intermediate steps that may not have been explicitly stated in the verses. Implied steps in the structure, when made explicit, may inevitably appear redundant, especially in enthymemes. Selvamony (1996: 112) quotes a verse from paḻamoḻi to show that latent propositions are a convention in Tamil didactic literature. We may also have to supply the missing or hidden presupposition, which when supplied, as Paul Bartha (2013) argues, turns the argument from example (paradeigma) into a syllogistic argument that is deductively analyzable (“Analogy and Analogical Reasoning”). It has earlier been seen that “example” is not mere rhetoric but is a logical step enabling argument to move from particulars to the universal and that the example is, on this account, invariably appended to the universal premise. Hiriyanna names this universal premise “general statement” (256) and claims that the prefixing of this to the example represents an evolutionary stage in the Indian syllogistic text. It ought to be observed that the general statement and the example following it had always been occurring together in the kāṇṭikai of tolkāppiyam and maṇimēkalai and also in the later works in Tamil devotional literature as it has been seen earlier in this chapter. The syllogistic text found in Sanskrit texts could not have been an independent development. This strengthens Selvamony’s claim of the Tamil provenance of kāṇṭikai, later known as avayava in the Sanskrit texts. kāṇṭikai in Tamil didactic texts shows that the reasoning by analogy is a way of arguing from particulars to the universal. What earlier began as example appended to a universal premise in kāṇṭikai later became elaborated into analogy and even into multiple analogies in a single verse providing an analogical reasoning. This peculiar

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aspect in the kāṇṭikai in Tamil didactic literature renders it inductive-deductive, a characteristic which Hiriyanna (257), as it has been seen earlier, claims for the syllogistic text in Sanskrit texts. Tamil didactic literature consists of 18 books, which are assigned by S. Ramakrishnan (2012), a historian of Tamil literature, to the period between 300 and 600 A.D. He finds tirukkuṟaḷ, nālaṭiyār, paḻamoḻi nāṉūṟu, mutumoḻik kāñci, and iṉṉā nāṟpatu as belonging to this period and tirikaṭukam, ēlāti, iṉiyavai nāṟpatu, and ācārakkōvai to a later period. He argues that tirukkuṟaḷ, nālaṭiyār, and paḻamoḻi nāṉūṟu were authored by the followers of Jainism, which is also the persuasion of writers earlier to him (Ramakrishnan). maṇimēkalai attributes its account of kāṇṭikai to āticinendra (Kanakasapai 218). Selvamony states that kāṇṭikai, which, in the period of tolkāppiyam, was generally used in disputations, later, when Buddhism and Jainism were in the ascendency, came to be mostly used for establishing the principles of those religions and the moral tenets (1996: 112). The progress from disputation to justificatory arguments with its attendant earnestness of persuasive purpose should obviously have perfected the kāṇṭikai form. It would therefore be legitimate to examine the Tamil didactic literature for recovering the structure of argument called kāṇṭikai and its variant forms, such as are discoverable in individual didactic verses. It is desirable to begin with the simplest and the most lucid of the Tamil didactic texts, ātticūṭi, a work which children traditionally learn as one of their initial lessons in language. This work consisting of 108 one-line verses is attributed to avvaiyār, a poetess held to be of the age of the poet, kampaṉ, variously assigned to the period from ninth to twelfth century A.D. The first verse in this work is “Be desirous of doing charity.” The latent argument in this verse can be put in the kāṇṭikai form as follows: (1) One should desire to do charity (mēṟkōḷ or claim). (2) There are things that one should desire to do and things that one should eschew (ētu or reason). (3) Charity is one among the desirable things or the world speaks of its desirability (eṭuttukkāṭṭu or example). (4) Everyone is under this dispensation (naṭai or application). (5) Therefore, one should desire to do charity (muṭipu or conclusion). The third member of this kāṇṭikai, that charity is one among the desirable objects of life, is a general statement, a warrant in the jurisprudential model, an absolute presupposition according to Collingwood. The acceptability of this presupposition is not questioned by the society. The second verse in ātticūṭi is “That which is to be contained by one is one’s anger.” The argument in kāṇṭikai form would be: (1) What is to be contained is one’s anger (claim). (2) There are things that one should care to contain (reason). (3) One’s anger is one among those one should care to contain; the world finds it as something to be contained (example). (4) Everyone is under this dispensation (application). (5) Therefore, a thing to be contained is one’s anger (conclusion). In both the verses, the third term is akin to the universal proposition, the acceptability of which the society does not scrutinize. If there is an example such as an oven that attests to the concomitance of smoke and fire, it would add force to the argument. That is how it functions as a backing to the warrant. These two verses are arguments that make use of warrants or general statements that have been taken

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for granted. A warrant-using argument can be said to be of the deductive type, and this is the predominant form of argument in Tamil didactic literature. Verse 22 of koṉṟai vēntaṉ, a work, which is also attributed to the poetess avvaiyār, is “A truer wealth than the one on one’s hand is education.” The kāṇṭikai form of this may work out to the following: (1) Education is a truer wealth than the material wealth that one has ready at hand (claim). (2) The standards to judge a thing’s worth are not its being visible and ready to hand (reason). (3) Apparent wealth is not true wealth, so says the discriminating world (example). (4) Everyone is under this dispensation (application). (5) Therefore, one should cherish education rather than material wealth (conclusion). Verse 29 of koṉṟai vēntaṉ is “One who seeks prosperity should seek to do farming.” In the kāṇṭikai form, it would be: (1) Those seeking prosperity should adopt farming (claim). (2) One adopts the sure means for his object (reason). (3) Prosperity and farming go together; it is so seen in the world (example). (4) It should be so for anyone (application). (5) Farming will ensure one’s prosperity (conclusion). Verse 70 of koṉṟai vēntaṉ is “The harm one does to another in the forenoon visits him in the afternoon.” This can be expanded in the kāṇṭikai form as: (1) One doing harm will soon suffer harm as its consequence (claim). (2) Every action has its own consequence (reason). (3) As per the law of consequence, one reaps what one sows (example). (4) Everyone is subject to this law (application). (5) Harm one does unfailingly visits the doer (conclusion). In these three verses of koṉṟai vēntaṉ, the unstated conclusions would respectively be that “one should prefer education to wealth,” “that farming never fails the farmer,” and that “one should avoid harming others.” A member of the kāṇṭikai remaining unstated or one or two members being redundant is not unusual to this form of argument. These are implied steps that legitimize the passage to the step following it in the argument. For instance, it would not be possible to reach the conclusion that “the harm one does unfailingly visits the doer” from the first member unless one adopts the premise in the third member, “as per the law of consequence, one reaps what one sows.” The argument is based on this “absolute presupposition.” To designate the “warrant” in the argument as “presupposition” is not to detract from its validity. It serves the purpose of putting this argument outside the pale of the empirical arguments. Like the third member, the second was also unstated in Tamil didactic literature. Selvamony discusses a truncated form of kāṇṭikai mentioned in tolkāppiyam – a form where the ētu or reason remains latent. This form was known as mutumoḻi. This can safely be taken as the way in which kāṇṭikai formalizes, on its own terms, the unstated member or the implied step in the argument. In verse 70 of paḻamoḻi nāṉūṟu, which he cites and explicates as an illustration, he finds both the ētu (reason) and the conclusion to be latent (Selvamony 1996: 111). He refers to a seventeenth-century work called ilakkaṇa viḷakkam which seeks to describe mutumoḻi, and he also refers to the commentary on this description by Gopala Iyer to establish that mutumoḻi was a form of kāṇṭikai. Interestingly, the commentary lists the members of kāṇṭikai as four and illustrates them. As this illustration and the list of members are a variation on the normal form of kāṇṭikai, it is worth mentioning them in full: “Claim: That there is fire in this hill. eṭuttukkāṭṭu or

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example: That it is as the oven. upanayam or explaining with reason: That where there is smoke there is fire. nikamaṉam or conclusion: Inferring that as there is smoke in this hill there certainly is fire” (Selvamony 1996: 118). In this description of Gopala Iyer, “reason” does not figure as a member, “example” reduces to pointing out a likeness, and upanayam, which in the normal form was seen to correspond to “application,” occupies, with its prefixed general statement, the place normally occupied by “example.” Selvamony refers to Vasubandhu of the fifth century A.D. (one among the authors who discussed kāṇṭikai in their Sanskrit texts written from the beginning of the first century A.D.), who uses a two-member kāṇṭikai, consisting of claim (cūttiram) and reason (ētu). An illustration given to describe these two members is “Sound is not eternal,” “As it is one that is made” (Selvamony 1996: 109). For passing from the premise to the conclusion of this two-member kāṇṭikai, a middle term such as “Anything that is made or created has to have an end” is needed. It can be surmised that as this middle term was a presupposition of that age, it was left unarticulated. mūturai is another didactic work attributed to avvaiyār. The second verse of this text is “The help done to the good one/Will persist as inscription on stone/But the help given to the unkind/Will be similar to that written on water.” This verse, which uses two positive analogies, is based on analogical reasoning. This would assume the kāṇṭikai form as: (1) The good remember the help they receive and the unkind will soon forget it (claim). (2) One’s nature always asserts itself (reason). (3) The good are grateful even as a lasting stone inscription (example). (4) People are like either stone or water (application). (5) The help done to the good does not go to waste (conclusion). “Milk, even when boiled does not lose its taste/those of unfriendly nature will never be friendly/Conch, even when burnt, becomes whiter than before/ The noble, even when they decline in prosperity, will continue to be noble” is the third verse of mūturai. The kāṇṭikai form of argument would cast it as: (1) Neither adversity will make the noble decline in their nobility nor the good of friendliness will make the unfriendly friendly (claim). (2) People act according to their nature (reason). (3) Character, not circumstances, decides one’s conduct, as in the case of milk when heated or conch when burnt (example). (4) The noble behave as milk and conch do (application). (5) Nobility persists even in adversity (conclusion). The construction of these analogies with contrasting strands, as they are in the sonata form, is a consciously practiced convention where logic merges with rhetoric. A well-known verse in the Tamil speaking world, the 14th of mūturai, is about the pretentious scholar who is analogized as a turkey cock imitating the dancing peacock: “Turkey cock that saw the peacock of the forest dancing/Imagined itself to be the peacock and/Spread its ugly feathers and danced/The poetic composition of the ignorant would be similar to that.” If put in the kāṇṭikai form, the argument would run as follows: (1) The ignorant imitating a scholar is ridiculous (claim). (2) The imitation will not compare with the original (reason). (3) Pretensions invite ridicule, as in the case of a turkey cock imitating the dancing peacock (example). (4) The pretentious do as the turkey cock does (application). (5) The pretentious ignorant will end up being ridiculed.

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naṟuntokai authored by ativīrarāma pāṇṭiyar of a period between twelfth and fifteenth century A.D. has the following one-line verse for its 30th aphorism: “Renown and infamy are of one’s own making.” Its kāṇṭikai form would be: (1) One’s deeds are the cause of one’s renown or infamy (claim). (2) One is known by one’s deeds (reason). (3) One’s deeds are certain to have their consequences (example). (4) Everyone is subject to this law (application). (5) It avails nothing to blame others for what one is responsible for (conclusion). A work by tuṟaimaṅkalam civappirakāca cuvāmikaḷ, a great scholar in Tamil known to have lived in the seventeenth century A.D., is naṉṉeṟi. The 18th verse of this didactic composition is about the efficacy of pleasant words: “By pleasant words alone this world, surrounded by the vast sea/Becomes happy, and never by harsh words. Listen, you wearing jingling bangles made of gold, /The sea does not swell by the hot rays of the sun/But heaves to the cool light of the moon.” The verse would assume the following kāṇṭikai form: (1) Pleasant words alone, not harsh ones, make people happy (claim). (2) People do not respond to harsh words (reason). (3) Pleasant words never fail, and harsh words never succeed, as even the sea does not swell by the hot rays of the sun but heaves to the cool light of the moon (example). (4) Pleasant words work like the cool light of the moon (application). (5) Pleasant words alone have the desired effect (conclusion). In verses using analogies, the third term of their kāṇṭikai is fully taken up by the example expanding or becoming multiple analogies. Unlike in the kāṇṭikai of the ancient times, the general statement in the elaborately analogizing later period verses, which this chapter identifies as the warrant or presupposition, remains unarticulated or left hidden. The verses are not arguments intended to establish a claim but warrant-using exhortations. The 400 verses of paḻamoḻi nāṉūṟu and another 400 verses of nālaṭiyār, which mostly adopt the analogical argument, were composed by saintly Jaina poets. Paul Bartha sees analogical argument to be related to both the Aristotelian “argument from example” and “argument from likenesses.” A quote from Aristotle’s rhetoric that Bartha gives says, “Enthymemes based upon example are those which proceed from one or more similar cases, arrive at a general proposition and then argue deductively to a particular inference.” Trying to show how argument from example (paradeigma) yields to analysis as a purely deductive form, Bartha says that the general proposition is the intermediate step and that it is to be regarded as a hidden presupposition. “This transforms paradeigma into a syllogistic argument with a missing or enthymematic premise. . ..” (Bartha). Analogies, in the opinion of Bartha, play a heuristic role aiding discovery, a justificatory role establishing plausibility, and are also pedagogically useful helping analogical cognition (Bartha). Verses from mūturai and naṉṉeṟi, which this chapter has quoted and discussed earlier, have each a latent general proposition which more or less answers to what Bartha calls “the hidden presupposition.” It would be worth seeing how this argument receives support from the analogical arguments of verses in paḻamoḻi nāṉūṟu and nālaṭiyār. In paḻamoḻi nāṉūṟu, the verses end with a proverb as a deft finale. These proverbs function as analogies or multiple analogies providing or indicating one of the terms of the argument. The 60th verse of paḻamoḻi nāṉūṟu is “Would it be proper to think that one who does not learn while he is young would

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cherish learning when he becomes old? There scarcely is a boatman who has recovered the fare after the passenger has left the boat. Nor is there anyone who has recovered the toll after the merchant is past the tollgate.” The argument in the kāṇṭikai form would be: (1) One can hardly learn past his youth (claim). (2) It is easy to bear hardship while still young (reason). (3) One’s youth is the time for learning – the boatman can hardly recover the fare from the passenger after the latter leaves the boat (example). (4) That one would care for learning past one’s youth is as impossible as recovering the toll and the fare (application). (5) Therefore, one should learn while he is still young (conclusion). These multiple analogies are proverbs that were already in existence, and the verse makes use of them as backing for the warrant, the third term of its argument. Verse 189 is “If one getting ridiculed by the ignorant gets angry with the person ridiculing he would be the mouse that burns itself by dragging the lighted wick of the oil lamp.” The argument would be: (1) He who is enraged by one ridiculing him is harming himself (claim). (2) Such anger would be a cause for more misery (reason). (3) Some provocations are best ignored – a mouse bent on dragging the burning wick is harming itself (example). (4) Getting provoked is similar to the doing of the mouse (application). (5) One should ignore ridicule by the ignorant (conclusion). Verse 173 of nālaṭi nāṉūṟu runs as follows: “The benefits of the evildoer’s wealth/The virtuous would never desire/Listen, ruler of the shores of roaring waves, /Ill-gotten wealth, though appearing to swell, will soon vanish.” The argument may be laid out as follows: (1) Evildoer’s wealth would fetch no real benefit (claim). (2) The prosperity it brings is apparent (reason). (3) Evil source breeds but evil; the virtuous shun such wealth (example). (4) Ill-gotten wealth is an evil source (application). (5) Therefore, one should not desire the benefits arising from evildoer’s wealth (conclusion). The verse uses no analogy. Nor does it append, as do most other verses, an illustration following the kāṇṭikai practice. Except for the address, the verse is a simple justificatory argument for the claim that the benefits of ill-gotten wealth are no benefits in the hands of the receiver. In a number of these arguments as presented here, one or two terms, especially “reason” (ētu), may appear redundant or would even be the cause for a certain circularity of argument. The attempt in the chapter is to work out the argument in terms of all the five members of a regular kāṇṭikai. It has earlier been seen that there was also a truncated form of the kāṇṭikai, and a four-member form was also described in Gopala Iyer’s commentary on ilakkaṇa viḷakkam cited by Selvamony. Discussing what he calls “Indian syllogism,” Hiriyanna argues “In a purely logical syllogism – unmixed with rhetorical appurtenances – it is admitted that the first two or the last two of the five members (avayava) may be dropped” (Hiriyanna 256). Verse 203 of nālaṭi nāṉūṟu is “Listen, ruler of the folds of mountains/The noble do not shake off their poor dependents/Even when bunches of fruits appear in succession/There is no branch that is tired of bearing them.” Its argument will assume the following kāṇṭikai form: (1) The noble will never say no to their dependent poor (claim). (2) Such is their nature (reason). (3) Noble persons’ charity never gets spent – none has seen a branch that is tired of sustaining its successive loads of fruits (example). (4) The rich are like the branches of trees (application). (5) Nobility lasts (conclusion).

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That the evil one has done is certain to visit him even in successive births is the theme of verse 101 of nālaṭi nāṉūṟu: “Even if left clueless amidst a herd of cows, the young calf is capable of seeking out its mother. In seeking out the doer, one’s sins of the earlier births are like the calf.” Put in the kāṇṭikai form, the argument would run as follows: (1) Sin does not cease with its doer ceasing to be (claim). (2) It attaches to the continuing soul (reason). (3) Whatever attaches to its begetter’s soul continues with him – the young calf seeks out its mother even if left clueless in a herd of cows (example). (4) Sin is like the young calf (application). (5) One cannot escape the consequences of one’s sins (conclusion). mutumoḻik kāñci consists of rich one-line verses having arguments that can be distributed among the five members of a kāṇṭikai, though these verses apparently are mere assertions involving no argument. The third decade speaks of things that the world refrains from ridiculing. The ninth in this series of ten is “The world does not ridicule the poor for not being liberal.” In kāṇṭikai structure it would be: (1) The discriminating will not ridicule the poor for not being liberal (claim). (2) There is that which is to be ridiculed and that which is not to be ridiculed (reason). (3) The discriminating do not ridicule inability in one (example). (4) The poor are among those with this inability (application). (5) Therefore, the world does not ridicule the poor for not being liberal (conclusion). The fifth series is about the greatest of virtues and the worst of vices. The fifth verse of this series is “There is no greater vice than pretending that one cannot afford to give what one can.” The kāṇṭikai structure of this would be: (1) Stealthily withholding what one can afford to give is the greatest vice (claim). (2) There are greater and lesser vices (reason). (3) Stealth compounds the vice of meanness (example). (4) Those who are mean thus compound their vice (application). (5) Therefore, there is no greater vice than stealthy meanness (conclusion). In all these verses, the identification or correspondence of the middle term (the general statement) with the warrant in the jurisprudential model has already been pointed out. It would be a mistake to take the verses as seeking to prove the claim they set up. Nor do the verses need to do that. This chapter has, following Toulmin, already questioned the presumption and applicability of universal standards of validity to arguments irrespective of the field of argument. kāṇṭikai is a form of argument and, with its analogies or examples, an inductive-deductive form that has its own standards of validity. In the analogical arguments of the verses, what is more interesting to follow is the way the analogies have been constructed to render a conclusion, implied or clearly articulated, plausible. A claim (kōḷ/mēṟkōḷ) in the kāṇṭikai is an assertion which plausibly, though not necessarily, follows from a factum probandum which the kāṇṭikai states through an analogy. That “the noble never say no to their dependent poor,” for instance, is the assertion (claim), which is invariably found to be the conclusion reworded. It follows from the third term of the kāṇṭikai, which forms the factum probandum and is a general statement or is a general statement followed by analogy. In most of the verses, the general statement is unarticulated, and its function is left to be performed by a deftly constructed analogy or analogies. It has been seen in this chapter that even a one-line aphoristic verse yields to being laid out in the five-member kāṇṭikai form. There had been a resilient logical tradition, continuing into the seventeenth century, of which kāṇṭikai could be the

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major form of argument, though in varying forms. For example, alternative technical terms like pakkam for kōḷ and mēṟkōḷ show that there had been varying trends or schools or perceptions within the Tamil logical tradition. In contemporary Tamil, the word pakkam continues to be used to refer to one of a set of rival contentions or claims – a fact that attests not only to the resilience of kāṇṭikai in its remnants but also to its having been popularly adopted.

Definition of Key Terms kāṇṭikai

kōḷ/mēṟkōḷ /pakkam ētu eṭuttukkāṭṭu

naṭai (upanayam)

muṭipu (nikamaṉam) Claim Datum

Warrant Backing Probative value Factum probandum Absolute presuppositions Exponent

Enthymeme

An ancient Tamil syllogistic form of argument, predating tolkāppiyam and having five members, usually translated as claim, reason, example, application, and conclusion. The first of the five members of the Tamil syllogistic form of argument, equivalent to “claim.” The second of the five members in the Tamil syllogistic form of argument, equivalent to “reason.” The third of the five members in the Tamil syllogistic form of argument, equivalent to a general statement followed by an example, usually translated “example.” The fourth of the five members in the Tamil syllogistic form of argument, which helps the argument to move from the general statement in the third member to the “conclusion.” The fifth member of the Tamil syllogistic form of argument, equivalent to “conclusion.” The second member in the jurisprudential form of argument arising from a fact or set of facts. The first member of the jurisprudential form of argument which is a fact or a set of facts giving rise to a claim. A general assumption which validates the claim in an argument. Anything that strengthens the warrant. The value that a fact has as proof advanced for the claim. The fact at issue that an argument seeks to prove. Collingwood’s term to refer to the basic beliefs of a society or period which it takes for granted. The concrete form in words that an abstract member (in a form of argument) such as “claim” or “warrant” or naṭai or eṭuttukkāṭṭu takes in an actual argument. An argument in which one or more propositions are left unarticulated.

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Summary Points • The ancient Tamil society was using a form of argument, which it generally called kāṇṭikai, from the times preceding tolkāppiyam in disputations. kāṇṭikai was used in the primal society (tiṇai) itself in arguments. • This form of argument, expressed as a syllogistic text, has five members – claim; reason for the claim; example, which consists of a general presumption and an illustration; application, which enables the passage from the example to the conclusion following it; and the conclusion. • Variants of this form of argument and also of its members, including a truncated form, are attested by texts of later periods, and it is also seen to have later migrated into Sanskrit texts on logic with lexical markers bearing witness to its Tamil provenance. • There has recently been at least one attempt, made by Nirmal Selvamony, to recover and reconstruct this form of argument. • Texts of Tamil didactic literature till about seventeenth century were constructing the arguments in its texts, adopting this Tamil form of argument as a template even though this logical tradition came to be used and was known as a hermeneutic tool. • The form and members of this syllogistic tradition in Tamil can be better understood and its members defined more sharply when they are studied in the light of jurisprudential method of argument. • Two members of the reconstructed kāṇṭikai, “example” and “application,” get problematized when the variants of the form are studied together. • The issue of validity of this form of argument can be resolved by a reference to Collingwood’s doctrine of absolute presuppositions. • When the texts of Tamil didactic literature are analyzed using this logical tradition, the analytical tool itself gets defined, and the logical presuppositions of the Tamil society are thrown into relief. • The application of this form of argument for studying the texts of Tamil didactic literature can thus be a contribution to attempts at recovering a nearly lost logical tradition of the Tamil society.

References Bartha, Paul. 2013. Analogy and analogical reasoning (3.2. “Aristotle’s Theory”). In The Stanford encyclopedia of philosophy (Fall 2013 edition). http://plato.stanford.edu/archives/fall2013/ entries/reasoning-analogy/ Bose, Ram Chandra. 1986. Hindu philosophy popularly explained – The orthodox systems. New Delhi: Asian Educational Services. Hiriyanna, M. 2013. Outlines of Indian philosophy. New Delhi: Surjeet Publications. Kanakasabhai, V. 2000. The Tamils 1800 years ago. tirunelvēli: The tirunelvēli South India Saiva Siddhantha Works Publishing Society Ltd., tirunelvēli.

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Llewelyn, John E. 1961. Collingwood’s doctrine of absolute presuppositions. The Philosophical Quarterly 11(42): 49–60. https://www.jstor.org/stable/2216679?seq¼1#page_scan_tab_ contents Ramakrishnan, S. 2012. tamiḻ ilakkiya varalāṟu (History of Tamil Literature). Chennai: New Century Book House (P) Ltd. Selvamony, Nirmal. 1990. The syllogistic circle in tolkāppiyam. 65th Session of the Indian Philosophical Congress. Madurai-Kamaraj University, Madurai, 27–29 Dec. Unpublished Paper. ———. 1996. tamiḻk kāṭci ṉeṟiyiyal (Methodology of tamiḻ philosophy). Chennai: International Institute of Tamil Studies. ———. 2000. The syllogistic circle in tolkāppiyam. Journal of Tamil Studies 57 & 58: 117–134. Toulmin, Stephen. 1964. The uses of argument. London: Cambridge University Press. vaṉmīkanātaṉ, kō. Trans. 2007. tirukkuṟaḷ uraikkottu (Collection of Commentaries of tirukkuṟaḷ with English translation of tirukkuṟaḷ verses). Ed. tā. ma. veḷḷaivāraṇaṉ. tiruppaṉantāḷ: Sri kācimaṭam. (The English translation. of all Tamil verses, except those from tirukkuṟaḷ, are by Thanga Jayaraman, the author of this essay)

The Logic of Late Nyāya: Problems and Issues

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Eberhard Guhe

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Operations on Properties and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Negation of a Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjunction of Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disjunction of Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sheffer Stroke Applied to Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conversion of a Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Composition of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Left- and Right-Restriction of a Relational Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theorems Related to Operations on Properties and Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Identities Concerning Iterated Absences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . De Morgan’s Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Quasi-Fregean Account of the Reference of Number Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Gaṅgeśa’s siddhāntalakṣaṇa-Definition of Pervasion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . An Analysis by Means of Quantifiers and Property Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toshihiro Wada’s Graphic Representations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

540 540 540 541 542 542 543 544 544 549 549 553 554 557 557 562 566 566

Abstract

The purpose of the present chapter is to demonstrate the utility of the propertytheoretic framework for a formal reconstruction of Navya-Nyāya logic which was introduced in Guhe 2016 (Comput Sci J Moldova 24/3 (72): 312–334). Some pertinent examples have been selected, namely the Navya-Naiyāyikas’ operations applied to properties and relations, their discovery of theorems related to these operations, their account of the reference of number words, and the definition of

E. Guhe (*) Department of Philosophy, Fudan University, Shanghai, China e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_33

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the concept of pervasion. Several alternative methods of formalization in secondary sources are noted and critically examined. Keywords

Indian logic · Navya-Nyāya · Intensionality · Property theories · Negation · Natural numbers · Definition of pervasion

Introduction A logical framework for interpreting Navya-Nyāya logic has been presented in Guhe 2016. Earlier and less elaborate versions of this approach are contained in Guhe 2008, 2014, and 2015. Its utility will be demonstrated here by giving an account of several operations on properties and relations in Navya-Nyāya. Some of them are also used by Bealer in order to explain how the denotation of a complex term [A]α can be determined from the denotation(s) of the relevant syntactically simpler term(s) (cf. Bealer 1982: 46f).

Operations on Properties and Relations Negation of a Property The two types of “absence” in Navya-Nyāya are explained in detail in Guhe 2016. Following Bealer, who names property operators after their corresponding first-order operators, both types of absence can be regarded as the result of “negating” a property term: “. . ., what is the most obvious relation between [Fx]x and [:Fx]x? As before, the second is the negation of the first.” (Bealer 1982: 47)

Mutual Absence The term [:FX]X may serve as a formal representation of the “mutual absence” (anyonyābhāva) of, i.e., of the “difference” (bheda) from an F (more accurately: from anything which is an F). The indefinite article has been added here in front of F in order to facilitate a smooth English translation. In Sanskrit there is no article. A phrase like “the mutual absence of a cloth” is commonly expressed by means of a compound (paṭānyonyābhāva) and the literal meaning would be “cloth-mutualabsence.” Similarly, “the relational absence of a pot” would be rendered as a compound which literally translates into “pot-relational-absence” (ghaṭasaṃsargābhāva). When asked to specify the absentees, the so-called “counterpositives” (pratiyogin) of these absences, a Navya-Naiyāyika might say “cloth” (paṭa) and “pot” (ghaṭa), where “cloth” and “pot” are usually meant in the sense of expressions which refer to any cloth or any pot, respectively. In order to emphasize that no reference to one particular cloth or pot is intended here, a Navya-Naiyāyika might say that in these cases the counterpositiveness is “limited” (avacchinna) by clothness or potness, respectively. On the other hand, by adding a demonstrative like “this”

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(etad) in front of “cloth” or “pot” he might indicate that these expressions are meant in the sense of singular terms. One can regard [:FX]x as a shorthand version of the following formalization which duly mirrors the fact that Navya-Naiyāyikas conceive of a mutual absence, i.e., of a difference, as a denial of an identity between the absentee and the locus of the absence: ð†Þ½:∃yðFy ^ x ¼ yÞx

Relational Absence The “relational absence” (saṃsargābhāva) of an F (more accurately: of anything which is an F) can be construed as a property which characterizes something as being devoid of (or: no locus of) anything which is an F. Sometimes it is just called “absence” (abhāva) or “absolute absence” (atyantābhāva). This property will be formalized here by means of the dyadic predicate L with the intended meaning “. . . is a locus of . . ..” L is supposed to be a more general occurrence relation than the Δ-relation in the sense that xLy might also be true if y is an urelement. Thus, one can infer xLy from xΔy, but not vice versa. Using the L-relation for the purpose of formalizing a relational absence is appropriate in cases where there is no specification of the occurrence relation which fails to subsist between the absentee and the locus of the absence. Thus, an unspecified relational absence can be formalized in the following way: ð‡Þ½:∃yðFy ^ xLyÞx If the relational absence is more specifically meant in the sense of a relational absence via contact or inherence, etc., one can replace L by the corresponding symbols for these relations (such as C or I ). Since the present formalization of relational absences has basically the same syntactic structure as a mutual absence, namely [:ϕ(x)]x, where ϕ(x):$∃y(Fy ^ xLy), one can also regard a relational absence as a negation, i.e., as the negation of the property “being a locus of an F” ([∃y(Fy ^ xLy)]x). Possessing a relational absence of an F means to be different from a locus of an F. So, a relational absence turns out to be a special case of a mutual absence. Both can be regarded as negations. Navya-Naiyāyikas see the essential difference between the two types of absence in the relation by which the absentee fails to reside in the locus of the absence. In the case of mutual absence, this relation is identity. In the case of relational absence, it is some kind of occurrence relation. This distinctive feature is duly mirrored in the present formalizations (†) and (‡), because they differ only with respect to the relations (L and =).

Conjunction of Properties Bealer introduces this operation in the following way: “. . ., what is the most obvious relation among the properties [Fx]x, [Gx]x and [Fx & Gx]x? As before, the third is the conjunction of the first two.” (Bealer 1982: 47)

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Navya-Naiyāyikas apply the same kind of operation when they talk about a pair of mutual absences (bhedadvaya). (Cf. Ingalls 1951: 63) There is even an expression for a generalized conjunction in Navya-Nyaya, namely kūṭa (“multitude”). It refers to a conjunction of two or more than two property terms (cf. Ingalls 1951: 65f and the reference to abhāvakūṭa in section “De Morgan’s Laws” below).

Disjunction of Properties This operation is called anyataratva (“the being either the one or the other”) in Navya-Nyāya. Ingalls notes the following example: vahnau parvatanirūpitavṛttitvajalahradanirūpitavṛttitvānyataratva – “There is in fire either occurrentness described by mountain or occurrentness described by lake.” (Ingalls 1951: 63) An explanation of the meaning of such relational expressions has been given in Guhe 2016: The fire as the first argument of the relation “x occurs in y” is here specified as the possessor or locus of the corresponding relational abstract, i.e., occurrentness, whereas the second argument functions as the “describer.” In this case, the describer is the disjunction (anyataratva) of two properties, namely “occurrentness described by mountain” and “occurrentness described by lake.” By means of the predicate symbols . . . Mx: “x is a mountain” and Lx: “x is a lake” . . . one can express these properties as [∃y(My ^ xL1y)]x and [∃y(Ly ^ xL1y)]x, respectively. The disjunction of both properties resides in a fire f iff f occurs on a mountain or on a lake (cf. Ingalls 1951: 63), i.e., . . . 



 fL ∃y My ^ xL1 y _ ∃y Ly ^ xL1 y x $ ∃y My ^ fL1 y _ ∃y Ly ^ fL1 y : Navya-Naiyāyikas also define the propositional operator “one or the other” (anyatara). (Cf. section “De Morgan’s Laws” below) The definition clearly indicates that “one or the other” is to be understood in the sense of the inclusive “or.”

Sheffer Stroke Applied to Properties In Mathurānātha Tarkavāgīśa’s Vyāptipañcakarahasyam (quoted in Ingalls 1951: 64f), this operation is named “conjoint absence” (ubhayābhāva). Maheśa Chandra characterizes it as “an absence due to prefixing ‘being both’”:. . .paṭaghaṭobhayatvarūpeṇa vobhayatvapuraskāreṇābhāvo . . . (BN: 14, 5f = Guhe 2014: 76) – “. . . or an absence due to prefixing ‘being both’ in the form of ‘being both, [i.e.,] cloth and pot’ . . .” Following Bealer, who names property operators after their corresponding firstorder operators, one can regard the Sheffer stroke applied to properties as the

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negation of the conjunction of properties. An example of such a property is the absence of both, cloth and pot, in a house where there is a cloth, but no pot. evaṃ gṛhe kevalasya paṭasya sattve ’pi ghaṭasyābhāvena paṭaghaṭobhayasyāpy abhāvo ’sty eva. ekābhāvenobhayābhāvasyāvas´yaṃbhāvitvād . . . (BN: 14, 8f = Guhe 2014: 76) – “So, when there is only a cloth in the house, there is absence of both, a cloth and a pot , because of the absence of a pot, because the absence of both is necessary on account of the absence of one.” The condition that there is a cloth but no pot in the house can be formalized as . . . ∃y(Cy ^ hLy) ^ :∃z(Pz ^ hLz) (where Cx is to be read as “x is a cloth,” Px as “x is a pot,” xLy as “x is a locus of y” and h as “the house”). Now, if there is a cloth, but no pot in the house, then it is not the case that there is a cloth and a pot in the house. This can be rendered as an implication with an alternative formalization of the consequent by means of the Sheffer stroke: ∃yðCy ^ hLyÞ ^ :∃zðPz ^ hLzÞ ! :ð∃yðPy ^ hLyÞ ^ ∃zðCz ^ hLzÞÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} ∃yðPy^hLyÞ"∃zðCz^hLzÞ

Since h denotes an urelement, one can apply (C) and substitute the consequent by an equivalent formula which expresses the fact that the house is a locus of the negation of the conjunction of the properties [∃y(Cy ^ xLy)]x (“being a locus of a cloth”) and [∃z(Pz ^ xLz)]x (“being a locus of a pot”): hΔ½∃yðCy ^ xLyÞ " ∃zðPz ^ xLzÞx The negation of each of the properties [∃y(Cy^xLy)]x and [∃z(Pz^xLz)]x yields the term for the corresponding absence, i.e., the absence of a cloth and the absence of a pot, respectively. Therefore, it makes sense to regard the negation of their conjunction, i.e., [∃y(Cy ^ xLy) " ∃z(Pz ^ xLz)]x, as a formal equivalent of the “conjoint absence” (ubhayābhāva) of a cloth and a pot.

Conversion of a Relation The converse of a relation term can be formally expressed by switching the order of the variables in the subscript. So, [xRy]yx is the converse of [xRy]xy (cf. Bealer 1982: 47). One can use an alternative formalization of the converse by means of the relation symbol R1, namely: [xR1y]xy Maheśa Chandra gives several examples (cf. BN: 12, 28f and 13, 21f = Guhe 2014: 72f): Applying the operation of conversion to locusness (adhikaraṇatā), substratumness (ādhāratā) and causeness (kāraṇatā), e.g., yields the converses occurrentness (vṛttitva), superstratumness (ādheyatā), and effectness (kāryatā). If one uses xLy to formalize “x is a locus of y,” then occurrentness, i.e., the converse of locusness ([xLy]xy), can be represented as [xL1y]xy.

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Composition of Relations This operation is a combination of Bealer’s conjunction operation and another one which he calls “existential generalization”: “[∃xFx] is the existential generalization of [Fx]x.” (Bealer 1982: 47) If R1  X  Y and R2  Y  Z, then the composition of R1 and R2 (with R1 as the first and R2 as the second member) is a relation R2 ○ R1  X  Z (“R2 after R1”) such that xR2 ○ R1z iffdf ∃y(xR1y ^ yR2z). The result of composing the corresponding relational abstracts [xR1y]xy and [xR2y]xy (in that order) is [xR2 ○ R1y]xy, the relational abstract corresponding to “R2 after R1.” Maheśa Chandra gives several examples of such composite relations, which he calls “indirect relations” (paraṃparāsaṃbandha). The following is, e.g., an instance of an application of this new operation which consists in composing the relation “being a substrate of” with itself: Suppose that a man stands in a house (i.e., the house is a substrate of the man) and he has tufted hair (i.e., he is a substrate of tufted hair). An indirect relation subsisting between the house and the tufted hair is then svās´rayapuruṣās´rayatva (BN: 11, 10 = Guhe 2014: 68) – “being a substrate of the man as its own substrate.” The relation itself without reference to any intermediate relatum (such as the man in the present case) is called svās´rayās´rayatva (cf. BN: 10, 4 = Guhe 2014: 65) – “being a substrate of one’s own substrate.” If one denotes the first-order relation “x is a substrate of y” by xRy, one may formalize the composition of R with itself as [xR ○ Ry]xy. Maheśa Chandra mentions also several types of non-composite, so-called “direct relations” (sākṣātsaṃbandha), such as contact, inherence, peculiar relation (cf. BN: 9, 17f = Guhe 2014: 63f) and paryāpti (cf. BN: 12, 3f = Guhe 2014: 71). In contrast to contact (saṃyoga) inherence (samavāya) integrates entities that cannot occur separately. It is a relation whereby (i) universals reside in substances, qualities, and actions, (ii) qualities and actions reside in substances, (iii) composite substances reside in their parts, and (iv) the so-called “ultimate particularities” (antyavis´eṣa), which account for the irreducible identity and distinctness of their substrates, reside in the individual atoms, souls, and mental organs, and in the unitary substances ether, space and time (cf. Frauwallner 1956: 246f). Peculiar relation (svarūpasaṃbandha) is the relation between (i) an imposed property or relational abstract and its loci and (ii) an absence and its locus. paryāpti subsists between the referent of a word for a nonzero natural number n and every group of n individuals (cf. also the broad description of these types of relations in Ingalls 1951: 74f).

Left- and Right-Restriction of a Relational Abstract

+

)

These operations are counterparts of the restriction of a binary relation R in set theory: A R = df {< x, y > |x  A ^ xRy} is the so-called “domain restriction” or “left-restriction of R to A.” R B = df {< x, y > |xRy ^y  B} is the so-called “range restriction” or “right-restriction of R to B” (cf. Borkowski 1976: 247).

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+

)

In Navya-Nyāya, similar restricting operations are applied to relational abstracts (cf. sections “Describers” and “Limitors”). Thus, [A(x) ^ xRy]xy or [x{x|A(x)} Ry]xy ([xRy ^B(y)]xy or [xR {x|B(x)}y]xy) may be regarded as the left-restriction (rightrestriction) of the relational abstract [xRy]xy to x’s (y’s) in the bracketed part which are A (B).

“Describers” By labeling something as a “describer” of a relational abstract [xRy]xy, the NavyaNaiyāyikas sometimes specify that it functions as an element of the range of the relation R. Since they had no ordered pair definition like the one by WienerKuratowski, they had to use another strategy to indicate that an expression like . . . (i) Δ[xRy]xy . . . is meant in the sense of aRb instead of bRa. They seem to “reformulate” the relational abstract as a kind of dependent property such that the locus of the property is the first member in < a, b >: adhikaraṇatāvṛttitāprabhṛtayo dharmāḥ sāpekṣāḥ. kimapy adhikaraṇaṃ vinā kutra vṛttitā bhaved ādheyaṃ ca vinā kasyādhikaraṇatā bhavet. tatas´ ca vṛttitādhikaraṇam apekṣate ’dhikaraṇatā cādheyam iti. (BN: 13, 21f = Guhe 2014: 73) – “Locusness, occurrentness etc. are dependent properties. Where should occurrentness be without some locus? And whose locusness should it be without a superstratum? And therefore [it has been said]: ‘Occurrentness depends on a locus and locusness depends on a superstratum.’ ” Actually, there is no real reformulation involved here. In Sanskrit, the property which derives from vṛttitā (“occurrentness”) is still called vṛttitā. But according to the preceding quotation, “occurrentness” is at least conceived of as a property when the NavyaNaiyāyikas try to express that a pair of individuals is an instance of a relational abstract. The dependent character of this property is owing to the fact that the second member of the relation from which the property derives is still an unspecified “something” and one might indicate this in the formalization by means of an existential quantifier: (ii) aΔ[∃y(xRy)]x If xRy is to be understood in the sense of “x occurs in y,” then (ii) can be read as “a is a locus of occurrentness in something.” Now, the second member of the ordered pair is added as a “describer” (nirūpaka), something which specifies the relation or restricts it in its range: yac ca yad apekṣate tat tannirūpitaṃ bhavati. ghaṭasya vṛttitā kutra. bhūtale ’nyatra veti sandeho bhūtale ghaṭasya vṛttitety uttareṇa nivāryate bhūtala ity anena hi vṛttitā kīdṛs´ī ti nirūpyate (vis´eṣato ’vadhāryate). ato bhūtalam adhikaraṇaṃ ghaṭasya vṛttitānirūpakam iti. etādṛs´yaiva yuktyādhikaraṇatādheyanirūpitā pratyetavyā. (BN: 13, 23f = Guhe 2014: 73) – “What depends on some x that is described by x. The doubt ‘Occurrentness of the pot with respect to what? With respect to the ground or something else’ is dispelled by the answer ‘Occurrentness of the pot with respect to the ground’, because by this [expression] ‘with respect to the ground’ is described ([or] specifically determined) of what kind the occurrentness is. Therefore

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the ground as the locus of the pot is the describer of the occurrentness. By such a rationale the locusness is perceived as being described by a superstratum.” “Describing” a relation is generally to be understood as a right-restriction of a relational abstract, and in order to indicate the order of the members of a pair instantiating a relational abstract, the latter is transformed into a property whose locus is the first relatum, while the second relatum functions as a describer: (iii) aΔ[∃y(xR {b}y)]x (“a is a locus of R-ness described by b.”) +

If a is an urelement or a set-like property, this is equivalent to ∃y(aR {b}y) according to axiom (C) and the latter can be proved to be equivalent to aRb. Likewise, (i) is equivalent to aRb on account of (C) if a and b are urelements or set-like properties. So, (i) and (iii) are equivalent and (iii) can be said to offer a solution to the problem to specify the order of the relata in an expression which states that a relational abstract applies to a pair of individuals. In (iii) the describer consists of only one individual. However, when the NavyaNaiyāyikas analyze a sentence like . . . (iv) “The mountain is a locus of fire,” . . . they might want to refer to fire in general as the describer of locusness resident in mountain. According to Ingalls the describer in this example can be said to be “fire” or “fireness”: “Where . . . (1) A mountain is a locus of fire . . . the Naiyāyika may analyze the situation by saying . . . (2) The locus1 (author’s note: This is Ingalls’s abbreviation for ‘locusness’) resident in mountain is described (nirūpita) by fire; he may also say that the locus1 is described by fire-ness (vahnitva-nirūpita). Fire or fireness may here be called the describer (nirūpaka) of the locus1 in mountain.” (Ingalls 1951: 46) It seems that the Navya-Naiyāyikas would rather use the expression “fireness” when they want to refer explicitly to fire in general as the describer instead of to a specific fire, i.e., the meaning of . . . (v) “The mountain possesses locusness described by fireness” . . . can roughly be rendered by means of the existential formula . . . (vi) ∃x(Fx ^ mLx) (where m translates into “the mountain,” Fx into “x is a fire” and xLy into “x is a locus of y”) . . . and the following should be a closer formal equivalent to (v) (presupposing the same symbolization key as before): (vii) mΔ[∃y(xLy ^ Fy)]x

“Limitors” A left-restriction is indicated by the word niṣṭha (“resident in”) in Navya-Nyāya: tatas´ ca kuṇḍe badaram iti vākyasya kuṇḍanirūpitavṛttitāvad badaram iti kiṃvā

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kuṇḍaniṣṭhādhikaraṇatānirūpitādheyatāvad badaram ity arthaḥ pratī yate. evaṃ badaravat kuṇḍam ity asya prayogasya kuṇḍaṃ badaranirūpitādhikaraṇatāvad iti kiṃvā kuṇḍaṃ badaraniṣṭhavṛttitānirūpitādhikaraṇatāvad ity arthaḥ paryavasī yate. (BN: 13, 8f = Guhe 2014: 74) – “And therefore the meaning of the sentence ‘There is dried ginger in the pot’ is cognized as ‘Dried ginger has an occurrentness described by pot’ or ‘Dried ginger has a superstratumness described by a locusness resident in pot’. So, the meaning of this usage, [namely] ‘The pot possesses dried ginger’, is included in ‘The pot has locusness described by dried ginger’ or ‘The pot has locusness described by an occurrentness resident in dried ginger.’ ” By means of the constant symbols . . . p: “the pot” and d: “dried ginger” . . . the sentence . . . (viii) “The pot has locusness described by an occurrentness resident in dried ginger.” . . . at the end of the preceding quotation can be formalized in the following way: )

(ix) pΔ[∃y(xLy ^ ∃z(y{d} L1z))]x

)

“Locushood” (more accurately: “being a locus of something”) is here a property ([∃y(xLy)]x) which derives from the relational abstract “locushood” ([xLy]xy). The describer “occurrentness resident in dried ginger” also derives from a relational abstract, namely the left-restricted “occurrentness resident in dried ginger,” i.e., “dried ginger’s occurrentness” ([y{d} L1z]yz), which is transformed into a property, namely “dried ginger’s occurrentness in something” ([∃z(y{d} L 1z)]y). This is the describer whereby the relation locushood is right-restricted to dried ginger as occurring in something, i.e., to y’s such that ∃z(y{d} L1z). An alternative expression which is frequently used in Navya-Nyāya to express a left-restriction of a relational abstract is “limited” (avacchinna) and the (abstract) noun which refers to the left-restricting individual (or property) is known as a “limitor” (avacchedaka). Actually, the expressions “limited” and “limitor” can also be used in a more general sense: avacchedakapadasya kvacid vis´eṣaṇam ity arthaḥ. avacchinnapadasya ca vis´iṣṭam ās´raya ity artho ’pi bhavati. (BN: 18, 22f = Guhe 2014: 89) – “In some cases the meaning of the word ‘limitor’ is ‘qualifier’. The meaning of the word ‘limited’ is ‘something qualified’ [or] ‘substrate’.” Only in the narrow sense of the word “limitor” (or “limited”) refers to a left-restriction. It is important to note that the relational abstract “counterpositiveness” (i.e., being the absentee of an absence) is associated with a twofold limiting operation in NavyaNyāya, one which specifies the instances of the absentee as loci of a property (functioning as a limitor) and another one which further specifies them as something which fails to reside in the locus of the absence by the limiting relation: tac )

)

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cābhāvasya vailakṣaṇyaṃ pratiyogitāvailakṣaṇyanibandhanam iti tad evocyate. abhāvasya pratiyogitā kenacit saṃbandhena kenacid dharmeṇāvacchinnā bhavati. yena saṃbandhena yena vā dharmeṇāvacchinnā bhavati sa saṃbandhaḥ sa ca dharmas tasyāḥ pratiyogitāyā avacchedako bhavati. yathā bhūtale samavāyasaṃbandhena ghaṭo nāstī ty ādau ghaṭābhāvasya pratiyogitā samavāyena saṃbandhena ghaṭatvena ca dharmeṇāvacchinnā. tatas´ ca tasyāḥ pratiyogitāyāḥ samavāyaḥ saṃbandho ghaṭatvaṃ ca dharmo ’vacchedakaḥ. ata eva samavāyena ghaṭo nāstīty asya samavāyasaṃbandhāvacchinnaghaṭatvāvacchinnapratiyogitāko ’bhāvo vartata ity arthaḥ. (BN: 14, 11f = Guhe 2014: 77) – “And this distinguishing feature of an absence is said to be this, [namely] ‘dependence on the distinguishing feature of the counterpositiveness’. The counterpositiveness of an absence is limited by a certain relation and by a certain property. That relation and that property are a limitor of this counterpositiveness by which it is limited. In ‘There is no pot on the ground by the relation inherence’ etc., e.g., the counterpositiveness of the absence of pot is limited by the relation inherence and by the property potness. Hence, the relation inherence and the property potness are a limitor of this counterpositiveness. Therefore the meaning of this, [namely] ‘There is no pot by the relation inherence’, is: ‘There is an absence which possesses a counterpositiveness limited by potness and by the relation inherence.’ ” In Maheśa Chandra’s example . . . (x) “There is an absence on the ground which possesses a counterpositiveness limited by potness and by the relation inherence,” . . . which can be understood as a statement of an absence of pots inhering in the ground, the counterpositiveness is limited by potness and inherence. It should be noted that the limiting relation of a counterpositiveness is the relation by which every absentee (= every first member of an ordered pair to which the counterpositiveness applies) fails to reside in the locus of the absence, whereas in the case of other relations than counterpositiveness the limiting relation conjoins the limiting individual or the instances of the limiting property and their respective loci. Since pots inhere only in their parts, but not in the ground, the absence under consideration is an absence of all pots. Even if there is a pot on the ground, the latter is a locus of an absence whose counterpositiveness is limited by potness and inherence. However, the ground is no locus of an absence whose counterpositiveness is limited by potness and contact if there is a pot on the ground. By means of the symbolization key . . . g: “the ground” Nx: “x is a relational absence” Px: “x is a pot” xPy: “x is an instance of the counterpositive of y” xIy: “x inheres in y”

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. . . (x) can be formalized as:

   (xi) ∃y Ny ^ gΔy ^ 8z zPy ! :zIg ^ xPy x ¼ ½Pxx One has to recall here the special sense in which a counterpositiveness is regarded as limited by a property in Navya-Nyāya: “If pot-ness is said to be a limitor of a counterpositive-ness, then such a counterpositive-ness is sometimes asserted to be indistinguishable from pot-ness.” (Matilal 1968: 76) Therefore, [x P y]x can be regarded as identical to [Px]x.

Theorems Related to Operations on Properties and Relations Identities Concerning Iterated Absences Guhe 2008 (147f) contains a proof of the following identity concerning iterated absences, which is endorsed by Mathurānātha (cf. Ingalls 1951: 71 and Matilal 2 1990: 152f): (Id) The relational absence (saṃsargābhāva) of the difference (bheda) from a pot is identical to potness. According to section “Mutual Absence,” the difference from a pot can be represented as . . . [:Px]x, where Px translates into “x is a pot.” In order to obtain a formal representation of the relational absence of the difference from a pot, one might specify Fy in (‡) (cf. section “Relational Absence”) as . . . y ¼ ½:Pxx : Then the relational absence of the difference from a pot (ghaṭabhedābhāva) can be expressed as [:∃y(y = [:Px]x ^ xΔy)]x (“being no locus of anything which is identical to the difference from a pot”). Since the relational absence of the difference from a pot is supposed to be identical to potness, both properties should be equi-locatable. Potness resides only in element-like individuals, i.e., [Px]x = [Ex ^ Px]x. If the difference from a pot were hyper-class-like, it would reside in all properly class-like properties, since [:Px]x = [:Ex _ :Px]x. However, since T1+ does not warrant the existence of a hyper-class-like property of all properly class-like properties, the difference from a pot can only reside in element-like individuals, i.e., [:Px]x = [Ex ^ :Px]x.

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If the relational absence of the difference from a pot were hyper-class-like, it would reside in all properly class-like properties, since [:∃y(y = [:Px]x ^ xΔy)]x = [:∃y(y = [Ex ^ :Px]x ^ xΔy)]x = [x Δ [Ex ^ :Px]x]x. However, since T1+ does not warrant the existence of a hyper-class-like property of all properly class-like properties, the relational absence of the difference from a pot can also only reside in element-like individuals, i.e., [:∃y(y = [:Px]x ^ xΔy)]x = [Ex ^ :∃y (y = [:Px]x ^ xΔy)]x. Thus, (Id) can be rendered as the following T1+ proposition and one can prove it in T1+: Theorem (Id) [Ex ^ :∃y(y = [:Px]x ^ xΔy)]x = [Ex ^ Px]x

The proof contains an application of the following instantiation of (C): 8x(Ex ! (xΔ[:Px]x $ :Px)) Proof of (Id): (A1) (C) (1st-order logic) (R2, R3) (A8, R1)

Ex ^ ::Px $ Ex ^ Px Ex ^ :xΔ[:Px]x $ Ex ^ Px Ex ^ :∃y(y = [:Px]x ^ xΔy) $ Ex ^ Px □8x(Ex ^ :∃y(y = [:Px]x ^ xΔy) $ Ex ^ Px) [Ex ^ :∃y(y = [:Px]x ^ xΔy)]x = [Ex ^ Px]x ▪

Maheśa Chandra states two other identities concerning iterated absences, namely the following reduction rules, which are referred to as (Id0 ) and (Id00 ) below: tathāhi dvitī yābhāvaḥ (ghaṭābhāvābhāvaḥ) pratiyogi(ghaṭa)svarūpas tṛtī yābhāvaḥ (ghaṭābhāvābhāvābhāvaḥ) prathamābhāva(ghaṭābhāva)svarūpa iti prathamābhāvasya (ghaṭābhāvasya) ghaṭa iva dvitī yābhāvo ’pi (ghaṭābhāvābhāvo ’pi) pratiyogī . (BN: 15, 27f = Guhe 2014: 81) – “So, the second absence (the absence of the absence of pot) is essentially identical to the counterpositive (pot). The third absence (the absence of the absence of the absence of pot) is essentially identical to the first absence (the absence of pot). So, the second absence (the absence of the absence of pot) is like ‘pot’ of the first absence (the absence of pot) a counterpositive (author’s note: The ‘second absence’ ghaṭābhāvābhāva is the counterpositive of the ‘third absence’ ghaṭābhāvābhāvābhāva.).” (Id0 ) The relational absence of the relational absence of a pot is identical to “pot.” (Id00 ) The relational absence of the relational absence of the relational absence of a pot is identical to the relational absence of a pot. In order to explicate the right side of (Id0 ) in an appropriate way, one might substitute “pot” (ghaṭa) by “being a locus of a pot” (ghaṭavattva), since this is common practice in Navya-Nyāya (cf. Matilal 21990: 115). After all, the property

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“being a locus of a pot” is equi-locatable with every pot. Even though the NavyaNaiyāyikas do regard expressions like ghaṭa and ghaṭavattva as interchangeable, this is not unproblematic, because a pot possesses potness, whereas the property “being a locus of a pot” does not. Since the relational absence of the relational absence of a pot is supposed to be identical to the property “being a locus of a pot,” both properties should be equilocatable. The property “being a locus of a pot” resides only in element-like individuals, i.e., [∃y(Py ^ xLy)]x = [Ex ^ ∃y(Py ^ xLy)]x. If the relational absence of a pot were hyper-class-like, it would reside in all properly class-like properties, since [:∃y(Py ^ xLy)]x = [:Ex _ :∃y(Py ^ xLy)]x. However, since T1+ does not warrant the existence of a hyper-class-like property of all properly class-like properties, the relational absence of a pot can only reside in element-like individuals, i.e., [:∃y(Py ^ xLy)]x = [Ex ^ :∃y(Py ^ xLy)]x. If the relational absence of the relational absence of a pot were hyper-class-like, it would reside in all properly class-like properties, since [:∃z(z = [:∃y(Py ^ xLy)]x ^ xΔz)]x = [:∃z(z = [Ex ^ :∃y(Py ^ xLy)]x ^ xΔz)]x = [x Δ [Ex ^ :∃y(Py ^ xLy)]x]x. However, since T1+ does not warrant the existence of a hyper-class-like property of all properly class-like properties, the relational absence of the relational absence of a pot can also only reside in element-like individuals, i.e., [:∃z(z = [:∃y (Py ^ xLy)]x ^ xΔz)]x = [Ex ^ :∃z(z = [:∃y(Py ^ xLy)]x ^ xΔz)]x. Thus, (Id0 ) can be rendered as the following T1+ proposition and one can prove it in T1+: Theorem (Id0 )
  Ex ^ :∃z z ¼ ½:∃yðPy ^ xLyÞx ^ xΔz x ¼ ½Ex ^ ∃yðPy ^ xLyÞx The proof contains an application of the following instantiation of (C):

 8x Ex ! xΔ½:∃yðPy ^ xLyÞx $ :∃yðPy ^ xLyÞ Proof of (Id0 ) (A1) (C) (1st-order logic) (R2, R3) (A8, R1)

Ex ^ ::∃y(Py ^ xLy) $ Ex ^ ∃y(Py ^ xLy) Ex ^ :xΔ[:∃y(Py ^ xLy)]x $ Ex ^ ∃y(Py ^ xLy) Ex ^ :∃z(z = [:∃y(Py ^ xLy)]x ^ xΔz) $ Ex ^ ∃y(Py ^ xLy) □8x(Ex ^ :∃z(z = [:∃y(Py ^ xLy)]x ^ xΔz) $ Ex ^ ∃y(Py ^ xLy)) [Ex ^ :∃z(z = [:∃y(Py ^ xLy)]x ^ xΔz)]x = [Ex ^ ∃y(Py ^ xLy)]x ▪

As noted by Matilal, the Navya-Naiyāyika Raghunātha Śiromaṇi rejected this identity. “Raghunātha, however, in his intensionalist vein, argued against the identification of x with ~ ~ x (author’s note: ‘~’ is Matilal’s abbreviation of ‘relational absence’). For, he thought, the notion of negation conveyed by the second can never be conveyed by the first, and hence it is difficult to think of them as non-distinct.” (Matilal

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1980: 5) For the same reason, Raghunātha would also not have been in favor of (Id), which was like (Id0 ) generally accepted in Navya-Nyāya (cf. Matilal 1980: 7). Raghunātha’s intuitions concerning properties are closer to Bealer’s system T2 (cf. Bealer 1982: 64f). In T2 axiom A8 of T1 is replaced by A8: [A]α = [B]α ! (A $ B). Moreover, T2 contains an axiom which coincides with Raghunātha’s argument against (Id0 ), namely A9: t 6¼ r (where t and r are non-elementary complex terms of different syntactic kinds). According to the present formalization techniques, the left side of (Id0 ) is the conjunction of [Ex]x with the negation of a property, whereas the right side should be interpreted as the conjunction of [Ex]x with a non-negated property. Hence, they belong to different syntactic categories and therefore A9 leaves no choice but to reject (Id0 ). In order to prove (Id00 ) in T1+ and any other reduction rule which states the identity of an uneven number of such relational absences to a single relational absence, it suffices to prove: (Id*) The relational absence of the property “being a locus of a pot” is identical to the relational absence of a pot. By adding one relational absence on both sides of (Id0 ), one can infer from (Id0 ) that the relational absence of the relational absence of the relational absence of a pot is identical to the relational absence of pot (where the pot in italics is supposed to be explicated in the sense of “the property ‘being a locus of a pot’”). On account of (Id*) the relational absence of “pot,” i.e., of the property “being a locus of a pot,” is identical to the relational absence of a pot, and this proves (Id00 ). Theorem (Id*): 
 :∃z z ¼ ½∃yðPy ^ xLyÞx ^ xΔz x ¼ ½:∃yðPy ^ xLyÞx The proof contains an application of the following instantiation of (C):

 8x Ex ! xΔ½∃yðPy ^ xLyÞx $ ∃yðPy ^ xLyÞ It is plausible to assume that neither of the members of the equivalence xΔ[∃y(Py ^xLy)]x $ ∃y(Py ^xLy) in this formula is true of any x which fulfills the condition :Ex, i.e., 8x(:Ex ! :∃y(Py ^ xLy) ^ :xΔ[∃y(Py ^ xLy)]x). (“Individuals which are neither set-like properties nor urelements are different from loci of pots and do not possess the property to be loci of pots.”) Hence, the equivalence xΔ[∃y(Py ^ xLy)]x $ ∃y(Py ^ xLy) can be applied unconditionally in this case. Proof of (Id*) (A1) (C) (1st-order logic)

:∃y(Py ^ xLy) $ :∃y(Py ^ xLy) :xΔ[∃y(Py ^ xLy)]x $ :∃y(Py ^ xLy) :∃z(z = [∃y(Py ^ xLy)]x ^ xΔz) $ :∃y(Py ^ xLy) (continued)

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(R2, R3) (A8, R1)

□8x(:∃z(z = [∃y(Py ^ xLy)]x ^ xΔz) $ :∃y(Py ^ xLy)) [:∃z(z = [∃y(Py ^ xLy)]x ^ xΔz)]x = [:∃y(Py ^ xLy)]x

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De Morgan’s Laws Ingalls refers to a definition of the propositional operator “one or the other” (anyatara) in Navya-Nyāya, which corresponds to one of de Morgan’s laws: If x is fire or water, then x “possesses the mutual absence to which the counterpositiveness is limited by a pair of mutual absences” (bhedadvayāvacchinnapratiyogitākabhedavat), namely, the mutual absence of fire and the mutual absence of water (cf. Ingalls 1951: 63). By means of the predicate symbols . . . Fx: “x is fire” and Wx: “x is water” . . . one can formalize these two absences as [:Fx]x and [:Wx]x, respectively. The limitor of the counterpositiveness of the mutual absence residing in x is the conjunction of both absences, i.e., [:Fx ^ :Wx]x. The counterpositive of the absence residing in x is therefore the complex predicate λx(:Fx ^ :Wx) (“neither F nor W”), so that the mutual absence residing in x can be represented as [:(:Fx ^ :Wx)]x. Hence, one can obtain the following definition of “one or the other” in the special case of the statement “x is fire or water”: Fx _ Wx :$ xL½:ð:Fx ^ :WxÞx The definiens can be rewritten as xΔ[:(:Fx ^ :Wx)]x, because the second argument of the L-relation is here a class-like property. Now, one can reasonably assume that 8x(:Ex ! :Fx ^ :Wx ^ :xΔ[:(:Fx ^ :Wx)]x). (“Individuals which are neither set-like properties nor urelements are different from fire, water and from loci of the difference from anything which is neither fire nor water.”) So, the definiens is equivalent to :(:Fx ^ :Wx) according to (C) and the definition proves to be a formulation of one of de Morgan’s laws in a special case. It should, however, be noted that in NKoś the definition is stated in a general form, i.e., anyatarat (“something which is the one or the other”) is defined as bhedadvayāvacchinnapratiyogitākabhedavat (“something which possesses the mutual absence to which the counterposiveness is limited by a pair of mutual absences”). (Cf. NKoś, s.v. anyatarat) So, it is really a law-like statement (similar to the tattvavat tad eva-rule), which one might appropriately formalize by means of schematic variables. The example of fire and water has just been chosen here in order to illustrate the definition. Ingalls mentions a different statement of this law by means of relational absences (cf. Ingalls 1951: 66): vis´eṣābhāvakūṭānāṃ sāmānyadharmāvacchinnapratiyogitākatvaṃ svī kriyate. (NKoś, s.v. abhāva) – “It is assumed that multitudes of specific absences have a counterpositiveness which is limited by a general property.” This version

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corresponds to the following propositional counterpart with generalized conjunctions and disjunctions: :A1 ^ . . . ^ :An $ :ðA1 _ . . . _ An Þ The Sanskrit original should, however, rather be understood as the statement of an identity of two property terms, namely the conjunction (kūṭa) of several relational absences a1, . . ., an and an absence whose counterpositiveness is limited by “a general property” (sāmānyadharma), i.e., the disjunction of the properties which limit the counterpositivenesses of the absences a1, . . ., an. This identity will be proved here for the special case of two relational absences, namely, [:∃y(Fy ^ xLy)]x and [:∃y(Gy ^ xLy)]x. Their conjunction is [:∃y(Fy ^ xLy) ^ :∃y(Gy ^ xLy)]x. The properties which limit the counterpositivenesses of the two absences are [Fx]x and [Gx]x, respectively. Their disjunction is [Fx _ Gx]x. If this limits the counterpositiveness of a relational absence, then the counterpositive of this absence is λx(Fx _ Gx) (“F or G”) and, hence, the absence is [:∃y((Fy _ Gy) ^ xLy)]x. So, the present formulation of one of de Morgan’s laws can be understood in the sense of the following identity: Theorem ½:∃yðFy ^ xLyÞ ^ :∃yðGy ^ xLyÞx ¼ ½:∃yððFy _ GyÞ ^ xLyÞx Proof (A1) (R2) (1st-order logic) (1st-order logic) (A1) (A1 = de Morgan) (R2, R3) (A8, R1)

(Fy _ Gy) ^ xLy $ Fy ^ xLy _ Gy ^ xLy 8y((Fy _ Gy) ^ xLy $ Fy ^ xLy _ Gy ^ xLy) ∃y((Fy _ Gy) ^ xLy) $ ∃y(Fy ^ xLy _ Gy ^ xLy) ∃y((Fy _ Gy) ^ xLy) $ ∃y(Fy ^ xLy) _ ∃y(Gy ^ xLy) :∃y((Fy _ Gy) ^ xLy) $ :(∃y(Fy ^ xLy) _ ∃y(Gy ^ xLy)) :∃y((Fy _ Gy) ^ xLy) $ :∃y(Fy ^ xLy) ^ :∃y(Gy ^ xLy) □8x(:∃y((Fy _ Gy) ^ xLy) $ :∃y(Fy ^ xLy) ^ :∃y(Gy ^ xLy)) [:∃y(Fy ^ xLy) ^ :∃y(Gy ^ xLy)]x = [:∃y((Fy _ Gy) ^ xLy)]x ▪

A Quasi-Fregean Account of the Reference of Number Words According to Maheśa Chandra, words for natural numbers refer to properties. So, “two” refers to twoness. There are two kinds of twoness: ayaṃ na dvau kiṃtu dvitvavān iti pratī ter dvau dvitvavān iti padayor arthavis´eṣāvadhāraṇaya navī naiḥ kas´cit paryāptināmakaḥ saṃbandhaḥ svī kriyate. paryāptiḥ paryavasānaṃ sākalyena saṃbandho ’rthād yasya yāvanta ās´rayāḥ santi tasya tāvatsv evās´rayeṣu militeṣv eva saṃbandhaḥ paryāptisaṃbandhena dvitvasaṃkhyā militayor eva

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dvayor vartate na tv ekaikasmin. evaṃ tritvasaṃkhyā militeṣv eva triṣu vartate na tv ekaikasmin dvayor vā. ata eva dvitvādayaḥ saṃkhyā vyāsajyavṛttaya (vyāsajya sarvam evādhāram adhikṛtya vartante) ity ucyante. evaṃ ca dvis´abdasya paryāptisaṃbandhena dvitvādhāratāpratī ter ekasya ca paryāptisaṃbandhena dvitvādhāratvābhāvād ayaṃ na dvāv iti pratī tir bhavati. samavāyasaṃbandhena punar dvitvasaṃkhyā dvayor ekaikasminn api tiṣṭhatī ti samavāyasaṃbandhena dvitvās´raya ity artham abhipretya dvitvavān iti prayogaḥ. tatas´ cāyaṃ na dvau kiṃtu dvitvavān iti vākyasyāyaṃ na paryāptisaṃbandhena dvitvavān kiṃtu samavāyasaṃbandhena dvitvavān ity arthaḥ. (BN: 12, 3f = Guhe 2014: 71) – “A certain relation called paryāpti is assumed by Navya-Naiyāyikas for the purpose of specifying the different meanings of two expressions, [namely] ‘two’ [and] ‘It possesses twoness’ [as part] of the cognition ‘This is not two, but it possesses twoness’. paryāpti is completion, a relation on account of thoroughness. The relation occurs in as many substrates of n collectively as there are on account of n’s meaning. By the paryāpti relation the number ‘twoness’ resides in 2 [things] collectively, not in each one. In the same way the number ‘threeness’ resides in 3 [things] collectively, not in each one or in two. Therefore the numbers ‘twoness’ etc. are said to have a joint occurrence. (They reside jointly with respect to all as a substrate.) And so, since for the word ‘two’ there is the cognition of substratumness of twoness via paryāpti relation and because of one [thing’s] absence of substratumness of twoness via paryāpti relation, there is the cognition ‘This is not two’. But since the number ‘twoness’ depends on each of two [things] via inherence relation, there is the usage ‘It has twoness’ with the intended meaning ‘It is the substrate of twoness via inherence relation’. And therefore the sentence ‘This is not two, but it possesses twoness’ has the meaning ‘This does not possess twoness via paryāpti relation, but it possesses twoness via inherence relation.’ ” The twoness which inheres in each of two things when someone regards them as a dyad corresponds to the old Vaiśeṣika understanding of numbers. According to Ingalls, “The ‘two-ness that inheres in each member of pairs’ corresponds to the Western ‘class of two members’.” (Ingalls 1951: 77) Although this comparison appears to be a little bit flawed, there is some truth in Ingalls’s claim that the twoness which resides by paryāpti in each dyad can be compared to Frege’s understanding of numbers: “This theory that numbers subsist by paryāpti in effect points out what Frege first pointed out in Europe in the nineteenth century.. . . The ‘two-ness that is related by paryāpti to the pairs and not to the members of the pairs’ corresponds to the Western ‘number two, the class of all classes of two members’.” (Ingalls 1951: 77) Since Maheśa Chandra regards the referent of “two” not as a class but as a property, it would be more appropriate to compare the twoness residing by paryāpti in each dyad to Bealer’s neo-Fregean analysis of natural numbers. Bealer interprets “12,” e.g., as “a property whose instances are all and only properties having 12 instances” (Bealer 1982: 124). If one regards “being an apostle” as instantiated by twelve individuals, then this property is one of the instances of “12”:

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being an apostle Simon Judas

being a month of the year

... January February

...

...

Bealer specifies his concept of natural number (including the number 0, which Maheśa Chandra does not refer to) by means of the following three definitions (cf. Bealer 1982: 121):



1. 0 = df [:∃u(uΔy)]y (0 = df the property of being a property with no instances.)    x uv 0 (the successor of 2. x ¼df ∃u uΔx ^ ∃v v Δu ^ y 5 ½wΔu _ w ¼ vw _

y

x = df the property of being a property with one more instance than the instances of x.) 3. NNx iffdf 8z(0Δz ^ 8y(yΔz ! y0 Δz) ! xΔz) (x is a natural number iffdf x is an instance of every property z such that 0 is an instance of z and the successor y0 of every instance y of z is also an instance of z.) Remarks • The twoness which resides by paryāpti “resides in 2 [things] collectively”: . . . militayor eva dvayor vartate . . . (BN: 12, 6f = Guhe 2014: 71) One may wonder what is to be understood by “collectively” (milita) here. If one wants to render Maheśa Chandra’s idea precise, one might do it in the same way as Bealer, who introduces properties as instances of the number n such that each of them has n instances. Alternatively, one might conceive of the entities which are instantiated by n individuals as classes of n elements. However, there is no reference to classes in the ontological framework of Navya-Nyāya. • Although Bealer’s use of the term “property” largely coincides with what Maheśa Chandra understands by a dharma, definition (1) seems to be an exception. Unlike the properties as instances of the property corresponding to 0 according to (1), a dharma should always have instances. But this is a minor problem, which does not affect the present formal reconstruction of Maheśa Chandra’s ideas, since he does not include the number 0 in his analysis of natural numbers. • In (2), upper index variables on the right side of a property term signify the free variables in the wff within square brackets which are not bound by any lower index variable of the property term. Moreover, y 5 z iffdf 8w(wΔy $ wΔz). _

It should be noted that Bealer defines a number n as a property whose instances are properties having n instances. So, (2) can be paraphrased in the following way: x0 is a property such that each of its instances is a property y which has as

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instances (i) the same instances as a property u which is an instance of x and (ii) an instance v which is not an instance of u. • Bealer’s definition of natural numbers (3) derives from a property version of Peano’s fifth postulate (cf. Bealer 1982: 121): ðP5Þ 8zð0Δz ^ 8xðxΔz ! x0 ΔzÞ ! 8xðNNx ! xΔzÞÞ (3) can be obtained from the implication NNx ! 8z(0Δz^8y (yΔz ! y0 Δz) ! xΔz), which is provable from (P5) by means of simple firstorder transformations: ðA2Þ 0Δz ^ 8xðxΔz ! x0 ΔzÞ ! 8xðNNx ! xΔzÞ ðA2, A1Þ 0Δz ^ 8xðxΔz ! x0 ΔzÞ ! ðNNx ! xΔzÞ ðA1Þ NNx ^ 0Δz ^ 8xðxΔz ! x0 ΔzÞ ! xΔz ðA1Þ NNx ! ð0Δz ^ 8xðxΔz ! x0 ΔzÞ ! xΔzÞ ðR2Þ 8zðNNx ! ð0Δz ^ 8xðxΔz ! x0 ΔzÞ ! xΔzÞÞ ðA3Þ NNx ! 8zð0Δz ^ 8xðxΔz ! x0 ΔzÞ ! xΔzÞ

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Since x  \{z | 0  z ^ 8x(x  z ! x0  z)} can be regarded as the set-theoretic equivalent of the consequent of the implication in the last line of this proof, it makes sense to turn the implication into an equivalence and use it as a definition of natural numbers.

Gan˙geśa’s siddhāntalaksana-Definition of Pervasion ˙ ˙ An Analysis by Means of Quantifiers and Property Terms Gaṅgeśa states his siddhāntalakṣaṇa-definition of pervasion in the following way: pratiyogyasamānādhikaraṇayatsamānādhikaraṇātyantābhāvapratiyogitāvacchedakāvacchinnaṃ yan na bhavati tena samaṃ tasya sāmānādhikaraṇyaṃ vyāptiḥ. (Goekoop 1967: 109) – “Pervasion is the co-occurrence of this (probans) with that (probandum) which is not limited by the limitor of the counterpositiveness of an absolute absence which co-occurs with that (probans) and which (= the absolute absence) does not co-occur with the counterpositive.” In order to understand this complex definition properly, it is important to note that tasya is here the correlative of yat (as part of the compound in the beginning), while tena is the correlative of yan. Moreover, what Gaṅgeśa means by “absolute absence” (atyantābhāva, sometimes also translated as “constant absence”) is not a new kind of

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absence. In Navya-Nyāya, the expression “absolute absence” (sometimes even without the prefixed atyanta) is used as a synonym of “relational absence” (saṃsargābhāva, cf. section “Relational Absence”): “Navya-nyaya logic uses two types of absence: (I) mutual absence (anyonyābhāva) or difference (bheda), by which the relation of identity is denied, and (II) constant absence (atyantābhāva or simply abhāva), by which a relation other than that of identity is denied.” (Goekoop 1967: 8) Further details will be explained below and then it should become clear that Gaṅgeśa’s formulation is an elaboration of the following core definition: (V0) H is pervaded by S iffdf H and S have common loci and H does not co-occur with the absence of S. One may wonder why Gaṅgeśa did not content himself with a much simpler formulation, such as the one suggested by Jinavardhana Sūri in his commentary on Śivādityamiśra’s Saptapadārthī: vyāptir yatra yatra sādhanaṃ tatra tatra sādhyam, yatra sādhyaṃ nāsti tatra sādhanam api nāstī ti lakṣanā. (SP: 69, 24) – “Pervasion is defined as: ‘Wherever there is the probans, there is the probandum. Wherever there is not the probandum, there is not the probans either’.” An obvious difference between Jinavardhana Sūri’s definition and Gaṅgeśa’s siddhāntalakṣaṇa is that in the latter the definiens does not contain the expressions “probans” (sādhana) and “probandum” (sādhya). Gaṅgeśa uses pronouns instead in order to safeguard his definition against the potential charge of circularity, because “probans” and “probandum” might be regarded as names of the relata of a pervasion relation. Moreover, according to Gaṅgeśa, the pervasion relation is associated with existential import, i.e., probans and probandum are supposed to co-occur somewhere. If one adds this information to Jinavardhana Sūri’s definition, one would not miss anything of the intended meaning of Gaṅgeśa’s definition, which can easily be expressed by means of the following first-order formula: (V1) ∃x(Hx ^ Sx) ^ 8x(Hx ! Sx), where Hx translates into “x is a locus of the probans” and Sx into “x is a locus of the probandum.” One might even simplify this expression by merely requiring the probans to occur somewhere, i.e., one might replace the first conjunct by ∃xHx, because ∃x(Hx ^ Sx) can be inferred from ∃xHx and 8x(Hx ! Sx). The most important reason for the kind of circumbendibus undertaken by Gaṅgeśa is that he wants to express the pervasion relation purely in terms of the elementary categories of Navya-Nyāya/Vaiśeṣika-ontology, which means, e.g., that he cannot identify pervasion with a proposition, i.e., the meaning of a declarative sentence. This is, actually, what the above-mentioned Jinavardhana Sūri is doing when he defines pervasion by means of an iti-clause. However, there are no propositions in the ontology presupposed in Navya-Nyāya. Ontologically speaking there is also no such referent of the formula (V1) from the perspective of the NavyaNaiyāyikas. Moreover, there is no direct ontological counterpart of quantifiers

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(“8” and “∃”) in Navya-Nyāya, so that Gaṅgeśa is forced to paraphrase quantifications in a clever way by means of property terms and relational abstracts. Occasionally, quantifications are merely implied in his definition. Gaṅgeśa can make use of relational abstracts such as “co-occurrence” (sāmānādhikaraṇya). Hence, he can allow himself to define pervasion as a special type of co-occurrence of probans and probandum. “Absences” (abhāva) as the ontological counterpart of negations (“:”) can be used to express the meaning of “x is not F” (“The mutual absence of F occurs in x”) or “F ’s do not occur in x” (“The relational absence of F occurs in x”). Gaṅgeśa exploits this device by specifying pervasion as a co-occurrence relation between probans and probandum, such that the probans does not co-occur with the absence of the probandum (cf. (V0)). There are, however, three problems involved in this preliminary definition and therefore Gaṅgeśa has to refine (V0) in certain respects: 1. Smoke is supposed to be pervaded by fire. Nevertheless, one might say that the smoke on a mountain, e.g., co-occurs with the absence of a certain fire such as the fire in a kitchen. The definition of pervasion should be safeguarded against this kind of mis-sorting of individuals (i.e., pairing, e.g., kitchen smoke with mountain fire), which is known as “sifting” (cālanī ya) in Navya-Nyāya (cf. Ingalls 1951: 60). 2. The probandum might be a universal property such as nameability (abhidheyatva). Everything is pervaded by nameability. Hence, the expression “absence of nameability” is non-denoting. Therefore, one cannot reasonably say that the probans does not co-occur with the absence of the probandum in this case. 3. Absences can occur non-pervasively in a locus. If contact with a monkey (kapisaṃyoga) occurs on a specific tree indexically referred to as “this tree” (etadvṛkṣa), there is a pervasion relation between “this-tree-ness” (as probans) and “contact with a monkey” (as probandum). In this case, the probans occurs also at the roots where there is no contact with a monkey, since the monkey is sitting on a branch of the tree. So, the probans might co-occur with the absence of the probandum if the absence is a non-pervasively occurrent property. Absence of contact with a monkey is a non-pervasively occurrent absence, because there is absence of contact with a monkey at the root of “this tree,” but not on the branch where the monkey is sitting. Ad 1) If one requires the probans of a pervasion relation not to co-occur with the absence of the probandum, it should be clear that the absentee includes all instances of the probandum. Normally, Navya-Naiyāyikas understand an expression of the form “absolute absence of F” in the sense of “absolute absence of anything which is F,” so that it seems to be sufficient to define pervasion as the co-occurrence of the probans with the probandum which is not the counterpositive of an absolute absence which co-occurs with the probans. However, as indicated by the existential quantification in (V1), the condition “co-occurrence of the probans with the probandum” need not apply to all instances of probans and probandum, so that one might be tempted to understand the expression “counterpositive” here as referring only to some instances

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of the probandum. In order to avoid this potential misunderstanding Gaṅgeśa specifies the probandum as not being limited by the limitor of the counterpositiveness of an absence which co-occurs with the probans. If the counterpositiveness of the absence referred to in his definition is limited by the property S-ness which applies to (or “limits”) all instances of the probandum S, it is clear that the absentee includes all instances of S. (V1) is too crude to exhibit Gaṅgeśa’s specification. Goekoop’s firstorder formalization of the siddhāntalakṣaṇa does this in an appropriate way: ðV2 Þ“ðExÞ½ðEyÞðAy ^ RyxÞ ^ ðEyÞðBy ^ SyxÞ^  ðExÞ½ðEyÞðAy ^ RyxÞ ^ ðyÞðBy ! SyxÞ” ðGoekoop 1967 : 116Þ “Ax” and “Bx” are to be understood in the sense of “x is an instance of the probans” and “x is an instance of the probandum,” respectively. Since probans and probandum can reside in a locus via different relations, Goekoop symbolizes the occurrence relations referred to by “R” and “S,” respectively. There is, however, no need to draw such a distinction here, because Gaṅgeśa does not express it in his definition. The first conjunct in (V2) renders the co-occurrence requirement. The second one states that there is no absence of any instance of the probandum in a locus of the probans, but without referring to entities like absences, limiting properties and relational abstracts like counterpositiveness. Nevertheless, if one relies solely on first-order methods, it is hardly possible to get a more accurate translation of Gaṅgeśa’s siddhāntalakṣaṇa than (V2). A slight improvement might be effected by shifting some of the embedded quantifications to the front in order to render the formalization more transparent. Moreover, one might use lower-case bold face variables in accordance with the property-theoretic framework introduced in Guhe 2016. Thus, by means of the symbolization key . . . Hx: “x is an instance of the probans” Sx: “x is an instance of the probandum” xLy: “x is a locus of y” . . . one can rewrite (V2) as follows: ðV3 Þ ∃x∃y∃zðHx ^ Sy ^ zLx ^ zLyÞ ^ 8x8yðHx ^ yLx ! ∃zðSz ^ yLzÞÞ Note that the following first-order equivalence allows us to shift the quantification “Ey” in the second conjunct of (V2) to the front (thereby turning it into a universal quantification): ‘ ð∃xβ ! αÞ $ 8xðβ ! αÞ, if x does not occur free in α: Ad 2) Since (V3) does not contain any reference to absences, the problem of nondenoting terms seems to be irrelevant to Gaṅgeśa’s definition of pervasion according to (V3). Now, Gaṅgeśa actually uses the expression “absence” (abhāva) in his definition, but he takes care not to talk about “the absolute absence of S,”

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i.e., [:∃z(Sz ^ xLz)]x, because :∃z(Sz ^ xLz) is false for any locus x if the probandum is a universal property. Whatever [:∃z(Sz ^ xLz)]x might denote, would be unlocatable then. But since it does not refer to any of the unlocatable entities whose existence is endorsed in Vaiśeṣika ontology (such as the ether), one has to regard this term as non-denoting in such a case. So, the following formalization, which contains the term [:∃z(Sz ^ xLz)]x, cannot yet be the final solution, but it can be rewritten in an appropriate way, as will be shown below:
 ðV4 Þ ∃x∃y∃zðHx ^ Sy ^ zLx ^ zLyÞ ^ 8x8y Hx ^ yLx ! :yL½:∃zðSz ^ xLzÞx Gaṅgeśa circumvents the use of non-denoting terms by means of the concept of counterpositiveness (pratiyogitā), a relational abstract which applies to an absentee and its absence. If the absentee of an absence z includes all and only instances of S, one can express this in Navya-Nyāya by saying that the counterpositiveness of z is limited by S-ness. The limitor functions here as a means to restrict the domain of the relation “x is a counterpositive of the absence y” to all S’s. “If pot-ness is said to be a limitor of a counterpositive-ness, then such a counterpositive-ness is sometimes asserted to be indistinguishable from pot-ness.” (Matilal 1968: 76) So, “being a counterpositive of the absence z” means to be S in this case. By introducing the predicate symbols . . . xPy: “x is an instance of the counterpositive of y” and Nx: “x is an absolute absence” . . . one can express this as:
   Nz ! z ¼ ½:∃zðSz ^ xLzÞx $ xPz x ¼ ½Sxx The consequent of this wff, i.e., the equivalence . . .   z ¼ ½:∃zðSz ^ xLzÞx $ xPz x ¼ ½Sxx . . . can be used to eliminate the potentially non-denoting term in (V4). One might, first of all, rewrite (V4) as: ðV5 Þ ∃x∃y∃z ðHx ^ Sy ^ zLx ^ zLy
Þ^
 8x8y8z Hx ^ yLx ^ Nz ^ yLz ! : z ¼ ½:∃zðSz ^ xLzÞx Then one can replace in (V5) the left side of the above-mentioned equivalence by the right side in order to obtain: ðV6 Þ ∃x∃y∃z ðHx ^ Sy ^ zLx ^ zLy  Þ^ 
 8x8y8z Hx ^ yLx ^ Nz ^ yLz ! xPz x 6¼ ½Sxx (V6) does not only show in what way Gaṅgeśa circumvents the problem of potentially non-denoting terms in his definition. It duly mirrors the fact that he

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uses property terms (especially relational abstracts) instead of quantifying determiners in order to resolve the ambiguity discussed in 1). A pure first-order formula like (V2) also expresses Gaṅgeśa’s intention to disambiguate the definition to this effect, but it does so by means of quantifiers, i.e., not in accordance with the technical language of the Navya-Naiyāyikas. Ad 3) Gaṅgeśa specifies the absence referred to in the definition as pervasive, i.e., it does not co-occur with the absentee in a locus which consists of parts, such that the probans is present in each of them, while the probandum is present in only some of them. So, the probans of a pervasion should not co-occur with a pervasively occurrent absence of the probandum. Gaṅgeśa’s solution can be integrated into (V6) by means of the predicate symbols . . . xLwy: “y occurs wholly in x” xLpy: “y occurs only partially in x” . . ., which one might regard as elementary, while xLy is supposed to be definable in terms of xLwy and xLpy: xLy iff df xLw y _ xLp y All that still needs to be done is to add an index in (V6) in order to make sure that it applies also to pervasion relations where the absence of the probandum is a nonpervasively occurrent property: ðV7 Þ ∃x∃y∃z ðHx ^ Sy ^ zLx ^ zLyÞ^ 
 8x8y8z Hx ^ yLx ^ Nz ^ yLw z ! xPz x 6¼ ½Sxx

Toshihiro Wada’s Graphic Representations Wada 2007 is certainly one of the most important contributions to research on NavyaNyāya. It abounds in insightful observations on the history of the school based on solid philological scholarship. Some basic understanding of the import of definitions and arguments in Navya-Nyāya can also be gathered from Wada’s graphs, although the method is probably less uncontroversial than the philological aspects of his work. These graphs consist of four types of edges which correspond to the relations (i) “x occurs in y,” (ii) “x is absent from y,” (iii) “x is a limitor of y,” and (iv) “x is a describer of y.” For (i) and (ii) the top-down orientation is essential: (i)

x

(ii)

x (iii)

y

y

x

y

(iv)

x

y

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Since Gaṅgeśa’s siddhāntalakṣaṇa-definition contains also the relation “x occurs wholly (or: ‘pervasively’) in y,” one might want to add a fifth type of edges in order to represent this relation as well: (v)

x

y

Apart from the fact that (v) is here taken into account (where Wada has only a straight line) the graphic representation Fig. 1 (see below) of Gaṅgeśa’s siddhāntalakṣaṇa-definition as shown in Fig. 1 is the same as in Wada 2007 (cf. 53). The boxes in Fig. 1 contain all the conceptual components of the siddhāntalakṣaṇa-definition. The “mutual absence” is not explicitly mentioned in the definition, but it is implied in the formulation na bhavati. In order to verify the applicability of Fig. 1 to genuine cases of pervasion, Wada specifies the content of each box in accordance with the example of the pervasion of smoke by fire (cf. Wada 2007: 55), as shown in Fig. 2 (see below). According to Wada, Fig. 1 also shows in what way Gaṅgeśa’s definition can be regarded as an elaboration of a definition which Śaśadhara seems to have endorsed: sādhanasamānādhikaraṇavyāpyavṛttyatyantābhāvāpratiyogisādhyakatvam (Wada 2007: 47f) – “Possessing the probandum (as a property of the locus of the probans) which is not the counterpositive of a pervasively occurrent absolute absence which co-occurs with the probans” Wada’s representation of Śaśadhara’s definition can be obtained from Fig. 1 by deleting the boxes for “a delimitor” and “counterpositiveness,” as depicted in Fig. 3 (see below). The triangle in Fig. 4 (see below), which is part of Fig. 1, but not of Fig. 3, is supposed to represent Gaṅgeśa’s trick to resolve the cālanī ya-ambiguity. Since this triangle is missing in Fig. 3, Wada argues that Śaśadhara’s definition fails to be applicable to the case of the pervasion of smoke by fire, because it admits the cālanī ya-instantiation shown in Fig. 5 (see below). Instead of the dotted line between “a mutual absence” and “fire (the probandum),” there is a solid line in the same place in Fig. 3. This signifies a violation of the relation which should hold between the corresponding boxes of the definition. The dotted line derives from the fact that the probandum “fire” is not different from the counterpositive of an absolute absence which co-occurs with smoke. The absence of mountain fire, e.g., co-occurs with kitchen smoke. By means of the cālanī yaprocedure, every smoke s can be paired with a fire f whose locus is different from that of s, so that ultimately all fires can be regarded as counterpositives of an absence which co-occurs with smoke. In order to eliminate this undesirable connotation Gaṅgeśa tries to express in his own way that there is not one locus of smoke where every fire is absent, as shown above. By means of quantification (cf. (V2)) or quantification in combination with property terms (cf. (V7)), one can appropriately render the intended meaning of his

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a probans x

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a mutual absence

a delimitor

counterpositiveness

a probandum y

a constant absence

a counterpositive

the locus of x Fig. 1 Wada’s diagram for Gaṅgeśa’s siddhāntalakṣaṇa-definition

smoke (the probans)

a mutual absence

mountainfire-ness

counterpositiveness

fire (the probandum)

a constant absence

mountain fire (a counterpositive)

a kitchen (the locus of the probans) Fig. 2 Wada’s diagram for the pervasion of smoke by fire according to Gaṅgeśa’s siddhāntalakṣaṇa-definition

a mutual absence

a probans x

a probandum y

a constant absence

a counterpositive

the locus of x Fig. 3 Wada’s diagram for Śaśadhara’s definition of pervasion Fig. 4 Gaṅgeśa’s trick to resolve the cālanī yaambiguity according to Wada’s diagram for Gaṅgeśa’s siddhāntalakṣaṇa-definition

a delimitor

counterpositiveness

a counterpositive

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a mutual absence

smoke (the probans)

fire (the probandum)

a constant absence

all fires, i.e., mountain fire, kitchen-fire etc. (a counterpositive)

a kitchen, an altar etc. (the locus of the probans) Fig. 5 A cālanī ya-instantiation according to Wada’s diagram for Śaśadhara’s definition of pervasion

definition. Since there is no indication of any quantification in Wada’s graphs, the entries in the boxes and their relations to the entries in other boxes can be interpreted in various ways. In Fig. 1 “a probans x” seems to refer to an instance of the probans, such as a specific smoke in the case of the probans “smoke.” Then “the locus of x” should be the locus of that particular smoke individual. But in Fig. 2, the content of the box for the probans is specified as “smoke (the probans),” which might refer to smoke in general. The box for the locus contains sometimes the name of only one locus of smoke (such as “a kitchen” in Fig. 2) and sometimes the names of several loci of smoke (such as “a kitchen, an altar etc.” in Fig. 5). So, why should one not specify the box for the locus in Fig. 1 (Wada’s graphic representation of Gaṅgeśa’s definition) in the same way as in Fig. 5 by entering the names of several loci of smoke? Then one would be entitled to construct the cālanī ya-instantiation depicted in Fig. 6 (see below). Upon closer examination the triangle in Fig. 4, which is supposed to suppress cālanī ya-instantiations, turns out to be rather vacuous: Every counterpositive is trivially endowed with the property “counterpositiveness” and the latter can be said to be somehow delimited. So, according to Fig. 1 the undesirable cālanī ya-connotation would be as much inherent in Gaṅgeśa’s definition as it is in Śaśadhara’s. Although Wada’s method has surely some practical value as a means to exhibit the basic structure of a vyāpti-definition, it is encumbered with ambiguities which Gaṅgeśa tries to eliminate. Its deficiency consists mainly in the fact that it lacks some device for representing quantifications. Quantifications are, however, sometimes implicit in Gaṅgeśa’s formulation and sometimes he paraphrases them by means of property abstracts. A first-order formalization like the one suggested by Goekoop is probably better adapted to the purpose of representing the logical content of Gaṅgeśa’s definition. By using property terms as an additional device, one can obtain a formalization which is still closer to the original Sanskrit formulation of the definition, as shown in the analysis attempted here.

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smoke (the probans)

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a mutual absence

fireness

counterpositiveness

fire (the probandum)

a constant absence

all fires, i.e., mountain fire, kitchen-fire etc. (a counterpositive)

a kitchen, an altar etc. (the locus of the probans) Fig. 6 A cālanī ya-instantiation according to Wada’s diagram for Gaṅgeśa’s siddhāntalakṣaṇadefinition

Summary • In Guhe 2016 the author outlined a property-theoretic framework called “T1 + .” In the present chapter, he demonstrates in what way T1+ can be used for the purpose of a formal reconstruction of Navya-Nyāya logic. • His analysis of some pertinent examples by means of T1+ includes the NavyaNaiyāyikas’ operations applied to properties and relations (negation, conjunction, and disjunction of properties, the Sheffer stroke applied to properties; conversion, composition, left- and right-restriction of relations, cf. section “Operations on Properties and Relations”), their discovery of theorems related to these operations (identities concerning negative properties and de Morgan’s laws, cf. section “Theorems Related to Operations on Properties and Relations”), their account of the reference of number words (cf. section “A Quasi-Fregean Account of the Reference of Number Words”) and Gaṅgeśa’s siddhāntalakṣaṇa-definition of pervasion (cf. section “Gaṅgeśa’s siddhāntalakṣaṇa-Definition of Pervasion”). • Moreover, the author discusses methods of formalization suggested by several other interpreters who have tried to render the technical language of the NavyaNaiyāyikas by means of some kind of symbolic notation.

References Bealer, G. 1982. Quality and concept. Oxford: Clarendon Press. Borkowski, L. 1976. Formale Logik. Berlin: Akademie-Verlag. Frauwallner, E. 1956. Geschichte der indischen Philosophie. Vol. 2. Die naturphilosophischen Schulen und das Vaisesika-System, Reihe Wort und Antwort 6/2. Salzburg: Otto Müller Verlag.

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Goekoop, C. 1967. The logic of invariable concomitance in the Tattvacintāmaṇi. Dordrecht: D. Reidel Publishing Company. Guhe, E. 2008. George Bealer’s property theories and their relevance to the study of Navya-Nyāya Logic. In Logic, Navya-Nyāya & Applications, Studies in Logic. Vol. 15, ed. M. K. Chakraborti et al., 139–153. London: College Publications. Guhe, E. 2014. Mahes´a Chandra Nyāyaratna’s “Brief notes on the modern Nyāya system of philosophy and its technical terms”. Shanghai: Fudan University Press. Guhe, E. 2015. The problem of foundation in early Nyāya and in Navya-Nyāya. History and Philosophy of Logic 36/2: 97–113. Guhe, E. 2016. “The absence of the difference from a pot is potness”– Axiomatic proofs of theorems concerning negative properties in Navya-Nyāya. Computer Science Journal of Moldova 24/3 (72): 312–334. Ingalls, D.H.H. 1951. Materials for the study of Navya-Nyāya logic, Harvard oriental series. Vol. 40. Cambridge, MA: Harvard University Press. SP. 1963. Śivāditya’s Saptapadārthī with a commentary by Jinavardhana Sūri, Lalbhai Dalpatbhai series. Vol. 1, ed. J.S. Jetley. Ahmedabad: Lalbhai Dalpatbhai Bharatiya Sanskrit Vidyamandir. NKoś. 1928. Nyāyakos´a or dictionary of technical terms of Indian philosophy. By Bhīmācārya Jhalakīkar, revised by Vasudev Shastri Abhyankar. Poona: Bhandarkar Oriental Research Institute. Matilal, B.K. 1968. The Navya-Nyāya doctrine of negation, Harvard oriental series. Vol. 46. Cambridge, MA: Harvard University Press. Matilal, B.K. 1980. Double negation in Navya-Nyāya. In Sanskrit and Indian studies. Essays in honour of Daniel H.H. Ingalls, ed. B.K. Matilal and J.M. Masson, 1–10. Dordrecht/ Boston/London: D. Reidel Publishing Company. Matilal, B. K.2 1990. Logic, language and reality. Delhi: Motilal Banarsidass. BN. 1891. Brief notes on the modern Nyāya system of philosophy and its technical terms. By Mahāmahopādhyay Maheśa Chandra Nyāyaratna. Calcutta: Hare Press. Wada, T. 2007. The analytical method of Navya-Nyāya, Gonda Indological studies. Vol. XIV. Groningen: Egbert Forsten.

Part IV Language

Logical Aspects of Grammar: Pa¯nini ˙ and Bhartrhari ˙

21

Raghunath Ghosh

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Grammar in Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grammar as an Aid to the Detection of Corrupt Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grammar as a Means to Justification of Word . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pāṇini and Bhartṛhari on Meaningfulness of Null-Class . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Grammarians on Flexibility of Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spontaneous Overflow of Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhartṛhari on Sentence-Holism (Akhaṇḍa-vākyārtha) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bhartṛhari on the Theory of Anvitābhidhāna . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sphoṭa Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divisions of Vāk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Epistemological Reflections of Bhaṛtrhari . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

572 573 574 574 578 579 580 580 581 582 584 584 587 588 591 592

Abstract

The present chapter addresses the problem that between language and grammar which one is prior and the relation between them with special reference to Indian tradition in order to understand the importance of grammar. Secondly, an address will be made to the chief contributions of the grammarians to the philosophy of language as such. By way of rounding off these questions some evaluative and critical points will be put forth. The present chapter will deal with the arguments behind the first problem mentioned earlier. Secondly, an address will be made to the reasonings for which they are considered as chief contributions of the grammarians to the philosophy of language as such. Grammar has been taken as an aid to detect corrupt word R. Ghosh (*) University of North Bengal, Darjeeling, West Bengal, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_18

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(apaśabda), formation of word, meaningfulness of null-class, akhaṇḍa-vākyārthavāda, anvitābhidhāna-vāda, division of Vāk, Sphoṭa theory, and some epistemological reflections of them. It should be borne in mind that all these theories are interrelated. There are certain reasonings which are related to each other. As the grammarians believe in Sphoṭa theory of Vāk, i.e., the primordial originating point of a word is Sphoṭa, it gives rise to the theory of holistic meaning which again is related to anvitābhidhāna theory. By virtue of being originated from one source its meaning comes into existence in one burst which is indivisible. Due to its indivisibility it becomes sentence-related contextualism, i.e., meaning of the sentence comes into being at the first flash giving rise to word-meaning afterwards. The grammarians do not always admit that their job is only to show the path of rectifying or purifying language. Ultimate goal of them is also to direct a man to reach in the realm of metaphysics which is beyond all cognitions. In Vākyapadīya grammar is taken as the entrance of heaven and also royal road to liberation. That is why they have given much importance to Vedas whether there is no room for corrupt word and misunderstanding of the statement. To Bhartṛhari liberation may not be considered as the ultimate goal to someone yet through the instrumentality of Vāk he attains a great poetic pleasure generated through the light of the lights ( jyotiṣāṁ jyotiḥ) of sound overcoming darkness (tamasaḥ parastāt), which is not at all ignorable in this world.

Introduction It is said in Indian tradition that it needs 12 years to know the ABC of grammar (dvādaśo hi varṣaiḥ vyākaraņaṁ śrūyate). It has also been said that one should be well-conversant with six vedāngas in which grammar (vyākaraņa) is one. From the above statements it can easily be presumed that grammar should be studied in a thread-bare manner for years together to do the justice to our written and verbal communication and it is the precondition of learning language as endorsed by Pataňjali in his Paśpaśā Āhnika of Mahābhāṣya. From this premise one can draw a conclusion that grammar is prior to language, since it is for the sake of the systematization of language. The present chapter addresses the problem that between language used in our day-to-day life in different provinces of our country and grammar which one is prior and the relation between them with special reference to Indian tradition. Secondly, an address will be made to the chief contributions of the grammarians to the philosophy of language as such. By way of rounding off these questions some evaluative and critical points will be put forth. In this context the term “Logic” is to be taken in the sense reasoning or argument. Normally logic seeks to determine good reasonings from the bad ones. An argument is an expression of an inference expressed in language. All the theories of Pāṇini and Bhartṛhari are well grounded with arguments and reasonings which may be called logical statement as opposed to illogical ones. In Indian philosophy it is very difficult to distinguish logic from epistemology and other branches of philosophy. If a theory is found “alone”(ekākinī) without the support of argument, it is not to be taken as a

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philosophical one (ekākinī pratijňā hi pratijňātaṁ na sādhayet- Sarvadarśanasamgraha, Bauddhadarśana). In this sense Pāṇini and Bhartṛhari have their own logic behind formulation of the theories. There is a deep relationship between logic and grammar, because grammatical formations as well as theories are grounded on logic. An illogical theory has no room in grammar which is evidenced through the theories of Pāṇini and Bhartṛhari. It is admitted by the scholars that even a half mātrā (proportion) cannot be spared in formulation of an aphorism without being justified by logic. Grammar provides us some logic behind accepting a word as “well-formed word” (sādhuśabda) and corrupt word (apaśabda), logical justification of the usages by the Eastern, Northern people, plutasvara-s, logical basis of human autonomy in expression (vivakṣā), logical account of null-class, sentence-holism (akhanḍavākyārtha), anvitābhidhānavāda, Sphoṭavāda, different division of Vāk, etc. Can we think of a philosophical theory which is not grounded on logic? If grammar is studied and logical aspect of its theories are properly known, it (grammar), one type of vedānga, can provide us perfection and all-round development including summum bonum by their logical basis. Had there been no logical basis of grammar, it would have been taken as a nonsensical Śāstra which is contradictory in terms. If it is a Śāstra, it can never be nonsensical. A Śāstra is so-called due to having some logical basis in it. That is why logic which is called ānvīkṣikī in Indian term is considered as the lamp of all Śāstras, means of all activities, locus of all morality or Dharma (“pradīpaḥ sarvaśāstrāṇām upāyaḥ sarvakarmaṇām/āśrayaḥ sarvadharmāṇāṁ śaśvadānvīkṣikī matā//’ Vātsyāyanabhāṣya” on Sūtra-1.1.1.). As there is no theory in India which is not based on arguments or logic, grammar cannot be an exception to it.

The Role of Grammar in Language At the outset we should consider the question: What is language? According to the grammarians, the root √bhāṣa can be applied to refer to something expressed through the verbal language as evidenced from the sūtra: bhāşa vyaktāyāṁ vāci (the root √bhāṣa, to express) which is applied in verbal communication. In other words, when something is expressed through alphabetical language but not symbolic one is called bhāşā (language) arising from the root √bhāşa. From this restricted view of the grammarians, other kinds of nonalphabetical language (symbolic language like bodily instruction of traffic police, etc.) cannot be taken as language in the true sense of the term. It is true that there are various modes of expression like symbolic etc., but the question of inclusion of these under language does not arise at all if the abovementioned view is taken for granted. The poets try to search language in mountain, fountain, rhythm, and pleasure (chander bhāşā ānander bhāşā). Even the description of the external features is also called bhāşā, e.g., sthitaprajñasya kā bhāşā (i.e., what is the language of a person who is steady in wisdom?). To the grammarians the bhāşā, however, is only spoken alphabetical language on account of the fact that they deal with only word or meaning of a sentence constituted by alphabetical language. The nonalphabetical language, i.e., symbolic etc., however fruitful may be in day-to-day communication, has no role in the philosophy of

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grammar, because it is grammar which provides us the criteria of a well-formed word (i.e., a word formed as per formulated rules of the grammar) in a sentence. A grammarian thinks it proper to formulate some rules essential for building a wellformed word (noncorrupt word) and sentence. Later we shall see that if a word or sentence is grammatically well formed, it must have some meaning, though referent may not be found in the external world.

Grammar as an Aid to the Detection of Corrupt Word Now a question may be raised between grammar and language which one is prior. In this context the language means language used in our day-to-day life, used in different regions of our country and some language used by duplicating a word (abhyāsa), but not the Vedic and other Āgamas (scriptural texts). One may think that the grammatical rules are formulated only to give an accuracy of such words or sentences. It is a grammatical rule which makes a word grammatically well formed. If not, it would be taken as a corrupt word (apaśabda), the usage of which may lead an individual to the path of harmfulness as endorsed by Patañjali in his Mahābhāşya. To him a corrupt word (duṣṭaśabda) arising out of misapplication of grammatical rules or mispronunciations of it particularly in the case of mantra may cause harm to the user or the performer of sacrifice. In the same way, this theory can be applicable to the secular world also. If a word is presented in a defective way due to some defect existing in the word itself or in its utterance, it creates a communication gap between the user of the language and the hearer, which may consequentially lead to misunderstanding, misbehavior etc., creating a lot of social problems. That is why it is said – “Duṣṭaḥ śabdaḥ svarato varņato vā mithyā prayukto na tamarthamāha/ sa vāgbajro yajamānaṁ hinasti yathendraśatruḥ svarato’ parādhāt// (Pāņinīya-śikşā 9/42 1957). That is, a corrupt word as regards accents or sound, employed improperly, fails to convey the sense intended. That thunderbolt of speech kills the performer of the sacrifice just as the expression Indraśatru on account of the error in the accent does. The proper use of grammatical rules thus can save us to form an ill-formed (corrupt) word and sentence. An individual becomes conscious of not using a grammatically incorrect or corrupt word and can save himself from the hazards of miscommunication etc.

Grammar as a Means to Justification of Word At the outset it can be pointed out that we come across many illiterate persons who do not know about the ABC of grammar, but having very good command over the language particularly if it is a mother tongue. It can be compared to a good inborn singer having no formal training and no grammar of music with the introduction of the notes like sā, re, gā, etc. or different systematic stroke (tāla) or rhythm (laya), which may be comparable to Chomsky’s concept of language as an innate capacity of the humans.

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In response to the above it may be said that one can have a natural command over his own mother tongue without knowing any grammar, but it is the function of grammar to frame rules for the sake of systematization of language, which is essential for a beginner who wants to learn language in a systematic manner. One can learn language conventionally from the seniors (vṛddha-vyavahāra) or by learning grammatical rules. It seems that grammar came into being afterwards the origination of language. Otherwise, grammar would have been originated just before the origination of language, which is impossible. That language is earlier than grammar is evidenced from the following facts. First, when Pāņini framed a particular aphorism, he incorporated a phrase like ‘iti prācām’ (this according to the inhabitants of the East) and ‘iti udīcām’ (this according to the inhabitants of the North) (‘Sonat prācām’ Aşţādhyāyī n.d.). From this incorporation particularly in a sūtra which is inevitable Pāṇini had tried to justify both the usages found among the Northern and Eastern people. Hence, the formulation of the grammatical rules is not arbitrary, but teleological in the sense that the conventional usage found differently in different society must be brought under some rules for the sake of their justification. Secondly, the grammarians have admitted a phenomenon of plutasvara (emotional language) as found in calling someone from a distance, in singing a song or in weeping state. The grammarians have defined it as “dūr-āhvāne gāne rodane ca pluto mataḥ.” Pāṇini has mentioned such plutasvaras in grammar in order to accommodate these usages under language (‘Pluta-pragŗhya aci nityam’ n.d.). If a particular language is prevalent in the society, people will call others, sing a song or weep or lament using the same language, which can never be denied. For this reason, it should be justified through some rules. Pāņini and others have taken such consideration in formulating an aphorism here. Thirdly, there is an aphorism in Pāņinian grammar – abhyāse carcaḥ’, (8/4/58) which is formulated to justify the usages like paṭpaṭākaroṣi, carcarāyate, pharpharāyate, etc. When a single term is used doubly to mean something as found in the abovementioned usages, it is technically called abhyāsa. In the cited examples the terms paţ, car and phar have been used doubly. These are used in the following manner: a) kathaṁ tvaṁ paţapaţākaroṣi? (Why are you talking instantly and without any hesitation?); b) Candanalepanam anabhyāsavaśāt lalāţaṁ carcarāyate (The pasting of sandal on the forehead becomes the cause of uneasiness due to nonhabituality); and c) Ganduṣa-jala-mātreņa sapharī pharpharāyate (In a handful water a small fish is creating the sound of movement). These usages are already in the language. The grammarian’s duty is to formulate rules so that all these usages constituted by non-Sanskrit words like paţpaţ, carcar, or pharphar may find their dignity in the language and should not be considered as apaśabda. After looking at the nature of the objects the grammarians have formulated the rules. Just as the words like dāra (wife), akṣata (unboiled rice), lāja (puffed rice), asū or prāṇa (vital organ), varṣā (rain), and sikatā (sand) are prescribed as having always plural number (nitya-bahuvacanānta) by the grammarians on account of the fact that the singular form of such words is not familiar. When there was a system of polygamy (having more than one wife) the word “dāra” (wife) with singular suffix

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was meaningless because there was none who had a single wife customarily. In the same way, the words “akṣata” and “lāja” (unboiled rice and puffed rice) mean not a single piece of such rice but a cluster. The words “asū” or “prāņa” means five vital organs (pañca prāṇāḥ), “varşā” or rain means not a single drop of water but a few and “sikatā” or sand does not mean a single piece at all. Our public usage is such that the singular usage of these words is meaningless and hence eternal plurality of them has been prescribed (“dārākşatalājāsūnāṁ bahutvañca”). In English language also such consideration of public usage has been taken into consideration. The word “deer” is prescribed by the grammarian as bearing always single number on account of the fact that a single deer never roams in the forest, but always in a body. The deer moves in the forest always in a group. From this it is proved that there is no need of using “deers” due to its superfluous nature. From the tradition a hearer can easily understand the term “deer” means “deers.” Moreover, the examples given in favor of sandhi and samāsa are taken from our public life. When the words “vidyālaya,” “vīņapāņi,” etc. are taken from our life, it is proved that these words are formulated to felicitate an individual for pronunciation. It is somehow difficult to utter “vidyā ālaya” (diving two words), “vīņā pāņau yasya” (dividing in three words). In order to simplify our communication, the grammarians have introduced the name for the concepts of sandhi, samāsa, etc., method of classifying them on various principles, method to get their meanings, and so on. The grammatical rules are framed in order to justify public usages which were prevalent in the society. Fourthly, it has been found in the grammar that when an aphorism is formulated, an example from the convention is put forth to show the relevance of the concerned rule. Had there been no prevalent language, no instance would have been possible. Let us see how an example from the convention as found in the society can help us to understand the aphorism formulated by a grammarian. Pāņini has framed the aphorism kartur ipsītatamaṁ karma (1/4/49), i.e., an object is that which is most desired by an agent. Now the question arises, why is the term “most desired” (īpsitatama) incorporated here? In order to reply the problem, the grammarians cited an example from the conventional usage. It has been exemplified –“Someone is taking rice with milk” (sa payasā odanaṁ bhuńkte) (‘Karturīpsitatamam karma’ n.d.). This is the usage already found in the society and hence the grammarians have tried to justify by inserting a suffix “tamap” used in the superlative degree to the term īpsita or desired. It signifies that for being an object (karma) it should be most desired (īpsitatama) by an agent, otherwise it will fail to act as an object or karma. Now in the present case between milk and rice the latter is more desired by an agent, because without rice an individual cannot survive. That is to say, it is rice which is prominent in the meal and milk is the medium of making it palatable. That is why, it receives the status of instrumental cause (karaņa) as evidenced from the third case-ending (payasā) and rice (odanam) attains the status of an object (karma). The example cited here was already in usage. Rather, it has been made use of just after the rule is formulated by Pāņini or any other grammarian. In any case, an example is always taken from the empirical world so that a logical mind can easily be convinced. Fifthly, the grammar is meant for the students willing to learn language in a perfect way. Those who know the language perfectly need not bother about

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grammatical rules. Even the veteran poets or seers are not always under the purview of grammatical rules. If an ordinary man violates grammatical rules and form a word, it would be treated as corrupt word (apaśabda) and the user of such corrupt word is called avyutpanna (i.e., having no sense of etymology), which is very much disgraceful for a man. But such is not the case with seers or poets. They have some liberty to use a word or sentence as per their “desire to speak” (vivakṣā) for the beautification of the poetry or any other reason. Even a grammatically incorrect words or sentences used by a seer or poet (ārṣa-prayogaḥ) can be overlooked after considering their necessity in expressing certain emotions, poetic or transcendental revelation for the seers. Moreover, the grammarians have shown their liberty not only to the case of ārşa-prayoga, but to the ordinary speaker also. To them grammatical rules cannot always be guiding principles to a speaker. Expression is a kind of art. If a speaker thinks that a particular pattern of usage would be more appropriate in a particular context or situation, he will be allowed to express as per his desire to speak called vivakṣā which has been sanctioned by the grammarians. In fact, an agent can use the sentence – sthālī pacati (pot cooks) or sthālyā pacati (cooking with the help of a pot) or sthālyāṁ pacati (cooking in a pot) as per his desire to speak. An agent’s desire is given prominence, which is beyond the ordinary grammatical rules (‘Vivakşāvaśāt kārakāņi bhavanti’ 1971). Grammar can accommodate all such so-called grammatically distorted sentences after considering the liberty of the speaker. In fact, language is to be spoken as per desire of the agent who may not even care for these grammatical rules. It may be found that the usage is much earlier which is later justified by grammar. In this context, one point of caution can be given to the speaker. Grammar justifies the usages found in ordinary society. If there is any word or language which is not found in the empirical world or convention, it cannot be justified by grammatical rules. Though they have given the priority of the desire of the speaker, it (vivakṣā) should have some limit. If an agent’s desire to speak (vivakṣā) contradicts the conventional usage, it cannot be justified or taken for granted. That is why vivakṣā (desire to speak) is metaphorized as house wife (kūlavadhū). Just as a house wife has got some liberty to maintain her family according to her own will without transgressing the social or ordinary restrictions, an individual has liberty to use language in a grammatically distorted manner without losing its communicative purpose or value (vivakşā tu kūlavadhūriva laukikīṁ maryādāṁ nātikramet). (“vastutastad anirdeśyaṁ nahi vastu vyvastithitam/sthālyā pacyata ityeṣā vivakṣā dṛśyate yataḥ//” n.d.) After all, an individual uses the language only to communicate something to other social beings. He has to use language in such a manner so that the hearer can understand his point, whatever it is grammatically distorted. It proves that the author or speaker has got autonomy of will. This type of autonomy or freedom is a kind of “restricted freedom,” but not an absolute one. If a speaker has autonomy to say any word for any meaning, it will create problem in verbal communication leading to the cessation of public usage of language (lokavyavahāra). That is why it is said that one should exercise autonomy keeping the pattern of public usage in view to make it understandable to others. If an expression does not tally with the public communication, it should not be used. It indicates that the grammatical rules are flexible depending on

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the desire of the speaker. This is an exceptional case that grammatical rules always justify the conventional usage, but sometimes to justify certain conventional usages grammatical rules are distorted. For the sake of autonomy of an agent such rigidity that language is prior to grammar has admitted in Indian tradition. Sixthly, the grammarians have admitted the six meanings of negation (nañ) on account of the fact that the negative particle is used in these senses in the society or in social custom. The negation is used in the senses of similarity (sādŗśya), e.g., a-brāhmaņa meaning “similar to Brāhmaņa” (brāhamaņa-sadŗśa), absence (abhāva), e.g., asukha meaning “the absence of happiness”(sukhasya abhāvaḥ), anyatva (differentiation), e.g., a-ghaṭa meaning “different from a jar” (ghaţād anyaḥ), less quantity (alpatā), e.g., akeśī meaning “less hairy” (svalpa-keśī), impropriety (aprāśatyam), e.g., asamaya meaning “improper time” (apraśasta-kāla), and conflict (birodha), e.g., asura meaning “conflict with the demons” (sura-birodhī) after considering different usages in the loka, i.e., in the convention in view (‘Tatsādŗśyam abhāvaśca tadanyatvaṁ tadalpatā/Aprāśastyaṁ birodhśca nañarthāh şaţ prakīrtitā//’ 1971). As all these senses of the negative particle are also available in the tradition, the grammarians have no other options than to accommodate these meanings in the grammatical aphoristic framework.

Pa¯nini and Bhartrhari on Meaningfulness of Null-Class ˙ ˙ That the grammarians like Pāṇini and Bhartṛhari have laid much emphasis on the “public usage” (loka-vyavahāra) is evidenced from the fact that they have admitted the meaning of the terms like vandhyāputra (barren women’s son’) and ākāśakusuma (sky-flower), though their referents are not found in the external world. Normally, the ordinary people understand some meaning as soon as the words are uttered and addressed to them. Considering this aspect, the grammarians have admitted that these words have got some ideational meaning (buddhyartha) though they lack referential meaning (vastvartha). The Naiyāyikas would say that these ideational meanings are tantamount to nonmeaningfulness of the words, because meaning cannot exist without reference. It is a matter of great controversy whether meaning remains in an idea of a human being or in really existent object. We cannot deny the claim of the realists if an entity does not remain in the real world, what is the use of language, which is nothing but a jugglery of words. The example which is available in the West, “The present king of France is bald,” is worth pondering here. As the words “the present king of France” does not refer anything due to not having kingship in France, the predicate portion yields any meaning. In the same way, the words like “barren woman’s son” (bandhyāputra), “sky-flower” (khapuşpa) etc. having no reference are taken to be meaningless. It is also true that sometimes through jugglery of words certain idea is projected to us. Though the world of reference is limited and easily identified, the ideational world is vast in which each and every thought can easily be accommodated. The utopian things which have no real existence in the world cannot be taken as meaningful. In the world of Naiyāyikas and Vaiśeṣikas there are seven members under

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which all existent objects can be accommodated. If any object which is not a member of this world of padārtha is believed to be existent in the world of our idea, it is not an existence in true sense of the term. If the Naiyāyikas allow the entry of the entities like vandhyāputra etc., many other utopian ideas fairies, Pankhîrāja horse (flying horse) etc. may come into existence in our ideas through the hole created by them, which would spoil the logical character of a system. Each and every philosophical school is bound by its categorical framework and hence Nyāya cannot allow such utopian ideas as a member of its world. Hence to Nyāya meaning remains only in the referent, but not in idea. Hence the ideational existence of an entity belongs to null-class. But the grammarians stick to their point that ideational meaning must be there, because people understand some sense when it is uttered. The grammarians have two points in their favor. First, the terms “vandhyā-putra” or “ākāśa-kusuma” are grammatically well formed. As per their axiom – each and every grammatically well-formed sentence is correct (grammatically) yielding some meaning. There is no grammatically well-formed sentence yielding no meaning at all. Secondly, the public usage “loka-vyavahāra” of such terms can never be ignored. This point of the grammarians has been admitted by the Buddhist logician, Dharmakīrti, after considering the merit of such statement. He said that when a sentence is uttered by a speaker, the meaning reflected in the intellect of the hearer is to be considered as valid, but no other considerations should be taken as the determinant of meaning (“yah arthah buddhau prakāśate vaktŗ-vyāpāra-vişaye/Pāmāņyaṁ tatra śabdasya nārtha-tattva-nivandhanam” – Pramāņa-vārtika – ¼). It is language through which we communicate and exchange our views. All types of cognition come through the use of language. The cognition, which does not come via language, is an impossible phenomenon. Even the cognition of absurd entities like barren woman’s son (vandhyāputra), hare’s horn (śaśaśŗnga), sky-flower (ākāśakusuma) etc. is attained through the usage of language.

Grammarians on Flexibility of Language Moreover, language is always flexible and growing, because it is normally created to express a particular thought. In other words, a special language is created to accommodate certain thoughts. Sometimes, many Indian terms have been angelized, e.g., “The students have gheraoed the Vice-Chancellor,” “The police have made a mild lāthi-charge” etc. In these cases, the terms “gheraoed” is formed through the verbification of the non-English term “gherao” and in this same way, the term “lāţhicharge” is formed with the English term “charge” with a Bengali word – “lāţhi.” These have become meaningful words, though they are formed with loan words from other language. In the same way, many Dravidian, Arabian words have entered into the world of Sanskrit. Hence it can be said that a word is not fixed but flexible. It is changed often seeing the usage of a particular sect or a section of people. As it is extendable or flexible, it can express many things. Though the abovementioned standpoint is true, an entity referred to by a word exists in the external world or intellectual world. In other words, all problems are

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related to the fact whether a word refers to something existing in the external world (vastvartha) or something existing in our intellect (budhhyartha). The former is also thought which has a corresponding fact to the real world while in the latter case there is some thought which may be product of our intellect which has nothing to do with the world of real. Indian philosophers are divided into two camps – in one side there are Naiyāyikas and Vaiśeşikas while on the other there are Buddhists and grammarians. The famous example is given in the following śloka which consists of four unreal objects giving some ideational meaning. It goes like this: “eṣa vandhyāsuto yāti khapuṣpa-kṛtaśekharaḥ/k ū rma-kṣīracaye snātaḥ śaśaśṛngo dhanurdharaḥ//” That is, here the barren woman’s son is going being adorned by sky-flower. He has taken bath in the milk of a tortoise and having a bow made of hare’s horn.

Spontaneous Overflow of Language From the above discussion it is to be noted that there is always and everywhere a class of people who cannot tolerate any discipline. They have no patience to bear with grammar, ethics, or logic. They prefer to express themselves or their feelings through natural words. As Patañjali has pointed, it is not necessary for us to go to the house of the grammarian and order for some words for our expressions, just as we go to a potter and order when we need it (“yathā ghaţena kāryaṁ karişyan kumbhakārakūlaṁ gatvāha- kuru ghaţaṁ kāryamānena karişyāmīti, na tadvacchabdān prayokşyamāno vaiyākaraņakūlaṁ gatvāha- kuru śabdān prayokşya kuru iti” 1417 B.S.). It is true that a lover does not care for grammar when pouring out his heart to his beloved; a mother does not need any grammatical accuracy while consoling the shocked mother due to the loss of her son. The expressions of the lover and the bereaved mother are more impressive and heartening though they are grammatically incorrect. All the knowledge that is necessary for us to communicate our thoughts to others is acquired unconsciously from the conversations of our elders (vŗddha-vyavahāra). Even in the matter of lending grace to a poetic composition, grammar is not of much use, for its restrictions often lead people to use harsh unmelodious words jarring to ears. In spite of this grammar is essential to know the proper word as opposed to improper word (apaśabda). Learning of correct language is possible by four appropriate means – reading, understanding, practising, and handling on to others. It is not admitted that the knowledge of words was to be attained by reading one by one. General and specific rules apply at once to many examples. These are divided into the artificial parts called roots, etc. In this way we can easily comprehend an exposition of many words. For this reason, the knowledge of grammar is essential.

Bhartrhari on Sentence-Holism (Akhanda-va¯kya¯rtha) ˙ ˙˙ According to the Advaitins, there are two types of sentence: bearing “divisible related meaning” (“samsṛṣṭārthaka”) and “indivisible meaning” (“akhaṇḍārthaka”). The former exists in the sentences like “Bring a cow” (“gāmānaya”), etc. because it

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gives an awareness of the verb “bringing” of an object “cow.” An entity called Candra remains in the expression: “The excellent shining entity is the moon” (“prakṛaṣṭaprakāśaḥ candraḥ”), which gives rise to meaning without making us aware of the relation (‘Atra vākyaṁ dvividhaṁ- samsŗşţārtham akhaṇḍārthaňca iti. Tatra samsŗşţāthaṁ saṁsarga-gocara-pratīti-janakaṁ, yathā gāmānaya ityādi. Akhaṇḍārthaṁ tu saṁsargā-gocara-pratīti-janakaṁ, yathā prakŗşţa-prakāśaḥ candraḥ ityādi’ 1993). The sentence cannot give the awareness of the moon, which has owned “the property of excellent shining” (“prakṛṣṭa-prakāśaḥ”) as its object. To the Vedāntins an individual who has seen the moon on the sky and does not know it as such has heard the term “moon” from a reliable person and asked him the question – “What is called moon?” (“kaścandraḥ”). On hearing such question, the reliable person told him that “the excellent shining entity is the moon” (“prakṛṣṭaprakāśaścandraḥ”). In this case the meaning of the sentence is not constituted with the relation of the meaning of the terms like prakṛṣṭaprakāśa, etc. The terms like prakṛṣṭaprakāśa, etc. are not apprehending relation between the meanings of the terms but through these the essence of the moon is known. It gives the indivisible meaning (akhaṇḍārtha) of the term “moon,” which is nothing but its essence. In the like manner, the sentences like “This is that Devadatta” (“so’yaṁ Devadattaḥ”) and “That art thou” (“Tattvamasi”) indicate the indivisible meaning and hence it cannot be described as a relational cognition or savikalpaka cognition. It is told by Bhartṛhari – “yathaika eka sarvārthapratyayaḥ pravibhajyate/dṛśya-bhedānukareṇa vākyārthavagamastathā//” VP-2/7. It is the conclusion of the grammarians that though one Brahman is manifested in the form of all objects in the transcendental or ultimate level (pāramārthika sattā) yet jar, cloth, etc. are taken as different in nature as the cognition of a jar or cognition of a cloth, etc. in the empirical or phenomenal level (vyavahārika sattā). In the like manner, the different terms of the indivisible sentence and their meanings are taken as phenomenal or empirical. The Advaitins have admitted the holistic meaning (akhaṇḍa-vākyārtha) in the sentences like “So’yaṁ Devadattaḥ,” “Tattvamasi,” etc. where the relation of the word-meaning does not come under the object of intention (tātparya). In these cases, the nature of Devadatta and the nature of Self are the objects of intention.

Bhartrhari on the Theory of Anvita¯bhidha¯na ˙ It is argued by Bhartṛhari that if it is said that term ākhyāta indicating verb is a sentence, it does not mean that only verb is a sentence. That is to say, if there is the usage of verb without using the kāraka-s or cases then also it would be taken as a sentence. In such cases the kāraka-pada-s are to be taken as understood, just as if the term pidhehi (close) is uttered, it would lead us to become aware of the understood meaning of the term dvāram (door). It is also called a sentence on assuming that the verb alone can express the remaining total meaning. It is told by Bhartṛhari (V.P. 2/326) (“vākyaṁ tadapi manyante yatpadaṁ caritakriyam/ākhyātaśabde niyataṁ sādhanaṁ yatra gamyate/ tadapyekaṁ samāptārthaṁ vakyaṁ ityabhidhīyate//” n.d.) that even the usage of the single word can express the whole meaning in spite of having the lack of expectancy

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(nirākānkṣa). In the commentary Harivṛṣabha on Vakyapadiya (2/326) it is said that the single word may be a verb or noun. The noun has taken the verb in its womb (caritakriyā or garbhīkṛtakriyāpada) (“garbhīkṛta-kriyāpadaṁ nāmapadaṁ vākyaṁ prayuñjate, carita-kriyāpadāt asmād arthāvagatiḥ nirākāńkṣā upajāyamānā dṛśyate” n.d.). As for example, if the word paraśunā (with the help of an axe) is uttered then the verb chinatti (cut down) is also understood. If the ākhyāta is taken as a sentence, then the verb must come as the meaning of the sentence (VP-2/414) (“kriyā kriyāntarād bhinnāḥ niyatādhārasādhanā/prakrāntā pratipatṛṇām bhedāḥ sambodhahetavaḥ//” n.d.). To describe a verb and noun as a sentence the ontological presupposition of anvitābhidhānavāda by the grammarians is to be admitted. If it is admitted that a verb or a noun has got capability of understanding the whole sentence, there is no other alternatives than to admit the meaning of rest of the word. Bhartṛhari in kārikā nos. 2/44–45 has agreed the theory of anvitābhidhāna that the meaning of the term is the meaning of the sentence and in order to show this Bhartṛhari has formulated the verse starting with sarvabhedānuguṇyam, etc. (“sarva-bhedānuguṇyaṁ tu sāmānyam apare viduḥ/ tadarthāntara-saṁsargād bhajate bhedarūpatām//bhedān ākāńkṣātastasya yā pariplavamanatā/avacchinatti sambandhastāṁ viśeṣe niveśayan//” n.d.) Moreover, the terms composing a sentence are mutually expectant no doubt, but among them the verb is prominent while others are secondary (VP- 2/46). Though the terms of a sentence are independently meaningless yet they are collectively meaningful. Just as letters and their parts are devoid of meaning by themselves, but are meaningful when combined, the same is the case with the sentence in respect of its being a mere combination of words. (Yathā sāvayāva varṇā vinā vācyena kenacit/arthavanta samuditā vākyamapyevamiṣyate//VP-2/54).

Sphota Theory ˙ Sounds are destroyable, remaining in three moments and having origination and destruction and hence the combination or amalgamation (saṁghāta) among them is not possible. For this reason, a power called Sphoţa which is different from sound (śabdātirikta) and revealer of meaning suggested by sound (śabdābhivyangārthābhidhāyaka) is to be admitted according to the grammarians. Such an indivisible Sphoţa gives rise to the actual meaning, but ordinary word used for the meaning is an auxiliary to the revealing the same. The etymological meaning of the term Sphoṭa is “sphuṭati arthaḥ asmāt,” i.e., that from which meaning is revealed. The sound (nāda) manifests Sphoṭa and Sphoṭa is the bearer of meaning. That is why sounds are said to be destroyable, momentary, etc. while Sphoṭa is a whole and eternal. Sphoṭa remaining in the internal region of the speaker is called sound (nāda), which is the cause of the utterance of the sound having successive existence. The uttered sound again awakens the same Sphoṭa remaining in the internal portion of the hearer. (VP-1/86). Hence, there are three levels of understanding – first there is sound (nāda) giving rise to Sphoṭa which again leads to the understanding of meaning. Just as sound (nāda) is having succession,

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nonsuccessive Sphoṭa seems to be having succession due to its manifestation by the sound (nāda) (VP-1/48). Bhartṛahari has drawn our attention toward this with the help of a metaphor. Just the moon being reflected in the disturbed flowing water of a river seems to be many, Sphoṭa being manifested by successive sound seems to be successive erroneously (V.P.1/49). Just as the special type of wood (araṇi) capable of producing fire through friction can illumine itself as well as others, the seed-like sound remaining in the intellect of the speaker can illumine itself and others like a jar, etc. Sphoṭa remaining in the intellect of the hearer, being manifested as a sound, awakens the Sphoṭa remaining in the intellect of the hearer. In this context, the sound-like fire being illumined can illumine the meaning. Just as fire in the form of burning property has got two characteristic features – capable of being illumined (grāhyatva) and being illuminator (grāhakatva), in the like manner all sounds have two types of power which becomes active in different times in different ways (grāhyaṁ grāhakatvaṁ ca dve śaktī tejaso yathā/tathaiva sarvaśarvanāmete pṛthagavasthite// V.P.1/55). Hence sound being illumined can manifest Sphoṭa. At last, it can be said that there is the necessity of sound which, having successive character, is capable of being heard in order to manifest Sphoṭa (VP-1/44). In the term ghaṭa each syllable gha and ṭa generated the cognition of Sphoṭa separately, because each syllable is the medium of indivisible Sphoṭa. When the first syllable gha is pronounced, Sphoṭa in the form of a jar is cognized in a nonmanifested form. As soon as the syllable ṭa is uttered, the cognition of a jar becomes clearer. The holistic cognition of the Sphoṭa in the form a jar is not divisible by gha and ṭa but an indivisible one having no parts. The cognition of the whole of Sphoṭa becomes clearer step by step. Sphoṭa is an initial sound (prākṛtadhvani) and dhvani or sound is reverberation (vaikṛtadhvani). Bhartṛhari does not take the initial sound as Sphoṭa itself, but a revealer of the same (Sphoṭa). Sphoṭa and initial sound is taken as a relation of revealed and revealer situation (vyangya-vyanjñaka-bhāvena tathaiva Sphoṭanādayoḥ- VP-1/97). Dhvani may be short (hrasva), long (dīrgha) due to variation of mode and time, etc. but Sphoṭa remains same being same with undivided time (Sphoṭasyābhinnakālasya dhvanikālātipātinaḥ/grahaṇopādhibhedena vṛttibhedaṁ pracakṣate//-VP,1/75). The initial sound is the revealer of Sphoṭa. The properties of initial sound are imposed in Sphoṭa. The reverberation being originated from the initial sound makes the awareness of Sphoṭa in tact (VP-1/75–77). There are three views regarding the relation between dhvani and Sphoṭa. First, both dhvani and Sphoṭa appear as identical (Sphoṭarūpāvibhāgena dhvanergrahaṇam īṣyate- VP-1/81) just as china rose ( jabākusuma) or any deep colored flower reflected in transparent jewel appears as identical in awareness (abhedapratīti). Secondly, dhvani is not capable of being known independently and hence it is known as the revealer of Sphoṭa (VP-1/81). Thirdly, one may have awareness of dhvani even without having that of Sphoṭa just as light is seen even without flame. Sphoṭa may be taken as prākṛta-dhvani and dhvani as produced by sound according to some scholars as admitted by Bhartṛhari (VP-1/102). Again, dhvani is not possible without Sphoṭa and hence he has mentioned that Sphoṭa is a kind of genre (jāti) and dhvani is a particular (vyakti) (VP-1/93).

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Divisions of Va¯k Vāk is divided into four after considering its vertical, subtle, subtler, and subtlest forms. It is accepted that one-fourth of the gross Vāk is uttered by human being with the help of his throat and the rest three-fourth is underlying in the internal part of the body which is below the throat and hence there is no question of producing any sound. Four forms of Vāk are Vaikharī, Madhyamā, Paśyantī, and Parā. The gross word remains in the region of throat and hence it is called Vaikharī which is audible to others (vaikharyā hi kṛto nādaḥ paraśravaṇagocaraḥ). That is, Vaikharī is a sort of vibration (nāda) which can easily be audible to others. Madhyamā Vāk remains in the region of heart and it is not audible due to its subtle nature. A speaker can hear such Vāk himself after closing his eyes. From this it is established by Bhartṛhari that there is certain sound called Madhyamā, though subtle in nature and remaining in region of heart, which is capable of being felt with the help of the closed eye, but not through ear. Paśyantī is more subtle remaining in the naval region of the body and it is revealed in the meditative state of a yogin as an object of determinate perception. It is in the form of bindu, nāda, and material element of the world. It means that an individual can feel the existence of subtler sound in the meditative state of a yogin. The last one Parā vāk is revealed in the meditative state of a yogin as an entity in the form of indeterminate perception. At this stage only the feeling or revelation of it can be realized but not in the form of gross sound. Though the latter grammarians had admitted the four types of Vāk yet Bhartṛhari has admitted as three-fold Vāk after incorporating Parā under Paśyantī (“Vaikharyā madhyamāyāśca paśyantyāścaitadadbhūtam/anekatīrthbhedāyāstrayyā vācaḥ paraṁ padam//” VP-1/143). Somānanda in his Śivadṛṣţi, a treatise based on Pratyabhijñā-darśana, had accepted four-fold Vāk. To him the Paśyantī type of Vāk cannot be a subtlest one and hence Parā Vāk has to be admitted separately.

Epistemological Reflections of Bhartrhari ˙ Bhartṛhari has admitted five types of pramāṇa-s, i.e., perception (pratyakṣa), inference (anumāna), testimony (āgama), habitual experience (abhyāsa), and unseen element (adṛṣṭa). Among these, āgama pramāṇa is taken as most trustworthy and dependable. It is said that the reasoning independent of āgama (or not dependent of āgama) cannot be taken as an evidence in favor of the existence of dharma (“na cāgamād ṛte dharmastarkeṇa vyavatiṣṭhate/ṛṣīnāmapi yaj jñānaṁ tadapyāgamapūrvakam//” VP-1/30). That is, only reasoning which is independent of āgama or scripture is of no use for establishing dharma. Even the wisdom of the seers becomes infallible as it is dependent of scriptures. In this context, the term dharma does mean only rituals but morality also. Whether something is moral or immoral is determined by scriptures. The reasonings based on scriptures are more powerful than those forwarded by an expert through inference. The argument based on āgama excels when it is seen that the conclusion attained by an expert individual through anumāna may be refuted by another individual having more expertise in anumāna (VP-1/34)

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(“yatnenānumito’pyarthaḥ kuśalairanumātṛbhiḥ/abhiyuktatarairanyairanyathaivopapādyate//” n.d.). Moreover, there are certain objects where perception, inference, etc. fail to reveal (“pratyakṣam anumānaṁ ca vyatikramya vyavasthitāḥ” VP-1/36). This point may be substantiated by the following arguments. If there is an object which is empirically verifiable, it can be revealed with the help of perception and inference. If something, on the other hand, is not empirically verifiable, there is no other way to depend on what is stated in the scriptures. The statements like – “There are fruits on the bank of the river” (nadyāstīre phalāni santi) and “there is fire on the mountain due to smoke arising out of it” (Parvato vahnimān dhūmāt) can easily be verifiable through perception and inference, respectively. But if it is said – “you should perform sacrifice if you are desirous of heaven” (svargakāmo yajeta), it, not being verifiable by perception and inference, must depend on the statement of the scripture for its reliance. One point of caution has been given by Bhartṛhari that just as a blind man cannot move in an uneven land with the help of his hands, an individual inclining to prove all objects with the help of reasoning receives the lack of success (“hastasparśād ivāndhena viṣame pathi dhāvatā/anumānapradhānena vinipāto na durlabhaḥ//” – VP-1/42). On the other hand, the super-sensuous and unknowable objects are capable of being known by the seer’s eye. It can never be contradicted with anumāna or inference (“atīndriyān asaṁvedyān paśyantyārṣeṇa cakṣuṣā/ye bhāvān vacanaṁ teṣāṁ anumānena na bādhyate//” VP-1/38). In such cases there is a prominence of knowledge of unseen factor earned through penance by the seers. When an expert jeweler can determine the jewel and its true nature at the same moment through his expertise vision, it cannot be explained through perception, inference, etc. In such cases the importance of abhyāsa pramāṇa is to be taken into account (“pareṣām asamākhyeyam abhyāsādeva jāyate/maṇirūpyādivijñānaṁ tadvidaṁ nānumānikam//” – VP-1/35). Bhartṛhari feels that the Vedas are the root of all Śāstra-s (“vidhātustasya lokānām angopānganivandhanāḥ/vidyābhedaḥ pratyayante jñānasamskārahetavaḥ// ” VP-1/10). That is, different forms of learning consisting of various important or unimportant literature of this world created by God are the causes of cognition and impression of the people. Meaning in general is emanated (vivarta) from Śabdabrahman which is beginingless and endless (“anādi nidhanaṁ Brahma śabdatattvaṁ yadakṣaram” – VP-1/1). All meanings are emanated (vivartate) but not transformed (na pariṇamate) from Śabdabrahaman just as Sankara admits emanation of all worldly objects from Brahman. There is a great difference between pariṇāma or transformation and vivarta or emanation. Transformation (pariṇāma) is the emergence of an effect which has the same sort of existence as its material cause (upādāna-sama-sattāka-kāryāpattiḥ). Emanation or vivarta is the emergence of an effect which has a different sort of existence from its material cause. A golden ring is the transformation of material cause, i.e., gold and hence the effect “gold” has got the same sort of existence in its material cause, which is also gold. But in case of vivarta a snake which is an effect has got a different sort of existence, i.e., rope, i.e., the material cause (upādāna-viṣama-sattāka-kāryāpattiḥ) (Adhvarīndra 2013). To Bhartṛhari the whole world, all activities, and all verbal behaviors are taken as the vivarta or emanation of the Śabdabrahman (“vivarttate’rthabhāvena jagataḥ prakriyā yataḥ” – VP-1/1). A class inhered with its meaning must naturally be

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understood by sounds (“yathātha-jātayaḥ sarvāḥ śabdākṛti-nivandhanāḥ.” –VP-1/ 15). Moreover, all behaviors connected with meaning are deeply rooted in sound (“artha-pravṛtti-tattvānāṁ śabdā eva nivandhanam.” VP-1/15). Such a science of sound has acquired much importance as a grammar. In other words, the true understanding of the sound is not possible without grammar (“tattvāvabodhaḥ śabdānāṁ nāsti vyākaraṇād ṛte” – VP-1/13). Bhartŗhari in his Vākyapadīya has highlighted the importance of grammar in metaphysical realm. To him Śabdabrahman is the ultimate reality. Vedas are the manifestation of Śabdabrahman which can alone be known through grammar (“Tad vyākaraņamāgamya paraṁ brahmādhigamyate” VP-1/22). The Ultimate Reality is known by some philosophers as Brahmā or Vişņu or Maheśvara, some describe it as an active Prakŗti and some as indifferent Puruşa. This ultimate reality is described by the grammarians as Vāk which, though undivided, non-dual, conscious, and having no stages, is imagined as having different stages like paśyantī, madhyamā, and vaikharī. In order to understand such vāktattva one should be well conversant with grammar and hence grammar has been taken as a medium of the realization of Śabdabrahman or Vāk, which has been endorsed by Bhartŗhari in the same text – “tasmād yah śabdasamskārah sa siddhih paramātmanah/Tasya pravŗttitattvajñastad brahmāmŗtamaśnute” (VP-1/132). That is, the impression of sound or formation of words leads us to substantiate the Ultimate Reality. One’s inclination for the study of grammar actually leads to the realization of nectar like Brahman. The concepts of righteousness and nonrighteousness are determined by the seers through the grammar but not through dry argumentation. The seers can realize what is dharma and what is adharma depending on the Śāstra which is named as vyākaraṇa (VP, 1/30). In other words, the righteousness and nonrighteousness are determined by the Ậgamas or Śāstras which are to be understood through the science of grammar (vyākaraṇaśāstra). The knowledge depends on Ậgama which again is dependable due to having noncorrupt words known through grammar and hence grammar is the staircase of our ascending toward the path of Ultimate Reality. The grammarians do not always admit that their job is only to show the path of rectifying or purifying language. Ultimate end of them is also to direct a man to reach in the realm of metaphysics which is beyond all cognitions (“taddvāram apavargasya vāńmalānāṁ cikitsitam/ pavitraṁ sarva-vidyānām adhividyam prakāśate//” VP-1/14). That is, through it the door of liberation is opened, bad words are rejected, and reveals the metaphysics of all pious learnings. The Vākyapadīya is taken as the earlier place for understanding word taken as a staircase to the entrance of heaven and also royal road to liberation (“idamādyaṁ padasthānaṁ siddhisopānaparvaṇām/ iyaṁ sā mokṣamāṇānām ajihmā rājapaddhatiḥ”// VP-1/16). To Bhartṛhari liberation may not be considered as the ultimate goal to someone, yet through the instrumentality of Vāk he attains a great poetic pleasure generated through the light of the lights (jyotiṣāṁ jyotiḥ) of sound overcoming darkness (tamasaḥ parastāt), which is not at all ignorable in this world. The beginningless and endless Śabdabrahman (anādi-nidhanaṁ Brahma) directs to the Sphoṭa which is taken as revealed by initial sound. The properties of initial sound are imposed in Sphoṭa. The reverberation being originated from the initial

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sound makes the awareness of Sphoṭa intact. Sphoṭa may be taken as prākṛta-dhvani and dhvani is produced by sound according to some scholars as admitted by Bhartṛhari (VP-1/102). Again, dhvani is not possible without Sphoṭa and hence he has mentioned that Sphoṭa is a kind of genre ( jāti) and dhvani is a particular (vyakti) (VP-1/93). Sphoṭa as an initial sound gives rise to the production of other sounds. Hence, the sentence-meaning is taken as an indivisible one (akhanḍavākyārtha) and from this the theory of anvitābhidhāna follows. Sentence-holism and anvitābhidhāna are connected with Sphoṭa theory or Śabdabrahman. The sentence-holism (akhanḍavākyārtha) means understanding of a sentence in a flash but not gradually through the understanding of the meaning of words. As meaning comes from the Sphoṭa, it reveals meaning in one go or flash when the sentence is completed. Due to having such metaphysical presuppositions Bhartṛhari has admitted audible sound (vaikharī) and also inaudible sounds, which justifies his vāktattva.

Conclusions It is already addressed that between language and grammar which one is prior and the relation between them with special reference to Indian grammarian tradition. Secondly, an address is made to the chief contributions of the grammarians to the philosophy of language as such. Grammar has been taken as an aid to detect corrupt word (apaśabda), formation of word, meaningfulness of null-class, akhaṇḍavākyārtha-vāda, anvitābhidhāna-vāda, division of Vāk, Sphoṭa theory, and some epistemological reflections of them. It should be borne in mind that all these theories are interrelated. As the grammarians believe in Sphoṭa theory of Vāk, i.e., the primordial originating point of a word is Sphoṭa, it gives rise to the theory of holistic meaning which again is related to anvitābhidhāna theory. By virtue of being originated from one source its meaning comes into existence in one burst which is indivisible. Due to its indivisibility it becomes sentence-related contextualism, i.e., meaning of the sentence comes into being at the first flash giving rise to wordmeaning afterwards. The grammarians do not always admit that their job is only to show the path of rectifying or purifying language. Ultimate end of them is also to direct a man to reach in the realm of metaphysics which is beyond all cognitions. The Vākyapadīya is taken as the entrance of heaven and also royal road to liberation. That is why they have given much importance to Āgama whether there is no room for corrupt word and misunderstanding of the āgamic statement. The beginningless and endless Śabdabrahman (anādi-nidhanaṁ Brahma) directly indicate the Sphoṭa which is taken as revealed by initial sound. The properties of initial sound are imposed in Sphoṭa. The reverberation being originated from the initial sound makes the awareness of Sphoṭa intact. Sphoṭa may be taken as prākṛta-dhvani and dhvani (noninitial sound) is produced by another sound according to some scholars as admitted by Bhartṛhari (VP-1/102). Again, dhvani is not possible without Sphoṭa and hence he has mentioned that Sphoṭa is a kind of genre (jāti) and dhvani is a particular (vyakti) (VP-1/93). Sphoṭa as an initial sound that gives rise to the production of other sounds. Hence, the sentence-meaning is taken as an indivisible

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one (akhanḍavākyārtha) and from this the theory of anvitābhidhāna follows. Sentence-holism and anvitābhidhāna are connected with Sphoṭa theory or Śabdabrahman. Due to having such metaphysical presuppositions Bhartṛhari has admitted both audible sound (vaikharī) and also inaudible sounds, which justifies his Vāktattva.

Definitions of Key Terms Sphoṭa:

Anvitābhidhāna:

The grammarians believe in Sphoṭa theory of Vāk, which is the primordial originating point of a word. For this reason, a power called Sphoţai which is different from sound (śabdātirikta) and revealer of meaning suggested by sound (śabdābhivyangārthābhidhāyaka), is to be admitted according to the grammarians. Such an indivisible Sphoţa gives rise to the actual meaning, but ordinary word used for the meaning is an auxiliary to the revealing the same. The etymological meaning of the term Sphoṭa is “sphuṭati arthaḥ asmāt,” i.e., that from which meaning is revealed. The sound (nāda) manifests Sphoṭa, and Sphoṭa is the bearer of meaning. That is why sounds are said to be destroyable, momentary, etc., while Sphoṭa is a whole and eternal. The theory of Sphoṭa gives rise to the theory of holistic meaning which again is related to anvitābhidhāna theory. By virtue of being originated from one source, its meaning comes into existence in one burst which is indivisible. Due to its indivisibility, it becomes sentence-related contextualism, i.e., meaning of the sentence comes into being at the first flash giving rise to word-meaning afterward. When the meaning of a sentence is grasped first and afterward the meaning of the words is taken into account accordingly, it is called the theory of

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Apaśabda:

Plutasvara-s:

Abhyāsa:

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anvitābhidhāna. It is Sphoṭa which reveals the holistic meaning of a sentence, which afterward gives rise to word-meanings. It is a grammatical rule which makes a word grammatically well formed. If not, it would be taken as a corrupt word (apaśabda), the usage of which may lead an individual to the path of harmfulness as endorsed by Patañjali in his Mahābhāşya. To him a corrupt word (duṣṭaśabda or apaśabda) arising out of misapplication of grammatical rules or mispronunciations of it particularly in the case of mantra may cause harm to the user or the performer sacrifice. In the same way, this theory can be applicable to the secular world also. If a word is presented in a defective way due to some defect existing in the word itself or in its utterance, it creates a communication gap between the user of the language and the hearer, which may consequentially lead to misunderstanding, misbehavior etc., creating a lot of social problems. The grammarians have admitted a phenomenon of plutasvara (emotional language) as found in calling someone from a distance, in singing a song, or in weeping state. The grammarians have defined it as “dūr-āhvāne gāne rodane ca pluto mataḥ.” Pāṇini has mentioned such plutasvaras in grammar in order to accommodate these usages under language. If a particular language is prevalent in the society, people will call others, sing a song, or weep or lament using the same language, which can never be denied. When a single term is used doubly to mean something, it is technically called abhyāsa. In the terms paţ, car and phar have been used doubly creating the terms paṭpaṭ, carcar, and

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Vaikharī Madhyamā, Paśyantī, and Parā:

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pharphar. These are used in the following manner: a) kathaṁ tvaṁ paţapaţākaroṣi? (Why are you talking instantly and courageously?). According to the Advaitins and grammarians, there are two types of sentence: bearing “divisible related meaning” (“samsṛṣṭārthaka”) and “indivisible meaning (“akhaṇḍārthaka”). The former exists in the sentences like “Bring a cow” (“gāmānaya”), etc. because itgives an awareness of the verb “bringing” of an object “cow.” An entity called Candra remains in the expression: “The excellent shining entity is the moon” (“prakṛaṣṭaprakāśaḥ candraḥ”), which gives rise to meaning without making us aware of the relation. In this case, the meaning of the sentence is not constituted with the relation of the meaning of the terms like prakṛṣṭaprakāśa, etc. The terms like prakṛṣṭaprakāśa, etc. are not apprehending relation between the meanings of the terms, but through these, the essence of the moon is known. It gives the indivisible meaning (akhaṇḍārtha) of the term “moon,” which is nothing but its essence. Four forms of Vāk are Vaikharī, Madhyamā, Paśyantī, and Parā. The gross word remains in the region of throat and hence it is called Vaikharī which is audible to others. That is, Vaikharī is a sort of vibration (nāda) which can easily be audible to others. Madhyamā Vāk remains in the region of heart, and it is not audible due to its subtle nature. A speaker can hear such Vāk himself after closing his eyes. From this, it is established by Bhartṛhari that there is certain sound called Madhyamā, though subtle in nature and remaining in region of

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heart, which is capable of being felt with the help of the closed eye, but not through ear. Paśyantī is more subtle remaining in the naval region of the body, and it is revealed in the meditative state of a yogin as an object of determinate perception. It is in the form of bindu, nāda, and material element of the world. It means that an individual can feel the existence of subtler sound in the meditative state of a yogin. The last one Parā vāk is revealed in the meditative state of a yogin as an entity in the form of indeterminate perception. At this stage, only the feeling or revelation of it can be realized but not in the form of gross sound.

Summary Points “Logic” is to be taken in the sense reasoning or argument. Normally, logic seeks to determine good reasonings from the bad ones. An argument is an expression of an inference expressed in language. All the theories of Pāṇini and Bhartṛhari are well grounded with arguments and reasonings which may be called logical statement as opposed to illogical ones. A grammarian thinks it proper to formulate some rules essential for building a well-formed word (noncorrupt word) and sentence. Later we shall see that if a word or sentence is grammatically well formed, it must have some meaning, though referent may not be found in the external world. Grammar is meant for justifying the public usages available in the society. It proves the meaningfulness of the null-class like sky-flower etc., as they are grammatically well formed. From these presuppositions, it follows the akhaṇḍavākyārtha-vāda, anvitābhidhāna-vāda, division of Vāk, Sphoṭa theory, and some epistemological reflections of them. It should be borne in mind that all these theories are interrelated. As the grammarians believe in Sphoṭa theory of Vāk, i.e., the primordial originating point of a word is Sphoṭa, it gives rise to the theory of holistic meaning which again is related to anvitābhidhāna (sentence holism) theory. By virtue of being originated from one source, its meaning comes into existence in one burst which is indivisible. Due to its indivisibility, it becomes sentence-related contextualism, i.e., meaning of the sentence comes into being at the first flash giving rise to word-meaning afterward. The grammarians do not always admit that their job

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is only to show the path of rectifying or purifying language. Ultimate end of them is also to direct a man to reach in the realm of metaphysics which is beyond all cognitions.

References ‘Atra vākyaṁ dvividhaṁ- samsŗşţārtham akhaṇḍārthaňca iti. Tatra samsŗşţāthaṁ saṁsargagocara-pratīti-janakaṁ, yathā gāmānaya ityādi. Akhaṇḍārthaṁ tu saṁsargā-gocara-pratītijanakaṁ, yathā prakŗşţa-prakāśaḥ candraḥ ityādi’. Br. Medhācaitanya (1993, sakābda) (Trs): Vedāntasāra, Kolkata p. 169. n.d. ‘Karturipsitatamam karma’ Aşţādhyāyī, Sūtra No 1/4/49. Commentary ‘Bālamanoramā’ by Bhattoji Diksit on the above sūtra. n.d. ‘Pluta-pragŗhya aci nityam’ Aşţādhyāyī, Sūtra No 6/1/125. n.d. ‘Sonat prācām’ Aşţādhyāyī, Sūtra No. 4/1/43. ‘Prācāmavŗddhāt phinvahulam’. Aşţādhyāyī, Sūtra No 4/1/160. ‘Udicam vrddhad agotrat’ Aşţādhyāyī, Sūtra No. 1/1/75. n.d. ‘Tatsādŗśyam abhāvaśca tadanyatvaṁ tadalpatā/Aprāśastyaṁ birodhśca nañarthāh şaţ prakīrtitā//’. A Higher Sanskrit grammar and composition, by Dr. Lahiri & Sastri, Calcutta, 179, 1971. ‘Vivakşāvaśāt kārakāņi bhavanti’, A Higher Sanskrit grammar and composition, by Lahiri and Sastri, 137, 1971. “garbhīkṛta-kriyāpadaṁ nāmapadaṁ vākyaṁ prayuñjate, carita-kriyāpadāt asmād arthāvagatiḥ nirākāńkṣā upajāyamānā dṛśyate”. Harivṛṣabha Commentary on Vākyapadīya-2/326. n.d. “kriyā kriyāntarād bhinnāḥ niyatādhārasādhanā/prakrāntā pratipatṛṇām bhedāḥ sambodhahetavaḥ//” VP-2/414. n.d. “sarva-bhedānuguṇyaṁ tu sāmānyam apare viduḥ/tadarthāntara-saṁsargād bhajate bhedarūpatām//bhedān ākāńkṣātastasya yā pariplavamanatā/avacchinatti sambandhastāṁ viśeṣe niveśayan//” VP-2/44. n.d. “vākyaṁ tadapi manyante yatpadaṁ caritakriyam/ākhyātaśabde niyataṁ sādhanaṁ yatra gamyate/tadapyekaṁ samāptārthaṁ vakyaṁ ityabhidhīyate//” (VP-2/326). n.d. “vastutastad anirdeśyaṁ nahi vastu vyvastithitam/sthālyā pacyata ityeṣā vivakṣā dṛśyate yataḥ//” Vākyapadīya-3/7/91. n.d. “yathā ghaţena kāryaṁ karişyan kumbhakārakūlaṁ gatvāha- kuru ghaţaṁ kāryamānena karişyāmīti, na tadvacchabdān prayokşyamāno vaiyākaraņakūlaṁ gatvāha- kuru śabdān prayokşya kuru iti”. Mahābhāşya (paśpaśā ānhika) Bengali Translation and elucidation by Dandiswami Damodar Ashram, Ramakrishna Sangha, Kolkata, 4th edition, 1417 (BS), p. 164. n.d. “yatnenānumito’pyarthaḥ kuśalairanumātṛbhiḥ/abhiyuktatarairanyairanyathaivopapādyate//” VP1/34. n.d. Adhvarīndra, Dharmarāja. 2013. Vedāntaparibhāṣā, Part-1, English translation and elucidation by Gopinath Bhattacharya and Prabal Kumar Sen, p. 100. Kolkata: University of Calcutta & Mahabodhi Book Agency. Pāņinīya-śikşā, 9/42 and also Mahābhāşya (Paśpaśā Ậhnika), Edited by K.C. Chatterjee, Calcutta, 11, 1957.

Bibliography A Higher Sanskrit grammar and composition, by Lahiri and Sastri, 1971. Adhvarīndra, Dharmarāja. 2013. Vedāntaparibhāṣā, Part-1, English translation and elucidation by Gopinath Bhattacharya and Prabal Kumar Sen. Kolkata: University of Calcutta & Mahabodhi Book Agency.

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Aṣţādhyāyīsūtrapāţhah. 2022. Chowkhamba. Samskrita series Office, Varanasi, 6th edition (samvat).ś. Das, Karunasindhu. 2002. Prācīn Bhārater Bhāṣādarśana. Kolkata: Progressive Publishers. Das, Karunasindhu. 2003. Vyākaraṇadarśane Vāgarthaprasangaḥ. Kolkata: Allied Publishers. Joardar, Koushik. 2017. Śabdabrahma. Kolkata: Levant Books. Mahābhāṣya Paśpasā. Bengali Translation and elucidation by Dandiswami Damodar Ashram, Ramakrishna Sangha, Kolkata, 4th edition, 1417 (BS), p. 164. Pāņinīya-śikşā, 9/42 and also Mahābhāşya (Paśpaśā Ậhnika), Edited by K.C. Chatterjee, Calcutta, 1957. Vākyapadīyam published by L.D. Bharatiya Samskrti Vidyamandir, Ahmedabad, 1st edition, 1984. Vākyapadīya Brahmakāṇḍa (Two parts), by Bishnupada Bhattacharya, Paschim Banga Rajya Pustak Parsad, First part, 1985, Second part, 1991. Vākyapadīyaṁ Brahmakāṇḍam. 1975. Chowkhamba, 3rd edition. Yogindra, Sadananda. Vedantasāra Bengali trs and elucidation by Br. Medhācaitanya 1993, sakābda.

Logic in tolka¯ppiyam

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logical Techniques in tiṇai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic as aḷavai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . aḷavai and Ethicality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic as tarukkam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . tarukkam and kāṇṭikai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . kāṇṭikai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Five Members of kāṇṭikai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mutumoḻi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . mutumoḻi and nayam/niyāyam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vākai . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Debate in the Village Assembly . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logic, Text, and tolkāppiyam . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

tolkāppiyam shows us that logic had an important place in the primal society (known as tiṇai). Such primal cultural practices as “tarukkam,” “vākai,” analogical reasoning, the use of the criteria of knowledge (“aḷavai”) in premarital life situations and in oral texts (such as mutumoḻi and kāṇṭikai), and also the primal social institution, namely, the assembly (avai) where public debates were held, were all part of the philosophical tradition of this society. Such philosophy embraced logic, which had rhetorical as well as epistemic functions. If rhetorical logic was persuasive (as in tarukkam) and contestatory (as in vākai in combat and in the assembly), epistemic logic (aḷavai) was validative. In fact, early Tamil logic N. Selvamony (*) Madras Christian College, Chennai, Tamil Nadu, India Central University of Tamil Nadu, Thiruvarur, Tamil Nadu, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_6

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was a complex discipline not easily distinguishable from philosophy (especially, epistemology and ethics) and rhetoric.

Introduction Unlike other grammar treatises of the world – Panini’s (4th c BCE) Ashtadhyayi, Dionysios Thrax’s (170–90 BCE) The Grammar, and Varro’s (116–27 BCE) On the Latin Language –tolkāppiyam (hereafter, tol.) is not devoted to the grammar of language alone. It deals with several matters such as (the grammar of) language, the primal society called “tiṇai,” the Inner (akam) and Outer (puṟam) aspects of life, analogy, emoting, poetics, and philosophy among others. In this regard, it belongs to no known textual genre. Besides being unique, it is also layered with material belonging to both the primal and state societies. In this respect also this text differs from the grammar texts of the rest of the world, in that the latter texts originated from state societies whereas tol. originated in a primal society (which warrants its antiquity) and later accommodated material from the state society also. Its primalness consists in its encyclopedic, anti-genre naṭure and also in its treatment of material belonging to primal society. Only a careful scrutiny can tell which material belongs to which age. What is distinctly primal about the text is the material pertaining to the primal society called “tiṇai.” In both the Inner and the Outer domains of life, logic has its own roles to play, and it is inseparably related to philosophy itself.

Logical Techniques in tinai ˙ Let us show how logic has been a part of philosophy in the Inner as well as Outer domains of life. If knowledge of truth is possible through logic-based philosophical vision (otta kāṭci, tol. III. 9. 112: 1), it is worthwhile considering the evidence of such knowledge. Of the three values, life (uyir), avoidance of evil (nāṉ), and steadfastness (kaṟpu), the ancient Tamil people knew that steadfastness was the highest through philosophical vision (tol. III. 3. 23: 1–3). Such philosophical knowledge enabled lovers to be steadfast in their love for each other. This precept pertaining to the Inner life is illustrated by a commentator with the help of an ancient Tamil song (kuṟuntokai 31). In this song, a female lover who intends to convey to her confidanté her love for a man draws her friend’s attention to the loose-fitting bangles on her arm. The confidanté infers the truth of love between them from the condition of her friend. Such inference (uyttukkoṇṭuṇarttal, tol. III. 9. 112: 23) is a logical technique (utti, tol. III. 9. 112) adopted for ascertaining steadfastness (in love). If the logical technique of inference alone is employed in the song pertaining to the Inner domain, the techniques of analogy (oppak kūṟaḷ, tol. III. 9. 112: 9) and inference are deployed in a situation of the Outer domain of life as evident in a song in puṟanāṉūṟu (214). In this song, the chieftain kōpperuñcōḻaṉ has decided to take his life by sitting until death, facing the north. He says that philosophical vision yields us the truth that

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we should do only good if we want the highest reward in our afterlife and that those who doubt this truth lack firmness of mind (kaṟpu). This truth is conveyed through the logical techniques of analogy and inference. The speaker draws our attention to the fact of everyday life, wherein it is quite possible that an elephant trapper may return with nothing less than an elephant, whereas the trapper of small birds may return merely empty-handed. As the elephant stands for the highest good we can possibly expect in our life after death, we are persuaded to infer that we need to seek the highest good and do only good in this life because that alone can bring us the reward we seek in our afterlife.

Logic as alavai ˙ Significantly, some logical techniques like analogy and inference are also aḷavaikaḷ or criteria for validating knowledge. Therefore, it is apt to dwell on the relation between aḷavai and logic. The term “aḷavai” derives from the verb, “aḷa,” to measure (Selvamony 1988–1989, 1996a: 147–165). But the nominal form, “aḷaviyal marapu” (aḷaviyal), refers to the logical tradition. In the verse where it figures in tol., it characterizes the utterances of the six personae – the seer, the confidant, the confidanté, the foster mother, the hero, and the heroine – in the premarital period (tol. III. 8. 186). Let us see which meaning fits this occasion. The term “aḷaviyal” occurs in the prosodical context (tol. III. 8. 1: 4; 8. 186), wherein it means “regulation regarding the number of lines in different kinds of stanzas” (Tamil Lexicon), and also in the speech-attribution context (tol. III. 8. 186: 3). The commentator pērāciriyar does not gloss the term, which occurs in the speech-attribution context. But iḷampūraṇar and aṭikaḷāciriyar do, and they take it to mean “by convention, mixing freely with others.” They do not give the meaning of “[poetic line] convention” because that is inapt here. We may note that the meanings given by iḷampūraṇar and aṭikaḷāciriyar are based on the verb form, “aḷavu,” meaning “to mingle” (Tamil Lexicon), and this word occurs twice in tol. (III. 1. 58 and III. 4. 5: 19). In the former instance, it does not refer to people mixing freely but what befits (tradition), and in the latter it does refer to people mingling freely. Now, if we take the phrase, “aḷaviyal marapu,” to mean “mingling with people freely,” it would mean that the other personae such as the mother, the father, the helpers, and others are unaffable. It would also denigrate the speaker-personae in the marital period: bards, actors, danseuse, latter wife, sages, and bystanders. This is not tenable. Further, affability is not a gainful virtue in the clandestine premarital stage in the lives of the central personae because they meet in trysts without the knowledge of their parents, relatives, and villagers. The confidanté cannot afford to be affable with a hero who postpones marriage endlessly. The foster mother has to be strict with a heroine who slips out of the house at the slightest pretext to meet her lover. The confidant (hero’s friend) is never seen being affable with the heroine. If affability is mingling with each other freely, we never find all the six mingling freely. Therefore, affability is not the proper meaning of the phrase, “aḷaviyal marapu” when it refers to the personae (tol. III. 8. 186: 3).

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The meaning “mixing freely” was attributed to the phrase under discussion, because the commentators regarded the word “aḷavu” in its verbal form. Instead, if we take it in the nominal form, we find that “aḷavu” means, among other things, “laws of reasoning,” “dialectics” (tirukkuṟaḷ 725; Tamil Lexicon; aḷavu, laws of reasoning + iyal, study ¼ aḷaviyal, study of the laws of reasoning, dialectics). If we take “aḷaviyal marapu” to mean logical tradition, the verse means that the six personae named in the verse, who are skilled in logical tradition, are the rightful addressers in the premarital stage of the life of the central personae. Because love affair is clandestine in the premarital stage, it can neither be shared with the heroine’s mother, father, and brother and other relatives nor can it be furthered by the latter personae. Dealing with such an affair requires logical acumen. Endowed with logical skills, the seer (pārppāṉ; pār, to see + p + p + āṉ, masculine suffix, pārppāṉ, one who sees, seer) weighed the pros and cons of all actions concerning the lovers. He usually played the role of the messenger (akanāṉūṟu 337: 7–10). In fact, in the primal society, he was the precursor of the ambassador of the state society. He took oral and written messages to lovers and chieftains and was probably compensated for his services (akanāṉūṟu 337: 7). Though writing is usually a characteristic feature of the state society, it was known in Tamil primal society, and it served the purposes of the ordinary people (Mahadevan 2003). This meant that the seer was literate and skilled in the art of writing on palm leaves. Besides literacy, he was skilled in the art of debate too. His rhetorical skills were sought after by chieftains who wanted to communicate messages to their opponents. Brevity, apparently, was the soul of his wit (puṟanāṉūṟu 305). He had to present his case cogently in a logical manner. The confidanté had to be clever, ingenious, and resourceful to deal with the heroine and the hero in the premarital stage. Of all the six personae, it was she who was described as the most skilled in scrutinizing situations and counseling (tol. III. 3. 36). Her favorite law of reasoning was “tradition” (ulakurai, tol. III. 3. 24: 8). When she disclosed the love affair between the heroine and the hero, she adopted a communicative strategy called “aṟattoṭunilai” (literally, “standing virtuously”), which employed one of the laws of reasoning called “ētīṭu” (tol. III. 5. 12; ētu, reason + īṭu, laying down ¼ ētīṭu laying down reason), validating a point with an adequate reason. During the premarital stage, the confidant admonished the hero by reminding the latter that he ought to be principled and steadfast. To the tiṇai people, one of the favorite metaphors for steadfastness is the land (which is regarded as the primary aspect of tiṇai). In the following song the confidant tells us that he infers this characteristic of the land. As we know, inference or karutal is one of the aḷavaikaḷ: tērōṉ teṟukatir maḻuṅkiṉum tiṅkaḷ tīrā vemmaiyoṭu ticainaṭuk kuṟuppiṉum peyarāp peṟṟiyil tiriyāc cīrcāl kulattil tiriyāk koḷkaiyuṅ koḷkaiyoṭu nalattil tiriyā nāṭṭamum uṭaiyōy kaṇṭataṉ aḷavaiyiṟ kalaṅkuti yeṉiṉim maṇtiṇi kiṭakkai mānilam uṇṭeṉak karuti uṇaralaṉ yāṉē

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(tol. III. 3. 11: 8; koḷkai: principle, proposition; karuti uṇartal ¼ inferring) (Even if the hot rays of the chariot-sun lose their heat, Even if the moon becomes uncontrollably hot, Even if the quarters of the earth shake, Unchanging and wholly noble is your lineage And unremittingly principled and committed to well-being art thou. If thou should be distraught at the mere evidence of appearance, I shall infer the lasting existence of this compactly lying wide earth. Trans. Nirmal Selvamony)

Both the confidanté and the foster mother excelled in the art of inference, which helped them understand the changes in the behavior of the heroine. When the confidanté felt that the hero should meet the heroine at a later time, she turned him away by appealing to tradition (ulakurai, tol. III. 3. 24: 8) or delayed the meeting by giving a reason by way of suggestion rather than direct statement (tol. III. 3. 24: 15). The foster mother inferred that her ward had been lovestruck from the following signs: strange body odor, eating less, growing thin, sleeplessness, and not being fond of making herself up (tol. III. 3. 25. iḷampūraṇar’s commentary). Though both the foster mother and the biological mother confirmed their doubts about the condition of their “daughter,” it is the former who related to the heroine more directly than the latter (tol. III. 3. 26. nacciṉārkkiṉiyar’s commentary). Therefore, it was the foster mother who would have more opportunities to reason out her speculations about the condition of the heroine than the biological mother, and hence the former was mentioned as one of the six personae skilled in logical conversations in the premarital stage of the lovers. The hero and the heroine deployed their logical skills in their exchanges with the confidanté and also in their own conversations. Stunned by the beauty of the heroine when the hero met her first, he was in doubt as to whether what he saw before him was a goddess or peafowl, or a human (tol. III. 3. 4; tirukkuṟaḷ 1081; 49). To dispel this doubt, the hero had to fall back on his textual knowledge as well as inferential power (tol. III. 3. 4. nacciṉārkkiṉiyar’s commentary). In a song in kuṟuntokai, to ascertain the fact that his lover’s tresses are more fragrant than any flower on earth, the hero challenges the bee (who probably has the best knowledge of flowers) to prove him wrong. In this exhortation, he asks the bee to be impartial by speaking about what it has perceived. Evidently, the criterion of knowledge referred to is perception (2: 2). More than anybody else, it was the heroine who emphasized “virtue” (aṟam) in the premartial stage. She invoked tradition (represented by “noble persons who let the wicked mend their ways on their own,” naṟṟiṇai 116: 1–2) when the village women gossiped about her character. She appealed to the public forum (which served as the court as well as a debating platform) in more than one song. In kuṟuntokai 25, we find the heroine speaking of the water bird as the witness to her premarital union with her lover and making an implicit reference to the tradition of the elders, namely validating an act in the public forum. In another song (kuṟuntokai 36), she refers to her lover’s heart as the public witness. She makes her case by showing the cause (ētu) for enduring the separation from her lover before marriage. She says that it was the Gloriosa superba (which belonged to her lover’s hill) she

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nurtured and fondled during his absence, which enabled her to endure the separation (kuṟuntokai 361). Evidently, she deployed inference quite ably in this stage of her life (kuṟuntokai 183). So far we have tried to show how the six personae (seer, confidant, confidanté, foster mother, hero, and heroine), the addressers in the premarital stage, are expected to be skilled in the logical tradition. The name “aḷaviyal” for logic derives from the word “aḷavu,” meaning criterion to validate knowledge and action. Later texts speak of ten such criteria: perception, inference, primary text, presumption, context, tradition, existence, nonexistence, and exclusion (maṇimēkalai 27: 14–56). Now let us see which of these figure in tol. (Selvamony 1988–1989: 14–15; 1996a: 147–165) and, when they do, how. When the hero meets the heroine for the first time, as mentioned earlier, his doubt about her stunning beauty is dispelled primarily by perceptual evidence (“kāṭṭal,” tol. III. 5. 52) like checking whether her feet are on the ground, whether her flowers have faded among others (tol. III. 3. 4; tirukkuṟaḷ 1081). While speaking of understanding qualities such as similarity, form, beauty, and so on, tolkāppiyar says that these can only be inferred (“neñcu koḷvana,” tol. III. 5. 52) rather than perceived. A third criterion is “uvamai” (analogy) and tol. has an entire chapter on it (tol. III. 7). In fact, tolkāpiyar himself uses this knowledge-criterion in his text. He tells us that a homeward bound hero does not tarry on his way because he has a birdlike horse that serves him like his own heart (tol. III. 4. 53). Besides serving the function of reason (ētu), analogy also gives us new knowledge about the manner of the hero’s return. Another rare use of analogy may be found in a rule pertaining to the heroine. In the premarital stage, she does not express her desire for the hero in words, but only through gestures and facial expression. Her desire oozes out of her body even as water does from a new mud pot (tol. III. 3. 28). The fourth criterion is “mutal nūl” (tol. III. 9. 96), the primary text composed by a sage who could perform an act without being overwhelmed by any aspect of the act itself (such as the end or instrument of act, tol. III. 9. 96). “iyalpu” is context, the knowledge-criterion that validates the utterances of various speakers (tol. III. 8. 204). ulakurai (tol. III. 3. 24: 8) is tradition, which compels the confidanté to turn away the hero who desires a meeting with the heroine. Yet another criterion of knowledge, “vantatu koṇṭu vārātatu uṇarttal” (tol. III. 9. 112: 7) consists in assuming a cause in order to explain an otherwise inexplicable fact or phenomenon. For example, when a person fasting by day does not lose weight, it may be presumed that he eats at night. This aḷavai can be differentiated from inference (neñcu koḷal). The latter proceeds from evidence (smoke in the hill) to fact (fire in the hill, which has to be explained), whereas in presumption, we proceed from fact (which has to be explained; not losing weight while fasting by day) to evidence (which explains it; eating by night) (Sundaram 1979: 94). The other criteria of knowledge are uṇmai (tol. I. 9. 25: 1; existence), vārātataṉāl vantatu muṭittal,” (tol. III. 9. 112: 6; non-existence), and “oḻiyicai” or exclusion. If “existence” consists in ascertaining something by virtue of its presence or inclusion or prior existence, “nonexistence” allows us to know something by virtue of its absence (tol. II. 2. 16. cuntaramūrtti, 221) and exclusion not by absence but by avoiding inclusion (tol. II. 7. 4, 5). Of all these criteria, perception, analogy, and inference seem to be the primordial ones.

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alavai and Ethicality ˙ So far we have tried to show how “aḷaviyal” or logic served epistemic purposes. But if we take a deeper look at the meaning of the term“aḷavu” (measure), we also learn that it connotes ethicality. Measure is setting limits. As a vital part of philosophy, “aḷavu” is not only setting limits to truthfulness of propositions but also to the rightness, goodness, and appropriateness of actions (if the latter three are the basics of ethics). In other words, aḷavu is as much about knowledge (ñāṉam) as about action (karumam/karma, from karu, to generate). On this basis, we may contrast logic nurtured by tol. with the logic in non-Tamil sources. The latter deals only with the epistemic aspects of “aḷavu” (validating criteria) and not with its ethics, whereas the former does with both. It must be noted that logic has progressively disengaged itself from the epistemic also but early Tamil logic is a complex entity in which the epistemic is an intrinsic part (cf. Matilal 1999: 1–2).

Logic as tarukkam Besides being known as “aḷaviyal,” logic was also known by other names. If it went by the name tarukkam (tol. III.1.53: 4) in the Inner domain, it did by the name “vākai” in the Outer. Even today “tarukkam” is the most common word for logic in Tamil, and it means “debate” in which one tries to exalt oneself (tarukku, to enlarge, to exalt oneself + am, nominal suffix ¼ tarukkam). Through debate one tries to show how one’s claim or argument is stronger than the opponent’s. According to tol. (III.1. 53), a lad who tries to woo a girl who has not yet responded positively will resort to tarukkam. We find, in an ancient Tamil song, such a strapling youngster smitten by the beauty of an unescorted lass. He follows her trying to get her attention and, if possible, lure her into a conversation. Here is his initial soliloquy followed by his words addressed to the girl: Here, this dear daughter of distressed parents; her will I try tarrying with wordplay. “Hello dear, listen: like a fine-plumed swan, like a beautiful peahen, like a messenger pigeon, is your great beauty, oh shy girl with captivating look of a deer, do you know or know not you discomfit the ones that look at you? ... As if distraught, you do not know how others suffer. You walk away silently; listen, you can’t be blamed, nor can your kin be who let you stroll in the street. Only the king, who did not insist on a drum message required when a tusker in rut was led to the waterhole, is to blame.” (kalittokai 56: 11–18; 28–34)

The term “wordplay” is significant. In fact, it is the essence of logic as tarukkam; it is more wordplay (collāṭal) rather than word game (col viḷaiyāṭṭu).

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A game (viḷaiyāṭṭu) has to bring about a result (vākai, game; from viḷai, result + āṭṭu, play ¼ viḷaiyāṭṭu, [result-oriented] play), whereas play (āṭal) does not necessarily move toward a result. Generally, tarukkam is all about wordplay that does not stint overstatement and provocative remarks. A lad who follows a pretty lass who does not respond to his advances can neither engage her in a dialogue nor get answers from her to his questions. All he can do is perform a verbal dance, which will not produce “results” (viḷai) like answer or reply. So, his words are but a failed attempt to make monologue dialogue. But the presence of the silent listener is essential to sustain the monological nature of the wordplay. As in the following song, the speaker (the male lover), poses questions that are provocative enough by virtue of their hyperbolic (tarukku) nature. The addresser exaggerates the beauty of the addressee, and advances accusing generalizations. His questions “Do you know or know not. . .?” presume that the girl is guilty of the offences he names. In fact, these are pseudo-questions because he does not really expect any answer from her. Assuming that the girl answers in the affirmative (that she knows that her beauty distresses other people in public places), the lad could cite that admission as evidence of her culpability. On the contrary, if she answered in the negative, he would still allege that she is insensitive to the suffering of others. However, to our surprise, he exculpates her and her kinfolk. Therefore, we know that he did not mean to say that she was guilty of wrongdoing and that his questions about the girl’s culpability are merely rhetorical. This logical practice, a part of the Inner life (akam) of the primal society, affirms the rhetorical nature of tarukkam.

tarukkam and ka¯ntikai ˙˙ The rhetorical or persuasive nature of the young man’s tarukkam also consists in trying to woo the girl, and, if possible, into a dialogue. Such tarukkam, strangely, appeals to ethics (tol. III. 1. 53: 3). To provoke the innocent girl who passes by, the lad makes it appear to her that her walking about the street is a crime. However, he finally exculpates her probably because he realizes that she is after all too young to fall in love or reciprocate it. But a careful consideration of his allegation reveals that the argument is based on a proposition that is phrased in the form of questions. Teasing it out of his questions, we have the following five-part argumentative text: When you walk about in public places, you are guilty of destroying the composure of male onlookers (proposition/kōḷ).

This proposition can be validated only if we advance the following reason: Because the beauty of young girls walking about in public places destroys the composure of the male onlookers (reason/ētu).

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To strengthen his argument (tarukkam), the young man attempts analogical reasoning: Whenever a beautiful young girl walks about unescorted in a public place, she disorients the male onlookers like a rutting tusker which is destructive in public places (illustration/ eṭuttukkāṭṭu).

The argument gains more force when the proposition is linked with the reason, and this yields the following: You walk about in public places unescorted and destroy, with your stunning beauty, the composure of the male onlookers (application/naṭai).

Now the young man has arrived at the following conclusion: Therefore, you are guilty of destroying the composure of the male onlookers when you walk about in public places unescorted (conclusion/muṭipu).

Though the abovementioned conclusion is what his argument led to, he generously exculpates the girl and finds the chieftain guilty.

ka¯ntikai ˙˙ The five-part argumentative, syllogism-like text underlying the youth’s argument is called “kāṇṭikai” (tol. III. 9. 103; Selvamony 1990a, 1996a: 100–119, 2000; Kandaswamy 1999) because it is a method of seeing. The last evidence for identification of kāṇṭikai with a type of syllogism is found in iḷampūraṇar’s [11th c ACE; aruṇāccalam 24] commentary of tol. [III. 9. 104]. Thereafter, the original meaning of the term was first restored and presented in the present author’s 1990 paper, “The Syllogistic Circle in tolkāppiyam.” Nevertheless, Kandaswamy’s 1999 essay, which takes kāṇṭikai to be a type of commentary using three syllogistic members (1999), neither cites nor disputes Selvamony’s “finding” that kāṇṭikai is a syllogistic text (1996a). It is an Old Tamil nominal form, which means, “seeing” (and its verbal form, “kāṇṭikā” means “look,” kalittokai 99: 9, 12, 15; its other cognate forms are “kāṇṭai,” you will see, paripāṭal 8: 85; kalittokai 12: 14; and “kāṇṭum,” we will see, puṟanāṉūṟu 173: 9. From the verbal base with the short vowel, “kaṇ” derives the form “kaṇṭikum” which means “we saw,” naṟṟiṇai 20: 1). As a kind of text which helps see the intended meaning clearly by arguing a proposition using reason and illustration and connecting them logically, kāṇṭikai is a method of seeing. Today “kāṇṭikai” (tol. III. 9. 103; 101: 2) is known as “mukkūṟṟu muṭipu” (Mahadevan and Shanmugasundaram 146), and this name literally means “three-part conclusion.” Obviously, the latter term is an inapt neology modeled on the Aristotelian three-member syllogism, which does not apply to the five-member “argumentative text” in tol. Further, the Aristotelian syllogism is a deductive form of argument with two premises and a conclusion:

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• Major premise: All mammals are warm-blooded • Minor premise: A whale is a mammal • Conclusion: A whale is warm-blooded The premise of the Aristotelian syllogism Aristotelian syllogism is never about a specific entity. It can only be in any one of the following forms: All. . . (universal affirmative)/No. . . (universal negative); Some. . . (particular affirmative)/Some are not. . . (particular negative). In contrast, the premise (proposition) of kāṇṭikai can be either particular or universal: This hill. . . (particular affirmative)/This hill is not. . . (particular negative); Sound is eternal (universal affirmative)/Sound is not eternal (universal negative). For this reason, we cannot describe the early Tamil form of argument as an inductive or deductive one. Therefore, the name “syllogism” (which is always deductive) is not appropriate to the Indian kāṇṭikai. Though kāṇṭikai has been in use even before the time of tol., the term itself is used in medieval and modern Tamilology, not in the sense of a syllogism-like argumentative form, but as a type of commentary. As opposed to the detailed commentary known as virutti urai, kāṇṭikai is used as a name for brief commentary. Such a definition of kāṇṭikai is found in the medieval Tamil grammar text called naṉṉūl (22). According to this text, kāṇṭikai is a device to bring out the inner meaning of cūttiram, and it consists of five parts: theme (karuttu), word meaning (patapporuḷ), example (kāṭṭu), question (viṉā), and answer (viṭai) (22). As a type of commentary, kāṇṭikai retains two members of the original argumentative text: proposition and example. But instead of arguing from the proposition to a conclusion, the commentary explains the meaning of the proposition. The four parts of the commentary, namely theme, word meaning, question, and answer, serve the purpose of explanation. We may note that the Aristotelian syllogism and Indian kāṇṭikai take for granted our understanding of the proposition. It may be surmised that kāṇṭikai came to be thought of as a type of commentary because the propositions were at times obscure or cryptic. Explanation of the proposition was adequate to the purpose of commentary as a prose genre. Commentary is not primarily an argument or debate though the latter may feature in it at times. However, it is evident that the expository text was confounded with the argumentative one, and this is most probably because the post-tol. grammarians and other scholars seem to have lost sight of the fact that the major function of kāṇṭikai was to argue the proposition and arrive at a conclusion about its assertion. Though the scholars of the post-tol. period did not call the argumentative text kāṇṭikai, they did make use of it either partially or wholly in literary and commentatorial tradition. This is apparent in Tamil didactic literature (see Jayaraman’s chapter in this section), the epics, and also in the commentaries of tol. Among the epics, nīlakēci and maṇimēkalai (29: 57–63) refer to the five-member kāṇṭikai without calling it by its Tamil name and also mention names of its parts. The former uses the Tamil names for three of the five limbs – pakkam (702: 5; kōḷ, 414: 4; 707: 4; mēṟkōḷ, 753: 2; 763: 3; kuṟi, 870: 4), ētu (831; ētuppōli, 367), and kāṭṭu (866: 2, instead of eṭuttukkāṭṭu) – while the latter does two (29: 57–63). cūttiram is the name for proposition in tol., but the equivalent pakkam also occurs in the text as a division of puṟattiṇai (tol. III. 2. 20, 21, 24, 35).

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All commentators of tol. have adopted the argumentative method in their commentaries, but it is only iḷampūraṇar who helped us revive the original meaning of the term “kāṇṭikai” (tol. III. 9. 104; Ibid.). It is quite plausible that he had access to some source, which preserved the original meaning of the term in his day. Unfortunately, this source is unidentifiable today. The object of the term kāṇṭikai has been known in Tamil tradition for more than 2000 years, but the term kāṇṭikai itself, in its original sense, has been preserved in tact only in tol. In marapiyal (chapter on traditions) of tol, we have two verses, which explicate it. paḻippil cūttiram paṭṭa paṇpiṉ karappiṉṟi muṭivatu kāṇṭikai yākum (tol. III. 9. 103). (kāṇṭikai is that which concludes in explicit terms the faultless proposition as it is laid down. Trans. Nirmal Selvamony) viṭṭakal viṉṟi virivoṭu poruntic cuṭṭiya cūttiram muṭittaṟ poruṭṭā ētu naṭaiyiṉum eṭuttuk kāṭṭiṉum mēvāṅ kamainta meyneṟit tatuvē (tol. III. 9. 104). (Fittingly does kāṇṭikai lead to truth, by employing reason, application and example to conclude the proposition that has demonstrated, in a detailed manner, without digression. Trans. Nirmal Selvamony)

The Five Members of ka¯ntikai ˙˙ The five distinct parts of kāṇṭikai are cūttiram, ētu, eṭuttukkāṭṭu, naṭai, and muṭipu. Each of these is a kaṇṭu, a part, which joins with the other parts to form a coherent argumentative text (nūl, tol. III. 9. 103, 104) not unlike an ornament (kaṇṭikai) of beads (kaṇṭukaḷ). Let us consider each of these briefly. The first member, cūttiram, is a compound word (cūḻ, to deliberate + tiram, firmness ¼ cūttiram, firm deliberation), which means, “that which is made after careful deliberation.” It must be pointed out that the term can also mean the textual form (nūṟpā) in which the text is couched. But in logic, the term means only the text and not its form. The proposition being a cūttiram, which has to be concluded in five steps, is a carefully laid down statement. When it expresses a position or view held by the debater, it is known as “kōḷ” (literally, that which is held; koḷ, to hold; koḷ > kōḷ; koḷ + k, euphonic augment + ai, nominal suffix ¼ koḷkai holding, principle, position), and when it expresses a belief, it is known as “matam” (from matu, strength; matu + am a nominal suffix ¼ matam, strength, position held strongly, belief, religion based on belief). Though the term pakkam is a synonym of cūttiram, it specifically refers to the locatee (or the locatable; “the unique characteristic of a singular locus,” Matilal 1999: 26–27; nilai) in a locus (“which may be a place, or a time or even an abstract object,” Matilal 1999: 26; nilam). The second member of kāṇṭikai is ētu, which means reason as well as cause. This term derives from the Tamil interrogative base, e (as in “evvaḻi,” tol. II. 2.10: 3; e, which + vaḻi, way ¼ evvaḻi, which way; e, which + t, formative + u, enunciative [Gnana Prakasar 405] ¼ etu, which; etu > ētu, which, cause). The proposition,

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“sound is not eternal because it is caused by effort,” states the reason why sound is not eternal. tol. refers to this type of reason as “iṉṉāṉ ētu” (tol. II. 2. 13: 7, cuntaramūrtti) wherein the reason is indicated by means of the case marker, “iṉ.” When ētu is cause, it goes by the name “kārakam,” and tol. denotes this as “ataṉiṉ ātal,” meaning “becomes because of” (tol. II. 2. 13, cuntaramūrtti). This is instrumental cause. For example, if “business made the British prosperous,” wealth was the direct cause of prosperity. Commentators of tol. also speak of three other types of cause: mutal, tuṇai, and nimittam. mutal is what Aristotle called material cause. For example, clay is the material cause of the pot. tol. denotes this type by the phrase “ataṉiṉ ātal.” The term “mutal” is also used in tol. in the sense of “the primary” for place-time, the preconditions for the existence/action of anything. The latter sense has to be distinguished from the former. The second is the concomitant cause (tuṇai, that which accompanies), which accompanies the action or process throughout the realization of the effect, as the wheel in pottery. The instrument necessary for producing an effect is such. The third, namely nimittam, is said to be the efficient cause, which brings about the effect, as the potter in pottery. The third member of kāṇṭikai, namely eṭuttukkāṭṭu or illustration, is a composite one consisting of a linking axiom (what Matilal calls “the general principle,” 1999: 4–5) such as “if X, then, Y” and an example. Both the linking axiom and the example serve the purpose of illustration. As the axiom has the force of both the preceding members, namely reason and proposition, it performs the function of linking the two halves of kāṇṭikai like a middle link (naṭu). Coming after the axiom, the example “holds up to show” or illustrates (eṭuttukkāṭṭu) an instance of the axiom. Being a member of the set to which the proposition belongs, the instance exemplifies the axiom and helps us see the meaning of the proposition distinctly. In this respect it is also nayam (>niyāyam; see the present author’s essay, “Logic in nīlakēci and maṇimēkalai” in this volume), an impartial middle entity (naṭu) between the proposition and the conclusion. Examples can be either affirmative or negative. Affirmative axiom is expressed as “If X, then, Y”: “Wherever there is smoke, there is fire.” Negation is expressed by the axiom, “If not-X, then, not-Y (whatever has no fire, has no smoke) and the analogy follows the affirmative or the negative axiom in order to strengthen the case: “like a fireplace” (where smoke entails fire), or “like a tank” (where smoke does not entail fire). Analogical eṭuttukkāṭṭu may be distinguished from comparison. Comparison looks for similarities between two known entities, say, the South Indian flute and the Hindustani flute. But analogy is comparison of a well-known entity with another which is not such for the purpose of explanation or clarification. By means of analogy (an example), one attempts to understand the unknown animal kavayamā (maṇimēkalai 27: 42) with the help of the known animal, namely the cow (tol. III. 6. 1. pērāciriyar commentary). However, analogy only suggests that it does not prove anything (cuntaram 92). But analogical knowledge is quite significant in primal society which did not aim at precision and accuracy always. The latter were, in fact, epistemic ideals of the state society. We are told that analogy was the oldest device for providing knowledge of an unknown entity and consequently, the oldest figure of

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speech too (ilakkuvaṉār 457). Probably, analogy (uvamai) is the source of inference (uyttuṇartal or karutal) and logic (aḷavai) itself. No wonder, this is the only epistemic device (which later became a figure of speech) treated in a separate chapter in tol. While discussing the conventions regarding implicit analogy, tol. rules that the heroine’s analogy will be based on what she knows from direct experience, whereas the confidanté’s will derive from any entity within her tiṇai, and the hero’s will be based on the breadth of his knowledge (tol. III. 7. 26, 27). This convention is legitimate if we remind ourselves that analogical knowledge is possible only if we have certain knowledge of an entity, which will form the basis for understanding the unknown or little known entity. For example, the heroine may use the vēṅkai tree as the basis of her analogy because the tree is found in her own garden but not the behavior of a crocodile because she may not have intimate knowledge about the latter. But such knowledge may be associated with the confidanté because she is expected to be knowledgeable about things not of her own house and garden but also of those of her whole tiṇai. Analogy, like “reason” (ētu), has been a part of the tradition of reasoning among the early Tamil people of primal society for a very long time. A careful look at the structure of kāṇṭikai will show us that these are the two members which are of vital importance. It is no accident that the commentator has to take us to the age of the hunter-gatherers in order to explain analogy. He has to invoke a time when there were still animals in the jungles the forest dwellers were not familiar with. They had to know the unknown creatures with the help of the known. Significantly, this is an epistemic or heuristic method rather than a device that served debates. Analogy, we may aver, is a very primitive method of knowing the world in which one lived. Its importance may not be fully appreciated by us at this point in the history of humanity, because there is next to nothing that is not “known” (probably only superficially though) about this planet now. Probably, analogy may now regain its full heuristic force when humans try to understand the Moon or Mars or some other heavenly body of which our knowledge is either nonexistent or meagre. We may imagine such a time on the earth itself when humans were still busy trying to understand their own surroundings by using the device known as analogy. Later on, we find that analogy loses sight of its original epistemic function and serves a different function called “embellishment.” Now, it became a branch of grammar called “embellishment” (aṇi ilakkaṇam). We do not find any such thing in tol. In the latter text, analogy is a basic method of knowing the world, even as “body language” (meyppāṭu) is of conveying that knowledge to others. For this reason, both deserved separate chapters for analytical treatment. The presence of primal analogy in the argumentative text shows the antiquity of the latter also. The next member applies the axiom in the illustration to a particular instance and hence it is called “application.” But the Tamil word “naṭai” connects this member with the example, which is the naṭu or the middle entity of the argument, serving the purpose of justification (nayam/niyāyam) in an impartial manner (naṭu, impartiality, being in the middle + ai, a nominal suffix ¼ naṭai, vestibule or iṭaikaḻi, a connective middle between two spaces, Tamil Lexicon IV, 2145; the middle member). Even as the example connects the general and the particular, naṭai also does much the same

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thing, and, for this reason it is also called “upanayam” (literally, secondary nayam) in later texts like maṇimēkalai (29: 57, 62, 109). In Matilal’s words, the fourth step is “the showing of the present thesis as a case that belongs to the general case, for reason or evidence is essentially similar to the example cited” (4). If example is nayam, application is upanayam. The fifth member establishes the locatee (fire) of the particular locus (namely hill) with evidence, leading to the conclusion (muṭipu) that there is fire in the hill. In fact, the conclusion is a restatement of cūttiram, the unvalidated proposition. But the restated proposition is validated by three members, reason, illustration, and application.

mutumoli ¯ So far we have explained the five members of a normative kāṇṭikai. But how did this argumentative text originate? In this regard, we might consider the type of text called “mutumoḻi” (literally, “old saying”), described by tol. in the following manner: nuṇmaiyum curukkamum oḷiyuṭai maiyum meṉmaiyum eṉṟivai viḷaṅkat tōṉṟik kuṟitta poruḷai muṭittaṟku varūum ētu nutaliya mutumoḻi eṉpa (III. 8. 174) (Subtlety, brevity, clarity, and simplicity being its uppermost [qualities], mutumoḻi, with its implicit reason, will conclude a particular proposition. Trans. Nirmal Selvamony)

Significantly, the verse cited above mentions three members of kāṇṭikai: poruḷ (short for cūttirapporuḷ), nutaliya ētu (literally, “intended” reason; implicit reason), and muṭipu (conclusion). Let us see how these three members are deployed in mutumoḻi. In fact, mutumoḻi means both proverb or maxim and the literary genre based on the proverb (tol. III. 8. 174. veḷḷaivāraṇar, commentary). Some of the texts composed in this genre are mutumoḻik kāñci, paḻamoḻi nāṉūṟu, mūturai, and the medieval text, paḻamoḻit tiruppatikam by appar. The proverb or the maxim does not contain the reason in itself, and this is why tolkāppiyar described reason as what is “intended” rather than stated. Aristotle also makes this clear when he says that the addition of reason (or cause) to the maxim yields the enthymeme (Cooper 150). In the Tamil tradition, usually the proverb (on which the text is based) is an analogy (the seed of the composition), which drives home a point (stated by the proposition or poruḷ) quite effectively by means of the qualities attributed to mutumoḻi by tolkāppiyar: subtlety, brevity, clarity, and simplicity. To drive home a point is to conclude it properly (ulakiyal poruḷ muṭipuṇarak kūṟiṉṟu, puṟapporuḷ veṇpā mālai 26). Consider the following song from paḻamoḻi (70): oṟkamṭām uṟṟa iṭattum uyarntavar niṟpavē niṉṟa nilaiyiṉmēl – vaṟpattāl

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taṉmēl naliyum paciperi tāyiṉum pulmēyā tākum puli (Even in abject poverty, noble persons persist in their nobility. Though crushed by hunger due to drought, a tiger will never eat grass. Trans. Nirmal Selvamony)

This song is based on the proverb, “pul mēyātu puli” (the tiger does not eat grass). The genre mutumoḻi builds up the following argument out of the proverb: Proposition (poruḷ): A noble person remains noble even when poverty-stricken. [Implicit Reason (ētu): Because noble persons uphold the value of steadfastness (kaṟpu)]. Illustration (eṭuttukkāṭṭu): Whoever is noble will not compromise steadfastness, like how a tiger will not eat grass even when famished. [Application (naṭai): A noble person is steadfast]. Conclusion (muṭipu): Therefore, a noble person remains noble even when povertystricken.

It may be noted that the song does not mention explicitly the reason why noble persons remain noble when poverty-stricken and why tigers do not eat grass in times of drought. It has the proposition, illustration, and conclusion. As the application builds up on the reason, the absence of reason will result in the absence of application too. The relation between nobility and steadfastness, we may note, is illuminated quite concisely and subtly. Instead of attempting to establish the steadfastness of noble persons in an argumentative manner as in “tarukkam” or “vākai,” mutumoḻi achieves the same purpose by exploiting the traditional knowledge of the people expressed in their sayings. We may like to think of mutumoḻi as a truncated form of kāṇṭikai like the enthymeme. But the enthymeme is a derivative of the deductive syllogism of the Western tradition, and Indian kāṇṭikai is not always deductive. Instead, we may also want to concede the possibility of the development of kāṇṭikai out of mutumoḻi, considering the fact that proverbs originate from preliterate primal societies. Therefore, mutumoḻi can be seen as a part of the preliterate logical tradition itself.

mutumoli and nayam/niya¯yam ¯ It must also be pointed out that mutumoḻi of tol. is the source of hundreds of justificatory sayings called “niyāyam,” deployed in texts for hermeneutic and rhetorical purposes (murukavēḷ 1960: 117; naṭēcak kavuṇṭar 1961: iii). Such a saying was originally denoted by the term nayam (tol. III. 5. 42: 4). Here are some instances of niyāyam (nayam) belonging to the genre mutumoḻi: 1. Hunting an elephant from behind a Calotropis bush (paḻamoḻi 62; niyāyak kaḷañciyam, 73; attempting the impossible). 2. No capital, no wages (paḻamoḻi 232; niyāyak kaḷañciyam 111; without cause, there cannot be effect).

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3. Putting butter on a peafowl’s head to catch it (paḻamoḻi 210; niyāyak kaḷañciyam 66, 67; without proper means, end cannot be attained). The argumentative texts, niyāyam and mutumoḻi, were employed in tarukkam, and in this regard, the latter anticipated some of the argumentative techniques of kāṭci, especially, stating one’s position (tol. III. 9. 112: 10; in the song from kalittokai we discussed earlier, the young man’s conclusion, namely it is the ruler who is to blame), agreeing with the opinion of others (tol. III. 9. 112: 11; invoking accepted ideas about good and evil), and application of analogy (tol. III. 9. 112: 9; the boy’s similes for the girl’s beauty).

va¯kai If tarukkam is a mode of logic deployed in the Inner life in tiṇai, “vākai” (which literally means “exaltation,” tol. III. 2. 19), as we mentioned earlier, is a logical mode in the Outer life of tiṇai society (vākai derives from vaku, to divide + ai, nominal suffix ¼ vakai, division; vakai > vākai, excess [that which surpasses]; surpassing [the opponent]; vaku, divide > vāku, side [paku, to divide > pāku, share, division, tol. III. 2.19:2] > vākku, side, word; vāku, side > vātu, argument [in which one tries to surpass the opponent]; vātu + am, nominal suffix ¼ vātam, argument). vākai is a kind of verbal duel, which is, in fact, a part of the combat, the representative praxis of the Outer life (puṟam). In this duel, the rival chieftains advance their arguments based on their respective premises with a view to crushing the argument of the opponent. Supposedly, the argument concludes when it reaches an invincible climax (tol. III.2.19). Since tol. is a grammar text, it does not give any examples of verbal duel. But examples may be found in various primal societies. We learn that some Inuit men in Canada’s Arctic settle their conflicts with song duels. “Each composed a song about his opponent that would be performed in a community feast. The composer of the cleverest song, the one the audience enjoyed the most, won the duel” (Zada; Bennet and Rowley; cf. Balikci 1970: 186–89). Like tarukkam, vākai also anticipates the argumentative logic embodied in polemics, apologetics, defences, poetic contests of the medieval age, the arguments in courts of law, the contestatory rhetoric in cinema, and political harangues (Bate). Perhaps the performative subgenre called “tarukkam” (sometimes, tarkkam) in the Tamil folk song and theater is as old as vākai. It must also be said that tarukkam in folk song and drama is a dialogical text, whereas tarukkam in kaikkiḷait tiṇai is a monological text. Further, it must also be pointed out that if vākai is a verbal duel in combat, it is also such in the public assembly (avai). Like tarukkam, vākai also embodies some logical devices and techniques. The proposition of kāṇṭikai has its precedent in vākai’s principle (koḷkai, tol. III.2. 19: 1), which was not always stated explicitly in tarukkam. The argumentative technique (utti), namely “detailed exposition through classification” (tol. III.9.112: 3), is foreshadowed by the subdivision of one’s claim into several units, and exalting each one of them (tol. III.2. 19). Both tarukkam and vākai are primal types of logic,

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which may be considered the sources of the later name for logic, “uṟaḻcci nūl” (literally, the text of contest, tol. III. 2. 20: 1. nacciṉārkkiṉiyar’s commentary). If “tarukkam” and “vākai” are not so much modes of validating either knowledge or action as forms of argument for establishing one’s position, aḷaviyal is logic that validates knowledge and action. We may distinguish “aḷavai,” “tarukkam,” and “vākai” with the help of their contexts in the text. If tarukkam and vākai are motivated by the competitive spirit, aḷavai is not. The competitive nature of the former two types of logic is also evidence of their rhetorical nature. aḷavai is content to advance reasons and evidences to strengthen the argument, whereas vākai and tarukkam seek to quell the opponent’s argument or claim.

Debate in the Village Assembly Logical argument employs the debating technique known as “utti” (from Ta. untu, to drive >unti, that which projects [Gnana Prakasar 349] >utti, that which thrusts forward like a pendant or a spot on the hood of a cobra; an opponent player, a technique that helps drive home a point; see pakavati 180; Selvamony 1996a: 86–97). tolkāppiyar lists 32 of them and says that they belong to “otta kāṭci” (tol. III.9. 112: 1), which connotes philosophy as logic. These techniques belong to logic and rhetoric as they did in ancient Greece. This is evident if we note that the argumentative techniques sought to persuade the opponent to accept a position or to convince the opponent of the strength of an argument (cf. Cooper 1960: 220) or to lead the opponent to admit defeat. The argumentative techniques, which were part of logic and philosophy, were inseparable from the public life of the people of the primal society known as “tiṇai.” If the debate culture in Greece was part of the life of the polis, a state society, in Northern India, it originated in the court of King Janaka (Matilal 1999: 31, 2). But in Southern India, it was already a part of the primal society itself. Philosophical debates in Tamil were held in the public forum called “avai” that often met under an important tree in the settlement or in a pavilion-like built structure called (Selvamony 1990b) or maṉṟam (Selvamony 1989) or potiyil (Selvamony 1993). The forum consisted of a president (talaivaṉ), members (avaiyattār), beneficiaries (avai ēṟuvōr), and the people of the settlement. The president was often the chieftain (Selvamony 1996b: 42–64) like titiyaṉ (akanāṉūṟu 331: 9–13), who convened a forum, heard the case, and pronounced the verdict. The members of the forum were respectable people of the settlement, and they gave their opinion on the cases that were tried and the points debated. The members had to possess the following traits: noble lineage, education, character, truthfulness, purity, impartiality, being devoid of jealousy, and covetousness (Selvamony 1996a: 47). The forum meant for hearing disputes consisted of the disputants, the president, the members, and the audience. Convened to test the quality of a newly composed text, the beneficiary of the forum was the author. When the forum sought to debate a proposition, the beneficiaries were the debaters. The audience were the people of the settlement whose presence authenticated or challenged the validity of the proceedings. In short, logic was nurtured by the assembly in the village.

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Logic, Text, and tolka¯ppiyam The village assembly was the forum where the poets and the scholars presented their oral texts for validation. This practice led to the categorization and theorization of texts (nūl; tol. III. 9. 100) subdivided in more than one way (tol. III. 8. 164–165; 9. 95; 9. 99–105). Of these, only the triadic categorization into proposition (cūttiram, tol. III. 9. 102), kāṇṭikai (tol. III. 9. 103–104), and commentary (urai, tol. III. 9. 105– 106) is pertinent to our discussion. If commentary deals with the meaning of the proposition, kāṇṭikai derives a conclusion from a proposition. These forms of text adopt one or more of the 32 argumentative techniques (utti) and avoid textual demerits which are ten in number (tol. III. 9. 100). Though tol. is also made up of cūttiraṅkaḷ (propositions), the latter are not concluded with evidence but expounded with commentary. The rules are stated (in tol.) in the following forms: (1) definition (I.1.1; III. 8. 201), (2) prescription (I. 2. 1; I. 4. 12), (3) proscription (I. 9. 2; 5), (4) proposition (I. 2. 13; II. 8. 11), and (5) quotation (I. 1. 6; 8). There are combinations of these also. For example, a cūttiram may combine definition and quotation (III. 1. 4) or the prescriptive and the proscriptive forms of the rule (I. 1. 30). However, these rules may be considered the most truncated form of the syllogism-like text, which has just one limb. For example, the rule that a consonant is articulated when it combines with /a/ (tol. I. 2. 13) may be regarded as a “proposition” as it implies the “reason,” because articulation is impossible without the vowel. The proposition may be further developed with an “illustration” (whenever a speech sound is articulated, it implies the use of the vowel /a/, as in /ka/), “application” (it is not possible to articulate the consonant without the vowel), and “conclusion” (therefore, a consonant cannot be articulated without the vowel /a/). If so, we may say that logic is found at the levels of form as well as content in tol. At the level of content, logic in tol. may be traced to such cultural practices of the primal society as the hyperbolic monologue of the male lover, the verbal duel of combatants, the public debate in village assembly, and the argumentative techniques employed therein. It is also indebted to the proverbs (mutumoḻi) out of which emerged, probably, the argumentative text called kāṇṭikai. As the form and method of truth affirmation, logic sought to establish a position intuited through inner vision (faultless kāṭci). This vision was not mere perception (kāṭci) but knowledge required for proper debate, based on inference, understanding, interpretative power, analogical reasoning, decorum (for respecting the opponent’s rights and dignity in debate), and even creativity (for coining new terms). These mental and moral powers make logic in tol. aḷaviyal, a discipline that is at once epistemic and moral.

Definitions of Key Terms aḷaviyal marapu ētu

tradition of knowledge-validation criteria; logical tradition (based on such criteria). reason.

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eṭuttukkāṭṭu kāṇṭikai

kāṭci kōḷ muṭipu naṭai

tarukkam utti uvamam ulakurai vākai

613

literally, “holding up to show.” Illustration. In kāṇṭikai, this member is a combination of an axiom and analogy. generically, a method of seeing; specifically, an argumentative text with five members (proposition, reason, illustration, application, and conclusion). literally, vision; perception; otta kāṭci, philosophy (which does not exclude logic). literally, “holding,” “grasp,” “position held.” In kāṇṭikai, the first member, proposition. conclusion, the last member of kāṇṭikai. in kāṇṭikai, application (which follows nayam/niyāyam or the analogical statement called eṭuttukkāṭṭu); also called upanayam (secondary nayam). monological verbal duel in which one exalts one’s case. argumentative technique. analogy. tradition, one of the criteria for validating knowledge. dialogical duel in which each party exalts its case in an analytical manner.

Summary Points • Logic in tol. originates in the cultural practices of the primal society. Some of these significant primal practices are tarukkam in daily life, vākai in combat, debates held in the public assembly (avai), the use of argumentative techniques (uttikaḷ) and the criteria of knowledge, and the pursuit of an inner vision. • tarukkam, a part of the Inner domain of tiṇai life, is contestatory in manner and employs logical techniques. Originally, it was monological, but later took dialogical form also. • vākai, which belongs to the Outer domain of tiṇai life, is dialogical in form and contestatory in its manner. It also uses logical techniques. • The debates held in the assembly of the primal society also anticipated some of the logical techniques of the later times. • There are evidences (in tol.) of the use of validative criteria (aḷavaikaḷ and uttikaḷ) in the Inner and Outer domains of tiṇai life. • Philosophy in the primal society was a pursuit of inner vision, and it continued well into the later times when the logical tradition that existed in oral form was reduced to writing. • Logic based on oral-textual tradition culminated in the logical genres called mutumoḻi and kāṇṭikai. • Some of the logical features of the primal cultural practices survive in the medieval and modern times.

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Dependency of Inference on Perception and Verbal Testimony

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Vinay P

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Scope of Pramāṇas (Means of Valid Cognition) and Mutual Dependency (Upajīvakatva) . . . Inference’s Dependency on Perception and Verbal Testimony . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Perception as Supportive (Upajīvya) for Inference and Others . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dependence of Inference on Verbal Testimony and Perception . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Natural Potency of Perception Over Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Factors of Potency of Perception Over Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ways of Dependency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Criteria for Inference’s Dependency on Perception and Verbal Testimony . . . . . . . . . . . . . . . . . . . . Can the Schemata Be Reversed? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Response to Vācāspati Miśra’s Objection Against Perception as Supportive . . . . . . . . . . . . . . . . . Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Indian philosophical schools heavily depend upon the means of valid cognition (Pramāṇas) to establish the findings of metaphysics and the stand of one’s school. Although there is divergence of opinion with the regard to the number of Pramāṇas, roughly three distinct means of valid cognition are acknowledged widely: perception, inference, and verbal testimony. Perception refers to sensory means of knowing, while inference is argument to arrive at some conclusion. Verbal testimony is a word of authority which propounds truth of one’s experience. In the present chapter, it is examined as to how inference depends upon perception and verbal testimony for its functioning and soundness. Various arguments forwarded by Dvaita thinkers (the school of Indian realism V. P (*) Faculty of Vedanta, Karnataka Samskrit University, Bangalore, Karnataka, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_25

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propounded by Śrī Madhvācārya, twelfth-century C.E.) are culled from the classical texts and presented. It is shown as to how any inference is subject to the Pramanya of perception and verbal testimony for its intrinsic soundness. The potency of perception, on natural stature as well as being supportive, is being presented here. When the argument at hand is about the sensory objects, the role of perception as supportive to inference is indispensible. In case of an apparent contradiction, the inference which is dependent loses soundness on empirical grounds. Similarly, if the argument is about the supra-sensuals like the virtue and vice, it cannot hope to contradict verbal testimony, since the latter is the actual means to know the supra-sensuals, and not mere inference. The factor of dependency and the grounds thereof with suitable examples have been presented in the present chapter. The working patterns of the Pramāṇas pertaining to dependency have been portrayed in the chapter. The views of different schools such as that of Vācāspati Miśra and others are presented on the subject and reviewed on the backdrop of great thinkers like Śrī vyāsatīrtha and others of Dvaita school. Keywords

Pramāṇas · Perception · Inference · Verbal testimony · Syllogism · Dependency (Upajīvakatva) · Supportive (Upajīvya) · Dvaita school · Śrī vyāsatīrtha · Vācāspati Miśra · Truth (Prāmāṇya) · Natural potency (Jātyā Prābalya) · PurvaMīmāṃsā · Advaita · Pratyakṣa · Anumāna · Śabda · Self-validity · Sublation · Śruti: Vedic texts Abbreviations

Het.pra. Nyā Pra Pur.Sū Ślo.vār Tat.cin.

hetvābhāsaprakaraṇam Nyāyāmṛta Prakāśa Purva-Mīmāṃsā Sūtras ślokavārtīkā tattvacintāmaṇi

Introduction Indian philosophy is of the tendency of establishing truth status of any supposed entity, be it ontological or epistemological, only on the solid ground of means of knowledge-styled Pramāṇas (“mānādhīnā meyasiddhiḥ” “Objects are established by Means of Knowledge” (Translation Mine)). It is quite to the strength of philosophical pursuit to strictly adhere to proof methods, rather than fabricating mere theories based on probability or mere possibilities. Although there is disagreement upon the actual differentiate number of Pramāṇas or proof/means of knowledge (“pramākaraṇaṃ pramāṇam” “That which is instrumental for valid cognition is Pramāṇa” (Translation Mine)) in exclusion to one another among the schools of Indian philosophy, all schools barring the heterodox comply with three Pramāṇas,

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namely, perception, inference, and verbal testimony, although more are recognized. Perception (Pratyakṣa) in general is the means for sensory knowledge, i.e., the means to knowledge arising out of five senses, namely, visual, auditory, of fragrance, taste, and touch (“indriyārthasannikarṣajanyaṃ jñānaṃ pratyakṣaṃ” “Perception refers to knowledge arising out of contact between senses and objects” (Translation Mine)). Inference (Anumāna) is in one way syllogism or formal logic (informal as well, being recognized as subjective and nor very important in logical examinations) which gives rise to inferential knowledge termed Anumiti (“anumitikaraṇam anumānam” “That which is instrumental in bringing out Inferential Knowledge is Inference or Anumāna” (Translation Mine)). Verbal testimony is valid words of an authority in right context which gives rise to valid cognition (śābdabodha) (“āptopadeśaḥ śabdaḥ” “The words of authority is Verbal Testimony” (Translation Mine)). Among these three, it is pertinent to examine as to which is more potent to than the others when apparent contradiction occurs or as to which is dependent on whom for its estimated functioning.

Scope of Pramānas (Means of Valid Cognition) and Mutual ˙ Dependency (Upajīvakatva) What is dependence? Dependency refers to the fact that a particular means of knowledge is dependent on other means of knowledge, in order to produce true cognition. (“anyādhīnayathārthajñānajanakatvaṃ” “Ability to produce true cognition, depending on another” (Translation Mine)). Thinkers of Dvaita Vedanta school take an important aspect into account before dwelling upon the Pramāṇas’ dependency. As counterparts to Pramāṇas, respectively, there can be two kinds of objects, viz., those which are sensual and those which are supra-sensual like virtue and vice. With regard to sensual ones like the experienced world, all the three Pramāṇas noted have a say. However, with regard to supra-sensuals, it is the verbal testimony exclusively which has the say, ruling out perception completely, but still entertaining inference as far as it collaborates verbal testimony. This distinction has to be clearly noted when examining the dependency of inference on perception and verbal testimony.

Inference’s Dependency on Perception and Verbal Testimony At the outset, the structure and nature of inference have to be noted in the context of Indian philosophy, where the onus is more or less upon the school of Nyāya. Inference is argument which is represented in five-limbed syllogism or even threelimbed at times. It is a form of verbal argument forwarded by a person to prove something. The five limbs (pañcāvayava) are: i. Proposition (Pratijñā) ii. Reason (Hetu)

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iii. Illustration (Udāharaṇa) iv. Subsumptive correlation (Upanaya) v. Conclusion (Nigamana) As an example, when proving the fire on a mountain with smoke as a proof to it, the syllogism forwarded is as: The mountain is fiery (proposition). As it has the smoke (reason statement). All that which has the smoke is fiery, like the kitchen (illustration). This mountain has such a smoke which is concomitant with fire (subsumptive correlation). v. Hence, the mountain is fiery (conclusion).

i. ii. iii. iv.

While forwarding an inference, one directs it to another to initiate a particular process through which the other one draws inferential cognition. Hearing each of the five statements, with the help of invariable concomitance and the presence of reason in the substratum, the hearer infers the presence of the probandum. In the abovestated example, upon hearing, the invariable concomitance between the fire and the smoke is realized. The presence of such a smoke too is recognized in the mountain. This makes him to invariably infer the presence of the fire in the mountain. In these five limbs, there are certain technical elements, namely: (A) Pakṣa: The substratum of the thing to be proved (in the above case, the mountain) (B) Sādhya: The thing to be proved or probandum (in the above case, the fire) (C) Hetu: Reason (in the above case, the smoke) (not to be confused with the second limb of the syllogism) (D) Vyāpti: Invariable concomitance(in the above case, between the fire and the smoke) (E) Sapakṣa: Similar instance for invariable concomitance (in the above case, the kitchen) (F) Vipakṣa: Contrary instance for invariable concomitance (in the above case, the pond) It should be noted that all these “A-to-F” elements have to be factual to make the inference sound. If not, the inference is mutilated and inefficient to produce valid inferential cognition (Anumiti) (the individual fallacies arising out of respective mutilation are seen in Het.pra of Tat.cin). These elements have to be established as factual beforehand of the inference. Hence, the moot question: through what means of knowledge are these elements established as factual before inference? It should be seen that as the means of knowledge or Pramāṇas are three in number, namely, perception, inference, and verbal testimony, one among them have to be chosen to answer the question. At the outset, it cannot be inference itself which serves the purpose of authenticating the elements of inference, as it gives rise to the fallacy of

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self-dependency. If another entity of inference is chosen as former to prove or validate the latter’s elements, the question bounces back to the former inference as well. It eventually results to reciprocal dependency fallacy, cyclic dependency fallacy, and infinite regress as well. Hence, perception or verbal testimony has to be the means of knowledge which a priori vindicates the soundness of the “A-to-F” elements of inference, making it sound. Thus, the inference, in this way, is dependent on perception or verbal testimony to validate it and also the soundness of its “A-to-F” elements. It can be quite logically accepted and reasoned that if any of these “A-to-F” elements of inference are sensual, the supportive means of knowledge shall be perception with regard to those elements. If any of those are supra-sensual, the onus of those is on verbal testimony.

Perception as Supportive (Upajīvya) for Inference and Others Dvaita school of thought points out several important reasons for treating perception as being supportive means of knowledge (Upajīvya-Pramāṇa). Śrī vyāsatīrtha questions as to why perception should be denied of efficacy and its consequence the truth, if at all, by the monists. He points out that perception has to be treated as all-important when it comes to things of convention, which are not supra-sensual. If at all the efficacy of perception has to be denied, it has to be denied only on the ground of solid reason or verbal testimony. However, inference and verbal testimony are dependent upon perception for their soundness and veridicality, since the very nature and structure of inference and verbal testimony are dependent upon perception. This may be seen as follows: the inference is five limbed and has components such as the substratum, probandum, reason, and illustration which are all established by perception. Even if it is argued that these are established by yet another inference, infinite regress apart, the inference is heard vocally, and this is a matter of auditory perception. Hence, if perception is denied of authenticity, the very fact of inference being heard or forwarded vocally has to be denied, which makes it lose its soundness. Hence, inference is dependent upon perception for its soundness. If not, it results in reciprocal dependence. The same argument holds good in case of verbal testimony too. The testimony is conveyed vocally and is also a matter of auditory perception initially (tayoḥ prāmāṇye tadvirodhena akṣasya aprāmāṇyaṃ, sati cā tasmin mānāvirodhena tayoḥ prāmāṇyamiti anyonyāśrayaḥ – Nyā., p. 276, “If those two are not authentic, perception would be invalid, their validity on the other hand is by non-contradiction of it, and hence reciprocal dependence” (Translation Mine)).

Dependence of Inference on Verbal Testimony and Perception Verbal testimony too is a supportive means (Upajīvya) for inference. This can be examined on many grounds. As seen before, inference is a vocal communication which is forwarded by an authority. If verbal testimony is denied of veridicality and

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its end the truth, the inference vocally forwarded too may not be believed at all. The presence of substratum, probandum, reason, invariable concomitance is also authenticated by verbal testimony. Hence, denying verbal testimony is self-contradictory. Even if the components of inference are established by perception, there are accepted arguments which are text-based and whose components are solely dependent upon verbal testimony. The above-stated arguments are forwarded by Dvaita thinkers to prove that inference is dependent upon perception and verbal testimony, for its soundness. Dvaita thinkers anticipate an objection to this theory. Reciprocal dependence can very well occur when perception is treated as supportive too. The same argument designed to prove that inference is dependent on perception can be juxtaposed to prove that the former is dependent upon the latter. Dvaita thinkers anticipate such an objection and remark that perception is independent of inference for its authenticity. The moment we perceive a pot as a pot, it goes without saying that the truth of such a knowledge is immediately grasped by the meta-knowledge of its experience. This forms the bedrock of the concept of self-validity (Svataḥprāmāṇya) (“akṣasya tu prabalasya prāmāṇyam anumānāgamavirodhāpekṣaṃ neti na anyonyāśrayaḥ” – ibid., “as the Truth of the potent perception is non-dependent on the absence of contradiction of inference and verbal testimony, there is no reciprocal dependence” (Translation Mine)). It may be seen here that such a self-validity of experience is even acceptable to the monist’s school (Advaita school) which follows the Bhāṭṭa School of Mīmāṃsā, where Kumārīla has argued elaborately on the concept of selfvalidity of knowledge (see Ślo.vār of kumārīla bhaṭṭa). A clarification is pertinent at this juncture. The inferential knowledge and verbal cognition are also self-valid. However, the structure of inference as a verbal communication and encompassing components makes it vulnerable to stand in need of an authentic perception. Inference is dependent upon perception for few other reasons as well. Perception is more potent than the inference by its very nature. This forms the concept of natural potency of perception styled in the Nyā as Jātyā prābalya of Pratyakṣa. Perception, by its very nature of being endowed with a unique structure, makes it more potent than the inference. The reason adduced for this is that perception has a unique capacity to grasp the finer details of a presented object by magnification and others, which is unseen in inference. For illustration, the inference can be of use to prove the presence of a hand or a limb in a person who is moving. However, inference stops and falls short to that extent. It cannot move further to make note of any finer details of the hand as such. However, the perception can take note of finer details like the lines and curves of a hand. This unique quality of perceiving the physical world in finer details makes perception potent than inference (“pratyakṣasya anumityāditaḥ prābalyaṃ ca tadagṛhītarekhoparekhādiviśeṣagrāhitvāt . . .” – Nyā, “The potency of perception over inference and others is due to grasping capacity of unique features such as lines and curves” (Translation Mine)). One more reason is adduced to prove the potency of perception over inference. Perception can ward off certain special cases of illusion, which are otherwise not sublated by mere inference. As an example, when a person is bewildered as to which direction is the east, any amount of inference without the perception cannot yield him the correct direction. It is only

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when a person perceives the sun rising, he realizes the correct direction, and the illusion of the east on the west is removed. Thus, perception has a unique capacity of removing certain special illusions, which mere inference cannot. This makes perception more potent than the inference, making the latter dependent or congruous to the former.

Natural Potency of Perception Over Inference The Dvaita thinkers forward one more inevitability to accept that perception is by its very nature more potent than inference. The Nyā demonstrates one more case where perception is more potent than inference by its very nature, barring any other reason. The Nyā contrasts perception and inference as the perception as “the fire is hot,” with the contrast of an inference trying to prove that the fire is cold. The latter inference is negated by the former perception in the real world. It is examined as to why the perception of “the fire is hot” is more potent than the inference of “fire is cold.” Dvaita thinkers point out that there is no other reason for the perception to be potent, than its natural potency over inference. Some of the other reasons may be the supportive nature of perception by grasping the substratum or grasping the counter-entity of negation. However, in the given case, the perception of “the fire is hot” has the fire for its object, which is not on technical grounds contradictory to the inference of “the fire is cold,” which too has the fire for its object. Hence, the criterion of substratum contradiction is ruled out in this case (dhārmigrāhakavirodha). The other criterion of counter-entity too does not hold water in this case. The perception is of the content of “being hot” which is not in strict technical terms of logic the counter-entity of “being cold,” which is altogether a different sensation. This too rules out the ground of grasping the contradictory counter-entity. Hence, to substantiate this real-world experience of contradiction between the two entities of knowledge, perception has to be accepted as naturally potent over inference (Nyā. Pra., pp. 308–309). The Dvaita thinkers take the support of a maxim of Purva-Mīmāṃsā to justify how the inference is dependent upon perception on a common plane, and not vice versa. Among the six deciding factors of hermeneutics noted in the Purva-Mīmāṃsā (Tātparyaṣaḍliṅgas), the prelude (Upakrama) and ending (Upasaṃhāra) are noted. In the third pāda of the third Adhyāya of Pur.Sū., there appears a seeming contradiction of certain passages with respect to the prelude and the ending (see Nyā. Pra., p. 309 for details). It is concluded in that section of the Purva-Mīmāṃsā that the prelude should be honored and the latter part has to be correctively interpreted so as to not contradict the former. In giving reason to this, the Purva-Mīmāṃsā school notes labdhātmatā or being of established nature” as a ground for the former to remain unaltered. Since “former” is a former to the latter, it has no contradiction yet at its time. But since the “latter” is a latter to the “former,” it stands already with a contradiction of the former. Hence, the former is established already with the contradiction, whereas the latter is not so. Hence, it is concluded there that the former is more potent than the latter due to the fact of being “formerly established.”

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The Nyā uses this logic to prove that it is the inference which needs to be rejected in face of a contradiction with perception. The reason of “being established already” is used here by the Dvaita thinkers in the context of dependency of inference over perception. Perception is always primary and precognition to the inference. Hence, whatever that is established formerly by perception is always “already established” without contradiction of the latter. On the other hand, a contradictory inference is standing with a former’s contradiction and is not fully ‘already established.” This hermeneutical logic of the Purva-Mīmāṃsā is used by Śrī vyāsatīrtha to show that perception is the supportive and inference is dependent upon it, the former being naturally potent.

Factors of Potency of Perception Over Inference When we take up the question as to why inference should be treated as dependent upon perception and verbal cognition, it boils down to the potency of any, over the other means when there is apparent contradiction. The factor of potency decides which is dependent upon whom. When there is no seeming contradiction, the realms are totally different, and there is no question or reason for dependency over other. The Dvaita school adduces one more reason to prove that inference is dependent upon perception and verbal testimony, in terms of potency of the latter two. The reason adduced is the quicker movement (śīghragāmitva) of the perception and verbal testimony as well in affirming facts than the inference. Perception is immediate and is very quick with any considerable delay in grasping an object, whereas in case of inference, it stands in need for reason, followed by the recall of invariable concomitance and others. The process is quite delayed. The affirmative and assertive power is more to perception than inference. This is an application of Śrī vyāsatīrtha by drawing upon the hermeneutical exegesis of Purva-Mīmāṃsā Adhikaraṇa of third Pāda of third Adhyāya, where the verbal testimony is proved to be more fast and immediate than inference (see Nyā. Pra., p. 320). (The exact details of Purva-Mīmāṃsā are not taken up here since it is beyond the purview of the present topic.) Another quite readily available reason adduced for the potency of the perception is the “futility by many contradictions” (bahubādhānupapatti). When there are two apparently opposing means of knowledge, the one with more objective details should not be mutilated for the obligation of the one with fewer objective details. The case of perception, inference, and verbal testimony is worth examining in this backdrop. The perception and verbal testimony supply a great number of details about a given object than mere inference which vindicates a few facts. If perception and verbal testimony are rejected as unauthentic with the opposition of inference, a greater number of details established by the former too need to be rejected. On the other hand, if the inference is rejected when there is a contradiction, only a few details are rejected, abiding by the law of parsimony. Hence, perception and verbal testimony stand not suffered by inference’s potency, whereas it is the inference which faces the brunt by perception and verbal testimony: (“bahubādhasya

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anyāyatvāt ... bahupramāṇavirodhe ca ekasya aprāmāṇyaṃ dṛṣṭaṃ śuktirājatādau iti ... iti pratyakṣasya jātyā upakramādinyāyaiśca prābalyam” “Manifold contradiction is illogical. . .when there is a contradiction of many means of knowledge, treating one as not authentic is seen, just as in the case of silver-on-nacre. . .thus, perception is potent by its nature and Upakrama and other Maxims as well” (Translation Mine) – Nyā., p. 277).

Ways of Dependency The next important question is about various ways by which inference, and even verbal testimony, is dependent upon perception. The factor of dependency lies on the fact that several elements of a given means of knowledge are dependent upon yet another means for their origin or ontological status. The Nyā provides a comprehensive list of the elements of inference and verbal testimony, which are dependent upon inference. The elements as such of inference are substratum, probandum, reason, invariable concomitance (others too such as Similar Instance and Contrary Instance), inferential knowledge (Anumiti), and the truth (Prāmāṇya) of inferential knowledge. All these are ontologically evident entities which need to be established as factual at the time of inference. Such an establishment can occur by perception and hence is inference dependent on it. Similarly, the elements of verbal testimony are the nature of verbal communication, the factuality of the connoted, semantic relation, semantic proximity, the unison of the sense between the prelude and the end, verbal cognition, and its truth. These too are objects of a real world which need to be established by perception. Hence, it is perception which is supportive to the inference and verbal testimony, which are dependents (“kiṃ ca upajīvyatvāt prābalyaṃ akṣasya . . .” “And also, perception is potent because of its supportiveness (Translation Mine) – Nyā. I).

Criteria for Inference’s Dependency on Perception and Verbal Testimony This brings up another natural question as to how the inference can be dependent upon another and on what criteria. Śrī vyāsatīrtha points out three criteria by which inference is dependent on other means of knowledge, viz., perception, verbal testimony, and even an another inference. The three aspects to be considered to weigh this are “grasp of the substratum,” “grasp of the reason,” and “grasp of the reason.” It may be noted here that these are three important components of an inference which need to be realized to eventually give rise to inferential knowledge. Śrī vyāsatīrtha quotes three examples for each of them: 1. (A) The pot is omnipresent (proposition). (B) As it has reality (reason). (C) Just as the ether (illustration).

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2. (A) The human skull is pure (proposition). (B) As it is an organ of a being (reason). (C) Just as the conch (illustration). (Note: This is an inference which is based on the religious concept of Hindu lore. Organs mutilated out of a being’s body are considered impure, like the human skull after a man is dead and exhumed. However, conch, which too is a shell of a worm, is considered pure as an exception. The above-noted inference makes use of this concept.) 3. (A) The mind is omnipresent (proposition). (B) As it is the substratum for the noninherent cause of knowledge (reason). (C) Just as the ether (illustration). In the first inference, the substratum, i.e., pot, is established prior to perception. However, the previously cognized pot in perception is a cognition of limitedness of pot which is contradictory to the omnipresence stated by the inference. Since the given inference is dependent upon this previous perception of pot, it cannot but its own root by saying that the limitedness of the pot is ignored. In this case, inference is dependent upon perception. In the second case, the probandum is purity. This concept is purely a religious concept which is evident in the text of Hindu lore. Thus, the argument of a thing being pure and impure is solely adjudged, in this context, by the textual authority. This being the case, there are references in the texts which declare that conch and other are pure, but not the skull (see Nyā. Pra., p. 330). If at all this inference has to move, it is on the concept of purity, which is in turn established by verbal testimony. Hence, in this case, inference is dependent upon verbal testimony and it cannot overrule it. In the third case, the mind is grasped as substratum for contact of mind and knowledge. In this case, the sole proof for the mind being so is inference of knowledge arising at every second. Since such a series and its components occur in mind and the mind is proved as atomic magnitude by this same inference, the former inference is dependent upon this inference. Hence, it cannot be rejected per se. Thus, it can be seen that inference is dependent upon perception and verbal testimony and even on another inference. The next issue with regard to dependency of inference over others is whether the dependency of inference over others is whether the dependency is limited to the ones which are particularly supportive object wise or is it general to the whole class of perception and verbal testimony. To make it more clear, inference has the components which may be grasped by visual perception in particular or maybe not. The substratum, reason, and probandum should be visually established. However, the auditory nature of the sentences of inference, its validity, and others are not visually established, but only by auditory perception. Hence, visual perception per se is not fully supportive to all elements of inference. Hence, it may be objected as to how the visual perception is supportive to inference, since all elements of inference are not supported.

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The Dvaita thinkers answer that any perception can be supportive to inference by the fact that it is of the same class as auditory. Although some elements are grasped and some are not by a given type of perception say visual, nevertheless visual perception still is of the same class as that of auditory perception which grasps other elements such as the sentence, structure, and its validity. It is a sufficient condition that both are perceptions and similar in that respect to make the given inference dependent upon it. The Nyā proves this with an example: in case of the inference on purity stated earlier, the verbal testimony is a proof so far as the purity of the conch and others are concerned. It has not got anything in particular to do with the purity stated in the inference. However, even though there is no direct instance of the verbal testimony being supportive to this particular inference, verbal testimony which speaks of purity in general serves the purpose, and inference is dependent upon it.

Can the Schemata Be Reversed? A subsequent question arises in this context. Inference and verbal testimony are stated to be dependent upon perception. The question is perception too can very well be argued to be dependent upon the other two. This objection takes a turn by posing a question: “the Nyāya and all other schools postulate the sense-organs as beyond perception (Atīndriya) and can be known only by verbal testimony or inference. This being the case, perception which arrives out of the sense-organs is also dependent upon other means of knowledge to produce cognition. This forces the situation that perception too is dependent upon others for its functioning.” This objection is answered by Dvaita thinkers with an argument that the argument is a confusion between the proof of existence and the method of functioning. No doubt the proof of existence of the sense organs is inference and verbal testimony, and not perception; however the functioning of the sense organs to produce cognition is not at all related to it being known by inference and verbal testimony. The sense organs produce cognition under given conditions, irrespective of the fact whether they are proved to be existent or not. There is no obligation for the sense organs to be proved as existent to function well (“yadyapi indriyaṃ anumeyaṃ, tathāpi tasya ajñātakaraṇatvāt na anumopajīvyā” “Senses are inferential, however, they work even without knowing them, and hence is not dependent upon Inference” (Translation Mine) – ibid). A central question arises whether the schemata of dependency can be reversed, i.e., to treat perception as dependent upon inference and not vice versa. This is answered by Dvaita thinkers by examining the method of functioning of the three means of knowledge. Perception arises out of the contact of sense organs with the objects. This stands in no need of inference or verbal testimony. On the other hand, inference functions on the ground of invariable concomitance and others which are established by perception of coexistence of probandum with the reason. Verbal testimony functions with semantic expectancy and others. These are all objects of

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perception. If perception is negated, these functions shall also be negated consecutively, which in turn negates inferential knowledge and verbal cognition. This is because inferential knowledge and verbal cognition are always of the same status of reality as its functioning components. This makes them dependent upon perception and not vice versa.

Response to Vācāspati Miśra’s Objection Against Perception as Supportive The Dvaita thinkers anticipate an objection, which was notably expounded by Vācāspati Miśra, author of the celebrated commentary, Bhamati. Vācāspati Miśra rejects the supportive nature of perception, when the case of an apparent contradiction occurs. Vācāspati remarks: “if the verbal testimony of the Śrutis are rejected due to contradiction of the supportive means of knowledge, let the sublative knowledge of ‘This is no silver’ be invalid, since it faces a contradiction of the earlier cognition as ‘This is silver’” (Vide ref. in Nyā., Vol. 1, p. 313). The contention of Vācāspati is that the perception’s capacity as a precedent cannot be a ground to prove its supportive nature over verbal testimony. Vācāspati feels that if the precedent nature of perception gives it the power to supersede the latter, then the precedent knowledge such as “this is silver” which is illusory in that case and yet precedent should have been more potent and not superseded by the sublative knowledge: “This is not silver.” It may be noted here that while Vācāspati clearly agrees that perception is not mere precedent but also supportive, he surprisingly denies its potency over verbal testimony. Śrī vyāsatīrtha notes this objection against the potency of perception and refutes Vācāspati’s arguments. Vācāspati seems to have mistaken the concept of supportiveness or Upajīvyatva. Supportiveness or Upajīvyatva is not being merely precedent or former in time to another means of knowledge. Upajīvyatva or supportiveness is to the means of knowledge whose soundness is the establisher of the soundness of another means of knowledge. In other words, a means of knowledge which depends upon another’s soundness to substantiate its soundness is said to have the latter as its supporter or Upajīvya. This suitably answers the objection of Vācāspati Miśra. Vācāspati had mistaken the ground of supportiveness of perception in the noted lines. Perception is supportive, not on the mere fact of being precedent, but on the fact of its soundness being the imparter (Samarpaka) of soundness to verbal testimony. In case of the cognition as “this is silver” which is illusory case, and the sublative knowledge as “this is not silver,” the former is not supportive, no doubt, not because it is precedent or otherwise, but because its soundness in no way imparts or holds ground for the soundness of a latter sublation: “this is not silver.” If at all the truth of the sublative knowledge had been lost due to the invalidity of the former perception as “this is silver,” it could have been treated as supportive or Upajīvya, which is not so. Śrī vyāsatīrtha quotes two such examples where perception is supportive to inference and where one verbal testimony which is supportive to another verbal testimony. If at all the inferential knowledge as “the mountain is fiery” has to be sound, the knowledge of invariable

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concomitance born out of perception of unstrayed coexistence of probandum and reason has to be true. If the truth of the knowledge of invariable concomitance is denied, the truth of inferential knowledge is also lost. Similarly, certain texts speak of omniscience of Brahman and his existence. If the veridicality of this is denied, even the veridicality of verbal testimony which speaks of identity of self and Brahman according to Vācāspati himself needs to be negated. This definition of supportiveness or Upajīvyatva forms the foundation of the theory of dependency of means of knowledge. Thus, among the three means of valid cognition, inference depends heavily upon perception and Verbal testimony for its soundness and functioning. At times, a particular inference can depend upon another inference too, which in turn should again necessarily depend on perception and verbal testimony. Verbal testimony stands not in need of inference for its veridicality, but still is dependent upon perception for its functioning. Perception is independent and does not stand in need of either inference or verbal testimony for this authenticity, by its very nature. This is one of the central concepts upon which the schools such as idealism and realism bank on for their respective stands. These relative dependencies of Pramāṇas effectively govern the further philosophical musings of Indian philosophy in many spheres such as metaphysics, ontology, and epistemology.

Definition of Key Terms 1. Pramāṇas: Means of valid cognition, that which brings about true cognition, such as perception, inference, and verbal testimony 2. Pratyakṣa: Perception, contact of the sense organs to bring out perceptual cognition. Roughly five in kinds, of visual, of auditory, of fragrance, of taste, and of touch 3. Anumāna: Inference. It is of two kinds, subjective and impersonal (syllogism). Impersonal has the structure of five limbs (see Avayava) 4. Avayava (Pancha-Avayava): The five limbs of a syllogism, viz., proposition (Pratijñā), reason statement (Hetu), illustration (Udāharaṇa), subsumptive correlation (Upanaya), and conclusion (Nigamana) (see respective terms) 5. Prama: Valid cognition 6. Indriya: Sense organs, five in number (if mind is considered so, six in number. If Sakshi is considered, seven in number) 7. Anumiti: Inferential cognition 8. Śabdabodha: Verbal cognition 9. Pratyakṣa-Jnana: Sensory cognition 10. Śabda (Pramāṇa): Verbal testimony 11. Validity (Prāmāṇya): The authenticity of any knowledge and even the means of knowledge instrumentally 12. Upajīvya: Supportive 13. Upajivaka: Dependent

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14. Dvaita school: School of Indian realism propounded by Śrī Madhvācārya (twelfth-century C.E.) 15. Advaita school: School Indian monism and idealism which includes thinkers like Vācāspati Miśra 16. Nyāya school: School of Indian logic 17. Pratijñā: Proposition or hypothesis statement in declarative form 18. Hetu: Reason statement or reason as the case may be 19. Udāharaṇa: Illustration or instance of invariable concomitance or coexistence of probandum and reason 20. Upanaya: Subsumptive correlation, statement of invariable concomitance in the present reason and the reason’s presence in the substratum 21. Nigamana: Conclusion of the hypothesis positively 22. Pakṣa: The substratum of the thing to be proved 23. Sādhya: The thing to be proved or probandum 24. Hetu: Reason Vyāpti, invariable concomitance 25. Sapakṣa: Similar instance for invariable concomitance 26. Vipakṣa: Contrary instance for invariable concomitance 27. Self-validity (Svataḥprāmāṇya): The nature of truth of knowledge being grasped by the meta-knowledge without aid 28. Purva-Mīmāṃsā: Indian school of Vedic exegesis or hermeneutics 29. Jatya Prabalya: Natural potency (of perception) 30. Sublation (Badha): Negation knowledge of the previously cognized 31. Reciprocal dependency fallacy: Mutual dependency of two in argument with respect to knowledge or existence 32. Cyclic dependency fallacy: Mutual dependency of more than two in argument with respect to knowledge or existence 33. Infinite regress: Infinite series of dependency of infinite sets with respect to knowledge or existence 34. Tātparyaṣaḍliṅgas: Six deciding factors of hermeneutics noted in the PurvaMīmāṃsā 35. Upakrama: Prelude 36. Upasaṃhāra: Ending 37. Labdatmataa: Being of established nature 38. Atīndriya: Supra-sensuals like virtue and vice 39. Śruti: Vedic texts

Summary Points • Objects are established on the basis of Pramāṇas or means of valid cognition. • Pramāṇas are essentially three in number, namely, perception (Pratyakṣa), inference (Anumāna), and verbal testimony (Śabda). • Dependency is to that means whose knowledge has its validity dependent on others.

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• With regard to sensual objects, perception is supportive and inference and verbal testimony are dependent upon it. • With regard to supra-sensual objects, verbal testimony is supportive and inference is dependent upon it. • Inference is five limbed with proposition (Pratijñā), reason statement (Hetu), illustration (Udāharaṇa), subsumptive correlation (Upanaya), and conclusion (Nigamana). • Inference has elements, namely, Pakṣa (substratum), Sādhya (probandum), Hetu (reason), Vyāpti (invariable concomitance, Sapakṣa (similar instance), and Vipakṣa (contrary instance) (“A-to-F elements”). • A-to-F elements have to be established a priori which makes inference dependent upon perception. • Perception is supportive (Upajīvya) to inference and even verbal testimony. • Verbal testimony can also be supportive (Upajīvya) to inference. • Perception is immediate and self-valid, making it supportive over inference. • Finer accuracy and special effectiveness to ward off certain illusions exclusively make perception naturally potent over inference. • Natural potency of perception over inference proved by real-life experiences. • Purva-Mīmāṃsā Maxims (hermeneutical logic) supports potency of perception over inference. • Factors such as immediacy, swiftness, hermeneutical logic, “futility by many contradictions,” and others contribute to perception’s potency. • Several ways of dependency by logical factors and semantic elements. • Criteria for dependency of inference on perception and verbal testimony such as “grasp of the substratum,” “grasp of the reason,” and “grasp of the reason” with examples. • Dependency extended to class. • Working patterns of perception, verbal testimony, and inference making the latter dependent upon the former two. • Irreversibility of the schemata of dependency. • Refutation of Vācāspati Miśra’s arguments and conclusion.

References Amoda, Nyā. 1991. 1st ed. Mantralaya: Brindavanam Office. Anuvyakhyana (with the Nyaya Sudha). 1982. 1st ed. Bangalore: Sri man Madhvasiddhanta Abhivriddhi Kaarinisabha. Bhagavad Gita (Gita Prasthana). 1993. Bangalore: Akhila Bharata Madhva Mahamandala and Ananda Tirtha Pratishthana, Second reprint. Bhamati. 1997. Delhi: Motilal Banarsidass Publishers Private Limited, First edition reprint. Brahma Sutra Sankara Bhasya. 1938. with the commentaries Bhamati, Kalpataru and Parimala, Edited with notes by Maha-M. Bombay: Anantkrisna Sastri, Nirnaya Sagara Press. Dasgupta, Surendranath. 1991. A history of Indian philosophy. Vol. 4. Delhi: Motilal Banarsidass Publishers Private Limited. Kosha, Nyaya. 1978. 4th ed. Poona: The Bhandarkar Oriental Research Institute.

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Nyā of Sri Vyasatirtha with Advaita Siddhi of Madhusudhana Saraswati. 1994. Edited by Prof. K.T. Panduragi. Bangalore: Dvaita Vedanta Studies and Research Foundation. Dr. Sharma, B.N.K.. 2000. History of the School of Vedanta and its Literature, from the earliest beginnings to our own times. 3rd rev. ed. Delhi: Motilal Banarsidass Publishers Private Dvaita Sch. Sharma, B.N.K., 2002. Philosophy of Sri Madhvacharya. Delhi: Motilal Banarsidass Publishers, Reprint. Srinivasa Chari, S.M. 1988. Tattvamuktakalapa. Delhi: Motilal Banarsidass Publishers. Tapasyananda, Swami. 1990. Mylapore, Madras: Bhakti Schools of Vedanta, Sri Ramakrishna Math, First impression. Tattva Cintamani of Gangesha. 1991. Varanasi: Sampurnananda Samskrita Vishwavidyalaya. Vartika, Sloka. 1993. 2nd ed. Varanasi: Rama Publications. Vinayacarya, P. 2006. Sri Madhvacarya’s Mithyatva-Anumana-Khandanam, (A refutation of the world’s non-reality syllogism) with the Tika of Sri Jayatirtha, Sri Vedavyasa Sanskrit Research Foundation.

Further Reading Advaita Siddhi of Sri Madhusudhana Saraswathi with commentaries Anuvyakhyana of Sri Madhvacharya Bhamati of Sri Vacaspati Mishra with commentaries Dr. Dasgupta, Surendranath. History of Indian philosophy. Vol. 4 Mithyatva Anumana Khandana with English Translation and Critical Study, Dr. Vinayacarya P. Nyā of Sri Vyasatirtha with commentaries Nyayasudha of Sri Jayatirtha Dr. Sharma, B.N.K. Nyā and Advaita Siddhi: an upto date critical reappraisal.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fallacies of Reason . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Relation Between the Pervaded and the Pervading (Entities in Invariable Pervasion) . . . . . . . . The Difference Between Inherent Cause and Efficient Cause . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Axiom that Negation of Absence Amounts to Presence of the Negated Entity . . . . . . . . . . The Complex Absence Caused by the Absence of the Attribute and the Absence of the Substantive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Encapsulation (Anugama) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Comprehension of the Relation of Identity Only with Regards to the Objects Cognizable Distinctively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Counter Entity of Difference Pertains Only to Relation (kevalānvayin) . . . . . . . . . . . . . . . . . . The Refutation of Non-difference Being a Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interpretation of the Sūtra “rucyarthānāṁ prīyamāṇaḥ” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Explaining the Import of the Verb Taking Cause-Effect Relation into Account . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The schools of Nyāya and Vyākaraṇa are mighty Śāstras among the traditional Indian knowledge systems. Both the Śāstras have a long and strong traditional lineage with vast literature, original and commentarial, delineating their unique doctrines, even on common topics between them, that are concluded after much systematic deliberation and in line with the ontology and tenets of the respective Śāstras. Such long traditions of knowledge have much dialectic and polemic engagement, thereby, mutually influencing the pathways the Śāstras traverse as the tradition moves ahead in time.

B. V. Venkataramana (*) Department of Philosophy, Evening Sanskrit College, Karnataka Sanskrit University, Bangalore, Karnataka, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_20

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Gaṅgeśopādhyāya, a radical thinker of the Nyāya school, not only changed the landscape of the ontology of his own school by bringing in more brevity in categorization and precision in expression, but also influenced the knowledge community then on greatly by giving them the gift of precision in expression through many technical axioms that he adopted for drawing conclusions through the thicket of gruesome complexities that epistemological topics dragged the scholars through. Incidentally, the scholars of other schools like Vyākaraṇa, Vedānta, Sāhitya, etc. borrowed the terminology, methodology, and axioms that the Navya-Nyāya school propagated for more incisive logical presentation of their doctrines and used them to refute the viewpoints of other schools, including Nyāya, and establish their own theories. This chapter throws light on how the Nyāya concepts, axioms, and language influenced the Vyākaraṇa scholars and gave them the tools for analytical and critical thinking, aiding them to establish their doctrines by defending the refutations from other schools and also refuting the viewpoints of other schools while claiming supremacy of their own theories. Keywords

Vyākaraṇa · Navya-Nyāya · Abhāva · Anugama · Abhedaḥ · Cause-effect · Efficient cause · Kevalānvyin · Invariable concomittance · Vyāpti · Pakṣa · Sādhya · Fallacies of reason · Śivasūtra · Pratyāhāra

Introduction The renowned saying goes as “the concepts of Kaṇāda and Pāṇini are aids for all Śāstras” (kāṇādaṁ pāṇinīyañca sarvaśāstropakārakam). As spelt therein, there is a lot of utility that the two Śāstras provide to all the schools of thought. However, when Gaṅgeśopādhyāya propounded the Navya-Nyāya-śāstra with his treatise, Tattvacintāmaṇi, from then on, it is the usage and influence of this Śāstra that is vastly seen in all other Śāstras and Darśanas. Its influence is clearly seen in all other later treatises such as those of Alaṅkāra (poetic aesthetics), Mīmāṁsā, and Vedānta for refutation of the conclusions of Navya-Nyāya, refutation of concepts of other schools of thought based on the ontological categories accepted in Navya-Nyāya using the very same style of improvisations (pariṣkāra) as used in Navya-Nyāya, for justification of one’s own tenets and so on. The Vyākaraṇa-śāstra is no exception to this influence. The influence of Navya-Nyāya is seen to be very heavy among the followers of the Vyākaraṇa School who succeeded in the period after Gaṅgeśopādhyāya. This chapter herewith attempts to explain this influence. A doubt may arise here pertaining to the statement of the Vyākaraṇa-mahābhāṣya by Patañjali that goes as “the regulation of words means knowing (the nature of words) as prescribed by the Vyākaraṇa-śāstra” seems to merely stipulate that the role of Vyākaraṇa involves in distinctly making known the proper verbal usages distinguished from the improper verbal usages through presentation of the division of

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words in to roots and suffixes. That being the case, with regards to a Śāstra that involves imparting merely the morphological process, how can there come about an influence of the propagator of the Navya-Nyāya-śāstra, the school that deals with topics such as the ontology, and epistemology, by the author of Tattvacintāmaṇi? This article endeavors to elucidate that. Apart from the Munitraya (the triad of exponents of Vyākaraṇa- Pāṇini, Vararuci, and Patañjali who authored the treatises Aṣṭādhyāyī, Vārttika, and Mahābhāṣya thereon), among others, Bhartṛhari, Kaiyaṭa, Vāmana, and Jayāditya are all predecessors of the author of Tattvacintāmaṇi. Hence, there is no question of the influence of Tattvacintāmaṇi in their works. Among the prominent exponents of Vyākaraṇa post the period of Gaṅgeśopādhyāya are Bhaṭṭojidīkṣita, Haridīkṣita, and Nāgeśabhaṭṭa. The influence of Navya-Nyāya-śāstra is seen in the treatises of these authors and in the commentaries thereon. The influence of the Navya-Nyāya-śāstra on their works may be categorized in two ways – the influence of the method of improvisation adapted by the Navya-Nyāya exponents and the influence of the means of valid cognition accepted by them. In what follows, we illustrate with examples how the fallacies of reason, the concept of encapsulation (anugama), the difference between the inherent cause and the efficient cause, and the discussion on invariable pervasion have helped the later grammarians to strengthen their arguments further. The discussions on double negation, identity relation, and the problem of counter positive in the case of kevalānvayi cases of pervasion have also been used effectively by the grammarians in their commentaries. Lastly, an example involving cause-effect relation in the verbal import is also presented.

Fallacies of Reason In the portions related to inferential cognition in the Navya-Nyāya texts a fivefold classification of fallacies of reason has been presented. They are: straying reason (Savyabhicāra), contradiction (Viruddha), counter-balanced reason (Satpratipakṣa), unknown reason (Asiddha), and sublated reason (Bādha). Even though the commenting on the Mahābhāṣya would have sufficed with no mention of the fallacies of reason, yet Mahābhāṣya was commented upon by Nāgeśa with its due enumeration. We present below three instances where Nāgeśa has used various reasons of fallacies while commenting upon the Mahābhāṣya. 1.1 While discussing “Now, here in (the reference of) ‘cow’, what is (exactly denoted by) the word” (atha goriti ko śabdaḥ – Paśpaśāhnika – Mahābhāṣya) first it was hypothesized as “it is the object that is of the form possessing dewlap, tail, hump, hoof and horns that (which is denoted by word “cow”) (kim yattatsāsnālāṇgūlakakudakhuraviṣāṇyartharūpam sa śabdaḥ)” and then refuted as he (Patañjali) said “no, that is indeed (what is known as) ‘substance’”(netyāha dravyam nāma tat). Following this, it was hypothesized as “then whether (that which is referred to as) ‘the white one’, ‘the black one’, ‘the tawny one’ that are (denoted by) the word (“cow”)” (kiṃ tarhi śuklo nīlaḥ kapilaḥ ityādikaṃ sa śabdaḥ) and refuted as he

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(Patañjali) said “no, that is indeed (what is known as) ‘quality’” (netyāha guṇo nāma saḥ). In the Mahābhāṣya, “no, that is indeed (what is known as) ‘quality’” is all that is mentioned. But, let us now see how Nāgeśa comments the statement reeling under the influence of training in Navya-Nyāya. The objection and its reply given in the Mahābhāṣya were commented as follows by Nāgeśa – “The inference goes as ‘white (colour of a cow, etc.) is not a sound, since (there is) the nature of quality (in it)’” (śuklādikaṃ na śabdaḥ guṇatvāt). It seems like that the author of the Bhāṣya intended to establish in white, etc. (colors) the difference from they being sound using the reason of the nature of quality being present in white color, etc. However, there is an apprehension from the Nyāya standpoint wherein sound is also categorized as a quality, that, since sound is also of the nature of quality, there arises the predicament of the absence of the probandum, namely, color being different from sound, which is present in sound, that is, absence of difference from sound is present in sound, there is presence of nature of quality, entailing in the reason being tainted with the fallacy of straying away from proving the probandum. Straying reason is defined as the presence of the reason in a substratum where what is intended to be concluded is not present, as in, “the hill has fire, since it is knowable.” Herein the reason in the form of “being knowable” is present in the location, such as, say, a pond, etc. where what is intended to be concluded in the given inference, namely, there being fire, is not actually present. Reasons of this sort do not prove what is intended to be concluded from the proposed inference. The conclusive tenet in this regard has been propounded by the author of Tattvacintāmaṇi that this is due to the straying reason obstructing the possibility of invariable relation between the reason and what it seeks to conclude, which in its turn is instrumental in giving rise to inferential cognition. Following this tenet the foregone apprehension was raised by Nāgeśa. The fallacy of straying reason may be resolved in two ways – by extricating the reason from the substratum where it is contended to be straying from, or by establishing the presence therein of that which is intended to be concluded by the inference. Thus, without the knowledge of averting the fallacy as propounded in the Navya-Nyāya of Gaṅgeśopādhyāya one will not be able to comprehend the resolving of the objection undertaken by Nāgeśa in the given context. That is, in that section, the fallacy of straying reason being in context – since the reason, namely, the nature being a quality, being present in the substratum of the absence of what is intended to be concluded (in white color absence of sound is intended to be concluded) in the inference – was eliminated by removing the reason from the substratum, or in other words, the nature of being a quality was removed from sound. According to the standpoint of Vyākaraṇa-śāstra sound is endowed with the nature of being a substance and not with the nature of being a quality. Therefore, there is no fallacy of straying reason (nature of being a quality) since there is actual absence of the reason in the form of the nature of being a quality in sound, which is the substratum of the absence of what is intended to be concluded (absence of sound) therein. Hence, white color, etc. are not sound, since they (white color, etc.) are of the nature of quality- is indeed true inference.

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1.2 Similarly, another fallacy of reason named as “unknown reason” (Asiddha) is propounded in the Nyāya-śāstra. That is again threefold as – non-establishment of reason in the ground of inference (Svarūpāsiddha), non-establishment of the locus (Āśrayāsiddha), and non-establishment of the one being pervaded (in the relation of invariable pervasion in an inference) (Vyāpyatvāsiddha). Therein, partial non-establishment of reason in the ground of inference (Bhāgāsiddha) is a type of fallacy of reason classified under Svarūpāsiddha. Bhāgāsiddha is a fallacious reason wherein the reason is not present in some parts of the ground of inference. This sort of a reason is also incapable of establishing what is intended to be concluded in an inference, for, in order to give rise to inferential cognition as conditioned by the universal nature of the entity that forms the ground of inference (that is when inferential cognition involves all and every instance of the ground of inference), the knowledge of the ground of inference as conditioned by its universal nature locating the reason in it (that is, the knowledge that the reason of inference exists in all and every instance of the ground of inference) is imperative. In the process of giving rise to inferential cognition as conditioned by the universal nature of what is intended to be concluded on the ground of inference (that is, the reason of inference present in the ground of inference establishing its invariably related pervading entity that is concluded from inference, being present in all and every instance of the ground bearing the reason of inference), the knowledge of the fallacy of partial non-establishment of reason in the ground of inference impedes by obstructing the cause of inference, namely, deduction (Parāmarśa) (Pakṣatā Gādādharī, Śāstramuktāvalī Series, page no. 42). Now we provide an example from the Mahābhāṣya. Three reasons were shown in the apprehended case of anubandha-s (markers) not forming part of the “aC”pratyāhāra (it is a set of vowels formed with the first 4 Śivasūtras [a i u Ṇ] [ṛ ḷ K] [e o Ṅ] [ai au C]), namely, owing to being in vogue (practice established by bygone scholars), owing to (the markers) being non-significant and since elision (of the marker) is predominant in the grammatical process. Thus results the inference – “Anubandha-s are not constituents of Pratyāhāra-s, since that is the practice in vogue, since the anubandha-s are non-significant and since elision of anubandha is a predominant grammatical process.” Herein, Nāgeśa poses the following apprehension – if the nasal vowel sound “a” in the “lAṆ”-pratyāhāra were also to be considered as an anubandha, then the third reason mentioned in the given inference, namely, elision of the anubandha being a predominant grammatical process would be tainted with the fallacy of being non-pervading (Laghu-śabdendu-śekhara – Sañjñā-prakaraṇa). In the given case, it is rather desirable that between the probandum and the reason that attempts to establish it, the probandum should be pervading and the reason should be non-pervading (that is, the reason should be pervaded by the probandum and not pervading the probandum). There would arise a common doubt as to what could be the objection if the third reason were not pervading in the inference. The resolution here is that the notion of reason being non-pervading is in relation to the coverage of the ground of reason in its universal aspect instead of reason being present only in a few instances, and not in relation to the invariable pervasion that the reason has universally that is being pervaded in all

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and every instance with the probandum. (That is, the reason (elision of anubandha being predominant in all grammatical processes) is not pervading all and every instance of the ground (all anubandha-s)). For, the nasal vowel sound “a” present in the “lAṆ”-pratyāhāra, belonging to the gamut of the all the pratyahara-s, that form the ground of the inference, is not predominantly elided in the grammatical processes. For, the factor of predominance in elision refers to trigger of grammatical process immediately succeeding the formation of the hal-pratyāhāra (all consonant sounds). In case of the elision of the nasal sound “a,” if the elision is not triggered in immediate succession to the accomplishment of the hal-pratyāhāra (a set of all consonants formed by the 10 Śivasūtras – [h y v r T] [l N] [ñ m ṇ ṇ n M][jh bh Ñ] [gh ḍh dh Ṣ] [j b g ḍ d Ś] [kh ph ch ṭh th c ṭ t V][k p Y][ś ṣ s R] [h L]) and instead (the anubandha) “a” is elided only after the labelling of “aC”-pratyāhāra is accomplished using the Sūtra “(a pratyāhāra is formed) by putting together the initial (sound) along with (one of the following) ‘it’ (the anubandha component that will be elided from formation of a set, but will be used for grammatical operations)” (ādirantyena sahetā 1.1.71). Therefore, the reason of “predominant elision” not being present to the sound “a,” which is one among the grounds of inference, (namely, the entire set of anubandha-s), there would arise the fallacy of partial non-establishment (of reason in the grounds of inference) – Bhāgāsiddhi. And thereby, as shown previously, the fallacy may obstruct the arrival at the inferential cognition, thereby the reason being incapable of proving the probandum (on the ground of inference) rendering the third reason propounded in the Bhāṣya to be inconsistent. Thus, if the nasal sound “a” in the “lAṆ”-pratyāhāra does not account as being an anubandha at all, then how can one get the “ra (nasal)”-pratyāhāra, is the intention expressed here. 1.3 In an inference, non-establishment of the reason as being pervaded by the probandum due to an intervening condition (upādhi) is termed as Vyāpyatvāsiddha. Upādhi refers to a condition which pervades the probandum but does not pervade the reason in an inference. For instance, in a fallacious inference that something has smoke, since there is fire, contact with wet fuel is the intervening condition (that obstructs the relation of pervasion between smoke and fire, wherein smoke is hypothesized to be pervading fire). For, contact with wet fuel pervades smoke since wherever there is smoke, wet fuel is observed to be present, which takes the role of what is intended to be concluded in the give inference, while it does not pervade, fire, which is the reason in the given inference. For, if red hot iron were considered to be the location of reason (fire), then the probandum (smoke) is seen to be absent. Hence, this reason is seen to be intervened by an (obstructive) condition disallowing it from proving the probandum. According to the Nyāya-śāstra, the upādhi obstructs inferential cognition by leading the reason to stray (from proving the probandum owing to fallacious invariable relation). Without this background the following discussion may appear incomprehensible. An instance herein, in the Bhāṣya (Mahābhāṣya) on the Sūtra (that enlists the sounds starting with) “a i u Ṇ,” there is an analysis of whether the sound “a” is one or (represents) many. Therein, the author of the Bhāṣya uses reasoning to arrive at the probandum that the sound “a” represents many (types of that singular) sound “a”

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as – The sound “a” has other forms, owing to the differences in its pronunciation (as nasal, not nasal, high pitched, etc.) and duration (of utterance as short, long, etc.) (ānyabhāvyaṃ tvakārasya kālaśabdavyavāyāt – Mahābhāṣyam). That is, the inference results as – the sound “a” is manifold, based on the differences in sounds and based on the duration of utterance of each of them. The annihilation of this inference is shown in the Bhāṣya itself as – “they (different forms of the sound ‘a’) may be like the Sun. Just as the same Sun placed in different substrata is made available simultaneously in different places,” (so is the case here) (ādityavatsyuḥ tadyathā eka ādityo anekādhikaraṇastho yugapaddeśapṛthaktveṣūpalabhyate. – Mahābhāṣyam). That is, an annihilating inference against the inference that concludes multiplicity of the sound “a” is spelt as – “the sound ‘a’ that is made available in different places (utterances) is the same one, owing to being recognized (again and again wherever it is come across), like the Sun.” However, this inference would have annihilated the inference that intends to prove the multiplicity (of the sound “a”), but it does not do so, since the reason that intends to prove the singularity (of the sound “a”) is clutched with an intervening obstructive condition or upādhi. This obstructive condition has been shown as follows by the author of the Mahābhāṣya – It is not that a single viewer simultaneously views the Sun as located in different substrata, but with the sound “a” is cognized in that way (that is, one and the same person simultaneously hears the sound “a” uttered from multiple sources) (naiko draṣṭā ādityamanekādhikaraṇasthaṃ yugapaddeśapṛthaktveṣūpalabhyate, akāraḥ punarupalabhyate. – Mahābhāṣyam). That is, the condition of being an object of cognition, such as is present in multiple substrata and has multiple viewers (perceivers), is the upādhi of the reason that annihilates multiplicity (of the sound “a,” instead of considering it as one entity reflected as many when uttered through multiple sources). It is justified that it (the condition of being an object of cognition, such as is present in multiple substrata and has multiple viewers (perceivers)) is indeed explicable as an upādhi, for, it pervades the probandum (of the inference, namely, singularity of the sound “a” based on an illustration of the only Sun reflecting in multiple substrata and appearing as many), since wherever there is the nature of an entity being singular therein is entity (has the tendency of) being available to the cognition of multiple perceivers, simultaneously in multiple substrata, (which is a condition) present in the Sun; and it does not pervade the reason (of inference, namely, differences in the sound (“a” in its varied pronunciations such as nasal, non-nasal, etc.) and timing (of pronunciation of “a”)), since there is recognizability (of the sound as being same as “a” pronounced earlier, elsewhere, etc.) in the sound “a” which is an object of cognition of a single perceiver while being simultaneously available through multiple sources (and hence does not succumb to the condition of requiring multiple perceivers also, thereby effecting its non-pervasion of the reason). Thus, the reason in the form of “being a recognizable entity” being tainted with the fallacy of upādhi, this reason does not give rise to the inferential cognition of singularity (of the sound “a”). Hence, the annihilating inference being inefficacious, it does not annihilate the inference that concludes multiplicity (of the sound “a”) as – the sound “a,” (etc.) are multiple, due to the

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differences in the sounds (of being nasal, high pitched, etc.) – this is the opinion of the Mahābhāṣya. From the foregoing illustration it is clearly known as to how a commentary is influenced by the tenets of Navya-Nyāya, even in regard to texts that are related to expounding purely (grammatical) processes. The impact of the Tattvacintāmaṇi text on these gets clearly established.

Relation Between the Pervaded and the Pervading (Entities in Invariable Pervasion) Those belonging to the Navya-Nyāya school accept that there is a relation between the pervaded and the pervading entities (that are related through invariable concomitance). It is of the nature that, where the pervaded entity is present, there the pervading entity is also present. So also, wherever the pervading entity is absent, there the pervaded entity is also absent. For instance, there is presence of the pervaded entity, namely, smoke in all instances where there is presence of the pervading entity, namely, fire. So also, wherever there is absence of the pervading entity, namely, absence of smoke, the pervaded entity, namely, absence of fire is also found. This doctrine has been accepted in the school of Vyākaraṇa vide the (treatise) Mahābhāṣya and its own doctrine is propounded on that basis. For instance, there is a discussion in the Mahābhāṣya as to whether sounds are meaningful or meaningless (in the commentary of [the pratyāhāra-sūtra [a i u Ṇ]). Therein, it has been proved that individual sounds are meaningful, for, the verbal roots such “iṅ” (the sound “ṅ” is elided and only “i” remains) in the sense of “to study,” composed of a single unit of sound are also seen as comprising some meaning. Among the reasons that have been employed to prove that individual sounds have meaning, is one reason that goes as “owing to the syllable also being meaningful.” The opinion of the commentary is that, since the group of sounds (a syllable) is meaningful, an individual sound (unit of the syllabic group) shall also be meaningful indeed. This viewpoint cannot be comprehended without the knowledge of the relation between the pervaded and pervading entities in invariable pervasion. The author of the Mahābhāṣya views the meaningfulness of the group (syllable) as the pervaded entity and the meaningfulness of the individual sound as the pervading entity. Or in other words, the viewpoint of the author of the Mahābhāṣya is that if there exists a pervaded entity, namely, meaning in the group of sounds (forming a syllable), then by all means the pervading entity, namely, individual sound (that forms part of the syllable) should be meaningful too.

The Difference Between Inherent Cause and Efficient Cause Three fold causes (of effects) have been accepted in the Navya-Nyāya-śāstra, as effect directly inhering in its cause (samavāyi-kāraṇa), effect indirectly inhering (through a property that exists in) the cause (asmavāyi-kāraṇa), and a totally

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non-inherent efficient cause (nimitta-kāraṇa) (Pratyakṣakhaṇḍa of Tarkasaṅgraha). Herein, the definition of directly inherent cause is stated as – that, inhering in which the effect comes into being, is the (direct or) inherent cause. That which inheres in the same entity as does the cause (of a related effect) or the effect and creates the effect is an indirectly inherent cause. Any cause that differs from these two causes is known as efficient cause. That is, the difference of directly and indirectly inhering causes has been accepted in the efficient cause. For instance, the directly inherent cause of pot is pot shreds. The indirectly inherent cause of the pot is the contact between the pot shreds. The difference of these two causes is found in a stick used to rotate the potter’s wheel and is concluded as an efficient cause of pot. This conclusion is actually limited to Nyāya-śāstra and a technical one. However, many results conducive to Vyākaraṇa were achieved by using this tenet. For instance, there is a Sūtra (aphorism) by Pāṇini – sārvadhātukārdhadhātukayoḥ (A 7.03.84). This sūtra regulates that when an aṅga (phonetic constituent) ends in one of the sounds among – i, u, ṛ, and ḷ followed by a pratyaya (suffix), be it sārvadhātuka or ārdhadhātuka (tiṅśit sārvadhātukam (3.4.113) – The verbal suffix which is tiṅ (one among the 18 verbal suffixes that start with ‘tip’ and end with ‘mahiṅ’) or śit (the suffix in which the sound ‘ś’ is elided, such as ‘śap’) is called as sārvadhātuka. Ārdhadhātukaṁ śeṣaḥ (3.4.14) – All verbal suffixes other than the sārvadhātuka-s are called ārdhadhātuka-s.), then the sound ends up being a Guṇa sound (a, e, or o). For example, when the verbal root “bhū” (“to be”) is added with the suffix “tṛc” (in the sense of agent of action, as in words like kartā, dātā, etc.) and the āgama (grammatical augment) “iṭ” (in between the verbal root and the suffix leading to the form – bhū þ i (“ṭ” elided) þ tā (from “tṛc”) joins in), then owing to this sūtra (sārvadhātukārdhadhātukayoḥ), the “u” gets guṇified, owing to which “bhū” becomes “bho” and then due to the following “‘i” “o” further gets substituted by “av” resulting into the form “bhavitā.” Going by the same method, with regard to the verbal root “śīṅ” (“to lie down”), when the suffix “tṛc” is added and the āgama (grammatical augment) “iṭ” joins in (between the verbal root and the suffix), then owing to this sūtra (sārvadhātukārdhadhātukayoḥ), the ‘ī’ of the verbal root śīṅ is guṇified to “e” resulting into “śe,” which further gets substituted by “ay” leading to the form “śayitā.” But, an objection that arises here that, since the verbal root “śīṅ” is one in which “ṅ” gets elided, and since the process of guṇa (the sound “ī“ transforming to “e” being applied to the constituent that precedes a sārvadhātuka or ardhadhātuka suffix, here “tṛc”) is impeded here by the sūtra “gkṅiti ca” (1.1.5) in connection with verbal roots in which the elided element happens to be “ṅ,” how does the guṇification come to play? This objection was answered by the Vyākaraṇa scholars by resorting to the tenet of Nyāya-śāstra as stated previously. That (the verbal root) “śīṅ” is instrumental to the (transformation of “ī” in “śī” (“ṅ” elided) into) guṇa sound (as “e”) indicates that the verbal root “śīṅ” participates in the result (of “ī” transforming in the form of) the guṇa sound (“e”). The verbal root that contributes to the effect is the direct inherent cause (of the effect in the form of guṇa sound), like, in the Nyāya-śāstra, once the effect comes in to being, the (direct inherent cause, namely,) the clay is destroyed

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and a pot gets created. Therein, clay happens to be the ditect inherent cause (of pot) and not an instrumental cause. Similarly, in the given instance also, since after the process of transformation into guṇa sound, the form resulting in the form of “śe” comes into being from (the verbal root) “śīṅ,” with regards to the effect in the form of a guṇa sound, the verbal root “śīṅ” is a direct inherent cause. Since the distinction between a direct inherent cause and an instrumental cause has been elucidated in the Nyāya-śāstra, the verbal root “śīṅ” can never be assigned as the instrumental cause of the guṇification. Thereby, since the guṇa sound does not have as its instrument the verbal root (ending in “ṅ”) wherein “ṅ” will be elided, the sūtra “gkṅiti ca” has no scope for application here and thereby we obtain the form (comprising of guṇa transformation) as desired.

The Axiom that Negation of Absence Amounts to Presence of the Negated Entity According to the scholars of Nyāya-śāstra, that “the negation of absence (of an entity) amounts to the presence of it” is an admitted axiom. For instance, the counter entity of the absence of a pot is the pot (itself). They assert that the negation of the absence of pot amounts to the presence of the pot. That is a common usage with them. Only by resorting to this axiom do they concur the definition of invariable concomitance (with real situations). For instance, invariable concomitance is defined in Nyāya as – lack of presence of the reason of inference in a locus that has the absence of the probandum of inference. In the case of an inference as “(an entity, say, a cloth) is one that has absence of pot, due to (presence of) cloth-ness,” the concurrence of the definition is noted as follows – Herein, the probandum intended from the inference is “absence of pot.” The absence of the probandum amounts to “negation of absence of pot” which is the same as the “presence of pot.” Thus, the locus of the absence of probandum is the one that possesses pot-ness, which is indeed, the pot. The presence of pot-ness is found in the pot and lack of such presence is found in the reason (of this inference, namely) cloth-ness. This is the method with which the concurrence of the definition of invariable concomitance as “non-presence of the reason of inference in a locus where the absence of the probandum is present” is achieved. Though this method appears to be totally limited to only the Nyāya-śāstra, some fallacies found in the context of Vyākaraṇa-śāstra were resolved by resorting to this method. As an illustration, with the sūtra “pūrvatrāsiddham” (A 8.2.1) (the sūtras running after 8.2.1 would be non-applicable in the domain prior to it, as though being invisible to that domain) it is indicated that the three quartets of the eighth chapter of Aṣṭādhyāyī (tripādī) are invisible for application to the seven chapters and the first quartet of the eighth chapter of Aṣṭādhyāyī (sapāda-saptādhyāyī). There are two standpoints concerning the invisibility (of the sūtras of tripādi in the sapādasaptādhyāyī). One standpoint is that the entire section of tripādī is as such invisible. The other standpoint is that the process prescribed in the tripādī section by the śāstra

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is invisible. Between the two views, those who advocate that it is the grammatical process that is invisible, are objected by the others with a reasoning that the process being most important component therein it can never be invisible. On the other hand those who advocate that the entire section of tripādī is invisible show some inconsistency in the stand point of those advocating invisibility of merely the grammatical process stated in the tripādī. Therein, if it be considered that the grammatical process is what is invisible (by the abovementioned sūtra), then it would imply that the sūtra indicates that one needs to impose that the grammatical processes would be triggered (by the sūtras in the tripādī) but such triggered process would not be visible (to the processes in the sapāda-saptādhyāyī). As an example, in case of derivation of a word which is at the stage of – “Manar rathaḥ” herein the sound “r” (found at the end of the term “Manar”) is bound to be elided based on the sūtra “ro ri” (A 8.3.14 – (When the sound “r” at the end of a pada (a meaningful word) is succeeded by the sound “r” at the beginning of the following word then the sound “r” at the end of the former word is elided and the vowel sound that precedes the sound “r” is elongated). It is to be noted that this Sūtra is placed in the tripādī section.) In this context, due to the sūtra “haśi ca” (A 6.1.114 – (When the sound “a” precedes the visarga (aspirate) sound (which stands as “r”) which is followed by a sound from the “haś”-pratyāhāra, (composed of the sounds ha, ya, va, ra, ṭa, la, ña, ma, ṅa, ṇa,na, gha, ḍha, dha, ja, ba, ga, ḍa, da, then the visarga (presented as “r”) is substituted by the sound “u.”) It is to be noted that this Sūtra is placed in the sapāda-saptādhyāyī section.) substitution of “r” to the sound “u” also finds scope (resulting the given instance in “manau ratha” and subsequently as “manoratha”). Between the two (processes), though it is to be regarded that elision of the sound “r” (according to the sūtra “ro ri”) shall remain invisible (owing to the sūtra’s location in the tripādī section of Aṣṭādhyāyī), in case of going with the standpoint that it is the process of the sūtra (in tripādī) that remains invisible, then it is to be agreed that the invisibility comes about after the process of elision is triggered. In such a case, if the elision (of the sound “r”) is done, then there shall be no result even if the process were declared as invisible. This is because, if after the elision of the sound “r,” the process was declared to be invisible, then since the location of substitution is lost where would the substitution (triggered by the Sūtra “ro ri”) occur? Thus, an objection is brought forth to the effect that may it be considered that the sūtra (in the tripādī) itself becomes invisible. This prima facie view is refuted based on the admitted axiom in the Nyāya-śāstra that “the negation of absence of an entity amounts to its presence” by the scholars of Vyākaraṇa who advocate that it is the processes prescribed in the tripādī that remain invisible (to the processes operative in the sapāda-saptādhyāyī section of Aṣṭādhyāyī) (Manoramāśabdaratnam – Sañjñāprakaraṇam – Pūrvatrāsiddhamiti sūtra). That is, after the sound “r” is elided, the process (triggered by the sūtra “ro ri”) is to be suspended (due to the consideration that the sūtra in tripādī section becomes invisible after effecting a process). Even when that is done the desired outcome is accomplished, for, elision means “not being visible” or “absence of visibility.” So, elision of the sound “r” means the sound “r” is absent from visibility. Thus, if it is stated that what is imposed is the invisibility (inoperativeness) of the

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absence of visibility of the sound “r,” it means that the negation of the absence of the sound “r” is being imposed. Going by (the Nyāya axiom that) negation of absence of an entity amounts to the presence of it, the negation of the absence of visibility of the sound “r” would amount to the presence of its visibility. And hence, since the visibility of the sound “r” that happens to be the location of substitution of the sound “u” (by the sūtra “haśi ca”) can be depicted even in the viewpoint of considering that invisibility (of the tripādī section) is applicable after the process triggered by the sūtras present therein and thereby there is no inconsistency (of inapplicability of the substitution due to lack of the location of substitution) herein. In this manner, the scholars of Vyākaraṇa establish their tenets based on the axioms admitted in the Nyāya-śāstra.

The Complex Absence Caused by the Absence of the Attribute and the Absence of the Substantive In the Nyāya-śāstra, absence of two things has been classified in different ways as – (a) absence of both entities under consideration jointly, when one of them is present (Ubhayābhāva), (b) complex absence caused by the absence of the attribute or the absence of the substantive (Viśiṣṭābhāva), (c) absence of both of the two entities under consideration (Anyatarābhāva), etc. Among them, there comes about the absence of both (entities under considered jointly) (Ubhayābhāva) with a view pertaining to two counter entities as “though one is present, both are not present (together).” For instance, even though (the property of) smell is present in the earth (element), it is possible to say that the Ubhayābhāva of smell and cognition (put together) is absent in earth. However, only when both counter entities of absence are negated (in a locus) that Anyatarābhāva is established, as in water, both smell and cognition are absent (and thus there is Anyatarābhāva of smell and cognition in water). Similarly, only when there is a person on the ground attributed with a stick that the usage “daṇḍin” (a person holding a stick) is feasible. When only the attribute in the form of the stick is present on the ground, since the substantive in the form of the person holding it is absent, then the complex absence (Viśiṣṭābhāva) caused by the absence of the substantive, which is cognized as “the one attributed with a stick is not present” comes to be established. So also, only when the person is present, then though the substantive is present, because of the absence of the attribute in the form of the stick, the Viśiṣṭābhāva that is caused by the absence of the attribute is established. This sort of Viśiṣṭābhāva has been adopted by the scholars of the modern school of Nyāya while defining Pakṣatā (that which constitutes the ground of inference). It is indeed not surprising then that this particular type of absence is widely used in the Nyāya-śāstra to fortify their arguments. However, this type of absence has been resorted to in Vyākaraṇa also to accomplish valid forms of usage. The manner of improvising (a process or a definition) was also resorted to accordingly. As an illustration – in the sūtra “aṭkupvāṅnuṁvyvāye’pi” Aṣṭādhyāyī – 8.4.2 – (This Sūtra follows the Sūtra 8.4.1 – “raṣābhyāṃ no ṇaḥ samānapade” – which indicates that

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within a word, when the sounds “r’ and “ṣ” are followed by the sound “n,” then it is substituted by the sound “ṇ.”), the words “samānapade” (within the same word) is followed in from the previous sūtra (8.4.1). This sūtra indicates that within the same word when the conditioning situation in the form of the sounds “r” and “ṣ” are present, then, the conditioned entity, namely, the sound “n” is to be substituted by the sound “ṇ”. In the case of the derivation of the word “rāmeṇa,” after getting rāma +ena ¼ rāmena, since the sounds “r” and “n” are present in the same word, the sound “n” is substituted by the sound “ṇ” following this sūtra. However, if the literal sense of the sūtra were taken as such, it would lead to the fallacies of over-pervasion and under-pervasion (of the conditional consequence in certain instances). As an instance, the compounded unit of the word “rāmanāma” would be with the fallacy of over-pervasion since the conditioning situation in the form of the sound “r” and the conditioned sound “n” are present in the same word, rendering the substitution as “ṇ” being effective. Similarly, in the word “Rāmāyaṇa,” while the conditioning situation in the form of the sound “r” is present, and the conditioned sound “n” is present in the form of a suffix, which is not a part of the same word, the substitution in the form of the sound “ṇ” ought not to have been affected, thereby leading to the under-pervasion of the condition in the given situation. Hence, the scholars of Vyākaraṇa, following the footsteps of the scholars of Nyāya, improvised the sense of the expression “within the same word” (in sūtra 8.4.2) as “not being a constituent of the word comprising the conditioned entity (namely, the sound ‘n’) while not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds ‘r’ and ‘ṣ’).” Therein, “the conditioned entity (namely, the sound ‘n’) not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds ‘r’ and ‘ṣ’)” is the attributive portion and “being the word” is the substantive portion. This way, the issue is resolved using the mechanism of Viśiṣṭābhāva that is developed in the school of Nyāya. This is done as follows – In the case of the word “Rāmeṇa,” even though there exists the attributive condition of “the conditioned entity (namely, the sound ‘n’) not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds ‘r’ and ‘ṣ’)’ in the suffix ‘-īna’,” since it is a suffix and not a word, it does not satisfy the substantive condition, the Viśiṣṭābhāva caused by absence of the substantive condition, that is absence of being a constituent of the (same) word, comes to be present here. Therefore (the substitution with), the sound “ṇ” (in the place of the sound “n”) does come about by all means and thereby there is no fallacy of underpervasion in this case.

Encapsulation (Anugama) Encapsulation is a well-known method in the Navya-Nyāya-śāstra that involves stating long improvisations (in definitions, etc.) very succinctly. Using the same style, the scholars of Vyākaraṇa also improvise their theories in order to discard what is not intended to be defined and to accomplish what is intended to be

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defined. As an instance, the sense of the expression ‘within the same word’ (in sūtra 8.4.2) is improvised to mean “not being a constituent of the word comprising the conditioned entity (namely, the sound “n”) while not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds ‘r’ and ‘ṣ’),” to avoid the fallacy ((of the Sūtra 8.4.2) not working out) in usages such as “Rāmanāma.” However, in the two instances of the terms “Rāmanareṇa” and “Paramāgnīdhreṇa,” the terms do not satisfy the condition of not being a constituent of the word comprising the conditioned entity (namely, the sound “n”) while not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds “r” and “ṣ”), since sound “r,” which is the conditioning entity (for substitution of the sound “n” with “ṇ”) is located in the same substratum (as the two conditioned sound of “n” as in “Rāma‘n’a”- and “Paramāg‘n’ī-”). Hence the interpretation of the condition of “not being a constituent of the word comprising the conditioned entity (namely, the sound ‘n’) while not being in the same substratum (being intrinsic to the word itself) as that of the conditioning situation (namely, the sounds ‘r’ and ‘ṣ’)” is redundant. For, the condition of not being a constituent of such a word (which is not in common substratum with the conditioning entity) is applicable in the cases of both the terms “Rāmanareṇa” and “Paramāgnīdhreṇa” and hence the objection that the sound “n” ought to be substituted with the sound “ṇ” herein. In order to resolve the problem arising with regards to the targeted (sound ‘n’), the modern scholars of Vyākaraṇa resort to the technique of Anugama. This is how it is done – “being unfragmented” is “not being a constituent of the word which comprises the conditioned entity (namely, the sound ‘n’), being attributed with the conditioning entity (namely, the sounds ‘r’ and ‘ṣ’).” “Being an attribute” herein refers to be with the relation of both (Ubhaya-sambandha) (features of) “(the sound ‘n’) being in succession of oneself (i.e., the sounds ‘r’ and ‘ṣ’)” and “being a constituent of a word that does not constitute oneself (i.e., the sounds‘r’ and ‘ṣ’),” together (being effective). The commentary Saralāṭippaṇī on Manoramāśabdaratnavyākhyā (on the Sūtra 8.4.2 – raṣābhyāṃ no ṇaḥ samānapade). In instances of “Rāmanara” and “Paramāgnīdhra,” when the sound “r” is considered to be applicable to the term “oneself” (in the given definition of akhaṇḍatva) then, the sound “n” in the words “nara” and “agnīdhra” do suffice the condition of succession (to the sound “r” in “Rāma” and “Parama”). The words “nara” and “agnīdhra” are the non-constituent words (of the words “Rāma” and “Parama”). But the sound “r” is a constituent of those (non-constituent words) also, namely, “nara” and “agnīdhra” which are constituents of the words which comprise the conditioned entity (namely, the sound “n”), being attributed with the conditioning entity (namely, the sound “r”) through the Ubhaya-sambandha (stated previously), which go to compose the words “Rāmanara” and “Paramānīdhra,” and hence there is no room for fallacy in the form of the sound “n” not being substituted by the sound “ṇ”. (Simply put, the sound “r” not only precedes the sound “n” in the two words but also succeeds it being a constituent of the words containing the sound “n” and hence this cancels out the substitution of the sound “ṇ” in place of the sound “n” which would have been the

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case if the sound “r” were only in precedence and not in succession to the sound “n” in the given words.) The foregoing illustration also elucidates the usage of the methodologies of NavyaNyāya-śāstra in Vyākaraṇa-śāstra. Similarly, the improvisations (in definitions) done by scholars of Vyākaraṇa to establish their doctrines using the technique of encapsulation is seen elsewhere too. Here is another illustration of this technique applied (in the context of Vyākaraṇa). There is a meta rule (paribhāṣā) (in the Vārtika of Vararuci) stated as “vyapadeśivadekasmin” ((usage is to be applied to) the one akin to that which is signified (by a term)). The meta rule indicates that when the primary referent of a usage is not available, then, for loss of choice, the single available referent itself should be treated as the primary referent. This is what is referred by the term “akin to the one that is signified (by a term).” Nāgeśa Bhaṭṭa elucidated with an illustration – “a distinct signification is vyapadeśa (naming of an entity) and the one that has it (such a name) is the vyapadeśin. Akin to the one which is referred by such a name (term) is referred as being vyapadeśavat. This is as in the usages of “being eldest, middle and youngest son” with reference to a single son. Similarly, the primary signification of a usage is to be done with the single entity (available for any signification at all)’ (viśiṣṭo’padeśo vyapadeśaḥ so’syāstīti vyapadeśī. Tasminniva vyapadeśivaditi. Yathā ekasminneva putre jyeṣṭhamadhyamakaniṣṭhatvādivyavahārāḥ bhavanti tadvat ekasminneva mukhyo vyavahāraḥ kartavyaḥ. – Paribhāṣenduśekharaḥ). Using this meta rule when the mere sound “a” is used in the sense of one’s offspring, then by extending the characteristic of a word ending in the sound “a” to the sound “a,” and the suffix “iñ” which is used in the context of words ending in the sound “a” (when the meaning of offspring of the being denoted by that word is intended to be derived) is applied (to the sound “a”), going by the sūtra “ata iñ” (4.1.95). For the query as to why the suffix “iñ” would not be (directly) applicable (as a suffix to the sound “a,” instead of using the extension mode), the modern Vyākaraṇa scholars respond here using the methodology of Nyāya-śāstra that with regard to the sound “a” the nominal stem itself is the sound “a” and not one that ends with the sound “a” and hence the context for suffix “iñ” (to be applicable) is not organically met with (since by rule, it is the word ending with the sound “a” that calls for the suffix “iñ” to be appended to a nominal stem to arrive at the meaning of the offspring of the one being referred by the nominal stem). And how they do it is by defining the characteristic of being an end (of a word) as a unit (of sounds) being endowed with the relation of the triad of the absences of being a constituent of oneself (the other sounds apart from the word ending sound “a”), being in succession to the constituent of oneself (the other sounds apart from the word ending sound “a”) and being in precedence to oneself (the other sounds apart from the word ending sound “a”) (vyapadeśivadekasminniti paribhāṣā – Nāgeśa’s Gūḍārthadīpikā – Commentary on Paribhāṣenduśekhara). In the current illustration, there is no other unit (of sounds apart from the word ending sound “a”) that may be referred to as “oneself” (in the given definition of the word ending in the sound “a” for the rule “ata iñ”). By some means, if that sole sound “a” itself were taken to be referred by the term “oneself,” then the sufficiency

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condition of “being in succession to the constituent of oneself” not being met (due to there being no sound in succession), the sound “a” does not account to be termed as a sound at the ending of a word, and thereby the word in the form of the sound “a” is not fit to be termed as one that ends with the sound “a,” leading to the consequence that (the suffix iñ) does not come to be applicable based on the Sūtra ‘ata iñ’ and hence the Vyākaraṇa scholars effect it by using the meta rule “vyapadeśivadekasmin” ((usage is to be applied to) the one akin to that which is signified (by a term)).

The Comprehension of the Relation of Identity Only with Regards to the Objects Cognizable Distinctively Usually, when two words come with the same grammatical case (vibhakti), they are experienced to be related to each other with identity. For instance, in the case of two words “nīlo ghaṭaḥ” (a blue pot), the verbal knowledge having as its object the identity in the form of “the pot is non-different from (that which is) blue” is experienced. This being so, when the question arises as to how come the relation of identity is not grasped from the group of words such as “ghaṭo ghaṭaḥ” (a pot is a pot) and “daṇḍavān daṇḍavān” (the one who wields a stick is the one who wields a stick) which are constituted with the same (nominal) case with regards to both words, then the doctrine that is used by the Nyāya scholars to resolve this query is that only when the forms of objects that are cognized (by the two words formed of the same case) are different that their relation of identity is cognized. Though the object signified herein by the two words “blue pot” is one and the same, by one of the words blueness is cognized while by the other word pot-ness is cognized and in this manner since the pot is cognized differently (on the utterance of the two words) there is cognition of their relation of identity. In case of the words, “ghaṭo ghaṭaḥ,” it is one and the same pot that is cognized with its feature of pot-ness (alone) by the usage of both the words and hence the cognition of their identity relation does not arise. This is the manner by which the scholars of Nyāya ward off an undesirable consequence. Similarly, the verbal knowledge with difference (between two objects) is accepted (in Nyāya-śāstra) when there is a difference cognized between the predicate and that which conditions the substantive (refer to discussion on absence in Dinakari). For instance, in the case of “the blueness of a pot.” There is a difference between the conditioning element of the pot, namely, the pot-ness and the predicate, namely, blueness. In the instance of “the pot-ness of a pot,” the conditioning element of pot is pot-ness and the predicate is also pot-ness, and hence going by the methodology of Nyāya-śāstra, the verbal knowledge of their difference does not arise. Using this same doctrine, the scholars of Vyākaraṇa establish their own doctrines. For instance, there are two schools of thought among the Vyākaraṇa scholars pertaining to the query of what is kāraka (participant in an action denoted by a verb in a sentence). According to one of the schools, being a kāraka indicates the nature of creating an action, while according to the other school, it indicates the nature of being related to an action. When there is evaluation between the two

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schools as to which of them is valid, considering the point of view that being a kāraka indicates the nature of being related to an action, then there would be incoherence in the usage such as, “kāraka-s are related to the volition of the performer (bhāvanā)” or that “kāraka-s are subject to be related to the volition of the performer.” This is because, the former usage would end up referring “those related to the action (kāraka-s) are those related to action (¼ volition of performer).” In that case, since both words do not give rise to different connotations of (the idea of) kāraka, the verbal cognition with difference (between the concept of kāraka and what it is predicated with) as its object does not arise, leading to the consequence that (even) a valid usage would come across as incongruent. So also, the latter usage (“kāraka-s are subject to be related to the volition of the performer”) would also end up referring “those related to the action (kāraka-s) are subject to be related to action (¼ volition of performer),” and owing to their identity, the verbal cognition having difference (between the substantive and the predicate) would not arise, leaving a valid usage to be incongruent. If being a kāraka is accepted to be indicative of the nature of creating an action, then there would be no inconsistency in both the usages (“kāraka-s are related to the volition of the performer (bhāvanā)” and “kāraka-s are subject to be related to the volition of the performer”). In the former usage, by one of the words (namely, kāraka) kāraka is comprehended as conditioned by being creator of action, while by the other term (being related to the volition of the performer) it is comprehended differently as that which is related to action, thereby there is congruence in the verbal cognition that has identity of the two objects kāraka and bhāvanā. In the second usage also, the conditioning element of the substantive is “being of the nature of creator of action” while the predicate is “the relation with action” whereby the verbal cognition of their difference is also coherent. Thus, the modern scholars of Vyākaraṇa justify that the viewpoint that being a kāraka involves the nature of creating actions is a better one (Vaiyākaraṇasiddhāntalaghumañjūṣā – Subarthaprakaraṇam – Dvitīyārthanirupaṇam).

The Counter Entity of Difference Pertains Only to Relation (kevala¯nvayin) There exists an axiom in the Nyāya school that the counter entity (negation) of difference is an entity that is always related with every other entity (kevalānvayin). The nature of an ever-relating entity (kevalānvayin) is defined as “the noncountering entity of absolute negation” (kevalānvayitvaṃ nāma atyantābhāvapratiyogitvam – Anumānakhaṇḍa of Tarkasaṅgraha). If the absence of an entity is not known to exist anywhere, then since such absence cannot be a counter entity of anything, it is accepted to be an ever-relating entity in the form of a non-countering entity of absolute negation. For example, since the property of being knowable is present everywhere, its absence is not known to exist anywhere. Hence the properties like being knowable, being nameable, etc. are termed as entities pertaining only to relation (since they are seldom unrelated with anything – ever-relating entities). In the same manner, all objects are subject to be counter entities of one or the other

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difference without fail and hence the counter entity of difference exists everywhere. Therefore, the absence of it (counter entity of difference) is not known to exist anywhere. Thereby, the property of being a counter entity of difference is also a non-countering entity of absolute negation and so according to the doctrine of Nyāya scholars, it (counter entity of difference) is also an ever-relating entity. Using this axiom (of the Nyāya school), the scholars of Vyākaraṇa prove their tenets. As an illustration, while discussing the meaning of the prefix “nañ” constituting a negative compound, there is this context of querying whether the Nañ-tatpuruṣa (privative compound) has the meaning governed by the sense of the first constituent word of the compound or by sense of the last constituent word of the compound or by the sense existing outside (but related to) the constituent words of the compound. Therein, if the negative compound was governed by the sense of the first constituent word, then in the case of compounds like “anaśva”(an-aśva ¼ not horse) wherein the meaning of the first particle “nañ” (meaning, “not”) is indicative of difference (of the given being from horse), the verbal cognition of such words would need to be taken as “difference from horse” or “one which is different from horse.” If the sense of the last constituent word of the compound were considered as governing the sense of the compounded word, then, the meaning of the first constituent word (the negative particle) is “difference” and the meaning of the last constituent word is “horse,” and the connection between the two would be accepted as the one countering each other and the verbal cognition would need to be explained as “horse in the counter entity of (its) difference.” The proponents (Vyākaraṇa scholars) state that between the two viewpoints, the better one is to consider that the sense of the first constituent word governs the compounded word, thereby refuting the other viewpoint basing their (argument) in this very axiom of the Nyāya scholars. It goes as, in the consideration that the sense of the last constituent word governs the meaning of the compounded word, since the verbal cognition (of the compound word “anaśva”) will need to be explained as “horse is a counter entity of difference (with some entity)” and since being a counter entity of difference is an ever-relating entity, and since the nature of being a counter entity of difference is a common one found to be present in everything, it is not possible such a common property to indicate difference of one object from the others. For, it is only a unique (uncommon) feature that is accepted to be a distinguisher (of one entity as distinct from others). Thus, even though a negative prefix may have been used (in the compound word), since the meaning of it as “being the counter entity of difference” does not distinguish anything, thereby (even with the utterance of the negative prefix) if what would be grasped would end up being the common notion of a horse, then the usage of the prefix “nañ” would be redundant – thus is the refutation of the Vyākaraṇa scholars.

The Refutation of Non-difference Being a Relation As per the Nyāya doctrine, two objects that are indicated by two words being used in the same case (vibhakti) should be connected with each other with the relation of non-difference. It is stated by the scholars using the following terms – Non-difference

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becomes evident from the expectancy pertaining to an object that is indicated by a word that is connected with its immediately preceding word, sharing the same case as itself. For instance, the objects that are indicated by the two words in the same (here, nominative) case as “blue jar” are connected with (the relation of) non-difference and it is validated by experience that the verbal cognition arising out of (such usage) is “the jar is non-different from that which is blue” as per the stand of the Nyāya scholars. However, based on several reasonings and for several reasons, the Vyākaraṇa scholars do not accept non-difference as a relation. In fact, they refute the point that non-difference is a relation using an axiom stated by the school of Navya-Nyāya itself. It goes as follows – the definition of inferential cognition stated by Gaṅgeśopādhyāya is – the cognition that arises from the knowledge of the reason being present in the ground of inference, characterized by (the prior knowledge of) invariable concomitance (relation between the reason and what is intended to be concluded on the ground of inference) (vyāpti-viśiṣṭa-pakṣadharmatā-jñāna) is inferential cognition. Therein, “that which is characterized by invariable concomitance” and “that which resides in the ground of inference” constitute the Karmadhāraya compound. On adding the suffix “tal” in the sense of “the state of being” to the term “pakṣadharma” (feature of the ground of inference) the word is formed as “pakṣadharmatā” (the state of being the feature of the ground of inference). On probing as to what meaning arises out of this word, the following explanation was provided by the crest-jewel of the school of Navya-Nyāya named Raghunātha Śiromaṇi – “There is an axiom which states that relation is expressed by a suffix that bears the sense of the state of being” when added after a compound word. Therefore, from the words (compounding to form rājapauruṣya), “kingly manliness” the relation between their senses is known as that of “owner and owned.” Similarly, in the instant, with the term “vyāpti-viśiṣṭapakṣadharmatā” there are two word senses signified, namely, “vyāpti-viśiṣṭaḥ” (the first word) and “pakṣadharmaḥ” (the last word). The relation between them is the co-location of the characteristics that condition the (two) word senses. And it is of the form of only co-location of “the characteristic of being endowed with invariable relation (vyāpti-viśiṣṭatva)” and “the characteristic of being a feature in the ground of inference (pakṣadharmatva) that is expressed by the suffix that bears the sense of ‘the state of being’ added to the word compounded as Karmadhāraya.” Following this commentary of the author of (Tattvacintāmaṇi)-dīdhiti, the Vyākaraṇa scholars also comment that “non-difference” is indeed not a relation at all.

˙ prīyama¯nah” Interpretation of the Sūtra “rucyartha¯na¯m ˙ ˙ There is a Sūtra stated by Pāṇini as “rucyarthānāṃ prīyamāṇaḥ” while explaining the label “Sampradāna” (recipient related to the action of giving). By this sūtra, the label of “Sampradāna” is ordained to the one which/who is related to the act of “liking,” indicated by verbal root “ruc” and others having the same sense as “liking.” For instance, in the example – “haraye rocate bhaktiḥ” (for Hari – likes – devotion), the verbal root used in the sense of “liking” is “ruc,” and the one related to it by “being liked” (Hari) is labelled by this Sūtra as “a recipient” and thereby is used in dative

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case, such is the situation. But, in case of a usage like “Hariṁ prīṇayati” ((X) likes Hari), Hari is not desired to be labelled as the recipient. That is why the usage of accusative case is seen (for “Hari” in the given Sanskrit sentence). But, since the verbal roots “ruc” and “prīñ” are synonymous, the question arises as to why is the label of “Sampradāna” not applied to the entity related with the action indicated by “prīñ,” while it applies to the entity related with the action indicated by “ruc.” It is not possible to reply therein without including the idea of “well-established relation,” etc. of the Navya-Nyāya techniques. Following is how the Vyākaraṇa scholars resolve this question. While there is synonymity between the verbal roots “ruc” and “prīñ” in the sense of “action that is conducive to be liked,” there is some granular difference also between them. The meaning of the verbal root “ruc” is “action that is conducive to be liked, conditioned by the relation of being the object (of liking).” In case of the verbal root “prīñ,” the meaning of the verbal root is “action that is conducive to be liked, conditioned by the relation of inherence (the object of liking is the substantive wherein the liking inheres).” In the instant Sūtra (“rucyarthānāṃ prīyamāṇaḥ”), the word “prīyamāṇaḥ” constitutes the verbal root “prīñ” used in the passive voice by adding the suffixes “laṭ” (referring to singular present tense) and “śānac” (referring to continuous state of action in passive voice). This brings the word to mean that the referent entity is the substantive of the result of the action generated by the entity referred by the verbal root. Thus, it entails that the labelling of an entity as “Sampradāna” by this Sūtra pertains to the referent entity is the substantive of the result of the action generated by the entity referred by the verbal root. Thus, while Hari is the substantive entity that is being liked related to both the verbal roots “ruc” and “prīñ,” since the meaning of the Sūtra is that when related to the verbal root “ruc,” an entity may be labelled as “Sampradāna,” where such referent entity is the substantive of the result of the action generated by the entity referred by the verbal root, it is only that substantive of (the act of) liking that is conditioned by the relation of being the object of (liking) that comes to be labelled as “Sampradāna.” In the usage “Hariṁ prīṇayati” (where the substantive of the act of liking, Hari, is in accusative case owing to being the karman (object) of liking and not Sampradāna (recipient) of the action of liking), Hari is the substantive of the action of liking being conditioned by the relation of inherence and not that substantive of (the act of) liking that is conditioned by the relation of being the object of (liking), and hence there is no scope for over application of the rule (of being labeled as Sampradāna) therein. In the same manner, the explanation of the Sūtra “ślāghahnuṅsthāśapāṁ jñīpsyamānaḥ” is done by the scholars of Vyākaraṇa by including the method of using well-established relation, etc.

Explaining the Import of the Verb Taking Cause-Effect Relation into Account There is a method adopted by the Nyāya scholars regarding verbal testimony, namely, accounting the cause-effect relation (between words in a sentence and meaning of the sentence). As an illustration, to the query as to how the verbal

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cognition of non-difference (between the meanings) of the expression “blue jar” arises, the Nyāya scholars connect some cause-effect relation (between them) as – with regards to the verbal cognition constituting some substance being the substantive conditioned by the relation of non-difference as its feature, the cause is the recollection of the meaning generated by (the utterance of) words with the relation of being the substantive. That is, wherever the recollection comes through with the relation of being a substantive, it is there that some substance becomes evident in the form of its feature that is non-different (from the substantive). It is thus that in the instance of “blue jar,” since the meaning as “jar” arises in the jar through the relation of (the jar) being the substantive, it is in the same substantive that the meaning as “one that is blue” arises as the non-different feature (in the substantive). “Jar-ness” is not the substantive (in this cognition), but comes across only as the feature (of the substantive, namely, jar). And hence, the meaning as “one that is blue” does not occur as non-different with regards to it (jar-ness). This is the well-known method followed by Nyāya scholars. Following the same method, the Vyākaraṇa scholars establish their theory by refuting the view point of the Nyāya scholars. For instance, according to the Nyāya school, with regards to usages like “Chaitra goes,” the verb indicates action. And that sense of action is considered as connecting with the entity signified by the nominative case in the form of a feature, with the relation of being the locus (of action). According to the Vyākaraṇa school the meaning of the verb is the doer of the action, and the doer of action connects with the entity signified by the nominative case. To prove that between the two views, the viewpoint of the Vyākaraṇa school is more valid, the method of accounting the cause-effect relation that is stated by the Navya-Nyāya scholars is only resorted. For instance, since the Nyāya scholars consider that the verb indicates action which is considered as connecting with the entity signified by the nominative case, then the number related to the action indicated by the verb must also be connecting with the entity signified in nominative case, since the verb intends to express the number (of entities involved) in the action. In that case, the causeeffect relation (regarding the number) needs to be explained as – with regards to the verbal cognition of the action indicated by the verb with the number as the feature of it, the recollection of the entity generated by the usage of nominative case ending is the cause. In the view point of the Vyākaraṇa scholars, on the other hand, the verb indicates the doer of action. Since the number related to (the doer (s) of) the action is expressed by the verb itself, the number related to the action indicated by the verb also needs to be connected to that (number pertaining to the doer(s) of the action) only. Thus, for the Nyāya view, the cause-effect relation needs to be explained as – with regards to the action indicated by the verb with the number as its feature, the recollection of the doer(s)/object(s) generated by the verb is the cause. In this manner, it is found that going with the view of the Nyāya school is cumbersome since it requires different expressions to indicate the cause-effect relation and the recollection of the entities indicated by them. On the other hand, in the case of the Vyākaraṇa school, the same expression generates the

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recollection of the same entities referred by the words and is thereby brief. The Vyākaraṇa scholars refute the stand of the other (Nyāya) school and establish that, thus, going by the brevity pertaining to the cause-effect relation, the viewpoint of the Vyākaraṇa school is superior. From this illustration, it is clearly evident how the doctrine of the Nyāya school is refuted by using their own approved methods. Improvisations are found in several instances based on the Nyāya methodology. For instance, while commenting on the Sūtra “arthavadhaturapratyayaḥ prātipadikam,” there arises a query as to what is the import of “arthavat” for, usually if the meaning of the term “arthavat” were taken as “one that has a meaning,” then the problem that arises is that, going by the sequence of the Aṣṭādhyāyī Sūtras, the following Sūtra, namely, “kṛttaddhitasamāsaśca” borrows the term “arthavat” to complete its purport, and since the words having kṛt and other suffixes are seldom found without a meaning attached to them, and thereby cannot be distinguished from them (the words that have meaning as stated in Sūtra 1.2.45), the adjective added to the Sūtra (1.2.46) would be futile. In order to justify the utility of the adjective, the interpretation of the term “arthavat” is improvised by Nāgeśa, influenced by the manner of improvisation in the Nyāya school, as follows –“being one endowed with meaning refers to being capable of giving rise to the cognition of meaning in the world, without any dependency on the usage of anything other than the case ending with the result of (the nominal stem that carries) this label” (as prātipadika). In that case, since the words using the kṛt and other suffixes give rise to cognition of meaning depending on the usage of the nominal stem that is other than the one to which the “sup” etc. nominal case ending suffixes are added beyond the result of (the nominal stem that carries) the label as prātipadika, and not without such dependence, thereby by the usage of the adjective “arthavat,” the words endowed with the kṛt and other suffixes also find justification (in obtaining the label of prātipadika, based on the adjective “arthavat”).

Conclusion In this chapter we have explained with illustrations how the Navya-Vaiyākaraṇas were influenced by the language, concepts, and the techniques of Navya-Nyāya. The forgoing are only some illustrations of improvisations/interpretations of definitions stated by the Vyākaraṇa scholars (based on methods and techniques put forth by the Nyāya school). From this, it is quite clear that all the scholars who followed the period of Gaṅgeśopādhyāya were influenced by him and followed his methods, axioms, and techniques in their own treatises. Acknowledgments This chapter was translated and edited from Sanskrit by Dr. Vaishnavi Nishankar.

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References Pande, Gopal Datta. 2004. Aṣṭādhyāyī of Pāṇini, elaborated by M.M. Panditraj Dr. Gopal Shastri. Chowkhamba Surbharati Prakashan Varanasi. Pendse, Narahari Sastri, ed. 1927. Kashi Sanskrit Series 5, Laghushabdendushekhara, Jai Krishna Das – Haridas Gupta, The Chowkhamba Sanskrit Series office, Benaras. Sharma, Raman Kumar. 1972. Mahabhashyam, (Paspashaanikam). Sharma, Pandit Shivadatta. 2007. Vyākaraṇamahābhāṣyam. Delhi: Chaukhamba Sanskrit Pratishtan. Tripāthī, Kedāranāth, ed. 2014. Tarkasaṃgrahaḥ. 7th ed. Varanasi: Motilal Banarasidas.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Language of the Navya Nyāya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief History of Nyāya . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unique Characteristics of the Navya Nyāya Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nirūpya-Nirūpakabhāva Relation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pratiyogi-Anuyogi Scheme . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Avacchedaka and Avacchedakatva . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Kinds of Absences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Various Kinds of Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representing Knowledge in Terms of Sāmānādhikaraṇyena and Avacchedakāvacchedena . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Technique of Using Double Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Advantages of the Navya Nyāya Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The Nyāya system, which is one of the six schools of Indian orthodox philosophy, can be broadly classified into Prācīna Nyāya and Navya Nyāya with respect to chronology. The older school of Nyāya discussed mainly about the objects of a valid cognition, namely Prameyās, while the neo-Naiyāyikās shifted their area of focus onto the means of valid cognition, i.e., the Pramāṇās. Hence, it is said that Prācīna Nyāya is Prameyā Pradhāna while Navya Nyāya is Pramāṇā Pradhāna. During this process of shifting, there evolved a methodology of language which was so technical yet precise that many other streams like grammar, law, O. G. P. K. Sastry (*) Department of Nyaya, National Sanskrit University, Tirupati, Andhra Pradesh, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_19

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and aesthetics adopted it. Words like avacchedaka, nirūpaka, pratiyogin, anuyogin, etc. were used frequently during philosophical debates and while defining new concepts. The current chapter deals with how such a methodological language evolved over a span of 15 centuries, ever since the Sūtras, the need for the development of such a technical language, and finally some of the unique characteristics of this technical language.

Introduction The Indian schools of philosophy have always upheld the importance of the four Puruṣārthās – the four human aims or objectives, viz Dharma, Artha, Kāma, and Mokṣa and have devised and paved ways for their achievement. Among these four, Mokṣa has been given the utmost importance as it is considered Nitya – eternal, while the other three are considered Anitya – noneternal or temporal. Hence any system of philosophy which discusses about the ways or methods to achieve these cardinal ethics has been termed as Vidyā. The word Vidyā comes from the root Vid – Jñāne which means knowledge. So, all those streams of knowledge which deal with the attainment of Puruṣārthas are called Vidyās. These are 14 in number and have been quoted in various Purānas and even the Mahābhārata. aṅgānivedāścatvāro mīmāṃsā nyāyavistaraḥ. purāṇaṃ dharmaśāstraṃ ca vidyāḥ aitāścaturdaśa. (Vayu Purana 61.73)

Yājñyavalka also talks about the same Vidyās in his Smṛti albeit using different words (Yājñyavalka Samhitā, Chap. 1, Verse 3) purāṇanyāyamīmāṃsādharmaśāstrāṅgamiśritaḥ. vedāḥ sthānāni vidyānāṃ dharmasya ca caturdaśa..

The Indian schools of thought are also classified as – āstika and nāstika, viz theist and atheist. In a general sense, theism is belief in the existence of God and atheism is not having belief in the existence of God as both the words find their origin from the Greek word theos meaning God. This meaning is however not accepted in the Indian schools of thought, for if it were to be accepted, then even Jaina darśana and Bauddha darśana would also be considered as theist as they (as in some sects) accept God; and Mimamsa, an āstika darśana, would have been considered a nāstika darśana as it does not accept the entity called “God.” Hence, it would be appropriate to use the words orthodox and heterodox as an English equivalent for the words āstika and nāstika. One who accepts the Vedas and its authoritative-ness is an āstika and conversely, one who does not believe in the authoritative-ness of the Vedas is a nāstika. As āchārya Manu quotes nāstiko vedanindakaḥ (Manu Smriti 2.11) – one who defiles or decries the Vedas is nāstika. The same can also be seen in the kiraṇāvalī commentary of nyāyasiddhāntamuktāvalī – nāstikatvamatra

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vedaprāmāṇyānabhyupagantṛtvam (Page 6). Also, it is essential to know that these systems of philosophy are called darśanas. The word darśana originates from the root dṛśir prekṣaṇe, meaning – to view or perceive. Broadly categorizing, there are six āstika and three nāstika darśanas. Nyāya-Vaiśeṣika, Sāṃkhya-Yoga, and Pūrvamīmāṃsā-Uttaramīmāṃsā fall under the categories of āstika darśanas. These are also known as the śaḍāstikadarśanas or simply śaddarśanas. The Cārvaka, Jaina and Bauddha schools of philosophy fall under the category of nāstika darśanas. It is to be noted that some authors do combine schools from both āstika and nāstika darśanas and eventually call them śaddarśanas, like Haribhadrasuri, the author of the famous work Śaḍdarśanasamuccaya, who enumerates the six darśanas as follows: bauddhaṃ naiyāyikaṃ sāṃkhyaṃ jainaṃ vaiśeṣikaṃ tathā jaiminīyaṃ ca nāmāni darśanānāmamūnyaho. (Verse 3)

The author, Haribhadrasuri, in his aforementioned work Śaḍdarśanasamuccaya gives the basic tenets of each of the abovementioned six philosophies and further makes note of another philosophy called Lokāyata, viz the Cārvāka school, and goes ahead and explains the basic doctrines of the Cārvākas. The fundamental principles of each of these darśanas are written in Sūtra form. The Nyāya Sūtras were written by Sage Gautama, The Vaiśeṣika Sūtras were composed by Kaṇādamuni, The Sāṃkhya Sūtras were composed by Kapilamuni, The Yoga Sūtras were composed by Patañjali, Jaimini composed the Mīmāṃsā Sūtras, and Bādarāyaṇa composed the Vedānta Sūtras. The foremost formulators of these Sūtras are considered Ṛṣis according to tradition and hence are addressed as Sage or Muni.

Language of the Navya Nya¯ya The language used in the latter texts of Nyāya bears very little resemblance to the language used in the earlier texts like the Sūtras, Nyāyabhāṣya, or the Pariśuddhi. To find out the reason for this evolution in the language, it is very essential and mandatory to know the historical evolution of the language in some of the main works of Nyāya, how the style and structure of language was in the earlier texts and at what particular/crucial point there appeared first signs of a change in the style or structure, and how, finally, an entirely new methodology of language was developed in due course of time to express ideas and views. A detailed description of the history of the Nyāya school of thought, is, however, not desired here, as it would itself turn out to be a mammoth chapter and secondly, the same has already been done by many authors previously and would ultimately result in redundancy and this would in turn result in shifting focus from the main topic to be discussed herewith thereby leading to arthāntaram. Hence, only important authors and their compositions shall be dealt with here.

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Brief History of Nya¯ya The Old School of Nya¯ya Due to the scientific inquiry of the human mind and the power of reasoning, the science of reasoning or logic existed in all the forms of Indian schools of thought, but in different ways. Although the term Nyāya was prevalent only from the time of the Bhāṣya, it was known earlier by various other names like pramāṇaśāstram, hetuśāstra, vādavidyā, tarkaśāstram, etc. Notable among them is the term – ānvikṣikī. Kautilya’s arthaśāstra considers ānvikṣikī as one of the four Vidyās – ānvikṣikī trayī vārtā daṇḍanītiśceti vidyā. There is a huge debate among historians regarding the dates and works of many of the Indian philosophers. Fixing the exact chronology is always a tough task for various reasons and Indian philosophical thought makes no exception. Firstly, Indian writers were primarily concerned about Vidyāṛṇa and very rarely did they mention about themselves or their eras. This makes the job of deducing the dates of authors and ascribing their works very difficult. Secondly, the nonexistence or nonavailability of works makes it even harder as much of the ancient śāstraic lore has perished in the web of time. The only way one can know of the existence of such works is by the references or quotations made by subsequent authors of the said works. Thirdly, there is always a scope for human error while making findings or while making transcriptions of various manuscripts. There have been numerous occasions in the past where a wrongly transcribed manuscript lead to a certain conclusion, only to be altered by latter historians after realizing the errata. And finally, finding new manuscripts and other material tends to change the existing history and hence a direct conclusion cannot be made, as history always tends to vary/update with the latter findings. Nevertheless, many authors, both Oriental and Occidental, have put in a lot of effort and years in shaping the history of the Nyāya philosophy and giving it a proper form. Indian schools of philosophy, now, have a concrete opinion when it comes to authors, their dates, and works and chronology, though minor disagreements still do exist due to the noteworthy contribution of authors like Vidyābhusana, Kuppuswami Śastri, Gopinath Kaviraj, Matilal, Keith, Randal, and Inghalls et al. Sometime around the second century A.D. the basic principles and tenets of this system were enumerated in the Sūtra form by Akṣapāda Gautama. Much debate exists whether Akṣapāda and Gautama are two different persons or the same person. Due to varying references about the authorship of the Nyāya Sūtras, some people consider them to be the same, whereas others are of the opinion that Akṣapāda was the individual’s name whereas Gautama or Gotama was the Gotra name (Kuppuswani Śāstri 2002, p. xii). Just as all the darśanas follow the Sūtra-BhāṣyaVārtika tradition, the Nyāya darśana has a bhāṣya written by Vatsyāyana and a vārtika written by Uddyotakara. The bhāṣya on Sūtra was composed by Vatsyāyana at around 400 A.D. The term Nyāya was first used by Vatsyāyana – kaḥ punarayaṃ nyāyaḥ? pramānirarthaparīkṣaṇaṃ nyāyaḥ (Nyāya Sutras 1-1-1). Vatsyāyana is also known as Pakṣila Svāmin and had criticized many doctrines of the Buddhists like the vijñānavāda

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and the śūnyavāda in his bhāṣya. This method of refutation and counter-refutation between the Naiyāyikās and the Buddhists set off a chain of events which has led to the development and evolution of logic in the Indian philosophical realm, without any sense of doubt. The exchange of thought between the two systems continued for almost a millennium and thus helped in shaping the system of logic and epistemology which is known to us today. Vatsyāyana mentions in his bhāṣya the methodology of the Nyāya darśana – trividhā cāsya śāstrasya pravṛttiḥ – uddeśo lakṣaṇaṃ parīkṣā ceti (Nyāya Sutras 1-1-2), wherein uddeśa is the mere introduction of objects, lakṣaṇaṃ is giving the definition, and finally parīkṣā is critical examination of the given definition. The views and ideas put forth by Vatsyāyana in his bhāṣya underwent harsh criticism by Dignāga, one of the greatest Buddhist logicians to have lived in India. His magnum opus – pramāṇasamuccaya heavily refuted the arguments put forth by Vatsyāyana in his bhāṣya. The vārtika by Uddyotakara is written as an answer to all the refutations put forth by Dignāga and other Buddhists of his period. Uddyotakara explicitly states this in the opening lines of his work – yadakṣapādaḥ pravaro munīnāṃ śamāya śāstraṃ jagato jagāda kutārkikājñānanivṛttihetuḥ kariṣyate tasya mayā nibandhaḥ

Akṣapāda, the foremost of sages, proponded a śāstra (body of doctrines) for the peace of the world; and I (Uddyotakara) shall write an expository treatise on it to remove the veil of error cast by quibblers; (Vidybhusana S.C. 1970, p. 125). Uddyotakara refers to Dignāga as Bhadanta – a revered Buddhist monk in his work, took up his criticism of the Nyāya definition of perception, and saved the Nyāya school from this Buddhist onslaught. The Nyāya conception of God was also initiated by Uddyotakara, thus cementing its stand in the realm of Indian philosophy as a theist school. Uddyotakara is believed to have flourished around 635 A.D. Next in line is the seminal work – NyāyavārtikaTātparyatīkā of Vācaspati Miśra, the multifaceted philosopher. Vācaspati Miśra is known for his proficiency in all the āstika darśanas and has written commentaries and glosses on almost all the schools of philosophy. His commentary on the śaṃkara bhāṣya called Bhamatī and the sāṃkhyatattvakaumudī on the sāṃkhyakārikās hold an esteemed place in the Indian philosophical literature. Due to his command and mastery over all the philosophies with equal ease, he is given titles like sarvatantrasvatantra and ṣaḍdarśanavallabha. In his Tātparyatīka, he took up all the criticism rendered by the Buddhists like Dignāga, Dharmakīrti, etc. and successfully managed to refute them and establish the Nyāya school. Vācaspati Miśra holds the distinct recognition of redeeming the forgotten work of Uddyotakara and successfully bringing it back to the mainstream. The Tātparyatīkā discusses several obscure portions of the Nyāya Sūtras and renders many difficult portions intelligible to the reader. For this effort, he is given an esteemed position in the literature of Nyāya and is respectfully called Tātparyācharya. Another important work by Vācaspati Miśra, with respect to Nyāya, is the nyāyasūcīnibandhana, wherein he gives the number of the Sūtras and also divides them into sections. The task of finalizing the text of the Sūtras, the

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topics dealt in them, and dividing the numbers of Sūtras with respect to each topic has been done efficiently in this. Vācaspati Miśra is supposed to have flourished in the latter half of the ninth century. Even the Tātparyatīkā was not spared by the Buddhists and they belted heavy criticism on this work. As an answer to all these criticisms, Udayana, one of the most prolific writers of the Nyāya school, composed the NyāyavārtikaTātparyatīkāpariśuddhi. Udayana was the first author to have written works on both the Nyāya and Vaiśeṣika schools. His Tātparyatīkāpariśuddhi is a commentary on Vācaspati’s Tātparyatīkā, while the Kiranāvali is a commentary on the Praśastapāda bhāṣya. But apart from this, he is attributed with several other works, the most important among them being – Nyāyapariśuddhi, Ātmatatvaviveka, otherwise known as the Bauddhadhikkāra, and finally the inimitable – Nyāyakusumāñjali. His contribution to the field of Nyāya is monumental and hence he is given the place of Ācārya. His Ātmatatvaviveka is a treatise composed to refute the four Buddhist theories and for the establishment of a permanent soul. His Nyāyakusumāñjali is a landmark, for it exclusively endeavors for the establishment of God. It is divided into five chapters and each chapter takes up one particular question and goes ahead to answer them. Udayana’s work also shows signs of a transition in the Nyāya methodology with the use of technical terms like Pratiyogitā (See Nyāyakusumāñjali 3.2) hinting at the onset of the Navya Nyāya age. The Sūtra, bhāṣya, vārtika, Tātparyatīkā, and Tātparyatīkāpariśuddhi are together known as the Nyāyapancagranthi. Other important authors of this period are Jayanta Bhatta, who authored the celebrated treatise – Nyāyamañjarī, an independent commentary on the Nyāya Sūtras. In this work, Jayanta criticizes the Sphoṭa theory and views of the Buddhists Kalyāna Rakṣita and Dharmottara. He also criticizes the definition of perception as specified by Dharmakīrti. Jayanta was a Kashmiri whose style was poetic, creative, and highly original. Bhāsarvajña, another important writer of this era, composed a work called Nyāyasāra and wrote an elaborate gloss on the same called Nyāyabhūṣaṇa.

The Parallel Vaiśesika School ˙ flourished right from the Sūtra period, the Vaiśeṣika school Just as the Nyāya School too had many authors producing original works. The Vaiśeṣika Sūtras were propounded by Kaṇāda, also known as Ulūka. Although many works may have been written after the Sūtras, none of them have survived today. Rare references about these commentaries have been found hinting at their existence in the past. The next important work in the chronology of the Vaiśeṣika school is that of Prasastapāda, who wrote an elaborate commentary on the Sūtras called the padārthadharmasaṃgrahaḥ commonly known as the Prasastapādabhāṣya. He is known to have lived in the fifth century. Vyomaśiva wrote a gloss on the bhāṣya named Vyomavati. Next in line is the celebrated work of Śridhara called Nyāyakandalī. Śridhara is said to have preceeded Udayana as Udayana has taken up many views of Śridhara and refuted them in his work. Although Udayana does not take Śridhara’s name directly, it becomes evident in the subcommentaries of Kiranāvali by Vadindrā and Vardhamanā. Udayana’s Kiranāvali is a very important contribution to the Vaiśeṣika

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school. Unlike the previous two aforementioned commentaries of Vyomaśiva and Śridhara which are full in nature, this work stops abruptly after two chapters. The first part is called Dravyakiranāvali, while the second is called the Gunakiranāvali. Post Udayana period, although there were a number of important works written in the Vaiśeṣika school by various authors, towering among them in terms of importance and recognition is the Nyāyalīlāvatī by Śrivallabha. This work is independent in nature and is written in a very technical style with poetic passages once in a while showing the poetic brilliance of the author. This particular work has been taken up by some of the Navya Nyāya authors like Vardhamānā, Pakṣadhara, etc. and commented upon. Śivadityā’s Saptapadārthī is also an important work in this period and definitely requires a special mention. It is to be noted that after the tenth century, there was a steady decline in the Buddhist school of thought and by the end of the twelfth century, there rarely was any author of relevance or reverence left in the Buddhist school and hence historians consider this time as the Decay of the Buddhists.

The New School of Nya¯ya The shift in the nomenclature of the Nyāya school from Prācīna to Navya began from the twelfth century. Many scholars attribute Gaṅgeśa Upādhyāya and his epoch-making work – Tattvacintāmaṇi as the turning point in the history of Nyāya. But recent discoveries like Nyāyasāra of Maṇikaṇta Miśra and Nyāyasiddhāntadīpa of Śaśidhara Miśra have proved that the Navya Nyāya style had been used before Gaṅgeśa himself. Nevertheless, the recognition obtained by Gaṅgeśa for his seminal work is unparalleled, to say the least. This work had become the focal point for all the authors after him for atleast a period of 400 years. Earlier scholars commented on the Sutrās and its related literature, but after Gaṅgeśa’s work, the interest shifted onto Tattvacintāmaṇi, although a handful of scholars still wrote glosses and commentaries on the older works of Nyāya. Gaṅgeśa hailed from Mithilā and he is considered to have lived in the last quarter of the thirteenth century and flourished literally in the first half of the fourteenth century. Beginning with Gaṅgeśa, Mithilā became the hot seat for studies in Nyāya philosophy for almost 150 years before it gradually shifted to Navadvipa in the fifteenth century after Vāsudeva Sārvabhauma established his school. Instead of following the age old practice of following the Sūtras, Gaṅgeśa overhauled the entire structure and divided his Tattvacintāmaṇi into four chapters, viz Pratyakṣa, Anumāna, Upamāna, and Śabda which consist of 47 broad topics (12 + 17 + 1 + 16). This method of division is the reason why the area of focus of the scholars of that era shifted from Prameya to Pramāna. The Anumāna chapter is the most recognized chapter of all and almost all the authors, post Gaṅgeśa, have commented on it. Conversely, the Upamāna part is the most ignored chapter and only a meager number of authors have taken up the task of writing a gloss on it (Rucidatta of Mithilā and Pragalbha of Bengal). Here it should be noted that during this era, there was also a shift in the opponent camp. In all the early works of Nyāya, Buddhists were the main opponents, but when it comes to the Tattvacintāmaṇi, the Prābhākara Mimamsakas became the prime opponents.

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After Gaṅgeśa, this new tradition of learning was kept alive by his own son, Vardhamāna, who was in no way inferior to his father in terms of intellect and proficiency. He wrote numerous commentaries on the works of previous authors like Gautama’s Sūtras, Udayana’s Kusumāñjali and Kiraṇāvalī, Vallabha’s Nyāylīlāvatī, and his own father’s Tattvacintāmaṇi with the name Prakāsa. Yajnapati Upādhāya wrote a commentary on the Tattvacintāmaṇi called Prabhā. This had been written on the Pratyakṣa, Anumāna, and Śabda chapters of the Tattvacintāmaṇi. Yajñapati’s work had been attacked by authors of the later years including his own student Jayadeva, also known as Pakṣadhara. Pakṣadhara was the supreme authority during his time and many students from all corners of the country flocked to read in his Tol. Under his tutelage, many scholars like Vāsudeva Sārvabhauma and Raghunātha Śiromaṇi continued the tradition of Nyāya philosophy. Pakṣadhara wrote the famous commentary Āloka on the Tattvacintāmaṇi and this commentary dominated all the other commentaries of that time and was taken up for study by students and scholars alike. Even to this day, this commentary is considered as one of the most standard commentaries on Tattvacintāmaṇi. Jayadeva had many pupils; noteworthy among them are Vāsudeva Miśra, Rucidatta, Bhagiratha, and Mahesa ṭhakkura from Mithilā and Vāsudeva Sārvabhauma, Raghunātha Śiromaṇi, etc. from Bengal. The conflict between Yajñapati and Jayadeva contributed immensely to the progress of the Navya Nyāya philosophy of that age as counters and refutations were poised from rival camps basing on subtleties and hair-splitting technicalities. Śaṅkara Miśra and Vāchaspati Miśra II were other important authors belonging to the Mithilā school. Śaṅkara Miśra wrote many commentaries including the famous Upaskāra, a commentary on the Vaiśeṣika Sūtras. Vāchaspati Miśra II on the author hand is credited with works like Nyāyatattvāloka, a commentary on the Nyāya Sūtras of Gautama, a commentary on Tattvacintāmaṇi, Nyāyasūtroddhāra, and Khandanoddhāra. After the immediate successors of Pakṣadhara, study of Nyāya began to wane in the Mithilā region and it gradually shifted to Navadvipa in Bengal. Here, studies in the intricacies of Nyāya and producing commentaries on the Tattvacintāmaṇi and other important works made sure that the Nyāya School reached its zenith. Bengal became a hub for studies in Navya Nyāya for almost another 200 years, starting with Vāsudeva Sārvabhauma. The son of Viśarada Maheśvara, Vāsudeva set out to Mithilā after completing his primary education under his father to study under Pakṣadhara. After reading many works and the Tattvacintāmaṇi under him, he was subjected to a traditional way of test – Śalākā Parīksā by his master himself, which he happened to pass with flying colors and the title Sārvabhauma was conferred onto him after this. After his completion Vāsudeva returned to Bengal and opened his school. He was fortunate as he had gifted students who would become famous in posterity. Noteworthy among them were Raghunātha Śiromaṇi, the composer of the famous commentary on Tattvacintāmaṇi called Didhiti, Chaitanya Mahāprabhu, the great Vaiṣṇava reformer of Bengal, Ragunandana – the expert on Śmṛti, and Krṣṇānanda – the expert on tantra. In his later years, Sārvabhauma shifted to Puri which was under the rule of King Pratāparudra and is said to have become an ardent follower of Chaitanya after coming in touch with him.

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Though there were a string of scholars who succeeded the Bengal school after Vāsudeva, it was Raghunātha who was the most famous of them all and who took the already great Navadvipa school to even greater heights. Hailing from a poor Brahmin family, Raghunātha lost his father at a very early stage. After coming in touch with Sārvabhauma, who made no mistake in recognizing the child’s sharp wit, Raghunātha learned under Sārvabhauma and was later sent to Mithilā to read under the age old Guru – Pakṣadhara. Stories of Raghunātha’s hostile welcome at the Mithilā School and how he outwitted the master when he was called Ekalochana (one-eyed) are still afloat in the traditional circles of the country. After completing his studies, he returned back to Bengal and was pivotal in the expansion of the fame of the Bengal school. Raghunātha was the most original thinker in the postGaṅgeśa period and has refuted many views of his teachers in his works. His commentary on the Tattvacintāmaṇi called Didhiti overshadowed all the existing commentaries of Tattvacintāmaṇi at that time and quickly got circulated throughout the country and scholars all over showed great interest in studying and examining it. Apart from the Didhiti, Raghunātha wrote many other commentaries on Udayana’s Kusumāñjali and Ātmatatvaviveka, Śrīharṣa’s Khaṇḍanakhaṇḍakhādya, and Vardhamāna’s Prakāśa. He also wrote an original work called Padārthatatvanirupanam which is considered original and bold. In this work, he had rearranged the categories of the Vaiśeṣikas and challenged some very important concepts of the Nyāya and Vaiśeṣika schools and is hence considered bold and rebellious. His independent works nañvāda and akhyātivāda are still studied in seminaries due to their original nature to this day. Raghunātha was also a skilled poet and his prowess is evident in his works. The next generation of authors wrote commentaries on the Didhiti, apart from the other important works of that era, famous among them are the Prākasa of Bhavānanda Siddhāntavāgisa, the commentary of Māthurānātha, the Jāgadisi of Jagadīśa Tarkālaṅkāra, and the Gādādhari of Gadādhara Bhattācharya. Bhavānanda wrote the famous Kārakachakram which is still studied by advanced students of Nyāya. Although Mathurānātha’s commentary on Didhiti has not come down to us, his commentary on the Tattvacintāmaṇi is considered a standard work and has been studied upon ever since it has been written. Even though innumerable commentaries have been written on the Didhiti, two commentaries stand towering over all the other commentaries, the Jāgadisi and the Gādādhari. Apart from writing various commentaries on previous works, Jagadīśa wrote an original work called Śabdaśaktiprakāśikā. This book is a very illumining work on the Śabda pramāna and deals with various topics related to it. Gadādhara on the other hand produced many commentaries and original works too. He is said to have authored 64 Vādas, although only a few have come down to the present generation. A compilation of his Vādas has been published by the name Vādavāridhi. His original works on the Śabdapramāna, the Śaktivāda, and the Vyutpattivāda are still read by students of Nyāya. Vyutpattivāda, particularly, holds the great distinction of being pursued even by students of grammar. The works of Mathurānātha, Jagadīśa, and Gadādhara form the crux of the material read by the students of Navya Nyāya. A peculiar style is followed in

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North India. After reading the primary texts like Tarkasaṅgraha and Siddhāntamuktāvalī, students progress further by reading the Vyāptivāda of Tattvacintāmaṇi along with Mathurānātha’s commentary. The Purvavādas (VyadhikaraṇaAvacchedakatvanirukti-Siddhantalakṣaṇa-Pakṣata-Avayavah) are read with the commentary of Jagadīśa. The Uttaravādas (from the Sāmanyanirukti to Bādhah) are read along with the commentary of Gadādhara. This particular method is implemented so that students are exposed to all the three different styles of the three authors and are well versed with their style, thought, and method of analyzing problems. Post Gadādhara, many authors have contributed to the further development of the Navya Nyāya school of thought. Books which are concise in nature and orient the student to the basics of Nyāya philosophy like Nyāyasiddhāntamuktāvalī by Viswanātha Panchānana and Tarkasaṅgraha by Annam Bhatta were authored and gained much prominence. Even to this day, these works are read by students to gain entry into the complex world of Navya Nyāya. The late eighteenth and the early nineteenth century mark the beginning of a new style in Navya Nyāya. Authors would write short but skillful technical notes on just certain portions and they came to be known as Krodapatrās. Although Krodapatrās are considered and argued by many to be of no intelligent addition to the existing Navya Nyāya literature, from the technical language point of view, they are the pinnacle. Concepts like Anugama, insertion of Paryāpti, and extremely technical usage of words like avacchedaka, pratiyogita, anuyogita, nirupti, and nirupaka embedded in long samāsas make it extremely difficult for even advanced learners to understand, let alone master them. Golokanatha’s Krodapatrā on the Sāmānyanirutki is a very famous work. Other important authors of this style include Kaliśaṅkara Bhatta, Dharmadutta Jha (Baccha Jha), Chandranārayana, etc. This tradition also reached the southern part of the country and scholars there too showed great interest in writing these Krodapatrās. Famous among them are Rāma Śāstri, the author of Satakoti (on the Satpratipakṣa), Kṛṣṇaṃbhatta, Śrikṛṣṇatatacharya, Gummaluri Sangamewara Śāstri, etc. The exhaustive literature of the Nyāya school of philosophy started with the Sūtras of Gautama but the end is certainly nowhere near. Thanks to the continuous efforts and research of scholars and enthusiasts, newer works and findings are continuously contributing to the expansion of an already comprehensive field of philosophy. The Nyāya school is a living proof and testament that Indian philosophy not only dealt with mysticism and monoism, but topics like logic, ontology, and epistemology were also discussed with great enthusiasm to even greater extent. The Realist school of Nyāya always tried to argue and prove about the existence of the world and its various materials. Due credit and recognition should be given to the rivals of this philosophy, primarily the Buddhists in the first millennium and the Mimāmsakās and the grammarians in the second millennium, for the improvement, development, and the evolution of this system. It was the constant questioning, the counters, and the refutations posed by the rivals which led to an intellectual war, thereby leading to the current shape and form of this school of thought.

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Is Gan˙geśa Really the Propounder of Navya Nya¯ya Language? Although it is widely accepted that Gaṅgeśa was the propounder of the Navya Nyāya language, it is important to look minutely into this matter and understand why only Gaṅgeśa has been attributed with devising this new methodological language when the same language style has been utilized by his predecessors and contemporaries like śaśidhara miśra, maṇikaṇṭha miśra in works like Nyāyasiddhāntadīpa, Nyāyaratna, etc. In what ways is the Tattvacintāmaṇi new from the previous works? The answer to this question cannot simply be “language or the style of the language” used in Tattvacintāmaṇi, for the same style has been used in other works like Nyāyasiddhāntadīpa and Nyāyaratna. So, how is this work of Gaṅgeśa different from the others? The question persists yet. The answer to this might not look as simple as it might seem and hence it requires a little deeper delving. Firstly, the shift of the focus from Prameyā to Pramāṇā can be predominantly seen in the work of Gaṅgeśa. This does not imply that previous works did not discuss, at all, about the means of valid knowledge. Works like Nyāyakusumāñjali, Nyāylīlāvatī did discuss on these topics. But their main subject matter was something else and whenever during the course of debate the topic of Pramāṇā sprung up, sufficient elucidation would be given. This was not the case in the work of Gaṅgeśa. The division of the TC into four chapters, viz pratyakṣa, anumāna, upamāna, and Śabda, is proof enough to show the highlighted importance of Pramāṇā in Gaṅgeśa’s view. The main area of focus is Pramāṇā. His analysis of Pramāṇās in the TC is lucid, descriptive, and pinpointed. During the course of analysis, he brings in the definitions of various other commentators belonging to various darśanas, and refutes them only after thorough examination. Another subtle yet important reason as to why Tattvacintāmaṇi is different from the previous works is the classification of the subject matter. Gaṅgeśa provides a lengthy classification of the various items he discusses in his work and places them under appropriate heads and subheads. In no previous work can such exhaustive classification be seen. Hence, this is also a very novel feature of the Tattvacintāmaṇi making it different from previous works.

Unique Characteristics of the Navya Nya¯ya Language Although the Navya Nyāya language boasts of innumerable unique properties, a detailed explanation of all such properties is the subject matter of at least tens of research theses. However, some important characteristics are described and hereby given below:

The Nirūpya-Nirūpakabha¯va Relation One of the most fundamental and recurring concepts in the Navya Nyāya terminology is the usage of Nirūpya-Nirūpakabhāva relation. This relation occurs usually

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between (but not restricted to) corelated properties (sasaṃbandhika-dharmas or sākāṅkṣadharmas). When a particular dharma is discussed and it gives rise to an innate expectancy or incites further enquiry, such dharmas are known as sākāṅkṣadharmas. When it is said, “Drona is the Guru,” there arises an innate expectancy as “of whom.” And the answer for that expectancy is “Ekalavya.” Hence words like Guru, ādharā, janya, pratibandhaka, etc. are corelated terms as they inherently give rise to enquiry about their other counterparts like śiṣya, ādheya, janaka, and pratibadhya, and only then does the default expectancy subside. The inherent properties in the above terms are connected using the relation of Nirūpya-Nirūpakabhāva. In the aforementioned example “Drona is the Guru,” since Drona is Guru, there exists a property Gurutvam (teacherness) in Drona and since Ekalavya is the disciple, there exists a property śiṣyatvam (disciple-ness) in Ekalavya and the Gurutvam in Drona is indicated by the śiṣyatvam in Ekalavya and vice versa too. Hence, these corelated properties Gurutvam-śiṣyatvam are interconnected using the relation called Nirupya-Nirūpakabhāva. So the knowledge “Drona is the Guru of Ekalavya” is represented as Drona- niṣṭha-Gurutva-nirūpitaśiṣyatvavān-Ekalavya in the Navya Nyāya language. The Nirupya-Nirūpakabhāva relation can also be used to link a dharma to a dharmi or vice versa too. The same example can also be represented as Drona-niṣṭha-Gurutva-nirūpaka-Ekalavya or Drona-nirūpita-śiṣyatvavān – Ekalavya using Navya Nyāya language. The Nirūpya-Nirūpakabhāva relation is also used to link an absence and its related counter-positiveness (pratiyogitā). When we say “absence,” there arises a default expectancy, i.e., “of whose/what?” Hence, if the absence is of a pot, then the pot is the counter-positive of the absence and hence the counter-positiveness has a relation nirūpakatva with the absence because the counter-positive and an absence are both interrelated. If pot is the counter-positive, there definitely arises an expectancy of an absence and an absence invariably gives rise to the expectancy of a counter-positive. Hence, the knowledge “absence of a pot” is represented as ghaṭaniṣṭha-pratiyogitā-nirāpaka-abhāva, meaning the counter-positiveness residing in a pot determined by absence. It is be noted that the Nirūpya-Nirūpakabhāva relation may be used not just to connect corelated/codependent properties but also between a property and the locus of the other corelated property. The knowledge “Stick is the cause of the pot” may be represented in Navya Nyāya language as ghaṭa-niṣṭha-kāryatā-nirūpitakārāṇatāvān-dandah wherein the codependent properties kāryatā-kārāṇatā are connected using the Nirūpya-Nirūpakabhāva relation. The same can also be represented as ghaṭa-nirūpita-kārāṇatāvān-dandah. It may be noted here that the relation nirūpitatva is connecting the pot and the property kārāṇatā. So from all the above expressions, the following conclusion can be made: • To link/connect two sākāṅkṣadharmas nirūpitatva/nirūpakatva relations can be used. • To link/connect a dharma to a dharmi nirūpakatva relation ought to be used, viz Drona-niṣṭha-Gurutva-nirūpaka-Ekalavya. • To link/connect a dharmi to dharma nirūpitatva relation ought to be used viz Drona- nirūpita-śiṣyatvavān – Ekalavya.

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Pratiyogi-Anuyogi Scheme The Navya Nyāya language is a complex web of technical terms like pratiyogin, avacchedaka, nirūpaka, etc. The term pratiyogin is used repeatedly in the Navya Nyāya texts and hence it is important that the proper meaning of that term is discussed. In the knowledge – “absence of pot,” the pot is the adjunct or the counter-positive of the absence. Annam Bhatta in his Tarkasaṅgraha says – “yasya abhāvaḥ sa pratiyogi.” So whosever absence is described in an absence becomes the counterpositive of the absence. In the aforementioned example, pot’s absence is being described and hence the pot is the adjunct of the absence. It is to be noted that whenever there is a pratiyogin, the expectancy of an anuyogin too instantly arises, for the two terms are interrelated, i.e., sasaṃbandhika or sākāṅkṣadharmas, and hence the absence becomes the anuyogin. As the adjunct here is with respect to an absence, it is known as abscential counter-positive. Once the concept of pratiyogi is known, it is very easy to understand the concept of Pratiyogitā or Pratiyogitva. Pratiyogitā or adjuctness is the property which resides in an adjunct. Hence in the example – “absence of a pot,” the adjunct of the absence is pot and the adjunctness (pratoyogitā) resides in the pot. It is to be noted that the term pratiyogin is not merely used in the context of an absence. The Navya Nyāya terminology allows the use of the term pratiyogin in the context of a relation too. In the knowledge – “There is contact of the pot with the ground,” pot is the superstratum and ground is the substratum and the relation between the two entities is contact (saṃyoga). Every relation has an adjunct and a subjunct, likewise the relation “contact” also has an adjunct, namely pot, and a subjunct, namely ground, in the aforementioned example. As the adjunct here is with respect to the relation, it is known as relational counter-positive. And hence, pratiyogitā can be broadly divided into two types – an abscential counter-positive (abhāvīya-pratiyogitā) and a relational counter-positive (sāṃsargīya-pratiyogitā). Also, to identify what is the superstratum and substratum in a given knowledge, the following rule is applied – “ādhārāstu anuyoginaḥ ādheyāstu pratiyoginaḥ,” i.e., the substratum becomes the anuyogin and the superstratum becomes the pratiyogin. Hence in the example – “there is contact between pot and ground,” the pot is the relational counter-positive as it is the superstratum and the relational counter-positiveness (sāṃsargīya-pratiyogitā) resides in the pot. Whereas in the knowledge “absence of a pot,” the counter-positive of the absence is the pot and hence the absential counter-positiveness resides in the pot. The need for adopting the pratiyogi-anuyogi scheme in the context of a relation needs to be further elaborated. Normally when we say “samyogena kunde badaram,” i.e., the berry is in contact with the vessel, the knowledge also gives us information about the relation between the two relata which is samyoga or contact. Now, this samyoga exists in both the relata, i.e., berry and vessel. Hence, there arises a question as to why the knowledge should be only “samyogena kunde badaram” and not vice versa as “samyogena badare kundam.” Why is it only said that the berry resides in the vessel and not vice versa? The Navya Nyāya methodology extends the usage of

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the pratiyogi-anuyogi scheme in the context of the relations and thus diligently answers the question. As elaborated earlier, every relation has a pratiyogianuyogi, and using the rule “ādhārāstu anuyoginaḥ ādheyāstu pratiyoginaḥ” that which is a substratum becomes the anuyogin and the superstratum becomes the pratiyogin. In the case of kunde badaram, the vessel is the substratum and not the berry and hence the knowledge samyogena kunde badaram is valid but not badare kundam. A key difference between an absential counter-positive and a relational counterpositive is that the absential counter-positive has an opposition (virodha) with its absence, which is not the case in a relational counter-positive. The counter-positive pot cannot exist on the same locus the absence of a pot exists as there is a virodha between pot and its absence. However, the relational counter-positive bears no such opposition with its relation. In the example “there is contact between pot and ground” the relational counter-positive “pot” does not possess any opposition with the relation contact. Once the concept of pratiyogi and pratiyogita is known, its corelated concept of anuyogin (subjunct) and anuyogita (subjunctness) can also be understood clearly. As mentioned earlier in the case of pratiyogitā, the anuyogitā is also of two types, viz abhāvīya (related to absence) and sāṃsargika (related to a relation).

Avacchedaka and Avacchedakatva Among all the technical terms or concepts used in the Navya Nyāya language, the concept of avacchedaka holds utmost importance. It is used recurringly and in multiple dimensions. It is with the help of the concept “avacchedaka” that many problems like ativyāpti, avyāpti, etc. are tackled while framing the lakṣaṇas. The term avacchadeka broadly means delimitor (avacchinatti vyāvartayati). The purpose of a delimitor is to distinguish or differentiate a particular entity from other entities. In the knowledge “absence of a pot,” the counter-positive of the absence is the pot and the counter-positiveness resides in the pot. This counter-positiveness is delimited by a property called pot-ness (ghaṭhatvam). Thus, this property pot-ness is called the avacchedaka of the pratiyogitā which resides in the pratiyogi – pot. And this avacchedaka property is connected with the pratiyogi by means of a relation called avacchina (limited). Thus, the knowledge “absence of a pot” is represented in Navya Nyāya language as ghaṭatvāvacchinnapratiyogitānirūpakābhāvaḥ, meaning the absence whose counter-positiveness is delimited by the property – pot-ness. Usually, a property or a relation is taken as an avacchadeka. In the knowledge “saṃyogena bhūtale ghaṭo nāsti,” the counter-positiveness residing in the pot is delimited by the property pot-ness which is known as the pratiyogitāvacchadekadharmah. At the same time the same counter-positiveness residing in the pot is delimited by the relation saṃyoga which is known as the pratiyogitāvacchadekasaṃbandhaḥ. Although an avacchedaka is typically denoted by a property or a

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relation, there are times when even space and time act as delimitors. In the famous example, “agre vṛkṣaḥ kapisaṃyogi mūle na,” i.e., there is contact of the monkey on the upper portion of the tree and not the lower portion, the space “agra” or the upper part of the tree becomes the delimitor for the occurrence of the contact, in the sense that it restricts or confines the existence of the contact of the monkey to a particular place in the tree (branch or the upper part). Likewise, the space “mūlam,” i.e., the lower part becomes the delimitor for occurrence of the absence of the contact. Similarly in the knowledge “utpattikāle ghaṭe gandho nāsti” the utpattikāla, i.e., the first moment becomes the delimitor for the existence of the absence of fragrance in the pot. Once the concept of avacchedaka is understood, avacchedakatva follows suit easily. Avacchedakatva resides in an avacchedaka (be it a dharma or saṃbandhaḥ) with a self-linking relation. Many recent authors have tried to present Navya Nyāya concepts pictorially as these representations aid in easier understanding of complex concepts. In the works of Joshi J and Kulkarni P.T (2014), Varakhedi S (2013), etc. the diagrams and pictorial representations of Navya Nyāya concepts help the reader in better understanding and comprehension of the technicalities.

Various Kinds of Absences It is well known that the Nyāya system accepts four abhāvas. Annam Bhatta in his Tarkasaṅgraha states abhāvaścaturvidhaḥ – prāgabhāvaḥ, pradhvaṃsābhāvaḥ, atyantābhāvaḥ, and anyonyābhāvaśceti. But apart from these four, there are other combinations of abhāvas used in the Navya Nyāya system to convey a specific meaning, viz anyatarābhāvaḥ, ubhayābhāvaḥ, viśiṣṭābhāvaḥ, etc. bhūtale ghaṭapaṭānyatarābhāvaḥ holds good only when both the counter-positives of the absence, i.e., the jar and the cloth do not occur in the substratum ground. If either jar or the cloth exist in the substratum – ground, then the cognition bhūtale ghaṭapaṭānyatarābhāvaḥ does not arise. But if either only a jar or a cloth occur on the ground at any specific point of time, then the cognition bhūtale ghaṭapaṭobhayābhāvaḥ may arise. But when both the counterpositives, i.e., the jar and the cloth occur simultaneously on the ground, then the cognition bhūtale ghaṭapaṭobhayābhāvaḥ may not arise. Another combination of abhāva is the viśiṣṭābhāvaḥ. This particular absence is dependent on the occurrence of either a qualifier or qualificand. The absential cognition daṇḍī nāsti means daṇḍaviśiṣṭapuruṣaḥ nāsti. Here, stick is the qualifier and man is the qualificand. The absential cognition daṇḍī nāsti shall not arise only when both the qualifier (stick) and the qualificand (man) exist in a locus. In all the other three cases, i.e., • Only the qualifier (stick) occurs but not the qualificand (man) • Only the qualificand (man) occurs but not the qualifier (stick) • Both the qualifier and the qualificand do not occur

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the specific absence – daṇḍī nāsti occurs. However, the first case is known as viśeṣaṇābhāvaprayuktaviśiṣṭābhāvaḥ, i.e., the specific absence promoted due to the absence of the qualifier, the second case is known as viśeṣyābhāvaprayuktaviśiṣṭābhāvaḥ, i.e., the specific absence promoted due to the absence of the qualificand, and the third case is known as viśeṣaṇaviśeṣyobhayābhāvaprayuktaviśiṣṭābhāvaḥ, i.e., the specific absence promoted due to the absence of both the qualifier and the qualificand. Now, it is essential to know why there exists a difference between anyatarābhāvaḥ and ubhayābhāvaḥ. For this difference to be known, it is mandatory to know the contradicted-contradictoriness scheme (pratibadhyapratibandhakabhāvaḥ) of an absence and its counter-positive. An absence is said to have a virodha with its counter-positive, meaning an absence and its counter-positive cannot occur in the same substratum. In the absence ghaṭapaṭobhayarābhāvaḥ, the counter-positive of the absence is ghaṭapaṭānyatarat, i.e., either a pot or a cloth. Therefore, as long as either a pot or a cloth, which are the counter-positives of the aforementioned absence, exist on a substratum, the absence ghaṭapaṭānyatarābhāvaḥ cannot arise in the same substratum. The logic however varies in case of ubhayābhāvaḥ. In the absence of ghaṭapaṭobhayābhāvaḥ, the counterpositive is ghaṭapaṭobhayam, meaning both the pot and the cloth. Hence, only when both the pot and cloth occur in a substratum, will the absence ghaṭapaṭobhayābhāvaḥ not arise. But if only a pot or a cloth occur on a substratum, then they do not oppose the rise of the abscential cognition ghaṭapaṭobhayābhāvaḥ. The rule “ekasatve api dvayaṃ nāsti” comes into play here for the rise of the aforementioned abscential cognition. Another very important absence is the vyadhikaraṇa-dharmāvacchinnābhāvah that is known to have been advocated by Soundala. Normally, in the case of cognition of an absence like ghaṭābhāvaḥ (absence of a pot), the counterpositive is the pot and the counter-positiveness (pratiyogitā) resides in the pot and this counter-positiveness is delimited by the pot-ness, which is the generic property residing in the pot. Here, both the counter-positiveness and the generic property pot-ness occur in the same locus, i.e., the pot itself and hence this absence is termed samānādhikaraṇadharmāvacchinnapratiyogitākābhāvaḥ, meaning an absence whose counter-positiveness is delimited by a property which occurs in the same locus as the counter-positiveness. Conversely, an absence whose counter-positiveness is delimited by a property which is nonconcurrent with the counter-positiveness is known as a vyadhikaraṇadharmāvacchinnapratiyogitākābhāvaḥ. The absential cognition “paṭatvena ghaṭah nāsti,” viz the “absence of a pot delimited by clothness,” is an example. Here, the counter-positive of the absence is pot and the counter-positiveness (pratiyogitā) resides in the pot itself. But, unlike the previous example, the counter-positiveness of the absence is delimited by clothness, a property which does not reside in the counter-positive pot. The key difference here being that in this particular absence the counter-positiveness resides in the pot but the property delimiting the counter-positive does not exist in it. Hence such a type of absence can

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exist even in a substratum where a pot can exist, thus making a vyadhikaraṇadharmāvacchinnākābhāvaḥ – kevalānvayi, i.e., universally existent.

Various Kinds of Relations In the Navya Nyāya language, saṃbandha or a relation holds a very important place and has been developed and crafted with utmost care. A relation possesses the following characteristics: • It always gives rise to a qualified knowledge, i.e., a viśiṣṭapratyaya. • It distinguishes the qualificand from other entities; this feature is known as itaravyāvartaka. • The relation of a qualifier with a qualificand narrows down the qualificand from other similar qualifiers. For example, in the sentence “Blue pot” the blueness narrows down the qualificand pot to only possess the quality blue color instead of any other color like yellow, green, etc. • It does not need another relation to arrive at a qualified knowledge; the nature and function of a relation itself is such that it always narrows a qualificand and hence does not require another relation. Hence the notion saṃsargasya saṃbandhāntarānavacchinnatvam. • For a verbal cognition to arise, it is mandatory that the entities that are the content of the verbal cognition exist prior to the Śabdābodha. However, this rule does not apply in the case of a relation. Take the sentence “nīlaḥ ghaṭaḥ” – the pot is blue, for a verbal cognition to arrive after the utterance from the said sentence; it is mandatory that the entities pot and blue must exist (in the form of padārthopasthiti) prior to the rise of the verbal cognition. However, the relation between the qualifier and the qualificand, which is abheda, in the current example, need not exist prior to the rise of the verbal cognition. Various relations have been accepted by the Nyāya logicians, noteworthy among them are the kālika-saṃbandha – a temporal relation, svarūpa saṃbandha – a selflinking relation, and daiśika saṃbandha – a relation described by space. When time itself acts as a relation, it is considered as kālika-saṃbandha and this is also why this relation is also known as kālasvarūpa saṃbandha. These relations have different categorizations. Relations can be classified as direct relations or indirect relations. Relations like contact and inherence are direct relations. These relations directly connect two or more entities. Indirect relations are those which do not directly connect two or more entities. For example, to perceive the color-ness, i.e., rūpatvajāti in a jar, the relation given by a Naiyāyika is saṃyuktasamavetasamavāyaḥ. Here, the organ eye does not have a direct relation with the generic character rūpatva, rather the eye has a contact with the jar, in which color is inhered and the generic character colorness is further inhered in the quality color. Hence, as the relation between the organ and color-ness is not direct, the relation is considered an indirect relation.

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Another way of categorizing is occurrence-exacting relations and nonoccurrenceexacting relations. In a relation, when the substratum (ādhāra) and the superstratum (ādheya) are knowable, then such a relation is known as a vṛttiniyāmaka saṃbandha and conversely when the substratum and the superstratum cannot be known, then such relation becomes a vṛtyaniyāmaka saṃbandha. The statement “saṃyogena bhūtale ghaṭaḥ asti” – there is a contact between jar and ground, gives a lot more information than just about the relation and its entities: • There is a relation between two entities, namely jar and ground. • The jar is the superstratum, i.e., the ādheya, and the ground is the locus, i.e., the ādhāra. • The jar is the qualificand as it is in the first case and the ground is the qualifier as it is in the seventh case. There exists a usual thumb rule that an entity which is used in the first case is the qualificand. • The relation also has an subjunct and an adjunct; the jar is the adjunct (saṃsargīya pratiyogi) and the ground is the subjunct (saṃsargīya anuyogi). When two hands are in contact and are directly perpendicular to the ground instead of one being on the other, the relation viz contact cannot signify what is the superstratum or the substratum. Such a contact is known as a nonoccurrenceexacting relation. More examples of nonoccurrence-exacting relations are tādātmya, viṣayatā, nirūpakatā, etc. Another very important relation widely accepted in Navya Nyāya language is paryāpti. The insertion of paryāpti to derive an exact meaning is one of most ingenious techniques developed by the Navya Nyāya logicians. This paryāpti is of two types: itaravārakaparyāpti and nyūnavārakaparyāpti. The relation of paryāpti (sufficiency) that avoids other irrelevant entities is called itaravārakaparyāpti, while the relation of paryāpti that puts a check on taking fewer entities than required is nyūnavārakaparyāpti. Mathurānātha, in his commentary on the Vyāptipañcaka, uses the technique of the insertion of paryāpti with utmost precision.

Representing Knowledge in Terms of Sa¯ma¯na¯dhikaranyena ˙ and Avacchedaka¯vacchedena Although knowledge is given a fourfold classification in Nyāya texts, it is further represented using prefixes like sāmānādhikaraṇyena and avacchedakāvacchedena. These terms give additional information about the knowledge being represented and are extensively used in Navya Nyāya texts. For example, there exists an inferential cognition sāmānādhikaraṇyena parvato vahnimān. Here, the subject is the hill and the probandum is fire. As the hill is the subject, it possesses a property called subjectness (Pakṣatā) and the property which delimits this subjectness is called pakṣatāvacchedaka, and the generic property hillness is the same

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in this case. Now, the additional information generated by this prefix sāmānādhikaraṇyena is that the probandum fire is to be inferentially established in any locus of the delimitor of the subjectness, i.e., hillness. If there are ten hills and fire is to be inferentially established in any of the hills, be it just one or more, the inferential cognition is termed as sāmānādhikaraṇyena anumiti. The only prohibition here being fire should not be inferentially established in all the ten hills. Conversely, if the probandum is to be inferentially established in all the locuses of the delimitor of the subjectness then such an inferential cognition is termed avacchedakāvacchedena anumiti. So, as long as the probandum is to be inferentially established in all the ten hills, the inferential cognition is called avacchedakāvacchedena anumiti. If the probandum is not desired in even one less locus, the anumiti loses it avacchedakāvacchedena tag. The critical logic here being that, in a knowledge, which is represented using avacchedakāvacchedena, a sense of pervadedness (vyāpakatva) is subtly understood. How and where this pervadedness is obtained and relates to is discussed in-depth by Gadādhara in his commentary on the Pakṣatā chapter of Anumānacintāmaṇi.

The Technique of Using Double Negation The Navya Nyāya writers specialize in the use of double negation to convey a subtle meaning. The first definition of Vyāpti has been given as sādhyābhāvavadavṛttitvaṃ, which is the nonoccurrence of the hetu in the locus of the absence of Sādhya. Here, it is evident that two negations have been utilized. The first being the absence of the Sādhya and the second being the nonoccurrence of the Hetu. Instead of using a double negation, if the definition of Vyāpti were to be given as sādhyavadvṛttitvaṃ after removing the two negations, then such a definition would overapply (ativyāpti) to the Hetu of a wrong inference like vahnimān Dravyatvāt as the hetu – dravyata exists in the locus of the Sādhya – fire. If a double negation is used, then the problem of overapplication may be resolved as the Hetu occurs even in the locus of the absence of the Sādhya. Similar instances of the use of double negation can be seen at numerous instances. Gadādhara, during his commentary on the Dīdhiti – Samavāyitaya Vāchyatvābhāva (Pg 1) in the Chaturdasalakṣaṇī, makes a statement – paṭādeḥ ghaṭādipadāvācyatvānupapattiḥ (Pg 4). In this statement, it is to be observed that instead of saying paṭādeḥ ghaṭādipadavācyatvāpattiḥ, Gadādhara uses a double negation to obtain a specific meaning. Another instance of the use of a double negation can be seen in the commentary of Gadādhara in the Pakṣatā chapter. While commenting upon the topic tadanyamātraliṅgakānumitīcchā, Gadādhara states the following – ālokāvṛttitvenājñāyamānadharmāvacchinna (Pg 121). Here too, the commentator has used a double negation instead of saying ālokavṛttitvenajñāyamānadharmāvacchinna. Gadādhara further explains as to why there is a need to use a double negation in that particular definition. Many more examples of this sort can be found in various texts of Navya Nyāya.

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Advantages of the Navya Nya¯ya Language The language of Navya Nyāya helps in describing the nature of entities with utmost precision and in areas of knowledge representation. The older texts of Nyāya dealt with the concept of definitions (Lakshanas), but the Navya Nyāya texts brought about a new dimension in the area of defining concepts through their rigorous approach and methodology. This helped a lot during the course of arguments with other schools of thought. A few examples of how the new robust language helped in the arguments is hereby given below. • While defining janya-pratyaksha (a producible perception, which in turn is essentially noneternal), the concept of avacchedaka helps in bringing a clearer understanding. It is said that a knowledge which is produced by a sense organ (indriya) is a janya-pratyaksha. But this definition would have the problem of overapplicability (ativyāpti) in inferential cognition (anumiti) as the inferential cognition is also produced by the sense organ – manas. This is because the logicians accept the contact of the mind and the soul to be the cause for every form of knowledge arising and since inferential cognition is also a type of knowledge, mind would definitely be the cause for it and hence the definition overapplies. To tackle this, the logicians update the definition of pratyakṣa as follows: indriyatvāvacchinnajanakatānirupitajanyatāśālijñānatvam. Although manas is the cause for inferential cognition, its causeness is not delimited by the property sense organ-ness but mind-ness only. The underlying logic here is that manas, being a sense organ, has the property sense organ-ness in it and also manastvam (mind-ness). The property delimiting the causeness residing in the mind with respect to inferential cognition is mind-ness and not sense organ-ness. Thus, the overapplicability is resolved as the definition says that the causeness has to be delimited by the property sense organ-ness. In the aforementioned example, the conflict has been resolved using the help of an avvachedaka – a delimitor, which is one of the most important aspects of Navya Nyāya language. • The Navya Nyāya language helps immensely while representing various types of knowledge. The knowledge “This is silver” could be both true and false. The knowledge is considered true when it is cognized after looking at silver and conversely it is considered false when it is cognizes after looking at an oyster shell (śukti). But in both the cases, the form of the knowledge remains “This is silver.” So, how are they distinguishable from one another? It is here that the Navya Nyāya language provides an easy way out. In a true knowledge, the qualifierness (prakāratā) resides in silverness (rajatatvam) whereas the qualifierness resides in the property oyster shell-ness(śuktitvam) for a false knowledge. To be even more precise, true knowledge is represented as rajatatva-niṣṭha-prakāratā-nirūpitarajata-niṣṭha-viśeṣyatākajñānam while a false knowledge is represented as śuktitva-niṣṭha-prakāratā-nirūpita-rajata-niṣṭha-viśeṣyatākajñānam. • Sometimes a pot is represented as “This is a pot” (ayaṃ ghaṭaḥ) or “This is a substance” (idaṃ dravyam) in the Navya Nyāya parlance. There is no ambiguity when a pot is being represented as “This is a pot,” but when it is represented as

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“This is a substance” there is a certain ambiguity which is resolved using the Navya Nyāya language. Since a pot is a substance, there is no harm in representing it as “This is a substance,” but what distinguishes it from the knowledge “This is a pot” is the qualifier. In both the examples, pot is the qualificand, but pot-ness is the qualifier in the knowledge as “This is a pot” whereas substance-ness is the qualifier in the knowledge “This is a substance.” Thus the accurate use of qualifier-qualificand structure helps in representing various kinds of knowledge unambiguously. • “saṃyogena ghaṭo nāsti” and “samavāyena ghaṭo nāsti” both talk about the absence of a pot, but the subtle difference between these two absences can be brought out using the Navya Nyāya language. A pot resides inherently in its constituent – kapāla and hence the absence saṃyogena ghaṭo nāsti occurs there; likewise a pot occurs with a relation – contact on the ground and hence the absence samavāyena ghaṭo nāsti occurs on the ground. Both these abscences are not interchangeable. The Navya Nyāya language helps in differentiating the two absences with the help of a samsargika-avacchedaka. In both the abscences, the abscential counter-positive is a pot, but in the first absence, the delimiting relation of the counter-positivenss (pratiyogitāvacchedakasaṃbandha) is saṃyoga whereas in the second absence it is samavāya. Thus, accepting a relation as a delimitor helps in differentiating between absences. The concept of anuyogin and pratiyogin help in providing great clarity while dealing with absences and relations. In the knowledge “a pot has contact with the ground” the pot is the pratiyogi and the ground is the anuyogi of the relation – contact which connects both the relata (pot and ground). Now this relation, namely contact, being a guna resides in the both the relata, viz pot and ground with the relation – samavāya. Now the question arises – since the relation contact resides inherently in both the pot and the ground, why is the knowledge only represented as bhūtale ghaṭaḥ and not vice versa, i.e., ghaṭe bhūtalam. The answer to this is given using the help of the terms anuyogin and pratiyogin. A relation has both a subjunct and an adjunct and whatever is the subjunct only can reside in some other substance and whatever is the adjunct will become base for a substance to reside in it. In the aforementioned example, a pot exists on the ground and not vice versa. Hence in the relation – contact, the pot is the pratiyogin and the ground is the anuyogin and hence the knowledge is only represented as bhūtale ghaṭaḥ and not ghaṭe bhūtalam.

Conclusion Apart from the abovementioned methods, there are many other techniques used in the Navya Nyāya texts to reach to a conclusion with pinpoint precision. The technique of Anugama, the technique of using the subtle difference between an upalakṣaṇa and a viśeṣaṇa to arrive at a particular meaning, the importance of parsimony (lāghava) and the role it plays to establish a certain theory, and the use of variety of problems like anyonyāśraya, ātmāśraya, cakraka, etc. to ascertain a

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certain concept are some of the innumerous methods employed in the Navya Nyāya texts to arrive at a certain conclusion. Careful crafting and evolution of this language is the reason why it has been adopted by various other streams of knowledge like grammar, Mīmaṃsā, etc. for if properly used, the degree of freedom provided by this technical language eradicates all forms of ambiguity.

Definition of Key Terms 1. Navya-Nyāya – The newer school of Nyāya usually associated with Gangesa of Mithila where the area of focus shifted from the Sutrā literature/prameyas to pramānas. The new school had a distinct style in language with the introduction of many technical terms thus differing a lot from the old school of Nyāya 2. Pramāṇā – means of a valid cognition/knowledge 3. Relation – sambandha 4. Nirupya-nirupakabhāva sambandha – a special relation predominantly used to link two co-related properties 5. svarūpa saṃbandha – a self-linking relation 6. vṛtyaniyāmaka saṃbandha – non-occurrence exacting relation 7. vṛttiniyāmaka saṃbandha – occurrence exacting relation 8. daiśika saṃbandha – a relation described by space 9. kālika-saṃbandha – a temporal relation 10. Pratiyogin – counter-positive of an absence 11. anuyogin – subjunct 12. Avacchedaka – delimiter 13. abhāvīya-pratiyogitā – abscential counter-positive 14. sāṃsargīya-pratiyogitā – relational counter-positive 15. Krodapatrās – short but skillful technical notes on certain knotty portions

Summary Points • Nyāya is one of the six schools of orthodox Indian philosophies. • Nyāya philosophy, with respect to chronology, can be classified into Prācīna Nyāya and Navya Nyāya, where Prācīna Nyāya focuses on valid knowledge, while Navya Nyāya focuses on the instruments of valid knowledge. • Buddhists were the main opponents during the older period of Nyāya and a chain of refutations in both camps went on for almost a thousand years. • During these debates and exchange of ideas, a technical methodological language was crafted for the sake of exact representation of thoughts and ideologies. • Gaṅgeśa brought about a new direction in the field of Nyāya with his groundbreaking work Tattvacintāmaṇi. After this seminal work, much of the Sutrā literature took a backseat and all focus was on Tattvacintāmaṇi and the new ideas and arguments put forth in it.

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• Post Gaṅgeśa, authors like Mathurānathā and Gadādhara adopted this new way of expression and took the Nyāya literature to even loftier heights. • The beginning of eighteenth century saw a new style of work, known as Krodapatras, which were scholarly notes written on particular knotty sections of the Nyāya literature in vogue. These krodapatras mark the zenith of the technical language. Authors used technical words like avacchedaka, nirūpaka, pratiyogin, anuyogin, etc. in long compound sentences and in the process strived to convey even the most delicate ideas with utmost precision. • This new language was immediately adopted in various other subjects like grammar, aesthetics, etc. Although the technical terms created a sense of panic initially, but once mastered, they gave authors an unending freedom in expressing their ideas with great exactitude.

References Bhatta, Annam. Tarkasaṅgraha. Daniel, H.H. 1951. Materials for the study of Navya- Nyāya logic. Inghalls: Harvard University Press. Gaṅgeśa. 1985. Vyadhikaraṇam. Tirupati: Kendriya Sanskrit Vidyapeetha. ———. 1988. Pakṣatā with the Dīdhiti, Gādādhari and the commentary Bhāvaprakaśikā by Prof NSR Tatacharya. Tirupati: Kendriya Sanskrit Vidyapeetha. ———. Tattvacintāmaṇi. New Delhi: Choukhambha Sanskrit Pratisthan. Guha, Dinesh Chandra. 1979. Navya-Nyāya system of logic. 2nd Revised ed. Motilal Banarasidas. Joshi, Jaideep, and Tirumala P. Kulkarni. 2014. Language of logic: Navyanyaya perspectives. Kaviraj, Gopinath. 1982. The history and bibliography of Nyāya – VaiśeṢika literature. Varanasi: Sampurnand Sanskrit University. Śastri, S. Kuppuswami. 1998. A primer of Indian logic. Mylapore, Chennai: Kuppuswami Research Institute. Shukla, Baliram. Navya-Nyāya ke Pāribhāṣik Padārth Part 1. Pune: Yoga Enterprises. Tarkavāgīśa, Mathurānātha. Vyāptipañcakarahasyam, Kashi Sanskrit Series 64. Varakhedi, Srinivasa. 2010. Tarkaśāstrapraveśikā. Hyderabad: Sanskrit Academy. ———. 2013. Knowledge representation: Navya Nyaya and conceptual graph. Chinmaya International Foundation. Vidyābhuśaṇa, Satish Chandra. 2002. Re-print. A history of Indian logic. New Delhi: Motilal Banarasidass Publishers Private Limited. ———. 2003. Nyāya Darśana of Gotama. 2nd ed. New Delhi: New Bharatiya Book Corporation. Viśvanātha Pañcānana with the commentary Kiraṇavalī of Sudarṣanācharyā. 2011. Nyāyasiddhāntamuktāvalī. Chaukhambha Sanskrit Sansthan.

The Importance of Śa¯bdabodha in Language Analysis

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Contents Definition of Keywords . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . What Is Śābdabodha? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Signifying Power (Śakti) of a Word and the Means to Know It . . . . . . . . . . . . . . . . . . . . . . . Where Does the Śakti Reside?: Analysis by Different Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Signifier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Śabda an Independent Pramāṇa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . How Does a Sentence Unfold Its Meaning? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Essential Factors for Śābdabodha . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Verbal Expectancy (Ākāṅkṣā) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Congruity (Yogyatā) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Proximity (Āsatti) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Role of Intentionality (Tātparya) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

In order to protect the Vedas, several linguistic theories covering various aspects of knowledge ranging from phonetics to understanding a sentential meaning were developed in India. A theory dealing with verbal cognition, known as śābdabodha in Sanskrit, is one of them. The discipline of understanding the verbal communication deals with various aspects of linguistic communication such as the nature of the relation between a word and its meaning, the reason behind naming an object, the process of arriving at the meaning of a sentence from its constituents, and various factors that contribute to this process. In this chapter, we present a bird’s eye view of the discussions presented by different schools in the Indian philosophical literature. R. Shukla (*) Department of Vaidic Darshan, Sanskrit Vidya Dharm Vigyan Sankay, Banaras Hindu University, Varanasi, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_22

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Keywords

Śābdabodha · Ākāṅkṣā · Yogyatā · Sannidhi · Anvitābhidhānavāda · Abhihitānvayavāda · Pravṛttinimitta · Tātparyagrāhaka · Śakti

Definition of Keywords śābdabodha: ākāṅkṣā:

yogyatā:

sannidhi:

anvitābhidhānavāda: abhihitānvayavāda:

pravṛttinimitta:

tātparyagrāhaka liṅga:

śakti:

The cognition generated by śabda and pramāṇa (means of valid knowledge), wherein śabda is pada (word) or dhvani (sound). The verbal expectancy. It is the characteristic of such a word which is not capable of giving rise to the verbal cognition in the absence of another word. The relation of one word-meaning to another one is technically known as yogyatā (congruity). This is one of the essential causes for śabdabodha. The words in a sentence must be uttered and also heard without much gap or interval between the utterances of words in order to generate verbal cognition. This is technically known as āsatti or sannidhi. The theory according to which the word communicates an object only when related to another (anvita). A theory according to which a sentence has no śakti to denote any meaning. The words/terms only have significative relation between them and their meanings. These words when connected with each other give rise to verbal cognition. It is a reason for the use of a word to denote the designated meaning. Mīmāṃsakas hold the view that the significance of a word lies in the universal of the object (jāti) and this serves as a pravṛttinimitta. Various methods to identify the speaker’s intention suggested with detailed interpretation and justification in different sections of different texts of Indian philosophical schools. The significative power in a word to communicate its meaning.

Introduction When we listen to a word, it immediately brings to our mind the object that is being referred to by that word. Of course, it goes without saying that the word heard by us should be in a language with which we are familiar, for it to make any sense to

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us. Though words are essentially sounds produced in some sequence, all sorts of sequences of sounds are not considered to be words. Only sounds produced in a particular sequence are considered meaningful words. What is the relation between a word and its meaning? How does a cluster of words that constitutes a sentence (vākya), generate meaning which is quite distinct from the mere words constituting a sentence? Since antiquity, various attempts have been made by Indian philosophers and linguists to answer such questions. An analysis of the semantic relations that exists between the words and their meanings, and how the group of words uttered in a particular sequence gives rise to the cognition that takes place on hearing them, etc., constitutes the branch of study called śābdabodha.

What Is Śa¯bdabodha? The word śābdabodha means the cognition generated by śabda. In śāstras, the word śabda is used in two contexts: (1) śabda as pada (word) or dhvani (sound) and (2) śabda as pramāṇa (means of valid knowledge) (imau tu vāsanājanyau jñānād api vinaśyataḥ| śabdo dhvaniś ca varṇaś ca mṛdaṅgādibhavo dhvaniḥ, Bhāṣāpariccheda verse-164, See Śarma 1911, 86., yathārthānubhavaś caturvidhaḥ pratyakṣānumity upamitiśābdabhedāt || in Tarkasaṃgrahaḥ Tripāthī 2014, 24. Also see śabdakhaṇda in Tarkasaṃgrahaḥ.). Every meaningful word uttered, though refers to a particular meaning, it doesn’t have the potential to produce a verbal cognition unless it is associated with at least one more meaningful word. Here the word “cognition” means not just an understanding of the meaning of a particular word but the entire thought and no word alone can communicate that; therefore, single word does not produce the cognition. Meaningful words put together in a proper order make what is called a sentence (vākya). The words constituting a sentence must not only be meaningful and uttered in a proper sequence but should also be compatible with each other in order to generate a verbal cognition. A sentence composed in aforesaid way results in a valid knowledge called pramāṇa. Indian traditional scholars have differing views on what constitutes a pada. According to the Nyāya view, a letter or a group of letters that is able to communicate a particular meaning can be called pada, whereas according to grammarians (Vaiyākaraṇas) only such a letter or the group of letters ending with a nominal (sup) or a verbal suffix (tiṅ vibhakti) is a pada (śaktam padam, in śabdakhaṇda, Nyāyasiddhāntamuktāvalī, Jere 2002, 381., suptiṅantam padam verse 1/4/14 in Aṣtādhyāyi). Hence, for Naiyāyikas, a pada defined by Vaiyākaraṇas becomes a vākya, because it contains two padas, that is, a stem and a suffix (prakrti and pratyaya) both having their own meanings. As mentioned earlier, a group of padas form a vākya (sentence) which produces cognition. In every cognition, three things are necessarily involved: viśeṣya (qualificand), viśeṣaṇa (qualifier), and a relation between them. For example, in the cognition parvato vahnimān (mountain is fiery), the relation between the mountain and the fire is saṁyoga (contact) and two relata parvata (mountain) and vahni (fire) are the viśeṣya (qualificand) and the viśeṣaṇa (qualifier), respectively. The

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cognition involves all these three. But what is newly known to a hearer when a sentence is uttered is only the relation between them, because the other two viz. the viśeṣya and the viśeṣaṇa are already known to him through the padas. Now the question is how does one know the meaning of a pada?

The Signifying Power (Śakti) of a Word and the Means to Know It When a sentence is uttered, all the words (including prakṛti and pratyaya of each word) contained therein correspond to their respective meanings independently. This happens because each and every word has got capacity to communicate its meaning. This capacity to communicate is the relation between a word and its meaning. A particular word communicates a particular object alone, and no other. It is because of the significative power (śakti) that a word is endowed with. Indian philosophers believe that this significative power in a word to communicate it’s meaning has been there since time immemorial. They have suggested many ways to get the awareness of this significative power of every word. These are called the śaktigrahopāyas (means to know the significative power of a word). There are six of them, viz. vyākaraṇa (grammar), upamāna (simile/analogy), kośa (dictionary), āptavākya (statement of a reliable person), vṛddhavyavahāra (conduct/act of an experienced/elderly person), vākyaśeṣa (supplementary sentence), vivrti (description of the word), and siddhapadasānniddhya (proximity of the known word) (śaktigrahaṃ vyākaraṇopamānakośāptavākyād vyavahārataś ca| vākyasyaśeṣād vivṛter vadanti sānnidhyataḥ siddhapadasya vṛddhāḥ || Nyāyasiddhāntamuktāvalī, Jere 2002, 359.). Vṛddhavyavahāra is the most important among all. The meaning of a particular word is primarily apprehended through its use in a particular context. When a speaker utters a word or a sentence, he holds something in his mind to be communicated to others. His desire to communicate to the other (parapratipattīcchā) creates the desire in him to speak (vivakṣā). Before uttering a word he should necessarily confirm that the word he is going to utter is capable of communicating what exactly he intends to communicate. This he does through vṛddhavyavahāra. We elaborate this vṛddhavyavahāra with a typical example. During his childhood, first the speaker (X) observes an elderly person (Y) engaging with the act of “going” (gamanakriyā), and no other act, every time after Y hears the word “gaccha.” Consequently, he confirms the śakti of expressing gamanakriyā in the word “gaccha.” And in due course X desires to express the same to others when he uses the word “gaccha.” On hearing a particular word (padajñāna), some of the hearers act accordingly but some do not. It indicates that some are endowed with the knowledge of the significative power (śaktijñāna) of the word in context, but others are not. The heard word helps hearer endowed with śaktijñāna to remember the meaning of that word and not others. It is because of the relation they have with each other (ekasambandhijñānam aparasambandhismārakam). The horse-rider is not remembered by seeing an elephant. This relation between a word and its meaning is known as śakti, and understanding this relation is called śaktijñānam (saṅketo lakṣaṇācārthe padavṛttiḥ, Śaktivāda, See Bhaṭṭācārya 1970, 1.).

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Where Does the Śakti Reside?: Analysis by Different Schools Where exactly is the śakti of a word located? This is a vexed question seriously discussed in various disciplines of Indian philosophy. Although it is a well-known fact that a word communicates what it signifies, Indian philosophers have expressed different views about the significance a word has. The two main disciplines that have seriously dealt with this issue are Mīmāṁsā and Nyāya.

Universal (Ja¯ti) as the Locus of Śakti Mīmāṃsakas hold the view that the significance of a word lies in the universal of the object ( jāti) and this serves as a pravṛttinimitta (a reason for the use of a word to denote the designated meaning). For example, consider the word ghaṭa (jar) whose pravṛttinimitta is ghaṭatvam (jar-ness). According to Mīmāṃsakas though the cognition produced by the word ghaṭa involves all the three: viśeṣya (ghaṭa), viśeṣaṇa (ghaṭatvam) and the relation between the two viz. samavāya (inherence), yet the śakti of the word lies in ghaṭatvam only. Since the jāti cannot be cognized without its particulars (vyakti), they argue that vyakti too is cognized with jāti. Hence, there is no need for accepting śakti in vyakti (ghaṭa). The basis for their argument is as follows: śakti, the significative power for a word to express its meaning, cannot be accepted to reside in an individual (vyakti), because it will amount to accepting infinite śaktis due to the infinity of vyaktis, and there is no criterion for choosing one particular individual as the locus of the śakti, out of the infinite individuals. This will lead to an error technically known as ananugama (nongeneralization). On the contrary, if the śakti is accepted in universal ( jāti) which is one in number with reference to many particulars of the same characteristics, then there will be no room for above error and we will be able to uphold the principle of parsimony. This theory proposed by Mīmāṃsakas is called Jātiśaktivāda (coditaṃ tu pratīyetāvirodhātpramāṇena, verse 1/3/10 in Śāstradīpikā,). Individual (Vyakti) as the Locus of Śakti As against to the theory of Jātiśaktivādins; Naiyāyikas hold the view that the word in a sentence dose not stand for only the universal belonging to the objects but also the individual objects themselves. Thus, the word, according to them, denotes the objects along with their universal. This doctrine is well known as Viśiṣṭaśaktivāda (vyaktyākṛtijātayas tu padārthaḥ, verse 2/2/65 in Nyāyasūtra.). In order to uphold their doctrine, the Viśiṣṭaśaktivādins (Naiyāyikas) need to proceed in two ways. First, they need to put forth the arguments in support of their own doctrine and also refute the objections raised by Jātiśaktivādis. Secondly, they should condemn the stands of opponents by showing some lacunae in their arguments. In response to the objection pertaining to the acceptance of infinite śaktis raised by Mīmāṃsakas, the Naiyāyikas argue that the infinity of śakti does not simply arise due to the infinity of its loci. One śakti can also reside in many particulars. The śakti will be many in number only when all the particulars are not denoted by the word at a time. For instance, the polysemous word Hari, though refers to many meanings like Lord Viṣṇnu, monkey, lion, etc., it conveys only one of

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them in one particular context (See entries under hariḥ in Śabdakalpadruma for all the references of the word hari. See in Radhakantadeva 1967). In other words, it has one śakti to convey one meaning and another śakti to convey the other. Such words having many śaktis are called nānārthakas (polysemous). But when the word denotes many particulars simultaneously, it carries only one śakti in it. For example, the word Puṣpavanta stands for Sun and Moon both together (sopaplavoparaktau dvāvagny utpāta upāhitaḥ| ekayoktyā puṣpavantau divākaraniśākarau|| verse 1/4/ 10 in Amarakoṣa 2001). It conveys both in all its occurrences. The use of this word for conveying either one of these two independently is not justified. Thus, the word Puṣpavanta, in spite of communicating more than one reference, is treated as ekārthaka (having one śakti) by the philosophers.

The Pravrttinimitta of a Word From the ˙above discussion, it is concluded that the multiplicity of the loci for śakti cannot be taken as the basis for arguing for multiplicity of śakti. So, the significance of the word can lie in many particulars also. As far as pravṛttinimitta is concerned, ghaṭatvam can be pravṛttinimitta only when it resides in the particulars denoted by the word ghaṭa. Any property, according to the Naiyāyikas, which resides in denoted particulars and is denoted by itself, can become pravṛttinimitta. Therefore, ghaṭatva can become pravṛttinimitta only when individuals (ghaṭas) are also signified by the word “ghaṭa.” Moreover, the universal alone cannot be conveyed by the word, for the significance of the word depends upon the nature of the cognition it produces. As mentioned above, in all the cognitions the universal, the particulars and the relation between two are necessarily involved. Nothing (except the relation between two words) can be grasped by verbal cognition if that is not signified by the word (According to Naiyāyikas, the relation between the meanings of two words (padārthadvayasaṃsarga) is not denoted by the word. But Mīmāṃsakas accept the significative power in the word to communicate that too.). Since the particulars are also grasped by the cognition, they must also be signified by the word. Rejoinder from Mīma¯msakas ˙ rejoinder by arguing that an object need not be signified Here Jātivādins set forth the by the only primary capacity called śakti but also by secondary capacities called lakṣaṇā (secondary/metaphoric meaning), etc. For instance, the sentence gaṅgāyām ghoṣaḥ (literally, a house in the river Gaṅgā) stands for the house on the bank of the river. Although, there is no word in the above sentence which has an ability to directly signify the bank, yet this meaning is communicated to hearer. Therefore, the secondary source named lakṣaṇā is admitted. In present case also, we may accept that though the particulars are not directly denoted, they are cognized through lakṣaṇā. Naiya¯yika’s Refutation of Mīma¯msakas ˙ Naiyāyikas dismiss the above arguments put forth by Mīmāṃsakas as follows: lakṣaṇā is not recommended unless there is a serious problem of logical inconsistency, impossibility, or anupapatti (argument against) in adopting the expressed

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sense of the word. In the above example, since it is impossible for a house to exist in the flow of river, which is the expressed sense of the word Gaṅgā, the secondary meaning bank is accepted. This problem is technically referred to as anvayānupapatti (argument showing the impossibility of connection between the word meanings). Moreover, only when a speaker intends to convey something special, does he choose to employ lakṣaṇā. For instance, in the example given above, the speaker intends to convey the proximity of the house to the river and the cold breeze, etc., that can be experienced in the house. Otherwise, he could have directly said gaṅgātīre ghoṣaḥ (house on the bank of Gaṅgā). Since the intended meaning of the sentence cannot be effectively conveyed, if we directly say “gaṅgātīre ghoṣaḥ,” the speaker chooses to use the form gaṅgāyām ghoṣaḥ. But to get the intended sense, obviously the secondary meaning of the existing word needs to be adopted. This anupapatti is known as tātparyānupapatti. In the case of question, none of these anupapattis is present which may lead to lakṣaṇā (Nyāyasiddhāntamuktāvalī, śabdakhaṇda, See in Jere 2002, 385–408.).

On Signifier So far as signifier is concerned, here too Indian schools differ among each other. The Naiyāyikas are of opinion that a word alone can signify an object (padaśakti/ padalakṣaṇā), whereas other schools like Vyākaraṇa, Mīmāṃsā, etc., opine that the sentence also signifies its meaning (samāsaśakti/ vākyaśakti/ vākyalakṣaṇā). Both Mīmāṁsakas and Vaiyakaranas on the one hand and Naiyāyikas on the other have put forth detailed arguments in support of their stands (A detailed discussion can be found in different texts of Nyāya, Vyākaraṇa and Mīmāṃsā like Nyāyasid-dhāntamuktāvalī with Dinakari, Paribhāṣenduśekhara, Paramalaghumañjūṣā, Śāstradīpikā etc. See in Śāstrī 1949). Since getting into the details of them is out of the scope of this chapter, we move on to other topics by simply mentioning that this is one of the core issues discussed in Indian philosophy.

Śabda an Independent Prama¯na ˙ Another important issue that is discussed in various schools of Indian philosophy is related to śabda as a distinct pramāṇa (means for valid cognition). śabda produces a valid cognition and hence is a pramāṇa. Here the word śabda does not simply mean a term/word but a sentence. This is the technical use (pāribhāśika-prayoga) of the word śabda. The validity of śabda is unanimously accepted by all the schools of Indian philosophy except the Cārvāka school. Cārvāka’s rejection of śabda as the source of valid cognition is totally based upon the inference which itself is not admitted to be a pramāṇa by them. Moreover, just by seeing some śabdas generating erroneous cognition, they rule out the validity of all śabdas. This argument set forth by them is a very weak one, for the śabda is valid not simply because it is a śabda,

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but because it is uttered by a trustworthy person. In fact, a sentence is called as śabda only when it is uttered by a trustworthy person (āptavākya), and not when it is uttered by a traitor (pratāraka). A sentence like “agninā siñcati” (sprinkles with fire) gives rise to erroneous cognition and cannot be called as śabda, for such statements would never be uttered by a trustworthy person (āptopadeśaḥ śabdaḥ, verse 1/1/7 in Nyāyasūtra, See Śāstrī 1922). Thus, the validity of śabda as a source of valid cognition cannot be ruled out. What can be debated perhaps is the question “whether śadba is to be admitted as an independent pramāṇa or can it be subsumed under any other pramāṇas like inference, etc.” The Indian tradition has dealt with this question at great length. Here we try to provide a brief glimpse of it. Among Indian philosophers, mainly Bauddhas and Vaiśeṣikas do not admit śabda as an independent pramāṇa. Śabda, according to them, can be clubbed with inference, for the cognition arising from śabda is akin to that arising from inference. For example, the fire on mountain is inferred on the basis of smoke that is perceived in it. In the same way, perception of śabda generates the cognition of its meaning, because word (śabda) and its meaning (artha) also have permanent relation (niyatasambandha) between them as fire and smoke (sādhya and hetu) have. If the niyatasambandha between śabda and artha is not accepted then there would be utter chaos, as any word can generate the cognition of any object. In practice, the cloth (paṭa) is not cognized by the word jar (ghaṭa). Hence, the cognition arising out of śabda does not differ in any way from the once that happens in an inferential process. Therefore, the two cannot be taken as independent means of knowledge. This and many other arguments were put forth by them in support of their stands. This issue has been thoroughly discussed by various ācāryas (teachers) like Praśastapāda, Vallabhācārya, Śañkara Mishra, Dignāga, etc. Apart from the original source works authored by the ācāryas mentioned above, a thorough discussion on the stands of Vaiśeṣikas and Bauddhas can be found in Nyāya texts like sūtras and their commentaries and commentaries thereon and also independent works like Nyāyamañjarī of Jayanta Bhatta, etc. Before refuting the opponent’s stand, the arguments offered by the opponents are presented. Sometimes, the refuters themselves offer additional arguments in support of opponent’s stands, to strengthen their arguments. Countering the arguments presented by the Vaiśeṣikas and the Bauddhas, Naiyāyikas state that the similarity of the niyata-sambandhas (between śabda and its meaning with reference to śabda-pramāṇa and between hetu and sādhya with reference to inference) is not a sound argument for clubbing śabda-pramāṇa with inference for the following reason. As mentioned above, the only sentential form of śabda is a valid cognition (pramāṇa). According to the Nyāya view point, the sentence has no śakti to denote any meaning. The words/terms only have significative relation between them and their meanings. These words when connected with each other give rise to verbal cognition. This theory is technically known as Abhihitānvayavāda. According to this, the words in a sentence signify their respective meanings to the hearer who is endowed with śaktijñāna (knowledge of the significative power) of those particular words. But with this does not end the process of verbal cognition. For just by knowing the meanings of each

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and every word contained in a sentence, the hearer will not come to know what exactly the speaker intends to convey. As said above, the meanings of the words are just remembered by hearing them, as they are already known to hearer through vṛddhavyavahāra, etc., and speaker will not intend to convey nor the hearer will intend to know what is already known to him. Here arises an obvious question – then what does exactly a speaker intend to convey when he utters a sentence?

How Does a Sentence Unfold Its Meaning? Abhihita¯nvayava¯da Abhihitānvāyavādins say that the relation between the meanings signified by the respective words in a sentence is the only thing to be conveyed by the speaker. The relation is not at all denoted by any of the words in a sentence. Therefore, it becomes the meaning of the sentence (vākyārtha). Here one may raise a question – if the relation is the meaning of the vākya then why it is said that the sentence has no significative power – śakti? Responding to this question, Naiyāyikas say that the relation is vākyārtha (the meaning of a sentence) not because of the fact that the sentence has significative power to convey it, but because it is conveyed by the ākāṅkṣā, the expectancy of the words for other words, which is the property of a sentence. Thus, the relation being conveyed by the property of the sentence, that is, ākāṅkṣā, is called vākyārtha. For example, the sentence grāmam gacchati (goes to a village) contains four words viz. grāma, am, gam, and ti. The word grāmam contains two parts viz. the word grāma, which conveys “village” only and the suffix am, which stands for the object-ness (karmatvam) of grāma. Similarly, the verb gacchati also contains two parts viz. the root gam, which signifies gamanakriyā “the action of going” and the conjugational suffix ti stands for “kṛti” the doer’s effort. These words having communicated their respective meanings simply retire. But the relation of grāma with karmatvam and so on is not yet conveyed, without which the sentencemeaning cannot be understood. Therefore, Naiyāyikas suggest that the relation (anvaya) between the objects signified by the words (abhihita) is conveyed by the ākāṅkṣā (expectancy). Returning to the inference, Naiyāyikas hold that since the sentence, not conveying any meaning, has no relation with any object, therefore, cannot be brought under an inference (anumāna). Moreover, the difference between verbal testimony and the cognition generated by inference is a common experience (lokānubhava) and cannot be simply sidelined. The nature/type of the cognition is identified through its recognition (anuvyavasāya). While introspecting the cognition generated earlier by the sentence, the hearer realizes that “I knew this from the sentence uttered by the speaker.” On the other hand, while recognizing the cognition produced by the hetu, he realizes that “I knew the fire on the basis of the perception of the smoke.” Thus, the difference between the two recognitions clearly proves the difference between the cognitions and thus, the means of the cognitions have to be naturally different from one another.

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Anvita¯bhidha¯nava¯da Other philosophers like Prābhākaras who do not agree with this theory of Abhihitanvāyavādins uphold the theory known as Anvitābhidhānavāda, according to which the word communicates an object only when related to another (anvita). This they claim is based upon the vyavahāra (practice/behavior). For instance, in the sentence under discussion grāmam gacchati, the word grāma refers not only to the village but the village related to its object-ness (karmatva) denoted by the suffix am which again conveys the karmatva related to the gamana kriyā (action of going) denoted by the root gam. Observing the conversation among the elderly persons in the home a child finds a word always used for conveying the related object and thus, he confirms the significance of the word in related object. The word, therefore, also conveys the relation between two objects. Moreover, it is universally known that no such object can be captured by a verbal cognition which the word does not convey through either śakti or lakṣaṇā. Otherwise, any Tom, Dick, and Harry would be cognized by the sentence in which the words do not occur. If the relation between the objects is also cognized then that must be signified by the word. The Prābhākaras put forth many other arguments to strengthen this theory (anumānataḥ parastād upamānaṃ varṇayanti tarkavidaḥ | vādiparigrahabhūmnā vayaṃ tu śābdaṃ puraskurmaḥ || tatra tāvat padair jñātaiḥ padārthasmaraṇe kṛte | asannikṛṣṭavākyārthajñānaṃ śābdamitīryate || verse 90,91 in Mānameyodaya, Sāstri 1912, 40.). Many other philosophical schools like Viśiṣṭādvaita vedānta, Dvaitavedānta, etc., also agree with this. Naiya¯yika’s Refutation of Anvita¯bhidha¯nava¯dins Criticizing the stand of Anvitābhidhānavādins, Naiyāyikas set forth the argument that if the related object like karmatvam in a given sentence is also conveyed by the word grāma then the other word the suffix am will be useless/redundant. One and the same object cannot be cognized twice in same verbal testimony unless there is a need. Arguments and counter-arguments are presented by the followers of these theories in their support. The debate between them forms yet another very important issue in Indian philosophy so far as the verbal testimony is concerned.

The Essential Factors for Śa¯bdabodha ¯ ka¯n˙ksa¯) Verbal Expectancy (A ˙ It was mentioned above that a group of the words having verbal expectancy (ākāṅkṣā) is called a sentence. The word ākāṅkṣā literally means the desire to know (on the part of hearer). This form of ākāṅkṣā is admitted by most of the ācāryas. But Gaṅgeśa the pioneer of the Navyanyāya school is of a view that the hearer’s desire to know has no role to play in giving rise to the verbal cognition. For without having the desire of knowing the meanings intended by the speaker, one can

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attain the cognition of the sentence-meaning. He, therefore, admits that the ākāṅkṣā is the characteristic of such a word which is not capable of giving rise to the verbal cognition in the absence of another word (abhidānāparyyavasānam ākāṅkṣā, in Tattvacintāmaṇi, See Tarkavagish 1990, 208.). In a given sentence grāmam gacchati, the word grāmam cannot communicate the complete meaning in the absence of the verb gacchati, and so is the case with gacchati as well. A kāraka (the word denoting the agent/instrument/object of an action) necessarily needs the verb and similarly, the verb also requires a word signifying any one of the objects mentioned above to convey the complete meaning. Hence, according to Gaṅgeśa, this syntactic expectancy of words is called ākāṅkṣā. Unless, this kind of expectancy among the words is there, the group of the words cannot get the form of a vākya and in turn, cannot give rise to the verbal knowledge. The ākāṅkṣā, therefore, is admitted as one of the essential causes for verbal cognition.

Congruity (Yogyata¯) It has been observed in practice that a sentence cannot generate the cognition if the relation between the objects signified by the words does not exist. For example, the sentence agninā siñcati – “One sprinkles with fire” – does not give rise to the valid knowledge, for the wetting activity with fire is not possible. From this, it is known that the cognition generated by a sentence can be valid, only when the relation referred to it is in existence. This relation of one word-meaning to another one is technically known as yogyatā (congruity), which too is admitted as an essential cause for śabdabodha (ucyate bādhakapramāviraho yogyatā, sā cetarapadārtha saṃsarge ‘parapadārthaniṣṭhātyantābhāvapratiyogitvapramāviśeṣyatvābhāvaḥ, in Tattvacintāmaṇī, See Tarkavagish 1990, 262.).

Proximity (A¯satti) Another instance is found where the words in a sentence if not uttered or heard within a certain time interval do not give rise to the cognition. It is, therefore, recommended that the words in a sentence must be uttered and also heard without much gap or interval between the utterances of words. This is technically known as āsatti or sannidhi (āsattiś cāvyavadhānenānvayapratiyogy upasthitiḥ, in Tattvacintāmaṇī, See Tarkavagish 1990, 286.). All these three ākāṅkṣā, yogyatā, and sannidhi being the properties of the sentence, the word-meaning, and the words, respectively, are the essential causes for generating the valid judgment. Language which is the main source of communication among human beings is made up of such sentences. While the concepts form a continuous space, the words are discrete in nature. Further the words are finite in number, whereas the conceptual space is infinite. Therefore, the words in every language are overloaded. Some words have to correspond to more than one meaning. Such words (nānārthaka) are responsible for creating ambiguity in natural language. In spite of this ambiguity

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and the variations in words, sentences, and languages, the speaker and the listener do not feel any difficulty in communication with one another. It is because the three factors described above viz. ākāṅkṣā, yogyatā, and sannidhi play an important role in communicating the appropriate meaning.

The Role of Intentionality (Ta¯tparya) The main problem that the hearer faces in verbal understanding is to identify the intended meaning, especially when a word signifying more than one meaning is used. As said above, the speaker utters a sentence keeping something in mind to communicate to the hearer. This desire on the part of speaker to communicate what he has in mind is known as tātparya (yatpadena vinā yasyānanubhāvakatā bhavet | ākāṅkṣā vaktur icchā tu tātparyaṃ parikīrtitam || Bhāṣāpariccheda, verse 84. See Śarma 1911, 50.). Unless there is any prescribed formula, it is next to impossible to understand other’s intention. For, to know anything, we need the means of our knowledge to be connected with it. For example, unless the jar is connected with our eyes, it cannot be perceived. Similarly, the speaker’s intention has to be connected with the means of our knowledge if it is to be known. We do not need to say that the desire of others can be known only through the inference, for which many hetus are suggested. The context is utmost important among them on the basis of which the speaker’s intention can be inferred. As far as the language used in dayto-day life is concerned, even the ordinary people do not feel any difficulty in finding out the context and cognizing the speaker’s intention. But when we deal with any unacquainted text, there arises a difficulty to exactly identify the intended meaning. Say for example, in ancient age, most of the scripture was written in sūtra-form. The sūtras, as the word itself indicates, are composed with very limited words but involve a vast meaning in them. The authors of sūtras have articulated entire śāstra in aphorism and the readers of similar intelligence did not find any difficulty in understanding. But with the passage of time, a need was felt to elaborate the meanings of sūtras as they seemed to be creating ambiguity and were subject to misinterpretation. In order to avoid this, number of bhāṣyas, vārtikas, commentaries, super-commentaries, and independent monographs came into existence to reveal the intention of original authors. Various methods to identify the speaker’s intention (tātparyagrāhaka liṅgas/pramāṇas) were suggested with detailed interpretation and justification in different sections of different texts by different Indian schools. The knowledge of speaker’s intention (tātparyajñāna) helps hearer to choose one out of many meanings denoted by a word and that alone gets relation with the meaning denoted by another word (See Śaktivāda, sāmānyakāṇda for more details.). Tātparyajñānam as the cause for giving rise to the sentential cognition is also an important issue seriously discussed in various schools of Indian philosophy. Among the philosophers, mainly the Naiyāyikas, the Vaiyākaraṇas, and the Mīmāṃsakas have developed a systematic methodology of cognitive analysis to make the meaning clearer and to avoid all possibilities of any kind of ambiguity in understanding.

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But in spite of all the efforts made by them, a need was felt to modify the existing systems and methods as they were not sufficient to resolve the ambiguities. This resulted into the rise of a new school of thought within the Nyāya school – Navyanyāya. It was the main turning point in the field of philosophy. This school of Indian philosophy took the level of logical thinking to great heights. It made a unique contribution by developing a typical methodology well known as pariṣkāraprakriyā, which was unanimously accepted by all the contemporary philosophers and the successors as well. In this method, wherever there is an ambiguity, the speaker removes these ambiguities by the use of certain technical terms that follow a fixed syntax. By using only a few technical terms like avacchedakatā, etc., they have elaborated the concepts in a way so that no ambiguity or unintended application takes place. Words were also brought in so as to avoid the possibility of any kind of misinterpretation and deviation at any stage. This attracted not only to the traditional scholars but also to the modern scientists and technologists, as they too have been facing the same problem of ambiguity and deviations in sentential understanding. The mechanism they have invented and applied to reach nearer to the fact is so mathematical and logical, as it became easily assessable and acceptable to the modern thinkers. But this is not all; new thoughts are still emerging and they have to be conveyed by the same stock of words which are limited in number. There is still need of enough churning.

Summary India has a unique contribution to the science of linguistic communication. While the generative grammars like Pāṇini’s Aṣṭādhyāyī provide a model for transforming the thoughts to speech, the theories of verbal cognition provide a model for understanding the speech. In this chapter, we discussed the verbal cognition arising from a word, the relation between a word and its meaning, various means to know the meaning of a word and views of different schools on the significative power of a word. A short summary of the arguments regarding the differences in views about the significative power among the Mīmāṃasakas and Naiyāyikas is provided. The concept of pravṛttinimitta is presented explaining its significance in the process of verbal cognition. Three important factors viz. expectancy, meaning congruity, and proximity that are essential in the process of verbal cognition are introduced. The role of intentionality in the cognition is highlighted and finally the two approaches viz. anvitābhidhānavāda and abhihitānvayavāda for arriving at the sentential meaning from the word meanings are presented. A brief mention is made of the problem due to the inherent ambiguity in communication that led to the development of the new school – Navyanyāya.

References Aṣṭādhyāyī of Pāṇini Bhaṭṭācārya, Gadādhara. 1970. Śaktivādaḥ (śabdakhaṇdagranthaḥ). Bombay: Sri Venkateswara Yantralaya.

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Jere, Atmārām, ed. 2002. Kārikāvalī of Viśwanāth Nyāyapañcānana Bhattācārya with the commentaries Muktāvalī, Dinakarī, Rāmarudrī. New Delhi: Rashtriya Sanskrit Samsthan. Radhakantadeva, Raja, ed. 1967. Śabdakalpadruma. 3rd ed. Varanasi: The Chowkhamba Sanskrit Series Office. Śarma, Mukund, ed. 1911. Bhāṣāpariccheda. Bombay: Nirnaya Sagar Press. Śāstrī, Rāma Miśra, ed. 1949. Śāstradīpikā A commentary of Jaimini Sutra. Kashi: The Medical Hall Press. Śāstri, Digaṃbara, ed. 1922. Śrīgautamamunipraṇītanyāyasūtrāṇi. Puna: Anandashrama Mudranalay. Sāstrī, Ganapati, ed. 1912. Mānameyodaya of Narāyaṇa Bhaṭṭa and Nārāyaṇa Pandita. Trivandrum: Travancore Government Press. Śrīmadamarasiṃhaviracita Amarakoṣaḥ. 2nd ed. Varanasi: Chaukhamba Publishers, 2001. Tarkavagish, Kamakhyanath, ed. 1990. Tattvacintāmani of Gaṅgeśa Upādhyāya with the commentary ‘Rahasya’ by Shri Manthuranatha Tarkavagisha. Vol. 4: Part 1. Delhi: Chaukhamba Sanskrit Pratishthan. Tripāthī, Kedāranāth, ed. 2014. Tarkasaṃgrahaḥ. 7th ed. Varanasi: Motilal Banarasidas.

Early Nyāya Logic: Rhetorical Aspects

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Theoretical Rhetoric in Ancient India, Debate in the Caraka Saṃhitā, Nyāyasūtra, and Aristotle’s Rhetoric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Aristotelian and Indian/Nyāya Rhetoric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rhetorical Origins of the Nyāya Method of Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preparing for Debate in the Nyāyasūtra and the Caraka Saṃhitā . . . . . . . . . . . . . . . . . . . . . . . . . Evaluation of Arguments in the Nyāyasūtra and the Caraka Saṃhitā . . . . . . . . . . . . . . . . . . . . . Fallacies and Defects Listed in the Nyāyasūtra and the Caraka Saṃhitā . . . . . . . . . . . . . . . . . . Analogical Relations: The Five-Part Method in the Nyāyasūtra and the Caraka Saṃhitā . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Effective Debaters: Aristotle’s Rhetoric and the Caraka Saṃhitā . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Ancient Examples of Nyāya Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Examples of Nyāya Vāda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Vāda-Type Arguments in the Brihadaranyaka Upanishad . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Vāda-Type Arguments in the Bhagavad Gita . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nyāya Vāda-Type Arguments in the Aṣṭāvakragītā and Brihadaranyaka Upanishad. . . . . . Nyāya Vāda-Type Arguments in The Debate of King Milinda (Milinda Panha) . . . . . . . . . . Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Nyāya philosophy emerged when Akṣapāda Gautama (first century CE), credited with authoring the Nyāyasūtra, redacted ideas originating from Medhatithi Gautama (fifth century BCE). However, many of Medhatithi’s ideas were K. Lloyd (*) Faculty of English, English Kent State University Stark, North Canton, OH, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_9

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“reproduced” in the Caraka Saṃhitā about 78 CE (Vidyābhūṣaṇa History 25), indicating that concepts from both Medhatithi and Nyāya were in circulation well before Akṣapāda’s redaction – including elements of the five-part approach to reasoning called “the Nyāya method.” The Caraka Saṃhitā, though mostly about Ayurvedic medicine, includes a very focused debate manual bearing profound implications for the history and study of “comparative” rhetoric, which attempts to look beyond ancient Greece for other origins and interpretations of rhetoric. Early studies of Robert Oliver (Communication and culture in Ancient China and India. Syracuse: Syracuse University Press. 1971) and George Kennedy (Comparative rhetoric : An historical and cross-cultural introduction. New York: Oxford University Press. 1998) identified Indian forms of rhetorical practice found in ancient literature. However, the presence of rhetorical manuals and terminologies shows that India, though it did not develop a concept exactly like Greek rhetoric, not only practiced debate but created theories about it, leading to an entire school of philosophy. To expand the knowledge of rhetorical terminologies, approaches, and contexts in ancient India, this chapter traces the case-based rhetorical origins and concepts that became Nyāya philosophy, focusing upon the Caraka Saṃhitā, the Nyāyasūtra, and other ancient sources. Keywords

Rhetoric · Comparative Rhetoric · Analogy · Debate · Persuasion · Nyāya · Nyāya Method · Caraka Saṃhitā · Nyāyasūtra · Aristotle · Paradigma · Enthymeme · Pratijn˜ ā · Hetu · Dṛṣṭānta · Vāda

Introduction Indian philosophy evolved through a process of textual interactions over time as schools of thought debated through texts that range over hundreds, if not thousands, of years. Commentaries and reformulations sometimes overshadow the originals. Such is the case with Nyāya philosophy, which bears some roots in the ideas of the traditionally ascribed author Medhatithi Gautama, a legendary figure from around 550 BCE, but came into its clearest explication through the redactions of Akṣapāda Gautama, first century CE, now credited with authoring the Nyāyasūtra (NS). As Vidyābhūṣaṇa observes, “It is by no means easy to determine. . .the real author of the Nyāyasūtra” (49). He traces two of the foundational principles, prameya (objects of right knowledge) and vāda (a debate or discussion), to Medhatithi Gautama, which seems reasonable, since these terms are related to logic and debate in very early Indian writings. Vidyābhūṣaṇa claims that Nyāya’s concepts of pramāṇa (types of right knowledge), the avayava (members of Nyāya arguments), as well as anyamataparika (an examination of contemporary philosophical doctrines) were added by Akṣapāda, thus creating the Sūtra as we know it. However, also according to Vidyābhūṣaṇa, Medhatithi Gautama’s ideas were “reproduced” in the Caraka Saṃhitā (CS) about 78 CE (History 25), which indicates

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that concepts from both Medhatithi Gautama and Nyāya were in circulation well before Akṣapāda’s redaction, including elements of the five-part method of reasoning referred to as “the Nyāya method” (often misidentified as the Nyāya “syllogism”). As Vidyābhūṣaṇa notes, “the Caraka Samhita, as far as we know, contains for the first time an exposition of the doctrine of syllogism under the name of sthāpanā (demonstration). . .” (Vidyābhūṣaṇa History 42). Matilal adds, “Early manuals for debate . . . are not extant”; however, “we have some crystalized versions of them,” the fullest of them being the CS. Though the bulk of the CS concerns medicine and health, it includes a section on debate that almost matches the opening sections of the Nyāyasūtra (NS) (Vidyābhūṣaṇa History 31–33; Matilal 12). Such a “crystalized” debate manual has implications for the history and study of rhetoric, in particular, “comparative” rhetoric, begun with the pioneering work of Robert Oliver and George Kennedy. Kennedy defined comparative rhetoric as “the cross-cultural study of rhetorical traditions as they exist or have existed in different societies around the world” (1). His approach, however, mostly focused on using rhetoric’s Greek terminologies to interpret cross-cultural practices. While he rightly compares some of the dialogue from the Brihadaranyaka Upanishad, detailed below, as indicative of persuasive speech as practiced in Ancient India, he does so without reference to either vāda (positive debate) or Nyāya (persuasion through the application of a five-part method of demonstration), the names for the Indian approaches to persuasion which influence its passages. Rhetoricians currently look beyond the founding assumptions of comparative rhetoric to the idea of “global rhetorics,” practices, terminologies, and approaches to persuasion unique to and defined by differing cultural traditions (Mao). As will be shown below, the CS is more rhetorical than philosophical, and it brings into relief rhetorical elements within Nyāya philosophy, tracing them closer to their rhetorical and cultural origins. As Ganeri observes, “the Nyāyasūtra begins a transformation in Indian thinking about logic” which has two results: “a shift of interest away from the place of argumentation with dialectic and debate” and “a correlated shift from case-based to rule-governed accounts of logical reasoning” (Handbook 321). The student of rhetoric must reverse this process, focusing on the debate-centered and case-based origins of the Sūtra. The term rhetoric encompasses at least two distinct meanings. It may refer to rhetorical theory, codified instructions in persuasive or instructive speech, and it also refers to rhetorical interpretation, analysis of speech acts in terms of methods, motives, audience, assumptions, etc. For this reason, this chapter first considers the CS and the NS together as identifying theoretical understandings of debate and persuasion in Ancient India and juxtaposes analogous elements of Greek rhetorical practice from Aristotle and Plato. Each culture works from similar dialogical coordinates, but expresses them in unique ways. Thus the term “analogous” is used carefully here. The goal of this juxtaposition is not to reveal how elements of Greek rhetoric exhibit in Indian contexts. The goal is to make the case that India developed its own unique methods of persuasion parallel to Greek practices, rhetorical only in the sense that they are analogous in function as (ideals for) persuasive speech.

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As Susan Jarratt notes in her book Re-Reading the Sophists, “‘Rhetoric’ at its most fruitful has historically functioned as a meta-discipline through which a whole spectrum of language uses and their outcomes as social action can be refracted for analysis and combination” (14). For this reason, the second section of this chapter analyzes rhetorically Ancient Indian-written dialogues to see what they confirm or add to Indian rhetorical theory. Specifically, section two traces elements of the CS and the NS as they find expression in ancient literary/mythological/philosophical texts. Just as one cannot understand Ancient Greek rhetoric from theoretical texts alone but must also analyze the many debates, discussions, and exchanges that inhabit contemporaneous texts, one must also interpret rhetorically theoretical and literary texts in Ancient India to begin to understand its rhetorical milieu.

Theoretical Rhetoric in Ancient India, Debate in the Caraka Samhita¯, Nyāyasūtra, and Aristotle’s Rhetoric ˙ Introduction The Nyāyasūtra is a philosophical treatise that includes some aspects of debate; its roots in rhetorical tradition are more clearly outlined in the ancient medical (Ayurvedic) text, the Caraka Saṃhitā, specifically in the Vimāna sthāna, adhyāya 8 (“diagnostics chapter, Sect. 8”). It describes methods for debate (vāda-vidhi) within that context: “If a person carries on a debate with another person versed in the same science, it increases their knowledge and happiness. If there were any misapprehension in a subject already studied it removes that apprehension, and there was no misapprehension in the subject it produces zeal or is further study” (Vidyābhūṣaṇa 28). Ganeri traces the origins of Nyāya to such “case-based reasoning” found in law and medicine, noting that “Perhaps something like this underlies a lot of the way we actually reason” (see also Lloyd “Re-Thinking” 376–77). He continues, “it was an attempt to capture this kind of reasoning that we should see the ancient logic of the Nyāyasūtra and indeed of the medical theorist Caraka” (Handbook 328). Its principles then are, like Aristotle says of rhetoric, “universal” (Rhetoric 1355b 8 p23), and the aspects of both blended into Nyāya, and later Navya-Nyāya, philosophy. As will be shown later, the two texts codified precursors of Nyāya reasoning found in all kinds of historical Indian texts.

Aristotelian and Indian/Nyāya Rhetoric As in the West, India drew a distinction between argumentation and elaboration, referring to the latter as alaṇkāra, the use of figurative language and embellishments in speech, and this term is a rough equivalent to a Medieval Western misinterpretation of the term rhetoric, when rhetorical practice was truncated to mere ornamentation (reflected in today’s dismissive “rhetoric vs. reality” dichotomy). Historically,

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this is a very limited view. Far from simple embellishment, Nyāya reasoning, in particular its five-part method, reflects methods of debate analogous to Aristotle’s rhetorical “enthymeme” and “example” (Lloyd “Culture” 76–77). As Aristotle notes in his Rhetoric, “rhetorical study, in its strict sense, is concerned with modes of persuasion. Persuasion is clearly a sort of demonstration, since we are most fully persuaded when we consider a thing demonstrated. The orator’s demonstration is the enthymeme, and this is, in general, the most effective of the modes of persuasion” (I.2.1355a 3–9, p 22). He goes on to note that the enthymeme is “a rhetorical syllogism, and example a rhetorical induction. Everyone who effects persuasion through proof does in fact use either enthymemes or examples” (I.2. 1356b 4–7 p 26). Similarly, the Caraka Saṃhitā identifies the precursor to Nyāya’s five-part method of reasoning as sthāpanā (“demonstration,” literally “stepping”), which in the Indian context is defined as “the establishment of proposition through the process of a reason, example, application, and conclusion” (Vidyābhūṣaṇa History 32). Traditionally in the West, enthymemes involve at least a claim and reason or a three-part syllogism from which one premise may be omitted because the audience need not be convinced of it (Rhetoric I. 2. 1357a line 23, p. 28; Burnyeat 100. See Lloyd “Culture” 81–84). The enthymeme’s definition has shifted over time (see Poster), and Aristotle’s discussion seems to assume the reader already knows what they are, but if one considers some sample enthymemes from Aristotle’s Rhetoric, they most often exhibit a claim/reason structure (see Lloyd “Rhetorical” 26–27). For example, in one section, he offers this if/then enthymeme: “if even the gods are omniscient, certainly human beings are not” (II 23 1397b 13–15 p 1440); this is clearly a claim and reason. By “example” (paradigma), Aristotle means arguing a point by applying a series of supportive instances, as when one lists three historical figures that asked for bodyguards and became despots to imply that this will be the case in a similar current situation (see Rhetoric I.2 1357b 28-1358a 1 p 30). What Aristotle meant by enthymeme and paradigma does not match exactly with Indian terminologies, but clearly one may apply the term rhetoric in a broad sense to the Indian context, given that it involves persuasive demonstration consisting of the application of a reason and example. Both rhetors and Naiyayikas explicate and create persuasive, demonstrative speech acts.

Rhetorical Origins of the Nyāya Method of Reasoning As is well known, the Nyāyasūtra describes reasoning in a five-part structure that differs from the three-part traditional syllogism in that it applies an analogical example in the place of a major general premise (Lloyd “Rethinking” 368; “Culture” 82). The five-part model also differs from Stephen Toulmin’s model of informal logic in that it has no “warrant,” which functions similarly to a major (general) premise (see Lloyd “Rethinking” 372–376; “Rhetorical” 29–32, 35–36).

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Hypothesis (pratijn˜ ā): Reason (hetu): Example (dṛṣṭānta)

The hill (pakṣa) is on fire (sādhya) Because there is smoke (hetu) Positive: Like in the hearth Negative: Unlike a lake

[Discussion – Nirnaya] Reaffirmation (upanaya): Conclusion (nigamana):

This is the case The hill is indeed on fire .

This approach is seventh in a list of 16 categories, concepts to be studied and understood for its use. One uses an example that shares a property relation with the first two elements. Over time Nyāya developed something similar to the general premise in a concept called vyāpti, an affirmation of the property relation applied in the argument. It was used, rather than a first premise, to explicate the second member of the avayava: The hill is smoky (vyāpti: “and where there is smoke there is fire”). Nirnaya is inserted in the middle to show its basic dialogic structure. One person offers the first three elements, discussion ensues, and if the interlocutors agree, they move on to the last two steps. The first two categories in the Nyāyasūtra provide a philosophical context for Nyāya reasoning – how we know and what we can understand. The first is pramāṇa (valid means of knowledge). The NS lists four subcategories of pramāṇa, while the CS lists them separately: Pratyakṣa (perception) CS:

NS I.1.4:

“that knowledge which a person acquires by himself through his mind conjoined with the five senses. Pleasure, pain, desire, aversion and the like are objects of the mind, while sound, etc., are objects of the senses.” (Vidyābhūṣaṇa History 33) “that knowledge which arises from the contact of the sense with the object”

Upamaya (comparison) CS: NS I.1.6:

“the knowledge of a thing acquired through its similarity to another thing” (Vidyābhūṣaṇa History 33) “the knowledge of a thing through its similarity to a thing previously known”

Anumāna (inference) CS: NS I.1.5:

“a reasoning based on the knowledge of connected facts, e.g. fire is inferred from the power of digestion.” (Vidyābhūṣaṇa History 33) “is knowledge preceded by perception, and is of three kinds, ‘from cause to effect,’ (s´esa-vat), effect to cause (nānyto-dristam), and ‘commonly seen’ (cha and).”

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Śabda (word) CS:

NS I.1.7:

“a combination of letters. It is of four kinds, viz. that which refers to a matter which is seen, that which refers to a matter which is not seen, that which corresponds to what is real, and that which does not correspond to what is real.” (Vidyābhūṣaṇa History 33) “the instructive assertion of a reliable person”

These examples illustrate the differing emphases of the two documents. Definitions in the NS are generally more technical and differ most concerning s´abda . The CS definition, “a combination of letters,” reflects Hindu linguistic theory, while the rest invites rhetorical discussion about both things that are and are not. The NS, focused on evident truth – veracity with the four pramāṇa – discourages such discussions by definition. “A combination of letters” reflects Hindu linguistics, where sound is just as important as meaning and one influences the other. Any combination of “letters” resonates in specific ways, thus the idea of the mantra (lit. “to protect the mind”). By implication, Nyāya-type arguments, properly constructed, intentionally resonate in the hearer. The CS includes the NS notion of s´abda in another term, aitihaya, or “tradition,” to include “reliable assertions,” such as from the Veda. The NS next lists the prameya (objects of valid knowledge), one of which – “pain” (and others by implication) – appears in the CS definition of pratyakṣa above, to differentiate objects of the mind from objects of the senses. Again, the NS takes a fuller view of the objects of knowledge, listing “the soul, body, senses, objects of sense, intellect, mind, activity, fault, transmigration, fruit, pain, and release” (NS I.1.9). This list indicates one of the key differences between the NS and CS. The CS says that its 44 categories are to be “studied for thorough knowledge of the course of debate,” (Vidyābhūṣaṇa History 31) while the NS connects study of the categories to mokṣa, liberation from reincarnation’s cycles of death and rebirth; as it notes in I.1.2: “pain, rebirth, activity, faults, and misapprehension – on the successive annihilation of these in reverse order, there follows release.” The CS focuses more on the rhetorical/dialogical aspects of debate, while the NS elevates the goal of persuasion to the highest pinnacle of Hindu thought, liberation from death and rebirth through the annihilation of misinterpretation and doubt, incorporating elements of Hindu Vais´eṣika philosophy. Not only did the NS shift the method toward a more formal logic, it shifted debate from a tool of human discourse and analysis to a method of enlightenment. Allied with Vedic thought, this enabled Nyāya to emerge as one of six orthodox Hindu philosophies. Yet even the CS also connects “the course of debate” (vāda mārga) with elements of Vais´eṣika, in its first six categories – dravya (substance), guṇa (quality), karmā (action), sāmānya (generality), vis´eṣa (peculiarity, “manifestation” vedabase .net), and samavāya (inherence). These terms work together as a philosophy of samavāya/ inherence, since that term incorporates the others in its definition: “a permanent relation between a substance and its qualities and actions in virtue of which they cannot exist separately” (Vidyābhūṣaṇa History 32). This perspective aligns naturally with the CS’s focus on debates about medical diagnosis. Inherence also relates to the NS’s developing concept of vyāpti.

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The Nyāyasūtra also elevates the pramāṇa from the CS’s “a source of knowledge” that provides a basis for the hetu to the four ways of knowing (all possible knowledge) that provide the structure for the whole five-part method. Matilal observes the following associations between the elements of the method and the pramāṇa: pratijn˜ ā: hetu: dṛṣṭānta upanaya:

“śabda pramāṇa or verbal testimony” “skeleton form of anumāna or inference” “perception [pratyakṣa] forms the main function” “similarity with the implication of upamaya or comparison” (Logic 23)

Various interpreters differ as to which parts align with which pramāṇa, but it remains clear that NS situates debate and persuasion and the five-part method, as expressions of our only reliable means of knowing about the world – establishing the five-part method – including the dṛṣṭānta, as the perfect method for inquiry, discussion, and agreement, as seen in the examples in section “Ancient Examples of Nyāya Reasoning.”

Preparing for Debate in the Nyāyasūtra and the Caraka Samhita¯ ˙ Both the NS and the CS include the following categories: saṁs´aya (doubt), prayojana (aim), dṛṣṭānta (example), vāda (discussion), jalpa (wrangling), vitaṇḍā (caviling), and chala (quibbling). What is implied is that the speaker goes through preparatory process before debate, beginning with doubt, establishing the purpose of inquiry, and searching for suitable analogical examples (Lloyd “Rethinking” 365). The speaker (rhetor) also prepares by understanding various rhetorical methods, friendly discussion (vāda), arguing to win ( jalpa), and arguing only to discredit a point of view (vitaṇḍā). Both texts allow use of all three, though the NS promotes vāda, encouraging knowledge of the other two for interpreting opponents or as a last desperate measure. The CS seems to favor jalpa. As Matilal points out, all three terms have Ancient Greek analogues in Socrates’ concepts of dialectic (vāda), eristic ( jalpa), and elenchus (vitaṇḍā) (Logic 20–21). The NS uniquely features these categories: siddhānta (conclusion), avayava (the five-part method referred to in the CS as “demonstration”), tarka (hypothetical reasoning), and nirṇaya (settlement), but the CS has roughly equivalent terms for all but tarka, as shown below. All four seem to be elements of the debate process – exposition via the five-part method, assertion of alternate hypotheses, discussion, and conclusions. It also includes hetvābhāsa (fallacy), jāti (“false similarity” – often mistranslated as “sophisticated refutation”), and nigrahasthāna (point of defeat), which point to its emphasis on formal, rule-based procedures, discussed below. Other categories unique to the CS stem from its medical roots are jijn˜ āsā (inquiry), vyavasāya (ascertainment), artha-prāpti (presumption), and sanbhava (originating cause). Four of these categories reflect the process of examination and diagnosis, while artha-prāpti has to do with surrounding assumptions – “if it is said

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that a person should not eat during the day, it is assumed he should eat during the night” (Vidyābhūṣaṇa History 33). The rest of the CS categories refer directly to rhetorical issues. Anuyojya (“censurable”) is “a speech ‘fraught with fault’ such as ‘using a general statement when a particular one is needed,’” while ananuyojya (“non-censurable”) is said to be, enigmatically, “the opposite.” Anuyoga (“interrogation”) is “an inquiry about a subject made by a person who studies it,” while pratyanuyoga (“re-interrogation”) is “an inquiry about another inquiry.” These terms reflect two steps in the debate process. In interrogation someone asserts “the soul is eternal” and a “fellow scholar” responds “What is the reason?” In re-interrogation, someone asserts that “the soul is eternal because it is non-produced” and the other asks “Why is it non-produced?” One seeks the reason and the other the reasoning (Vidyābhūṣaṇa History 33), revealing the CS’s dialogic emphasis and reflects processes evident in the examples in the next section.

Evaluation of Arguments in the Nyāyasūtra and the Caraka Samhita¯ ˙ The next two categories reflect rhetorical concerns shared in many times and cultures, not to mention writing classrooms and public speaking courses. Vākyadoṣa (“defect of speech”) finds expression in “Inadequacy” (saying too little) “Redundancy” (including irrelevance and repetition) “Meaninglessness” (“grouping of letters without any sense”) “Incoherence” (“combination of words which do not convey a connected meaning”) “Contradiction” (“opposition to the example, tenet, or occasion”)

In contrast, vākya pras´aṃsā (“excellence of speech”) is “speech freed from inadequacy, fraught with expressive words, and is otherwise uncensurable,” “applauded as excellent, perfect, or meritorious” (Vidyābhūṣaṇa History 34). In the CS, debate always occurs in an “assembly,” which may be learned or foolish; in either case, the winner is decided based on “erudition” and “eloquence” (which includes lack of fallacy) (Vidyābhūṣaṇa History 29), rhetorical elements added to NS’s emphasis on avoiding fallacy or trickery. Both the CS and the NS include chala (quibble), which the CS identifies rhetorically as “a speech fraught with cunning, plausibility and diversion of sense.” The NS offers, as expected, a more technical definition: “opposition to a proposition by the assumption of an alternative meaning” in respect to a “term,” “genus,” or “metaphor” (i.e., confusing literal and metaphorical sense and vice versa) (NS I. 2. 10 p24).

Fallacies and Defects Listed in the Nyāyasūtra and the Caraka Samhita¯ ˙ The CS category list includes three types of ahetu (“non-reason or fallacy”), all false equivalencies (sama means “same”):

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prakaraṇa-sama (lit. “question same,” i.e., “begging the question”) saṁs´aya sama (lit. “doubt same,” an assumption based on doubt) “that which is the cause of the doubt is regarded as dispelling the doubt” varṇya sama (lit. “subject same” or “balancing the subject”), when “the example is not different from the subject in respect to their questionable character.”

Like in the NS, fallacies involve a misappropriation of the five-part method and categories. The CS offers these examples: Problem in the hetu and pratijn˜ ā prakaraṇa-sama: “the soul is eternal because it is distinct from the body: the body is non-eternal, and the soul being heterogeneous from the body must be eternal” Problem related to doubt (saṁs´aya) saṁs´aya sama : “it is doubtful whether a person who has studied a portion of the science of medicine is a physician; this person has studied a portion of the science of medicine: hence he is a physician” Problem in the dṛṣṭānta: varṇya sama: “the intellect is non-eternal because it is intangible, as sound. Here the eternality of the intellect is as questionable as that of sound”

As noted above, the NS also lists fallacies, hetvābhāsa, as its thirteenth category (I.2.4), all fallacies involving the hetu or reason. These include the “erratic,” “contradictory,” “equal to the question,” the “unproved,” and the “mistimed.” The erratic is a reason leading to more than one conclusion. The contradictory is a reason that “opposes what is to be established.” “Equal to the question” is the same as prakaraṇa-sama above. The “unproved” is a reason that needs establishing as much as the question is supposedly solved. The CS includes the NS’s fifth fallacy, as its own category, atī ta kāla (“mistimed”). Clearly, its definition is more rhetorical: “when that which should be stated first is stated afterwards” (Vidyābhūṣaṇa History 35). The NS definition is more technical and obscure: “the reason which is adduced when the time is passed in which it might hold good” (NS I.2. 9). The NS includes only one more type mistaken debate, jati, or “futility,” which is offering an objection “founded on mere similarity or dissimilarity” (NS I.2.18). As expected, the CS elaborates further rhetorical categories. While upalambha involves censuring an argument, parihara is when one recognizes and corrects one’s own defective argument. The other terms reflect similar shifts. In pratijn˜ ā-hāni one abandons a proposition, and in abhyanujn˜ ā one accepts as valid the opponent’s arguments. In hetuvantara an argument goes wrong when one advances the wrong reason, and in arthāntara one advances entirely the wrong subject matter. Both category lists end with nigrahasthāna “point of defeat,” which the NS identifies as “an occasion for rebuke” when “one misunderstands, or does not understand at all” (NS I.2. 19). The CS more rhetorically notes that nigrahasthāna “occurs when a disputant suffers defeat at the hands of his opponent. It consists in the disputant misapprehending, or being unable to apprehend, something repeated thrice in an assembly the members whereof have apprehended it” (Vidyābhūṣaṇa History 35).

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Analogical Relations: The Five-Part Method in the Nyāyasūtra and the Caraka Samhita¯ ˙ As is evident, the CS, while including 44 categories rather than Nyāya’s 16, differs in some interpretation of terms but harmonizes easily with Nyāya. They differ on why one studies the categories, however. Nyāya sets the categories as objects of philosophical reflection; in the CS, one studies its categories “for a thorough knowledge of the course of debate.” Also, in contrast to the NS, which defines vāda as peaceful or fruitful debate, the CS uses vāda as a general term, applying the term sandhāya for peaceful debate. The text nonetheless equates sandhāya with vāda vidyā, the “science” of debate, illustrating some ambivalence about the term. Both texts distinguish peaceful debate from jalpa, which the CS identifies as a “debate for the purpose of defense or attack,” and vitaṇḍā, which the CS defines as “a perverse debate for the purpose of mere attack” (Vidyābhūṣaṇa History 31). The CS lists them all as subcategories of vāda, and most of its rhetorical advice is rather jalpic. Interestingly, the CS describes exactly the five-part Nyāya method as a rhetorical demonstration, implying its pre-NS status. The CS defines the term hetu, the “reason,” as an application of one of the four pramana – perception (pratyakṣa), comparison (upamaya), inference (anumāna), or word (s´abda). As it notes, the hetu is “the source of knowledge such as perception,” etc. (32). The Nyāya method applies all four of the pramāṇa, but the hetu takes on a more technical meaning as the reason for the specific claim being debated (pratijn˜ ā). Both texts refer to supporting the hetu and pratijn˜ ā with an “example” (dṛṣṭānta), using almost exactly the same language, calling it “the thing about which an ordinary man and an expert entertain the same opinion” (Vidyābhūṣaṇa History 32; NS I.1.25). The NS involves two dṛṣṭānta, the affirmative example, “known to possess the property to be established,” and the negative example, which does not (NS I.1.36, 37). While commonly an example is a supporting instance of general principle, the example here is analogical. The CS notes that the example “describes the subject, e.g. hot as ‘fire,’ stable as ‘earth,’ etc. , or just as the ‘sun’ is an illuminator so is the text of the Samkhyas.” As Ganeri puts it, “a perceived association between the symptoms of one case provides a reason for supposing there to be an analogous relation in other, resembling cases” (History 328; see also Lloyd Rhetoric Review 372–376). Interestingly, Aristotle also traces rhetoric’s origins to case-based reasoning: “Rhetoric’s function is not simply to succeed in persuading, but to discover the means of coming near such success as each particular case will allow” (Rhetoric I. 1. 1355b 10–12 p 23). Analogy is a four-part relation tied through at least one shared characteristic, as illustrated through the dṛṣṭānta in the excerpt above, “the ‘sun’ is an illuminator so is the text of the Samkhyas”: Sun

The text of the Samkhyas Illuminates

The world

The reader

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This analogical relation is also evident in the dṛṣṭānta of an argument in the CS detailed below: Soul

Ether Non-product (eternal)

Humans

Elements

The dṛṣṭānta’s relation to the reasoning is thus rhetorically analogical. While the NS refers to the five-part method of reasoning as the avayava, its seventh category, the CS refers to it, its ninth category, as noted above, as “demonstration” or sthāpanā – which it defines as “the establishment of a proposition [pratijn˜ ā] through the process through the process of a reason, example, application and conclusion” (Vidyābhūṣaṇa History 32). It offers a sample demonstration (sthāpanā, lit. “causing to stand,” “arranging,” “supporting”): The soul is non-eternal . Because it is a non-product . Just as ether being a non-product is eternal . The soul being similar to ether is a non-product . Therefore the soul is eternal .

And counter demonstration: pratisthāpanā (prati means “return” or “answer”) The soul is non-eternal . Because it is cognized by the senses . Just as a pot, which being cognized by the senses is non-eternal . The soul similar to a pot is cognized by the senses . Therefore the pot is non-eternal .

While this analogical relation is not as apparent in Nyāya’s fire/smoke example, these two arguments appear almost exactly in its first commentator Vātsyāyana’s examples of vāda (NS I. 2. 42 p.19). Just as in the Nyāya method, the counter demonstration serves as a “rejoinder” (uttara) to the first proposition, and debate involves defense of and possible adherence to one of the positions. Nyāya arguments serve a similar function to Aristotle’s enthymeme: “There are two kinds of enthymemes. One kind proves some affirmative or negative proposition; the other kind disproves one” (Rhetoric II 23 1396b 25–27 p. 142). Interestingly, the NS adds a negative example to the method (a “hare’s horn” as a nonexistent thing I. 2. 42), much as in Aristotle’s description.

Effective Debaters: Aristotle’s Rhetoric and the Caraka Samhita¯ ˙ Unlike the NS, but similar to Aristotle’s Rhetoric, the CS discusses the characteristics of an effective debater, which it lists as “erudition, wisdom, memory, ingenuity, and

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eloquence.” It identifies three types of respondents (paraḥ): “superior, inferior, or equal.” And it identifies two types of assemblies (pariṣad), “learned” and “ignorant,” each of which may be “(a) friendly, (b) indifferent or impartial, and (c) hostile or committed to one side.” It recommends against arguing with a superior debaters or arguing in hostile assemblies. Nonetheless, it outlines methods for arguing in front of ignorant audience or someone who is respected but not eloquent, basically by stringing together “word bolts” and insults. The main focus, however, is upon vāda, “a discourse between two parties agreeably to the scriptures and in a spirit of opposition on a subject such as ‘whether there is rebirth or there is no rebirth’” (Vidyābhūṣaṇa History 31). The CS shares compelling rhetorical advice, which centers on winning the debate, a sentiment not shared by the NS, which elevates vāda and denigrates jalpa, i.e., arguing to win. In this sense, the CS leans toward an Aristotelian model of rhetoric, though there is no evidence of active connection. As Matilal notes, “it is fairly certain that no translated version of either the Rhetoric or any part of the Organon, in any Indian language existed at the time” (Logic 3). In any rhetorical situation, the orator must analyze the audience. However, the CS elaborates this process differently than the rhetoric, noting that the debater, especially in front of biased or indifferent audiences, should gear his message specifically in regard to the opponent’s individual rhetorical skills: If superior (a superior debater), do not challenge If inferior, immediately defeat Weak in scriptures, quote long passages Devoid of erudition, employ “unusual words and phrases” Memory “not sharp,” “defeat with crooked and long-strung word bolts” Devoid of eloquence, defeat “through jeering imitations of his half-uttered speeches” Shallow knowledge, “put to shame on that account” Irritable temper, throw “into a state of nervous exhaustion’ Timid, excite his fear Inattentive, defeat by putting him “under the restraint of a certain rule” (Vidyābhūṣaṇa History 30)

One should not, however, as part of these expedients of debate (vādo pāya) go too far in provoking the opponent, for this may lead to unintended results and loss: “Even in a hostile debate one should speak with propriety, an absence of which may provoke the opponent to say or do anything.” This emphasis on winning is a far cry from the NS view of vāda, and even the practice of vāda as expressed in various literary documents, but it expresses clearly how speakers might have proceeded in much public debate, and clearly shows why the Caraka allowed jalpa as a type of vāda. There is no attempt in either the NS or CS to distinguish, as in Aristotle, forensic, deliberative, and epideictic rhetoric (Rhetoric I.3. 1353b), and neither distinguishes audience by age group (Rhetoric II.12. 3389), but the Caraka does directly address that the speaker should seek an audience “endued with attentiveness, erudition, wisdom, memory and eloquence” and that in that context one should “speak with great care marking the merits and demerits of one’s opponent.”

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Conclusion This discussion reveals that though Aristotle’s Rhetoric, the CS, and NS all focus on persuasive speech, they do so using different terminologies and approaches unique to their cultures and time periods. Nevertheless, Nyāya reasoning is analogical to Aristotelian reasoning in their shared emphasis on persuasive speech and desire for a method of its conveyance: Nyāya

Aristotelian Rhetoric Method of demonstration

India

Greece

In analogies, all elements and relations must remain distinct. Indian rhetoric is not Greek rhetoric, but they fill analogous cultural roles. As shown above, the CS includes and highlights rhetorical dimensions of what became Nyāya reasoning. Its “demonstrations” always involved speaking to an assembly, and it offers very candid jalpic advice on how to win arguments. Its descriptions reveal the dialogic roots of the five-part demonstration. While Aristotle refer to the speaker’s ethos or character in a cryptic manner, noting that, “Persuasion is achieved when the speech is so spoken as to make us think him credible” (Rhetoric 1.2.1356a 4–6), the CS describes more actualized characteristics of the orator (“erudition, wisdom, memory, ingenuity, and eloquence”) and, like the Rhetoric, applies audience analysis to debate theory. The CS also traces debate to its case-based origins reflecting discussions among physicians seeking effective diagnoses and insights. While Nyāya strips some of this rhetorical context, it elevates the status of rhetorical demonstration to that of release from the cycles of death and rebirth. Its more focused 16 categories make clear that argumentation involves preparation and insight and that dialogue can lead to enlightenment. Over time it defines more clearly how the method works. Both sources codify widespread and pre-existing methods of reasoning, as shown below.

Ancient Examples of Nyāya Reasoning Introduction For the scholar of rhetoric, dialogues between Socrates and the Sophist Gorgias reveal as much or more about Greek rhetoric than the definitions and codifications in Aristotle’s Rhetoric. Ultimately, it is about how we say, not just what we say. Many dialogues appear in Indian literature, I which participants apply informal rules of reasoning and debate in mostly unconscious and culture-bound, but readily identifiable, ways. In these contexts, both CS and NS reasoning are evident, as are their modes and motives. The Nyāya method appears in informal dialogues, out of order, and rarely with both positive and negative examples. Always at some culminating point in a dialogue, it bridges the perspectives and experiences of interlocutors.

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While it may be anachronistic to call early informal versions of the five-part method Nyāya arguments, they are definitely Nyāya-like arguments and cultural precursors to the later codifications of the CS and NS. These Nyāya-like arguments (claim/reason/analogy) appear even in the earliest Indian writings, including the Brihadaranyaka Upanishad, considered the oldest of the Upanishads (seventh century BCE) (Ganeri History 309). They usually occur at key argumentative junctures as analogical capstones to the statements surrounding them. Roy M. Perrett calls them “knowing episodes,” which consist of “awareness or experience” that is the “end-product of a perceptual or inferential episode” (320). Their presence confirms that the writers of the Caraka Saṃhitā and the Nyāyasūtra were codifying long-standing and widespread practices. As Vidyābhūṣaṇa notes, though the earliest version of the Ayurveda dates from about 550 BCE (the same time Medhatithi Gautama is said to have taught and debated), its sections on debate were added by the redactor Caraka “in whose time they were widely known” (26). In each of the excerpts below, we find evidence of Nyāya-like reasoning. Discussions (vāda) begin when one interlocutor expresses saṁs´aya “doubt,” and together speakers begin to establish the prayojana (aim). One speaker usually functions as a teacher (guru) who offers various pratijn˜ ā, hetu, and dṛṣṭānta in response to the questions and concerns of the respondent (s´eṣa). Most significantly, Nyāya vāda functions just at the NS suggests; conversations lead interlocutors intentionally toward mokṣa, liberation from doubt and misapprehension, and ultimately cycles of death and rebirth. In this context, Indian rhetoric, rather than focused on winning someone over to one’s point of view, functions as a type of revelation, and respondents are not so much “convinced of” as convinced to look for and find the truth within themselves. Speaker’s also exhibit the CS characteristics of “erudition, wisdom, memory, ingenuity, and eloquence.”

Early Examples of Nyāya Vāda Dialogues appearing in literary/mythological sources relate more to Nyāya vāda than vāda defined by the CS. As noted above, the Nyāyasūtra indicates three types of arguments that use the Nyāya method, vāda (lit. “kindly speak” or “please speak”), fruitful discussion; jalpa (lit. “mad talks,” “gossip,” or “playful speech”), arguing to win; and vitaṇḍā (“counterarguments”) arguing just to argue (Literal Translations Vedabase .net; English translations Vidyābhūṣaṇa). The Nyāyasūtra identifies only vāda as “fruitful” because it uses the Nyāya method to seek mutually discovered truth. As Vātsyāyana notes, in Nyāya, “Discussion (vāda) is the adoption of one of two opposing sides. What is adopted is analyzed in the form of the five members [the Nyāya method], and decided by any of the means of right knowledge [pramāṇa] while its opposite is assailed by confutation. . .” (NS I. 2. 42). Vāda, as noted above, also implies liberation from doubt, misapprehension, and the cycles of life and rebirth (NS I.1.2.).

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As Jonardon Ganeri also notes in The Handbook of the History of Logic, this following excerpt from ancient Debates of King Milinda highlights a contemporary definition of vāda: Then the king said, “Venerable sir, will you discuss with me again?” “If your majesty will discuss as a scholar [vāda], yes; but if you will discuss as a king, no.” “How is it then that scholars discuss?” “When scholars discuss there is a summing up and an unraveling; one or other is shown to be in error. He admits his mistake, yet he does not become angry.” “Then how is it that kings discuss?” “When a king discusses a matter and advances a point of view, if anyone differs from him on that point he is apt to punish him.” “Very well then, it is as a scholar that I will discuss. Let your reverence talk without fear.” (Debates 35)

This “unraveling” (nibbeṭhnam) involves “revealing commitments, presumptions and faulty argument,” and the “summing up” (niggaho) involves a censure of one person’s arguments (Ganeri 310). The person censured, however, withdraws because he admits that his position cannot stand up to the arguments of the other. The motive is “maieutic” – “helping each side to clarify the nature of their commitments and the presuppositions on which their positions depend” (Ganeri 311). These elements parallel CS’s descriptions of pratijn˜ ā-hāni, abandoning a proposition, and in abhyanujn˜ ā, accepting as valid the opponent’s arguments.

Nyāya Vāda-Type Arguments in the Brihadaranyaka Upanishad Such discussion was not limited to scholars. In the Brihadaranyaka Upanishad, we find a Nyāya vāda-type passage in a dialogue between a husband, Yajnavalkya, and his wife, Maitreyi, who asks him to teach her about immortality. I exemplify how Nyāya reasoning appears in Ancient Indian persuasive speech, in this case mirroring the guru/s´eṣa relationship while also depicting a woman seeking and being taught insights into Hindu theology. Yajnavalkya offers his wife a claim reason and analogy to clarify the relation of the individual self to the universal self: Na hāsya udgrahaṇāyeva syāt, yato yatas tv ādadī ta lavaṇam eva evaṁ vā ara idam mahad bhūtam anantam apāraṁIdam mahad bhūtam anantam apāraṁ vijn˜ āna-ghana eva (BU 4: 12) As a lump of salt thrown in water dissolves and cannot be taken out again, though wherever we taste the water it is salty, even so beloved, the separate self dissolves in the sea of pure consciousness, infinite and immortal. Separateness comes from identifying the Self with the body, which is made up of the elements; when this physical identification dissolves, there can be no more separate self. (Easwaran’s translation 102)

We can easily adapt these remarks to a claim, reason, and analogy formulation, revealing its underlying structure:

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• Pratijn˜ ā : “the separate self dissolves in the sea of pure consciousness, infinite and immortal” • Hetu : [Because] “when this physical identification dissolves, there can be no more separate self” • Dṛṣṭānta : “As a lump of salt thrown in water dissolves and cannot be taken out again, though wherever we taste the water it is salty” The Nyāya method asserts and clarifies, inviting the respondent to apply previous perceptions to the new situation, reflecting Hindu notion of truth as something sharable because human experience is universal. We also witness the importance of the dṛṣṭānta, which the NS 1.1 36 says is “a familiar instance which is known to possess the property to be established.” Both the claim/reason and example involve the analogical property of dissolution, and the experience of one’s inability to take salt out of water parallels here our inability to see oneself outside of the self, the unified energy that creates and sustains each of us. Maitreyi’s response illustrates the dialogic nature of the method when she asks for clarification, “I am bewildered, Blessed One, when you say there is then no separate self” (BU 4.13). This is the process the NS calls nirnaya, discussion (see the smoke/ fire example above). She is asking for more explanation for the hetu. Eventually their discussion leads to agreement, nigamana (“binding minds”).

Nyāya Vāda-Type Arguments in the Bhagavad Gita We see evidence of the Nyāya method also in the discussion between the prince Arjuna and his cousin Krishna, actually an avatar of God, in the ancient Bhagavad Gita, part of the ancient epic Mahabharata: aapuuryamaaNamachalapratishhTha .n samudramaapaH pravishanti yadvat .h . tadvatkaamaa yaM pravishanti sarve sa shaantimaap noti na kaamakaamii (BG 2:70–71) “Even as all waters flow into the ocean, but the ocean never overflows, even so the sage feels desires, but is ever one in his infinite peace. For the man who forsakes all desires and abandons all pride of possession and of self reaches the goal of peace supreme.” (Translation Juan Mascaró) • Pratijn˜ ā : “the sage feels desires, but is ever one in his infinite peace” • Hetu : “For the man who forsakes all desires and abandons all pride of possession and of self reaches the goal of peace supreme.” • Dṛṣṭānta: “Even as all waters flow into the ocean, but the ocean never overflows. . .”

In this case, Arjuna responds with the category the NS calls tarka (if/then reasoning), asking for clarification as to how this infinite peace would apply in his situation – facing an impending battle. The proceeding discussions serve as nirnaya for this and related issues.

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In the final chapter of the Gita, after the avatar Krishna reveals his full divine nature to Arjuna, he offers another Nyāya-like argument in response to Arjuna’s questions about karma, this time in exact formulation: saha-jaḿ karma kaunteya sa-doṣam api na tyajet sarvārambhā hi doṣeṇa dhūmenāgnir ivāvṛtāḥ (BG 18:48) “A man should not abandon his work even if he cannot achieve it in full perfection; because in all work there may be imperfection, even as in all fire there is smoke.” (Trans. Juan Mascaró) • Pratijn˜ ā : “A man should not abandon his work even if he cannot achieve it in full perfection” • Hetu : “because in all work there may be imperfection.” • Dṛṣṭānta: “even as in all fire there is smoke”

In context, Arjuna is seeking clarification on renunciation of action, i.e., karma yoga, or action without regard to consequences or reward. Krishna responds as guru to s´eṣa using Nyāya reasoning to encourage Arjuna to persevere. Arjuna, having experienced a revelatory knowing episode, enters the battle.

Nyāya Vāda-Type Arguments in the Astāvakragītā ˙˙ and Brihadaranyaka Upanishad. Dated perhaps around 400–500 BCE, the Aṣṭāvakragī tā or “Song of Ashtavakra,” a classical Advaita Vedanta scripture, similarly features a Nyāya-like dialogue between the sage Ashtavakra and Janaka, king of Mithilā, here in exact Nyāya form: [The “true seeker”] understands the nature of things. His heart is not smudged by right or wrong, as the sky is not smudged by smoke.

Again, we can diagram its Nyāya reasoning: Pratijn˜ ā: [The “true seeker”] understands the nature of things. Hetu: [Because] His heart is not smudged by right or wrong, Dṛṣṭānta: as the sky is not smudged by smoke (AG 4:3).

Sage and king also dialogue in what in Nyāya vāda as fellow searchers for truth. Though this passage, part of an extended monologue, has no immediate response, its Nyāya formulation is most clear and in order, illustrating its use as a knowing episode. The ancient Brihadaranyaka Upanishad, with which this discussion began, features the same king, Janaka, in dialogue with the philosopher Yajnavalkya, who had earlier entered a debate contest in which he won many cattle. Yajnavalkya

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admitted at one point that his motive was both to impart wisdom and to win the cattle, illustrating a combined CS and NS perspective. Here the philosopher uses Nyāya reasoning to further explain previous arguments. In the 13th verse of the fourth chapter, Sect. 3, Yajnavalkya notes that “In the dream world, the shining one, attaining higher and lower states, puts forth innumerable forms. He seems to be enjoying himself in the company of women, or laughing, or even seeing frightful things. . ..” He follows this with various observations, concluding with the following Nyāya-like knowing episode: 14 ārāmam asya pas´yanti, na taṁ pas´yati kas cana . . .. 17 sa vā eṣa etasmin buddhānte ratvā caritvā dṛṣṭvaiva puṇyaṁ ca pāpaṁ ca, punaḥ pratinyāyam pratiyony ādravati svapnāntāyaiva. 18 tad yathā mahāmatsya ubhe kūle anusaṁcarati, pūrvaṁ cāparaṁ ca, evam evāyam puruṣa etāv ubhāv antāv anusaṁcarati, svapnāntaṁ ca buddhāntaṁ ca. (BU 4.3 14, 17–18). 4.3.14 The drama of the mind is witnessed in dream, as it is in waking, but the director of the drama is somewhere else. He is not to be observed either in waking or in dream. (Trans. Krishnananda). 4.3.17 “After enjoying himself and roaming in the waking state, and merely seeing (the results of) good and evil, he comes back in the inverse order to his former condition, the dream state (or that of profound sleep). 4.3.18 As a great fish swims alternately to both the banks (of a river), eastern and western, so does this infinite being move to both these states, the dream and waking states.” Pratijn˜ ā: [The “shining one”] moves through both “the dream and waking states” “merely seeing” Hetu: [Because] “the director of the drama is somewhere else. He is not to be observed either in waking or in dream” Dṛṣṭānta: “As a great fish swims alternately to both the banks (of a river), eastern and western”

The Nyāya reasoning is evident in this passage, though in an early, more complex, and informal form. This passage again illustrates Nyāya’s rhetorical function as an argumentative knowing episode bridging the experiences of the speaker and listener. One can disagree with the conclusion of a Nyāya-like argument, but rarely if ever does one not understand it, a power it would lack without the dṛṣṭānta. Janaka responds by giving the speaker “a thousand cows” (BU 4.3. 33) and a request for more teaching – a definite confirmation of nigamana.

Nyāya Vāda-Type Arguments in The Debate of King Milinda (Milinda Panha) The following passage, again from The Debate of King Milinda (Milinda Panha, compiled first century CE), a Pali (a Middle Indo-Aryan language) Buddhist text, a Buddhist philosopher debates with the Greek Seleucid king. It illustrates the

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influence of both vāda and the Nyāya method in its Nyāya-like dialogues between the Buddhist philosopher Nāgasena and the Indo-Greek King Milinda (Menander I Soter, entury BCE). This text indicates their use and influence outside Sanskrit/ Hinduism just past their codification in the NS. They function as a bridge across cultural divides – a symbolic meeting of Greek and Indian civilizations. This excerpt between Nāgasena and Milinda features a precise example of Nyāya vāda which also shows the dialogic origins of Nyāya reasoning: [Nāgasena:] “When wisdom has accomplished its task then it disappears; but [the wise person’s] understanding of impermanence, unsatisfactoriness and soullessness [what he gained through wisdom] does not disappear.” [Milinda:] “Give me an illustration.” [Nāgasena:] “As a man who wants to write a letter at night would have a lamp lit and then write the letter. Then he would put out the lamp, but though the lamp had been put out the letter would remain.” (DKM II.3 p 44)

Nāgasena’s argument clearly applies Nyāya reasoning: Pratijn˜ ā: Wisdom will disappear Hetu: Because it has accomplished its task of revealing understanding Dṛṣṭānta: Like the lamp and letter

The surrounding discussion involves the topic of death and rebirth, and Milinda’s response indicates his assent to the lamp analogy because he returns to the previous topic: “Does he who will not be reborn feel any painful feeling?” (DKM II.3 p 44). In this example we can trace clearly the rhetorical origins of the Nyāya method as the speaker presents a claim and reason and the respondent asks for an analogical example. The dṛṣṭānta, the connecting point between the speaker and hearer, originates at the hearer’s request. This adds another dialogic and rhetorical dimension to the CS’s concepts of anuyoga (“interrogation”) and pratyanuyoga (“reinterrogation”), when an interlocutor asks for the reason or the reason for the reason. Here the speaker offers both, and the interlocutor asks for the illustrating dṛṣṭānta. It is used here to level a power hierarchy and elucidate intercultural dialogue. Clearly, by the second century CE, Nyāya vāda was established among various schools of thought, and what is significant in this later instance is that we have evidence of its use in an intercultural exchange, as well as between husbands and wives, guru and s´ eṣa, and philosopher and king.

Definitions Nyāyasūtra: The key sourcebook for Nyāya philosophy, redacted by Akṣapāda Gautama in the (first century CE). The book was based on ideas originating from Medhatithi Gautama (fifth century BCE) that were reproduced in the Caraka Saṃhitā (78 CE) (Vidyābhūṣaṇa History 25),

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Caraka Saṃhitā: A medical (ayurvedic) text (78 CE) with a section on debate codifying many terms and concepts that appears in the later Nyāyasūtra. Its focus is more on rhetoric (persuasion and eloquence) than the Sūtra, which develops the concepts into a philosophy of logic. Pramana: The four ways humans can access the world according to the Caraka Saṃhitā and. Nyāya philosophy: The four are perception (pratyakṣa), comparison (upamaya), inference (anumāna), and authoritative word (s´abda). Avayava: In the Caraka Saṃhitā and Nyāya philosophy, the avayava (literally “members” or “limbs”) is the five-part method of debate, the Nyāya method. The Caraka Saṃhitā promotes it as an effective mode of debate used to win arguments. The Nyāyasūtra lists it as the seventh of sixteen categories of philosophical study and suggests its use as a mode of liberation from death and rebirth (mokṣa). Pratijn˜ ā: The first element and hypothesis in the Nyāya method, literally “promise,” “allegation,” or “declaration.” Hetu: The second element or reason in the Nyāya method, literally “means,” “motive,” or “logic.” Dṛṣṭānta: The third element or analogical example in the Nyāya method; usually it is put in both positive and negative form, literally “standpoint.” Upanaya: The fourth element of Nyāya method. It is applied if the first three elements are not questioned and the interlocutors agree that the formulation is true to the case at hand, literally to “attain” or to “procure.” Nigamana: The conclusion or fifth element of the Nyāya method, literally “binding the mind.” Vāda: A type of fruitful discussion where interlocutors exchange ideas on a level playing field and without fear of retribution. The object is to seek sharable truth, or knowing episodes, while resisting positions based in desire or fear. Nyāya philosophy distinguishes it from jalpa (seeking to win only), vitaṇḍā (arguing just to argue), or chala (quibbling). The Caraka Saṃhitā, more focused on the practice of debate, makes less distinction between jalpa and vāda . Sthāpanā (“demonstration” literally “stepping”): Which in the Indian context is defined as “the establishment of proposition through the process of a reason, example, application, and conclusion” (Vidyābhūṣaṇa History 32). Rhetoric: Traditionally the study and practice of persuasion and/or eloquence. Aristotle refers to it as a techne (an art) that can be learned and used to find the best available means of persuasion in any given situation. Modern definitions extend its meaning to other types and forms of communication, even those without direct persuasive intent. Enthymeme: Aristotle’s term for a rhetorical argument; since it is the “counterpart” of dialectic, it usually mirrors a three-part syllogistic form, but may only appear as one or two parts if an audience is familiar with the premises. Enthymemes are used when conclusions are debatable, whereas syllogisms are used when premises are established as certain.

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Paradigma: The Greek word often translated as “example” used by Aristotle and other ancient writers. Aristotle noted that arguments must employ either paradigma or enthymemes or both. Ethos: A term used by Aristotle to refer to the virtue or character or the speaker as expressed in speech or writing.

Conclusions The historical examples illustrate the five-part Nyāya method in action, with claim, reason, and example offered resulting in both interrogations and acceptance. We witness the dialogic and rhetorical origins of the avayava driven by the needs and questions of the interlocutors. The early date of these dialogues confirms that both the CS and NS codified pre-existing argumentative praxis, and they confirm that the discursive goals of Indian rhetoric are more revelatory than persuasive. Knowing episodes involve application of experience to shared perspectives, so the function of rhetoric in the Indian context is not to persuade someone through language, but rather to point through language and image to something within the respondent that resonates with the whole of life. Nyāya vāda in Ancient India was used to dispel doubt and misapprehension and to fan the flame of understanding that burns within each one of us. The Caraka Saṃhitā’s references to the Nyāya method, as well as commentary on debate practices, illustrate a more pragmatic dimension than Nyāya philosophy. It not only reveals and codifies the Ancient Indian rhetorical (persuasive) origins for the terms and concepts, it also sets debate practices in a less esoteric context. Rather than studying and applying the 16 Nyāya categories as part of the process of release or moksa, the focus is on winning, on finding ways to best one’s opponent, and creating the most positive diagnosis in case reasoning. In this way, the Caraka Saṃhitā is more closely related to Aristotle’s Rhetoric. They share ideas about the motives for persuasion while differing significantly on what elements of the rhetorical situation to stress according to their Indian and Greek contexts. Nyāya’s blend of philosophy and rhetoric sidestepped the split between them begun by Socrates that continues to this day. Nyāya then offers rich dimensions and nuance to Western concepts of both rhetoric and philosophy. As Beth Daniell notes in her JAEPL article, “To the Contrary,” “Rhetoric is social. Always connected to a rhetorical situation, human language rises out of identity and culture, and its development depends on who is allowed to speak or write, about what, how freely, and in which forms” (57). Though the Caraka Saṃhitā and, to a greater extent, the Nyāyasūtra codified a widespread method of debate, it would have disappeared if not for its efficacy in communication and practical uses. Indeed, Nyāya arguments continue into present Indian culture (see Lloyd “Learning”), and the work of Matilal, Ganeri, and others is restoring it to some of its precolonial status. Most significantly, Nyāya vāda offers an alternative model of argument as mutual truth seeking needed in the polarized conflictual models of argumentation so common today. As Paul Heilker quotes Johan Galtung as saying, “violence is present

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when human beings are being influenced so that their somatic and mental realizations are below their potential realizations” (49); Nyāya’s method, in focusing on the liberation of interlocutors, moves beyond persuasion to such realizations. Sanskrit Definitions from spokensanskrit.de dictionary http://spokensanskrit.de/ index.php?script=DI&beginning=0+&tinput=prati&trans=Translate&direction=AU Sanskrit transliteration from The Brihadaranyaka Upanishad by Swami Krishnananda Chapter II “Fourth Brahmana: The Conversation of Yajnavalkya and Maitreyi on the Absolute Self.” Sanskrit Text of Bhagavad Gita from The Bhagavad Gita in Sanskrit and English ITRANS Encoded Sanskrit Text of the Bhagavad Gita, from perhaps 400 BC. c.v. http ://hpb.narod.ru/tph/TPH_BHAG.HTM Sanskrit Brihadaranyaka Upanisad (Janaka dialogue) from The Brihadaranyaka Upanishad by Swami Krishnananda The Divine Life Society. http://www.swamikrishnananda.org/brdup/brhad_IV-03.html Translation of Brihadaranyaka Upanisad (Janaka dialogue) from Shankara’s Bhashya. Our Avaita Philosophy Ashram. http://www.upanishads.kenjaques.org. uk/#Brihadaranyaka%20Upanishad Revised 26th April 2014.

References Aristotle’s Rhetoric. 2004. Trans. Roberts, W. Rhys. New York: Dover Publications. A Hypertext resource compiled by Lee Honeycut. Last modified: 9/27/11. http://rhetoric.eserver.org/aristotle/ rhet1-1.html. The Ashtavakra Gita. Trans. Thomas Byron. Boston: Shambala, 1990. Print. The Bhagavad Gita. Trans. Juan Mascaró. New York: Penguin, 2003. Print. Burnyeat, M.F. 1996. Enthymeme: Aristotle on the rationality of rhetoric. In Essays on Aristotle’s rhetoric, ed. Amelie Oksenberg Rorty, 88–115. Berkeley: University of California Press. Print. Daniell, Beth. 2014–2015. To the contrary. JAEPL. 20:52–59. Print. The debate of king Milinda. Trans. Bhikkhu Basala. Buddha Dharma Education Association. Penang Malaysia: Inward Path, 2001. Web. www.buddhanet.net Heilker, Paul. 2014–2015. Coming to non-violence. JAEPL 20:52–59. Print. Ganeri, Jonardon. 2004. Indian logic. In Handbook of the history of logic, Greek, Indian, and Arabic Logic, ed. Dov M. Gambay and John Woods, vol. 1. Amsterdam: Elsevier North Holland. Print. Ganeri, Jonardon. 2001. Philosophy in classical India. New York: Routledge. Print. Jarratt, Susan. 1998. Rereading the sophists: classical rhetoric refigured. 1st Ed. Carbondale, IL: Southern Illinois University Press. Kennedy, George A. 1998. Comparative rhetoric: An historical and cross-cultural introduction. New York: Oxford University Press. Print. Lloyd, Keith. 2007a. Rethinking rhetoric from an Indian perspective: Implications in the Nyāya Sutra. Rhetoric Review 26 (4): 365–384. Print. Lloyd, Keith. 2007b. A rhetorical tradition lost in translation: Implications for rhetoric in the ancient Indian Nyāya Sutras. In Advances in the history of rhetoric, vol. 10. College Park: American Society for the History of Rhetoric. Print. Lloyd, Keith. 2011. Culture and rhetorical patterns: Mining the rich relations between Aristotle’s enthymeme and example and India’s Nyāya method. Rhetorica 29 (1): 76–105. Print. Lloyd, Keith. 2013. Learning from India’s Nyāya Rhetoric: Debating Analogically through Vāda’s Fruitful Dialogue. Rhetoric Society Quarterly 43 (3): 285–299. Print. Matilal, Bimal Krishna. 1985. Logic, language and reality. Indian philosophy and contemporary issues. Delhi: Motilal Benarsidass. Print.

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Matilal, Bimal Krishna. 1998. In The character of logic in India, ed. Jonardon Ganeri and Heeraman Tiwari. New York: State University of New York Press. Print. Mao, LuMing. 2003. Reflective encounters: Illustrating comparative rhetoric. Style 37 (4): 401–425. Print. Oliver, Robert T. 1971. Communication and culture in Ancient China and India. Syracuse: Syracuse University Press. Print. Perrett, Roy W. 1999. History, time and knowledge in Ancient India. History and Theory 38 (3): 307–321. http://www.jstor.org. 15 Oct 2005. Poster, Carol. 1992. A historicist recontextualization of the enthymeme. Rhetoric Society Quarterly 22 (2): 1–24. JSTOR. Web. 31 May 2012. The Upanisads. 2nd ed. Trans. Eknath Easwaran. Canada: Nilgiri Press, 2007. Print. Vidyābhūsana, Satish Chandra. 1988. A history of Indian logic. Delhi: Motilal Banarsidass. Print. Vidyābhūsana, M. M. Satista Chandra. 1930, 1990. In The Nyāya Sutras of Gautama, ed. Nanda Lal Sinha. Delhi: Motilal Banarsidass Publishers. Print.

Logic of Diagrams

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Reetu Bhattacharjee, Mihir Kumar Chakraborty, and Lopamudra Choudhury

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Brief History . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Euler Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venn Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venn-Peirce Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Shin’s System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hammer’s System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spider Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Systems of Choudhury and Chakraborty . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensions of Venn Diagram Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Venni . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vennin . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Vennio1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Square of Opposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Venn-Peirce diagram system has been extended with the incorporation of individuals and absence of individuals. Three types of diagram-logic system are presented, and the soundness and completeness results are claimed with respect to appropriate semantics. Traditional square of opposition has been extended with singular proposition and its negation. Pictorial representation of the corners of the

R. Bhattacharjee · L. Choudhury School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India M. K. Chakraborty (*) School of Cognitive Science, Jadavpur University, Kolkata, West Bengal, India Former Professor of Pure Mathematics, University of Calcutta, Kolkata, West Bengal, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_46

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hexagon thus obtained is analyzed. Finally, nine-cornered and five-cornered diagrams are presented and discussed.

Introduction During past few decades, there has emerged a branch of research that may be termed broadly “logical studies with diagrams.” In these research diagrams have not been simply used as aids, rather it has been established that diagrams constitute the language itself and through transformation of diagrams by syntactic rules valid conclusions (represented by diagrams again) may be obtained. This branch of research has its origin in the works of Euler (1768), Venn (1881), Peirce (1933), and many others. These scholars, particularly Peirce, went quite far in this direction. Recently Shin wrote a thesis (The logical status of diagrams (Shin 1994)) in the year 1994 through which studies in this domain have been revived. Subsequently many researchers like Hammer (1995), Allwein and Barwise (1996), Howse et al. (2001, 2005; Gil et al. 1999), Stapleton et al. (2005, 2009; Stapleton 2007), Burton et al. (2016), Ma and Pietarinen (2017a, b), and many others have joined the area, and conferences on the logic of diagrams have been organized regularly. Another type of diagrams has re-entered into the research interest since recent past, namely, “the square of opposition.” Traditionally, the square represents relations between categorical propositions by Aristotle denoted by A, E, I, and O. Its history dates back to Boethius in the middle ages based on the formulation of Apuleius (Sharma 2012). Afterwards many other logicians took part and created the square in different manners (Sharma 2012). However, Parson’s diagram (vide section “Square of Opposition”) survived the test of time and is generally used in the textbooks. The square of opposition is a simple geometric figure expressing some fundamental ideas about cognition. It is based on Aristotle’s philosophy and has been fascinating people for two thousand years. The three notions of opposition presented in the square can be applied to analyze and understand such diverse subjects as reasoning about mathematical objects, perceptions of reality, speech acts, moral reasoning and reasoning about possibility. (Quote from the back-cover of the book “The Square of Opposition – A general framework for cognition” (Béziau and Gillman 2012))

Interestingly the resurgence of interest in diagram studies pertaining to both the traditions has occurred almost during the same years and independently. The first world conference on diagrams was organized in 2000. On the other hand, the first congress on the square of oppositions was held in 2007. The current authors delved into the area around the beginning of the last decade and have contributed in both types of diagrammatic research (see the references (Bhattacharjee et al. 2018; Bhattacharjee et al 2019a; Bhattacharjee et al. 2019b; Choudhury and Chakraborty 2004, 2005, 2012, 2013, 2016)). In this chapter a summary of their work has been presented. Their contribution in the first kind of diagrams consists in primarily enhancing Shin-diagrams by the incorporation of individuals as well as absence of individuals and introduction of the notion of open

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universe. Depending upon the three types of interpretation they offered to the notion of absence of individuals (vide section “Extensions of Venn Diagram Systems”), three systems of diagram-logic, namely, Venni (Venn-diagrams with individuals and absence of individuals), Vennin (Venni with non-classical interpretation of absence), and Vennio1 (diagram-system with open boundary), were proposed, soundness and completeness have been established. The detail will be presented in section “Extensions of Venn Diagram Systems.” As regards the second kind of diagram-studies, namely, square of opposition, the contribution of Choudhury and Chakraborty lies in addition of two more corners SA (singular assertion) and SN (singular negation) and depiction of all the six corners A, E, I, O, SA, and SN by diagrams. What is novel in this study is the presence of individuals and their absence. Subsequently a nine-cornered and a five-cornered diagram have been emerged in a very natural way. In this chapter their work has been presented with some modifications; the detail may be obtained in section “Square of Opposition.” Section “Brief History“ deals with a brief history of logical studies with diagrams. There are some concluding remarks and indications to future directions of research in section “Conclusion.”

Brief History In this section a very brief history of logical studies with diagrams will be presented. One should keep in mind that history of diagram is vast and it is not possible to accommodate all the relevant studies in this chapter. Only few of the most important cases which are mainly based on Euler or Venn diagrams are presented here.

Euler Diagrams Leonhard Euler introduced his circles to represent four categorical sentences (Figs. 1, 2, 3, and 4). These diagrams were published in 1768 in Lettres a une princess d’Allemagne (Euler 1768). Fig. 1

Fig. 2 B

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Here, Figs. 1 and 2 represent the two universal statements “All A are B” and “No A is B,” respectively. Figures 3 and 4 represent the two particular statements “Some A is B” and “Some A is not B,” respectively. The main strength of Euler diagrams lies in the representation of universal statements. The diagrams representing universal statements are very intuitive. No additional conventions are needed to understand the meanings of these diagrams. For example in Fig. 1, the circle representing A is inside the circle representing B, so one can easily understand that everything that is A is also B by seeing this diagram. While using these diagrams to represent the premises of syllogism (which consist of only universal premises) the conclusion crops up. For example let us consider the following syllogism “Barbara.” All M are P All S are M ∴All S are P Now the diagram in Fig. 5 represents the first premise “All M are P.” To represent the second premise “All S are M,” we introduce the curve S inside the curve M and get the resultant diagram in Fig. 6. Figure 6 directly gives us the conclusion “All S are P.” Euler diagrams lose its intuitiveness when particular statements are considered. Both the particular statements are represented by overlapping circles, only the position of the letter A determines whether it is affirmative or negative. For “Some A are B,” letter A is in the intersecting region of the circles A and B (Fig. 3). For

Fig. 3 B

Fig. 4 B

Fig. 5

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Fig. 6

Fig. 7 M

P

Fig. 8 S

S

M

(I)

P

M

S

P

(II)

M

S

M

P

(III)

Fig. 9

“Some A are not B,” letter A is in the region of the circle A which is not included in B (Fig. 4). Combining these diagrams leads to the most important defect of Euler diagrams that is the difficulty in combining more than one piece of information. Let us consider the syllogism “Ferio.” No M is P Some S are M ∴ Some S are not P Now the diagram in Fig. 7 and the diagram in Fig. 8 represent the first and second premise, respectively. To combine these diagrams Euler suggested the three possible cases as shown by Fig. 9. It is problematic to read off the conclusion “Some S are not P” from these three diagrams. In fact (ii) is the only diagram that gives the direct visual of the conclusion but how to discard others.

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Venn Diagrams

Fig. 10







Venn’s diagrams first appeared in his 1880 paper “On the diagrammatic and mechanical representation of propositions and reasonings” (Venn 1880). Later, in 1881, a more elaborated description about these diagrams appeared in “Symbolic Logic” (Venn 1881). When two terms, say ‘A’ and ‘B’, are taken in consideration, there are exactly four combinations that could be formed between these terms. These are A
B, A \ B, B
A, and
(A [ B). Venn’s main criticisms of Euler diagrams were that Euler diagrams were incapable of representing more than one combination among two terms at a time. In Euler system instead of representing all combinations between two terms the actual relation among them were shown separately, that is, A
B ¼ 0 or A \ B ¼ 0, etc. So any uncertainty regarding the relation between A and B could not be represented by Euler’s diagrams. This is the reason why adding more than one information becomes difficult in this system. To overcome this defect Venn proposed two solutions. First, all the possible combinations among two terms A and B were drawn in one diagram. This diagram was called “primary diagram.” Thus all the possible combinations between A and B were represented by each compartment of the primary diagram as in Fig. 10. This primary diagram does not convey any information about the actual relation between the terms A and B. In the next step a new syntactic device was added to represent information about the relations between the two terms. Venn used “shading” to represent “emptiness” of a certain compartment. So in Venn system, universal propositions “All A are B” and “No A is B” were represented by the diagrams in Figs. 11 and 12, respectively.

B

A

Fig. 11

A

B

Fig. 12

A

B

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Surprisingly in (Venn 1881) Venn was silent about how to represent particular propositions. In 1883, Moktefi and Shin (2012) pointed out that Venn suggested to use “bars” to represent “non-emptiness” of certain compartment. But Venn did not provide any graphical illustration to explain how to use this new device.

Venn-Peirce Diagrams In the previous section it was pointed out that Venn diagram system could not represent particular propositions. Another problem with Venn’s system was that it could not represent disjunctive information. Charles Sanders Peirce pointed out both of these problems and resolved them by means of the following three devices: (i) Venn’s shading was replaced with a new symbol ‘o’ to represent emptiness. So the universal propositions “All A are B” and “No A is B” are represented by Figs. 13 and 14, respectively. (ii) A symbol ‘x’ was introduced to represent non-emptiness. Particular propositions “Some A are B” and “Some A are not B” were represented by Figs. 15 and 16, respectively. Fig. 13

A

B Ο

Fig. 14

A

B Ο

Fig. 15

A

B X

Fig. 16

A

B X

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A

Fig. 17

B X

Fig. 18

A

ο

Ο

ο x x

B ο

ο

x ο

Fig. 19 A

B ο

x

A

B ο

x

(iii) For disjunctive information a linear symbol ‘—’ was introduced which connects pairs (‘o’, ‘x’) or (‘o’, ‘o’) or (‘x’, ‘x’). The information “Some A are not B or All B are A” was represented by the diagram in Fig. 17. Peirce probably replaced Venn’s shading with the symbol ‘o’ so that it would be easy to represent disjunctive information by connecting respective symbols. Although by making this improvisation Peirce increased the expressive power of the system, but it also loses the visual clarity of Euler’s system. For example the diagram in Fig. 18 represents the information “Either all A are B and some A are B or no A is B and some B are not A.” This example was given by Shin to show the complexity of Peirce’s diagrams (Shin 1994, p. 23). Shin also mentioned that Peirce was aware of this complexity problem and suggested an alternative way to represent disjunctive information. Peirce’s suggestion was to replace the diagram in Fig. 18 by the diagram in Fig. 19, where each compartment of the diagram (Fig. 19) gave only conjunctive information and the relations among the compartment were disjunctive. The outer circle of the two intersecting circles in each compartment represented the universe. Peirce also introduced six rules to derive one diagram from another. He was probably the first person to discuss rules of transformation in a diagrammatic representation system. But most of these rules were incomplete and confusing. There was also no clear distinction between syntax and semantics. But nevertheless it was a very important contribution. Later, most of these rules were improvised and used by Shin in her own diagrammatic system.

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A

B ⊗

A

B ⊗

Fig. 20

Shin’s System Sun-Joo Shin modifies Venn-Peirce diagrams to achieve an increase in expressive power without a severe loss of visual clarity. Shin introduces two diagram systems, namely, Venn-I and Venn-II (Shin 1994). In these systems A rectangle, that is, ‘ ’, is used to represent the universe. A closed curve, that is, ‘ ’, is used to represent a monadic predicate. Venn’s shadings is retained to represent emptiness. Peirce’s ‘x’ is used to represent N non-emptiness although in both Venn-I and Venn-II ‘x’ is presented as ‘ ’. N (e) Peirce’s connecting line is used to represent disjunction between ’s. Also in Venn-II system this line represents disjunction between two diagrams.

(a) (b) (c) (d)

So now in Shin’s system Peirce’s complicated diagram in Fig. 18 is represented by the diagram in Fig. 20. Shin also presents each of these two systems as standard formal logical system defining syntax and semantics separately. She provides six transformation rules for the system Venn-I and ten transformation rules for the system Venn-II. Shin is the first person to provide such logical systems which are solely based on diagrams and prove that these systems are both sound and complete.

Hammer’s System Hammer’s system (Hammer 1995) is an extension of Shin’s system (Venn-I). Just like Shin, Hammer presents a standard formal logical system defining syntax and semantics separately. He also proves the soundness and completeness of the system. The main difference between Hammer and Shin’s system is that Hammer presents this system in a more mathematical way. Syntax and semantics are more precisely defined in Hammer’s system. Shin’s completeness theorem for a finite set of diagrams is extended by Hammer and Danner. They present a general completeness theorem for infinite set of diagrams.

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Spider Diagrams Spider diagram systems are consisted of both Euler and Venn diagrams. The first spider diagram system which was published in 1999 (Gil et al. 1999) used only Venn diagrams. This system is called SD1 (Stapleton 2005). SD1 system is an extension of of Shin’s Venn-II. N The symbol ‘ ’ is used N to represent only non-emptiness in Shin’s system. Even though there are two ‘ N ’s in the same region r, it will mean that the region r is non-empty. But in SD1 ‘ ’ is replaced by ‘spiders’ where distinct spiders represent existence of distinct objects. For example, consider the diagrams D1 and D2 in Fig. 21. D1 is a diagram from Shin’s system and it represents the information “All ANare B and there are some N B that are not A.” In the diagram D1 if one replaces two ‘ ’ symbols with one ‘ ’ it will still convey the same information. The diagram D2 is a spider diagram and it represents the information “All A are B and there are at least two distinct elements in B that are not A.” The next Spider system, called SD2 (Stapleton 2005), is an extension of SD1 system, where it is possible to express the cardinality of a set. If spiders and shading are placed in the same region of an SD2 diagram, then it represents exact number of elements that the set has. For example, the diagram in Fig. 22 represents that there are exactly three elements in A that are not B. N The point to be noted is that in a Venn-II diagram, if shading and ‘ ’ are placed in the same region, then the diagram is considered to be an inconsistent diagram. Similarly, placing spider and shading in the same region leads to contradiction in SD1 system.

A

B

A

B • •

⊗ ⊗

D1

D2

Fig. 21

Fig. 22

A

B • • •

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Dogs

Cats

Mammals

Cats

web

web

D1

D2

Fig. 23 Fig. 24

P a

The system ESD2 is an extension of SD2 system where both Euler and Venn diagrams are used. The spiders used in both SD1 and SD2 system are called existential spiders. There is another kind of spiders called the “constant spider” (Stapleton 2005). Constant spiders are analogous to constant in first-order logic. These spiders look like a square block instead of dot and are always labeled. For example consider the diagrams D1 and D2 in Fig. 23 where D1 represents that “web” is either a cat or a dog, but not both and D2 represents that “web” is a cat and a mammal and that all cats are mammals (this example was given by Stapleton in 2005).

Systems of Choudhury and Chakraborty In 2004, Mihir Kumar Chakraborty and Lopamudra Choudhury introduced the diagram system Venni which extended the works of Shin and Hammer by incorporating names of individual and absence of individual (Choudhury and Chakraborty 2004). In Venni, Fig. 24 represents that an object ‘a’ is in P or has the property P and Fig. 25 represents that the object ‘a’ is not in P or does not have the property P. The property P is represented by the closed curve and the universe of discourse is represented by the rectangle. In this system Fig. 25 can also be represented by an equivalent diagram in Fig. 26, where a represents the absence of the individual a. The cue to the notion of absence comes from the Indian logical systems (NavyaNyaya). Datta, 1991 like Nyaya-thinkers the authors believe that in one’s cognition absence of an individual is directly perceived in a locus just as one perceives the

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Fig. 25

P a

Fig. 26

P a

Fig. 27

presence of an individual in a locus. One does not generally infer the absence of something by observing its presence somewhere else. Figure 26 depicts the absence of the individual a in P directly as Fig. 24 depicts the presence of a in P. Whereas in Fig. 25 the information “a is not in P” is derived by seeing a outside P, that is, “a absent in P” is a kind of derivative from the information depicted in Fig. 25. Also cognitions of the absence of an object a and the absence of another object b are different. Depiction of these absences by a and b leads toward perceiving this difference in a straightforward way. By incorporating the notion of absence clutter in a diagram can be reduced. For example, both the diagrams in Figs. 27 and 28 represent that “a is not in (A  C)
B” ( and – denote, respectively, the intersection and relative complement of regions). The broken line in Fig. 28 represents the exclusive disjunction. Now Figs. 27 and 28 are equivalent but Fig. 27 is obviously less cluttered than Fig. 28 and, cognitively, Fig. 27 represents the information much more efficiently than Fig. 28. This clutter in diagrams increases if more than one individual are considered. For example, both the diagrams in Figs. 29 and 30 represent that “a is not in ((B  C)
A) and b is not in (A
B
C) and c is not in ((A  C)
B).” Again the digram in Fig. 29 represents the information much more efficiently than Fig. 30.

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Fig. 28

Fig. 29

Fig. 30

By incorporating the notion of absence in the system, three interpretations have been generated. The first one is classical where Figs. 25 and 26 are equivalent. The corresponding system is Venni. In the second interpretation Figs. 25 and 26 are not equivalent. From absence of a in the set P one cannot necessarily infer that a is in –P, the complement of P. Examples of such a situation will be discussed in section “Vennin .” In the third interpretation property, more specifically positive ascription of a property plays a fundamental role. The bounding rectangle will be dropped in this representation depicting unboundedness of the universe of discourse. More details will be discussed in section “Vennio1 .” Choudhury and Chakraborty also proposed a system where the universe is in flux (Choudhury and Chakraborty 2012). The details about this system will be given in section “Vennio1 ” also.

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Extensions of Venn Diagram Systems In this section we will introduce three formal diagrammatic systems which are based on the three interpretations of absence.

Venni Although the diagram system Venni was already established by Choudhury and Chakraborty in 2004, there were several gaps. Completeness and soundness of this system were claimed but not proven. In this section, a modified Venni system will be established where it would be possible to remove these gaps and prove completeness and soundness.

The Diagrammatic Language The Primitive symbols or the alphabet of the language consists of the following: : Rectangle; representing the universe : Closed curve; representing monadic predicate : Shading; representing emptiness x: Cross; representing non-emptiness a, b, c,. . .: Names of individuals (finitely many) a, b, c,. . .: Absence of individuals named a, b, c, . . . A, B, C,. . .: Names for closed curve or labels (finitely many) — : Line connecting crosses (x’s) and rectangles - - -: Broken line connecting individuals (a’s) Among them the following items are diagrammatic objects: 1. 2. 3. 4. 5.

Shading ( ) Cross (x) Names of individuals (a, b, c,. . .)   Names of individuals with bars a, b, c,. . . Sequence of crosses (x’s) connected by — [line connecting crosses, in short lcc (x—x—x—)] 6. Sequence of individuals (a’s) connected by - - [broken line connecting individuals, in short lci (a- - -a- - -a- - -)] for each name ‘a’ The following figures are not considered as lcc or lci. x

x

x

x x

x

x

x x

a

a

a

a a

a

a

a a

Each x in an lcc is called an x-node. Each a in an lci is called an a-node. x-nodes and a-nodes are simply called nodes if there is no confusion. A single ‘x’ and a single ‘a’ are degenerate cases of lcc and lci, respectively.

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Any closed curve without any diagrammatic object is called a blank closed curve. Definition 3.1.1. Diagram A rectangle containing finitely many closed curves or diagrammatic objects or both are the building blocks of basic diagrams (see Fig. 31). Besides, there may be a diagram which is a sequence of such basic diagrams joined by lines. Of these diagrams Fig. 31(iii), (v), and (vi) are well formed. Definition of wellformed diagrams is given in Definition 3.1.2. A basic region is the space enclosed by a rectangle or a closed curve. A basic region included by a closed curve A shall also be denoted by A whenever necessary. The rectangle and the closed curves together divide the space within the rectangle into disjoint spaces. Each such space is called a minimal region. A region is the union of some minimal regions. Example 3.1.1 In Fig. 32, the diagram D has three basic regions A, B, and the basic region enclosed by the rectangle. Four minimal regions (
A)  (
B), A  B, A  (
B), and (
A)  B. A region is the union of some of this minimal regions, for example, ((A  B) + (A  (
B))) is a region (+,  and – denote, respectively, the join, intersection, and relative complement of regions). Let r be a region obtained by the union of minimal regions m1, m2,. . ., mn. Then it is said that r has a x-sequence or r has an a-sequence if and only if each mi contains exactly one node of that sequence and none of the nodes fall in any minimal region outside r. Two x-sequences or a-sequences are said to be identical if and only if nodes of one sequence occur in exactly the same minimal regions where the nodes of the other sequence occur. Definition 3.1.2. Well-Formed Diagram (wfd) There are three types of well-formed diagrams.

Fig. 31 Fig. 32

D

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Fig. 33

Type-I 1. A single blank closed curve within a rectangle with a single label attached to it is a type-I diagram. 2. A single closed curve having a label with one or more diagrammatic objects inscribed within it or outside it but within the rectangle is a wfd provided the conditions i–iv are satisfied. (i) Two nodes of an lcc or lci will not occur in the same minimal region. (ii) Two identical x-sequence or a-sequence will not occur in a diagram. (iii) Two single a’s will not occur in the same minimal region. (iv) If shading occurs in a minimal region, it should cover the entire minimal region. Following are some examples (see Fig. 33). Type-II A type-II diagram is a diagram with more than one closed curve within a rectangle that can be ordered in a sequence C1, C2,. . ., Cn (say) such that the closed curve Ci divides all the minimal regions obtained from the closed curves C1, C2,. . ., Ci–1 in exactly two minimal regions and which satisfy the conditions i–vii: (i) The minimal regions so formed may have or may not have entries of diagrammatic objects. (ii) The closed curve should not pass through the signs x, a, or a or labels, A, B,. . .. (iii) Labels should be attached to each closed curve and different labels for different closed curves within the same rectangle. (iv) The diagram may contain lcc or lci with the restriction that two nodes of the same lcc or lci will not appear in the same minimal region. (v) Two single a’s will not occur in the same minimal region. (vi) If there is shading in a minimal region, it has to cover the entire minimal region. (vii) Two identical x-sequence or a-sequence will not occur in a diagram. Following are some examples (see Fig. 34). Venn’s method of constructing overlapping closed curves is followed here. It is established that for any number n >1 it is always possible to draw a Venn diagram satisfying the conditions stated in the definition of type-II diagrams. Type-III If D1, D2,. . ., Dn, n  2 are of type-I or type-II diagrams, then the diagram D0 resulting from connecting them by straight lines (written as D1 – D2 – . . . Dn) is a wfd of type-III. Each Di is called a component of the diagram D0 (see Fig. 35).

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Fig. 34 Fig. 35

Fig. 36

Note that by a proper part of a type-III diagram D is meant a diagram obtained by dropping some of the components of D. Henceforth by diagram we shall mean wfd. Now Fig. 31(i), (ii), and (iv) are not well-formed diagrams for the following reasons: (i) In Fig. 31(i), two nodes of lci are in the same minimal region. (ii) In Fig. 31(ii), let us consider the two curves as C1 and C2 (Fig. 36). Now we obtain two minimal regions from C1 and the curve C2 divides both of the minimal regions in three parts. Moreover there is no label. (iii) In Fig. 31(iv), shading does not cover the entire minimal region. Definition 3.1.3. Counterpart Relation The counterpart relation is a relation between the regions of any two diagrams defined as follows.

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Fig. 37

Two basic regions are counterparts if and only if they are regions enclosed by curves having the same label. If r and r0 are regions of diagram D and s and s0 are regions of diagram D0 , r is the counterpart of s and r0 is the counterpart of s0 , then r + r0 is the counterpart of s + s0 , r  r0 is the counterpart of s  s0, and r – r0 is the counterpart of s – s0 , where +,  and – denote, respectively, the join, intersection, and relative complement of regions. By – r we shall denote the relative complement of a region r with regard to the rectangle. It follows that if r is the counterpart of s, then – r is the counterpart of – s. Example 3.1.2 In Fig. 37, the diagram D1 has two closed curve P and Q. The counterparts of the basic region P are in the diagram D3 and in the diagram D5. The counterparts of the basic region Q are in the diagrams D2 and D3. The diagrams D2 has four curves A, Q, R, and S out of which we talk about the counterparts of the basic region Q. The basic region A has the counterparts in the diagrams D3 and D5. The basic regions R and S have no counterparts in any of the diagram. The diagram D3 has three curves P, Q, and A, and we talk about their counterparts. The diagram D4 has one curve B and the counterpart of the basic region B is in the diagram D6. The diagram D6 has two curves B and C where C does not have any counterparts in any of the given diagrams. The region P – Q has x in the diagram D1 and its counterpart region in the diagram D3 is without any diagrammatic object. The region Q – P has a in the diagram D1 and its counterpart region has a-sequence in the diagram D3. The region A – Q in the diagram D2 and its counterpart region in the diagram D3 both have no diagrammatic objects. The region Q – A has no diagrammatic objects in the diagram D2 and its counterpart region in the diagram D3 has a node of an a-sequence. One of the minimal regions that constitute the basic region P in the diagram D1 has a x, and counterpart region of P in the diagram D5 has shading and part of a x-sequence. The basic region A in the diagram D2 has no diagrammatic objects and its counterpart in D5 has x-sequence. The basic region B has shading in D4 and its counterpart has no diagrammatic objects in D6. Definition 3.1.4. Identity Two type-I/II diagrams are identical if and only if (i) the set of labels used in the two diagrams are same. (ii) A region in one is shaded if and only if its counterpart in the other is shaded.

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Fig. 38 Fig. 39

(iii) A region of one has a x-sequence if and only if its counterpart in the other has a x-sequence. (iv) A region of one has an a-sequence if and only if its counterpart in the other has an a-sequence. (v) A region in one has a if and only if its counterpart in the other has a. (This definition is a modification of Hammer (1995)). Two type-III diagrams are identical if and only if each component of one is identical with some component of the other. Example 3.1.3 In Fig. 38, the diagram D1 and the diagram D2 are identical. Identity of two diagrams does not depend on the shape and size of the basic regions. Let D (Fig. 39) be a type-II diagram having two closed curves. Let the minimal regions of D be m1, m2, m3, and m4. Now m1, m2, m3, and m4 represent, respectively, the regions A  ð
BÞ, A  B, ð
AÞ  B and ð
AÞ  ð
BÞ:

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These are all the possible combinations of the basic regions A, B, and their complements –A, –B taken two together, that is, for two curves we have 22(¼ 4) minimal regions and 22(¼ 4) combinations. Suppose D has three curves A, B, and C. Then all possible combinations of the basic regions A, B, C, and their complements –A, –B, –C taken three together are A  ð
BÞ  C, A  B  C, ð
AÞ  B  C, ð
AÞ  ð
BÞ  C, A  ð
BÞ  ð
CÞ, A  B  ð
CÞ, ð
AÞ  B  ð
CÞ, ð
AÞ  ð
BÞ  ð
CÞ: Definition 3.1.5. Normal Form Let a type-II diagram D have n curves C1, C2, . . ., Cn (n  2). The diagram D is said to be in its normal form if it satisfies the following properties: (i) There are 2n minimal regions in D. (ii) All possible combinations of the basic regions C1, C2,. . ., Cn and their complements –C1, –C2,. . ., –Cn taken n together should be represented by these 2n minimal regions. Note that from the first two properties it follows that a combination should be represented by exactly one minimal region. It is clear that for any diagram with n closed curves there exists a unique Venn diagram which is in the normal form. Example 3.1.4 Let us consider the following diagrams (Fig. 40). These diagrams have the same number (¼ 3) of basic regions but varying number of minimal regions. It may be noted that only diagram D1 is in the normal form. The diagram D2 has 5 (6¼23) minimal regions, that is, it violates the first property of normal form. The diagram D3 has 8 (¼ 23) minimal regions but the combination ((
A)  (
B)  C) is not represented, that is, it violates the second property of normal form. Also the combination (A  B  (
C)) was represented by two minimal regions in the diagram D3. The diagram D4 has 10 (6¼23) minimal regions, that is, it violates the first property of normal form. Note: (i) A type-I diagram is in normal form by default. (ii) A diagram is in normal form if and only if it is wfd.

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Fig. 40

Fig. 41

Rules of Transformation Like rules of Natural Deduction method of logics, there are introduction and elimination rules in the present system. Besides there are other rules. Introduction Rules (for Closed Curves, a, a, and x) For Closed Curves Let D be a type-I/II diagram. It may be transformed into a diagram by introducing a closed curve in D, obeying the restrictions 1–6 below: 1. The newly introduced curve should have a new label. 2. The newly introduced curve should divide all the minimal regions in D, into exactly two parts. It is to be noted that if a minimal region is shaded, then it is divided into two minimal regions where both of them are shaded. Example 3.1.5 In Fig. 41, the diagram D0 is obtained by introducing the closed curve C in the diagram D. 3. The newly introduced curve should not pass through the already existing labels in D.

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Fig. 42

Fig. 43

4. If there is an x=a=a in a minimal region m of D, the new curve, say B, should be drawn in such a way that neither B passes through x=a=a nor the existing x=a=a appears in the intersection of the minimal region and the new curve, that is, m  B. Example 3.1.6 Figures 42 and 43 are the examples of wrong introduction of closed curve B in the diagram D to get the diagram D0 . 5. (a) If a minimal region m included in D has a node of a x-sequence and a new closed curve B is introduced in D, then x should be added in the region m  B and a line should connect the new x with one of the end x’s of the existing x-sequence. Example 3.1.7 Suppose one wants to transform the diagram D (Fig. 44) into a new diagram D0 by introducing a closed curve B in the diagram D. Then the minimal regions A – C, C – A, and
(A + C) in the diagram D are divided into two parts due to the introduction of the closed curve B (see Fig. 45). The minimal region A – C into (A – C) – B and (A
C)  B. The minimal region C – A into (C – A) – B and (C
A)  B. The minimal region
(A + C) into
(A + C) – B and
(A + C)  B. We now add x in the minimal regions (A – C)  B, (C
A)  B, and
(A + C)  B (Fig. 46). Finally the new x is connected with one of the end of x’s of the existing x-sequence (Fig. 47). Figure 47 is the diagram D0 . The diagrams in Figs. 45 and 46 are just intermediate steps to demonstrate how to get the diagram D0 (Fig. 47) from the diagram D (Fig. 44). One

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Fig. 44

A

C X

X

X

Fig. 45

C

A X

X

B

X

Fig. 46

C

A X

X X

B

X X

X

Fig. 47

C

A X

X X

B

X X

X

cannot get the diagrams (Figs. 45 and 46) from D by applying the rule of introduction of closed curves. Note that when the diagrams in Figs. 48 and 49 cannot be obtained also as the new x’s are connected with the middle node of the existing x-sequence in Fig. 48 and the new x’s are connected with both of the end nodes of the existing x-sequence in Fig. 49.

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Fig. 48

Fig. 49

a

a

C

A

C

A

a a

a

a

a

a

B a

D

D’

Fig. 50

5. (b) If a minimal region m included in D has a node of an a-sequence and a new closed curve B is introduced in D, then a should be added in the region m  B and a broken line should connect the new a with one of the end a’s of the existing asequence. Example 3.1.8 In Fig. 50 we obtained the diagram D0 by introducing the closed curve B in the diagram D. 6. If a minimal region m included in D has a and a new closed curve B is introduced in D, then a should be added in the region m  B. Example 3.1.9 Suppose we want to transform the diagram D (Fig. 51) into a new diagram D0 by introducing a closed curve B in the diagram D. Then the minimal

28

Logic of Diagrams

743

Fig. 51

Fig. 52

Fig. 53

regions A – C and C – A in the diagram D are divided into two parts due to the introduction of the closed curve B (see Fig. 52). The minimal region A – C into (A – C) – B and (A
C)  B. The minimal region C – A into (C – A) – B and (C
A)  B. We now add a in the minimal regions (A
C)  B and (C
A)  B (Fig. 53). Figure 53 is the diagram D0 . The diagram in Fig. 52 is just an intermediate step and we cannot get this diagram from D (Fig. 51) by the rule of introduction of closed curve B. Example 3.1.10 If D has more than one diagrammatic object, then to introduce a new curve in D one should keep in mind all the four restriction mentioned above. In Fig. 54 the diagram D0 is obtained by introducing the closed curve D in the diagram D. For a Let D be a type-I/II diagram such that there is no a-sequence in D. Then D may be transformed into a diagram by introducing for each a in D an a-sequence such that each minimal region of D has a node of this a-sequence.

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Fig. 54

Fig. 55

Example 3.1.11 In Fig. 55 the diagram D0 is obtained by introducing a-sequence in the diagram D. For a Let D be a type-I/II diagram such that there is no a in a minimal region m of D. Then D may be transformed into a diagram by introducing a in the minimal region m of D if (i) m is shaded or (ii) a portion (may be whole) of – m, the complementary region of m has a-sequence. Example 3.1.12 In Fig. 56 the diagram D0 is obtained by introducing a in the shaded minimal regions of D. Example 3.1.13 In Fig. 57 the diagram D has a-sequence in the region ((A
B
C) + ((A  B)
C) + (C
A
B)). The diagram D0 from the diagram D by introducing a in a portion of the complementary region of ((A
B
C) + ((A  B)
C) + (C
A
B)). Here three applications of the rule of introduction of a have been used.

28

Logic of Diagrams

745

Fig. 56

Fig. 57

For x 1. Let D be a type-I/II diagram such that there is no x-sequence in some region r of D. Then D may be transformed into a diagram by introducing a x-sequence in the region r of D if there is an a-sequence in r. 2. Let D be a type-I/II diagram such that there is no x-sequence that spreads over the whole rectangle. Then D may be transformed into a diagram by introducing a x-sequence in D such that each minimal region of D has a node of this x-sequence. Example 3.1.14 In Fig. 58 the diagram D0 is obtained by introducing x-sequence in the same region which has a-sequence in the diagram D. Example 3.1.15 In Fig. 59 the diagram D0 is obtained by introducing x-sequence in the diagram D such that each minimal region of D has a node of this x-sequence. Extension Rules (for Lcc, Lci, and Components) For Lcc and Lci Let D be a type-I/II diagram containing a x-sequence or an a-sequence in some region r, then it may be transformed into a diagram by introducing x or a in a

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Fig. 58

Fig. 59

minimal region m (which is outside of r) connecting it with one of the end nodes of the existing x or a-sequence with a line or a broken line, respectively. Example 3.1.16 In Fig. 60 x is introduced in D in any of the minimal regions (A  D)
B
C, (A  C  D)
B, (A  C)
B
D, A  B  C  D, (A  B  C)
D, (B  C)
A
D, (B  C  D)
A, B
A
C
D, D
A
B
C, (D  C)
A
B,
(A + B + C + D). The diagram D1 is obtained by introducing x in the minimal region D
A
B
C in the diagram D and connecting this new x with one of the end x’s of the existing x-sequence with a line. Similarly diagrams D2 and D3 are obtained. D4 is obtained by multiple applications of the rule. However, the diagram D0 cannot be obtained from D using the rule of extension of lcc as the new x in the minimal region D – A – B – C is connected with one of the middle node of the existing x-sequence (Fig. 61). Example 3.1.17 In Fig. 62 the diagram D0 is obtained by introducing a in the minimal regions (A  B)
C, (B  C)
A, and C
A
B in the diagram D and connecting them with one of the end a’s of the existing a-sequence with a

28

Logic of Diagrams

747

Fig. 60

Fig. 61

Fig. 62

broken line. Here we have used three applications of the rule of extension of lci. For Components A diagram D may be transformed into a diagram D0 – D or D – D0 by connecting any diagram D to D0 by a line ——. Example 3.1.18 In Fig. 63 the diagram D – D0 is obtained by connecting the diagram D0 to D.

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Fig. 63

Fig. 64 Fig. 65

Elimination Rules (for Lcc, Lci, Shading, a, and Closed Curve) For Lcc and Lci 1. A type-I/II diagram D may be transformed into a diagram by eliminating entire x-sequence (or a-sequence) from D. Example 3.1.19 The diagram D1 is obtained from D by eliminating the x-sequence or the diagram D2 from D by eliminating the a-sequence or the diagram D3 from D by eliminating both x-sequence and a-sequence (Fig. 64). Note that the entire sequence is to be deleted be it an lcc or lci. One cannot get the diagram D0 from D (Fig. 65).

28

Logic of Diagrams

749

Fig. 66

2. If a type-I/II diagram D contains a sequence of x’s (or a’s) with more than one node and some nodes fall in shaded region, then it may be transformed into a diagram by eliminating those nodes in the shaded region and preserving the remaining nodes in a chain. After eliminating the nodes and preserving the chain if we get two identical lcc or lci, then only one such lcc or lci will be retained. We can call this rule as the rule of elimination of the part of lcc (or lci). Example 3.1.20 The diagrams D1 and D2 are obtained by eliminating the nodes of the x-sequence and a-sequence which falls in the shaded regions of D, respectively, and preserving the remaining nodes in a chain. But as there are two identical lci’s in the diagram D2 we retain only one and get the diagram D3 (Fig. 66). The diagram D0 is obtained from D by multiple application of the rule of elimination of the part of lcc and lci (Fig. 67). 3. If a type-I/II diagram D contains a sequence of a’s with more than one node and one of the nodes falls in a minimal region containing a, then it may be transformed into a diagram by eliminating that node and preserving the remaining nodes in a chain. After eliminating the nodes and preserving the chain if we get two identical lci, then only one such lci will be retained. This rule is also called the rule of elimination of the part of lci. Example 3.1.21 The diagram D0 is obtained from D by eliminating the nodes of the a-sequence that falls in the regions containing a, that is, B
A
C and C
A
B, and preserving the remaining a-nodes in a chain (Fig. 68). Here the rule of elimination of the part of lci has been applied twice. For Shading A type-I/II diagram D may be transformed into a diagram by eliminating shading from any minimal region of D. Example 3.1.22 The diagram D0 is obtained from D by eliminating the shading from the minimal region A
B (Fig. 69).

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Fig. 67



Fig. 68

For a A type-I/II diagram D may be transformed into a diagram by eliminating a from any minimal region of D. Example 3.1.23 The diagram D0 is obtained from D by eliminating the a from the minimal region B
A (Fig. 70). For Closed Curve For a type-I diagram closed curve elimination is not allowed since then the diagram obtained will not be well-formed. A type-II diagram D may be transformed into a diagram, say D0 , by eliminating a closed curve, say B, provided the following conditions 1 and 2 hold: (1) For all regions r, if B  r is shaded, then r – B is also shaded and vice versa. (2) For all regions r, if B  r has a, then r – B also has a and vice versa.

28

Logic of Diagrams

751

Fig. 69

Fig. 70

If such conditions are satisfied, then D0 is obtained through steps i–v below by a series of constructions D1 to D4. (i) B is eliminated. Let the resulting diagram (not necessarily wfd) be D1. (ii) A diagram D2 is obtained thus – if D1 is in normal form, D1 is D2. If not, then we reduce D1 to its normal form D2 (This reduction is always possible, see Definition 3.1.5). (iii) All the diagrammatic objects of D1 are transferred into the respective counterpart regions in D2. The resulting diagram is D3. (iv) If more than one node of a x-sequence/a-sequence or two a’s fall within the same minimal region of the diagram D3, then only one such node or a is to be retained. The resulting diagram is D4. (v) If two identical lcc or lci occur in the diagram D4, then only one such lcc or lci will be retained. The resulting diagram is D0 . Example 3.1.24 The diagram D1 is obtained from the diagram D by eliminating the closed curve B. The closed curve B can be eliminated as it satisfies the first condition, that is, both the regions B  (D
A
C) and (D
A
C) – B are shaded. Now D1 is not in normal form as there are three curves but number of

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Fig. 71

minimal region is 10 (6¼23 ¼ 8). So D1 is reduced to its normal form D2. All the diagrammatic objects of D1 are transferred into the respective counterpart regions in D2, the resulting diagram is D3. In diagram D3, there are two nodes of b-sequence in the minimal region (A
C
D). Only one node of the b-sequence is retained to get the diagram D0 . So, by eliminating the closed curve B from the diagram D, one gets the diagram D0 (Fig. 71). Unification Rule For Type-I or Type-II Diagram Diagrams D1 and D2 may be transformed into a diagram by uniting them in one diagram, obeying the steps 1 and 2. 1. All the closed curves of D2 of which there were no counterparts in D1 are introduced in D1 to obtain the diagram D. 2. All the diagrammatic objects of D2 are drawn in the respective counterpart regions of the diagram D, obeying the rule of introduction of closed curve. The resulting diagram is D0 . (That is, if a region r in D2 has a x-sequence and if c(r), the counterpart region of r in D0 , has the minimal regions m1, m2,. . ., mn within it due to the introduction of curves, then each of the mi has a node of the abovementioned x-sequence with lines connecting them. Similarly for a-sequence. If a minimal region m in D2 has shading or a, then c(m), the counterpart region of m in D0 , is divided into two parts due to introduction of closed curve and thus each part has shading or a, respectively.)

28

Logic of Diagrams

753

Fig. 72

3. If two identical lcc or lci occur in the diagram D0 , then only one such lcc or lci will be retained. If two a’s occur in the same minimal region of the diagram D0 , then only one such a will be retained. The resulting diagram is called the Unification of two diagrams and is denoted as Uni(D1, D2). One can observe that the operator Uni is commutative. Also the operator Uni is associative. So, the process of unification can be extended to n number of diagrams, that is, Uni(D1, D2,. . ., Dn). Example 3.1.25 Two diagrams D1 and D2 are united to obtain the diagram Uni (D1, D2). First the closed curves C and D are introduced in D1 to obtain the diagram D as C and D are closed curves of D2 of which there are no counterparts in D1. Next all the diagrammatic objects of D2 are drawn in the respective counterpart regions of the diagram D, obeying the rule of introduction of closed curve. The resulting diagram is Uni(D1, D2) (Fig. 72). Before going any further it will be better to explain clearly what is meant by “all the diagrammatic objects of D2 are drawn in the respective counterpart regions of the diagram D, obeying the rule of introduction of closed curve.” Suppose a closed curve A is introduced in the diagram D2 and the newly obtained diagram is called as D3 (Fig. 73). The diagram D3 is obtained from D2 by using the rule of introduction of closed curve. Thus restrictions 1–6 mentioned in section “Introduction Rules (for Closed Curves, a, a , and x)” are all satisfied. Now if the diagram Uni(D1, D2) (Fig. 72) is considered, one will see that all the diagrammatic objects of D3 are in the respective counterpart regions of the diagram Uni(D1, D2). Figure 74 represents the diagram Uni(D2, D1). If the two diagrams, Uni(D1, D2) (Fig. 72) and Uni(D2, D1) are compared, then one will see that both the diagrams have the same set of labels. Moreover, the diagrammatic objects in the diagram Uni (D2, D1) are also in the respective counterpart regions of the diagram Uni(D2, D1). Thus both the diagrams are identical, that is, operator Uni is commutative. For Type-III Diagram with Type-I or Type-II Diagram If D1–D2 is a type-III diagram and D3 be a type-I/II diagram, then they may be transformed into the type-III diagram Uni(D1, D3) – Uni(D2, D3), by unification rule.

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Fig. 73

Fig. 74

Example 3.1.26 Two diagrams D1 – D2 and D3 are united to obtain the type-III diagram Uni(D1, D3) – Uni(D2, D3) (Fig. 75). For Type-III Diagram with Type-III Diagram If D1 – D2 and D3 – D4 be two type-III diagrams, then they may be transformed into the type-III diagram Uni(D1, D3) – Uni(D2, D3) – Uni(D1, D4) – Uni(D2, D4), by unification rule. Clearly this rule can be extended to the unification of type-III diagrams with any number n1, n2 of components. Example 3.1.27 Two diagrams D1 – D2 and D3 – D4 are united to obtain the type-III diagram Uni(D1, D3) – Uni(D2, D3) – Uni(D1, D4) – Uni(D2, D4) (Fig. 76). Rule of Splitting Sequences Let D be a type-I/II diagram containing a x-sequence or an a-sequence in a region r. Let m1, m2,. . ., mn be all the minimal regions contained in r. Then D may be transformed into a type-III diagram D1 – D2 – . . . – Dn such that (i) Each Di has the counterparts of all the basic regions of D. (ii) Each Di has only one x or an a in the counterpart regions of mi (1  i  n).

28

Logic of Diagrams

755

Fig. 75

Fig. 76

(iii) Any other diagrammatic object in any region s of D will be present in the counterparts c(s) in each Di. (iv) If identical lcc or lci occur, then only one such lcc or lci will be retained in each Di. Example 3.1.28 The diagram D1 – D2 – D3 – D4 is obtained by splitting the x-sequence in the region ((A
B
C) + ((A  B)
C) + (B
A
C) + ((B  C)
A)) of the diagram D (Fig. 77).

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Fig. 77

Fig. 78

Example 3.1.29 The diagram D1 – D2 – D3 is obtained by splitting the a-sequence in the region ((A
B) + (A  B) + (B
A)) of the diagram D (Fig. 78). Rule of Excluded Middle (a) If D is a type-I/II diagram such that there is a minimal region m containing no diagrammatic object, then it may be transformed into a type-III diagram D1 – D2 such that (i) Both D1 and D2 have the counterparts of all the basic regions of D. (ii) The counterpart of m in D1 is shaded and the counterpart of m in D2 has a x. (iii) All the other diagrammatic objects remain the same in D1 and D2 as in D. Example 3.1.30 The minimal region B – A in D has no diagrammatic object. The diagram D1 – D2 is obtained from D by using the rule of excluded middle-(a) (Fig. 79). The diagram D3 – D4 (Fig. 80) can also be obtained from D by applying the rule of excluded middle-(a) in the minimal region
(A + B).

28

Logic of Diagrams

A

757

B a

A

a

B a

A

a

B a

a

x

D1 − D2

D Fig. 79

A

B a

A

a

B a

a x

D3 − D4 Fig. 80

(b) If D is a type-I/II diagram such that there is no a-sequence in D and there is a minimal region m such that m has neither shading nor a then D may be transformed into a type-III diagram D1 – D2 such that (i) Both D1 and D2 have the counterparts of all the basic regions of D. (ii) The counterpart of m in D1 has an a and in D2 has a. (iii) All the other diagrammatic objects remain the same in D1 and D2 as in D. This rule is applicable for any name of the individual. Example 3.1.31 By applying the rule of excluded middle-(b) in the minimal regions –(A + B + C), (B
A
C), ((B  C)
A), and (C
A
B) of the diagram D (Fig. 81) one can get the diagrams in Figs. 82, 83, 84, and 85, respectively. Here this rule is applied for the name of individual a. One can apply the rule of excluded middle-(b) on the diagram D (Fig. 81) for any name of individual except for ‘b’.

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Fig. 81

Fig. 82

Fig. 83

Rule of Construction A wfd D1 – D2 – . . . – Dn may be transformed into a diagram D if each of D1, D2, . . ., Dn can be transformed into D by some of the previously mentioned rules (sections “Introduction Rules (for Closed Curves, a, a, and x),” “Extension Rules (for Lcc, Lci, and Components),” “Elimination Rules (for Lcc, Lci, Shading,  a, and Closed Curve),” “Unification Rule,” “Rule of Splitting Sequences,” and “Rule of Excluded Middle”). If rules like introduction rules (for closed curve, a, a and x), extension rules for lcc/lci, elimination rules (for lcc, lci, shading, a and closed curve), rule of splitting sequences, and rule of excluded middle are considered, then one will see that applications of this rules are only mentioned for type-I/II diagrams. For any typeIII diagram D these rules are applied to its components and the final diagram, say D0 , is obtained from D by applying the rule of construction. The rule of extension of

28

Logic of Diagrams

759

Fig. 84

Fig. 85

Fig. 86

components can be applied to any of the components of D and the diagram D0 is obtained by rule of construction. For example, let us consider the type-III diagram D where D  D1
D2
D3 (Fig. 86). The diagram in Fig. 90 is obtained from the component D1 (Fig. 87) of the diagram D by following steps i–iii below. (i) Introduce a in the shaded regions (Fig. 88) (ii) Eliminate x-sequence (Fig. 89) (iii) Eliminate shading (Fig. 90)

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The diagram in Fig. 96 is obtained from the component D2 (Fig. 91) of the diagram D by following steps i–v below. (i) Eliminate x-sequence (Fig. 92) (ii) Eliminate shading (Fig. 93) (iii) Eliminate closed curve C (Fig. 94) Fig. 87

Fig. 88

Fig. 89

Fig. 90

28

Logic of Diagrams

761

(iv) Introduce a (Fig. 95) (v) Eliminate a-sequence (Fig. 96) The diagram in Fig. 101 is obtained from the component D3 (Fig. 97) of the diagram D by following steps i–iv below. (i) (ii) (iii) (iv)

Eliminate b (Fig. 98) Eliminate closed curve C and D (Fig. 99) Introduce a (Fig. 100) Eliminate a-sequence (Fig. 101)

Fig. 91

A

B

C

Fig. 92

a

A

x

B

C

Fig. 93

x

a

A

B

C

a

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Fig. 94

A

B

a

Fig. 95

A

B a

a a

Fig. 96

A

B a

a

Thus from all the components of D we get the diagram in Fig. 102 and this diagram is the diagram D0 . So by applying the rule of construction we can get D0 from D (Fig. 86). Note that during these steps the same rule may have been used more than once. Let us see how we can get another diagram from D (Fig. 86). The diagram in Fig. 106 is obtained from D1 (Fig. 87) by following steps i–iv below. (i) Eliminate x-sequence (Fig. 103) (ii) Introduce a-sequence (Fig. 104) (iii) Eliminate part of a-sequence (Fig. 105)

28

Logic of Diagrams

763

Fig. 97

Fig. 98

Fig. 99

A

B

a

(iv) Connect diagrams by applying the rule of extension of components (Fig. 106) The diagram in Fig. 108 is obtained from the component D2 (Fig. 91) by following steps i and ii below. (i) Extend a-sequence (Fig. 107) (ii) Connect diagrams by applying the rule of extension of components (Fig. 108) The diagram in Fig. 110 is obtained from the component D3 (Fig. 97) by following steps i and ii below. (i) Apply excluded middle rule-(b) for the name of individual b (Fig. 109) (ii) Connect diagrams by applying the rule of extension of components (Fig. 110)

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Fig. 100

A

B a

a a

Fig. 101

A

B a

a

Fig. 102

A

B a

a

Fig. 103

Thus from all the components of D we get the diagram in Fig. 111 and this diagram is the diagram D0 . So by applying the rule of construction we can get D0 from D (Fig. 86). Note that in the above example also the same rule may have been used more than once.

28

Logic of Diagrams

765

Fig. 104

Fig. 105

Fig. 106

Definition 3.1.6. ρ-Equivalence Let D and D0 be two wfds. Then D ρ D0 holds if and only if there is a sequence of diagrams D1(D), D2, . . ., Dn(D0) such that Di+1 is obtainable from Di (i ¼ 1, 2,. . ., n-1) by one of the previously mentioned rules (sections “Introduction Rules (for Closed Curves, a, a, and x),” “Extension Rules (for Lcc, Lci, and Components),” “Elimination Rules (for Lcc, Lci, Shading, a , and Closed Curve),” “Unification Rule,” “Rule of Splitting Sequences,” “Rule of Excluded Middle,” and “Rule of Construction”). Diagrams D and D0 are said to be ρ-equivalent if and only if D ρ D0 and D0 ρ D. Note that since the unification rule requires at least two diagrams for being applicable, in the definition of ρ this rule does not apply. Inconsistency Rules Definition 3.1.7. Inconsistent Diagram 1. A type-I/II diagram D is said to be an Inconsistent diagram if either of the conditions i–vii hold. (i) There is a minimal region m in D such that it has both shading and a x. (ii) There is a minimal region m in D such that it has both shading and an a.

766 Fig. 107

Fig. 108

Fig. 109

Fig. 110

Fig. 111

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767

There is a minimal region m in D such that it has both a and an a. Every minimal region in D is shaded. Every minimal region in D has a. Some minimal regions in D have shading and rest of the minimal regions in D has a. (vii) ‘a’ occurs in more than one minimal region in D. 2. A type-III diagram D is said to be an Inconsistent diagram if all the components of D are inconsistent in the sense (1). 3. Any diagram D which is ρ-equivalent to an inconsistent diagram of kinds (1) and (2) is an inconsistent diagram. 4. No other diagram is inconsistent. (iii) (iv) (v) (vi)

Example 3.1.32 Here are few examples of inconsistent diagrams of kinds (1) and (2) (Fig. 112). Example 3.1.33 The diagram D1 can be obtained from the diagram D2 – D3 by following steps i–iii below (Fig. 113). (i) Extend a-sequence of the component D2. (ii) Extend a-sequence of the component D3. (iii) Since D1 is obtained from both D2 and D3, D1 is obtained from D2 – D3 by applying rule of construction. Again one gets D2 – D3 from the diagram D1 by applying the rule of splitting of a-sequence. Thus the diagram D1 is ρ-equivalent to the inconsistent diagram D2 – D3 and therefore D1 is an inconsistent diagram. The diagram D4 and the diagram D8 are also ρ-equivalent to the inconsistent diagrams D5 – D6 – D7 and D9 – D10 – D11 – D12, respectively. Thus D4 and D8 are inconsistent diagrams (see Figs. 114 and 115). From the diagram D8 in example 3.1.33, we can say that if a diagram has two asequences having no nodes in any common minimal region, then that diagram will always be an inconsistent diagram. Definition 3.1.8. Consistent Diagram 1. If a type-I/II diagram D is not inconsistent, then D is consistent. 2. A type-III diagram D is said to be a consistent diagram if and only if at least one of the components of D is consistent. Inconsistency Rules (i) An inconsistent diagram may be transformed into any diagram. Example 3.1.34 From the inconsistent diagram D using the inconsistency rule one can get any diagram, for example, one gets the diagram D0 (Fig. 116).

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(ii) A type-III diagram may be transformed into a diagram by dropping an inconsistent component from it. Example 3.1.35 From the type-III diagram D1 – D2 – D3 using the inconsistency rule one can drop the inconsistent component D2 and get the diagram D1 – D3 (Fig. 117).

Fig. 112

a a

a a

a a

a

a

D1

a a

D2–D 3

Fig. 113

Fig. 114

a

a

a

a

a

a

Fig. 115

a

D9

a a

a

a

D8

a

D10

D11

D12

28

Logic of Diagrams

769

Fig. 116

Fig. 117

Definition 3.1.9. Provability or Syntactic Consequence A diagram D is provable from a non-empty, finite set Δ of diagrams (Δ ‘ D) if and only if there is a sequence of diagrams D1, D2, . . ., Dn(D) such that each diagram is either a member of Δ or is obtainable from earlier diagrams in the sequence by one of the rules of transformation. Any rule may be written as D ‘ D0 where D0 is obtained from D by any of the rules. Clearly, if D1 ρ D2, then D1 ‘ D2 holds but not the converse. This is because if D2 is obtained from D1 by using the inconsistent rule then we can say that D1 ‘ D2 but not D1 ρ D2. Diagrams D1 and D2 are syntactically equivalent if and only if D1 ‘ D2 and D 2 ‘ D 1.

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Semantics Definition 3.1.10. Model A model is defined to be a triple (U, I, h), where (i) U is a non-empty set. (ii) I is a function assigning subsets of U to all regions of all wfds such that 1. I(r) ¼ U, whenever r is a basic region enclosed by a rectangle, 2. I(r) ¼ I(s), whenever r and s are two counterpart regions, 3. If r and s are two regions of a diagram D, then I(r + s) ¼ I(r) [ I(s), 4. If r and s are two regions of a diagram D, then I(r  s) ¼ I(r) \ I(s), 5. If r and s are regions of a diagram D, then I(r – s) ¼ I(r) \ I(s). (iii) h is a function assigning objects h(a) of U to the names of individuals a. Note that since h is a total function from the set of the names of individuals to U, if there is a name ‘a’ in the language, then there is a corresponding object ‘h(a)’ in U. It follows from 5 and (Definition 3.1.3) that I(–r) ¼ U \ I(r). Definition 3.1.11. True in a Model Let M ¼ (U, I, h) be a model. A type-I/II diagram D is True in M (denoted by M ⊩ D) if and only if the following conditions i–iv are satisfied.
















(i) If r is shaded, then IðrÞ ¼ 0 (null set). (ii) If r ( m1 + m2 + . . . + mn) has x-sequence, then IðrÞ 6¼ 0 [Iðm1 Þ 6¼ 0 or I ðm2 Þ 6¼ 0 or . . .or Iðmn Þ 6¼ 0] (or being inclusive). (iii) If r ( m1 + m2 + . . . + mn) has an a, then h(a) is an element of the universe and hðaÞ  IðrÞ ½hðaÞ  Iðm1 Þ or hðaÞ  Iðm2 Þ or . . . or hðaÞ  Iðmn Þ ( or being exclusive). (iv) If a is in r, then h(a) 2 = I(r), that is, h(a)  U\I(r). If D is a type-III diagram and let D  D1
D2
. . .
Dn. Then M ⊩ D if and only if M ⊩ Di for at least one of the components Di. Definition 3.1.12. Semantic Consequence Let Δ be a set of diagrams and D be a diagram. D is a Semantic Consequence of Δ and written as Δ  D if and only if D is true in every model in which every member of Δ is true. A rule is sound if and only if for every instance D ‘ D0 of the rule, D  D0 holds. Theorem 3.1.1 Soundness Theorem For any set Δ [ {D} of diagrams, if Δ ‘ D, then Δ  D. Theorem 3.1.2 Completeness Theorem For any finite set Δ of diagrams and any diagram D, if Δ  D, then Δ ‘ D.

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The proofs of soundness and completeness theorems have been omitted here. These may be obtained in a way similar to the corresponding proofs of the system Vennin which is published in (Bhattacharjee et al. 2018).

Vennin Vennin is the non-classical diagrammatic system based on the second interpretation of absence. In this system, although the digram in Fig. 119 follows from the diagram in Fig. 118 but not the converse, that is, if we see a in A that does not necessarily imply that a is in – A (compliment of A). A situation where ‘a 2 = A’ does not necessarily imply that ‘a 
A’ was first mentioned in Choudhury and Chakraborty’s paper “On representing Open universe” (Choudhury and Chakraborty 2012). In this paper they discussed the situation mostly in the context of open universe (to be discussed later in section “Vennio1”). That this situation may also arise when the universe is bounded was discussed in their work (Choudhury and Chakraborty 2013). As we mentioned earlier, in (Choudhury and Chakraborty 2012) the assumption ‘a 2 = A’ does not necessarily imply that ‘a 
A’ was mostly justified in a universe which is open. We would like to point out another context where this situation was explained from the information theoretic point of view. To the query “is a in A?” there may be two responses “a is in A” or “a is not in A.” If the answer is “a is in – Fig. 118

A a

Fig. 119

A a

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A” then it will mean that the answerer knows the location of a which is outside of A. So “a is in – A” is a much stronger information than “a is not in A,” it requires the answerer to have knowledge about the location of ‘a’. But there may be the case where the answerer is uncertain about the exact location of ‘a’ and even if he/she has an idea about the location of ‘a’ he/she may not be able to locate ‘a’ there. Another point is that while “a is not in A” is a negative information, “a is in – A” is a positive information. By the depiction of the absence of a in the diagram we intend to reflect the negative information. This interpretation may also be understood from another angle which is the angle of recursion theory and Turing computability. A set A is recursively enumerable but not recursive if and only if there is an algorithm which when run to determine whether a  A or not will terminate with an answer “yes” if a is in A but will continue for ever if a is not in A. So, for such sets A, given any element ‘a’ of the universe, if a  A, then it would be locatable by a Turning Machine (TM). But if a2 = A, that is, a  A, no TM would locate ‘a’ in the complement – A of A. In this sense a  A does not imply a 
A. However, we take a more general stance in the second interpretation for a. From a  A, there may arise the following possibilities. (i) ‘a’ exists in the complement – A and is locatable by a TM. (ii) ‘a’ exists in the complement – A and is not locatable by any TM. (iii) ‘a’ does not exist at all in the universe of discourse. Case (ii) is the case of above mentioned strict recursive enumerability. Case (iii) may arise when ‘a’ is an empty term such as “Pegasus.” It is known that Free logic deals with empty terms, but comparison between our approach and that of Free logic is beyond the scope of this study. Here we will only mention the differences between Vennin and Venni regarding the rules of transformation and semantics.

The Diagrammatic Language In the language there is no difference between Vennin and Venni. These two systems use the same primitive symbols. The definition of well-formed diagrams also remains the same here. Rules of Transformation One major change between Vennin and Venni is that there is no introduction rule for a in Vennin . So from the diagram D (Fig. 120), there is no rule to get the diagram D0 (Fig. 121). The reason for dropping the introduction rule for a in Vennin is as follows. In Venni, if there is a name of individual ‘a’ in the language, then it definitely has a referent somewhere in the universe. So the diagram D0 (Fig. 121) always follows from the diagram D (Fig. 120). But this is not the case in Vennin . In Vennin if there is a name of an individual ‘a’ in the language, then there will be exactly three possibilities.

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Fig. 120

A

a

Fig. 121

A a

1. The name a has a referent and it is locatable in the universe. 2. The name a has a referent but it is not locatable in the universe. 3. a is an empty term, that is, it does not have any referent. For the case 2 and case 3 one cannot say that the diagram D0 (Fig. 121) follows from the diagram D (Fig. 120). So in Vennin the introduction rule for a is dropped as according to this rule one can introduce a-sequence to any diagram D such that each minimal region of D has a node of this a-sequence. The second difference between Vennin and Venni can be found in the definition of the inconsistent diagram. In Vennin the definition of inconsistent diagram is given as follows. Definition 3.2.1. Inconsistent Diagram 1. A type-I/II diagram D is said to be an Inconsistent diagram if either of the conditions i–v hold. (i) There is a minimal region m in D such that it has both shading and an x. (ii) There is a minimal region m in D such that it has both shading and an a. (iii) There is a minimal region m in D such that it has both a and an a. (iv) Every minimal region in D is shaded. (v) ‘a’ occurs in more than one minimal region in D. 2. A type-III diagram D is said to be an Inconsistent diagram if all the components of D are inconsistent in the sense (1).

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3. Any diagram D which is ρ-equivalent to an inconsistent diagram of kinds (1) and (2) is an inconsistent diagram. 4. No other diagram is inconsistent. In Venni if there is a name of individual ‘a’ in the language, then it definitely has a referent somewhere in the universe thus the following type-I/II diagrams are considered to be inconsistent diagrams (Definition 3.1.7). 1. Every minimal region in D has a (Fig. 122). 2. Some minimal regions in D has shading and rest of the minimal regions in D has a (Fig. 123). But these two diagrams in Figs. 122 and 123 are consistent diagrams in the system Vennin as the name ‘a’ may be an empty term. All the other rules of transformation remain the same for Vennin and Venni. Definition 3.2.2. Provability or Syntactic Consequence A diagram D is provable from a non-empty, finite set Δ of diagrams (Δ ‘ D) if and only if there is a sequence of diagrams D1, D2, . . ., Dn(D) such that each diagram is either a member of Δ or is obtainable from earlier diagrams in the sequence by one of the rules of transformation.

Semantics Definition 3.2.3. Model In the system Vennin model is also defined to be a triple (U, I, h), where U and I remain the same as in Venni. But in this system, h is a partial function assigning objects h(a) of U to some of the names of individuals a. Since h is a partial function h(a) may not be defined for some a. Definition 3.2.4. True in a Model In Vennin , a type-I/II diagram D is True in M (denoted by M ⊩ D) if and only if the conditions i–iii, as mentioned in Definition 3.1.11, along with the following condition iv are satisfied. Fig. 122

A

B a

a

a a

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Fig. 123

775

A a C

a a

B a a

(iv) If a is in r, then either h(a) does not exist in U, or if h(a) exists in U, then h(a) 2 = I(r), that is, h(a)  U\I(r). Consider the diagrams D1, D2 in Fig. 124. Consider a model M ¼ (U, I, h) such that h(a) is not defined. So in that case both M ⊩ D1 and M ⊩ D2 holds. Thus these two diagrams are consistent diagrams in Vennin system (both of these diagrams were inconsistent diagrams in Venni). Definition 3.2.5. Semantic Consequence Let Δ be a set of diagrams and D be a diagram. D is a Semantic Consequence of Δ and written as Δ  D if and only if D is true in every model in which every member of Δ is true. Now let us look at the following Fig. 125. One can find a model M ¼ (U, I, h) such that h(a) is not defined. So the diagram D1 is true in the model M but D2 is not. Thus one cannot necessarily say that D2 follows semantically from D1. Proofs of the following theorems may be found in (Bhattacharjee et al. 2018). Theorem 3.2.1 Soundness Theorem For any set Δ [ {D} of diagrams, if Δ ‘ D, then Δ  D. Theorem 3.2.2 Completeness Theorem For any finite set Δ of diagrams and any diagram D, if Δ  D, then Δ ‘ D.

Vennio1 Before stating the system Vennio1 let us re-visit one important aspect of the previous systems. In both the systems Venni and Vennin the rectangle of the diagram is interpreted as a universe and it is taken to be non-empty. In this universe objects pre-exist without specification of any positive property that each object possesses. In these systems although Fig. 126a is not a wfd, Fig. 126b is.

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A

B a

a

A

B a

a a

C

D1

a

a

a D2

Fig. 124

A

B a

A

B a

a

a D2

D1 Fig. 125

P a

(a)

a

(b)

Fig. 126

The first is not accepted since following the tradition of first-order logic, there has to be at least one predicate in the language and in a well formed formula. On the other hand the second is accepted because in the universe an object named ‘a’ may remain without any predication.

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The system Vennio1 is based on the third interpretation of absence. In this interpretation properties are fundamental, an object exists only with regard to a property. Properties with empty extension or no existential import are also accepted. The syntactic language or the diagrams do not accept to have a bounding rectangle representing universe since no pre-existing objects without at least one specified property is accepted in this ontology. But it will necessarily have a collection of closed curves drawn according to Venn’s method. Interpretation of these curves will be in a collection C of sets (that may include the empty set) without there being any circumscribing, non-empty universal set. This, in turn, implies that there is no absolute complementation Pc of a set P, a complement is always relative to another set Q (say), namely, Q \ P. There is nothing corresponding to Fig. 126b in the new system. One may argue that Fig. 126b can be interpreted as ‘a’ has the property “non-P,” thus there should not be any problem in accepting the stance that each object must have a property. We think that “non-P,” in general, represents exclusion, not the ascription of some property. We want positive ascription of a property. For some properties P, “non-P” might be ascribed as another word in a language, for example, if P is “intelligent,” then non-P turns out to be “dull.” But in most cases such duals “P” and “non-P” will not cover up all objects. In our case the possibility of this kind of a situation would be quite high. Thus the main differences of the system Vennio1 with Venni and Vennin lie in 1. While in the latter two the universe of interpretation is non-empty, in the first one [C the union of collection of the sets of interpretation may be empty. 2. Any name, if it has a referent, must have at least one positively ascribed property. The motivation for this diagram system comes from an earlier work (Choudhury and Chakraborty 2012) by Choudhury and Chakraborty. They proposed to develop a diagram system adopting the assumption that the universe is open, in other words it is in a flux. It is continually changing; new objects may appear and existing objects may disappear. Besides, along with the standard objects, unknown, fictitious, and logically absurd objects should have representations in the diagrammatic language. The language may also accommodate empty names. That work (Choudhury and Chakraborty 2012), however, has several lacunae. In the system Vennio1 it has been possible to remove them to a great extent though not totally. We call this system Vennio1 since this may be considered as a step toward developing the full import of the notion of open universe that we intend to call Vennio as a continuation of (Choudhury and Chakraborty 2012).

The Diagrammatic Language Primitive Symbols In this section the main difference is that while in both Venni and Vennin , there is a symbol ‘ ’, the rectangle that represents the universe, in Vennio1 there is

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no rectangle and objects exist only with regard to a positively ascribed property. The diagram cannot have a bounding rectangle since it will give the impression that objects can exist in the region outside the closed curves, but inside the rectangle, and this region does not represent a positively ascribed property. Another difference is that to form the type-III wfds, in both Venni and Vennin , the type- I/type-II wfds are connected with ‘—’ representing inclusive disjunction between the diagrams. Unlike Vennio1 , in both of these systems the rectangle itself forms a boundary of the diagrams. In Vennio1 the symbol ‘| |’ is used just like brackets are used in first order logic, to remove ambiguities. Type-III wfds are thus formed by connecting type-I/type-II wfds with ‘—’ and ‘| |’ (Fig. 127). So the primitive symbols of the system Vennio1 are as follows: : Closed curve; representing monadic predicate : Shading; representing emptiness x: Cross; representing non-emptiness a, b, c,. . .: Names of individuals (finitely many) a, b, c, . . .: Absence of individual named a, b, c,. . . A, B, C,. . .: Names for closed curve or labels (finitely many) —: Line connecting crosses (x’s) and | | - - -: Broken line connecting individuals (a’s) | |: used to disjoint two type-I or type-II diagrams with the help of a ‘—’ sign. Definition 3.3.1. Region, Basic Region, Minimal Region In Vennio1 , a basic region is only the space enclosed by a closed curve. In both Venni and Vennin, the space enclosed by a closed curve and the space enclosed by a rectangle are considered to be basic regions. In Vennio1, only the disjoint spaces within the closed curves are called minimal regions. In both Venni and Vennin, the disjoint space enclosed by a rectangle, but outside all the closed curves, is also considered as minimal region. So if there are n closed curves, we have 2n
1 minimal regions in a diagram in Vennio1, whereas in the other two systems number of minimal regions is 2n.

Fig. 127

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Well-Formed Diagram (wfd) The diagram in Fig. 128a is a wfd in both Venni and Vennin. But the corresponding diagram, Fig. 128b, is not a wfd in Vennio1 as there is an object with the property A or there is an object in the universe without the property A. This sentence becomes meaningful in the systems Venni and Vennin but not so in the system Vennio1 since the objects without any specified property cannot exist. Similar situations arise for other diagrammatic objects. So whether it is an type-I or type-II diagram, the diagrammatic objects will not occur outside the closed curve. In both Venni and Vennin , a type-II diagram is a diagram with more than one closed curve within a rectangle that can be ordered in a sequence C1, C2,. . ., Cn (say) such that the closed curve Ci divides all the minimal regions obtained from the closed curves C1, C2,. . ., Ci–1 and the rectangle in exactly two minimal regions. In Vennio1, a type-II diagram is a diagram with more than one closed curve that can also be ordered in a sequence C1, C2,. . ., Cn (say) such that the closed curve Ci divides all the minimal regions obtained from the closed curves C1, C2,. . ., Ci–1 in exactly two minimal regions. But there should be exactly one minimal region which is inside Ci but outside the curves C1, C2,. . ., Ci–1 (Fig. 129). Normal Form In Vennio1 , if a type-II diagram D has n curves C1, C2,. . ., Cn (n  2), then the diagram D is said to be in its normal form if it satisfies the following properties:

A

A x

(a) Fig. 128

Fig. 129

x

x

(b)

x

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(i) There are 2n – 1 minimal regions in D. (ii) All possible combinations of the basic regions C1, C2,. . ., Cn taken n together should be represented by these 2n – 1 minimal regions. It may be recalled that in the systems Venni and Vennin the number of minimal regions of a type-II diagram is 2n where n is the number of closed curves (n  2). This is due to the fact that the region within the rectangle and outside the closed curves is also taken into consideration.

Rules of transformation Introduction Rules For Closed Curves In all the three systems one can introduce closed curves in a type-I/II diagram D obeying the restrictions mentioned in section “Introduction Rules (for Closed Curves, a, a, and x).” Although the restriction 1 and the restrictions 3–6 are similar for all the systems, due to the fact that there is no rectangle in Vennio1 , implication of restriction 2 is different in Vennio1 . In Venni and Vennin it was adequate to say that “the newly introduced curve should divide all the minimal regions in D, into exactly two parts.” In Fig. 130, when one introduces the curve C in the diagram D, it (curve C) divides all existing four minimal regions into exactly two parts. Whereas in Vennio1 when the curve C is introduced in the corresponding diagram D0 of D, it (curve C) not only divides all existing three minimal regions into exactly two parts, it also creates a minimal region which is inside C but outside the curves A and B (Fig. 131). This minimal region (C
A
B) is actually the minimal region created by dividing the minimal region (
(A + B)) in two parts in Venni and Vennin . As (
(A + B)) is not a minimal region in Vennio1 , it is not any more sufficient to say that the curve C only divides the existing minimal regions in two parts. One also has to mention that C creates another minimal region which is outside the existing curves in D0 but inside C.

A

B

A

B

C D Fig. 130

D1

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For a Just like closed curves, in all the three systems, a can also be introduce in a typeI/II diagram [section “Introduction Rules (for Closed Curves, a,  a , and x)”]. But since there is no notion of absolute complement in Vennio1 , the condition (ii) in the introduction rules for a has to be changed to “a portion (may be whole) of region r has a-sequence, where r is the union of all the minimal regions in D other than m.” In Fig. 132 the diagram D2 is obtained from the diagram D1 by the rule of introduction of a. No Introduction rule for a In Venni, a type-I/II diagram D0 may be obtained from a type-I/II diagram D by the rule of introduction of a-sequence (Fig. 133) (section “Introduction Rules (for Closed Curves, a, a, and x)”). The reason is that, in Venni, if there is a name of an individual, then it has a referent somewhere in the universe. This is not the case in Vennin or Vennio1 . We have already discussed the reason for Vennin in section “Rules of Transformation.” In Vennio1, neither D1 nor D2 follows from D (Fig. 134).

A

A

B

B

C D1⬘

D⬘ Fig. 131

A

B

a

a D1

Fig. 132

A

a

B

a D2

a

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Fig. 133

Fig. 134

D1 is not a well-formed diagram in Vennio1 system. Although D2 is a wfd in Vennio1 , it does not necessarily follow from D because of the following reasons: 1. The name a might be an empty term. 2. The name a may have a referent but it may exist in a member of C which is not assigned to A, B, or C. (For more detailed explanation check section “Semantics”) For x In Venni and Vennin, universe being non-empty, a type-I/II diagram D may be transformed into a diagram by introducing an x-sequence in D such that each minimal region of D has a node of this x-sequence (Fig. 135). In Vennio1, C can be a collection of empty sets. So in Vennio1, we can introduce x-sequence in a region r in a type-I/II diagram only when there is an a-sequence in r. Extension Rules for lcc and lci In all the three systems one can extend x-sequence as well as a-sequence in a diagram D [section “Extension Rules (for Lcc, Lci, and Components)”]. But in Vennio1, neither x-sequence nor a-sequence can be extended outside the regions created by the closed curves in D, otherwise it will form a non-well-formed diagram. So one gets the diagram D2 from the diagram D1 by rule of extension of x-sequence in Venni and Vennin (Fig. 136) but cannot get the diagram D4 from D3 in Vennio1 (Fig. 137).

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A

A

x

x

D

D⬘

Fig. 135 D0 is obtained from D by the rule of introduction of x-sequence

A

B

x

A

B

x

x

x

x C

C

D1

D2

Fig. 136

Fig. 137

A

x

B

A

x

x

x

x C

B

C

D3

D4

Elimination Rules for Closed Curve Like the extension Rules for lcc and lci similar situation arises in case of elimination rule for closed curve. In Vennio1 , after eliminating a closed curve if any of the nodes of an x-sequence/a-sequence is outside the region created by the remaining closed curves, then the entire sequence is omitted otherwise a non-wfd will result.

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Other Rules of Transformation Rules namely extension rule for components, elimination rules for lcc, lci, shading, and a, unification rule, rule of splitting sequences, rule of excluded middle, rule of construction, and inconsistency rules remain same for all the three systems. Definition 3.3.2. Inconsistent Diagram As mentioned earlier, in Venni if there is a name of individual, say a, in the language then the individual a is in the universe. Thus if every minimal region in a type-I/II diagram D has a (Fig. 138a), then D is an inconsistent diagram. For similar reason, if some minimal regions in D has shading and rest of the minimal regions in D has a (Fig. 138b), then D is also an inconsistent diagram. But both of these diagrams are consistent diagrams in the systems Vennin as a may be an empty term. For similar reason, diagrams in Fig. 139 are consistent diagrams in Vennio1 . Since the universe in the systems Venni and Vennin is considered to be non-empty, if every minimal regions in D is shaded (Fig. 140), then D is an inconsistent diagram. Whereas, in Vennio1 , D0 will be a consistent diagram.

Semantics Before giving the formal definition of models of Vennio1 – diagrams it is necessary to clarify some differences of present semantics with that of Venni and Vennin systems. In both Venni and Vennin, a model is a triple (U, I, h). But in Vennio1, instead of the non-empty set U we take a collection C of sets. If a system has n number of labels, then this C should have n number of sets at the most.

Fig. 138

Fig. 139

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Fig. 140

Fig. 141

A a

D




Suppose we see the absence of a in A in a diagram D (Fig. 141). In Venni system, h being a total function, the referent for the name a, that is, h(a) exists in U. As h (a) 2 = I(A) the interpretation of A, it exists in the complementary set I(
A), that is, U \ I(A). In Vennin system, h being a partial function, h(a) may not exist in U. But if it does exist in U, h(a) 2 = I(A) implies that it exists in the region U \ I(A). Thus, in both of these system, the object h(a) (if exists) not being in I(A) immediately implies that it belongs to the complementary set U \ I(A), and this complement does not represent a positively ascribed property. But in Vennio1 an object exists only with regard to a positively ascribed property. In this system it is necessary to have the collection C of sets where number of sets does not exceed the number of labels that the system in consideration has. So if one observes the absence of a in A in a diagram D (Fig. 142) and the object h(a) exists in [C (see Definition 3.3.3.), then hðaÞ  IðAÞ implies that it exists in some member I (S) of the collection C for some label S other than A. Thus, h(a) will exist in a positively ascribed property. Definition 3.3.3. Model A model to be a triple ðC , I, hÞ, where (i) C is a non-empty collection of sets not exceeding the number of labels of the system (however, some or all the members of C may be null set).

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A

Fig. 142

a

D

(ii) I is a function assigning members of C to all regions of all wfds such that 1. I(r) ¼ I(s), whenever r and s are two counterpart regions. 2. if r and s are regions of a diagram D, then I(r + s) ¼ I(r) [ I(s). 3. if r and s are two regions of a diagram D, then I(r  s) ¼ I(r) \ I(s). 4. if r and s are regions of a diagram D, then I(r – s) ¼ I(r) \ I(s). (iii) h is a partial function assigning objects h(a) of [C to the names of individuals a. Since h is a partial function h(a) may not be defined for some a. Definition 3.3.4. True in a Model Let M ¼ ðC , I, hÞ be a model. A type-I/II diagram D is True in M (denoted by M ⊩ D) if and only if the following conditions are satisfied.
















(i) If r is shaded, then IðrÞ ¼ 0 (null set). (ii) If r (m1 + m2 + . . . + mn) has x-sequence, then IðrÞ 6¼ 0 [Iðm1 Þ 6¼ 0 or Iðm2 Þ 6¼ 0 or . . .or Iðmn Þ 6¼ 0] (or being inclusive). (iii) If r (m1 + m2 + . . . + mn) has an a, then h(a) is an element of [C and hðaÞ  IðrÞ ½hðaÞ  Iðm1 Þ or hðaÞ  Iðm2 Þ or . . . or hðaÞ  Iðmn Þ (or being exclusive). (iv) If a is in r, that is, a is in all the minimal regions in r, then either h(a) exists in [C but hðaÞ 2 = IðrÞ or hðaÞ does not exist in [C . Definition 3.3.5. Semantic Consequence Let Δ be a set of diagrams and D be a diagram. D is a Semantic Consequence of Δ and written as Δ  D if and only if D is true in every model in which every member of Δ is true. The rule of introduction for a is sound in Venni but not so in Vennin and Vennio1. Also the reasons of unsoundness are not the same in the latter two systems. We have already discussed the reason for Vennin in section “Semantics.” Let us discuss the reason for Vennio1 by using an example. Let us consider the diagrams in Fig. 143, diagram D2 is obtained from the diagram D1 by the rule of introduction of a-sequence [section “Introduction Rules (for Closed Curves, a,  a, and x)”].

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Fig. 143

X

X D1

D2

Fig. 144

X

X D3

D4

both C 1 and C 2 are collection of three sets I(A), I(B) and C, For both models IðA
BÞ 6¼ 0. h1(a) does not exist in [C 1 and h2(a) exists in [C 2 but h2(a) is in C.




(i) (ii) (iii) (iv)







In Venni if there is a name of individual a in the language, then there is always an object h(a) in U as h is a total function. Suppose D1 is true in an arbitrary model M and IðA
BÞ 6¼ 0. Then D2 is also true in M as IðA
BÞ 6¼ 0 (I(r) ¼ I(s) whenever r and s are two basic regions that are labeled by the same label) and h(a)  U, that is, h (a)  I((A
B) + (A  B) + (B
A) + (
(A + B)) [as I(r) ¼ U, whenever r is a basic region enclosed by the rectangle (see section “Semantics”)]. Since M is arbitrary, D1  D2 holds. Thus the rule of introduction of a-sequence is sound in this system. In the system Vennio1 the diagrams corresponding to D1 and D2 are D3 and D4, respectively (Fig. 144). Suppose D3 is true in the model M1 ¼ ðC 1 , I1 , h1 Þ and M2 ¼ ðC 2 , I2 , h2 Þ such that

In both M1 and M2, D4 is not true. Therefore, D3 ⊭D4 : Thus the rule of introduction of a-sequence fails to be sound in this system. Similar situations arise in case of introduction of x-sequence. Let us consider the diagrams in Fig. 145, diagram D2 is obtained from the diagram D1 by the rule of introduction of x-sequence [section “Introduction Rules (for Closed Curves, a,  a, and x)”]. Now in Venni and Vennin, U is a non-empty set and I(r) ¼ U whenever r is a basic region enclosed by a rectangle. If M is an arbitrary model such that M ⊩ D1, then D2 is also true in M as I((A
B) + (A  B) + (B
A)) ¼ I(r) ¼ U and U is non-empty. So, the x-sequence rule is sound in both Venni and Vennin .

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Fig. 145

x

x

x x

D1

D2

Fig. 146

x

D3

x

x

D4










In the system Vennio1 the diagrams corresponding to D1 and  D2 are D3 and D4, respectively (Fig. 146). One can find a model M where C ¼ 0 and IðAÞ ¼ IðBÞ ¼ 0. Now M ⊩ D3 but M ⊩D4 . Therefore, D3 ⊭D4 . Thus the rule of introduction of x-sequence fails to be sound in this system. Theorem 3.3.1 Soundness Theorem For any set Δ [ {D} of diagrams, if Δ ‘ D, then Δ  D. The proof of the soundness theorem can be found in (Bhattacharjee et al. 2019b). Theorem 3.3.2 Completeness Theorem For any finite set Δ of diagrams and any diagram D, if Δ  D, then Δ ‘ D. The proof of the completeness theorem can be found in (Bhattacharjee et al. 2019b).

Square of Opposition The traditional square of opposition is a diagram representing categorical propositions, namely, A: All men are mortal (universal affirmation) E: No man is mortal (universal negation) I: Some men are mortal (particular affirmation) O: Some men are not mortal (particular negation)

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The usual pictorial representation which is due to Parson (Sharma 2012) has been the most popular one (Fig. 147). The relations between (A, O) and (E, I) are contradictory, between (A, E) is contrary, and between (I, O) is subcontrary. The relations between (A, I) and (E, O) are subalternation. The general form of the above mentioned propositions are (A) “all P are Q,” (E) “no P is Q,” (I) “some P are Q,” and (O) “some P are not Q.” However, in logical discourse (and also in everyday conversation) there is another type of sentences, namely, “a is Q” (singular affirmative or SA) where a is a proper name. Its negation is “a is not Q” (singular negation or SN). The relationships between the pairs of corners, namely, contradictory, contrary, subcontrary, and subalternation, however, do not hold if the existential import for the subject and predicate terms P and Q are not presumed. For a discussion on this problem we refer to (Sharma 2012). SA propositions are considered in traditional Aristotelian logic as subsumed under type A “since in every singular proposition the affirmation or denial is of the whole of the subject” (Tadeusz 1955) and from this respect cardinality of P does not matter. This assumption has faced serious criticisms dating back from the thirteenth century in favor of considering SA and A propositions as separate types (vide (Khomskii 2012)). Firstly as indicated in (Tadeusz 1955), negation of a universal proposition becomes particular whereas by negating a singular proposition we obtain another singular proposition. Secondly, two propositions are contrary if they cannot be true together but can be false together. Propositions of the forms “All P are Q” (A) and “No P is Q” (E) are contrary. Now, if P is a singleton a, one gets a proposition of the form, namely, “a is Q.” In this case the counterpart of “No P is Q” turns out to be “No a is Q” which is equivalent to “a is not Q” and this is contradictory to “a is Q.” Thus in case of singleton P, sentences “All P are Q” and “No P is Q” cannot be false together and hence the

Fig. 147

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treatment of SA and A propositions alike misses this difference. This does not, of course, mean to say that singular affirmatives have no contraries. For a detail study on the peculiarities of singular propositions, we refer to (Tadeusz 1955). In this paper Czezowski Tadeuz considered above three types of categorical propositions. He has used the phrase “This P” instead of a proper name in order to enable the singular proposition to enter the opposition square. We quote from (Tadeusz 1955): “a distinction ought to be made between singular and universal propositions and that trichotomy into universal (All P is Q), singular (This P is Q) and particular (Some P is Q) propositions should be introduced in place of the customary dichotomy according to quantity, into universal and particular propositions.” Thus emerged a hexagon of opposition (Fig. 148) as an extension of the traditional square as given below. The new relations involving SA (singular affirmation) and SN (singular negation) are the following: SA – SN : contradictory A – SA : subaltern SA – E : contrary SA – I : subaltern SA – O : subcontrary

SN – A : contrary E – SN: subaltern SN – I : subcontrary SN – O : subaltern

Various types of diagrammatic representations at the four corners of the square have been studied in (Bernhard 2008) of which we pick only the Vennrepresentation, namely, Fig. 149. A glance at the counterpart regions immediately show the relations between the pairs of corners as claimed before to be valid. Bernhard in (2008) rightly mentions that “The representation of the four categorical propositions by different diagram systems allows a deeper insight into this structure.” We would like to enhance the diagram by adding corners SA and SN using methods adopted in Venni system. ‘a’ may be considered as the proper name for “This P” and a for “absence of This P.”

A

Fig. 148

E

SN

SA

I

O

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Fig. 149

Fig. 150

Rectangles are used as we have adopted before to represent the universe. Thus the diagram corresponding to Fig. 148 would be Fig. 150. Since two more corners are now added in diagram in Fig. 150 (namely, SA and SN), it would be imperative to see the links between the representation of new (singular) propositions and their relations with the existing square. We follow

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Bernhard’s method in this regard with Venn diagram as base (Bernhard 2008). Following Czezowski (Tadeusz 1955), depicting the singular name “This P” (or this object which is P) by the letter ‘a’ within the circle (P) we have the following all possible pictures corresponding to Gergonne relations (Bernhard 2008). While the sentence “This P is Q” is true in Fig. 151(I) it is false in Fig. 151(II). Thus representations of SA and SN have no diagram common and together exhaust all possibilities, hence these nodes are contradictory. It is to be noted that this second picture is equivalent to the picture at the corner SN of Fig. 150 with respect to the classical interpretation of a. On the other hand the node SA and E cannot be true together since then we get Fig. 152 which say that P \ Q is empty and contains ‘a’. From our cognitive stand point this cannot be. But SA and E can be made false by the situation depicted in Fig. 153. So, SA and E are contrary. Similarly, SA and O are sub-contrary since while they can be true together, they cannot be false for in that case, we have the diagram in Fig. 154. Figure 154 depicts a contradiction. We can similarly show pictorially the subalternate relationship between some other pairs. Note: In all the above pictures a stands for a name for “This P.” All the explanations are based on the assumptions that a is P, and P is non-empty. We, however, are interested in more general cases when ‘a’ is an unqualified proper name which stands for an object without any qualifying property. This will be evident in

P

a

Q

P

Q

a

(II)

(I) Fig. 151

Fig. 152

P

a

Q

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Fig. 153

P

a

Q X

Fig. 154

a

subsequent pictures. In the case of open universe, that is, universe without boundary (cf. section “Vennio1 ”), of course, we shall assume that when an object is mentioned, it is mentioned with at least one qualifying property. Pictorially any letter ‘a’ is placed always within some closed curve(s) in the case of open universe. In the remaining part of this section singular affirmation and singular negation would be dealt with only. The distinguishing characteristics of these statements would be only considered. However, toward the end, for better visual clarity, Euler diagrams would be used in some cases. So, considering one simple object named ‘a’ and one property named ‘P’, there can be the following possibilities: a  P, a 2 = P, a  P, a 2 = P: According to the classical interpretation of absence a 2 = P and a 2 = P are equivalent to a 
P and a 
P, respectively, –P being the absolute complement of P with regard to an universe represented by a rectangle. The picture, in this case, becomes Fig. 155. Let us now recall the second interpretation of the absence of a, that is, a. In this case the principles to be followed are (i) a is in P implies a is in –P (the complement of P). (ii) a is in P does not necessarily imply a is in –P.

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Fig. 155

Fig. 156

(iii) a is in P and a is in P are contradictory. (iv) Whereas a cannot be present in more than one minimal regions, a can. If a in P is true or equivalently, a in P is false it becomes uncertain whether a is in the complement of P (relative to some universe) or not. Thus besides true (T) and false (F), a third category “uncertain” (U) steps in. The relation expressed through the traditional square of opposition is one of truthfalsity relation. The four assertions a is in P, a is in P, a is in –P, and a is in –P now enter into a relationship as shown in the diagram in Fig. 156. The situation in Fig. 156 may have a classical analogue by taking two disjoint properties P and Q and adopting the first (classical) interpretation of a. To depict disjoint P and Q, we shift from Venn to Euler diagram (Fig. 157). The similarity in Fig. 156 and Fig. 157 may be marked. This similarity indicates that the second interpretation of a may be understood in terms of the first interpretation of a and considering two mutually disjoint properties. So, at this step, complement of P is not treated as the entirety outside P but which is known.

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Fig. 157




Fig. 157 is classical because of the equivalences of pairs of diagrams at the two lower corners of the square. Yet one can see that uncertainty creeps in. For example, if a  Q is true, then a  P is uncertain. That is, from the information that a is not in Q one is not certain about whether a is in P or not. Now, if the above mentioned classical equivalence is not stipulated, one can observe the nine different possibilities as shown in Fig. 158. The diagram of opposition for all the nine cases mentioned above is now summed up in Fig. 159 (only the respective numbers are shown) with bold line representing contradictory relation, normal line representing contrary relation, and dotted line representing subcontrary relation. However, all the relations are not shown. In the context of open universe the notion of absence becomes more significant. The motivation for open universe has been discussed in section “Vennio1”. When an object appears in the universe it appears along with a property being positively ascribed, that means every object must exist within a closed curve. When an object a disappears from the extension P of some property, a appears in P. However, this dynamics is neither depicted in the diagram nor captured in the formal theory that has been developed. We only consider a static slice of the changing universe in which absolute complement of a set is not meaningful. Pictorially this needs only to remove the boundary rectangle from the diagram and carry out necessary adjustments. It may be noticed that the categorical propositions A, E, I, and O do not need the existence of a universal set for their cognitive import. Difficulty arises only in case of representing negations of singular propositions in the classical way that is by not using a sign for “absence of a.” Since we have incorporated a, for representing that a is not in P, existence of the boundary rectangle is not required either. In case of open universe absence of a in P does not imply that a is in the complement of P simply because the complement does not exist as the universe is open. But absence of a in P that is a in P implies two possibilities: a is in some region Q such that P \ Q ¼ 0 or a is nowhere which in turn means that a is absent also in any other disjoint closed curve Q whenever Q is depicted. Thus, the assumptions here are:

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Fig. 158 Fig. 159

1. The boundary of the Universe is unknown. 2. a and a (absence of a) are to be present always within some extensions (represented by closed curves).

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Fig. 160




3. a cannot be present in two disjoint locations (represented by minimal regions). 4. a can be present in two disjoint extensions and even in all known mutually disjoint locations (represented by mutually disjoint regions). 5. a is in region P implies that a is in region Q and P \ Q ¼ 0 or a is not depicted at all. Figure 160 shows the mutual relationship between various pairs with one individual ‘a’ and two disjoint predicates P and Q in open universe (following the same convention with lines and numbering as in Fig. 159). The difference between Fig. 160 involving open universe and that of closed universe (Fig. 159) may be noted. There are five cases instead of nine. As in Fig. 159, this picture also is incomplete, the relations between all pairs are not depicted. Interested readers may try to complete.

Conclusion Study of diagram logic has taken a turn toward what is now called abstract syntax. This kind of work is bent on abstract algebraic presentation of diagram in terms of mathematical tuples containing abstract entities like “inside region,” “outside region,” etc. (Howse et al. 2001, 2005). In our opinion this approach goes against

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the very purpose of this branch of research, namely, pictorial representation of the concepts and to convey the meaning as directly as possible. Also by diagrams communication may be pursued among subjects of varied linguistic cultures. Euler diagram in this respect scores the highest. However, to express more contents the extreme simplicity of the diagram can not be retained, one needs symbolic pictures that require interpretation. It follows that a trade-off or balance is sought for between simplicity and expressiveness. Introduction of shades and cross, that is, a mix of Venn and Peirce, has been very elegant and useful in this respect. Euler-Venn-Peirce symbiosis often gives good results. Our presentation appears to be quite successful in this regard. There has been some statistical research on the usefulness of the notion of absence of an object ‘a’ and its representation by ‘a’ (Stapleton et al. 2016, 2017). We believe that this type of statistical experiment should be carried out further on subjects with different cultural backgrounds before coming to a final conclusion. Our future research has one direction aimed toward this end. Besides, the issue of decidability of the systems Venni, Vennin, and Vennio1 is also a very important issue that needs thorough investigation.

References Allwein, G., and J. Barwise, eds. 1996. Logical reasoning with diagrams. New York: Oxford University Press. Bernhard, P. 2008. Visualizations of the square of opposition. Logica Universalis 2: 31–41. Springer. Béziau, J.Y., and P. Gillman, eds. 2012. The square of opposition: a general framework for cognition. Bern: Peter Lang. Bhattacharjee, R., M.K. Chakraborty, and L. Choudhury. 2018. Venn diagram with names of individuals and their absence: a non-classical diagram logic. Logica Universalis 12: 141 . Springer. Bhattacharjee, R., M.K. Chakraborty, and L. Choudhury. 2019a. A diagram system extending the system venn-II, presented at eighth Indian conference on logic and its applications, IIT Delhi. Bhattacharjee, R., M.K. Chakraborty, and L. Choudhury. 2019b. Vennio1: A diagram system for universe without boundary. Logica Universalis 13: 289–346. Springer. Burton J., Chakraborty M., Choudhury L., Stapleton G. 2016. Minimizing Clutter Using Absence in Venn-ie. In: Jamnik M., Uesaka Y., Elzer Schwartz S. (eds) Diagrammatic Representation and Inference. Diagrams 2016. Lecture Notes in Computer Science, vol 9781. Springer, Cham. https://doi.org/10.1007/978-3-319-42333-3_9 Choudhury, L., and M.K. Chakraborty. 2004. On extending Venn diagram by augmenting names of individuals. In Diagrammatic representation and inference, ed. A. Blackwell et al., 142–146. Berlin: Springer. Choudhury, L., and M.K. Chakraborty. 2005. Comparison between spider diagrams and Venn diagrams with individuals. In Proceedings of the workshop Euler Diagrams 2005, INRIA, Paris, pp 13–17. Choudhury, L., and M.K. Chakraborty. 2012. On representing open universe. Studies in Logic 5 (1): 96–112. Choudhury, L., and M.K. Chakraborty. 2013. Singular propositions and their negations in diagrams, published in the proceedings of DLAC 2013. In CEUR workshop proceedings, Vol. 1132. http:// ceur-ws.org/

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Choudhury, L., and M.K. Chakraborty. 2016. Singular propositions, negation and the square of opposition. Logica Universalis (Copyright 2016 Springer International Publishing), 10 (2–3):215–231. Datta, S. 1991. The ontology of negation. Jadavpur studies in philosophy, in collaboration with K. P. Bagchi and Co., Kolkata. Euler, L. 1768. Lettres ‘a une Princesse d’Allemagne. St. Petersburg: l’Academie Imperiale desSciences. Gil, J., J. Howse, and S. Kent. 1999. Formalizing spider diagrams. In Proceedings of the IEEE symposium on visual languages (VL 99), Tokyo, pp 130–137. Hammer, E. 1995. Logic and visual information. Stanford: CSLI Pubs. Howse, J., F. Molina, J. Taylor, S. Kent, and J. Gill. 2001. Spider diagrams: a diagrammatic reasoning system. Journal of Visual Languages and Computing 12 (3): 299–324. Howse, J., G. Stapleton, and J. Taylor. 2005. Spider diagrams. LMS Journal of Computation and Mathematics 8: 145–194. London Mathematical Society. Khomskii, Y. 2012. William of Sherwood, singular propositions and the hexagon of opposition. In The square of opposition. A general framework for cognition, ed. J.Y. Béziau and P. Gillman, 43–59. Bern: Peter Lang. Ma, M., and A. Pietarinen. 2017a. Graphical sequent calculi for modal logics. Electronic Proceedings in Theoretical Computer Science 243: 91–103. https://doi.org/10.4204/EPTCS.243.7. M4M9 EPTCS 243. Ma, M., and A. Pietarinen. 2017b. Gamma, graph calculi for modal logics, Synthese. Springer. https://doi.org/10.1007/s11229-017-1390-3. Moktefi, A., and S.J. Shin. 2012. A history of logic diagrams. In Handbook of the history of logic, vol. 11, 611–682. Amsterdam: Elsevier. Peirce, C.S. 1933. Collected papers of C.S. Peirce. Vol. iv. Cambridge, MA: Harvard University Press. Sharma, S.S. 2012. Interpreting square of oppositions with the help of diagrams. In The square of opposition. A general framework for cognition, ed. J.Y. Béziau and P. Gillman, 174–192. Bern: Peter Lang. Shin, S.J. 1994. The logical status of diagrams. Cambridge, UK: Cambridge University Press. Stapleton, S. 2005. A survey of reasoning systems based on Euler diagram. In Proceedings of the first international workshop on Euler diagrams, Vol. 134, 127–151. Stapleton, G., & Masthoff, J. 2007. Incorporating negation into visual logics: A case study using Euler diagrams. In P. Maresca, Y. Khalifa, & X. Li (Eds.), Visual languages and computing 2007 (pp. 187–194). Skokie: Knowledge Systems Institute. Stapleton, G., J. Howse, and J. Taylor. 2005. A decidable constraint diagram reasoning system. Journal of Logic and Computation 15 (6): 975–1008. Stapleton, G., J. Howse, J. Taylor, and S. Thompson. 2009. The expressiveness of spider diagram augmented with constants. Journal of Visual Languages and Computing 20: 30–49. Stapleton, G., A. Blake, L. Choudhury, M. Chakraborty, and J. Burton. 2016. Presence and absence of individuals in diagrammatic logics: An empirical comparison. Studia Logica (2006) 82:1–24. Stapleton, G., A. Blake, J. Burton, and A. Touloumis. 2017. Presence and absence of individuals in diagrammatic logics: An empirical comparison. Studia Logica 105: 787–815. Springer. Tadeusz, C. 1955. On certain peculiarities of singular propositions. Mind 64: 392–395. JSTOR. Venn, J. 1880. On the diagrammatic and mechanical representation of propositions and reasonings. The London, Edinburgh, and Dublin Philosophical Magazine and Journal of Science 10: 1–18. Venn, J. 1881. Symbolic logic. 1st ed. London: Macmillan.

Part V Comparative

Abhinavagupta on Śa¯nta Rasa The Logic of Emotional Repose

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rasa in the Absence of Emotions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Serenity Emerges from Cessation (Nirodha): Position I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position II: Serenity Emerges from Dispassion (Nirveda) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Position III: Serenity Emerges from Self-Awareness (Ātmajñāna) . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Phenomenology of Savoring Serenity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter explores the ways Abhinavagupta, an eleventh-century Kashmirian polymath, establishes the experience of serenity (śānta) as one of the appraised emotions called rasa. Beyond the issue of whether serenity can be the savoring of rasa, this chapter explores various models from classical Hindu and Buddhist philosophies that establish serenity in order to contextualize the phenomenology of experiencing serenity. For Abhinava, this experience is not a mere negation of emotions but a positive experience. And to establish his argument, Abhinava explores the ways absence is analyzed in Hindu and Buddhist traditions. One of the central problems of aesthetics that overlaps metaphysics is whether the experience of serenity is identical to the experience of liberation. Abhinava paves his path through the middle, without collapsing this experience to the mystical experience of the Brahman or to common everyday experiences. By rejecting the argument that serenity is a product of cessation or that dispassion evolves into serenity, Abhinava argues that serenity emerges from self-awareness.

S. Timalsina (*) Religious Studies, San Diego State University, San Diego, CA, USA © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_49

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Keywords

Abhāva · Abhinavabhāratī · Abhinavagupta · Ānandavardhana · Ātmajñāna · Bharata · Bhāva · Dhvanyāloka · Locana · Lokottara · Nirodha · Nirveda · Rasa · Śānta · Sthāyī bhāva · Viśrānti

Introduction Classical Hindu rasa aesthetics rests on emotions, their commixture, and their appraisal. Central argument for rasa is that similar to relishing sweet and sour dishes, people savor emotions when they are harmonized and aesthetically evaluated. Furthermore, it is not just the positive emotions such as erotism, heroism, comic sense, or wonder, argument goes; even the negative ones such as disgust, fear, or fury can be aesthetically evaluated and savored. This, however, is not to say that every layman is entitled to savoring rasa, as it requires a connoisseur for it. As it comes to the savoring of śānta or serenity, there are unique problems that even the rasa theorists could not agree upon. Some do not consider serenity as a distinctive rasa. Even among those who consider this as a separate rasa, there is no agreement upon what counts as enduring bhāva for the surge of serenity. Furthermore, the very notion of sthāyī bhāva as enduring emotion is to be questioned here, for all the portions to be presented in the following sections demonstrate śānta is not an exalted form of some emotion, infused with some other emotions. If what is meant by śānta is a form of tranquility, savored by the enlightened beings such as the Buddha, what type of savoring would there be? Even in the rasa paradigm, if this experience parallels the mystical experience of the state of liberation, what factors would there be to separate these two modes of aesthetic and mystical experiences? In order for me to engage some of these issues, it is first required to determine the classical parameters in which the savoring of śānta has been delineated. And for that, this essay will rest on arguments upon concepts surrounding the position of Abhinavagupta, a prominent eleventh-century Kashmiri polymath. The real challenge here is in deciphering rather than tracing Abhinava’s words to read what he has said without words, or what he suggests (dhvanita) without a literal statement. Abhinavagupta’s arguments in defense of serenity (śānta) as a rasa, in addition to the standard eight rasas, are scattered in two of his commentarial works: Abhinavabhāratī (ABh) upon the Nāṭyaśāstra of Bharata and Locana upon Ānandavardhana’s Dhvanyāloka. In my reading, the savoring of serenity the way Abhinava has argued is categorically distinct from the savoring of other rasas. There is an even deeper resemblance of this experience with the liberating experience as that which is being evaluated for the emergence of serenity is not akin to other enduring emotions, whether someone were to follow Abhinava or the other aestheticians. Most of the classical rasa theorists present serenity as transcending all emotions. This chapter will conclude with a position that the state of serenity needs to be appraised for it to be rasa; however, the determining factors for its surge are not emotions. Rather than tracing back to śānta, this essay considers it as

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the foundational being, the state of the self within its own immanence, having the latency for the surge of any of the rasas. And what constitutes liberation in the Trika and Pratyabhijñā paradigm is not the blissful state that contrasts everyday experience, not pure consciousness that negates its being in the world, and not a pure being that juxtaposes the manifold. The intention of this chapter is therefore to read śānta as a philosophical consequence of the aforementioned schools and not as an aberration.

Rasa in the Absence of Emotions Even if śānta is not an emotion like the others, it is the foundation for the emotions to emerge. This understanding contrasts with those that define śānta as emerging from the lack of emotions. This is not to claim emotionality of śānta but merely arguing that experiencing serenity does not need to be juxtaposed with emotional being. This understanding rests on my reading of Abhinava’s treatment of self-experience and serenity. But before addressing this position, it is necessary to delineate the presuppositions that Abhinava rejects. Of primary concern are the ways in which negation is understood in different philosophical schools: to begin with, if serenity emerges from the state of cessation, whether this cessation is confirmative, negation of x as (i) confirming the substrate of x, or (ii) it confirming something else (~x ¼ y), the confirmation of y, or if it is a form of absolute negation – the negation of x not confirming anything. Although Abhinavagupta does not cite distinct philosophers when addressing śānta rasa, it is clear when one reads his commentaries that he anticipates Nyāya/Mīmāṃsā and Sautrāntika/Mādhyamika presuppositions regarding negation. Reading the Abhinavaguptian logic of aesthetics demands reading between the lines, to explicate the positions that he critiques without identifying them. And this takes one to reconsidering negation even though Abhinavaguptian aesthetics of serenity do not rest on evaluating negative contents. From the Nyāya standpoint, there are two types of absence – relational absence or saṃsargābhāva and difference or anyonyābhāva/bheda. The first or relational absence is threefold: antecedent absence (prāgabhāva) or the absence of effect prior to its emergence; consequent absence (pradhvaṃsābhāva) or destruction; and absolute absence (atyantābhāva). The second type of absence, difference or bheda, relates to the absence of x in y. It is crucial to understand the structure of negation in the classical philosophical context in order to contextualize serenity as an aesthetic savoring. For, only when one keeps the structure of absence in mind, he can contextualize the absolute absence of emotion, the cessation that in all likelihood is the Prāsaṅgika interpretation of serenity, in contrast to the absence of the flux of emotion in itself being a form of emotion, or the termination of the turmoil revealing the surface wherein the emotions surge. The position that Abhinava maintains and the one that he reconciles with are also divided on interpreting serenity, whether this serenity is antecedent to the surge of emotions – a position cited without crediting to anybody – or, Abhinava’s position, of a consequent absence, the return of the mind to the serene state, liberated from emotional flow.

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The first issue, in light of the above conversation on absence, is to locate the phenomenal content of serenity so that one can determine whether it is a distinct form of being or presence, or a sheer absence that one confronts when experiencing serenity. When engaged his interpretation of “dispassion” (vairāgya), Abhinava explores both the possibilities of reading dispassion as having a positive content or the lack thereof. By creating a hierarchy of dispassion where the first is simple frustration/depression and the second the exalted form of self-recognition, Abhinava argues that serenity is a presentive consciousness that stems from the enduring mode of self-awareness. Thus, to argue that what lies at the heart of evaluating serenity is its phenomenological content. And the fundamental divide is whether this is an exalted or elevated form of absence or a positive experience of directly encountering the self. Questions associated with this premise are whether serenity can be aesthetically appraised, whether an underlying basic enduring emotion evolves into rasa, or if śānta is an absence of emotion. Above all, it asks whether sheer lack can be considered as phenomenal content, or if absence of content is itself the content that is transformed into rasa. Some scholars believe that Abhinava failed to reconcile his aesthetics with his metaphysics. His treatment of serenity as emerging from self-awareness while keeping it within the fold of aesthetic experience, separating it from the experience of self-realization, does make one wonder. Gerow (1994), for instance, argues, “śānta rasa represents a challenge to Abhinava’s philosophical position, as well as to his aesthetics.” The problem begins when a vertical split between the aesthetic and the mystical is anticipated or when one assumes that the serenity experienced by a liberated being is qualitatively distinct from that savored in the ordinary sense. If read Abhinava closely, he is not only merging the aesthetic with the mystical, he is also keeping the everyday experience within the domain of self-realization. This is not a philosophy that contrasts the world with the recognition of the self. Nevertheless, one can argue, Abhinava does categorize these experiences, aesthetic and mystical, as transcendent or outside the scope of the sensory faculties (lokottara). This argument, however, does not prove that these two experiences are categorically different, quite the opposite. Even this stratification is merely a process of bracketing so that the most intricate modes of experience are revealed without supporting a dichotomy. To begin with, saṃsāra and nirvāṇa are not two opposites in his philosophical paradigm. This issue will be pursued further in the last section. Nonetheless, Abhinava does not make the experience of serenity identical with the recognition of the self. Even while they both are qualitatively similar, liberation stands for termination of all the afflictions (kleśa), while the aesthetic experience of serenity relies merely on a transitory escape from these factors. From the TrikaPratyabhijñā perspective, liberation is nothing but absolute freedom, and experiencing serenity is merely one of its consequences. In essence the aesthetic experience of serenity is a window through which one can measure the state of liberation. Everyday experiences reoccupy the mental landscape even after savoring serenity. In contrast, self-realization is considered final, with no afflicting emotions further conditioning the subject. By shifting his attention from cessation and the factors that condition experience to making self-awareness an enduring mode of being that

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culminates into the savoring of śānta, Abhinava is guiding us from the metaphysics that explains liberating experience merely in negative terms. For Abhinava, any emotion needs to be aesthetically evaluated for it to evolve into rasa. It needs to have camatkāra, the reflexivity intrinsic to consciousness. And this thesis provides the platform for a departure from nirodha-based aesthetics that lack reflexive evaluating modes for the appraisal of emotions. It is this reflexive mode embedded with savoring that allows Abhinava to establish self-awareness as the foundational mode. To feel serenity is not just feeling serene but enjoying serenity, according to Abhinava. In this paradigm, the very surge of serenity becomes savoring in the form of rasa. Most fundamentally, serenity is not a resting ground or metaphoric cremation ground of emotions but a transitory station for the emotions to repose and be stimulated again. While Abhinava explains serenity as the resting station, he agrees with the argument that this may be the fountain for emotions to spring forth. If serenity were a final repose to be experienced by liberated beings alone, this would not be yet another rasa, an experience accessible to all the connoisseurs. Śānta rasa, as a window for liberating experience, also retains its higher status by having self-recognition as the mode of experience or emotional state that transforms into rasa. Noteworthy also is that the experience of the self is not a cognitive mode oriented toward an object but a unique kind of reflexive consciousness. At the same time, the experiencing that is equated with the self is also an emotional state. If bereft of emotion, this experiencing of the self could not be elevated into rasa. Abhinava’s formulation that the very self-experience functions as the enduring state that transforms into śānta is problematic for several philosophical schools. For instance, when Nyāya philosophers maintain the state of liberation as a total termination of suffering, this liberated state is described in terms of negation. This is a qualitatively distinct state to that which Abhinava maintains by classifying selfexperience as the basis for the surge of śānta. In a total lack of consciousness, the liberated state that the Nyāya philosophers have maintained, no aesthetic judgment is possible. On the other hand, although the Advaita of Śaṅkara maintains pure consciousness as the basis to which the self-realized subject returns to, even being in this state cannot be complemented with the aspect of “savoring.” What makes Abhinava’s thesis unique is he makes the exposition on śānta the foundation for explaining the phenomenal content of the liberated state, arguing that the self is constantly savoring its own being while at the same time liberating experience is immanent and embodied. Furthermore, Abhinava flattens the ground, making aesthetic judgment of serenity accessible to all the connoisseurs even while maintaining that this is transcendent (lokottara) and in that regard akin to liberating experience. There apparently is an overlap between everyday experiences, aesthetic, and mystical ones, and Abhinava finds the savoring of serenity a bridging point. The only departure is, rather than making serenity a place to return to, this essay is reading it as a state filled with the potential for the surge of emotions. It is not the lack of feelings or their surge in various forms of emotions that defines liberation. From the Abhinavaguptian perspective, it is the absolute freedom (svātantrya), being in the state of Bhairava, as what liberation is all about. Everyday experience is not

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subordinate due to its emotionality or objectivity. It rather is the limitation, a separation from the realm of experience, that makes subjects finite. Absolute freedom allows liberated subjects to exercise their being in the world, savoring feelings and emotions without being conditioned by them. If śānta is understood as repose, a tranquil state that leads to rest, reading from Latin repausāre, this concept of reposing makes it possible to keep returning to this state. The concept of śānta is captured in Abhinava’s analysis of viśrānti, repose. This essay concurs with Arindam in explaining viśrānti as a polymorphic term. As he suggests, repose is both an ontological and epistemological category, and the term is used to refer to the self, natural fullness, self-savoring creative leisure, freedom, and also the absolute citi or consciousness deified (Chakrabarti 2006, p. 294). This essay is exploiting the freedom aspect of viśrānti in arguing that subject’s freedom to experience repose is not final and can continue to savor this state. When read this as qualitatively similar to the state of liberation, this freedom of experiencing the world is not to be terminated from the free subjects. Along these lines, just as the absolute self retains the potential to give rise to the manifold, so also śānta has the potential for the surge of emotions hidden. Rather than making śānta emotionally empty, this reading makes it pregnant with all the future expression of emotion. To say that serenity is the most natural of all the emotional states is also to say that other emotional states are its transformation. This is similar to saying that white light contains all wavelengths of visible light. Furthermore, if aesthetic savoring is yet another expression of bliss (ānanda), this cannot be expunged from the self, with bliss being its inherent nature. If it is the repose to the self, the reflexive anchoring of the self within itself, that constitutes joyousness, this is the same joyousness that is expressed in savoring rasa. Serenity, along these lines, is the foundation of all emotions, and there is something inherent to it that, given the circumstances, emotions can evolve. The hypothesis above is merely an extension of the position that Bharata has outlined: svaṃ svaṃ nimittam āsādya śāntād bhāvaḥ pravartate| punar nimittāpāye tu śānta eva pralīyate||. (Nāṭyaśāstra, Chap. 6, p. 335)

Corresponding emotions emerge from the basis of serenity when appropriate conditions are met, and when those conditions are removed, they dissolve back to serenity again.

What is the catalyst for the surge of serenity? According to Abhinava, it is “the cognitive state that lacks the surge of any specific mode” (anupajātaviśeṣāntaracittavṛttirūpa. Locana, p. 391). In other words, a non-specified cognitive mode, or mental state lacking specific direction, functions as the source or the enduring mode (sthāyibhāva) for the surge of serenity. The general translation of bhāva as emotion fails here, as “a non-specified or non-directional state of the mind” is not what one anticipates when addressing emotion. To be discussed in the next sections, if serenity is the emergent property of the very absence of emotions, or even the lack of

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cognitive activity, they do not fit to be called emotional state. Also noteworthy is, for Abhinava, this still is a “mental modification” (cittavṛtti) that surges into the savoring of serenity. Bhāva, in these accounts, is simply a mode of consciousness, a particular becoming. And this is neutral as it comes to emotional or cognitive or volitional modes. This conversation started addressing absence. When engaged what the classical aestheticians considered as the enduring mode (sthāyibhāva) for the surge of serenity, the discussion of absence becomes essential. What is it that is lacking and what the lack is like when evaluated the qualitative state of experiencing non-being are relevant questions in addressing serenity. According to the passage of Bharata cited above, all the modes of being, or the expressions of emotions, spring forth from this foundational serenity. If this is the case and if serenity is an extension of some lack, the tension of a lack giving rise to something phenomenal, the state of śānta being the source for the surge of different bhāvas, needs to be resolved. If the state of serenity is from which bhāvas unfold, this would be the state having an antecedent absence (prāgabhāva) or the absence of effect prior to its emergence. On the other hand, if this is the state that emerges upon cessation of emotions, this would have a “consequent absence” (pradhvaṃsābhāva) of emotions. When doing mere phenomenology, there may not be any qualitative difference in the state no matter the particular type of absence. This, however, is crucial metaphysically. It is not the same to say it is serenity from which all the emotions spring forth as it is to say it is in serenity that all emotions dissolve. The reading of śānta in this essay along the lines of Bharata parallels the opening and closing or unmeṣa and nimeṣa, the two metaphors for the cosmogony and cosmic dissolution in the Spandakārikā (I.1). If the cosmic unfolding of the self is a constant cyclical process, emotionality of the self is likewise the same. There is no need for a final suspension of the eternal rejuvenation, splashing into the surge of rasas, even if some are bitter and sour. Aesthetic savoring can therefore be even better than yogic savoring, as in the first the subject is actively evaluating, appraising his cognitive and emotional state while undergoing the surge of emotions. It cannot be said the same about the second.

Serenity Emerges from Cessation (Nirodha): Position I Abhinava cites one position that the enduring emotion for śānta is the cessation of all the cognitive modes (sarvacittavṛttipraśama. Locana, p. 390). According to this perspective, the absence of passion or aversion functions as subsequent emotions (anubhāva) to complement the emergence of serenity. Abhinava’s objection, as mentioned in the introductory section, is that if this absence of cognitive functions is considered to be an absolute negation, it will not amount to being a bhāva, a cognitive/emotional modification of consciousness. On the other hand, if this is a relational absence, confirming y by means of negating x, Abhinava finds this identical to his own position. By addressing negation, Abhinava not only distinguishes his aesthetics of serenity from that of the Buddhist philosophers, he also affirms his model of liberation as having positive content. Both the Buddhist and

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Hindu aestheticians maintain that savoring of serenity mirrors the penultimate experience of liberation. However, Abhinava rejects the experience of cessation (śama) as an enduring mode for the surge of serenity. He interprets this as absence, a sheer non-being that lacks the content to be savored. In essence, the difference relates to the qualitative state of serenity that mirrors the qualitative state of liberating experience. To say that there is a phenomenological content in cessation would be tantamount to saying that the absence of cognitive content is somehow revealed to consciousness. If the argument is that cessation as a cognitive state is given to consciousness, this would contradict the position that the state of nirvāṇa lacks any inherent nature (svabhāva). Abhinava’s critique of cessation needs to be read in this light. There are other problems embedded in the position that cognitive or emotional cessation transforms into serenity. If this state of cessation were to function as an enduring mode or emotion for the emergence of serenity, there needs to be a distinctive mode of consciousness to evaluate this mode, and the subject savoring serenity could not merely be experiencing cessation. In absence of any cognitive function, this appraisal seems impossible. If the state of serenity is nothing but the cessation of cognitive functions, there literally is no distinction between cessation and serenity. Furthermore, the absence of particular cognitive functions cannot in themselves be constitutive for the emergence of something positive, as serenity is a positive experience and the lack of passion and aversion cannot be considered transient factors in the surge of rasa. Abhinava argues that positive qualitative content is what determines psychological states, including the states of deep sleep or being unconscious, as these states are determined on the basis of deep breathing or collapsing (Abhinavabhāratī, Chap. 6, p. 333). The argument is, in absence of any positive content, the state of serenity cannot be confirmed. Noteworthy in this conversation is all the examples he gives are intersubjective, as these relate to subjects determining the psychological states of some other subjects. In the case of cessation, or even more when in savoring cessation in the form of serenity, no external symptoms are visible. And if this identification is not even based on the very subject having experience, referring to the savoring of serenity, this state cannot be subjectively confirmed either. The issue of the absence of experience in cessation also appears in Locana where Abhinava argues that if the absence of cognitive functions is negative in nature, i.e., referring to absolute absence, no cognitive function can correspond to this state, including the aesthetically evaluated experience of serenity. As far as enduring an emotion is concerned, if cessation is explained in terms of lack, this absence cannot be transformed and thereby judged to be aesthetically serene. Abhinava therefore takes the option that negation in the cognitive state of cessation is reciprocal, affirming y by means of the negation of x. If interpreted along these terms, Abhinava collapses this model within his own (Locana, p. 390).

Position II: Serenity Emerges from Dispassion (Nirveda) Unlike cessation of the cognitive function, Abhinava considers dispassion as an active positive emotional state, a state in which the subject actively rejects the sway of the external stimuli that condition his subjectivity. Abhinava, however,

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understands dispassion (nirveda) in two different ways. The first is the emotion that can be better understood in terms of aversion or apathy. This relates to an acute state of frustration. There is no problem with this state being emotional and being complementary to other enduring emotions. The other, dispassion, Abhinava argues, is the culmination of self-awareness, the insight that keeps the subject lacking any desire to be stimulated by the external objects. This is where the understanding of bhāva in terms of enduring emotion collapses. Abhinava rejects the first model of dispassion while subsumes the second model within his own system. The first position is more compatible with the Advaita of Śaṅkara, while the second position relies on the Pratyabhijñā system. This is how Abhinava dialogues with other systems even without mentioning them for once. When dispassion is interpreted in the first sense, or in terms of apathy, Abhinava synthesizes this position that makes dispassion as the enduring mode that evolves into the savoring of serenity: tṛṣṇānāṃ viṣayābhilāṣāṇāṃ yaḥ kṣayaḥ sarvato nivṛttirūpo nirvedaḥ tad eva sukhaṃ tasya sthāyibhūtasya yaḥ paripoṣo rasyamānatākṛtas tad eva lakṣaṇaṃ yasya sa śānto rasaḥ|. (Dhvanyaloka Locana, p. 390)

The absolute cessation of the cognitive mode of desire or the wanting of something, a resignation [of the mind from being engaged] is in itself a joyous state. The savoring of serenity is characterized by the cultivation of this very enduring emotion which is caused by making it an object of sustained savoring.

According to this position, just like passion or fear transforms into the savoring of beauty or horror, there should be a mechanism to evaluate the elevated state of dispassion, and this is what amounts to the savoring of serenity. It is where dispassion functions as enduring mode or enduring emotion and serenity becomes its appraised state combined with other subsequent and transient emotions. The same dispassion functions as enduring emotion for the savoring of serenity, and this is the same experience that leads to liberation. There is then a categorical difference between experiencing dispassion and the experience of liberation, and this can be mapped by the gap between dispassion and serenity, an aesthetically evaluated emotion. There are key differences in the liberating mode of experience and this savoring. However, this has not come to conversation in Abhinava’s discourse on serenity. The starkest case is the qualitative state of liberating experience in contrast to aesthetic judgment: there is no subject-object dichotomy in evaluating or savoring the experience of liberation. However, as an aesthetic experience, serenity is savored and is objectified. Nevertheless, according to Abhinava, every single aesthetic experience is lokottara or outside of the sensory realm, something that is not quite captured by the sensory modalities. Basically, there is something it is like in the savoring of rasa that is not an object of phenomenological analysis which makes qualitative comparison impossible. Abhinava at this juncture demonstrates that there is yet another dimension to dispassion, as self-realization transforms into dispassion in its highest state.

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Abhinava argues the experience that leads to the savoring of serenity is not frustration or depression. There is another dimension of serenity, a positive one, that drives the subject in actualizing the foundational being of oneself. Yet again the conversation on aesthetic experience overlaps the discourse on liberation: for Abhinava, liberating experience is positive, and has phenomenal content, reflexivity itself becomes its basis, and in liberation the self is savoring its own being. Abhinava derives his interpretation of dispassion (nirveda) from the Vyāsa-bhāṣya of the Yogasūtra. The text says, “dispassion is the very culmination of self-realization” ( jñānasyaiva parākāṣṭhā vairāgyam | YS I.16–17). This makes dispassion a twofold psychological state. From the perspective of emotion, it relates to aversion from the world that culminates into the savoring of serenity. However, following the second interpretation, nirveda is a cognitive state, where selfrealization functions as a catalyst for achieving the higher state of nirveda. The fundamental divide therefore rests on the ways the term nirveda is explained. Abhinava argues: tataś ca tattvajñānam evedaṃ tattvajñānamālayā paripoṣyamāṇam iti na nirvedaḥ sthāyī kin tu tattvajñānam eva sthāyi bhavet |. (Abhinavabhāratī, p. 334)

With this, the very recognition of the reality becomes nourished in a chain of recognizing reality and so dispassion will not be the enduring emotion, it will be the recognition of reality.

Abhinava, however, does not collapse the nirveda model with his own. Even though he acknowledges that dispassion can be understood in two different ways, he is aware that the advocates of dispassion who apply the terminology of nirveda may not come to conformity with his interpretation. This is where he separates the two terms of nirveda and vairāgya: nirvedho hi śokapravāhaprasararūpaś cittavṛttiviśeṣaḥ |. (Abhinavabhāratī, p. 335)

Nirveda is the specific state of the mind that has the expanse of the flow of grief.

vairāgyaṃ tu rāgādīnāṃ pradhvaṃsaḥ |. (Abhinavabhāratī, p. 335)

Dispassion (vairāgya) is the cessation of passion etc.

Even with this distinction, the deeper meaning of vairāgya as a culmination of self-realization, that Abhinava seeks, is not acquired. Nevertheless, the definitions above demonstrate a radical difference in the terminology of nirveda and vairāgya: the first has a positive being, a phenomenal content, of negative experience, such as

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grief, while the second simply refers to the destruction, absence of passion. One should not be too quick to equate this with cessation though, as what Abhinava anticipates in this definition is the type of negation that confirms y in negation of x. Even when the cognitive mode has been negated, its foundation remains unchallenged. If this is a negation that confirms difference, basically the negation of x implying the presence of y, rasa experience would not be empty of its own content, or empty of having its own svabhāva, self-regulating nature. On the other hand, if this is a form of absolute negation, this would make the experience of śānta a form of absolute negation. This is where the conversation crosses the boundary of aesthetics and enters interdisciplinary metaphysics. As a consequence, how one interprets negation not only relates to describing everyday experience, this also concerns the ways one comprehends the penultimate goal of liberation. It has been addressed earlier that both Buddhist and Nyāya philosophers describe the state of liberation in terms of negation, as duḥkhadhvaṃśa, the final termination of suffering. Unlike the Buddhist counterparts, the negation of the cognitive functions or of consciousness does not translate into the negation of the self for the Naiyāyikas. For them, consciousness is not an inherent nature of the self, but rather, it is one of the secondary tropes or properties that the self can exist even in its absence. As far as the enduring nature of consciousness is concerned, Nyāya position contrasts that of the Advaita of Śaṅkara. Nevertheless, in both Nyāya and Advaita accounts, understanding nirveda in positive terms (negation of x confirming y) makes it possible to trace back to the self that has its inherent nature (svabhāva). In all accounts, the foundational problem is that if there is nothing to savor, no emotion or cognitive content to be appraised, that state cannot be rasa, as it violates the very notion of relishing. Particularly from the perspective of cessation, if there is no enduring self, there is not even a subject savoring rasa. It is the phenomenal content, something being savored, that makes savoring savoring. It is the qualitative state that makes one rasa distinct from the other, and if there is nothing positive to appraise in serenity, this may likewise be the case with savoring eroticism or humor, and the categorization of rasa would be meaningless. Abhinava responds to this position with a caveat. On the one hand, there is something enduring, something having luminous quality in all rasa experiences including the savoring of serenity. On the other hand, this savoring relates only to the initial surge of rasa, as its qualitative state cannot be savored once the rasa overpowers the subjective gaze since rasa itself is considered “beyond the scope of the sensory faculties” (lokottara). As if a person merging into a deep lake, a subject merges within rasa experience, and there is no duality, no subject-object relation, no appraisal, while remaining merged in this pure fluidity. Even for Abhinava, the establishment of śānta as a positive experience refers only to its early stage of emergence where the savoring begins and the enduring emotions transform, but not in their exalted states. Abhinava rejects having any representational content in the exalted state of experiencing śānta, or for that matter, any other rasa. The descriptions of savoring in this essay, the difference that has been maintained on rasa, and even the analysis of the internal structure, all rest on the initial stage before all the differences collapse and the experience that determines

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two horizons of subject and object collapses into one, as in this singular experience, there is just the savoring of being and awareness. In other words, the positivity of rasa experience is a description of the foreground and not the essential core. Even then, Abhinava’s objections to the negative hermeneutics of serenity stand. Experiencing serenity, or for that matter experiencing anything, cannot ever be confirmed as having a negative content. An experience, whether that corresponds to positive or negative entity, still stands as positive, has its own self-determining nature (svabhāva).

Position III: Serenity Emerges from Self-Awareness (A¯tmajn˜a¯na) Three enduring emotions or states of being (bhāva) – cessation (śama), frustration (nirveda), and dispassion (vairāgya) – have so far been examined as the potential candidates to elevate into the savoring of serenity. According to Abhinava, none of these qualify to be so. Even when he seems to confirm one of the above models, he is either collapsing the model within his own or interpreting the categories differently. Abhinava ultimately establishes his own position that the enduring state of being, enduring psychological state (sthāyībhāva), is “recognizing reality” (tattvajñāna). What he means by “recognizing reality” is self-realization (tattvajñānaṃ ca nāmātmajñānam eva | Abhinavabhāratī, p. 336). It is by making self-realization the cause for savoring serenity that Abhinava ties the two systems of aesthetics and Pratyabhijñā together. This, however, raises various questions. The first glaring question is the inclusion of self-awareness (ātmajñāna) as the enduring bhāva. This does not even count in Bharata’s list of enduring emotions, and Abhinava is well aware of this issue. Next, the translation of bhāva as emotion fails here as what type of emotion would selfrealization be? All in all, it is in addressing śānta rasa that one encounters a real cultural shift where cognitive and emotional states, mental and somatic states, all fall under the category of bhāva. In his paradigm, bhāvas are the modes of the self or consciousness, and these include both cognition and emotion. Another issue here is that self-realization cannot be of the same category to cognizing objects. When cognizing external entities, there is a fundamental difference in the cognizing self and the cognized objects. In the case of knowing the self, it cannot be an object, even for itself, for the reason that when objectified as the other, it no longer is the subject (paro hy evam ātmā anātmaiva syāt | Abhinavabhāratī, p. 336). Knowing the self is unique in the sense that there is no difference in the awareness between the cognizing self, the consciousness that is being self- given, and the reflexivity that is grasping consciousness itself. On the contrary, if self-awareness were to be similar to knowing external objects, this would not be the knowledge of the self in any account. That which is projected to be the self, the objectified subjectivity, is not the self. Keeping these two premises in mind, Abhinava argues: tena ātmaiva jñānānandādiviśuddhadharmayogī parikalpitaviṣayopabhogarahito ‘tra sthāyī |. (Abhinavabhāratī, p. 336)

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Therefore, the enduring [emotion for the surge of serenity] is the very self that is associated with the pure attributes such as consciousness and bliss and is bereft of enjoying the conceptualized objects.

Accordingly, when in contact with the external stimuli, the subject enjoys the simulated objects, its own copies that resemble the externals. And it is in this outward gaze that the self deviates from its immanence and finds itself differentiated, manifesting in the modes of volition, cognition, and action. Any and all modes of consciousness that make the subject hidden from its own reflexivity make the self associate with the transient bhāvas. This self-awareness is therefore not similar to other enduring emotions because those others endured briefly, may be conditions for the surge of rasa, but are not fundamental to the self. Self-awareness, however, is always there: it is due to orientation outward that this consciousness remains obscure. Because the self is nothing but consciousness and since consciousness in this paradigm is reflexive, this self-awareness of the self is the basis of all other experiences. The same way, the savoring of serenity is the basis for the experience of all the rasas. Abhinava therefore argues that “this is the foundation for all other emotions and is therefore most enduring of all the enduring emotions” (sakalabhāvāntarabhittisthānīyaṃ sarvasthāyibhyaḥ sthāyitamaṃ . . . | Abhinavabhāratī, p. 336). This is where the savoring of serenity becomes synonymous to resting on the self. Following the model presented by Bharata, this foundation is not expunged of all emotions. On the contrary, this foundation is filled with latency for the self to actualize all other modes of emotion, as has been epitomized in his statement, “all bhāvas spring forth from the state of serenity” (śāntād bhāvaḥ pravartate). This leads to another question: is there something intrinsic to egoity, something unique to the self, that constitutes self-experience as distinct from other experiences or that gives an ontological status to self-experience? This question characterizes the discourse on the inherent nature (svabhāva) of the self or the lack of self-nature, a classical debate between the Hindu and Buddhist philosophers expanding for over a millennium. With Abhinava’s analysis of the savoring of serenity, this question emerges again, as what constitutes the savoring of serenity is not as straightforward as one might assume. Being or the lack of inherent nature is at the core of defining the experience of serenity, and the question is also relevant phenomenologically: whether or not there is something like savoring serenity? Also embedded is the question, whether or not the self deviates from its primordial immanence in savoring serenity? Abhinava’s position, as mentioned earlier, is that the appraised emotional state, even if that makes double appraisal with the basic emotions themselves being appraised, only endures in the preliminary phase of the surge of rasas, not just serenity but all other rasas. It is in this initial ground, the early exposure of emotion, that serenity is determined as serenity. For when the experience overpowers and eventually erases the horizons of subject and object, all that remains is the surplus, or the overflow, of rasas. Abhinava actually classes subjectivity as a postscript of this singularity. Or all differences merge within this outpouring of emotions appraised and savored in terms of rasa. This now reverses the paradigm, and rasas become

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primary, more intimate to the self, and bhāvas become the post-experiential analytical state of experience. This convergence of self-experience and the savoring of serenity resolves perennial issues embedded with the Pratyabhijñā system. What is confirmed from the discourse above is that the very self is the foundation for all the enduring emotions. Not just that, this very self becomes yet another enduring bhāva in savoring serenity, and this is where the issue of translating bhāva as emotion becomes problematic. From this perspective, there is marginal difference in the self being aware of itself and evaluating its immanence in terms of the savoring of serenity. This also goes without saying then that the self in its most pristine form is not necessarily expunged of the latencies for emotions, and when the episodes emerge, there is no categorical difference in cognitive and emotional modes. This comes from my reading of consciousness in terms of luminosity or prakāśa that is accompanied by reflexivity or vimarśa. This reflexivity or vimarśa has something like it, a qualitative state, a stickiness (√mṛś) the sense of which is not contained by reflexivity. It integrates feeling, it incorporates what it is like, and it involves savoring or bliss (ānanda). Consciousness or citi in this paradigm is not passive. The dynamism of consciousness involves being in the world, and this very engagement is the mechanism that allows the inherent tendencies embedded within consciousness to unfold in the form of emotions. When Abhinava says that the evaluative aspect or the savoring of rasa rests only on the initial ground of emergence, what it implies is that self-awareness is not yet another mode of consciousness that objectifies. This actually is the consciousness, in its complete immanence, that lacks directionality. And since it is not directed toward any object, it has no external object to grasp or otherwise that would not be selfknowledge. The appraisal of emotions – the cognitive mechanism that transforms the enduring emotion into rasa – in the case of experiencing serenity is the very self, as there is no difference in consciousness that grasps the self and the self itself.

The Phenomenology of Savoring Serenity The issue common to the savoring of any rasa, and in particular of savoring serenity, is its intentionality. On one hand, any emotional appraisal requires something to be appraised, some phenomenal content, while on the other hand, Abhinavaguptian aesthetics not only establishes rasa experience as lokottara or transcending the sensory realm, but he also confirms again and again that the innermost core of rasa experience cannot be described. Even the confirmation of positive being of savoring, along these lines, is based on the initial exposure of rasa. As it comes to conceiving and describing the savoring of serenity, it becomes all the more difficult. For one, even the enduring mode (sthāyin) is self-awareness, a state of consciousness that lacks representation. If serenity is described as the repose of all emotions, the resting of all the cognitive functions, this negation, an empty void of emotional surge, cannot have any positive content, something for the mind to objectify.

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Abhinavagupta is particularly conscious that this experience can be interpreted as a mere lack. While Abhinava does not explain this enduring mode in negative terms – as it is the very self-awareness that functions in his aesthetics as the basis for the savoring of serenity – it is not just the Buddhist aesthetes, including Ānandavardhana, who hold the idea that the removal of passion, etc. is the basis for the emergence of śānta. In this platform, however, the enduring emotion would be delightful. Ānandavardhana thus does not terminate the possibility of having the savoring of serenity appraised. What one confront is the possibility that there is no objective horizon (if considered the cessation model), or there is not even a subject evaluating the rapturous surge of emotions when engaging Buddhist perspective. As long as there is mind, there always is something for it to grasp, whether in terms of being or absence. Abhinava subsumes mind and subjectivity in all-embracing aham, but this is not the subject or the ego of everyday convention. This is the totality experiencing itself immanently. In his discourse on rasa, Abhinava occasionally eludes to this state but refrains from merging two disciplines of aesthetics and metaphysics. The mind and mentation, the ego and its other, are all dissolved in the mode of experiencing, whether the experience is mystical or aesthetic. Abhinava differentiates these two states based on subjective evaluation. Nevertheless, even the aesthetic experience transcends the cognitive horizon in its depth. The perplexity here is what would then be the structure of phenomenality and what form of intentionality can one assign to this state of savoring? If intentionality is understood as what Frega would have, “a mode of presentation,” there does not appear to have any modality for serenity to uncover. Following Kriegel’s (2002) model, when experiencing blue, there exists a representational content that stands for something blue alongside a phenomenal character that there is something in it that makes the experience of blue blueish. From Abhinava’s perspective, there seems to be different layers of the savoring of serenity. In its initial surge, the subject is capable of evaluating the experience, ergo finding it worth savoring. However, when the rasa experience becomes intense, it takes over the horizons of subject and object, making it impossible to describe. The following statement of Abhinava is particularly insightful to peel off the layers of this experience, as he nonetheless provides a scaffolding for this savoring to be determined: uparāgadāyibhir utsāharatyādibhir uparaktaṃ yad ātmasvarūpaṃ tad eva viralombhitaratnāntarālanirbhāsamānasitatarasūtravad yadāhitatatsvarūpaṃ sakaleṣu ratyādiṣūpa-rañjakeṣu tathābhāvenāpi sakṛdvibhāto ‘yamātmeti nyāyena bhāsamānaṃ parāṅkukhatātmakasakaladuḥkhajālahīnaṃ paramānandalābhasaṃvidekatvena kāvyaprayoantarmukhāvasthābhedena gaprabandhābhyāṃ sādhāraṇatayā nirbhāsamānam lokottarānandā-nayanaṃ tathāvidhahṛdayaṃ vidhatta iti |. (Abhinavabhāratī, pp. 340–341)

What is vividly manifest is the essential nature of the self, that is transparent even when being colored by stimulation or passion that color [the self] following the maxim that ‘the self is simultaneously manifesting’ while having its essential nature being covered in accordance with the stimulants such as passion. The essential nature of the self is similar

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to a shining white thread manifesting in the gaps of sparsely threaded gems. [This manifestation] constitutes the heart such that it generates extra-sensory bliss by being one with the state of inward facing that is free from all the nets of suffering, having the character of facing outward and manifesting in the homogenous form of being one with the awareness of encountering the absolute bliss by means of poetry and drama.

The above description delineates a two-tier strategy to determine the phenomenal content of savoring serenity. From this perspective, the description of serenity, or for that matter the savoring of any rasa, is based on initial capturing or objectifying of the state. This is to say that both the qualitative state and its evaluation are possible in this phase of origination. Rasa experience, along these lines, surges and overflows the barriers of the body-mind so that it fails to determine its horizons but rather the experiencing subject soaks into it or submerges in the surplus of bliss. But this also results in saying that this experience cannot be represented, as it is not conceptualized by the mind. Embedded with this statement is that the constitution of the transcendent object and the immanence of the ego is subsequent to this ground. It is the non-dual experience wherein the two horizons are carved. This also confirms the foundational being wherein the description in terms of luminosity (prakāśa) and reflexivity (vimarśa) do not apply. If this non-dual state were to confirm human subjectivity, it is in this very foundation that emotions endure in their latency. Rather than adopting Abhinava’s approach to complete the analysis of the surge of serenity, this essay proposes a return to Bharata’s poignant statement that had some proponents in classical times. For Bharata, serenity is not just a resting ground without return but a reservoir teeming with latency for the exuberance to overflow. And it is in this fullness that the cognitive horizon is carved. Every cognitive instance along these lines retains some form of savoring, and every instance of being therefore is infused with rasa. The more the experience moves to its pristine form, the less it is captured by the scaffolding of concepts. What one calls śānta is therefore merely based on the way the experience reveals itself to a subject in its first exposure (Locana 390–94). Since the heightened state of serenity coupled with savoring is theorized as extra-sensory (alaukika) or transcendent (lokottara), it makes sense to acknowledge that the aspects of this experience can be determined after the experience subsides and the two banks of subject and object manifest again. In other words, if rasa experience is determined in its constitutive phase, this can also be determined when the intensity of experience phases out and the constitutive elements are analytically given. But as far as the core of experience is concerned, no judgment of the state is possible. There is no horizon given, and in the lack of the grasping subject and grasped object, no appraisal possible. Finite human subjectivity and the limits of cognition, then, are constituted in this very foundation of serenity. This is why self-awareness plays as the foreground for its savoring. Whether Abhinava wrote his epistemology to confirm his aesthetics or his aesthetics to establish his epistemology, it is in the analysis of serenity that both epistemology and aesthetics collapse their disciplinary boundaries and endeavor to confirm the most intricate core of human experience.

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Definitions of Key Terms Abhāva: absence. Classical Indian philosophers analyzed absence primarily as relational absence or saṃsargābhāva and difference or anyonyābhāva/bheda. The first or relational absence is threefold: antecedent absence (prāgabhāva) or the absence of effect prior to its emergence; consequent absence (pradhvaṃsābhāva) or destruction; and absolute absence (atyantābhāva). The second type of absence, difference or bheda, relates to the absence of x in y. Ātmajñāna: self-realization. Abhinava uses this term in the context of analyzing serenity, arguing that self-realization forms the basis for the experience of serenity. Lokottara: transcending the commonsense world or transcending the realm of sensory experience. Abhinava uses this term to explain the experience of śānta. Nirodha: cessation. Classical Buddhist and Hindu yogic practices focus on nirodha to pacify the fluctuations of the mind. Some have argued that nirodha functions as the basis for the surge of śānta. Nirveda: dispassion. Some classical philosophers have argued that nirveda forms the basis for the emergence of śānta rasa. Rasa: elixir, savoring. Rasa is the most fundamental theory based on emotional analysis and fusion of emotions to ground aesthetic experience. Śānta: the experience of serenity. This in the aesthetic context is considered the ninth rasa. Sthāyī bhāva: enduring emotion. Every rasa experience is unique in the sense that it has, for its basis, a distinctive enduring emotion that evolves into rasa when combined with other emotions and appraised accordingly. Viśrānti: resting. In Trika system, this term refers to the returning of selfconsciousness to its core and reside in its immanence. Abhinava uses this category to explain the experiencing of śānta.

Summary Points 1. Rasa is a key term to describe aesthetic experience that evolves from evaluating the commixture of emotions. While majority of rasa theorists advocated for eight rasa, Abhinavagupta and his followers argued that the aesthetic experience of serenity counts as an additional rasa. 2. Śānta is the ninth rasa in classical Sanskrit aesthetics. Since all rasas have one or another central emotional core called sthāyī bhāva, the issue of what becomes the enduring emotion for the emergence of śānta becomes one of the central issues in the classical philosophy of aesthetics. While Abhinavagupta explores the models based on cessation or the suspension of all the mental fluctuations and dispassionbased models, he argues that none of these satisfy to be the enduring emotion for the surge of śānta. He argues on this ground that it is the very self-realization that becomes the catalyst for the surge of serenity.

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3. Nirodha, common to early Buddhist and Hindu yoga practices, refers to the cessation of the fluctuations of the mind. While in the Buddhist context the experience of nirodha is simply to suspend all the mental activities, in the Hindu context, this only relates to suspending the factors that are blocking the access to the self-illuminating consciousness or the self. 4. Abhinava explains dispassion in both ways: as a catalyst for self-actualization and as the product of self-actualization. While Abhinava objects to dispassion being the basis for the experience of serenity, he reconciles his own position with the second one, i.e., dispassion is the product of self-realization. 5. One of the major arguments of Abhinava in analyzing śānta is that selfexperience is at the core of experiencing serenity. He compares the self with the thread and our emotional life as the color beads. By exploiting this metaphor, we can argue that there are interruptions in each of the mental fluctuations but at the same time, even during those mental fluctuations, self-experience is merely covered and not absent.

References Abhinavabhāratī. Abhinavagupta. See Nāṭyaśāstra of Bharata Muni. Chakrabarti, Arindam. 2006. From Vimarsha to Vishrama: You, I, and the tranquil taste of freedom. In Abhinavagupta: Reconsiderations, ed. Makarand Paranjape and Sunthar Vishuvalingam. New Delhi: Samvad India Foundation. Gerow, Edwin. 1994. Abhinavagupta’s aesthetics as a speculative paradigm. Journal of the American Oriental Society 114 (2): 186–208. Kriegel, U. 2002. Phenomenal content. Erkenntnis 57 (2): 175–198. Locana. The commentary of Abhinavagupta upon Dhvanyāloka. See Dhvanyāloka.

Convergence and Divergence of Nyāya and Tattvavāda (Dvaita) Theories of Logic

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of Cognition: An Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Concept of Inferential Cognition in the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Inference According to the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constituents of Inferential Cognition in the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Anumāna in the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Types of Vyāpti: Positive and Negative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hypothetical Logic in the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fallacies of Inference in the Two Schools . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Indian epistemology has delved deep into the analysis of logic as a component of cognition. It is considered as an essential constituent or, rather, an essential process of inferential cognition. Most Indian schools of thought, except the materialists called Cārvākas, recognize inferential cognition as a kind of cognition. The varied schools developed conclusive theories of inference, each with their own unique contributions, resulting in a vast body of literature, often polemical, in this field of analysis. This chapter focuses on providing an introduction to the various constituents and the logical process of inferential cognition according to the Indian intellectual tradition using illustrations, both classic and contemporary, for elucidation. It particularly focuses on the doctrines of the Nyāya and Dvaita Schools of V. Nishankar (*) Research Scholar, Mumbai, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_27

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philosophy in this regard and aims to objectively bring out the points of convergences and divergences in the theories of logic of the two schools and tries to provide the background for the divergences to clarify the reasons for the differences in their view points. Keywords

Theories of logic · Traditional Indian epistemology · Yukti-shastra/śāstra · Inferential cognition · Inference · Naiyayikas/Naiyāyikas · Dvaitins · Anumiti · Anumana/anumāna · Paramarsha/parāmarśa · Convergence and divergence · Fallacies of reason · Erroneous cognition · Deduction · Invariable relation · Vyāpti · Pervasion · Vyaptigraha/vyāptigraha · Hypothetical logic · Tarka

Introduction Cognition is an invariable experience of every living being. Survival, sustenance, progress, and prosperity of beings depend to a great extent on their cognitions of things, persons, and events that they encounter. Philosophers both in the east and west have deliberated on the form, classification, and means of cognition and documented their epistemological conclusions as their conclusive theories or siddhāntas. Traditional Indian schools of philosophy, particularly the Nyāya School, had epistemology as a favorite field of study. Generations of scholars have participated in evolving the polemic literature in this topic and engaged in dialectic deliberations. This was the prestigious field of study – the pramāṇa-śāstra – that gauged the intellectual supremacy of schools of thought. The study of logic or yuktiśāstra was therein a hot topic, and the dialectical indulgence in this subject gave rise to terse and even terser treatises in a highly technical lingua franca. In the Indian epistemological world, logic – yukti – belongs to the domain of inferential cognition, anumiti. Most Indian schools of philosophy, from both theistic and atheistic categories, accept inferential cognition as a form of cognition. The schools of philosophy relevant to the discussion in this chapter, namely, Nyāya and Tattvavāda, accept inferential cognition as a classification of cognition along with perceptual cognition and verbal testimony. The Nyāya School additionally propounds comparative cognition as a form of cognition, which according to the Tattvavāda or Dvaita School, as it is popularly known, is a combined cognition of verbal testimony, perception, and inference. Following the kinds of cognition, the kinds of means of cognition acceptable to both schools are perception, inference, and cognition of words. Additionally, the Naiyāyikas accept cognition of similarity as the means of comparative cognition. All schools of Indian philosophy accept the nomenclature of the final cognition as pramiti and the means of the final cognition as pramāṇa. In the Dvaita School, the preferred names are kevala-pramāṇa and anu-pramāṇa, respectively, for the final cognition and its means. These are the first instances of doctrinal convergences and

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divergences between the Nyāya and the Tattvavāda Schools related to their epistemological theories. Both the Nyāya and the Dvaita Schools have a long academic tradition of philosophy spanning over centuries. Both schools have generated and attempted to improvise the definitions given to them by the predecessors with regard to every philosophical element of discussion. An attempt to derive the convergences and divergences of the views of these schools regarding the definitions and classification of the epistemological elements related to logic becomes a little intriguing due to this evolving nature of theories in the philosophical schools. We may handle this situation by discussing the basic ideology regarding the logical concepts of these schools in this chapter.

Concept of Cognition: An Overview Logic or yukti is a tool of cognition. Cognition means knowing. One may know things rightly or wrongly. In other words, cognitions may be true or false. True cognitions are known as knowledge – pramā or yathārtha-jñāna. False or erroneous cognitions are called fallacies – apramā or ayathārtha-jñāna. Knowing an object as it actually is is a true cognition of the object. For instance, knowing a rope as a rope is knowledge of rope. Knowing an object, unlike what it actually is, is erroneous cognition. For instance knowing a rope as a snake is erroneous cognition of rope. Commonly, present events and objects and past events and objects are constituents of what can be known. Knowing future events and objects require supernatural capacity of cognition in the knower. Knowing an object or an event in the present is experience. Remembering the experience when the mental impressions of past experiences are somehow triggered is memory. While another person’s past experience may be one’s present experience, memory is the kind of cognition which involves only one’s own past experience. Thus, even when one says “I remember that X had two cars 10 years ago,” she is only remembering her own past experience of having known that X had two cars 10 years ago and not the past experience of X owning two cars 10 years ago. Cognitions are of two types – immediate and mediate – aparokṣa-jñāna and parokṣa-jñāna. Mediate cognitions are those that involve an intermediate cognition to arrive at the final cognition. Perceptual cognition is an immediate cognition as it is arrived at merely by the contact of the sense organ with the object of cognition. Verbal or sentential cognition is a mediate cognition as cognition of words and their meaning are essential to arrive at the sentential cognition. Inferential cognition is also mediate in nature as there have to be some known elements that can give rise to a new cognition while applying some logic. So, logic is essentially the process of arriving at inferential cognition. This intellectual process involves not one but several intermediate cognitions. There are at least three objects involved in the logical process of arriving at inferential cognition – the reason that promotes deduction, the conclusion that results from

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applying the logical process based on the reason, and the ground of deduction. Of these, the object in the form of reason is invariably related to the conclusion, which is why its cognition is capable of giving rise to the definite knowledge of the object in the form of conclusion. What is meant by “invariably related” is that wherever the object of reason is present there, the object of conclusion also is invariably found to be present. For instance, in the classic illustration of smoke and fire, when fire in not visible and accompanied by smoke (which is visible), cognition of smoke is the reason on the basis of which fire is concluded to be present in the same place. This is because smoke is invariably related with fire, and wherever smoke is present, fire is definitely present. A person, who knows this relation of pervasion between smoke and fire, logically concludes that fire is present in a place where smoke is perceived. The relation between the reason of deduction and ground of deduction is incidental. In fact, the ground may vary from time to time. But the conclusion deduced each time from the reason remains the same in each of these grounds owing to the relation between the reason and conclusion being invariable. For instance, in case of smoke and fire, all places where fire is not visible and smoke is visible are grounds of deduction of fire from smoke, like a hill (with fire on the hind side and hence not visible). Logicians belonging to all schools of thought agree on the basic process of arriving at inferential cognition. There are at least two major cognitions involved in arriving at inferential cognition: • There is a trigger in the form of cognizing the reason or mark in a particular location or situation which is invariably related to the conclusion, which is not directly cognizable at the moment of cognition of the trigger. • This trigger brings about the remembrance of the invariable relation between the reason and the conclusion at the moment of cognition of the trigger. – The cognition of invariable relation between the two objects prerequires cognition of the two objects. – The cognition of invariable relation between two objects prerequires cognition of the relation of the two objects: • The cognition of the invariable relation of the two objects comes about by cognizing multiple instances of the coexistence or co-relation of the two objects. • The cognition of the invariable relation of the two objects comes about by the lack of cognition of even one instance of lack of the coexistence or co-relation of the two objects. Some instances of inferential cognition are provided below: • X hears the bell ring at the closed door. X concludes that there is some “person” at the door, and it turns out that X “sees” that there is a person at the door. This happens every time; only the “person” is variable; nevertheless there is a “person.”

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• Y is sitting in the living room. Y smells something burning in the kitchen. Y concludes the food and the pan on the lit stove are charred. Y rushes to the kitchen. Lo! They are charred. There are many instances; many times unnoticed ones, when inference is employed and some conclusive cognition is arrived at, while the object of conclusive cognition itself is not perceivable at the time of arriving at such cognition. Young mothers, caretakers of pets, etc. more or less run their routines with their young ones or pets, at the mercy of inferential cognition. It is so also with the doctors diagnosing their patients and so on. Inferential cognition is often likened to ūha – “assumption” or “guess” – where too some logic is employed, but conclusions are not always conforming to the proposition or are coincidentally conformed. These are semblances of inferential cognition where the validity of the invariable relation between the components in such guesses is still shaky.

Concept of Inferential Cognition in the Two Schools Knowing the outlook of the Nyāya and Dvaita Schools about inferential cognition gives an insight into the means of the convergences and divergences that these schools have regarding inferential cognition. According to Tarkasaṅgraha, the primer of Indian logic, inferential cognition is the cognition caused by parāmarśa – deduction. Parāmarśa is a complex cognition. Firstly, it constitutes the cognition of a mark or reason as found on a particular ground – pakṣa. This cognition is known as pakṣadharmatā-jñāna, which literally means “the cognition of the ground as possessing an attribute (which is the reason of inference).” Secondly, the cognition of invariable pervasive relation between the reason of logic and the conclusion of logic forms part of parāmarśa. This cognition is known as vyāpti-jñāna, which literally means “cognition of pervasion.” In fact, the latter cognition is latently present in the mind of the person involved with the logical process. That is, the person knows that the reason, hetu, and the conclusion, sādhya, of the logic are invariably related to each other with the relation of pervasion known as vyāpti. Which means that the cognizer knows that wherever the reason is cognized, the conclusion will invariably be present, as the conclusion always pervades the reason. The remembrance of this cognition of the invariable relation of pervasion is triggered when the mark of logic is cognized on a ground. This reminiscent cognition is known as vyāpti-smṛti, the remembrance of pervasion. On the strength of the knowledge of the pervasive relation, from the cognition of the mark on a ground, the conclusion is also cognized remotely as being there on the ground, through the remembrance of the invariable relation between the reason and conclusion. The logic may be presented as follows:

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• It is known that the mark/reason and conclusion are invariably related • It is found that the reason and ground are incidentally related • Hence, it is deduced that the conclusion and ground are also incidentally related This may be demonstrated with the help of the popular smoke and fire example: • B knows that smoke and fire are invariably related • B cognizes that there is smoke on a hill • Hence, B deduces that there is fire on the hill What is to be noted here is that the Nyāya School contends that the reason and the conclusion are located on the same ground spatially and/or temporally, which makes the logical deduction possible. In other words, pakṣadharmatā-jñāna, triggered by the cognition of the hetu on the pakṣa, and vyāpi-smṛti, comprising the cognition of the pervasion between the hetu and the sādhya, give rise to the final cognition of the sādhya on the pakṣa. (Vyāpti-smṛti is caused by a prior vyāpti-jñāna, which constitutes the cognition of vyāpti between the hetu and the sādhya.) The Dvaita School does approve of this theory of logical deduction. But, the divergence therein is in the denial of ground being common to both the reason and the conclusion, always. The contention of the Dvaitins is that while sometimes the reason and the conclusion in an inference may be located on a common ground, it is not a necessary condition for inferential cognition to arise. There are instances where the reason and conclusion are located on different loci, spatially and/or temporally, and yet are invariably related with each other and give rise to a valid inferential cognition. The cognition of the reason being located in an appropriate locus (spatial and/or temporal) to give rise to the inferential cognition of the conclusion located on its locus is sufficient to conduct the process of inference. The location of the reason on an appropriate locus (spatial and/or temporal) is known as samucita-deśādau vṛttitvam of the hetu. Thus, while according to Nyāya tradition vyāpti-jñāna and pakṣadharmatā-jñāna give rise to inferential cognition, for the Dvaita School vyāpti-jñāna and samucitadeśādau vṛttitva-jñāna pertaining to the hetu give rise to inferential cognition. This is stated clearly by the seer-philosopher Śrī Jayatīrtha of the Dvaita tradition in his work Pramāṇa-paddhati with the words – anumānasya dvayaṁ sāmarthyam. Vyāptiḥ, samucita-deśādau siddhiśceti. na tu pakṣadharmatā-niyamaḥ. These constituents of inference will be dealt in detail in the following sections of the chapter.

Inference According to the Two Schools The Nyāya School confers parāmarśa with the status of being the means of inferential cognition. The definition of inference (the means of inferential cognition) in the Tarkasaṅgraha is – anumiti-karaṇam anumānam – the means of inferential cognition

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is inference. To elucidate further, the definition of anumiti is given as parāmarśajanyaṁ jñānam anumitiḥ – inferential cognition is the cognition generated from deduction. Thus, if anumānam ! anumitiḥ and anumitiḥ

parāmarśaḥ,

then anumānam = parāmarśaḥ According to the Dvaita School, the standard definition of inference is nirdośopapattiḥ (nirdośa-upapattiḥ) anumānam – defect-free reason(ing) is inference. That which justifies the cognition of conclusion in a deduction is upapatti. That is nothing but the reason in an inference. When not tainted with any fallacy, the reason is capable of giving rise to defect-free inferential cognition. When it is said that the defect-free reason is the instrument of inferential cognition, what it entails is that the reason is the justifier (upapādaka) of the existence of the conclusion, even though the conclusion is not directly cognizable in an instance. Being a relative entity, the justifier invariably is related to that entity which it justifies (upapādya). This invariable relation between the justifier and the justified is vyāpti – pervasion. Thus, when it is mentioned that the upapādaka is the instrument of inferential cognition, it implies that, in the relation of pervasion between the upapādaka and the upapādya, the upapādaka is the prime instrument of inferential cognition and hence is likened to inference. It is indeed the cognition of the upapādaka that triggers the entire logical process.

Constituents of Inferential Cognition in the Two Schools The universally accepted constituents of anumiti are pervasion in the form of invariable concomitance, vyāpti, and reason being located on the ground – pakṣadharmatā. Counterfactual reason – tarka – is also considered as a constituent of anumiti in the Dvaita School. Universally, all schools of thought that accept inferential cognition as a form of cognition agree that vyāpti is the most important constituent of anumiti. There are divergences in their opinions regarding the kinds of invariable relations that are logically possible. However, with regard to the other two constituents, there are diverse views, majorly between the Nyāya and the Dvaita Schools. The case of the ground of inference has already been mentioned in the foregone passages. Hence, it is not discussed again further in this chapter. There is divergence

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in the outlook of the two schools regarding the treatment of tarka. While Dvaitins confer the status of being a constituent of inference to tarka, according to the Nyāya School, it is accounted as a fallacy of inference. Firstly, the discussion on vyāpti is undertaken here. The definition of vyāpti in the Tarkasaṅgraha is sāhacarya-niyamaḥ vyāptiḥ. The word niyama means a rule or that which is invariable. The word sāhacarya means association. And vyāpti means pervasion. Thus, the invariable association between two entities is termed as vyāpti. The two entities that are involved in this invariable relation are known as vyāpya, the pervaded entity, and vyāpaka – the pervading entity. The invariable relation between the pervading entities that pervades the pervaded entity ensures that if the pervaded entity is present, then invariably the pervading entity also exists there. Being the cause of deduction of inferential cognition in this way, the pervaded entity that triggers the logical process is called as the hetu – reason. And being an entity to be proven to exist through such reason, the entity deduced is called as sādhya – conclusion. Thus, in the invariable relation of pervasion, hetu is vyāpya and sādhya is vyāpaka. At this point it is important to note that in the relation of pervasion, one entity is pervading the other entity that which is pervaded. Obviously, the scope of the pervaded is lesser than or equal to the pervading entity, but seldom more than the scope of the pervading entity. Hence, the relation of pervasion between the pervaded and the pervading entities may be either unilateral (in case the pervasion is non-equal) or mutual (if the pervasion is equal). However, if the pervasion is not equal between the pervading and pervaded entities, then the relation cannot be mutual also. This is because, logically of course, the scope of the pervaded entity can never exceed the scope of its pervading entity. That is why the pervaded entity is capable of giving rise to the cognition of the entity pervading it, while the vice versa may not happen. This may be explained with the help of the classic example of fire and smoke. Between smoke and fire, smoke is always accompanied with fire. Rather, the existence of smoke is dependent on the existence of fire as smoke is caused by fire. Fire, on the other hand, may occur independent of smoke and may be associated with smoke in some instances and not in others. The instances of occurrence of smoke are lesser than the instances of occurrence of fire. Thus, it may be said that smoke is pervaded by fire and that fire is pervading the smoke. That is how, to a person who knows the invariable relation between smoke and fire, the cognition of smoke will always give rise to the cognition of fire. The relation of pervasion exists between the two entities in this order of cognition. The cognition of fire may not give rise to the cognition of smoke always, as the relation is not invariable in that order of cognition. The relation of pervasion does not exist between smoke and fire in this order of cognition. For instance, on seeing the blue flame of a gas stove burning, one may easily not be reminded of smoke! In the Dvaita School, vyāpti is defined as avinābhāvaḥ. Here, the prefix “a” refers to “not,” “vinā” means “without,” and “bhāva” means existence. Literally, this translates to “pervasion is existence (of one entity) not without (the other)!.” This boils down avinābhāva to mean “nonexistence (of one entity) without (the other).”

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This is nothing but unconditional relation between the two entities, namely, the justifier and the justified, the justifier being the reason and the pervaded entity and the justified entity being the conclusion and the pervading entity. While this definition of vyāpti does not seem to be very different from the definition of vyāpti in the Nyāya School, it does have a radical deviation from the basic tenet of vyāpti relation regarding the loci of the hetu and the sādhya. The Nyāya School confines the niyama – the invariability of association between the reason and the conclusion to one common locus. In his auto-commentary on Tarkasaṅgraha, its author Annambhaṭṭa states this explicitly – sāhacaryaṁ sāmānādhikaraṇyam – association is (the state) of having a common locus. This condition breaks the element of unconditionality in the relation between the hetu and the sādhya according to the Dvaita School, which defines vyāpti as unconditional relation. The existence of the pervaded and the pervading entities in pervasion is not confined to a common locus here. The existence of the hetu and sādhya are not conditioned by simultaneity of time or space. So also their relation becomes unconditional in one sense. The only condition of pervasion in this school is the unconditional relation between the pervaded and pervading entities! Sometimes, the hetu and the sādhya may be spatially and temporally related, as in the color and taste of a mango. Sometimes they may be connected only spatially or only temporally, as in the case of thunder and rain, where rain follows thunder in time. There may also be cases where the hetu and sādhya may be invariably related only in succession of time or space; nevertheless, they are invariably related. The instance of rain on the hill and the swelling of the river on the plains on the succeeding day have already been discussed in this light. This definition of unconditionality strengthens the invariability in the relation of pervasion and widens the scope and opportunity of inferential cognition covering a wider range of the cognitive experiences of people. Since the stand against pakṣadharmatā and for samucita-deśādau vṛttitva has already been discussed and illustrated, the types of inference is taken for discussion ahead.

Types of Anumāna in the Two Schools The logical process of deduction may be used for arriving at inferential cognition for one’s own sake or for the sake of others when one is already enlightened with the cognition. According to the Nyāya School, anumāna is nothing but parāmarśa that constitutes vyāpti-jñāna and pakṣadharmatā-jñāna. Pakṣadharmatā-jñāna is available through perception, another inference, verbal testimony, or memory of a past event. Vyāpti, on the other hand, is acquired gradually through multiple observations based on which the invariable relation between two entities comes to be imprinted in the observer’s memory. For the sake of arriving at inferential cognition for one’s own sake, it is necessary to know the process of acquiring the cognition of vyāpti.

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The process of acquisition of vyāpti is called as vyāptigraha. There are two conditions that acquisition of vyāpti needs to fulfill to confirm the invariable pervasion relation between, say, A and B, where A pervades B (mutually or otherwise): • There are multiple cognitions of concomitance/association between A and B, such that whenever B is cognized, then A is cognized. This condition of having multiple cognitions is called as bhūyodarśana. • There is not a single cognition of lack of concomitance/association between A and B, i.e., there is never a case that B is cognized as dissociated with A. Even one instance of cognizing B dissociated with A is sufficient to break the logical relation of pervasion between A and B. The lack of cognition of discrepancy (of association) between A and B is called vyabhicāra-adarśana. For instance, X sees smoke in several instances like the incense stick, a lighted camphor, a campfire, a barbecue grill, a fire sparkler, a festive cracker, a steam engine, a fire accident site, and so on. X notices that in every instance, fire accompanies smoke. X gets an understanding that if there is smoke, then there is a possibility of fire there. With increase in the number of instances of seeing smoke, always along with fire, X gradually grasps the invariability of their association. As X does not come across an instance where smoke is seen without fire, there is now a confirmed cognition regarding the invariable relation between smoke and fire. This cognition, like any other cognition latently registered in the mind, occurs as memory when triggered on the sight of smoke. Since the initial cognition comprises of the invariable relation between smoke and fire, the trigger in the form of sight of one of the constituents of the cognition, nevertheless, tugs along the entire cognition constituting the smoke, the fire, and their invariable relation. Thus, suppose X sees smoke fuming out of a neighbor’s kitchen, X is reminded of fire. Initially, cognition of fire occurs as a possibility. X goes out and actually finds that fire is indeed there with the smoke. Now the possibility is converted to definite cognition of invariable relation of smoke with fire. Suppose X also had an understanding that wherever there is fire, there is smoke based on some limited instances; even a single instance of fire seen without smoke, say, in a flaming gas stove, is sufficient to break down the previous cognition of invariable relation of fire with smoke in X. In fact this revelation also puts X through a mental check of whether smoke also would exist somewhere dissociated with fire, just as fire is found dissociated with smoke. On analysis of why smoke does not exist without fire, X finds out that smoke is caused by fire in a particular condition. When fire is in contact with wet fuel, it gives rise to smoke. If the fuel is dry, then there is no smoke. Thus, practically, fire is the cause of smoke. Hence, smoke cannot exist without fire. But, one cannot say that wherever there is fire, smoke exists there. Discrepant instances like a red hot iron or coal disprove the possibility of such pervasion.

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This goes to mean that in the consideration of invariable relation of smoke pervading fire, there is a limiting condition (upādhi) that disallows such pervasion, namely, contact with wet fuel (ārdra-indhana-saṁyoga). In a vyāpti, it is already known that the sādhya is the pervading entity and the hetu is the pervaded one. Ideally, that which pervades the sādhya ought to pervade the hetu for vyāpti to be error-free. So if “>” were to mean “pervades” and “>/” were to mean “does not pervade,” ideally: sādhya > hetu Z > sādhya ! Z > hetu With the upādhi in a vyāpti, this does not happen. Technically, upādhi pervades the sādhya but does not pervade the hetu, thereby rendering the pervasion erroneous. That is, upādhi > sādhya upādhi >/ hetu Hence, sādhya >/ hetu ! no vyāpti In the instance of smoke pervading fire, smoke is the sādhya, fire is the hetu, and contact with wet fuel is the upādhi. Wherever there is smoke, there is contact of fire with wet fuel. Thus, the upādhi pervades the sādhya. Wherever there is fire, there is contact of fire with wet fuel is not true. There is fire in a live coal, which is not a wet fuel. Here, the upādhi does not pervade the hetu. So vyāpti is not acquired in this order of pervasion between smoke and fire. Thus, one is not reminded of smoke on seeing fire, but the sight of smoke invariably reminds one of fire. Śrī Jayatīrtha gives a very lucid and convincing account of vyāptigraha in his Pramāṇapaddhati. The Nyāya and the Dvaita Schools more or less agree on the process of vyāptigraha. Both schools agree that tarka – counterfactual reasoning – is helpful in resolving any doubt that may arise regarding the discrepancy of pervasion between sādhya and hetu. In his Nyāyasiddhānta-muktāvalī Viśvanātha Pañcānana opines that where there is clarity about invariability of relation and non-possibility of discrepancy in a particular instance, vyāpti may be acquired in a single instance itself. Hence, cognition of multiple instances of concomitance between sādhya and hetu need not be considered as a necessary condition for acquisition of vyāpti. The divergence between the Nyāya and Dvaita Schools regarding vyāptigraha is with regard to its application. The utility of acquisition of vyāpti is that this cognition occurs to the cognizer as a memory when there is a trigger for it to be aroused. The memory of vyāpti acquired in an earlier instance is applied elsewhere in arriving at the final inferential cognition. For instance, vyāpti between smoke and fire may be acquired

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through the perception of their association in several instances, like the neighbor’s kitchen, a campfire, etc. When smoke is seen from the rear side of a hill, based on the vyāptijñāna, the final inferential cognition that “the hill has fire” is arrived at later. The smoke and fire of the neighbor’s kitchen is different from the smoke and fire on the hill. On what basis does one accept that the case of vyāpti in the instant fire and smoke conform to that of one’s past experience when vyāpti is said to be acquired? It is here that there is a divergence between the theories of Nyāya and Dvaita Schools. According to the Naiyāyikas, during vyāptigraha, along with the cognition of the entities in pervasion, their generic property is also cognized. This generic property is common to all entities of those two classes. Thus, through the relation of generality (sāmānyalakṣaṇā-pratyāsatti), all the past, present, and future entities belonging to the generic classes of the sādhya and the hetu are cognized. That is, while cognizing the invariable relation between smoke and fire, their generic properties, namely, smoke-ness and fire-ness, are grasped that bring into the ambit of cognition of smoke and fire during vyāptigraha, all past, present, and future smokes and fires. This enables vyāpti between smoke and fire grasped in one instance to be applied in any other instance. This proposition is unacceptable to the Dvaitins. They do not agree with contact through the generic class of entities as a kind of contact. They propose a more direct solution to the problem posited. It is true that the smoke and fire in the different instances are not the same. But, the similarity (sādṛśya) between the entities in the state of vyāptigraha and those that are encountered later govern the vyāpti to be applied during those later instances. The smoke on the rear side of the hill is similar to the smoke in the neighbor’s kitchen. Thus, on seeing smoke emanating on the hill – the pakṣa – the cognizer recognizes it as smoke similar to the one perceived in the neighbor’s kitchen and deduces that fire similar to the neighbor’s kitchen fire exists on the hill. So much about svārthānumāna – inference for one’s own sake. With regard to parārthānumāna –inference for others’ sake – there is vast divergence in the opinions of the Nyāya and Dvaita Schools. The Naiyāyikas have a water tight prescription of a five-limbed syllogism – pañcāvayava-vākya – for imparting inferential knowledge by an accomplished cognizer to a seeker or to others. According to them such syllogism packages the constituents of parāmarśa, the means of instructing the seeker (or the opponent in case of a debate) with inferential cognition when uttered by the accomplished cognizer, now an instructor. The five limbs of syllogism are stated and illustrated as follows: Proposition – pratijñā – the hill has fire Reason – hetu – as it has smoke Illustration – udāharaṇa – wherever there is smoke, there is fire, as in the kitchen Application – upanaya – it is so here (the hill has smoke) Conclusion – nigamana – hence it is so (the hill has fire)

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The number of limbs that make a syllogism varies in the different schools of thought. The Advaitins contend that either the first three or the last three limbs are sufficient to give rise to inferential cognition. The Buddhists opine that only the second and the third limbs are sufficient. The Dvaitins and Viśiṣṭādvaitins deny that a syllogism with a fixed number of limbs is needed to impart inferential knowledge to others. The number of limbs would rather depend on the capacity of the listener in grasping the inference which is imparted. In some instances, even a single limbed syllogism may serve the purpose, while others may need seven or eight or ten.

Types of Vyāpti: Positive and Negative When two entities are related, logically, their absences/negations are also related. On the basis of whether the relation of the entities in vyāpti is considered positively or negatively, vyāpti is classified into two types: anvaya-vyāpti – pervasion between the sādhya and hetu, sādhya pervades hetu, wherever hetu exists sādhya exists vyatireka-vyāpti – pervasion between negation of hetu (hetvabhāva) and negation of sādhya (sādhyābhāva), hetvabhāva pervades sādhyābhāva, wherever sādhyābhāva exists hetvabhāva exists The vyāpti between smoke and fire is of the former type. It is characterized by the cognition – wherever there is smoke there is fire. The vyāpti between absence of fire and absence of smoke is of the latter type, characterized by the cognition – wherever there is no fire there is no smoke. The cognition is validated by a positive illustration in the former case such as the kitchen and in the latter case by a negative illustration such as a lake. For an anumāna to be valid, the vyāpti between its constituents needs to be a confirmed one. The confirmation of vyāpti cannot be done on the ground of inference – the pakṣa. This is because the entire logical process is triggered only to prove that a particular pakṣa is the locus of the sādhya. Obviously, vyāpti cannot be validated on the very ground where its pervading constituent, namely, sādhya is sought to be proven. Hence, it is the illustration that comprises of an instance from a previous valid cognition that confirms the validity of the vyāpti. When in an instance, there is no positive illustration to validate vyāpti, then the validation could be accomplished by providing a negative illustration wherein the invariable relation between the absences of the sādhya and the hetu are confirmed, and thereby their positive pervasion is indirectly proven. Based on these two types of vyāpti, the mark in an inference, the liṅga, is classified into three categories in the Nyāya School: Absolutely positive mark – kevala-anvayi liṅga – where the liṅga being a constituent of only a positive vyāpti can give rise to inferential cognition

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Absolutely negative mark – kevala-vyatireki liṅga – where the liṅga being a constituent of only a negative vyāpti can give rise to inferential cognition Positive cum negative mark – anvaya-vyatireki liṅga – where the liṅga being a constituent of both positive and negative forms of vyāpti can give rise to inferential cognition The types of the liṅgas or hetus mentioned in this classification depend on the kind of validating illustrations that those liṅgas are related with. There are instances where vyāpti can be confirmed only through some positive illustration, while there are others in which vyāpti can be confirmed only through negative illustration. And again there are instances in which both kinds of illustrations are available. It is based on these conditions that the liṅgas, and thereby the anumānas formed of them, are, respectively, classified as kevala-anvayi, kevala-vyatireki, and anvaya-vyatireki. The classic case of vyāpti between fire and smoke is one with the positive cum negative mark. As this has already been mentioned in the foregone passage, the other two types of liṅgas are discussed below. In the Nyāya School, there is a need to accept the kevala-anvayi form of vyāpti due to the following contentions of theirs: • • • •

If any fictitious entity forms part of a cognition, then the cognition is fictitious. Cognitions constituting fictitious entities are erroneous or fallacious. Fallacious inferences give rise to fallacious inferential cognitions. All objects of the world are capable of being named and being cognized. That is, all objects are nameable and knowable. In fact, being nameable and knowable are the common characteristics for the entirety of worldly objects. • An entity that is not nameable or knowable is a fictitious one. Thus, an inference involving the absence of being nameable or absence of being cognizable, etc. is fallacious.

These contentions entail that in an inference involving “being nameable” and “being knowable” such as “pot is nameable, as it is knowable,” their vyāpti can only be asserted positively, as it can be validated only with a positive illustration. So, it may be said, “pot is nameable, as it is knowable, as is the case with a cloth.” But, an inference involving the negations of “being nameable” and “being knowable” cannot give rise to a valid cognition as both these negations are fictitious in nature. It is not possible to find a real illustration for a fictitious entity. Thus, inference involving negations is not possible in the given instance. And so, this is a case of an inference comprising absolutely positive liṅga. An instance of absolutely negative inference is “all live bodies have souls, as they have life breath, etc.” Since the entirety of live bodies forms the pakṣa of this anumāna, there is no entity available for confirming this pervasion in the form of a positive illustration. In such cases, only the negative form of vyāpti, involving the absences of the sādhya and the hetu, gains validation through an instance that illustrates the invariable relation of the absences of the sādhya and the hetu. In the given instance, the negative pervasion may be illustrated as “wherever there is no

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existence of souls, there is no existence of life breath, etc., as in a pot.” This goes to prove that wherever there is life breath etc., there is existence of souls. Hence, all living entities have souls. All schools of thought agree with anvaya-vyāpti being a form of vyāpti. Dvaitins and Viśiṣṭādvaitins do not approve of the kevala-anvayi and kevala-vyatireki categories of vyāpti. Dvaitins contend that fictitious entities being a part of an anumāna does not make the anumāna fictitious. Such entities are acceptable in an inference as the counter entities of their absences. This being the case, practically no difference remains between anvaya-vyāpti and kevala-anvayi-vyāpti. Hence, the classification of kevala-anvayi is futile. In the case of kevala-vyatireki anumāna also, the vyāpti is actually between the sādhya and the hetu. It is only for want of validation through illustration that a negative instance is sought. That being the case, vyatireka-vyāpti may simply be accepted as an aid in confirming the anvaya-anumāna instead of categorizing a separate classification of anumāna as kevala-vyatireki anumāna. In fact, vyatireka-anumāna goes against Naiyāyika’s tenet of anumāna. According to the Nyāya School, parāmarśa is the instrument of anumiti and parāmarśa is a complex cognition of pakṣa being the locus of such a hetu which forms part of a vyāpti of the hetu with a sādhya. In vyatireka-anumāna, pakṣadharmatā involves hetu and vyāpti involves the absence of hetu, whose locus cannot be the pakṣa. This digression in the constituents of parāmarśa technically renders vyatireka-anumāna unsound for the process of anumāna proposed by Naiyāyikas. One would definitely agree that in the case of anvaya-vyatireki vyāpti, the anvaya form of vyāpti is sufficient for a valid anumāna and the vyatireka-vyāpti is redundant. Thus, according to Dvaitins, the liṅga, and thereby the vyāpti comprising it, is only positive. Negative vyāpti is an aid for validation of positive vyāpti rather than being a classification of vyāpti in its own right.

Hypothetical Logic in the Two Schools Hypothetical logic or counterfactual reasoning is an important tool of debate. It is also used to reinforce one’s logic or deduction when inference is employed for giving rise to inferential cognition for others’ sake. While both Nyāya and Dvaita Schools are convinced with the efficacy of hypothetical logic in the process of reasoning, there is divergence in the way in which it is classified and treated in the two schools. Hypothetical logic is called as tarka. It is quite amusing that one of the appellations of the body of theories of the Nyāya School is Tarkaśāstra! When in a debate or while using inference for others’ sake, the opponent/the seeker of inferential cognition doubts or negates the conclusion of the proponent/ instructor of inferential cognition, then hypothetical logic or tarka may be employed to settle the opposition/doubt. Usually, the reason in an inference is mutually acceptable to both parties as it is the reason that triggers the inference by being cognizable.

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An illustration of tarka would enable the further discussion about the topic to be more clear. The following is an argument between a proponent and an opponent based on the instance of proving fire on the hill based on the sight of smoke on the hill. Proponent The hill has fire as it has smoke. Opponent The hill has smoke, no doubt. But let it be contended that the hill does not have fire. Proponent If there were no fire on the hill, there would be no smoke on the hill. There is smoke on the hill that you too agree with. Hence, there is fire on the hill. Opponent Alright, I agree that there is fire on the hill. The presence of smoke is not compatible with the absence of fire as fire is the cause of smoke. That is how smoke and fire are invariably related. Negation of the cause automatically entails the negation of its effect. Since negation of smoke is not desirable to the opponent, the opponent is compelled to shun the argument that let smoke be and fire not be there on the hill. Tarka involves: • Hypothesizing the negation of the condition as proposed by the opponent/seeker of inferential cognition; the condition being the presence of sādhya in an appropriate locus. • Proponent/instructor countering the negation by professing the inevitable undesirable consequence of such negation; the consequence being the negation of hetu in an appropriate locus. • The consequence being undesirable, the opponent/seeker of inferential cognition shuns from negating the consequence; the negation of hetu being impossible as presence of hetu is cognizable. • The opponent/seeker is thereby compelled to give up hypothesizing the negation of the condition undertaken in the first place, to avoid the undesirable consequence. • Thereby the doubt/opposition is eliminated, and the conclusion professed by the proponent/instructor is validated. Tarka primarily comprises of five constituents: • Invariable relation between the hypothesized conclusion and hypothesized reason, such as negation of fire and negation of smoke – āpādakasya āpādyena vyāptiḥ • Non-opposition by a counter logic – pratitarkeṇa apratighātaḥ • Undesirability of hypothesized reason – āpādyasya aniṣṭatvam • Concluding into the contrary (of what the opponent/seeker proposed/doubted about the sādhya) – viparyaye paryavasānam • Unfavorable condition for the other person (compulsion to accept the conclusion contrary to the opponent’s argument in a debate) – parasya-ananukūlatvam

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Both Naiyāyikas and Dvaitins concede that tarka is a benefactor of valid inferential cognition. However, going by their ideals, Naiyāyikas classify tarka as a category of erroneous cognition along with doubt (saṁśaya) and misapprehension (viparyaya). This is so because, though hypothetically, tarka does involve fallacious conditions including negating the presence of hetu on the pakṣa. Tarka is a constituent of inference in the Dvaita School. Anumāna is classified into two categories based on the purpose for which it is framed, namely, anumāna to establish the proponent’s point (sādhanānumāna) and anumāna to refute the opponent’s point (dūṣaṇānumāna). The latter type of anumāna is further classified into two types – anumāna to expose the errors in the opponent’s point (duṣṭi-pramitisādhanam) and hypothetical logic to rebut the opponent’s point by counterfactual reasoning (tarka). The Dvaitins opine that tarka does not merely comprise of hypothesizing a fallacious condition. The process of counterfactual reasoning is completed only by concluding with the agreement of the condition contrary to the fallacious condition hypothesized. The logical process, put together, acts as an instrument of inferential cognition. Hence, it is a classification of anumāna as well as a benefactor of anumāna. A detailed account of the constituents of tarka is provided in Pramāṇapaddhati of Śrī Jayatīrtha.

Fallacies of Inference in the Two Schools It was initially discussed that cognitions can be true or erroneous. This is true of inferential cognition also. The final cognition ends up being erroneous owing to some fallacy in the instrument of such cognition. Such fallacies occur owing to some impairment in the logical process or in the weak argument of a losing disputant in a debate. Logic has two utilities. Firstly, it is a tool that aids in arriving at inferential cognition in routine life. Secondly, it is a weapon with which an opponent is countered and attacked in debate. It is established that inferential cognition is arrived at by passing through a process involving multitude of other cognitions. That goes to show the multitude of possibilities of impairment that may occur in the logical process, at the end of which befalls erroneous inferential cognition. Fallacies in the logical process are also manifold depending on the eventual result that they generate. The Nyāyasūtra of Gotama enlists these as entities whose knowledge is essential to liberate one from mundane existence. They are: • Discussion (vāda) – The tenets of the loser in an argument are proven to be fallacious in vāda. • Wrangling (jalpa) – The disputation that solely aims at victory in an argument by hook or crook, without any doctrinal conviction. • Cavil (vitaṇḍā) – The disputation that solely aims at offending and attacking whatever is presented in an argument.

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• Fallacies of reason (hetvābhāsa) – The tainted reasons that lead to lack of cognition, or erroneous cognition, or give rise to doubtful cognition. • Quibble (chala) – The deliberate misinterpretation of the proponent argument with an alternative meaning of the words used in the argument. • Futile objections (jāti) – The opponent’s objections while well being aware of the efficacy of the proponent’s argument with the mere aim to gain victory against the distracted proponent. • Point of seizure (nigrahasthāna) – Misinterpretation or lack of comprehension by the opponent in an argument that compels the frustrated proponent to quit the debate. Among these, all entities except the fallacies of reason are applied in the context of debate. There are further internal classifications within these categories of fallacies. Fallacies of reason are applicable in case of debate as well as while arriving at routine inferential cognition, both in case of inference for one’s own sake and for the sake of others. Fallacy in anumāna is termed as hetvābhāsa (appearing like valid hetus) in the Nyāya School, while it is identified as upapattidoṣa (error in propriety/justification) in the Dvaita School. Naiyāyikas contend that all fallacy, whether they pertain directly to hetu or otherwise, are attributed to the hetu, as it is the hetu that ultimately gives rise to inferential cognition. Even if the hetu is not fallacious, it is tainted with the error pertaining to any of the erroneous components of inference owing to its connection with such component. This is not acceptable to the Dvaitins. According to them the component of inference that is actually erroneous should be accounted for the error, and thus they classify the fallacies of inference based on the components which are particularly tainted in each case, as pratijñā-doṣa (fallacy of proposition), hetu-doṣa (fallacy of reason), and dṛṣṭānta-doṣa (fallacy of illustration). The Nyāya School propounds five types of hetvābhāsas. Their names vary in the old and the new Schools of Nyāya but are similar in categorization. They are presented with their illustrations below: • Savyabhicāra – Straying reason (In the old school – Anaikāntika – Erratic reason) – Sādhāraṇa (nonconclusive reason) – Hill is fiery as it is knowable – Asādhāraṇa (reason exclusive to the ground of inference) – Sound is eternal due to soundness – Anupasaṁhārī (nonexclusive reason) – Everything is non-eternal due to being knowable • Viruddha (contradictory reason) – Sound is eternal as it generated • Satpratipakṣa (counterbalanced reason) – Sound is eternal as it is audible, like soundness (is eternal and audible) Countered by the inference – Sound is non-eternal as it is generated, like a pot (In the old school – Prakaraṇasama – Balancing the controversy) • Asiddha – Unknown reason (In the old school – Sādhyasama – Unproved reason)

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– Āśrayāsiddha (non-establishment of the locus) – Sky-lotus is fragrant due to being a lotus, like the lotus flower – Svarūpāsiddha (non-establishment of reason in the ground of inference) – Sound is a quality as it can be seen – Vyāpyatvāsiddha (non-establishment of being pervaded) – Hill has smoke as it has fire The conditional adjunct that prohibits pervasion here being – wet fuel • Bādhita (sublated reason) – Fire is not hot, as it is an entity This cognition is sublated later by the tactile perception of fire being hot. (In the old school – Kālātītaḥ – Mistimed Reason) The broad classification of upapatti-doṣas in the Dvaita School comprises of defect of objects (artha-doṣa) and defect of speech (vacana-doṣa). Therein the fallacies that directly involve defect in objects are two – contradiction, virodha, and incongruity, asaṅgati. Those fallacies that are related to speech, and thereby to the related object, are omission of essential, nyūna, and addition of the redundant – ādhikya. These four fallacies of inference are, respectively, applicable in cases of both debate and routine inferential cognition. In the context of routine inferential cognition, fallacies may pertain to proposition (pratijñā), reason (hetu), and illustration (dṛṣṭānta), as mentioned earlier, depending on the particular component of inference tainted in each instance. The following is the illustrated classification of the fallacies in routine inferential cognition in the Dvaita School: • Virodha (contradiction) – Fire is not hot – Pratijñā-virodha (contradiction to proposition) – The hill is not fiery • Pramāṇa-virodha (contradiction by proof) – Prabala-pramāṇa-virodha (contradiction by a stronger proof) • Bahutvena prabala (quantitatively stronger) – The hare’s horn is real, as it has hornness • Svabhāvena prabala (intrinsically stronger) – The cow may be touched by one’s foot (it is contradictory to Vedic scripture) – Pratyakṣa-virodha – Fire is cold – Anumāna-virodha – God does not exist as He is not seen – Āgama-virodha – The cow may be touched by one’s foot – Samabala-pramāṇa-virodha (contradiction by an equally strong proof) • Samānānumāna-virodha (contradiction within the inference) – The lake has fire as there is smoke • Anumānāntara-virodha (contradiction with another inference) –Sound is eternal as it is audible (counter inference) – Sound is non-eternal as it is generated • Svavacana-virodha (contradiction by one’s own statement) – Apasiddhānta (contradiction to doctrine of one’s own school) – When a Naiyāyika says sound is eternal (as against the school’s doctrine in the process of countering others)

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– Jāti (self-contradiction) • Svavākya-virodha (contradiction within one’s own sentence between sentential components) – Sound is eternal as it is generated • Svakriyā-virodha (contradiction to one’s speech by one’s action) – Saying – Fire is cold, seeking fire for heat • Svanyāya-virodha (contradiction between one’s speech and accepted principles) – When a Naiyāyika says sound is eternal (as against the School’s doctrine on one’s own accord) – Hetu-virodha (contradiction to reason) • Asiddhi (non-establishment) (non-apprehension of the reason in an appropriate ground) – Pramityasiddhi (non-establishment of cognition) – Usage of “due to smoke” as a reason while not knowing the distinction between smoke and vapor – Āśrayāsiddhi (non-establishment of locus) • Siddhasādhana (proving the proven) – is not a defect when arriving at inferential cognition for one’s own sake, but a defect when it is for others’ sake as in saying a person convinced about the hill being fiery that “the hill is fiery” • Asadāśraya (fallacious locus) – like Sky-flower, is not a defect if there are no other defects – Vyāpyatvāsiddhi (non-establishment of being pervaded) – The hill has smoke as it has fire The condition that prohibits pervasion here is wet fuel. • Avyāpti (non-pervasion) – Sādhya-tadabhāva-sambandha (pervasion with both conclusion and its absence) – The hill has smoke as it has fire – Sādhyābhāva-mātra-sambandha (pervasion with the absence of conclusion alone) – The hill has water as it has smoke – Ubhaya-sambandhābhāva (non-pervasion with both conclusion and its absence) – Sound is eternal as it has soundness – Dṛṣṭānta-virodha (contradiction to illustration) • Sādhya-vaikalya (lacking conclusion) – The mind is non-eternal as it has material form, like the atom • Sādhana-vaikalya (lacking reason) – The mind is non-eternal as it has material form, like movement • Ubhaya-vaikalya (lack of both) – The mind is non-eternal as it has material form, like the ether Similar to the classification of virodha as related to pratijñā, hetu, and dṛṣṭānta, asaṅgati, nyūna, and ādhikya are also classified in the three said categories as follows: • Asaṅgati (incongruity) – Pratijñā-asaṅgati (incongruity to proposition) – Fire pervades smoke

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– Hetu-asaṅgati (incongruity to reason) – The hill has no fire – Dṛṣṭānta-asaṅgati (incongruity to illustration) – All things are nameable as they are knowable • Nyūna (omission of essential) – Pratijñā-nyūna (omission of essential in proposition) – Merely uttering “due to smoke” – Hetu- nyūna (omission of essential in reason) – Merely uttering “the hill” – Dṛṣṭānta-nyūna (omission of essential in illustration) – Merely uttering “as in” while illustrating • Ādhikya (addition of the redundant) – Pratijñā-ādhikya (addition of the redundant in hypothesis) – The hill possessing a particular luminance has fire as it has smoke – Hetu-ādhikya (addition of the redundant in reason) – The hill has fire due to smoke and due to possessing a particular luminance – Dṛṣṭānta-ādhikya (addition of the redundant in illustration) – The hill has fire as it has smoke like in the case of the kitchen possessing a particular luminance Some fallacies of inference share some illustrations in common; however, the perspectives with which the illustrations are viewed account for the variation in the defects discussed. The exhaustive list of fallacies of inference in the Dvaita School make up for the voluminous classification of points of seizure – nigrahasthānas in the Nyāya School, 22 in all, listed as follows: (i) (ii) (iii) (iv) (v) (vi) (vii) (viii) (ix) (x) (xi) (xii) (xiii) (xiv) (xv) (xvi) (xvii) (xviii) (xix) (xx) (xxi)

Pratijñā-hāni – Loss of proposition Pratijñāntara – Change of proposition Pratijñā-virodha – Contradiction to proposition Pratijñā-sannyāsa – Renouncing proposition Hetvantara – Change of reason Arthāntara – Change of object Nirarthaka – Futile Avijñātārtha – Unknown object Apārthaka – Absurdity Aprāptakāla – Untimely Nyūna – Deficit Adhika – Excess Punarukta – Redundant Ananubhāṣaṇa – Speechless state Ajñāna – Ignorance Apratibhā – Lack of timely response Vikṣepa – Discontinuation Matānujñā – Agreement with the opposition Paryanuyojyopekṣaṇa – Ignoring the point of seizure Apasiddhānta – Contradicting one’s own doctrine Hetvābhāsa – Fallacies of reason

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Against this list of points of seizure of the Nyāya School, the Dvaita School presents its list of six nigrahasthānas and claims that the rest of the list presented by the Naiyāyikas finds inclusion in the fallacies of inference accepted in the Dvaita School. The six points of seizure of the Dvaitins have only two additions to the four classifications of fallacies of inference detailed in the foregone section. The additional points are: • Saṁvāda – Admission of the disputed tenet • Anukti – Speechlessness Through the foregone discussion, it is evident that theories of logic in the Indian epistemological pantheon have been a subject of intricate analysis and perennial debate. There have been convergences and divergences, more of divergences rather, aligned to the varied basic tenets of each school of thought. While the components of inferential cognition, namely, the reason, the ground, the conclusion, the illustration, etc., remain common to all philosophers, everything else pertaining to their definition, their classification and categorization, the logical process, etc. finds divergence in the different philosophical schools.

Conclusion Cognition is an inevitable experience of all beings of the world. The world, which provides the experience of cognition, is quite baffling with its share of sensorial, super-sensorial, and non-sensorial (yet cognitive) objects in it. Whether the world is as is apprehended is a matter of debate among philosophers. Many schools of thought, including philosophy and science, strive to encapsulate the entirety of world and experiences of its beings summarily by forming systems that would explain appropriately as to what experience is, what is experienced, how experience happens, and what its variations are. Philosophically, both Naiyāyikas and Dvaitins are realists and dualists. That accounts for the many convergences in their theories. However, their perspectives about God, world, and its beings differ vastly. The ultimate purpose of life of beings according to both the schools is to attain cessation from the cycles of birth and death. But the experience of beings in such state is more or less envisaged in the Nyāya School as a sorrowless state attained through knowledge of all entities (practically through omniscience), while it is believed by Dvaitins to be as revealed in the sacred scriptures. Then, for the Naiyāyika, inference becomes a vital tool that explains the experiences that are supersensory, including the proof of existence of God. The Vedas and other scriptures are used for reference and validation of what is cognized by perception or inference. Even with regard to sensory experiences like that of cognition of color in an object, Naiyāyikas churn out categories of entities like inherence relation (samavāya), particulars (viśeṣa) using inference, to handle the supersensory entities or such connections among entities. Inference is practically the

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epistemological savior of the Naiyāyikas, which is why they expend a lot of energy in developing the theories pertaining to inference, which sometimes end up being quite artificial. For an illustration, one may refer to the logical process of five-limbed syllogism giving rise to inferential cognition discussed in this chapter. For the Dvaitin, perception and Vedas are intrinsically superior as means of cognition. Without perception, even verbal cognition of Vedas is not attainable. Hence, any cognition that contradicts perception is not valid according to this school. For things that are supersensory, perception is not plausible as a means. In such instances the all-encompassing Vedas are accepted as authoritative in the Dvaita School. Inference, then, is dependent on the other two cognitions and is rather an aid to perception and verbal testimony here that fills the gaps in comprehension of cognitions by resorting to a logical means. That is why the theory of logic in this school is analyzed and presented in light of common experiences that any common person will be able to relate to and could use it as a tool to resolve seeming logical lapses in perception and verbal testimony. The status that inference has in the epistemological systems of the two schools is different which in turn propagates the divergences in the formulation of their logical theories. This chapter aims to provide an introduction to the theories of logic of the two schools while broadly presenting the convergences and divergences therein. A study of the vast dialectic literature that spanned over centuries of exchange, debate, and improvisation will reveal the extent of intellectual exercise that had gone behind this logical endeavor.

Summary • Cognition, its classification and its means, is an important matter of discussion among the various schools of philosophy in India. They vary from each other with regard to the number and types of cognition accepted. So also do their definitions and further classifications. This chapter focuses on discussing about logic as a tool of inferential cognition in the Nyāya and Dvaita Schools and the points of convergences and divergences between the two schools in this regard. • Both Nyāya and Dvaita Schools accept the taxonomy of cognition as perceptual, inferential, and verbal. In addition, the Nyāya School propounds comparative cognition as an independent category. • While all schools call cognition as pramiti and its means as pramāṇa, the Dvaita School calls pramiti as kevalapramāṇa and pramāṇa as anupramāṇa. • The means of inferential cognition or anumiti is inference or anumāna. In the Nyāya School, parāmarśa – deduction – is anumāna. It is a complex cognition of the ground of inference attributed with the reason of inference as qualified by the cognition of invariable relation of pervasion between the pervading and pervaded entities involved in the inference. In the Dvaita School, anumāna is defect-free reasoning.

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• The invariable relation or vyāpti between the pervading (vyāpaka) and pervaded (vyāpya) entities in an inference is related to each other by invariable coexistence on common grounds spatially and/or temporally according to the Nyāya School. The Dvaita School does not emphasize on spatial and/or temporal coexistence of the vyāpya and vyāpaka in vyāpti. Invariability is the only condition of vyāpti between them which they may have, being in appropriate locations and times that vouch such invariability between them. • The mark (liṅga), which is the reason in an anumāna, is of three types according to the Nyāya School – absolutely positive, absolutely negative, and positive cum negative. According to Dvaitins, the mark is actually positive but may be substantiated using a negative inference when there is no positive illustration to prove the same. So, negative mark is not an independent category of the mark or inference but only an aid for the positive inference. • The limbs of syllogism in an inference are five according to the Naiyāyikas. The Dvaitins find the number and sequence of limbs of syllogism as artificial and prescribe that as many limbs of syllogism that serve the purpose of inference in each case need to be administered. • Counterfactual reasoning (tarka) is classified as an erroneous cognition in the Nyāya School due to the element of fallacy involved in the imposition of absence of reason in its very presence. The Dvaita School assigns it the status of being a constituent of anumāna as the very intention of counterfactual reasoning is to prove the conclusion, though by using an argument that contradicts the facts to start with, but ends up countering the false imposition and proving the conclusion, all the same as inference. • All the fallacies in anumāna are attributed to the reason (hetu) of the inference in the Nyāya School. The Dvaitins classify fallacies of inference into three categories, namely, those pertaining to the reason, to the proposition, and to the illustration. Broadly, the fallacies of inference are classified in to five categories in the Nyāya School and four categories in the Dvaita School. • The points of seizure are enumerated to be 22 in the Nyāya School, while the Dvaita School broadly classifies them as six.

Definitions of Key Terms Nyāya definitions and unspecified ones are from Tarkasaṅgraha and Dvaita definitions are from Pramāṇapaddhati. 1. Anumānam – Inference Nyāya definition (anumiti-karaṇam anumānam) – The means of inferential cognition is inference. Dvaita definition (nirdoṣopapattiḥ anumānam) – Defect-free reasoning is inference. 2. Vyāptiḥ – (Invariable) pervasion

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Nyāya definition (sāhacarya-niyamaḥ vyāptiḥ) – Invariable coexistence is pervasion. Dvaita definition (avinābhāvaḥ vyāptiḥ) – Existence (of one entity) not without the (other pervading entity) is pervasion. 3. Parāmarśaḥ (equivalent to anumānam in the Nyāya school) – Deduction vyāpti-viśiṣṭa-pakṣadharmatā-jñānaṁ parāmarśaḥ – Deduction is the complex cognition of the ground (of inference) as possessing an attribute (namely, the reason) that is qualified by (the prior cognition of invariable) pervasion (between the reason and conclusion). 4. Pakṣaḥ – Ground (of inference) sandigdha-sādhyavān pakṣaḥ – The (entity) having a doubtful conclusion (pervading entity) is the ground (of inference). 5. Tarkaḥ – Counterfactual reasoning Nyāya definition (vyāpyāropeṇa vyāpakāropaḥ tarkaḥ) – Imposition of the pervading entity (that is undesirable to the opponent in an argument as it is counterfactual) through imposition of the pervaded entity is counterfactual reasoning. Dvaita definition (kasyacid-dharmasya aṅgīkāre arthāntarasya āpādanaṁ tarkaḥ) – Counterfactual reasoning is causing to accept another object (which is undesirable to the opponent due to being counterfactual) on acceptance of an object (by the opponent, which is opposed to reason).

References Athalye, and Bodas, eds. 1963. Tarkasangraha of Annambhatta. Pune: BORI. Granthamālā, U.V. 1978. Nyayaparisuddhi by Sri Vedanta Desika, Anumānādhyāya, Madras. Madhavananda, Swami. 1942. Vedānta-paribhāṣā of Dharmarāja Adhvarī ndra. Chapter II Inference. Belur Math/Howrah/West Bengal: Swami Vimuktananda. Shastri, Narayanacharana, and Shvetavaikuntha Shastri, eds. 1995. The Kashi Sanskrit Series – 212: Nyayasiddhantamuktavali of Vishvanatha Pancanan. Varanasi: Chaukhamba Sanskrit Sansthan. Vaishnavi Rao, R. 2004. Dvaitavedānte anumānaṁ tadaṅgāni ca, Thesis submitted to Rashtriya Sanskrit Vidyapeeth, Tirupati. Varakhedi, Shrinivasa. 2011. The path of proofs-Pramāṇa-paddhati of Śrī Jayatī rtha. Chapter II Inference. Manipal: Manipal University Press. Tr. Mahamahopadhyaya Satisa Chandra Vidyabhushana. 1913. Vol. III: The Nyaya Sutras of Gotama. In The sacred books of the Hindus, Book I Chapters 1 & 2, ed. B.D. Basu. Allahabad: Panini Office.

The Opponent: Jain Logicians Reacting to Dharmakīrti’s Theory of Inference

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Marie-Hélène Gorisse

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Jaina Contribution to the Composition of an Inferential Reasoning . . . . . . . . . . . . . . . . . . . . . . The Canonical Form of an Inferential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Evolution of the Form of an Inferential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Divergences Between Jains and the Buddhist Dharmakīrti . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Jaina Contribution to the Evaluation of an Inferential Reasoning . . . . . . . . . . . . . . . . . . . . . . . . Prabhācandra Against the Solution to the Problem of Induction of the Buddhist Dignāga . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Jaina Solution: A Perceptual-Like Grasp of Universals Named “Tarka” . . . . . . . . . . . . . . Prabhācandra’s Defense of the Jaina Solution Against Dharmakīrti’s Conception . . . . . . . . The Jaina Contribution to the Formal Structure of an Inferential Reasoning . . . . . . . . . . . . . . . . . . Dharmakīrti’s Theory of Ontic Foundations for Valid Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . Jaina Amendments to Dharmakīrti’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Jaina Novelty: From Non-apprehension to Negative Inferential Statements . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The goal of this chapter is to present the contribution made by Jaina philosophers in the conception of inference in the classical paradigm of philosophy in India. In this perspective, the attacks and defenses that Jaina philosophers in the lineage of Akalaṅka performed against the Buddhist thinker Dharmakīrti are precious witnesses, not only of the development of the Jaina conceptions but also of their contribution to the general constitution of the framework of debate. This chapter is moreover conceived to clarify the consequences of argumentative, epistemic, M.-H. Gorisse (*) Faculty of Arts and Philosophy, Ghent University, Ghent, Belgium e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_15

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and more formal considerations at important transition periods of the development of these conceptions of inference. Keywords

Inference · Truth-preserving argument · Jainism · Akalaṅka · Māṇikyanandin · Prabhācandra · Buddhism · Dharmakīrti · Discernment of universals · Theory of knowledge · Theory of argumentation · Logic · Negation · Necessity · Evidence

Introduction In the Indian framework of philosophy in the classical period, inferential reasoning is presented as the tool to acquire new knowledge through reasoning. Its stated form is in turn conceived as a truth-preserving argument by means of which it is possible to bring awareness to somebody else, that is to say to convince another. What is more, a remarkable peculiarity of this framework is that the different traditions share the belief that from this truth-preserving argument, it is possible to define the standards of an ideally organized rational discussion the outcomes of which are necessary true statements. From this, philosophers from a plurality of obedience, especially from Buddhism, Jainism, and Nyāya-Vaiśeṣika, offered decision procedures on the correctness (or incorrectness) of such arguments and they succeeded in developing a common inter-doctrinal framework of argumentation. This chapter presents the Jaina contribution on these issues, especially as it is found in the work of Prabhācandra, a scholar active in the turn of the tenth century (980–1065) who wrote the Sun [that opens] the Lotuses of the Knowable (Prameyakamalamārtaṇḍa, henceforth PKM). The PKM is a commentary on the Introduction to Philosophical Investigation (Parī kṣāmukham, PM), a treatise composed by Māṇikyanandin (ninth century). In turn, the PM is a methodical and aphoristic presentation of Akalaṅka’s philosophy (720–780). This lineage of Jaina authors challenged the conceptions of the famous Buddhist philosopher Dharmakīrti (seventh century) and got engaged in a discussion with Buddhist authors when developing their own theories of inference and argumentation. In this dynamic, Akalaṅka is the one who offers “a doctrine of type of knowledge (pramāṇa) typical of Jains” (Dixit 1971, p. 143). But Akalaṅka’s presentation is very concise and unsystematic. It therefore became the task of later thinkers, such as Vidyānanda (ninth century) and Māṇikyanandin to present a more structured version of Akalaṅka’s innovative theory, and of Prabhācandra to present this in a developed way with precise references to the discussions with other schools, see Soni (2014). In his investigation on inference, Prabhācandra especially quotes Dharmakīrti’s Essay on Knowledge (Pramāṇavārttika, PV) and Auto-commentary on the Essay on Knowledge (Pramāṇavārttikasvavṛtti, PVsV). The first section of this chapter is a presentation of the form of this truthpreserving argument, focusing on the discrepancies between Dharmakīrti’s conception and the conception of selected Jaina authors in the lineage of Akalaṅka. The

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investigated discrepancies concern more precisely the distinction between psychological and formal considerations in an attempt to bring the stated form of an inference to its minimal form. The second section consists of an investigation into the decision procedures according to which an argument is granted as being a truth-preserving argument, or not. In this dynamic, an argument is considered as being probative if and only if the “evidence-property” it features is shown to be necessarily concomitant with the “target-property.” Here again, Jaina philosophers oppose the Buddhist theories and have to defend themselves against Dharmakīrti’s criticisms. The third section aims at a general presentation of the types of relation between the evidence-property and the target-property, focusing on Jaina amendments to Dharmakīrti’s theory, as well as on their innovative treatment of negations.

The Jaina Contribution to the Composition of an Inferential Reasoning First of all, inference is the means to acquire new knowledge through the careful examination of what can be concluded with certainty from previously acquired knowledge. In the Indian paradigm of philosophy, a classic example is to infer that something is not permanent from the already-known fact that this same thing is a product: first, because by definition, every product has been created and is therefore conceived as having parts that have been arranged; second, because philosophers from the different schools agree on the fact that an arrangement cannot be permanent. In other words, inference is the means to proceed from the recognition that a knowledge statement is granted, to the recognition that another knowledge statement is also necessarily granted on the basis of it.

The Canonical Form of an Inferential Reasoning According to the Aphorisms on Logic (Nyāyasūtra, NS) of Gautama, a Naiyāyika text of the second century CE considered as the root text for logical considerations in classical India, the only good way to express such a truth-preserving argument consists more precisely of a group of five statements: NS.1.1.32. [Statement of] the thesis, the evidence, the account, the application and the conclusion are the constituents [of inference] (pratijn˜ ā-hetu-udāharaṇa-upanayanigamanāny avayavāḥ), (Jhā 1984, p. 355).

Following this pan-Indian tradition, Māṇikyanandin introduces in PM.3.65 the stated form of an inference using the same five-statement example put forward in the Naiyāyika tradition: [Thesis] Sound is impermanent (pariṇāmī s´abdaḥ). [Evidence] Because it is a product (kṛtakatvāt).

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[Account] Whatever is such (a product) is alike (it is impermanent), such as a pot (ya evaṃ sa evaṃ dṛṣṭo yathā ghaṭaḥ). [Application] This (sound) is a product (kṛtakas´ ca ayaṃ). [Conclusion] Therefore it is impermanent (tasmāt pariṇāmī iti), (Goshal 1940, p. 127). To explain, the thesis is what is to be inferred. It consists of the ascription of a property, called the “target-property” (sadhya), to a given object, called the “locus” (pakṣa). And the evidence-statement is the tool to infer this. It pertains to stating that another property, called the “evidence-property” (hetu), is already granted as being ascribed to this given object. As for the account, it consists both of the indication of the necessary relation that holds between the evidence-property and the targetproperty, as well as iterating an example. This form of argument is an extended version of the modus ponens: P, P ! Q Q And can be reconstructed as follow: [Thesis] a1 is Q. [Evidence] Because a1 is P. [Account] Whatever is P is also Q, as in a2. [Application] This is how a1 is P. [Conclusion] Therefore a1 is Q. In this Naiyāyika presentation followed by Jains, it seems that both bottom-up and top-down perspectives are used. More precisely, the bottom-up approach consists of the three first steps. It proceeds from conclusion to premises and insists on the justifications to be produced in order to legitimize the claim of a given thesis. As for the top-down approach, it consists in the three last steps. It proceeds from premises to conclusion and insists on what can be concluded from given previous knowledge. According to the Nyāya, both these focuses bring different inputs and are required steps in order to ensure the adhesion of the interlocutor, since it has been said: NS.5.2.12. Defective is [the inference] in lack of any of its constituents (hī nam anyatamena apy avayavena nyūnam), (Jhā 1984, p. 1757).

Evolution of the Form of an Inferential Reasoning Even though they regularly use the five-statement form of reasoning during debates, the other traditions claim that only a subset of the five-statements is sufficient in order to bring the adhesion of the interlocutor. A brief survey of the evolution of the form of the truth-preserving reasoning makes it evident that there has been a decision to bring it to its minimal form – that is to say to keep only the elements having a probative value. This, in turn, regularly implies a decision to discriminate and treat separately psychological conditions from more formal ones. In this process, Shah (1967, p. 282), and Mookerjee (1935, p. 364), remark that the Jaina approach is the

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one that goes the furthest. More precisely, the old Nyāya advocated a ten-membered reasoning in which explicit psychological conditions, like doubt and desire to know, are required. Then the NS brought it to five conditions linked only with structure of the argument. In this canonical form, redundancy has been pointed. Especially, the Buddhist Dharmakīrti claimed that the statement of the account accompanied either by the statement of evidence or by the statement of application is sufficient. Jaina philosophers, particularly since Siddhasena’s seventh century Guide of Logic (Nyāyāvatāra, NA), are opposed to the Buddhist conception and recognize only the statement of the thesis and of the evidence as necessary argumentative moves, see Balcerowicz (2008, pp. 59–64), although they recognize that up to ten argumentative moves can be useful for pedagogical purposes. In Māṇikyanandin’s terms: PM.3.46. These (account, application and conclusion) may be for the understanding of those who have little knowledge and for this purpose may be discussed only in the Śāstra, but these are quite unfit to be used in logical discussions (bāla-vyutpatty-arthaṃ tat-traya-upagame s´ā stra eva asau na vāde, anupayogāt), (Goshal 1940, p. 111).

A noticeable aspect of the Jaina approach is their rejection of the statement of the account, that is to say of the statement that “whatever is P is also Q, as in a2.” This statement is a pivotal one, since it contains the indication of the necessary relation that holds between the evidence-property and the target-property. But this statement is also a problematic one, since it contains the displaying of an example. In classical India, there were major controversies between those who considered the indication of an example as probative, and those who did not. The idea behind the requirement of an example can be traced back to the need to show that there is at least one case other than the case under consideration in which the evidence-property occurs when the target-property also occurs. This need can be explained in the following way: if the object under consideration is the only locus of the evidence-property, one can never be sure that there is an essential (and not only an accidental) relationship between the two properties. The classical example is that “sound is eternal, because it is audible.” In this example, since sound is the only audible thing, it is not possible to find any other example of another audible thing that would be eternal also. Therefore, the argument is not a convincing one, because nothing is known about the relationship between audibility and eternality and that it is as well possible to claim that “sound is non-eternal, because it is audible.” What is more, the idea behind the requirement of an example can be traced back to an old theory of drawing inferences from paradigmatic examples. For example, in order to know whether grains of rice are cooked or not, it is not required to taste each of them. Indeed, tasting a few of them is sufficient to infer from their state the state of the other ones, because they are all under the same conditions. Therefore, rejecting the step of the account, which includes the indication of an example, represents a step further towards mere formal considerations. But this was only possible from the Jaina perspective, because only they could take the liberty of rejecting the need to indicate the special relationship between the two inferential properties. Indeed, according to Jaina philosophers, this special

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relationship is known thanks to another, noninferential, type of knowledge. The causes, modalities, and consequences of this position are considered in the next section.

Divergences Between Jains and the Buddhist Dharmakīrti In this picture, Jaina philosophers hold that in order to be ensured of the adhesion of the interlocutor in a debate in which he always performs the best argumentative move, it is sufficient to state the thesis and the evidence-statement. In the abovementioned example, stating the evidence pertains to stating that sound is a product. The fact that there is a probative value of the statement of evidence is clear, since evidence is either the very justification of a claim or the trigger that prompts a conclusion. More precisely, since everybody agrees that whatever is a product is also impermanent, then agreement on the fact that sound is a product is the very trigger (justification) of the agreement on the fact that sound is impermanent. But how is there a probative value of the statement of the thesis? Jaina philosophers have to answer this question, since the Buddhist Dharmakīrti, following Dignāga, holds that the thesis statement does serve a purpose, namely it indicates the goal of evidence, yet it is in no way a necessary part of an inference as a type of knowledge: PV.4.15. The statement of that [thesis] which is powerless is explained as having the goal of evidence (hetvartha) as its object (hetv-artha-viṣayatvena tad-as´akta-uktir ī ritā), translated in (Tillemans 2000, p. 30).

And Prajñākaragupta (ninth century) to comment: PVBh.488.5–6. But for us, the presentation of what is to be established is [regarded] as having as [its] object the goal of evidence and is not [regarded] as having a probative value (asmākaṃ tu yo ’numeya-nirdes´aḥ sa hetv-artha-viṣayatvena na sādhanatvena), (Tillemans 2000, p. 30).

This conception challenges the Jaina theory according to which if one does not state the fact that the target-property is ascribed to the object under discussion, then evaluating an inference would be like seeing an archer hitting a target without knowing whether it was the intended target or not. The first mention of this argument is found in Siddhasena’s NA: NA. 14. [. . .] the pronouncement of the thesis has to be made here as showing the domain of the evidence-property (tat-prayogo’tra kartavayo hetor gocara-dī pakaḥ), translated in (Balcerowicz 2008, p. 59). NA.15. Otherwise, for [a person] to be apprised, who is confused regarding the domain of the evidence intended by the proponent, the evidence might appear to be suspected of being contradictory, just like. . . (anyathā vādy-abhipreta-hetu-gocara-mohinaḥ, pratyāyyasya bhavedd hetur viruddha-ārekito yathā), (Balcerowicz 2008, p. 59). NA.16. . . .for a person watching an archer’s skill, the archer who hits without the specific mention of the target [is endowed with both] skill and its opposite (dhānuṣka-guṇa-

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saṃprekṣi-janasya parividhyataḥ, dhānuṣkasya vinā lakṣya-nirdes´ena guṇa-itarau), (Balcerowicz 2008, p. 60).

But for Dharmakīrti, iterating the thesis is one condition of the possibility of the mere performance of an inferential process, exactly in the same way as doubt and desire to overcome this doubt. For example, doubt is a condition of possibility of inference, since it is not plausible that someone who directly sees an elephant is in position to infer the existence of this elephant from the sound of its trunk. Focusing on the Jaina account, it is noteworthy that a specific Jaina condition of the possibility of inference, like any indirect cognitive process, is for the epistemic agent to be in a special state of awareness called “destruction-cum-subsistance” (kṣayopas´ama). In this state of awareness, the karma which obstructs the inherent capacities of the soul has been only partially removed. In other words, these psychological conditions are a fact. But if stating them was a formal requirement, then stating a truth-preserving argument would be an infinite process. Indeed, it is always possible to imagine new psychological conditions. Therefore, these have to be stated separately and not to be considered as part of the argument. With this, Dharmakīrti reminds us that to be a condition of possibility of a probative tool does not equal being a probative tool. What is more, the statement of the thesis cannot be of probative relevance, simply because it is precisely what is to be proved by means of inferential reasoning; if it had a probative strength, the argument would be a circular one. Māṇikyanandin stands against Dharmakīrti’s position from another line of argument and state in PM.3.36 that Buddhist philosophers cannot disagree on the necessity of the statement of the locus, which is part of the thesis, because otherwise they cannot substantiate their own doctrine. Indeed, it is important to state the locus in order to deal with a particular instance of the ascription of a property to a given object and not with the universal relationship between two properties. And this is especially important for the Buddhists following Dignāga, for which one requirement for an evidence to be correct (as developed in the next section) is that it should be present in this locus. But the Buddhists do agree on this much, since they do recognize that either stating the evidence or stating the application is required and that both of them include a mention of the locus.

The Jaina Contribution to the Evaluation of an Inferential Reasoning Prabhācandra Against the Solution to the Problem of Induction of the Buddhist Dignāga Now that the main characteristics of the form of a convincing argument are understood, the next step is to recall that such arguments are displayed in philosophical debates on an inter-doctrinal level, and that decision procedures are needed for ensuring the fact that an agreement between the different participants on the fact that a given inferential reasoning is (or is not) a truth-preserving argument can be

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reached. Such a decision procedure takes the following form: the assertion of a stated inference is traditionally followed by a regulated debate focusing on the characteristics of the evidence-property. This debate aims either at proving that the piece of evidence is a probative one or at refuting that it is. And the statement of evidence is considered as probative if and only if the evidence-property is necessarily concomitant with the target-property. In other words, logical treatises tackle the following question: how is it possible to know that the two inferential properties are necessarily, and not only accidentally, linked? How to be certain that “being a product” ensures “enduring changes” not only in a subset of every possible situations? With respect to this problem, traditionally called the problem of induction, Jaina philosophers engaged in discussions with both Buddhist and Naiyāyika philosophers. First of all, the question whether it is possible to know that all Ps are also Q from the knowledge that all a1, a2, . . ., an are Ps that are also Q is phrased the following way in the Indian tradition: Is there a necessary concomitance (vyāpti) between the evidence-property P and the target-property Q? The Jaina answer to this question is that the only necessary and sufficient condition to be ensured of the presence of a necessary concomitance is precisely to know that the evidenceproperty has the characteristic of being “impossible otherwise” (anyathānupapatti) than in the presence of the target-property. In Māṇikyanandin’s words: PM.3.15. Evidence is characterised by being inseparably connected with the target-property (sādhya-avinābhāvitvena nis´cito hetuḥ), (Goshal 1940, p. 91).

In (Balcerowicz 2003, p. 343), Balcerowicz shows that this new Jaina conception of what counts as good evidence is to be traced back to a lost treatise, the Torment of the Triple Characteristic (trilakṣaṇakadarthana), an eighth-century work of Pātrasvāmin directed against the theory of the triple characteristic of evidence developed by the Buddhist Dignāga (480–540), in whose lineage Dharmakīrti stands. Dignāga’s theory of the triple characteristic of evidence is an attempt to discriminate between accidental and necessary relationships by stating that an evidence is a good one if and only if it is: (i) Present in the case under consideration (pakṣa-dharmatva) (ii) Present at least in one similar case (sapakṣa-sattva) (iii) Absent in dissimilar cases (vipakṣa-asattva), translated in (Hayes 1988, p. 238). More precisely, the first clause prevents cases in which the necessary relationship is not instantiated in a particular case. This requirement comes from the fact that the final aim of inference is to ensure knowledge concerning the particular case. What is more, the third clause prevents cases of inconclusiveness, that is to say cases in which there is a counterexample. As for the second clause, it is meant to avoid a situation in which one of the means used to check the relevance of the necessary relationship is absent. This clause is the Achilles’ heel of Dignāga’s theory and requires detailed comments. Indeed, this clause has been widely

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criticized by classical Indian philosophers, because it prevents situations that are formally valid. More precisely, the only way to defeat someone who claims “this is impermanent, because this is audible” is either to challenge the premises by showing that this is not audible (first condition) or to challenge the relationship between being audible and being impermanent, which can be done only by showing that there is at least one case in which it is true that something is both audible and not impermanent (third condition). But nothing about the relationship between the two properties can be concluded from a case in which it is not true that something is audible (second condition). In other words, it can perfectly be the case that there is a necessary relationship between the evidence-property and the target-property when there is no similar case. If Dignāga did not accept it as a situation ensuring the presence of a good evidence although it is a formally valid situation, it is because he had in mind persuasiveness: a situation without a similar case is not persuasive, as said earlier, because when the evidence is present only in the object under consideration and in no other object, it is harder to know whether the connection between the two properties is an accidental one or an essential one. What is more, this requirement is the sign that Dignāga could not yet free himself from the old model of inference from sampling, that is to say inference from a paradigmatic case. In conclusion, even if Indian philosophers at that time made regular breakthroughs with regard to formal considerations, they never explicitly formulated the requirement of formal validity and they were trying to define certainty by means of persuasiveness. Facing this tradition, Prabhācandra, like many philosophers from different schools from the eighth century onwards, argues that the three signs are neither sufficient nor necessary in order to be assured of the correctness of the inferential evidence, and that Dignāga’s theory lacks a theory of the relevant relationship between evidence-property and target-property, see (Gorisse 2014). For example, when commenting on Māṇikyanandin’s PM.3.15, Prabhācandra shows that there are situations in which the three signs indeed qualify the evidence-property, and yet the evidence-property is not probative: PKM.357.1–3. The three characteristics [described by Dignāga] should absolutely not define evidence, because they can be the case even when it is incorrect, as in “these fruits are ripe, because they bloom on one [and the same] branch, like this fruit [that is also ripe]”, as well as in “this Devadatta is stupid, because he is the son of this [man], like this other son of this [man] [who is also stupid]” (nanu trairūpyaṃ hetor-lakṣaṇaṃ mā bhūt “pakvāny etāni phalāny eka-s´ā khā-prabhvatvād upayukta-phalavat” ity ādau “mūrkha-ayaṃ devadattas tat-putratvād itara tat-putravad” ity ādau ca tad-ābhāse api tat-sambhavāt), (Shastri 1912, p. 357).

The last argument can be reconstructed as follows: [Thesis] Devadatta is stupid. [Evidence] Because he is the son of this man. [Account] Whoever is the son of this man is stupid, like this other son of the same man. In this argument, the evidence-property is not probative, because lesser cognitive abilities can be due to other factors than genetic and educational ones linked to this

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precise man, as for example if there was a lack of oxygen for too long a period in parts of the brain during childbirth. Such a cause of lesser cognitive abilities is accidentally and not essentially connected with the evidence-property, namely the fact of being a son of this man. Yet, the three marks of good inferential evidence are present, since: (i) The evidence-property (being the son of this man) is ascribed to the object (Devadatta). (ii) The evidence-property is ascribed to similar objects (another man has the same cognitive abilities and is also the son of this man). (iii) And the evidence-property is not ascribed to dissimilar objects (no man of different cognitive abilities is the son of this man). The traditional Buddhist reply to this attack consists in saying that in such an example, the third condition is not fulfilled, because it could perfectly be the case that someone with different cognitive abilities is the son of this man. What is new with Prabhācandra’s attack is that he then uses the Buddhist defense to claim that this latter equals defending what ultimately counts is that the evidence-property cannot be thought of otherwise than in the presence of the target-property and that therefore, Buddhist philosophers in fact agree that the Jaina “impossibility otherwise” is ultimately the only relevant criteria for the correctness of evidence. In other words, what the Jains are aiming at when attacking the Buddhist theory of the triple characteristic is to show that verifying that the evidence-property is well ascribed to the subject under consideration (pakṣa), to similar ones (sapakṣa), and not to dissimilar ones (vipakṣa), is not relevant in the evaluation process of the validity of evidence, because only the nature of the latter’s relation with the target-property is insightful. This has been put forward by Akalaṅka and further developed by Prabhācandra in his Moon [that opens] the lotus of logic (Nyāyakumudacandra, NKC), see Nyayacarya (1991), a commentary he wrote on Akalaṅka’s Three short [treatises] (Laghī yastraya, LT), see Jain (1939).

The Jaina Solution: A Perceptual-Like Grasp of Universals Named “Tarka” Moreover, a peculiar feature of the Jaina tradition is that this impossibility for the evidence-property to be present otherwise than in the presence of the target-property, or, in other words, what is to be known in order to be assured that the two inferential properties are necessarily concomitant, is known by a separate cognitive process called “tarka.” Tarka is a perception-like grasping of universals, not particulars. In this context, the word can be translated as “discernment of universals.” In Māṇikyanandin’s words: PM.3.19. This (necessary concomitance) is ascertained by discernment of universals (tarkāt tan-nirṇayaḥ), (Goshal 1940, p. 93).

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This has been commented by Prabhācandra: PKM.178.16–18. It has been said that necessary concomitance (vyāpti) cannot be grasped by perception. This too is a mere (ineffective though true) statement. Necessary concomitance is based on a type of knowledge called ūha (an expression equivalent to ‘tarka’) that rests on the strength of apprehension and non-apprehension. Neither the infinity of individuals nor the deviation in place and so on suffice to obstruct the acceptance of that (necessary concomitance) (yat ca uktam na ca vyāpti-grahaṇam adhyakṣatah iti ādih tadā api uktim atram, vyāpteḥ pratyakṣa-anupalambha-bala-udbhūta-ūha-ākhya-pramāṇāt prasiddheḥ. na ca vyaktī nām ānantyaṃ des´a-ādi-vyabhicāro vā tat-prasiddher bādhakaḥ), (Shastri 1912, p. 178). PKM. 179.19. It is the knowledge of the whole class that is labeled otherwise as ūha (tarka) (sarva-upasamhāreṇa pratipattis´ ca nāma-āntareṇa ūha eva uktaḥ syāt), (Shastri 1912, p. 179).

It is clear that necessary concomitance cannot be known by customary perception, since perception deals only with particulars, and that even the biggest list of particular instances would not suffice to reach certainty. What is more, human beings do perform inferences involving objects which cannot be perceived. Therefore, in a situation of necessary concomitance of the two inferential properties, knowledge of their necessary concomitance does not come from the apprehension of these two as particulars, nor from a mere repetition of these apprehensions, but from the recognition of universal features in the two inferential properties. This is what “discernment of universals” means. More precisely, apprehension is here “not merely observing things together [. . .]. Apprehension is realizing that if something with certain properties exists, something else with certain properties must also exist,” in (Chakrabarti 2010, p. 274). This discernment of universals can be conceived of as a sign of a Jaina perspective in the sense that the possibility of knowing universals in a particular situation can be linked with the fact that the Jaina epistemological theory of particular-in-universal facilitates the epistemic access to one from the other. More precisely, according to this theory an object is a complex having both an existent universal aspect and an existent particular aspect. Therefore, in the same situation in which one grasps fire and smoke, one can also grasp “fireness” and “smokeness.” Jaina and Buddhist conceptions of universals are different ones, since according to the first, a universal is a distinct type of real entity, whereas it is the mere conceptual exclusion of dissimilarities for the latter. What is more, it seems that the very possibility of this special cognitive process is linked to the fact that Jaina metaphysics allows for the soul to possess extra-mundane faculties by means of removing further karma. Indeed, as pointed by Shah, the Jaina authors themselves were conscious of the difficulty to explain how discernment of universals can give us certitude, since in his Investigation on Knowledge (Pramāṇamī māṃsā, PMī), see Mookerjee (1970), Hemacandra (1088–1173) writes that: PMī.36. At the time of knowing the necessary concomitance between two properties, a man attains the status of a mystic (tasya api vyāpti-grahaṇa-kāle yogī iva pramātā sampadyata iti), translated in (Shah 1967, p. 262).

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This observation links the Jaina discernment of universals with the Naiyāyika extraordinary perception called “sāmānya-lakṣaṇa,” as well as with the yogic perception propounded by the Buddhist Prajñākaragupta, see Bagchi (1953).

Prabhācandra’s Defense of the Jaina Solution Against Dharmakīrti’s Conception The next step for Prabhācandra in his PKM is to prove that conceiving a separate type of knowledge in order to explain our knowledge of the necessary concomitance is a necessary move. He has to prove it, because Dharmakīrti challenged this fact in his PV, when he claims that knowledge of the necessary concomitance is already acquired during the process of inference itself, thanks to the very concept of “effect.” Dharmakīrti’s thesis as quoted in PKM.350.7–8 is the following: PV.1.34. Smoke is the effect of fire in conformity with the nature (dharma) of an effect. The presence of this (effect) in the absence of this (cause) transgresses this (conformity to the nature of an effect) having this cause (kāryaṃ dhūmo hutabhujaḥ kārya-dharma-anuvṛttaḥ. sa bhavaṃs tad-abhāve tu hetumat-tāṃ vilaṅghayet), (Shastri 1912, p. 350; Gnoli 1960, p. 22).

Therefore, in both approaches it is possible to have a firm knowledge of the fact that smoke cannot be conceived of without fire. But Prabhācandra defends the usefulness of the discernment of universals, since this is what supports other types of knowledge when he writes: PKM.352.19–22. If the agreement (reliability) of discernment of universals (tarka) is in doubt, how can there be inference that is free from doubt? In the absence of that (faith in inference), how can it be proven that perception as a whole is reliable and separate it from what is not reliable? Therefore, whoever wants inference to be doubt-free should admit a doubt-free type of knowledge for the relation between the probans and the probandum (tarkasya saṃvāda-sandehe hi kathaṃ nissandeha-anumāna-utthānam? tad-abhāve ca kathaṃ sāmastyena pratyakṣasya aprāmāṇya-vyavacchedena prāmāṇya-prasiddhiḥ? tato nissandeham anumānam icchatā sādhya-sādhana-sambandha-grāhi pramāṇam asandigdham eva abhyupagantavyam), translated in (Chakrabarti 2010, p. 272).

To explain, how is the being-an-effect of a given property to be established in the Buddhist approach, in which only perception and inference are granted? Indeed, according to Buddhists, no other type of knowledge can give us certainty. But knowing the being-an-effect of a given property equals knowing a universal relationship between two properties. Therefore, it cannot be established by a customary perception of particulars. And it cannot be known by inference neither, because this would lead to infinite regress, as witnessed in the following schemata: ½. . . C, C ! ðA ! BÞ A, A ! B B

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This schemata indicates that in order to infer B, it is necessary to know that both A and A!B are the case, where “A!B” represents the necessary concomitance between A and B. But if the fact that “A!B is the case” is known when it is known that A is the effect of B, and if this in turn is known by means of another inference, then this second inference would also imply the knowledge of a necessary concomitance that would be known by means of another inference, and so on. This has been further developed in Vādi Devasūri’s Commentary on the Explanation of the Nature of Universal and Contextual Knowledge (Pramāṇanayatattvālokālaṃkāra, PNT), a twelfth-century commentary on the PKM, see Bhattacharya (1967). Therefore, only the Jaina tradition of Prabhācandra can escape the problem of infinite regress, and the validity of inference is established if the necessary concomitance is known by a separate cognitive channel, like the discernment of universals. To conclude, the Jaina theory of discernment of universals that makes it possible to know the one mark of evidence is a challenging one. One of its main consequences is the fact that it transfers the problem of induction to a second-order level. In turn, another consequence is the fact that inferential rules are not falsifiable anymore. But this also means that what ultimately grounds inference is a personal experience not subject to decision procedures in rational investigation and discussion.

The Jaina Contribution to the Formal Structure of an Inferential Reasoning Dharmakīrti’s Theory of Ontic Foundations for Valid Reasoning What is more, although they do offer an external means to ground inference, Jaina philosophers also follow and propose amendments to Dharmakīrti’s attempt to tackle the question of the criteria of a good inference from an internal perspective, because this theory is a breakthrough in the understanding of the precise functioning of the inferential process. More precisely, it has been shown that Dignāga’s theory is not appropriate to evaluate the correctness of a necessary concomitance, because it lacks a proper theory of the relevant relationship between two inferential properties. In order to achieve one, the Buddhist Dharmakīrti drops the second condition of the theory of the triple characteristic of evidence and accepts as good evidence only the ones that are “naturally” connected. In Katsura’s words, Dharmakīrti thus provides “the ontic foundation for valid reasoning” (Katsura 1992, p. 223). In this theory, natural connections that enable inference are of two kinds: essential and causal. The first type of natural relation is the essential (sva-bhāva) relationship between a natural (pervaded) property and its pervasive property. The traditional example both in Buddhism and in Jainism is the relationship between “being a Sissoo-tree” and “being a tree.” This type of necessary concomitance defines a type of inference related to class identity, which ensures absolute certainty since they are cases of analytic inclusion of a class within another. Dharmakīrti’s theory points to the fact that it is not accidental that whenever there is a Sissoo tree, there is also a tree, this is due to the very nature of the tree. Therefore, it is correct to infer the presence of a tree from the presence of a Sissoo. This relation is not symmetric, since it is incorrect to infer the

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presence of a Sissoo from the presence of a tree, for there might be another type of tree. As for the second natural relation, it is the causal (kārya-kāraṇa, or tadutpatti) relationship between an effect and its cause, for example, between fire and smoke. From the presence of smoke on a remote hill full of tigers, this is correct to infer the presence of fire on it, even though nobody went there to see the fire. This is also nonaccidental, since it is due to the very nature of smoke, which is an effect of the fire. This cause-effect relationship is the canonical model for the presentation of inference schemata. For Dharmakīrti, this relation is not symmetric either, due to the fact that the causal process might involve different temporal points, therefore speech on future events, as in the case of the emergence of a sprout from a seed. Indeed, if a soil has the property of “possessing a seed,” it is not possible to be sure that it will be “possessing a sprout,” because it is not certain that no impediment will block the potency of the given cause to produce its effect, nor that all the conditions required for the production of the effect at stake are present.

Jaina Amendments to Dharmakīrti’s Theory Jaina philosophers received this theory and improved on it, by arguing that much more types of evidence are to be granted. The first Jaina critic of Dharmakīrti was Akalaṅka, who initiated a discussion on the status of cause, predecessor, and successor as correct evidence. Then, Māṇikyanandin offered a comprehensive list of the types of inferential evidence. He granted six situations in which the presence of a necessary concomitance is unquestionable, namely when the evidence-property is (i) a property pervaded (vyāpya) by the target-property; (ii) an effect (kārya) of it; (iii) a cause (kāraṇa) of it; (iv) a predecessor (pūrvacara) of it; (v) a successor (uttaracara) of it; or (vi) a coexistent (sahacara) with it. The first divergence is that Dharmakīrti considers that only the effect, and not the cause, can serve as evidence in a valid inference, because there might be impediments blocking the potency of the cause. Contrarily to this, Jaina philosophers recognize cause as a correct inferential evidence in the following example: PM.3.67. There is shadow here, because there is an umbrella here (asty atra chāyā chatrāt), (Goshal 1940, p. 128).

In order to legitimate this, Māṇikyanandin offers a more finely grained definition of a “cause” as being what already consists of the totality of conditions for the emergence of the effect. In other words, as what already ensures the fact that the prerequisite that nothing is blocking its potency is fulfilled. As for Dharmakīrti, a closer look at his texts, especially PVsV 1.7.1, reveals that, given appropriate restrictions, he also considers it possible to draw an inference in which a cause is used as good evidence, provided the fact that the conclusion of the inference has the status of a potentiality. Indeed, he denied the status of inferential evidence to cause, mainly in order to avoid talk about

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future events. In this respect, noteworthy is the fact that Jaina philosophers use only examples in which a cause and its effect are synchronous. The second disagreement concerns worldly regularities. Following this line of argument, Jaina philosophers argue that two phenomena do not need to be essentially or causally related in order to be necessarily concomitant. First, this can be the case between two copresent properties, for example, between the taste and the color of a fruit. Indeed, it is sufficient to know the color of a fruit to infer its taste (PM. 3.70. asty atra mātuliṅge rūpaṃ rasāt, (Goshal 1940, p. 129)). For a study of this example, see Gorisse (2015). Second, this can be the case between two properties whose presence is separated by a time-interval, for example, the order of apparition of the stars in the sky. Indeed, it is sufficient to know that the Pleiades are rising in order to know that Aldebaran will rise soon (PM.3.68. udeṣyati s´akaṭaṃ kṛttikā-udayāt, (Goshal 1940, p. 128)). For a study of this example, see Clavel (2014). And this is due to a worldly regularity by means of which the rising of the stars is something predictable. In the first case, the Buddhists disagree and argue that the taste and the color of the mango are both effects of the same stage of ripeness of the fruit. Therefore, this situation can be traced back to causality and essence only. The second case seems more robust to criticism, since neither essence nor causality is fit to explain properties whose existence is separated by a (discontinuous) time interval, therefore Buddhist philosophers are unable to rephrase this situation in terms of their acknowledge relationships and have to accept an extra category, or to deny that the Pleiades-inference is a correct one. But here again, the Buddhists conception of causality is strong enough to enable them to argue that both the risings of the stars are co-effects of the same causal conditions, namely a given state of the sky. One important teaching of this discussion is that the last divergence seems to be the sign that Buddhist philosophers ground inference upon a necessary relation, whereas a universal relation is sufficient for the Naiyāyika and the Jaina conceptions. The Buddhist position is an insistence on the fact that predictions are possible and that practical certainty is effective as a guideline for everyday behavior, but that this does not equal scientific certainty, which is more demanding. It is possible that the star Aldebaran disappears. As a consequence, the inference “Aldebaran will rise soon, because the Pleiades has just risen” would not be true anymore. On the contrary, no tree might exist anymore, it will not change the fact that if there is a Sissoo here, it is entirely impossible that there is no tree here. This inference remains true whatever the empirical situation might be. In this discussion, only the link between a natural property and its object is a necessary one. This explains the fact that Dharmakīrti rephrases the causal relation in terms of the essential relation, saying that the set of conditions enabling the presence of an effect is included in, or is essentially connected to, the set of conditions enabling the presence of its cause. To come back to the main argument, it seems that Naiyāyika and Jaina philosophers on the contrary are not seeking necessity. But whereas this is the case with Naiyāyika philosophers, Jaina thinkers do seek necessity, even if they do so on a different level. More precisely, the reason of their acceptance of the regularity of worldly phenomena as sufficient grounds for inference is due to the fact that the regularity of worldly phenomena that they grant is strong enough to ensure necessity even in these cases.

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More precisely, in Jaina cosmogony, it is considered that after the universe is destroyed, it manifests itself again, endures, is again destroyed, and so on in an infinite circle of manifestations. In this way, even if the Pleiades die, their nature is such that at the next manifestation of the universe, they will again be followed by Aldebaran. Hence, the presence of a necessary concomitance means that in every context, there is another accessible context in which the relationship between the two inferential properties holds. This is in this precise sense that the search for necessity does not invalidate inferences based on worldly phenomena thanks to the regularity granted in Jaina cosmology.

A Jaina Novelty: From Non-apprehension to Negative Inferential Statements From a logical point of view, the next interesting step in these treatises is the introduction of negations, since this prompts a reflection concerning the structure of truth-preserving arguments. Up to now, such a reflection had been restricted to the indication that essential relations are asymmetric. This asymmetry is due to the fact that the two inferential properties do not have the same scope and that one is included in the other. It is possible to infer the presence of a tree from the presence of Sissoo, but not the reverse. Causal relations are also asymmetric. In this case, the asymmetry is linked with the problem of the possibility to perform a speech on future events. Dharmakīrti pointed out that it is possible to infer the past presence of a seed from the actual presence of a sprout, but not the reverse. And Jaina philosophers insist on the fact that, given appropriate restrictions, the reverse is possible, because it is possible to infer the future presence of a sprout from the actual presence of all the conditions (and conditions of conditions) of its emergence, including the presence of a seed. Here, Jains have an external comprehensive – “God-eye” – view, therefore nontemporal, on the causal chain. As expected, the introduction of negations in this framework led to the introduction of shifts. Indeed, psychological considerations aside, it is possible to infer the absence of a Sissoo from the absence of a tree, but not the absence of a tree from the absence of a Sissoo. And it is possible to infer the absence of a sprout from the absence of a seed, but not the absence of a seed from the absence of a sprout (unless appropriate restrictions are introduced). In other words, the situation described earlier is reversed. Jaina philosophers are the first ones who tackle this issue and who offer a classification of types of evidence taking into account these shifts. Interestingly, they have developed this through a side-way, since the departure point of these considerations is a criticism of Dharmakīrti’s conception of non-apprehension as a type of inferential evidence. More precisely, the introduction of negations in this framework primarily comes from the observation that humans do not only want to infer presences and do not only infer from presences. Knowledge of absences also is useful in the inferential process, be it in the premises or in the conclusion. In this dynamic, Dharmakīrti introduced a third type of inferential evidence, named “non-

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apprehension” (anupalabdhi). This type of evidence, which accounts for the fact that it is not accidental that whenever there is no tree, there is also no Sissoo, is not directly linked with a natural connection, but it does ensure necessity too, because it can be considered as a subtype of the essential and causal types of evidence. To sum up, when introducing non-apprehension as a type of evidence, Dharmakīrti’s program is quite specific: he intends to prove the possibility of knowing absences from inference. This, in turn, enables him not to commit himself to the existence of a third kind of means of knowledge besides perception and inference, see Kellner (2003). The main difference between Dharmakīrti’s account and the Jaina one by the time of Māṇikyanandin is that “non-apprehension” is not listed as good evidence, but as part of the general form the linguistic display of an inference might have. Indeed, Jaina philosophers inherit Dharmakīrti’s theory, but not his original problem, since they do not mind positing extra types of knowledge. From this, they can restructure the theory and shape it in order to tackle other issues. And they do so in such a way that non-apprehension becomes primarily conceived as a negative premise in the stated form of an inference. In other words, the Jaina focus is on non-apprehension as a negation, that is to say as a linguistic device usable to reverse the truth value of a sentence. This focus on the relationship between negative (or affirmative) premises and negative (or affirmative) conclusions led them to single out four forms an inference might have, namely: (i) (ii) (iii) (iv)

Affirmation of the thesis when compatible evidence is known Negation of the thesis when incompatible evidence is known Negation of the thesis when compatible evidence is not known Affirmation of the thesis when incompatible evidence is not known

In this classification, a different set of types of evidence is considered as prompting valid inferences in each form. More precisely, in the first form, there are especially six situations in which the presence of a necessary concomitance is unquestionable: PM.3.59. In the case of affirmation [of the thesis], there are six kinds of cognition of [evidence] compatible [with the target-property], namely [the evidence can be a property] pervaded (vyāpya) [by it], an effect [of it], a cause [of it], a predecessor [of it], a successor [of it] or [a property] co-existent [with it] (aviruddha-upalabdhir vidhau ṣoḍhā vyāpa-kāryakāraṇa-pūrva-uttara-sahacara-bhedāt), (Goshal 1940, p. 122).

This is the by-default form investigated until now. And the six same situations ensure the presence of a necessary concomitance in the second form as well: PM.3.71. In the case of negation [of the thesis], the types of cognition of evidence incompatible [with the target-property] are the same [as the types of cognition of evidence compatible with the target-property in the case of affirmation of the thesis] (viruddha-tadupalabdhiḥ pratiṣedhe tathā), (Goshal 1940, p. 129).

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In the third form, this state of affairs changes: PM.3.78. In the case of negation [of the thesis], there are seven kinds of non-cognition of [evidence] compatible [with the target-property], namely [the evidence can be the targetproperty] itself, a [property] pervading (vyāpaka) [it], an effect [of it], a cause [of it], a predecessor [of it], a successor [of it] or a [property] co-existent [with it] (aviruddhaanupalabdhiḥ pratiṣedhe saptadhā svabhāva-vyāpaka-kārya-kāraṇa-pūrva-uttarasahacara-anupalambha-bhedād), (Goshal 1940, p. 131).

The first difference from the previous forms is that “the target-property itself” is added to the list of correct evidence. This difference is explained by the fact that, in reasoning aiming at inferring an absence from a noncognition, knowledge is gained through the inference from “I do not know the presence of the target-property” to “I know that the target-property is absent.” But in the affirmative form, nothing would have been gained by means of the inference proceeding from the knowledge of an evidence-property to the knowledge of itself. The second difference is that only pervasive properties are good evidence to infer the absence of their respective pervaded properties, and not the other way around, since knowing that there is not a tree is sufficient to know that there is no Sissoo, but knowing that there is no Sissoo is not sufficient to know that there is no tree, for there might be an oak. As for the fourth form, involving a combination of negations, an inference can rely on only three types of evidence: PM.3.86. In the case of affirmation [of the thesis], there are three kinds of non-cognition of [evidence] incompatible [with the target-property], namely the non-cognition of [evidence] incompatible with [the target-property] itself, an effect [of it], or a cause [of it] (viruddhaanupalabdhiḥ vidhau tredhā viruddha-kārya-kāraṇa-svabhāva-anupalabdhi-bhedāt), (Goshal 1940, p. 133).

In this form, one cannot draw as many types of correct inferences as in the other forms, because there exist situations in which “non–non-A” does not equal “A.’” It does so only in very specific situations as “all things possess several aspects, because something having only one aspect is never found” (anekānta-ātmakaṃ vastv-ekāntasvarūpa-anupalabdheḥ), (Goshal 1940, p. 134). This theory is developed in detail in Gorisse (2017). In our context, suffice is to know that the focus is more and more on the linguistic form and that rules concerning this form can be found, for example, a rule for imbricated cognitions, which is equal to saying that the necessary concomitance relation is transitive. A last important remark is that the material implication used by contemporary logicians is not a good candidate to express in a formal way the relationship between the evidence-property and the targetproperty. Indeed, material implication has several properties, among which is symmetry. But epistemological concerns such as the fact that no discourse should be made about future events prevent such a symmetry. These considerations are really at the junction between logic (recognition of certain patterns, and rules describing them, such as transitivity, types, and function of negations),

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epistemology (what a person can know), and argumentation (how to convince a given interlocutor). What is more, any attempt towards a formal representation of these theories should keep in mind that as many logical connectors would be needed as there are types of evidence.

Conclusion After the adoption of the theory of ontic foundations by the different traditions, the core of logical analyses in the classical Indian hall of philosophical debates was centered on the following questions: what are the exact properties between which the necessary concomitance holds? And how are they naturally connected, that is to say, which type of necessary concomitance does hold? Once these questions are answered, the deduction can go on. Obviously, the problem of an agreement on the decision procedures is still open, both in the framework of Jaina philosophers of Akalaṅka’s lineage and in Dharmakīrti’s framework. In the latter, this is because debaters can still disagree on what can be called an “effect” of something. Therefore, both the Buddhist and the Jaina conceptions only transferred the question of the establishment of the validity of the evidence-property from the realm of inference and inferential rules, to a second-order realm. At this stage, Jaina philosophers took one step further and offered one of their greatest contributions to solve this problem. More precisely, they argue that facing such a situation in which no agreement can be reached, it is at least possible to agree to disagree. From this, they develop a theory making explicit and legitimizing the divergences between the claims of the different traditions, by offering a theory of the parameterization of assertions, both at the epistemic and ontological level. This is especially developed in the theory of angles of analysis (nikṣepavāda) and in the theory of viewpoints (nayavāda). For epistemological approaches to this theory of context, see Balcerowicz (2002) and Trikha (2012); as for a hermeneutical approach, see Flügel (2009). Noteworthy is the fact that the same Jaina traits are present both in these theories of context and in their theory of inference, namely, a Jaina insistence on linguistic considerations; second, a belief in the underlying rationality of the world; and third, a belief in the possibility of an all-comprehensive perspective, which is connected to a will to exhaustivity in their classifications.

Definitions of Key Terms Target-property (sādhya) Evidence-property (hetu)

Necessary concomitance (vyāpti)

Property whose ascription to the object under discussion is to be inferred. Property by means of which it is possible to infer the ascription of the target-property to the object under discussion. Relation between the evidence-property and the target-property, according to which the first cannot occur without the latter.

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Impossibility otherwise (anyathānupapatti)

Discernment of universal (tarka)

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Defining characteristic of a probative evidence-property, according to which it is not possible for a probative evidence-property to be conceived in a situation devoid of the corresponding target-property. Perception-like grasp of universal features.

Summary Points Inference is the means to convince interlocutors in an inter-doctrinal philosophical debate, through a truth-preserving argument. It consists in a group of five statements (thesis, evidence, account, application, and conclusion), which combines both a bottom-up and a top-down presentation of the modus ponens. In the attempt to bring it to its minimal form, which leads to distinguishing between psychological and formal considerations, Jaina philosophers grant only thesis (a1 is Q) and evidence (because a1 is P). On the one hand, this pertains to a relevant rejection of the need of the account that involved the statement of an example (whatever is P is also Q, as in a2); but on the other hand, their requirement of the thesis is successfully attacked by Dharmakīrti. An argument is granted as being truth-preserving if the evidence-property P is shown to be necessarily concomitant with the target-property Q. For Dignāga, the evidence-property is necessarily concomitant with the targetproperty if it is possible to prove that the evidence is present in the case under consideration, in similar cases and in no dissimilar cases; but the Jains show that this is not sufficient, nor necessary. For the Jains, the evidence-property is necessarily concomitant with the targetproperty if it is known as being “impossible otherwise” by a separate type of knowledge, a direct discernment of universals; but Dharmakīrti refutes the need for postulating such an extra type of knowledge. For Dharmakīrti, the evidence-property is necessarily concomitant with the targetproperty if it can be shown that there is an essential or a causal relationship between them; and Jaina philosophers follow and propose amendments to this theory. The structure of truth-preserving arguments becomes a concern in Jainism when negations are introduced, as the expression of inferences involving absences. A general survey of the types of relation between the evidence-property and the target-property reveals that there are different relations and that most of them are transitive, but not symmetric, due to the presence of epistemological concerns.

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Shah, Nagin J. 1967. Akalaṅka’s criticism of Dharmakī rti philosophy: A study. Ahmedabad: L.D. Institute of Indology. Shastri, Mahendra Kumar. 1912/1990. Prabhācandra’s Prameyakamalamārtaṇḍa, Garib Dass Oriental Series. Vol. 94. Delhi: Sri Satguru Publication. Soni, Jayandra. 2014. Prabhācandra’s status in the history of Jain philosophy. International Journal of Jaina Studies 7-9: 510–536. Tillemans, Tom J.F. 2000. Dharmakī rti’s Pramāṇavārttika and Prajn˜ ākaragupta’s Pramāṇavārttikabhāṣya. Wien: Verlag des Österreichischen Akademie des Wissenschaften. Trikha, Himal. 2012. Competing world views: Perspectivism and polemics in the Satya-śāsanaparīkṣā and other Jaina works. Journal of Indian Philosophy 40 (1): 25–45.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Buddhism and the Catuṣkoṭi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Early Denials of the PEM and PNC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Catuṣkoṭi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . First-Degree Entailment . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Proof-Theoretic Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nāgārjuna and the Buddha’s Silence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Paradox . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Jainism and the Saptabhaṇgi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Anekānta-Vāda . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . and the Saptabhaṇgi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K3 and LP . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Plurivalent Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Definitions of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Technical and Historical Details Concerning Some Paraconsistent and Paracomplete Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . FDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . K3 and B3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . LP and H3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

The Principles of Excluded Middle and Non-Contradiction are highly orthodox in Western philosophy. They are much less so in Indian philosophy. Indeed, there are logical/metaphysical positions that clearly violate them. One of these is the Buddhist catuṣkoṭi; another is the Jain saptabhaṇgi. Contemporary Western G. Priest (*) Departments of Philosophy, The CUNY Graduate Center, New York, NY, USA University of Melbourne, Parkville, VIC, Australia © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_50

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logicians have, however, investigated systems of “non-classical” logic in which these principles fail, and some of these bear important relationships to the catuṣkoṭi and the saptabhaṇgi. In this chapter, we will look at these two principles and see how these may inform and be informed by those systems. Keywords

Catuṣkoṭi · Saptabhaṇgi · Buddhism · Jainism · Nāgārjuna · Principle of Excluded Middle · Principle of Non-Contradiction · Many-valued logic · Plurivalent logic · FDE · LP · K3

Introduction Aristotle enunciated and defended two important principles: the Principle of Excluded Middle (PEM) and the Principle of Non-Contradiction (PNC), which may be expressed as follows: • PEM: Every statement is either true or false. • PNC: No statement is both true and false. He does this in the Metaphysics, not the Analytics, which is where we find Aristotle’s logical writings. However, these two principles have been cornerstones of Western logic ever since. True, there have been some who have balked at them. Oddly enough, Aristotle himself, in the somewhat notorious Chap. 9 of De Interpretatione, at least appears to reject the PEM. And, though the interpretation is contentions, Hegel appears to reject the PNC in his Logic. However, those who have problematized the principles are lone historical voices. It is fair to say that the PEM and PNC are still orthodox in contemporary logic. However, in the twentieth century, some Western logicians have certainly challenged these principles (see Priest (2008: 7.6–7.9)). Indeed, the century saw the development of formal (mathematical) logics which reject these principles: certain kinds of “non-classical” logics. These logics and their properties are now well understood. Nearly everything that Aristotle argued for has been rejected – or at least seriously challenged in the two millennia since he wrote. Why the orthodoxy of Aristotle’s views concerning the PEM and the PNC has lasted so long is an interesting question, which we must leave for historians to ponder. Matters in India are notably different. There have been defenders of the PEM and PNC, such as logicians in the (Hindu) Nyāyā School, and the Buddhist logicians Dignāga and his successor Dharmakirti (fl. c. sixth c. CE). However, logical/metaphysical thinking which rejects both of these principles is much older. We find this in, for example, the Buddhist catuṣkoṭi and the Jain saptabhaṇgi. Of course, how to understand these ideas is a contentious scholarly matter. Moreover, these thinkers did not have the resources of contemporary mathematical logic at their disposal. However, there are contemporary mathematical logics which naturally do justice to

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such ideas – though those who invented them did not do so in response to anything in Indian thought. In this chapter, we will look at the catuṣkoṭi and saptabhaṇgi, and the formal logics in question, seeing how this meeting of minds works. Putting these two things together can benefit both. It shows that the Indian thinking can be put on a rigorous mathematical basis, and so give the lie to anyone who would take such thinking to be confused or irrational. Conversely, the Indian ideas can show that the logical systems are no mere formalism, but can be seen as encoding profound metaphysical views of the world.

Buddhism and the Catuskoti ˙ ˙ Early Denials of the PEM and PNC But let us start with some denials of the PEM and PNC in Indian thought which predates Buddhism and Jainism. The earliest Vedic text, the Ṛg Veda (? 1500– 1200 BCE), contains what appear to be denials of the PEM. Thus, in describing the origins of the cosmos, it says (Koller and Koller 1991: 6): There was neither non-existence nor existence then; there was neither the realm of space nor the sky which is beyond. What stirred? Where? In whose protection? Was there water, bottomless deep?

The denial is also to be found in one of the most famous parts of Hindu philosophy: neti, neti (literally: not, not), meaning not this, not this, or neither this nor that. Thus, in the Bṛhadāraṇyaka Upaniṣad (? c. 700 BCE) we read (Koller and Koller 1991: 22): This Self is simply described as “Not, not”. It is ungraspable. For it is not grasped. It is indestructible, for it is not destroyed. It has no attachment and is unfastened; it is not attached, and yet it is not unsteady. For it, immortal, passes beyond both these two states (in which one thinks) “For this reason I have done evil,” “For this reason I have done good.” It is not disturbed by good or evil things that are done or left undone; its heaven is not lost by any deed.

The passage is most naturally read as saying that one must reject all claims about the Self: It is neither this nor not this. So we have a denial of the PEM. But at the same time, it does endorse claims about the Self, for example, that it is immortal. So we have a denial of the PNC, at least implicitly. A denial of the PNC also appears to have been found explicitly in the writings of the Hindu Ājivika sect which flourished for a while after about the fifth century BCE. Their texts are now lost, but in Abhayadeva’s commentary on the SamavayāṅgaSūtra, we find (Jayatilleke 1963: 155): These Ājivikas are called Trairāṣikas. Why? The reason is that they entertain (icchanti) everything to be of a triple nature, viz. soul, non-soul, soul and non-soul; world, non-world,

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world and non-world; being, non-being, being and non-being, etc. Even in (api) considering standpoints they entertain a three-fold standpoint such as the substantial, the modal and the dual.

Finally, the philosopher Sañjaya (dates uncertain, but probably sixth or fifth century BCE) was also known for denying the PEM and PNC – indeed, for denying that anything is true, false, both, or neither, as a way of rejecting all views (see Raju (1954: 694)). Clearly, what is behind these various views are very distinctive metaphysical positions, to the effect that reality or, at least, parts of it are either under- or overdetermined. However, this is not the place to go into these matters.

The Catuskoti ˙ ˙ With this background, let us now turn to the Buddhist catuṣkoṭi (four corners/points). According to this, a claim can be true, false, both, or neither. These are the four koṭis in question. So the principle is something like a “principle of excluded fifth.” That all four of these possibilities could be in play must have been a commonplace view by the time of the historical Buddha Siddhārtha Gautama (Pāli: Gotama) (fl. c. sixth c. BCE), since it appears to be taken for granted in some of the sūtras – though there seems to be no connection between the third and fourth koṭis and specific Buddhist doctrines at this point. Thus, in the Agivacchagotta Sutta we find the following (Ñāṇamoli and Bodhi 1995: 591). Note that a Tathāgata – literally (one) thus gone – is someone who has achieved enlightenment: “How is it, Master Gotama, does Master Gotama hold the view: ‘After death a Tathāgata exists: only this is true, anything else is wrong’?” “Vaccha, I do not hold the view: ‘After death a Tathāgata exists: only this is true, anything else is wrong.’ ” “How then, does Master Gotama hold the view: ‘After death a Tathāgata does not exist: only this is true, anything else is wrong’?” “Vaccha, I do not hold the view: ‘After death a Tathāgata does not exist: only this is true, anything else is wrong.’” “How is it, Master Gotama, does Master Gotama hold the view: ‘After death a Tathāgata both exists and does not exist: only this is true, anything else is wrong.’?” “Vaccha, I do not hold the view: ‘After death a Tathāgata both exists and does not exist: only this is true, anything else is wrong.’ ” “How then, does Master Gotama hold the view: ‘After death a Tathāgata neither exists nor does not exist: only this is true, anything else is wrong’?” “Vaccha, I do not hold the view: ‘After death a Tathagata neither exists nor does not exist: only this is true, anything else is wrong.’”

Vaccha asks about the status of an enlightened person after death and runs through the four possibilities of the catuṣkoṭi. And both he and the Buddha take the partition for granted. Neither does the Buddha say “Don’t be silly, Vaccha. It makes no sense

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for something to be both true and false or neither true nor false.” Neither does he say “Aha, Vaccha, you are missing another possibility.” So they both seem to accept that this partition is exclusive and exhaustive. It is true that the Buddha refuses to endorse any of the four koṭis. Why he does so is moot. Some sūtras have the Buddha continuing by saying something like: “Look, I’m telling you how to improve your life. You shouldn’t be worrying about all this metaphysical nonsense.” Others have him saying that none of the four possibilities “fits the case.” What this means is less than clear. Some commentators (e.g., Siderits and Katsura (2013: 302)) suggest that since the person no longer exists we have a case of reference failure. The problem with this is that standard accounts of reference failure do not take us outside the catuṣkoṭi. Thus, for example, Frege takes such sentences to be neither true nor false, and Russell takes them to be simply false. These views are built into so-called free logics of different kinds. (On these matters, see Priest (2008: 7.8 and Chap. 13).) Whatever the Buddha meant, the refusal to endorse any of the four koṭis certainly prefigures the later Buddhist view that there is a fifth possibility. We will come to this in due course.

First-Degree Entailment So much for the Buddhist catuṣkoṭi. Let us now turn to an appropriate formal logic. This is a logic called “First Degree Entailment” (FDE) – don’t ask. Modern logic avoids all the irrelevant complexities of a natural language by dealing with formal languages: languages with regular grammars and no ambiguity. A relation of logical consequence is defined on sentences of such a language. Let us stick with a simple propositional language. (It is easy to extend the techniques to more complex languages.) The simplest sentences of the language are called propositional parameters, and more complex sentences are constructed from these in an iterative process using sentence-connectives. In FDE, these are ^, _, and :. You can think of them as meaning “and” (conjunction), “or” (disjunction), and “it is not the case that” (negation), respectively. A standard way of defining a consequence relation is by giving the language a semantics. A semantics for FDE can be set up in a number of different ways. Here, I describe a 4-valued semantics, where the connection with the catuṣkoṭi is at its most obvious (see Priest (2008: Chap 8)). An interpretation for the language specifies values for every propositional parameter. In a 2-valued semantics, there are just: true and true only, and false and false only. We can write these as t and f, respectively. In FDE, there are two more: both true and false and neither true nor false. We can write these as b and n, respectively. The four values form a structure that mathematicians call a lattice. A lattice can be depicted in the form of a diagram called a Hasse diagram. The diagram for the lattice we are dealing with here is as follows and is often called the Diamond Lattice:

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t ↗

↖

b

n ↖

↗ f

The catuṣkoṭi is evident. Given an assignment of values to the propositional parameters, this is extended to an assignment of values to all sentences of the language recursively, by the following conditions. If A has the value t, :A has the value f, and vice versa. b and n are “fixed points” for negation. That is, if A has the value b, so does :A; if A has the value n, so does :A. The value of a conjunction is the greatest lower bound of the values of the conjuncts, that is, the greatest thing less than or equal to the values of both conjuncts. What that means is that one goes down the arrows of the Diamond Lattice till one gets to the first thing that is less than or equal to both of them. Thus: • If A has the value t and B has the value b, A ^ B have the value b. • If A has the value b and B has the value n, A ^ B have the value f. For disjunctions, one just goes up the arrows instead, giving the least upper bound. So: • If A has the value t and B has the value b, A _ B has the value t. • If A has the value b and B has the value n, A _ B has the value t. Finally, we may define validity. Write this as . In a many-valued logic, some of the values are said to be designated. (One may think of these as the values that represent some kind of truth.) An inference is invalid if it is possible that (i.e., there is an interpretation in which) the premises have a designated value and the conclusion does not. It is valid if it is not invalid. That is, whenever all the premises have a designated value, so does the conclusion. In FDE, the designated values are t and b (since these are the two ways in which something can be true). It is a relatively simple matter to determine whether particular inferences are valid or invalid. This can safely be left to the reader. I will give some examples of valid inferences in the next section. Let us just note that we do not have the following: • A  B _ :B • A ^ :A  B The second inference is usually, now, called Explosion. The first has no standard name, but symmetry suggests Implosion. For the failure of Implosion, take A to have the value t, and B to be have the value n – in which case, B _ :B has the value n. For the failure of Explosion, take B to

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have the value f, and A to be have the value b – in which case, A ^ :A have the value b. Clearly, these two inferences are versions of the PEM and the PNC respectively. The first says that you have B _ :B whatever else you have. The second says that if you have A ^ :A, you will have everything, including crazy things such as “the Moon is a cube,” “12 ¼ 17,” etc. So you cannot have A ^ :A. (One might think that the PNC would be expressed by the inference A  : (B ^ :B), but it is not. If in an interpretation the value of B is b, then :(B ^ :B) holds in the interpretation. Indeed (B ^ :B) ^ :(B ^ :B) holds!)

A Proof-Theoretic Characterization This relation of logical consequence can also be characterized by a set of rules of inference (see Priest (2019)). FDE is characterized by (that is, FDE is sound and complete with respect to) the following rules: A B A^B

A A_B A ::A

A^B A

B A_B :ðA _ BÞ :A ^ :B

A^B B A

B

⋮

⋮

A_B

C C C :ðA ^ BÞ :A _ :B

Premises are above the line; conclusions are below; a double line indicates that an inference goes both ways; and a line above a formula means that the rule discharges this assumption. That is, the final conclusion does not depend on this assumption. Thus, in the third rule for disjunction, A is used to deduce C, and B is used to deduce C. We then infer C depending on the premise A _ B, but not on A and B themselves.

Na¯ga¯rjuna and the Buddha’s Silence Let us return to the Buddha’s silence. This was picked up by Nāgārjuna (fl. first or second c. CE). His text, the Mulamadhyamakakārikā (MMK, The Fundamental Verses of the Middle Way), was the foundational text of all later (Mahāyāna) Buddhisms (see Garfield (1995), Priest (2013), and Siderits and Katsura (2013)). In this, Nāgārjuna argues that there is nothing which exists with intrinsic nature. That is, everything is empty (śūnya) of svabhāva (self-being/nature). The catuṣkoṭi plays an important role in this. The arguments often start by assuming that something or other has svabhāva. It then divides the matter up into the four cases of the catuṣkoṭi and shows that none of them can hold. Hence, we have a four-way reductio

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of the assumption. The argument is one of reductio ad absurdum, not reductio ad contradictionem. It cannot be ad contradictionem because the third koṭi is one that explicitly allows for the possibility of a contradiction. But there are many things that are more absurd than some contradictions. The claim that you are a frog is more absurd than that the liar sentence (“this sentence is false”) is both true and false. And Nāgārjuna has to show only that there is some consequence of the assumption that is absurd – or maybe just unacceptable to his opponents, since many of the arguments are ad hominem. Anyway, at MMK XXII: 11–12, Nāgārjuna picks up the Buddha’s silence concerning the status of the enlightened person (Tathāgata) after death. There, we have the following (translations from the MMK are from Garfield (1995)); note that the catuṣkoṭi is referred to by its Greek translation, tetralemma): ‘Empty’ should not be asserted. ‘Nonempty’ should not be asserted. Neither both nor neither should be asserted. They are used only nominally. How can the tetralemma of permanent and impermanent, etc., Be true of the peaceful? How can the tetralemma of the finite, infinite, etc., Be true of the peaceful?

Given that the catuṣkoṭi is an enumeration of all the things that can be said, the implication would appear to be that nothing can be said about the status of the Tathāgata. The state of affairs concerning the Tathāgata is simply ineffable. In fact, we may apply the machinery of the catuṣkoṭi to states of affairs themselves. To do this, we must think of sentences, not as expressing propositions, but as referring to states of affairs. For each (possible) state of affairs, A, there is a corresponding negative state of affairs, :A. (So, corresponding to the state of affairs that the Tathāgata exists is the state of affairs that the Tathāgata does not exist.) Similarly, if A and B are states of affairs, there are conjunctive and disjunctive states of affairs A ^ B and A _ B. Now, states of affairs are not the kind of things that are true or false, but the kinds of thing that exist or do not. So we must now think of the four values of the catuṣkoṭi as follows: • • • •

The value of A is t: A exists and :A does not exist. The value of A is f: A does not exist and :A exists. The value of A is b: both A and :A exist. The value of A is n: neither A nor :A exists.

And now, we have a fifth possibility, ineffability. Call this value e. Clearly, it must be distinct from the others, since if we can say that A exists or does not, we can say something about it, and so it cannot be ineffable.

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How does the value e work? It would seem that if A is ineffable, so is :A, and so are A _ B and A ^ B. So if the value of A is e, so is the value of anything of which it is a part. The designated values are those that preserve existence, t and b. So e is not designated (see Priest (2018: Chap. 5)). Hence, we have a 5-valued logic, which we may call FDEe. A system of rules of proof which are sound and complete with respect to these semantics is obtained by replacing the rules for _-introduction with the rules of weak _-introduction: A B† A† B A_B A_B where C† is any formula which contains all the propositional parameters of C (see Priest (2019)).

Paradox We are not quite finished with the catuṣkoṭi yet. In Buddhist philosophy, there is a standard distinction between conventional reality (saṃvṛiti-satya) and ultimate reality (paramārtha-satya). (“Satya” may be translated at truth or as reality. The former is the more usual translation.) Conventional reality is the world we are familiar with, our Lebenswelt. Ultimate reality is the way that things actually are behind these appearances., This is, exactly, a contentious point of Buddhist philosophy, and Nāgārjuna is less than explicit on what he takes it to be, but at MMK XXII: 16a, b, he tells us that: Whatever is the essence of the Tathāgata This is the essence of the world.

And it is clear that “the world” is a reference to ultimate reality. It would seem, then, that this also is ineffable. This point is made explicitly at MMK XVIII: 9: Not dependent on another, peaceful. Not fabricated by mental fabrication. Not thought, without distinction. That is the character of reality.

Ultimate reality is beyond conceptualization (without distinction); it is conventional reality that is created (fabricated) by conceptualization. Indeed, that ultimate reality is ineffable is a common view in many later Buddhist schools, such as Yogācāra and Chan (Zen) (see Priest (2014b)). But if ultimate reality is ineffable, there is an obvious issue. Nāgārjuna is himself talking about ultimate reality and so conceptualizing it – as are those who follow him in this matter. So this reality would seem to be effable and ineffable.

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The point troubled many Buddhist philosophers after the PNC had generally come to be accepted. Thus, take, for example, the Tibetan Māhāyana philosopher Gorampa (1429–1489). He is as clear as his Māhāyana predecessors that the ultimate is ineffable. He says in his Synopsis of Madhyamaka v. 75 (quoted in Kassor (2013)): The scriptures which negate proliferations of the four extremes [cf. of the catuṣkoṭki] refer to ultimate truth but not to the conventional, because the ultimate is devoid of conceptual proliferations, and the conventional is endowed with them.

But he also realizes that he is talking about it. Indeed, he does so in this very quote. Gorampa’s response to the situation is to draw a distinction. Kassor describes matters thus (2013: 406): In the Synopsis, Gorampa divides ultimate truth into two: the nominal ultimate (don dam rnam grags pa) and the ultimate truth (don dam bden pa). While the ultimate truth. . . is free from conceptual proliferations, existing beyond the limits of thought, the nominal ultimate is simply a conceptual description of what the ultimate is like. Whenever ordinary persons talk about or conceptualize the ultimate, Gorampa argues that they are actually referring to the nominal ultimate. We cannot think or talk about the actual ultimate truth because it is beyond thoughts and language; any statement or thought about the ultimate is necessarily conceptual, and is, therefore, the nominal ultimate.

It does not take long to see that this hardly avoids contradiction. If all talk of the ultimate is about the nominal ultimate, then Gorampa’s own talk of the ultimate is about the nominal ultimate. Since the nominal ultimate is effable, Gorampa’s own claim that the ultimate is devoid of conceptual proliferations is just false. So, what does Nāgārjuna himself say about the matter? Nothing. Why? We can only conjecture. It cannot be because he failed to notice the situation. It would stare in the face of a much lesser philosopher. However, Nāgārjuna is working in the context of the catuṣkoṭki, the third koṭki which explicitly allows for contradictions to be true. Perhaps, more wisely than some of his successors, he simply took the situation to provide a counterexample to the PNC (see, further, Priest (2018)).

Jainism and the Saptabhangi¯ ˙ The Aneka¯nta-Va¯da Let us now move from Buddhism to Jainism. This is traditionally taken to have been founded by Mahāvira (fl. fifth or sixth c. BCE), a rough contemporary of the Buddha. Jainism has a very distinctive metaphysics, captured in the doctrine of anekāntavāda – nononesidededness, as it is sometimes translated. The Jains believed that truth was not the prerogative of any one school. The views of Buddhists and Hindus, for example, may disagree about crucial matters, such as the existence of an individual soul; each has, nonetheless, an element of truth in it. This can be so because reality itself is multifaceted (see Ganeri (2002: 5.2)).

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Reality is a complex, with a multitude of aspects, and each of the competing views provides a perspective, or standpoint (naya), which latches on to one such aspect. As Siddhasena (fl. fifth or sixth c. CE) puts it in the Nyāyāvatāra, v. 29 (Matilal (1981: 41)): Since a thing has manifold character, it is comprehended (only) by the omniscient. But a thing becomes the subject matter of a naya, when it is conceived from one particular standpoint.

On its own, each standpoint is right enough, but incomplete. To grasp the complete picture, if indeed this is possible, one needs to have all the perspectives together – like seeing a cube from all six sides at once. It follows that any statement to the effect that reality is thus and such, if taken categorically, will be, if not false, then certainly misleading. It is better to express the view with an explicit reminder that it is correct from a certain perspective. This was the function with which Jain logicians employed the word “syāt.” In the vernacular, this means something like “it may be that,” “perhaps,” or “arguably,” but in the technical sense in which the Jain logicians used it, it may be best thought of as something like “In a certain way. . .” or “From a certain perspective. . .” (Matilal 1981: 52; Ganeri 2002: 5.5). So, instead of saying “An individual soul exists,” it is better to say “Syāt an individual soul exists.” This is the Jain method of syād-vāda. It is worth nothing that, according to Buddhism, reality is also multifaceted in a certain sense. As we have already seen, it is standard in Buddhist philosophy that reality has both a conventional aspect and an ultimate aspect. However, it is generally agreed that ultimate reality, as the name suggests, is what is really there, and conventional reality is, in some sense, less real. Hence, this is quite different from the Jain view that the different aspects of reality are equally real. Moreover, there was no attempt to aggregate these two realities into a compound picture as, we are about to see, the Jain saptabhaṇgi does.

. . . and the Saptabhangi¯ ˙ Given this background, we can now understand the Jain saptabhaṇgi (sevenfold division). A sentence may have one of seven truth values, or, as it may be put, there are seven predicates that may describe its semantic status. The matter is explained by the twelfth-century theorist, Vādideva Sūri, in Pramāṅanaya-tattvālokālaṁkāra, Chap. 4, vv. 15–21 (Battacharya 1967): The seven predicate theory consists in the use of seven claims about sentences, each preceded by ‘arguably’ or ‘conditionally’ (syāt) [all] concerning a single object and its particular properties, composed of assertions and denials, either simultaneously or successively, and without contradiction. They are as follows: (1) Arguably, it (i.e., some object) exists (syād esty eva). The first predicate pertains to an assertion.

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(2) Arguably, it does not exist (syād nāsty eva). The second predicate pertains to a denial. (3) Arguably, it exists; arguably it does not exist (syād esty eva syād nāsty eva). The third predicate pertains to successive assertion and denial. (4) Arguably, it is non-assertable (syād avaktavyam eva). The fourth predicate pertains to a simultaneous assertion and denial. (5) Arguably, it exists; arguably it is non-assertable (syād esty eva syād avaktavyam eva). The fifth predicate pertains to an assertion and a simultaneous assertion and denial. (6) Arguably, it does not exist; arguably it is non-assertable (syād nāsty eva syād avaktavyam eva). The sixth predicate pertains to a denial and a simultaneous assertion and denial. (7) Arguably, it exists; arguably it doesn’t exist; arguably it is non-assertable (syād esty eva syād nāsty eva syād avaktavyam eva). The seventh predicate pertains to a successive assertion and denial and a simultaneous assertion and denial.

A perusal of the seven possibilities indicates that there are three basic ones, (1), (2), and (4), and that the others are compounded from these. (1) says that the statement in question (that something exists) holds from a certain perspective. (2) says that from a certain perspective, it does not. (4) says that from a certain perspective, it has another status, nonassertable. Exactly what this is is less than clear. We will return to the matter in a moment. Let us call these three values t, f, and i, respectively. In understanding the other possibilities, we hit a prima facie problem. Take (3). This says that from some perspective the sentence is t, and from some perspective it is f. That is intelligible enough, but unfortunately, it would seem to entail both (1) and (2). If it is true from some perspective and false from some perspective, it is certainly true from some perspective. The solution is straightforward, however. We have to understand (1) as saying not just that the sentence is true from some perspective, but as denying the other two basic possibilities: It is t from some perspective, and there are no perspectives from which it is f or i. (3) is now to the effect that there is a perspective from which the sentence is t, a perspective from which it is f, and no perspective from which it is i. In fact, all the seven cases now fall into place. Thus understood, each of the three basic possibilities may hold or fail – except that they cannot all fail, since there must be at least one perspective. Hence, there are 23 – 1 ¼ 7 values. But what are we to make of the value i? A natural possibility is that i means both true and false. That is essentially how Vādideva Sūri glosses case (4) in the quotation above. Unfortunately, he also glosses i as unassertable. So, the status of i is more like neither true nor false. Which is the most plausible interpretation of i in Jain logic, all things considered, is a moot point. Stcherbatsky (1962: 415), Bharucha and Kamat (1984), and Sarkar (1992) argue that i is most plausibly interpreted as both true and false. Ganeri (2002: 5.6; 2002: Sect. 1) favors neither true nor false. We may leave scholars to debate the matter. In what follows, we will take both possibilities into account.

K3 and LP We may now turn to matters of contemporary formal logic. Concentrate, first, on the basic logic with three values, t, i, and f. If i means both true and false, we can think of

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this as the value b of the FDE semantics. And we then get an appropriate 3-valued logic simply by ignoring the value n in the 4-valued semantics. This gives a logic known as LP. Alternatively, if i means neither true nor false, we may think of it as the value n. We then obtain an appropriate 3-valued logic simply by ignoring the value b (both as a value and as a designated value) in the 4-valued semantics. This is a logic known as K3, usually referred to as strong Kleene (see Priest (2008: Chap. 7)). We might therefore picture the two logics thus. LP is on the left; K3 is on the right: t " b

t " ¼ i ¼

" f

n " f

Notice that as far as the two lattices go, there is now nothing to distinguish between b and n. (The lattices are isomorphic.) The difference between the two logics lies only in the fact that b is designated, and n is not. Proof theoretic characterizations of these two logics can be obtained using the rules: : B _ :B

A ^ :A B

(The first rule means that B _ :B can always be added as a line in a deduction. If it is an assumption, it is immediately discharged.) Adding the first to the rules of FDE gives the logic LP, which validates the PEM. Adding the second to the rules of FDE gives the logic K3, which validates the PNC. This is exactly as one would expect, since the first uses the value b but rules out the value n, and the second uses the value n but rules out the value b. Adding both rules to those for FDE – ruling out both b and n – delivers a system of rules for classical logic (see Priest (2019).

Plurivalent Logic Now, as we have seen, the saptabhaṇgi allows statements to take any combination of our three basic values (except the combination with none of them). Formally, this can be handled with a construction called plurivalient logic (see Priest (2014a)). In a plurivalent logic, sentences may have one or more of the available values. In the present case, they may have any number of the values t, i, and f – except none of them. (Technically speaking, then, an assignment of values is not a function, but a one-many relation – or equivalently, a function whose values are nonempty subsets of {t, i, f}.) An interpretation assigns some values to propositional parameters, and the values of compound formulas are then determined recursively by computing all possible combinations. (That is, the values are computed point-wise, as mathematicians say.)

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Thus, suppose that A has the values t and f, and B has the values t and i. We compute the result of combining these under the rules for LP or K3. (As noted, it does not matter whether one thinks of i as b or n.) We might draw the result in the following table (the values of A are in the left-hand column; the values of B are in the top row): Ŀ

t

i

t f

t f

i f

As the matrix values show, the possible values for A ^ B are t, i, and f. ( f occurs twice, but that is irrelevant.) So, these are the values assigned to A ^ B. (If we draw up a table of this kind, there will be one to three columns and one to three rows, depending on how many values each formula has.) We do the same thing for disjunction. Thus, suppose that A has the values t and f, and B has the values t and i. The result of combing these is shown in the following table: ŀ t

t f

i

t t t i

The possible values for A _ B are t, and i. So these are what A _ B is assigned. To compute the values of :A, we simply negate all the values of A. Thus, suppose that A has the values t and i. The result of negating these is shown in the following table: A

t

i

¬A

f

i

Hence, :A has the values f and i. The definition of validity is now given in a natural way. Say that a formula is designated under the new regime if at least one of its values is designated under the old. That is, one of its values is t or, if i is b, i. Then an inference is valid in the plurivalent logic, and if whenever all the premises are designated in this sense, so is the conclusion. The consequence relation for plurivalent LP is the same as that of LP itself. Hence, it is characterized by the same set of rules. The consequence relation for plurivalent K3 is, in fact, the same as FDE and so is characterized by the rules for this (see Priest (2014a)). Notice, then, that even though K3 validates Explosion, plurivalent K3 does not. To see this, suppose that A has the values t and f, and that B has just the value f. Then :A has the values f and t (that is, the same as A; the order does not matter). And A ^ :A has the values t and f. So, A ^ :A is designated, but B is not. I note that one could equally apply the plurivalent construction to the four values of FDE or the five values of FDEe (though I know of no Indian texts which suggest or countenance this). In the first case, we obtain a 24–1, that is, 15-valued logic. In

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the second case, we obtain a 25–1, that is, 31-valued logic! Plurivalent FDE is characterized by the same set of rules as those of FDE itself, and Plurivalent FDEe is characterized by the same set of rules as those of FDEe (see Priest (2014a)).

Conclusion We have now looked at the Buddhist catuṣkoṭi, the Jain saptabhaṇgi, how these work, and what underlies them. We have also seen how the ideas can be built into some contemporary nonclassical logics. This makes it clear that the catuṣkoṭi rejects the PNC and the PEM. If i is n, the saptabhaṇgi rejects the PEM and the PNC; if i is b, it rejects only the PNC. Of course, using the techniques of contemporary logic to interrogate ancient Indian texts is anachronistic. But the anachronism is not a pernicious one. Contemporary logicians, in fact, do exactly the same to ancient and medieval Western texts. Thus, if one browses the issues of the journal History and Philosophy of Logic, one will find many examples of this. Here are a few more. Versions of ontological argument for the existence of God have been given by a number of philosophers, including Anselm, Descartes, and Leibniz. These arguments are often analyzed with the tools of modern logic (see many of the papers in Oppy (2018)). Another argument is, there have been many attempts to analyze Hegel’s dialectics using the techniques of modern logic (see many of the papers in Marconi (1979)). Finally, one can find analyses of views of, among others, Berkeley, Kant, and Hegel, which employ the tools of contemporary logic in Priest (1995). Moreover, it is clearly sensible to investigate something using the tools one has at one’s disposal, even if they were not available at the time when the thing to which the tools are applied was proposed/discovered. In biology, it is silly not to use a microscope if one is available. In logic, it is silly not to use the tools of modern mathematics if they are available. At any rate, what we have seen concerning the PEM and PNC is this. These principles have been high orthodoxy in Western logic/philosophy. However, the principles were being challenged by Indian thinkers at the same time when (or just before) Aristotle was fixing them into orthodoxy in the West. Contemporary Western logicians have now cast doubt on both the PEM and the PNC. In particular, they have constructed systems of logic in which they fail. Moreover, some of these systems – constructed in ignorance of the relevant parts of Indian thought – provide just what is needed for a rigorous development of these profound and Ancient Indian ideas.

Definitions of Key Terms Principle of Excluded Middle: Principle of Non-Contradiction:

A logical principle according to which statements are either true or false. A logical principle according to which statements are not both true and false.

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Catuṣkoṭi:

Saptabhaṇgi:

Anekānta-Vāda: Many-valued logic: Plurivalent logic: FDE: LP: K3: Explosion: Implosion: Paraconsistent logic: Paracomplete logic:

G. Priest

A logical/metaphysical principle deployed by Buddhist philosophers, according to which statements may be true, false, both, or neither. A logical/metaphysical principle deployed by Jain philosophers, according to which statements may be true, false, or “non-assertible,” or any combination of the three. A Jain principle according to which reality is multifaceted. A logic in which statements may take one of more than two values. A logic in which statements may take more than one of the available values (at the same time). A 4-valued logic in which the values are: true, false, both, and neither. A 3-valued logic in which the values are: true, false, and both. A 3-valued logic in which the values are: true, false, and neither. The inference A ^ :A ‘ B. The inference A ‘ B _ :B. A logic in Explosion is not valid. A logic in which Implosion is not valid.

Summary Points • The Principle of Excluded Middle (PEM) and the Principle of Non-Contradiction (PNC) are highly orthodox in Western philosophy. • However, they have been rejected by some important Indian philosophical traditions. • Buddhist philosophers deploy a logical/metaphysical principle called the catuṣkoṭi, according to which statements may be true, false, both, or neither. • Jain philosophers deploy a logical/metaphysical principle called the saptabhaṇgi, according to which statements can be true, false, or “non-assertible” – sometimes interpreted as both truth and false, sometimes interpreted as neither true nor false – or any combination of the three. • Contemporary logicians have investigated systems of logic in which both the PEM and the PNC fail. • One of these is the system FDE, a system of many-valued logic, which is based on the four possibilities of the catuṣkoṭi. • The technique of plurivalent logic allows statements to have more than one value (at the same time). This can be used to construct systems which encode the ideas of the saptabhaṇgi.

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• The invention of these logics had nothing to do with Indian philosophy. However, putting the Indian ideas together with the modern logical constructions can benefit both. • The formal logic shows how the Indian ideas can be put on a rigorous mathematical basis. Conversely, the Indian ideas can show that the logical systems are no mere formalisms, but can be seen as encoding profound metaphysical views of the world.

Appendix: Technical and Historical Details Concerning Some Paraconsistent and Paracomplete Logics A logic where Explosion fails is called paraconsistent. A logic where Implosion fails is often now called paracomplete. Classical logic is neither paraconsistent nor paracomplete. Not all paraconsistent and paracomplete logics are many-valued logics. In this appendix, I will discuss a few points of technical and historical interest concerning some that are.

FDE The logic FDE is the core of a family of logics called relevant logics. It is both paraconsistent and paracomplete. It was invented/discovered by the US logicians A. R. Anderson and N. D. Belnap in 1962. The main concern of relevant logic is that if A entails B, A should be relevant to B. If a logic satisfies Explosion or Implosion, this is obviously not the case. The 4-valued semantics was invented/discovered a little later, by J. M. Dunn. (For discussion and references, see Anderson and Belnap (1975), Chap. 3.) One way of setting up the semantics of FDE is as follows. The language contains a set of propositional parameters, P, and the connectives, ^, _, and :. An interpretation is a binary relation ρ  P  {0, 1}. Given an interpretation, truth and falsity are assigned independently to all formulas as follows. ⊩+ A means that A is true; ⊩ A means that A is false. If p  P: • ⊩+ p iff pρ1 • ⊩ p iff pρ0 Then: • • • • • •

⊩+ :A iff ⊩ A ⊩ :A iff ⊩+ A ⊩+ A ^ B iff ⊩+ A and ⊩+ B ⊩ A ^ B iff ⊩ A or ⊩ B ⊩+ A _ B iff ⊩+ A or ⊩+ B ⊩ A _ B iff ⊩ A and ⊩ B

If Σ is a set of formulas, then Σ  A iff for all ρ: if ⊩+ B, for all B  Σ, then ⊩+ A.

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We may define a conditional, A  B, as usual in classical logic. As is easy to check, both  A  A and A, A  B  B fail – the first, because A may have the value n; the second, because A may have the value b. Full relevant logics add a new conditional, !, to the language and give it an appropriate (and more complex) semantics. These inferences hold for !. As is clear, the relational FDE truth/falsity conditions are exactly those of classical logic (though, in the case of classical logic, the falsity conditions are redundant). The definition of validity is also exactly the same as that of classical logic. If an interpretation is a total function (that is, it relates every p to exactly one member of {0, 1}), then it is an interpretation of classical logic. Hence, FDE expands the possibilities (interpretations) countenanced by classical logic. Given a relational FDE interpretation, there are obviously four possibilities for a formula, A: • • • •

⊩+ A and ⊮+ A ⊮+ A and ⊩ A ⊩+ A and ⊩ A ⊮+ A and ⊮ A

If we write these four possibilities as t, f, b, and n, then the relational truth/falsity conditions deliver the Diamond Lattice and its operators, as may easily be checked. And relational validity is equivalent to preserving the values t and b. Hence, the relational semantics and the 4-valued semantics are equivalent (see Priest (2008: 8.4)). The system FDEe was introduced in Priest (2018) and, unlike the other logics mentioned in this essay, was motivated by Buddhist considerations.

K3 and B3 If in the relational semantics one requires that for no p, pρ1 and pρ0, then, as is easy to check, this is so for all formulas. The semantics is then one for the logic K3. K3 is paracomplete, but not paraconsistent. As indicated, the 3-valued version of the semantics is obtained by taking the right-hand side of the Diamond Lattice of 2.3. K3 was invented/discovered by the US mathematician S. C. Kleene in 1938 (see also his book (1952: §64)). Kleene was concerned with partial recursive functions. The value of such a function may not be defined. Hence, if f is such a function, the equation f(i) ¼ j may be neither true nor false. Hence, Kleene calls the value n “undefined.” If we replace the value n by the value e of 2.5, the resulting logic is often called “weak Kleene logic,” but it is better called Bochvar Logic (B3), since it was invented by the Russian logician D. A. Bochvar in a paper in Russian in 1938. (An English translation appears as Bochvar and Bergmann (1981).) Like K3, B3 is paracomplete, but not paraconsistent. Bockhvar interprets the value e as nonsense. (So, for the connectives: nonsense-in, nonsense-out.) As the title of the paper indicates, he takes

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sentences involved in paradoxes of self-reference, such as that involved in Russell’s paradox, {x: x 2 = x}  {x: x 2 = x}, to be nonsensical. A rule system that is sound and complete with respect to B3 can be obtained from that of K3 by removing _-introduction, and adding: A† A _ :A (Recall that A† is any formula containing all the propositional parameters of A.) See Priest (2019).

LP and H3 If in the relational semantics one requires that, for every p, pρ1 or pρ0, then, as is easy to check, this holds for all formulas. The semantics is then one for the logic LP. LP is paraconsistent, but not paracomplete. As indicated, the 3-valued version of the semantics is obtained by taking the left-hand side of the Diamond Lattice of 2.3. The logic LP was invented/discovered by G. Priest (1979). Like Bochvar, he thought of the value b as applying to paradoxical sentences. (He calls the value paradoxical.) But unlike Bochvar, he read it as both true and false – and so as a species of truth. To round out the picture: B3 may equally be obtained from LP by replacing the value b with e – since e is not designated. However, if we then take e to be designated, we obtain a logic usually now often called “Paraconsistent Weak Kleene,” though it would be better called Halldén logic (H3), since it was invented/discovered by the Swedish logician Sören Halldén in 1949. Like LP, H3 is paraconsistent, but not paracomplete. As the title of Halldén’s work indicates, he interprets the middle value as nonsensical, like Bochvar. Why he takes the value to be designated is somewhat opaque, however. A sound and complete system of rules for H3 can be obtained by taking the rules for LP, deleting the rule for ^-elimination, and replacing it with: A

:A A†

A^B A _ B†

A^B A† _ B

See Priest (2019).

References Anderson, A.R., and N.D. Belnap. 1962. Tautological entailments. Philosophical Studies 13: 9–24. Anderson, A.R., and N.D. Belnap. 1975. Entailment: The logic of relevance and necessity. Vol. I. Princeton: Princeton University Press. Battacharya, H.S, ed. and trans. 1967. Pramāṇa-naya-tattvālokālaṁkāra. Bombay: Jain Sahitya Vikas Mandal.

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Bharucha, F., and R.V. Kamat. 1984. Syādvāda theory of Jainism in terms of deviant logic. Indian Philosophical Quarterly 9: 181–187. Bochvar, D.A., and M. Bergmann. 1981. On a three-valued logical calculus and its application to the analysis of the paradoxes of the classical extended functional calculus. History and Philosophy of Logic 2: 87–112. Ganeri, J. 2002. Jaina logic and the philosophical basis of pluralism. History and Philosophy of Logic 23: 267–281. Garfield, J., trans. 1995. The fundamental wisdom of the middle way. New York: Oxford University Press. Halldén, S. 1949. The logic of nonsense. Uppsala: A. B. Lundequistska Bokhandlen. Jayatilleke, K.N. 1963. Early Buddhist theories of knowledge. London: George Allen and Unwin. Kassor, C. 2013. Is Gorampa’s “freedom from conceptual proliferations” dialetheist? A response to Garfield, Priest, and Tillemans. Philosophy East and West 63: 399–410. Kleene, S.C. 1938. Notation for ordinal numbers. Journal of Symbolic Logic 3: 150–151. Kleene, S.C. 1952. Introduction to metamathematics. Amsterdam: North Holland Publishers. Koller, J.M., and P. Koller, eds. 1991. A sourcebook in Asian philosophy. Upper Saddle River: Prentice Hall. Marconi, D. 1979. La formalizzazione della dialettica. Turin: Rosenberg & Sellier. Matilal, B.K. 1981. The central philosophy of Jainism (Anekānta-Vāda). Ahmedabad: L. D. Institute of Indology. Ñāṇamoli, Bikkhu, and Bikkhu Bodhi, trans. 1995. The middle length discourses of the Buddha. Somerville: Wisdom Publications. Oppy, G. 2018. Ontological arguments. Cambridge: Cambridge University Press. Priest, G. 1979. Logic of paradox. Journal of Philosophical Logic 8: 219–214. Priest, G. 1995. Beyond the limits of thought. Cambridge: Cambridge University Press. 2nd edn, Oxford: Oxford University Press, 2002. Priest, G. 2008. Introduction to non-classical logic. Cambridge: Cambridge University Press. Priest, G. 2013. Nāgārjuna’s Mūlamadhyakamakārika. Topoi 32: 129–134. Priest, G. 2014a. Plurivalent logic. Australasian Journal of Logic 11: 1. http://ojs.victoria.ac.nz/ajl/ article/view/1830. Priest, G. 2014b. Speaking of the ineffable. . .. In Nothingness in Asian philosophy, ed. J. Lee and D. Berger. London: Routledge. Chapter 7. Priest, G. 2018. The fifth corner of four. Oxford: Oxford University Press. Priest, G. 2019. Natural deduction for systems in the FDE family. In New essays on Belnap-Dunn logic, ed. H. Omori and H. Wansing, 279–292. Berlin: Springer. Raju, P. 1954. The principle of four-cornered negation in Indian philosophy. Review of Metaphysics 7: 694–713. Sarkar, T. 1992. Some reflections on Jaina Anekāntavāda and Syādvada. Jadavpur Journal of Philosophy 2: 13–38. Siderits, M., and Katsura, S., trans. 2013. Nāgārjuna’s middle way. Boston: Wisdom Publications. Stcherbatsky, F.T. 1962. Buddhist logic. New York: Dover Publications.

Imperative Logic: Indian and Western

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Contents Definition of Key Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Possibility of Imperative Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of Imperatives in Indian Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Soundness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Study of Imperatives in Western Tradition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Reflections on the Logic of Imperatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Negation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Disjunction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Some Further Survey on the Logic of Imperatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Concept of Validity in the System of Vranas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Unconditional Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conditional Prescription . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Argument 1* . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Content-Validity of Josh Parsons3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Statements having imperative intents are found in plenty in the context of formal argumentative discourses. Commands of any form are commonly accepted as expressing a “prescription” as distinct from “proposition,” though the possibility of their crossing over is not ruled out. The lingering question remains about the M. Sanyal (*) University of Calcutta, Calcutta, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_52

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logical status of prescriptions as constituents of imperative arguments. Discussion of Vidhivākya (imperative) is extensively found in the Mīmāṁsā system and it has its pragmatic application in recent days in a very robust form. Since twentieth century, imperative arguments are considered as more than a possibility in the West. Its early history starts with Aristotle’s practical syllogism, passes through the thoughts of the Stoics, Leibnitz, Hume, and finally has a primary line of development, i.e., becomes a formal logic for normatives in Ernst Mally’s Principle of Inheritance of Obligation. Logicians take different stands in explaining the inferential property of an argument involving commands at par with that property of an inference of classical propositional logic. The present chapter undertakes a survey of the logic of imperatives both from the Indian and the Western points of view. Keywords

Vidhi · Niṣedha · Bhābanā · Niyog · Imperative · Prescription · Validity · Preposcription · Imperassertion

“Off with their heads!” shouted the Queen of Hearts. Lewis Carroll, Alice’s Adventures in Wonderland. Cambridge, Press of John Wilson and Son

Definition of Key Terms Vidhi – Niṣedha – Bhābanā – Niyoga – Imperative – Prescription – Validity – Preproscription – Imperassertion –

Injunctive statement Prohibitive statement That which causes the execution of an action Appointed task Command statement Ought statement The property of an argument that the truth of the premises guarantees the truth of the conclusion according to logical rules A representation of either the truth-condition of an assertion, or the compliance-condition of a command A generalization of both assertions and commands

Introduction The Queen of Hearts was aware of the fact that words have immense power and boundless significance, mainly in conversational context. And there is no exception of the fact in the real world also. So a great deal of attention has to be paid to language even in the field of logic. In Aristotelian logic (Aristotle 1901, 1989),

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language has been given a specific shape in order to enable us to determine the validity of arguments. In two-valued logic, ordinary inferences contain declarative sentences. But there are different types of inferences, which do not contain only declaratives. It rouses inevitably a doubt about the capacity of logic as a study of methods and principles of valid reasoning. Living examples of such cases are found in our daily life where arguments often contain only imperative sentences or a mixture of both declarative and imperative sentences. In the present chapter, an attempt has been made for giving an account of imperative logic developed and nurtured both in West and East.

Possibility of Imperative Inference In the past few decades, imperative logic has been developed both logically and philosophically. It is the speech-act theory introduced and nurtured by Austin-Searle legacy (Austin 1961, 1962; Searle 1971) that identifies the illocutionary act of a linguistic presentation as the act which differentiates between sentences that are descriptive/declarative/assertive, and those that are prescriptive/directives/prohibitives. What is the status of an imperative sentence? Imperatives are treated by a handful of thinkers as expressions of emotional reactions. Carnap (1935) treated imperatives as value judgements having meaning in psychological contexts. Ayer (1960, 1961) gave imperatives the status of an utterance that impels a person to do something. So, such a sentence both expresses and elicits emotions. Stevenson (1960), however, offered a status to imperatives that is higher than that of emotive expressions, though he acknowledged that imperatives simply express the attitude of the agent. Here, rationality plays a role, because imperatives are motivated by rational belief. This overall attitude of treating imperatives as pragmatically less important than declaratives may be the reason why possibility of imperative inference has been doubted over a period of years. Though imperative sentences were not given the status of a regular sentence, thinkers were aware of a kind of normative sentence (Portner 2007, 2016; Hamlin 1987). Since early twentieth century, thinkers acknowledged the use of deontic sentences such as “No entry,” “One ought to perform one’s duty,” or simply “Stop!”. Normative sentences are either deontic or imperative. According to R. M. Hare (1952), moral or deontic judgements share all their important logical features with imperatives. It is against the view of Hare that P.T. Geach (1958) argued in favor of the distinction between deontic and imperative. The distinction is not merely that the deontic sentences express direct obligations, and imperatives are often indirect or neutral; rather they possess different kinds of logical features. Deontic sentences, in most cases, contain the words like “ought,” “must” and have moral elements while imperatives are mostly devoid of such usage, and can never be treated like moral statements (Hamlin 1987). As the present discussion focuses upon imperative sentences, let us start with different types of imperative sentences.

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“Keep your mobile outside the room” (command). “Consult physician before taking the medicine” (warning). “Have a safe journey” (expressive imperatives). “Press the right button to open the door” (informative imperative).

Commands of any such form is commonly accepted as expressing a “prescription” as distinct from “proposition.” Aristotle’s (1962 reprint) distinction between practical and theoretical syllogism may be counted as the initial acknowledgement of imperatives, because he considered the premise of practical syllogism as a should-sentence. He recognized the conclusion of such a syllogism to be an action, though not as an imperative enjoining an action. The reference to legal modalities is found in Leibniz. Hume’s discussion of factual and evaluative judgement gives a hint to imperatives. There are evidences of some references of discussion on imperatives in Stoic philosophy (Mates 1953). One such reference is found in a paper “On Commands” by a stoic philosopher Chryssipus (Barnes 2012). Some thinkers (e.g., Follesdal Dagfinn and Risto Hilpinen 1981) recognize imperative sentences as deontic ones. Some others (e.g., Carnap 1960) recognize deontic sentences as hidden imperatives. Von Wright (1971) is an important name in this context, who applied modal logic to construct a logic of norms or deontic logic. The norms may be rules of logic, mathematics, etc., or directives, which are called “technical norms.” Prescriptions are the third variety of norms, which includes commands, permissions, and prohibitions (Hintikka 1981). The attempt to treat imperatives as deontic propositions is also evident in the writing of Miguel PerezRamirez (2003). There is, however, a strong opinion about the distinction between the two, viz., deontic logic treated as an extension of modal logic, and imperatives as completely outside the scope of modal logic. It is mainly found in the writing of R. M. Hare (1971). Let us start with an example of imperative inference: Help the distressed. This is a man in distress. Therefore, help this man.

Williams (1963), Wedeking (1970), Harrison (1991), and Hansen (2008) are some thinkers who endorse the denial of such inferences. The main reason for such denial is that distinctive imperatives have conflicting permissive presuppositions. So imperatives are generally treated neither as the premise nor as the conclusion of an argument. Bernard Williams’ example is: Do X or do Y. Do not do X. Therefore, do Y.

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Unlike declaratives, the commands expect that the world should be modelled accordingly. But here, the combination of two premises clearly indicates either a contradiction or a change of mind of the command-issuing agent. Williams denies that there is logical relation between the imperatives in ordinary life, though he admits such a relation in philosophical writings. For example, Read all the books of poems by Rabindranath Tagore. So, read Gitanjali.

According to Hansen, the term “so” is the motivating force to do some action. It cannot be replaced by the term “therefore,” implying thereby that the second sentence is not derived from the first. Gary A. Wedeking also rejected the possibility of imperative inference. He tried to establish his view by proving the defects of Castaneda’s attempt to accept such inferences. According to Wedeking, employment of the inferential words, viz., “for,” “since,” “therefore,” “so” has the purpose of connecting an imperative with other utterances, which are imperatives or indicatives. But the strange fact is that Castaneda’s argument has the paradoxical result that imperatives may occur in arguments only as conclusions. Let us cite some examples. (i) “Since she is too busy right now, please sit down and wait” (ii) “This is the best thing for John to follow, so follow it.” Wedeking further points out that Castañeda’s (1957) argument is based on the supposition that the terms like “since,” “therefore,” etc. are definitely inferential words. Though it is generally accepted that these terms are inferential when they are applied in factual assertions, doubt remains regarding their nature in the case of so-called imperatives. Wedeking undertook a detailed analysis to show that the existence of conditional imperative does not imply that commands are inferable. He also went to the extreme to say that a conditional imperative is not imperative in the strong sense. The generally admitted definition of logical inference in the traditional sense is that only those sentences which have the capability of being true or false can feature as premise or conclusion of an inference. The non-reductionist thinkers – being not in favor of reducing imperatives into indicatives for the sake of inference – do not accept this claim. Peter B. M. Vranas, who is a founder of an exclusive system of imperative logic, states that a process of reasoning can very well begin by endorsing an imperative sentence. According to Vranas, to endorse an imperative sentence is to believe that the sentence is binding, i.e., obligatory. Let us take an example. In an examination, there are four questions, and there are two instructions: (i) Answer exactly three out of four questions. (ii) Answer at least one even-numbered question.

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Here, someone will automatically obey the second instruction if she obeys the first one. To accept the first instruction as binding is to accept the second one also being endorsed. We shall come to the Western theories of imperatives and, in that context, the discussion of non-reductionism in section “Study of Imperatives in Western Tradition.”

Study of Imperatives in Indian Philosophy Study of imperatives is distinctly evident in the Indian context (Potter 1995). In recent times, surveys on knowledge representation and reasoning in the computational domain are mainly based on the study of human actions guided by instructions (Srinivasan and Parthasarathi 2014). First-order logic is often used in this respect. Among different schools of Indian philosophy, the Mīmāṁsā system offers an analysis of imperative sentences, where actions, guided by instructions, play an important role (Motilal 1986). It is necessary to be acquainted with this Indian school not only as a philosophical background of the new development of the study of imperative statements, but, more importantly, as a fundamental study of imperatives (Monier 1993). Unless one knows the meaning of a command-statement, one cannot proceed in constructing a logic of such statements. There are five types of Vedic sentences, of which only the first is in the imperative form: (i) (ii) (iii) (iv) (v)

Vidhi or normal injunctive statements (dictating one to perform actions) Mantra or hymns (recited during sacrifice) Nāmadheya or titles of the sacrifice (account of names of sacrifices) Niṣedha or prohibitions (prohibiting the performance of an action) Arthavāda or corroborative statements (encouraging performance of positive imperatives and discouraging performance of prohibited actions)

In Jayantabhat‚‚ta’s Nyāyamañjarī (1895), proper characterization of an imperative is said to be difficult. It even raises the question about the necessity of an imperative (Bhattacharya 1978). The importance of imperative, however, is that knowledge acquired from imperatives cannot be accessed by any other instrument of knowledge. When somebody utters an imperative “Svargakāmo yajeta” (One who desires heaven should perform a sacrifice), then it is implied that there is a connection between “Svarga” and the ritualistic sacrifice made for achieving it. This connection is known only by the excellence of the imperative itself. To quote Śabarasvamī, (Jha 1933) “Upadeśa iti viśiṣṭasya śabdasyoccāraņaṁ” (Śabarabhāṣya, p. 29) (Śabarasvami 1976). Now the activity of sacrifice cannot be self-initiated. So, it requires an agent who is conscious and who has knowledge. Such an agent is available in case of empirical imperatives, but not in case of Vedic Vidhivākyas which have no speaker. Hence, the only way of explaining the workability of imperatives is by analyzing their meaning. There is a great controversy regarding the principal qualificand (mūkhyaviśeṣya) of imperative sentence (Motilal and Sen 1988). According to the Mīmāṁsakas, in a

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prescription, every morpheme has a specific meaning, and one or other paraphrase is used to make the meaning explicit. It is a fact that the Mīmāṁsakas are more concerned about the explanation of Vidhivākyas (imperatives/prescriptions) in the context of ritualistic sacrifice. In case of the prescriptions, the verbal endings are the linguistic elements which are associated with prescriptions, i.e., the endings denoting the optative (liṅ), the imperative (lot‚), the gerundive (tavya), and the Vedic subjunctive (let‚). Another thing to note is that an imperative points both to the person to whom the command is given, and to the action that is supposed to be produced by that command. The Mīmāṁsakas consider the same rule to be applied to both positive and negative commands (Verse-85, NM). The imperative “Do not kill” (“Na hiṃsyāt”) is directed to the angry attacking person as well as to the prevention of the act of attack. One may think of a critical comment here that prevention (of a bad action, viz., murder) is not an action. So, the negative imperative is not a command statement proper. The answer is, just as the negative term “non-brahmin” designates a person (positive entity), so also the negative command works as a positive designator. Jayantabhat‚‚ta, in Nyāyamañjarī (2008), has discussed in detail about the meaning of a prescription (Vidhi). It is relevant to focus upon the issue, because it provides the foundation upon which a system has been built to represent the use and application of imperatives from Indian perspective. Jayantabhat‚‚ta referred to “activity,” which is the consequence of the testimonial knowledge of the verb, as one of the many alternative meanings of a command-statement. The same view is also found in Bhartŗhari’s (1968) Vākyapadīya (2/414). It is said that the meaning of a sentence is that what is to be executed, i.e., the action (Verse-13, Chapter-V, NM, Chakrabarti 1989). The Bhāṭṭa Mīmāṁsakas argue that until and unless a person understands his connection with the activity, he cannot perform the ritual. Why is an action important ? The reason is, whenever an instruction is given, it is given for being executed. The agent is important here, because only by performing the action, the person becomes an agent. The imperative is authoritative (prāmāņyavākya) in nature. The Bhāṭṭa Mīmāṁsakas think that whatever causes the execution of an action (bhāvanā), is the meaning of the command-statement (Verse 25, Chapter-V, NM). When an instruction is given, the expected result is at that moment “yet to be achieved.” That which is “yet to be achieved” cannot be the meaning of the instruction, because future is not certain. So the Bhāt‚‚tas consider bhāvanā (not to be confused with motivation) as the meaning of the statement. It is something that is conducive to the execution of the expected result. The causative verbal noun bhāvanā (“causing to be”) was introduced into Mīmāṁsā hermeneutics by Śabara. The term is a causative verbal noun which denotes the undertaking of an activity by a person. This undertaking directed to an object is designated by the verbal ending (ākhyāta) words. Kumārila introduced another linguistic force “śabdabhāvanā,” implying thereby that the person listening to the sentence becomes aware of the compulsion to perform the action indicated by the verbal root. But as this linguistic force operates only on the linguistic level, it has no direct influence upon the actual performance. Therefore,

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another objective force called “arthabhāvanā,” that is produced by “śabdabhāvanā,” and which produces the intended result, is also admitted. Āpadeva (1929) explains in detail the distinction between “word-efficient-force” and “end-efficient-force” (Mīmāṁsānyāyaprakāśa, pp. 39–40) in the following way. The first one is a characteristic of exhortative forms only and works at the psychological and linguistic level, compelling the person concerned to perform the prescribed action. It contains an expectation for an object, an instrument, and a procedure. But this linguistic force is incapable of producing the actual performance of the action. So the objective force was introduced as Ārthībhāvanā. For example, in the case of the imperative “Svargakāmo yajeta” (One who desires heaven should perform a sacrifice), the meaning is apprehended in the following way: “Through sacrifice one should cause heaven to occur” (Śābdībhāvanā). “One should cause an activity that leads to heaven, and that has the sacrifice as its instrumental cause” (Ārthībhāvanā).

For Maņḍana Miśra (1922), an imperative for performing a certain action produces in the listener the cognition that the prescribed act is a means for achieving something that is desirable (Iṣt‚asādhana) (Bhatta 1994). In his Brahmasiddhi and Vidhiviveka (Miśra and Miśra 1907), he attempted to show that it is the knowledge of achieving the expected result that inspires the agent to persevere. The sense of duty is born out of the knowledge of the means of reaching the expected result. An objection has been raised in verse-27 (NM) that in the statement “Svargakāmo yajeta,” the verb itself initiates the ritualistic sacrifice and nothing called “Bhāvanā” is present there. There is a serious debate between the Bhāt‚‚tas and the Prābhākaras regarding the meaning of verbal endings (Freschi 2012). The Prābhākaras claim that the entire Veda is an instrument of knowledge, not by virtue of prescribing means of attaining some end, but by virtue of pointing something that is to be done irrespective of the result of that activity. Rāmānujācārya (1956) speaks of “unprecedented” duty (Tantrarahasya IV 11.1). According to the Prābhākaras, bhāvanā is not the meaning of the exhortative suffixes. The suffixes denote something to be done and accordingly they denote action. The Vedic injunctions do not describe reality, rather they prescribe what is to be done. This knowledge of the “ought” cannot be procured by any other means. Accordingly, it is called “apūrva,” i.e., “unprecedented” by any other means of knowledge. Rāmānujācārya calls this Prābhākara view of treating Vidhi as Niyoga(appointed task) (Tantrarahasya IV 11.2). The motive behind performing an action is not the consequence of that action, though every action must have some consequence. Unlike the Bhāṭṭa theory, he postulates one action which is instrumental to the realization of both the unprecedented duty and the result (Tantrarahasya IV 11.7.5). It solves the problem that arises when one says that until and unless a person understands his connection with sacrifice, he cannot perform the rituals. In this view, therefore, the prescription is treated as an authoritative sentence. It is by virtue of the power of the words that the action is known to be primary or important. Words do not primarily aim at conveying the result/consequence of the

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prescribed action (Verse-17, NM). When an action is done, the result appears automatically. The appointed task (Niyoga) (Verse-65 NM) is something in whose presence the hearer thinks “I am obligated for performing this work” (Chatterjee 1980). The other name of “Niyoga” is “duty” (Kartavya), i.e., something that has to be performed, which is the nature of some duty. One performs it out of a sense of obligation. It is said (Verse-76, NM) that the knowledge of the commanded task is possible by the presence of the suffix (liṅ). When one utters “Do the sacrifice,” the dictate itself is authoritative and requires nothing else for prompting someone to perform a particular action. In one sense, the imperative points to both the agent and the action in the following way: (a) Who is given the order? (b) Which order is given? Śālikanāth Mishra (1904), in Prakaraṇapañcikā has shown – with the help of some arguments – that duty is different from achieving the purpose/end (Sharma 1987). What is important is that one has to remember that in every prescribed action, there is a purpose or goal for successful execution of the instructions. Indian thinkers have given much attention to it, and those who rejected the idea also have sufficient reason in favor of their view. But this issue is not given importance in the Western theories, not even in the full-fledged theory of Peter Vranas. That the purpose is playing a role behind the execution of an action is not mentioned, rather the attitude found there is that the success of such a statement is more or less presupposed. Another pertinent question may arise here. Do all command statements have immediate consequence? For example, statements like “Satyam vada” (speak the truth), “Pitṛdevo bhava” (show respect to your father) do not mention any direct consequence that results from obeying such imperatives. Any negative answer to the question may deny the relevance of imperative statement. A strong defense in favor of an affirmative answer may be found in Pantñajali’s Mahābhāṣya (2006) where derivation of conclusion from an imperative is clearly marked. For example, prescriptions expressing prohibitions in respect of some items distinctly show permission of the same act in respect of some other items. Tad/jathā/bhakṣyaniyamena/abhakṣyapratiśodho/gamyate— pañca/pañcanakhā/bhakṣyāh/ , ity/ukte gamyate/ – etat atah /anye abhakṣyāh/ iti Ι Abhakṣyapratiśodhena/ vā / bhakṣyaniyamah –tad/jathā abhakṣyo/grāmyakukkutah abhakṣyo/grāmyaśūkarah ity/ukte gamyate/- etat āraņyo/ bhakṣyah/ iti Ι (Mahābhāṣya, p. 39)

For example, from the injunction “Five among the five-nailed animals may be eaten,” it follows that other five-nailed animals are not to be eaten. Again, from the prohibition “Domestic cocks and boars should not be eaten,” it follows that wild cocks are boars may be eaten. Keeping in mind the issue of goal/purpose as well as the issue of consequence, a formalism named MIRA (Mīṁāmsa Inspired Representation of Actions) has been

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initiated by Bama Srinivasan and Ranjani Parthasarathi for interpretation (syntax) and evaluation (semantics) of action-prescribed imperatives. It considers the agent’s intention of achieving the goal. Gradually, this system has taken the shape of a formal system of imperative logic inspired by Mīmāṁsā system, but with no distinct line of thought either of the Bhāṭṭas or of the Prābhākaras. Srinivasan and Parthasarathi have referred to several interpretation rules, viz., those of conjunction and exclusive disjunction. The motive is to derive an injunctive statement from a set of injunctive statements. The inspiration is derived from the event of Vedic sacrifice, where the performance is conducted by following a specific order comprising of six criteria, viz., Śrutikrama, Arthakrama, Pāt‚hakrama, Sthānakrama, Mukhyakrama, and Pravrittikrama. In case of conjunctive statements, the connectivity between two components is taken care of. The Mīmāṁsā system identifies two types of connection between two sentences as vākyaikavākyatā and padaikavākyatā. The former takes place between two injunctions having separate verbal forms and being mutually expectant of each other. The second one shows connectivity between the narrative sentence and injunction (Pandurangi 2006, p. 373). Injunction (Vidhi) is classified into five types: 1. Principal injunction (Utpattividhi): Injunctions enjoining an unknown entity, an auxiliary, or a procedure. 2. Injunction enjoining auxiliaries (Guņavidhi). 3. Restrictive injunction (Niyamavidhi): Injunctions making one method mandatory, out of two or more methods which are available for reaching a goal. 4. Exclusive injunction (Parisaṅkhyāvidhi): Injunctions excluding one items from two items which are simultaneously present. 5. Injunction setting forth result (Phalavidhi): Injunctions that indicate results. For example, “One who desires heaven should perform fire-sacrifice.” The recent formalism followed the introduction of five types of injunctions. The intention of achieving the goal, when expressed and added to an unconditional imperative, forms a conditional imperative statement. It often combines two actions involving the element of temporality. In this interpretation, imperatives are treated either as conditional or as unconditional. From another perspective, imperatives may be affirmative or negative. Conditional imperatives often speak of goal, reason, or sequence of actions. Imperatives are expressed sometimes in terms of binary connectives, viz., conjunction, mutually exclusive disjunction, implication, etc. Let i and m be two imperatives (a) (b) (c) (d)

Conjunction: i Ʌ m [Do i and do m] Disjunction: i _ m [Do i or do m] Sequence of action: i ¼> im [Do i, then do m] Ground for performing an action: Ʈ ! r φ [If Ʈ then φ] (where Ʈ is a ground for an action to be performed indicated by the imperative φ)

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(e) Imperative regarding actions to be performed for achieving a goal: φ! p ө [Do φ to do ө] (if φ is an imperative indicating an action such that when performed, it leads to the goal ө) Three values of imperatives have been suggested, viz., “S” (satisfaction), “V” (violation), and “Gn” (absence of goal). Let us take an example to illustrate the ascription of values, Take a pen to write.

S is the evaluation if the intention to reach the goal of writing is present and the action is performed. V is the value ascribed to the imperative A if the said intention is present and the action is not performed. Gn is the ascribed value if the intention is not present, irrespective of the performance of the action. This system has introduced the third value “absence of goal” (which is the same as absence of intention to reach the goal) in place of “avoidance” introduced by Vranas, and, unlike Vranas, it enjoys the facility of applying three values both to the unconditional and conditional imperative. The syntax consists of a language of imperatives which includes a set of imperatives I such that {i1, i2, . . .. in}, a set of reasons R {r1, r2, . . .. rn}, and a set of purpose in terms of goals P {p1, p2, . . .. pn}. B is a set of binary connectives such that B {^, _ , !r, !i, !p}. Unconditional and conditional imperative formula are represented as Fa and Fb. There are formations rules and several deduction rules including introduction and elimination rules in respect of the connectives. The semantics has been developed in respect of imperatives enjoining goals (φ ! pө), reason (τ ! rφ), and temporal actions (i1 ! i i2), respectively. The operation for conjunction is given as SɅS¼S SɅV¼VɅS¼VɅV¼V S Ʌ Gn ¼ Gn Ʌ S ¼ Gn Ʌ Gn ¼ Gn The operation for disjunction is S v V ¼ V v S ¼ S v Gn ¼ Gn v S ¼ S S v S ¼ V v V ¼ V v Gn ¼ Gn v V ¼ V Gn v Gn ¼ Gn The system presented the procedure for the deduction process with the help of formation and transformation rules. The deduction rules are proved to be valid. It

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implies that if the premises which are propositions are true, and the premises which are instructions are satisfied, then the conclusion must be true (if it is a proposition) or satisfied (if it is an instruction). The deduction rules also help in the derivation of other rules. By repeated application of deduction rules, a conclusion ψ can be deduced from a set of premises φ1, φ2, . . .. φn. It is shown in the following way:

The system has successfully handled different varieties of uses of the connectives, viz., conjunction and disjunction. It appears that it is in a better position for explaining action representation in respect of imperatives in comparison to other systems. MIRA is based on the principle that actions are performed with the intention of reaching the goal, which is not found in the system of Vranas. So unlike this system, the operations of imperatives in MIRA may vary when imperatives are joined with connectives. In an article entitled “A formalism to specify unambiguous instructions inspired by Mīmāṁsā in computational settings,” Srinivasan and Parthasarathi have discussed the types of Bhāvanā (preraņā and pravŗtti) after the Bhāṭṭas. Soundness and completeness of this system have been proved to show that any imperative provable by MIRA formalism (2014) is also satisfied during the performance of action. In proving soundness, it attempts to show that the deduction of a conclusion from a set of premises is valid in terms of the values held by the premises and conclusion. The theorems which are proved for this purpose are as follows:

Soundness Theorem 1 Let and be imperative or propositional formulas. If , then holds. The proof for soundness includes one inductive step and proofs for each of the deduction rules.

Completeness Theorem 2 Let

and be imperative or propositional formulas. If , then the property of a plan is expresse holds. The proof for completeness is constructed on the basis of induction and being supported by action performance tables and deduction rules.

Study of Imperatives in Western Tradition Let us take note of the study of imperatives in Western tradition by remembering the possibility of imperative inference in section “Possibility of Imperative Inference.” Once one is convinced of the possibility of imperative logic, one can pursue any of the two approaches to undertake a survey of imperative logic in West.

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One is the reductionist approach, the other is the non-reductionist one. The reductionist approach admits the possibility of imperative inference, but it accepts the possibility of translating imperative sentences into declarative one, thereby transforming imperative inference into declarative inference, and explaining it in the structure of classical two-valued logic. The non-reductionist approach, on the other hand, treats imperative sentences as fundamental units, and it is in favor of constructing a separate logic for imperative inference. Two things must be noted at this juncture. First, those who interpret imperatives in terms of truth or falsity translate an imperative in the following way: Do it. Let it be done.

Second, imperatives are often viewed from different perspectives, viz., the perspective of the speaker or the perspective of the action. We can refer to Ernst Mally’s (1926) Principle of Inheritance of Obligation. He was the first thinker who distinguished between judging and willing. According to him, when a given state of affair, viz., p takes place, then it may be expressed as “p ought to be the case.” The principle of obligation is:

Using the monadic operator ‘!’ to express the obligation, we can show the following principles which are deducible in Mally’s axiomatic system: P ! !P !P ! P Therefore, P  !P: This system allows obligatory statements to be reducible to statements having truth-values. There is, however, a confusion regarding the nature of “!,” viz., whether it should be treated as material or strict implication. Dagfinn Follesdal and Risto Hilpinen have shown that the above derivation is responsible for making deontic logic trivial. Jorgen Jorgensen (1938) rejected the possibility of imperative logic as a separate system. He adopted the perspective of the speaker. In his interpretation, the property of the statement “such and such action is to be performed” is attributed to the person issuing the command. He admitted the imperative factor of a sentence, but this imperative factor cannot be accounted for unless it is translated into the indicative factor. An imperative inference can be called valid if it is translated into a declarative inference. The following is an example of such translation suggested by Jorgensen. Keep your promises. This is a promise of yours. Therefore, keep your promise.

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In order to be assessed as valid, this argument may be translated as: All promises are to be kept. This is a promise of yours. Therefore, this promise is to be kept.

According to Jorgensen, the imperative factor is translated as “is to be.” This system studies imperative from speaker’s perspective. The imperative factor is a kind of quasi property which, when translated, is ascribed to an action when the person who makes a command wants the action to be performed. Interestingly, it is comparable to the view of Udayana (Mohanty 1966; Udayana 1957) who says that Vidhi expresses āptābhiprāya (intention of the authoritative person). It is then only that the translated inference is evaluated as true or false. It is, however, a dilemma referred to by Jorgensen that leads to the construction of a new logical system for imperatives. We shall come to it later. The perspective of action is exhibited in the theory of Hofstadter and McKinsay (1939), where the satisfaction criterion shows the satisfaction of action. Hofstadter and McKinsey distinguish between two types of imperatives, viz., fiats and directives. Directives are direct commands, viz., “Shyam, don’t drive when you are drunk.” Fiats are reference-less commands, viz., “Let the show start.” In this system, the directives are translated into fiats. An imperative is correct if what is commanded “ought” to be the case. It is satisfied if what is commanded “is” to be the case. So, satisfaction is used almost in the analogous sense of the term “truth.” This is the reason why the connectives have been used analogously as they are used in classical logic with a little variation in the symbols. The different operators, viz., the complementary “─,” the sum “++,” the product “x,” the conditional “!!” are functional here along with the unary operator “!,” and the operator of translation from an imperative to an indicative used here is “˂.” Here, a formal language has been constructed which is an extension of Carnap’s language of LI. This is an early attempt at treating imperative logic as different from classical logic. Karl Menger (1939) considers the commands and wishes as neither asserting nor denying anything, but as doubtful statements. The symbolic forms for assertion, negation, and doubtful adopted by Menger are M+, M─, Mo, respectively. The interpretation of “I command p” is “Unless p, something unpleasant will happen”

Under this circumstance, if A stands for all unpleasant things that will happen, then the whole interpreting sentence may be symbolized as “Cp  (p! A).” Menger attempts to show that propositional logic is inadequate for explaining commands and wishes. The problem with this many-valued logic of imperatives is the introduction of the doubtful sentence. If a command is doubtful, then not only the essence of command is lost, but the command also becomes powerless.

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Kurt Grelling (1939) accepts that imperatives are sentences, though not in the “logical” sense of being a true or false sentence. Here is an example, where an imperative sentence is drawn as a conclusion from an argument containing an imperative sentence: All promises are to be kept. This is a promise. Therefore, this promise is to be kept.

The two premises and the conclusion may be symbolized as “x,” “y,” and “z,” respectively. The schematization of the whole argument may be represented as: From x ought to be and y, it follows that z ought to be true.

This illustrates an expansion of the meaning of logical implication, which is evident in one of his axioms: ðp&ðq ! rÞÞ ! ðp&ðOq ! OrÞÞ The attempt has been criticized by Karl Reach (1940) because of the vagueness of the idea that imperatives are sentences not in the usual sense. The counterexample of “A” is hinted at by Reach, where a true conclusion can follow from something false. If “x” is rephrased as “z,” and “y” is rephrased as “non-z,” then the conclusion, i.e., “z” can follow from “z and non-z.” Hence, the following argument may be constructed:

The conclusion is counterintuitive, as it means that if we do something which we ought not to do, then we ought to do it. In Alf Ross (1941), imperative is studied from the perspective of action. The question whether imperative can be a constituent part of a logical inference prompted Ross to redefine logical inference. Initially, he defines inference as a movement of thought, where one begins with one or more propositions and reaches one conclusion whose truth is contained in the premises. But it is realized gradually that the concepts of truth and falsity alone are not sufficient for defining an inference. The validity of an inference, where one imperative is inferred from the other, does not depend upon corroboration by facts; it rather depends upon formal rules of deduction. Ross agrees with Jorgensen in distinguishing between imperative and indicative properties of imperative sentence. Ross has made an important contribution by connecting logic of satisfaction with logic of validity. In his presentation, the

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fulfilment of imperatives in terms of “being in force” has pointed out the issue of performance of an action, which has been given much importance by later logicians. Validity for imperative is defined by the possession of the psychological “state of acceptance” by the person, who acts upon the imperative. But though the issue of satisfaction is spoken of, still an imperative inference in this system is understood by mapping the satisfaction values to truth-values. Truth/ falsity is assigned to the propositional content of the imperative, I  !ðSÞ I is the imperative and S is its propositional content. The interesting part is that Ross speaks of a problematic translation of imperatives to indicatives, which is expressed in the following way : Post the letter. Therefore, post the letter or burn it.

This argument may be translated as: The letter has to be posted. Therefore, the letter has to be posted or the letter has to be burned.

This inference is obviously found to be counterintuitive. This paradox plays a vital role in changing the logical approach to the reductionist analysis of imperative inferences. Logicians started to view imperative inferences as independent of declarative ones. This thought paves the way for constructing a separate logic for imperatives. The lingering question is – how to construct a full-fledged system of imperative logic? Eminent thinkers in this venture are Beardsley, Chris Fox (2008), and Peter B.M. Vranas. Beardsley adopted the speaker’s perspective and interpreted imperatives in respect of the relationship between the imperative and the person making the imperative. An imperative that is either uttered or written, expresses the desire of the speaker/writer which is known as satisfaction of the imperative. Here, the speaker expects the command to be satisfied, which is called pragmatic implicate. Under no circumstance can the imperative (e.g., “Window be shut”) and a so-called corresponding indicative (“The window is shut”) be correlated. A similar perspective is found in the analysis by A. J. Kenny (1966) also, though with a little variation in developing the idea. Important contribution of Beardsley (1944) is found in analyzing compound imperatives. Unlike classical logic, where the uses of common connectives, viz., “and” and “or” are both treated as “conjunction” and “choice” as “disjunction,” Beardsley uses these terms occurring in imperative sentences, not only as “conjunction” and “choice,” but also as consequence. For example: Finish homework and take rest. Practice Yoga or you will fall sick.

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Here, in the first conjunctive command, the event of taking rest is prescribed only as a consequence of the command regarding completion of homework. The second command involving disjunction also indicates that the hearer will fall sick as a consequence, in case the command of practicing Yoga is not obeyed. Hare’ s approach was that of a non-reductionist who accepted the possibility of imperative logic. The distinction between “descriptor” and “dictor” speaks of the distinction between indicative and imperative in some unique way. It may be illustrated with the help of two examples: Open the window. You are going to open the window.

Both the sentences have one “descriptor,” which is the common element, viz., “your opening the window in immediate future.” The “dictive” function of the sentence may be illustrated by showing the mood of the sentences: Your opening the window in immediate future, please. (2.1) Your opening the window in immediate future, yes.

The rule for validity of imperative inference is this: No imperative conclusion can be validly drawn from a set of premises which does not contain at least one imperative. This criterion, along with the distinction mentioned before, does not however provide sufficient ground for identifying the uniqueness of the system of imperative logic.

It may be a reason why one should consider Hector-Neri Castañeda’s view (1970, 1975), who offered a non-reductionist explanation for a “semi-logic” of imperative. He introduced a broader sense of a logic of norms, which are marked by several characteristics. He admitted a “kinship” between imperatives and normatives (norms include imperatives); though there is a marked distinction, in the sense that unlike norms, imperatives can never feature either as premise or as conclusion in reasoning. Imperatives have a distinct mode of combining the subject with the predicate. Nicholas Rescher (1966) initiated a logical system of commands which have less applications as compared to imperatives. According to him, conditional imperatives are often used to make purely factual assertions. For example, “If you want to visit the mountains in winter, go to Darjeeling in January.” A command, however, has several elements, viz., the source, the recipient, the mooted action, execution timing, and period of the command-in-force. He laid down three conditions for the validity of commands. A command is either terminated or unterminated. A command inference is invalid if the indicative and/or command premises of the imperative syllogism are either true or (in the case of command premises) terminated, i.e., successfully acted upon, though the command conclusion is unterminated. He introduced the criterion for patent validity of imperative inference with the help of the concept of command decomposition.

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Non-reductionist systems discussed so far have not provided any full-fledged system for imperative inferences. The major non-reductionist systems of Ross, Hare, and Castañeda have not offered a clear definition of the concepts of validity or satisfaction for imperative arguments, both pure and mixed. Castañeda’s semi-logic lacks a mature logical structure, and Rescher has formulated a system for commands, and not for imperative sentences. There are, however, evidences of full-fledged systems of imperative logic presented by some logicians like Chris Fox and Peter B. M. Vranas. We may refer to the following imperative sentences which are concise forms of respective imperative inferences: 1. 2. 3. 4.

Buyers of different appliances are provided with orders on how to apply them. Taxpayers are provided with instructions on how to fill up income-tax forms. Children are sometimes given conflicting orders by their seniors in any field. Students are sometimes provided with conflicting suggestions by others on what courses to select. 5. Everyone is suggested with sets of dietary guidelines, and so on. An imperative inference is often more straightforward than the declarative inferences. Directives and disinterested advice feature in the language as evidence of imperative. It is the relevancy and usefulness of imperative inference that count as evidences for endorsing the possibility of imperative inferences in human life. A non-reductionist approach, where imperatives are not translated into indicative propositions, is found in the writings of Chris Fox (2012). Here, the speaker is given importance in dealing with judgement criterion. He presented a proof-theoretic formalization of imperatives. Here, the temporal order of actions is incorporated. The formalism advocated here incorporates the use of future tense modal operator and agentive property. The indicative proposition possesses the property of having truth and falsity as values. The imperative is considered to be satisfied if the corresponding indicative proposition is true. An example may be used for illustrating the formalism: Open the window. [imperative]

This imperative is satisfied if it is true that “In the future the agent opens the window.” This criterion of satisfaction of imperatives is applicable to imperatives involving unary and/or binary connectives. By mapping the truth of the propositional factor into the satisfaction of the imperative, this formalism cannot, however, deal with the use of “and” in two different contexts. It fails to distinguish between the following cases: 1. “Take the bag and take the umbrella.” 2. “Finish your pending works and go to sleep.”

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In “1,” the sequence of the actions does not matter, while in “2,” the sequence is important. The second action is supposed to follow the first one. Fox has dealt with this issue by bringing sequence into imperatives. He also introduced a formalism based on the intuition which occurs out of satisfaction of the command and coherence of the commanding authority. Here, in the first structure, the imperatives are defined with basic imperatives (Ib) and conditional imperatives (Ic). The second structure consists of a classical proposition and the third structure of this syntax is a judgement. Judgement helps in determining the set of imperatives. The imperatives may be commanded, satisfied, or incoherent. A better non-reductionist approach is found in the writing of Peter Vranas (2010). After speaking about the necessity and inevitability of imperative inferences in our everyday life, he delves into the question of validity of the imperative inferences, which cannot be explained in the classical model. Imperatives may be commands, suggestions, requests, instructions, or some such similar expressions. They are treated as fundamental, having a distinct character of their own. The basic difference between indicative and imperative sentences, according to Vranas, is that while the first one expresses a proposition which is either true or false, the second one deals with a prescription which is neither true nor false. A prescription may be conditional or unconditional. He introduced three measures to study prescriptions. They are either satisfied, or violated, or avoided. A prescription is defined as an ordered pair of logically incompatible propositions. An unconditional prescription is an ordered pair with satisfaction as the first member and violation as the second member: I ¼< S, V > In the case of conditional imperative, the prescription is a set of three values, viz., satisfaction, violation, and avoidance. This set constitutes a partition of the set of all possible worlds, i.e., they are mutually exclusive and jointly exhaustive. Thus, given any two, the third one is the complement of the union of the remaining two. The condition is treated as context, which is the union of the set of satisfaction and that of violation. A conditional prescription, e.g., “If you trust him, wait for him” is (i) Satisfied – if you trust him, and wait for him. (ii) Violated – if you trust him, but don’t wait for him. (iii) Avoided – if you don’t trust him, no matter whether you wait for him or not. Different connectives in conditional imperative also function accordingly. A conditional imperative may have a propositional antecedent and imperative consequence. The antecedent is evaluated as true or false, and the consequence enjoys the three alternatives, viz., satisfaction, violation, and avoidance. Let us be familiar with the illustrations of all connectives:

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Negation Do not trust him I ¼< S, V > ─I ¼ ─ < S, V >¼< V, S >

Conjunction The conjunction of two prescriptions is the prescription in which the context is the union of the contexts of the conjuncts and in which the violation is the union of the violation sets of the conjuncts. < S, V > & < S’ , V’ >   ¼ ─ < V U V’ > , < V U V’ >> C ¼ S U V; C’ ¼ S’ U V’ ¼ ─ < V U V’ > , < V U V’ >> Disjunction, conditional, and biconditional connectives are also likewise interpreted. A set of prescriptions is inconsistent exactly if the conjunction of the prescriptions is omniviolable (i.e., both unsatisfiable and unconditional). This is a natural deduction system, where validity of an inference is defined in terms of satisfaction. Satisfaction is further defined by the criterion of bindingness, and also interpreted in terms of obedience of the imperative. The proofs for soundness and completeness have been provided here. The general definition of validity given by Vranas is as follows: An argument is valid (i.e., its premises entail its conclusion; equivalently, its conclusion follows from its premises) exactly if, necessarily, every fact that sustains every premise of the argument also sustains the conclusion of the argument. One of the consequences of a theory of validity, which he names as (D2) is: (D2) An argument is valid, only if, necessarily, if its premises merit endorsement, then its conclusion merits endorsement.

In order to get an uniform desideratum, Vranas tried to define validity of both types of arguments – declarative and imperative – in terms of meriting endorsement. In case of declarative arguments, meriting endorsement is the same as being supported by a true proposition, which is guaranteed by some fact, while in the case of an imperative argument, it is the same as being supported (sustained) by some reason (fact). “Reason” is an umbrella term, which accommodates reasons for acting, feeling, believing, etc.

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A prescription’s meriting endorsement means being supported by a reason. It is the same as favoring its satisfaction-proposition over the violation-proposition. This is unquestionably asymmetric in nature. Meriting endorsement may be interpreted in two ways, viz., meriting endorsement either jointly or separately. A cross-species imperative merits endorsement jointly, exactly if some fact sustains both the proposition and the prescription; and it merits endorsement separately, exactly if some fact sustains the proposition and some (same or different) fact sustains the prescription. Hence, by definition, the first one entails the other. Using these two meanings of meriting endorsement, we get the following two interpretations of (D2): (D2J) An argument is valid only if, necessarily, if its premises merit endorsement jointly, then its conclusion merits endorsement. (D2S) An argument is valid only if, necessarily, if its premises merit endorsement separately, then its conclusion merits endorsement.

According to Vranas (2011), there is a distinction between “pro-tanto” (prima facie) endorsement and “all-things-considered” (undefeatedly supported by some fact/reason) endorsement. They may be explained respectively in this way: (D2P) An argument is valid only if, necessarily, if its premises merit pro tanto endorsement (sustained by some fact), then its conclusion merits pro tanto endorsement (sustained by some fact). (D2A) An argument is valid only if, necessarily, if its premises merit “all-thingsconsidered” endorsement (undefeatedly sustained by some fact), then its conclusion merits “all-things-considered” endorsement (undefeatedly sustained by some fact).

As (D2A) speaks of undefeated reason, (D2) is to be better understood as (D2A). It is now time to apply this definition to the cross-species imperative argument in the following way: Definition 3 A cross-species imperative argument is valid exactly if, necessarily, every fact that guarantees the (declarative) premise of the argument supports the (imperative) conclusion of the argument. The usability of this definition is shown by the following theorem: Equivalence Theorem I – (1) the cross-species imperative argument from the proposition P to the prescription I0 is equivalent to the pure declarative argument from P to the proposition that some fact whose existence follows from P undefeatedly supports I0 . (2) Equivalently, P entails I0 exactly if P entails that the fact that P is true undefeatedly supports I0 . Hansen, in his writings (2014), has however attempted to show that some rules introduced by Vranas seem to be problematic. Charlow (2014) developed a theory of modal noncognitivism that admits an indirect relation between imperatives and modal sentences. Instead of being

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connected to propositions, the imperatives are regarded as a property of plans. An imperative of the form !φ informs the agent what to plan. A plan describes a set of actions which are known as action descriptors. “Requirement” and “tolerance” are the concepts on which the semantics is based, and these are related to the concepts of “necessity” and “possibility” in modal logic. In case of ! :ф, it expresses a property when it is tolerant of :ф. Charlow proposes the following structure as a semantic methodology for dealing with object language. (a) Retain the core of traditional model-theoretic semantics: in a sentence ф, if the entity e meets the condition specified by the sentence, then ф receives a positive verdict relative to e. (b) Declaratives place conditions on sets of possible worlds. Such sets are abstract representations of an agent’s information. (c) Imperative sentences place conditions on sets of action-descriptors. Such sets are abstract representations of an agent’s plan. A possible world is a set of atomic sentences. Propositions are sets of sets of atomic sentences. Atoms are what are true at that world. An atomic declarative p holds at a set S of possible world if and only if for every wϵS, pϵw. We can have a recursive table as follows: S╞p iff wϵS : pϵw S╞:ϕ iff wϵS : fwg j6¼ ϕ S╞ðϕ ^ ψÞ iff S╞ϕ and S╞ψ ||ф|| is the set of possible world at which ф holds. When ф is declarative, S╞ ф iff S is the subset of ||ф||. Here, only a cognitive element is added: p holds at S iff at every world compatible with S, p holds iff S “believes” or “accepts” p. In case of imperative: (i) Imperatives tell a person what to plan. (ii) Imperative of the form !ф tells this by encoding the property a plan has if it is decided on ф. Its semantic function is to specify plans along the expected lines. The property of a plan is expressed by an imperative. That property has connection with how the imperative tells the agent to plan. In case of !ϕ, it is interpreted as “the property a plan has if it is decided on ϕ.” But this account never identifies the semantic value of an imperative !ϕ with the property of being decided on ϕ. It is said that an imperative !ϕ expresses the property a plan _ has, just if ^ requires ϕ. ½½!ϕ ¼ λϕ ^ requires ϕ:

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Here, this function is the meaning of the imperative !ф. The plans are sets of actiondescriptor. In case of an imperative !ф, the plan must prefer ф worlds to :ф worlds. When the plan gets positive verdict, i.e., 1, then it meets this condition. This is the sense in which an imperative encodes the property which a plan has if it is decided on ф. Many thinkers point out that imperatives express a property of an action guiding semantic parameters. Charlow speaks of a close connection between semantic value of an imperative and its proposal of a plan, and in this sense, his account is better than the account of Portner who speaks of a secondary role of semantical value of an imperative. It, however, remains a question whether such a model can justifiably explain the issue of validity of imperative inferences though the model of modal noncognitivism may rightly interpret an imperative sentence. Josh Parsons, in his article, “Command and Consequence” also raises the issue of content-validity as a criterion to solve the problem of imperative consequence. According to him, the content-validity has three virtues. The first virtue is called “general,” which is applicable to arguments that are either (i) purely indicative ones, or (ii) purely imperative ones, or (iii) mixed arguments of both indicatives and imperative ones. The second is the conservative virtue, which is applicable to arguments of all indicatives. The third one, that is the most important virtue, is called “adequate,” which involves a test to determine an imperative inference as valid. He argues that if someone utters all of the premises and yet refuses to utter the conclusion of the argument, then the argument is invalid, otherwise it is valid. The concept of possible world has been introduced by Parsons to explain the validity of an imperative argument. Propositions are sets of worlds. So the truthconditions of a sentence S is the set of worlds at which S is true. The complianceconditions of an imperative S is the set of worlds at which S is complied with. The main concern of Parsons is to interrelate the term “content” with the indicative and its truth-condition on the one hand, and imperatives and its compliance-condition on the other. The content of the indicative sentence “You attack at dawn” is the set of worlds at which the addressee attacks at dawn, and the content of the imperative sentence “Attack at dawn” is the very same set. So, both indicatives and imperatives have propositions as their contents. According to him, an argument is content-valid if and only if the contents of the premises jointly entail the content of the conclusion. It means that a collection of propositions p1. . .pn entail a proposition q if and only if there is no possible world that is a member of each of p1. . .pn and not a member of q. A presposcription is a representation of either the truth conditions of an assertion or the compliance conditions of a command. It is represented in terms of a matrix. A preposcription is a set of pairs of possible worlds. The first member of each pair which is to say, the worlds shown at the left of the matrix, heading the rows has to do with truth conditions, the second member of each pair which is to say, the worlds shown at the top of the matrix, heading the columns has to do with compliance conditions. The presposcription “Attack at dawn if the weather is fine” can be shown in the following matrix.

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Wfa Wf* W*a W**

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Wfa √ √ √ √

Wf*

√ √

W*a √ √ √ √

W**

√ √

Parsons has also introduced the concept of imperassertion, which is a generalization of both assertions and commands. An assertion is an attempt to influence or constrain the interlocutors’ beliefs, and a simple command is an attempt to influence or constrain the interlocuters’ intentions. Different theories discussed so far focus upon different perspectives in their study of imperatives. In the computational domain, actions are understood as a change of situations. Different attempts have also been made (Ciabattoni 2015) to formalize such action representation, where imperative logic is viewed not from the perspective of philosophy and logic. In the computational action representation schemes, one can find situation calculus, intuitionistic linear logic (ILL), and dynamic logic (DL). In several cases, the natural language instruction in agents is interpreted with the help of the “theory of intention” which is used by DL. The propositional dynamic logic (PDL) also has done some work here. The initial purpose of PDL was to explain the behavior of computer programs (V.R. Pratt 1976; M.J. Fischer and R.E. Ladner 1979). But later on, it concentrated upon the agent’s knowledge and action. PDL is concerned with both the propositional language and language of actions (van Benthem 2012). The theory of intention of Cohen and Levesque (1990) introduced a theory of rational actions with four basic modal operators along with some temporal operators for actions. Here, explicit reference to the agent carrying an action is given. It also considers the perspective of the action. The whole picture of these different theories is an evidence of a steady development of study of imperative logic in West.

Reflections on the Logic of Imperatives Logic of commands, as we have seen, have been studied in both Indian and Western traditions extensively. The connection between syntax and semantics found in Mīmāṁsā system sounds better than the said connection as conceived by Vranas. It is definitely an achievement on the part of the Indian system that ascribes all the three values both to unconditional and conditional imperatives, while the Western counterpart of the logic of imperatives ascribes three values to the conditional imperative, and only two values (S and V) to the unconditional imperative. That is the reason why a similarity is evident between the independent system of logic of imperatives conceived specially by Vranas in case of unconditional imperatives and classical two-valued logic. It may be explained in the following way. For Vranas, context is the condition of a conditional prescription. The context is the union of its satisfaction and violation sets. The avoidance therefore is the negation of its context. In case of unconditional prescription, there is no condition,

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but there is context. Vranas has denied (2008) the possibility of the third value, i.e., avoidance (page 23 of the present paper). Vranas lays down (2008) two specific reasons for lending support to three possible values of imperative statements. The first one is the elaboration of three values – satisfaction(S), violation(V), and avoidance(A) of a prescription. The second one refers to conditional imperative statements which generally mix propositions with prescriptions. In speaking of the second reason, therefore, Vranas has in his mind, the cases of conditional prescriptions whose antecedent is a proposition and consequent is a prescription. Perhaps this observation that the possibility of three values is strongly supported by the cases of conditional prescription is responsible for considering unconditional prescriptions as comfortably having two values – satisfaction and violation. This keeps imperative logic – partially though – much closer to classical two-valued logic. But imperative logic as a distinct system – as viewed by Vranas – deserves a better formulation where both the conditional as well as the unconditional prescriptions can be explained in terms of three values. There are two reasons in favor of such formulation: (a) Prescription, as distinct from proposition, essentially excludes the possibility of ascription of truth-falsity, and this is what isolates imperative logic from standard two-valued logic. This role of a prescription remains the same both in conditional and unconditional statements. (b) If conditional prescription is essentially a matter of three possible satisfactionvalues, then it is by virtue of its prescriptive part (which is normally the consequent, but not always so) and not the propositional part which allows the three possibilities. The propositional part does not give conditional imperative a special status of having three satisfaction-values. Hence, the imperative has three values. In the vocabulary of the system formulated by Vranas, therefore, the imperatives may be studied in the following way: A prescription ¼ ˂ S, V, A > or ˂ ˂ S, V>, A > context ¼ ðS U VÞ avoidance ¼ ~ ðS U VÞ Conditional Prescription If you love him, help him. You love him and help him (satisfied). You love him but do not help him (violated). You do not love him (avoided).

Vranas has shown how these three values have been used to explain a conditional prescription involving binary and unary connectives. We can represent the unconditional prescription using the identical symbolic form, instead of limiting it to an

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ordered pair of S and V, i.e., ˂ S, V >. It is possible now to explain unconditional prescription with illustrations of all connectives in terms of three values. Let us now attempt to present the case of unconditional prescription involving similar connectives:

Negation Unconditional prescription – Help her. Negation – Don’t help her. You don’t help her (satisfied). You help her (violated). You remain indifferent (avoided).

Note that this state of indifference is not the same as being unmindfully indifferent to a passerby who may need some help. I may be indifferent to her, because I am mentally engaged with other issues at that moment. But the present case of indifference is a state of conscious indifference after hearing somebody giving me the instruction “help her.” The other point to note is that there is a difference between violation and avoidance, though in both the cases the resulting action is the absence of help. In case of violation, it is a deliberate withdrawal of help, while in case of avoidance it is total noninvolvement in the situation. Of course, both the reactions are the lived experience of the agent, to whom the command is passed.

Conjunction Unconditional prescription: “Trust me and touch me.” You trust me and you touch me (satisfied). You do not trust me or you do not touch me or both (violated). You are simply present as a stranger who denies all acquaintance (avoided).

Note that in the case of avoidance, the presence of the person for whom the imperative is uttered is important. This presence is accompanied by an awareness of the conjunctive imperative without having a deliberation to violate it. So it is not the same as violation, though it appears to be so. In fact, in understanding an imperative statement, it is not enough to depend only on physical observation of the worldly affairs. Unlike descriptive or declarative proposition, it connects us with the total attitude of the agent – utterer or hearer – of the imperative statement.

Disjunction Unconditional prescription – “Write to me or talk to me.”

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You write to me or you talk to me (satisfied). You do not write to me and you do not talk to me (violated). You are simply present as a stranger who denies all acquaintance (avoided).

Here the case is the same as found in conjunction. The illustrations offered, if found cogent, show nevertheless the distinction between imperative logic and standard two-valued logic in a sharper way. This lacuna of treating conditional and unconditional imperatives differently is absent in the Mīmāṁsa-inspired system developed by Srinivasan and Parthasarathi. The third value called “Gn” stands for absence of intention to reach the goal is applicable to both conditional and unconditional imperatives, viz., (i) If one desires to get a job, then one ought to fill up a form. (ii) One should take healthy food. In both the cases, the third value indicates absence of intention, irrespective of the performance of action. In the first case, it is a question of choice of the person to desire a specific job. In the second case, one may lack the intention. This has no reference to abnormal cases, viz., he is suffering from terminal disease or he is a mental patient. Rather, the absence of intention may be found in case when he is either enough confident about the goodness of his health or he doubts about the meaning of healthy food. Another semantical issue, closely connected with imperative, is that of the connection between an imperative and the action intended to be performed. The marked distinction between a descriptive and an imperative is that, unlike the former, an imperative gets significance by being connected to an action, which is to be performed either immediately or in suitable situation. An action involves minimum three things: (i) The intention of the speaker (ii) The belief-state of the hearer (iii) The connection between (i) and (ii) There is no reflection of these aspects in the system of Vranas. Here the theory presupposes the successful performance of the action prescribed. Josh Parsons (2013) introduced the concept of possible worlds in order to explain the truth-conditions of propositions as well as the compliance-condition of a command or prescription. This system refers to the issues of intention and belief in the context of imperassertion which is a generalization of both assertion and command. Constraining the interlocutor’ beliefs is the objective of an assertion, while constraining the interlocutor’s intention is the objective of a command. But here the reference to intention and belief is made in the context of explaining the validity of imperative argument. The formalism called MIRA attempts to interpret and evaluate the actionprescribed imperatives. Here the explanation of an imperative considers the imperative, not either as a premise or as a conclusion of an argument, but as it occurs

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independently in a context. The primary importance is given on the agent’s intention of achieving the goal. In fact, the interpretation of the Mīmāṁsa system is successful in unearthing the significance of an imperative where the command is given both to the person and the action expected to be produced by the command. It is evident both in the Bhāṭṭa view of bhāvanā (Ramaswami 1951) as well as the Prābhākara view of apūrva. A Vidhivākya has no significance apart from its capacity of being executed, and successful execution has an unfailing bond with a purpose or goal. It is the tie of intention and belief which is considered as primary by the Mīmāṁsakas and is fully reflected in the MIRA interpretation of values of imperatives, viz., S, V, and Gn, each speaking of presence or absence of intention to reach the goal.

Some Further Survey on the Logic of Imperatives In both Indian and Western study of imperatives, the role of speaker-hearer or instructor-agent has been treated as inevitable. The Mīmāṁsā theory of Vedic Vidhivākya that denies any human speaker/instructor entertains the concept of intention from the point of view of the hearer. Researchers in India often undertake critical analysis of imperatives regarding the issue of validity. Comments are also made upon the cases where the instructions are found “obeyed” or “complied with,” even if there is no human agent directly involved in the execution of the action. Let us refer to such discussions in the context of study of logic of imperative in the West.

The Concept of Validity in the System of Vranas 1) Vranas has introduced the concept of satisfaction-validity which is generally framed in this way: The argument from I(premise) to I´(conclusion) is satisfaction-valid exactly if s(satisfaction of premise) entails s´(satisfaction of conclusion).

In order to reflect upon his criterion of validity1, it is necessary to refer to some definitions: Definition I

Definition II

A pure imperative argument is valid, i.e., satisfied, exactly if, necessarily, every reason that supports the conjunction of the premises of the argument also supports the conclusion of the argument. A reason supports a prescription exactly if it favors the satisfaction over the violation proposition of the prescription.

Vranas introduced a model of obedience and bindingness to account for validity. Question may be raised about acceptability of obedience as the proper criterion of

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validity in the context of imperative arguments. An example of a conditional prescription may be used to illustrate the point. Conditional prescription – If you find her, inform her. You find her and inform her (satisfied). You find her but don’t inform her (violated). You don’t find her (avoided).

According to the general model, if the premise is obeyed, then the conclusion is obeyed. This obedience-condition is not fulfilled only if the prescription is violated. So obedience is to be understood as non-violation. Applying this rule to pure imperative argument, we get the result: If you meet me, inform me and help me. ——————————————————————————————————————— So, if you meet me, inform me.

It is obedience-valid in this way: [ (meet me, inform me, help me)v(don’t meet me)! (meet me, inform me)v(don’t meet me) ]

Now, in order to consider the claim of obedience as a criterion of validity, we may start with formalization of obedience in the following way. Before that, let us remember the interpretation of obedience: A prescription ¼ ðS v A v VÞ obedience ¼ ~V ¼SvA

Unconditional Prescription It will be better to start with an unconditional prescription: “Shut the door”

Here the definition of obedience in terms of the disjunction can be evaluated with reference to two disjuncts (S v A) separately: The first disjunct: There is a serious problem regarding whether apparent satisfaction or so-called “carrying out” the command really means obedience. Let us suppose that the utterance of the command in this example is followed by the event

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of shutting of the door, yet it could be false that it is obeyed. We can offer different possible analyses of the event as follows: (i) The addressee might not have listened to the command at all, but was going to close the door anyway, and did it of her own accord. (ii) The hearer might have heard the command, and intended to ignore it; but while walking by the doorway, accidentally shut it. (iii) The hearer might have wanted to do as she was told, but when she actually shut it, she might have done it because of the dust which was coming through the door. Now, while (i) and (iii) may be borderline cases, (ii) is a clear case where mere “carrying out” of the action does not constitute obeying. (2) The second disjunct: Serious objection comes from a case of avoidance. Let us take another example: A person is given the command “Speak the Truth” and the person remains silent. Here the command is avoided, because the person did not allow any occasion of speech. It is a clear case of voluntary avoidance, which has sufficient empirical reason to be considered as a case of disobedience. Hence, the problem persists on both sides and the definition fails.

Conditional Prescription In the case of conditional prescription, avoidance is a constituent of the said definition: If you love him, pray for him ¼ L!P Obedience ¼ (L & P) v ~L

Here the first disjunct of this nonexclusive disjunction is by itself strict non-violation. It does not rule out the difficulty we have seen in the case of unconditional prescription. Again, it is debatable whether the second disjunct leads to non-violation/obedience in the same way as in the case of the first one. Now, avoidance indicates the falsity of the antecedent of a conditional prescription and thereby leads to obedience of the command. So, it never directly causes obedience. The resulting obedience is apparent or vacuously fulfilled. This apparent naïve presence of avoidance yields an alarming effect when it plays the same role in leading to disobedience: Avoidance ¼ ð ~L ! PÞ fobedienceg ¼ ð ~L ! ~PÞ fdisobedienceg

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Because of its inconspicuous role, obedience may be comfortably understood as neither violated nor avoided, which is the same as satisfied. A prescription ¼ ðS v V v AÞ Obedience ¼ ð ~V&~AÞ ¼ S ðby applying the rule of disjunctive syllogism separately to the two conjunctsÞ: Hence, the notion of obedience-validity is not unquestionably acceptable. It can be further shown that the satisfaction criterion is also not beyond suspicion. 2) Validity of imperatives is arguably defined by Vranas (2016) in terms of meriting endorsement. But it is hard to endorse2 the general definition of argument validity applied in the case of imperative arguments. In case of declarative arguments, meriting endorsement is the same as being supported by a true proposition, which is guaranteed by some fact; while in imperative argument, it is the same as being supported (sustained) by some reason (fact). Reason is an umbrella term which accommodates reasons for acting, feeling, believing, etc. Let me start with one of the consequences of the theory of validity which he names as (D2): (D2) An argument is valid, only if, necessarily, if its premises merit endorsement, then its conclusion merits endorsement. Vranas refers to three issues as consequences of (D2): (i) Reasons and support (ii) Meriting endorsement, jointly versus separately (iii) Pro-tanto versus all-things-considered endorsement The first one is acceptable, the second one is proposed to be amended, and as the second one has impact on the third issue, the third one also needs to be modified in this respect. There are two interpretations of “meriting endorsement,” viz., meriting endorsement either jointly or separately. A cross-species imperative argument is one that comprises of both proposition and prescription, and it merits endorsement jointly, exactly if some fact sustains both the proposition and the prescription. Such an argument merits endorsement separately, exactly if some fact sustains the proposition and some (same or different) fact sustains the prescription. It follows that the first one entails the other by definition. These two interpretations of “meriting endorsement” help us to get the following two interpretations of (D2): (D2J) An argument is valid only if, necessarily, if its premises merit endorsement jointly, then its conclusion merits endorsement.

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(D2S) An argument is valid only if, necessarily, if its premises merit endorsement separately, then its conclusion merits endorsement. Now as meriting endorsement jointly entails meriting endorsement separately, Vranas suggests that (D2) is the same as (D2J). That (D2J) is the sole interpretation of (D2) may suffer a setback, however. It is possible to show, with the help of two counterexamples, that (D2J) also fails as an interpretation of (D2). The logical rules allow the following arguments to be valid, but in each of the cases, the conclusion does not merit endorsement, while the premises do it jointly. These counterexamples are as follows:

Argument 1* (i) Feel good if you dance You dance Therefore: You feel good.

(ii) You feel good if you dance Dance Therefore: You feel good.

The first premises in both the forms are hypothetical propositions which speak of an event of feeling good that is associated with a dance performance. The second premises in both cases speak of a dance performance. All of them may be supported by fact. Still the addressee may not feel good, because none of the first premises bears any undefeated fact as a support. How is (D2) then interpreted? The phrase “supported by some reason” may be interpreted in two ways. It may be “pro-tanto” (prima facie) endorsement or it may be “all-things-considered” (undefeatedly supported by some fact/reason) endorsement. Accordingly, there are two ways of understanding (D2J): (D2JP): An argument is valid only if, necessarily, if its premises merit “pro-tanto” endorsement jointly, then its conclusion merits “pro-tanto” endorsement. Equivalently: an argument is valid only if, necessarily, if some fact sustains every premise of the argument, then some fact sustains the conclusion of the argument. (D2JA) : An argument is valid only if, necessarily, if its premises merit “all-thingsconsidered” endorsement jointly, then its conclusion merits “all-things-considered” endorsement. Equivalently: an argument is valid only if, necessarily, if some fact undefeatedly sustains (i.e., its conjunction with any fact sustains) every premise of the argument, then some fact undefeatedly sustains the conclusion of the argument. Now the general definition of validity given by Vranas is as follows: General definition: An argument is valid (i.e., its premises entail its conclusion; equivalently, its conclusion follows from its premises) exactly if, necessarily, every fact that sustains every premise of the argument also sustains the conclusion of the argument.

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According to Vranas, both (D2JP) and (D2JA) are consequences of the general definition. As (D2JA) speaks of undefeated reason, (D2) is to be better understood as (D2JA). Vranas has developed some definitions which are special cases of this general definition. Definition 3 – A cross-species imperative argument is valid exactly if, necessarily, every fact that guarantees the (declarative) premise of the argument also supports the (imperative) conclusion of the argument. It is an important contribution of Vranas that he speaks of validity of imperatives in terms of meriting endorsement. Meriting endorsement is totally defined in terms of being sustained by some facts. These facts, however, are different in the cases of indicative and imperative sentence. In case of the former, a fact is a state-of-affair, and more specifically a physical object or event. On the other hand, the fact relevant in the context of imperative argument is not a factual one. Now his criterion of validity (definition 3) may be tested with the help of a simple example: Dance. So, you dance if it rains.

The standard rules of validity allow the argument to be valid. But it is not clear from the premise whether there is undefeated support of reason to dance. In the given example, the satisfaction-proposition (dance- performance) is favored over the violation-proposition (absence of dance-performance), and it sustains the premise. In case of the conclusion, it is rain followed by dance-performance which sustains the proposition. In the premise, however, the relevant fact does not contain any occurrence of rain. Hence, the argument no longer remains valid according to the definition 3 given by Vranas. For the sake of validity of such arguments, definition 3 may be reframed as follows: Definition 3* A cross-species imperative argument is valid, exactly if, necessarily, every fact that guarantees the (prescriptive) premise of the argument is either the fact or constitutes an essential element of the total undefeated fact/reason in favor of the (descriptive) conclusion of the argument.

Content-Validity of Josh Parsons3 The insufficiency of content-validity in respect of standard view has been elaborately shown by Parsons. Some examples are cited where content-valid arguments are found to be intuitively invalid. There are two sets of examples of content-valid arguments along with their “evil twins.” He claims that the invalidity of the evil twins cannot be shown in classical view. Example 1 (A1) Attack at dawn if the weather is fine! (A2) The weather is fine.

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Therefore, (A3)Attack at dawn! Example 2 (B1) Flee for your life only if our plans are betrayed! (B2) You flee for your life. Therefore, (B3) Let our plans be betrayed! Example 3 (I1) Flee for your life only if our plans are betrayed! (I2) Flee for your life! Therefore, (I3)Our plans are betrayed. Example 4 (Y1) Attack at dawn if the weather is fine! (Y2) Let the weather be fine! Therefore, (Y3) You attack at dawn. According to Parsons, Examples 1 and 3 are valid arguments, while examples 2 and 4 are the evil twins of examples 1 and 3, respectively. Taking “I” for “imperative” and “A” for “assertive,” the issue may be made clear by observing the symbolic representation of the arguments. Example 1 and 2: I ð p ! qÞ AðpÞ Therefore, Iq: Example 3 and 4: I ð p ! qÞ IðpÞ Therefore, AðqÞ: Though the symbolic forms are same, the respective inner structures of the arguments are enough to show their disparities. According to Parsons, the insufficiency of the standard view is that it cannot account for the invalidity of argument B and argument Y, which are said to be the evil twins of A and I, respectively. It is quite

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understandable why the first premise in each argument is to be regarded as an imperative instead of making the imperative either as an antecedent or as a consequence. Now, according to the standard view, the consequence of a conditional statement can be derived validly from the antecedent being asserted independently as a separate premise. But it depends upon the sameness of the antecedent of the conditional statement and the antecedent which is asserted independently. Sameness of two propositions may be understood syntactically or semantically. Syntactical sameness requires combination of same words. In case of argument A, the proposition “the weather is fine” is the antecedent of the conditional statement and the same proposition is asserted separately. But in case of argument B, they differ. Flee for your life (Antecedent of the conditional statement). You flee for your life (Antecedent alone as an independent proposition).

These two sentences are syntactically different. This syntactical difference is closely connected with their semantical difference, as they are two different kinds of speech act, and more importantly, they perform different kinds of communication. It indicates that they differ in their meanings also. So they are semantically different. If they are both syntactically and semantically different, they do not serve the purpose of modus ponens argument. The case is similar with I and Y arguments. Y is claimed to be the evil twin of I, and the invalidity of Y is veiled under its symbolic representation. But the fact that it is not so may be easily seen in the following way. The weather is fine (Antecedent of the conditional proposition). Let the weather be fine (Antecedent alone as an independent proposition).

Hence, Y argument does not serve the purpose of modus ponens. Second, in fact neither B nor Y have been found to be valid in the standard view. The reason for their supposed validity (not being invalid) is not that they observe the rules but rather that the rules are not applicable in their cases. Hence, the insufficiency of the standard view cannot be shown in this way. The concept of imperassertion includes the notion of beliefs and intentions. Possible worlds are thought to be linked with a person’s belief-intention set. Belief is interpreted in terms of doxastic possibility, while intention is interpreted in terms of intentional possibility. Now, a successful imperassertion in respect of a conditional imperative involves both the imperassertion made by the speaker and the belief-intention set of the hearer. In case of the standard view, validity of an argument is not essentially connected with the truth of the premises. It goes very well with the falsity of the premises. We can only say that in a valid deductive argument, if the premises are true, then the conclusion cannot but be true. The case is, however, the same with imperative argument. If in a valid imperative argument the premises are complied with, then the conclusion cannot but be complied with. As in the case of a valid indicative inference, the truth of the premise/s is often “accepted as true,” so also in the case of a valid imperative inference, the compliance condition of the

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premise/s is to be “assumed.” It cannot be taken for granted. It is better understood when the issue of a person making a command and another person to whom the command is issued is considered. So, here also, the premise is “supposed to be” complied with. It is to be delicately handled.

Conclusion The study of imperatives in Indian context has been marked by successful development of a new system inspired by the thoughts of Mīmāṁsā as an Indian philosophical school. The structure of an inference involving Vedic sentences is wonderfully used in the mundane context involving ordinary imperatives. The system is found useful in AI planning (Srinivasan and Parthasarathi 2012) and in special education (Srinivasan and Parthasarathi 2013). The use of evaluation of imperatives is demonstrated in the domain of robotics (Srinivasan and Parthasarathi 2014). There is a beautiful illustration (Srinivasan and Parthasarathi 2017a, b) of various formalisms (Hofstadter & McKinsay, Chris Fox, Vranas, Situation Calculus & STRIPS, Intuitionistic Linear Logic, Propositional Dynamic Logic, MIRA) with an example of Grimms’ fairy tale “Little Red Riding Hood.” In the recent article, the authors claim that the application of MIRA formalism may be successfully extended to the fault tolerance system and AI chatbots. It is claimed to be successful in solving computational problems which deal with instructions. Eventually it is to be noted that recently, the Mīmāṁsā principles have also been studied as deontic principles and a new deontic logic has been introduced. Attempts have been made to use a proof-theoretic method to apply the principles to deontic arguments. It is especially interesting to find out whether the examples of conflicting obligations, found in the Vedas, can be covered successfully by the principles of this new deontic logic. Equally important is the study of imperatives in the West. The gradual development of the logic of imperatives has shown a threadbare analysis of an imperative sentence and its relevance in conversational context. The explanation behind the idea of assertions and commands being special cases of imperassertion perhaps lies in the area of Philosophy of Mind. Assertions are concerned with beliefs, while commands are connected with intentions, as the latter is closely connected with the Philosophy of Action. It is thus a beautiful blending of Philosophy of Mind with the Philosophy of Action which is found in the development of logic of commands. The concept of imperassertion as found in Parsons’ writing is praiseworthy in tying the knot among Logic, Philosophy of Mind and Philosophy of Action. Perhaps, in near future, it may help in unearthing different layers of the logic of imperative.

Notes 1. Sanyal, Manidipa (2009) “The Fabric of Commands,” talk delivered at Chennai in The Congress (International) on Logic and its Applications, The Institute of Mathematical Sciences, Chennai Mathematical Institute.

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2. Sanyal, Manidipa (2019) “Dance if it rains,” Philosophy Study, October issue, pp 571–577 Wilmington, USA. 3. Sanyal, Manidipa; Basu, Debirupa (2019) “Attack at Dawn if the Weather is Fine” – talk delivered at Prague in the International Seminar on Logic, Methodology, Philosophy of Science and Technology (CLMPST), Czech Institute of Technology, Czech Republic. I am beholden in a large measure to my teacher, Professor Prabal Kumar Sen for his valuable guidance in understanding and writing the philosophical thought of Mīmāṁsā school.

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Ciabattoni, A., E. Freschi, F.A. Genco, and B. Lellmann. 2015. Mīmāṁsā deontic Logic: prooftheory and applications. In Automated Reasoning with analytic tableaux and related methods, 323–338. Berlin: Springer. Cohen, P.R., and H.J. Levesque. 1990. Intention is choice with commitment. Artificial Intelligence 42: 213–261. Fischer, M.J., and R.E. Ladner. 1979. Propositional dynamic logic of regular programs. Journal of Computer Systematic Science 18 (2): 194–211. Follesdal, Dagfinn and Risto Hilpinen. 1981. Deontic logic: An introduction. Deontic logic : Introductory and systematic readings, Ed. By Risto Hilpinen, Synthese Library 33, Dordrecht, Boston, London: D. Reidel, p. 1–35. Fox, Chris. 2008. Imperatives: A logic of satisfaction. University of Essex, England. Fox, Chris. 2012. Imperatives: A judgemental analysis. Studia Logica 100 (4): 879–905. Freschi, Elisa. 2012. Duty, language and exegesis in Prābhākara Mīmāṁsā, Jerusalem studies in religion and culture. Leiden, Boston: Brill. Geach, P.T. 1958. Imperative and deontic logic. Analysis 18 (3): 49–56. Grelling, Kurt. 1939. Zur Logik der Sollsatze. Unity of Science Forum, p. 44–47 (also in Moutafakis, Nicholas J.(1975) Imperatives and their Logics, New Delhi, Sterling, p. 36. Hamlin, H.L. 1987. Imperatives, 18. Great Britain: Basil Blackwell. Hansen, J. 2008. Is there a logic of imperatives?. Deontic logic in computer science, Twentieth Europeon summer school in logic, language and information, Germany. Hansen, J. 2014. Be nice ! how simple imperatives simplify imperative logic. Journal of Philosophical Logic 43 (5): 965–977. Hare, R.M. 1952. The language of morals. Great Britain: Oxford University Press. Hare, R.M. 1971. Practical Inferences. London: Macmillan. Harrison, Jonathan. 1991. Deontic and imperative logic. In Logic and ethics, ed. P.T. Geach. Dordrecht: Kluwer Academic Publishers. Hintikka, Jaakko. 1981. Some main problems of deontic logic. Deontic logic: Introductory and systematic readings, Ed. By Risto Hilpinen, Synthese library 33, Dordrecht, Boston, London: D. Reidel, p. 59–104. Hofstadter, A., and J. McKinsey. 1939. On the logic of imperatives. Philosophy in Science 6 (4): 446–457. Jayanta Bhat‚‚ta. 2008. Nyāyamañjarī, 5th Ahnika, Sen, Prabal Kumar (Trans.), Kolkata: Sanskrit Book Depot. Jayanta Bhat‚‚ta. 1895. The Nyāyamañjarī of Jayanta Bhat‚‚ta ed. by Gangādhara Śāstrī Tailaǹga. Vizianagaram Sanskrit Series 10. Benares: E.J. Lazarus and Co. Jhā, Gaṅgānātha ed. 1933–1934–1936. Śabara-bhāṣya, trans. Into English, Index by Umesha Misra, Gaekwad’s Oriental Series 66, Baroda: Oriental Institute. Jorgen, Jorgensen. 1938. Imperatives and logic. Erkenntnis 7: 290. Kenny, A.J. 1966. Practical inference. Analysis 26 (3): 65–75. Mally, Ernst. 1926. Grundgesetze des Sollens, Elemente der Logik des Sollens, 229–324. Graz: Leuschner-Lubensky. Mates, Benson. 1953. Stoic logic, 19. Berkeley: University of California Press. Motilal, Bimal Krishna. 1986. Perception: An essay on classical Indian theories of knowledge. Oxford: Clarendon Press. Motilal, Bimal Krishna, and P.K. Sen. 1988. The context principle and Some Indian controversies over meaning. Mind 97 (385): 73–97. Menger, Karl. 1939). A logic of the doubtful on optative and imperative logic Reports of a Mathematical Colloquium 2 Notre Dame University, Indiana University Press, Notre Dame pp 53–gercll. Miśra, Śālikanātha. 1904. Prakaraṇapañcikā, Ed. By Śāstrī, Mukuṇḍa, Chowkhamba Sanskrit Series, Benares.

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Srinivasan, Bama, and Ranjani Parthasarathi. 2017a. A survey of imperatives and action representation formalisms. Artificial Intelligence Review 48: 263–297. Srinivasan, Bama, and Ranjani Parthasarathi. 2017b. Action representation for robots using MIRA. 2nd international conference on knowledge engineering and applications. Stevenson, C.L. 1960. The emotive meaning of ethical terms. In Logical positivism, ed. A.J. Ayer. Glencoe: The Free Press. Udayana. 1957. Nyāyakusumāñjali. Ed. By Padmaprasāda Upādhāya and Dhuṇḍirāja Śāstrī. Chowkhamba Sanskrit Series 30, Varanasi. van Benthem, J., H. van Ditmarsch, J. van Eijck, and J. Jaspars. 2012. Logic in action. http://www. logicinaction.org/docs/lia.pdf von Wright, George H. 1971. A new system of deontic logic. In Deontic logic: Introductory and systematic readings, Synthese Library 33, ed. Risto Hilpinen, 105–120. Dordrecht, Boston, London: D. Reidel. Vranas, Peter B.M. 2008. New foundations for imperative logic I: Logical connectives, consistency and quantifiers. Noûs 42: 529–572. Vranas, Peter B.M. 2010. In defense of imperative inference. Journal of Philosophical Logic 39: 59–71. Vranas, Peter B.M. 2011. New foundations for imperative logic: Pure imperative inference. Mind 120: 369–446. Vranas, Peter B.M. 2016. New foundations for imperative logic III: A general definition of argument validity. Synthese 193 (6): 1703–1753. Springer. Wedeking, Gary A. 1970. Are there command arguments? Analysis 30: 161–166. Williams, Bernard A.O. 1963. Imperative inference. Analysis 23: 30–36.

Part VI Modern

Later Nyāya Logic: Computational Aspects

34

Amba Kulkarni

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Syntax of Navya Nyāya Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Graphical Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conceptual Graphs for NN Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Computational Parsing of an NN Expression . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Segmenter for NN Expressions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constituency Parser for NNE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semi-Automatic Parsing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Translating NN Expressions into Conceptual Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

932 933 936 936 941 944 946 946 949 951 952

Abstract

In this chapter we describe the computational aspects of the Technical Language of Navya-Nyāya. Navya-Nyaya is an off-shoot of the early Nyāya philosophy. It deviates from the Nyāya philosophy in three major ways. First, instead of prameyas the discussions are centered around the pramāṇas. Second, the Navya-Nyāya has adapted the Vaiseṣika ontology. And finally it has introduced a few concepts expressed through an unambiguous technical terminology that brings in a clarity in the communication removing the inherent ambiguities of a natural language. In this chapter the syntax of expressions involving this technical terminology is described, followed by a scheme based on Conceptual Graphs of Sowa for their graphical rendering. Finally a computational algorithm is described that renders the graphs corresponding to the Navya-Nyāya expressions semi-automatically.

A. Kulkarni (*) Department of Sanskrit Studies, University of Hyderabad, Hyderabad, Telangana, India © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_12

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Introduction The words in any language are countably infinite. The concepts they correspond to, however, form a continuum. Naturally the words – both content words and function words – are prone to be over-loaded. This leads to ambiguity in a Natural language. In spite of having ambiguity, human beings do not find much difficulty in language communication. This is mainly because of the shared background knowledge. However, the scientific work and philosophical discussion demand precision. Indian logicians who were engaged in philosophical debates realized the need for expressing communications in an unambiguous way. Their efforts culminated in a new school Navya-Nyāya “Neo-Logic” which emerged as an off-shoot of the Nyāya school of philosophy. Seeds of Navya-Nyāya (NN) School are found in Udayana’s work (eleventh century). Gaṅgeśa (twelfth century) in Tattvacintāmaṇi introduced a few technical terms and provided a well-defined scheme to express a cognitive structure using these technical terms. Later on Raghunātha (sixteenth century), Jagadīśa (late sixteenth century) and Gadādhara (seventeenth century) made significant contributions to this field enriching it further. The philosophers and the logicians in the West also needed precision. This need gave rise to the development of formal languages. These formal languages have a very limited vocabulary with precise meaning. Propositional calculus, for example, uses only three words viz. “and,” “or,” and “not.” Later the vocabulary was enhanced by introducing universal and existential quantifiers, the modal logic extended these further to include modal operators and Montague introduced generalized quantifiers. But in spite of all these developments, we hardly see any use of these formal languages in the field of Humanities and Social sciences. On the other hand the technical language of Navya Nyāya is built on top of the classical Sanskrit. In this language, technical words of Navya-Nyāya are intermixed with the original words in such a way that the resulting expression is unambiguous. Thus, a Navya-Nyāya expression has the complete power of expressibility of a Natural language at its disposal. And probably, it is because of this reason that we find use of technical language of NN in various fields of Humanities such as Vyākaraṇa “grammar,” Sāhitya “literature,” Mīmāṃsā “exegesis,” etc. The main purpose behind the development of such a language was to describe the cognitions either perceived through sense organs or through the verbal utterances, and to deduce inferences from the valid cognitions. Bhattacharya (1990, 130) rightly observes. “Thus this language could be used in every sphere where cognition, belief, doubt, and other epistemic and doxastic factors play an essential role. This explains why this language could be used universally in the humanities, where the epistemic factors predominate.” In addition to describing the cognitive structures, because of the precision it provides, this language was also used to provide the definitions (lakṣaṇa) of scientific terms in various disciplines. Heavy usage of these technical terms to define various grammatical concepts and to bring in precision in the communication by the sixteenth-century grammarians such as Bhaṭṭoji Dikṣita and later by Kaunḍabhaṭṭa

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and Nāgeśa distinguishes modern grammatical texts (Navya-vyākaraṇa) from the old ones. Good understanding of this technical language of NN is thus necessary to understand the texts of Indian origin in various disciplines. Another important reason for understanding the technical language of NN arises because of its relevance in the age of information technology. With the advent of computers and the age of information revolution, in the West there was an upsurge in the field of Knowledge Representation (KR). Several schemes for knowledge representation were designed and proposed. Woods (1975) defines the properties of a Knowledge Representation language as A KR language must unambiguously represent any interpretation of a sentence (logical adequacy), have a method for translating from natural language to that representation, and must be usable for reasoning. The technical language of NN is designed precisely for representing the cognition – both perceptual and the verbal one. The well-defined description of the cognition gives a scheme to translate from natural language to the technical language of NN, and it is also used very extensively, in the Nyāya school of Indian logic, for reasoning. Briggs (1985) points out, with examples from the śābdabodha (verbal cognition), how the NN expressions are close to a typical KR scheme such as the Semantic Net. Two major problems we envisage in understanding the technical language of NN. The first problem is due to the mode of presentation. Traditional learning in India was through oral communication. Sanskrit is very rich in compound formation. This feature of Sanskrit has been utilized to its full extent by the Indian logicians in describing cognitive structures using the technical language of NN. Such expressions are typically exceptionally long, many-a-times one compound running into pages. While the oral transmission of knowledge and all serious debates could sustain these long compounds, modern scholars not trained in oral tradition found it difficult to understand these long expressions. As a result, since as early as twentieth century we find use of diagrams to represent NN expressions. The second problem is related to the syntax and semantics of the technical language of Navya Nyāya. While Ingalls (1951), Matilal (1977), Shaw (1980), and Mohanty (2000), to name a few, worked towards understanding the concepts and comparing them with the western logic counterparts, Bhattacharya (1990) and Ganeri (2008) explored the underlying syntactic structure and the grammar as well. In what follows, we first describe the syntax of the NN expressions and then give an overview of pictorial representation of NN. Finally we describe how this knowledge of syntax of NN expressions and the computational tools for analyzing Sanskrit texts are put together to mechanically render the NN expressions graphically.

Syntax of Navya Nyāya Expressions An NN expression involves a small number of technical terms together with a nonlogical vocabulary (Matilal 1968). Ganeri (2008) in the informal description of the NN classifies these technical terms into six categories.

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1. Primitive Terms Nouns such as ghaṭa “pot,” bhūtala “ground,” gandha “smell,” etc., are the primitive terms. 2. Abstract Functor A derivational suffix “tva” or “tā” (-ness or -hood) that maps a noun to an abstract noun is termed as an abstract functor. 3. Relational Abstract Expressions Relational abstract expressions are derived from relation-denoting terms by adding a “tva” or “tā” (-ness or -hood) suffix. For example, pitṛ “father” is a relation-denoting term. By adding “tva” suffix, it changes to pitṛtva “fatherhood,” a relational abstract expression. 4. Conditioning Operator The conditioning operator nirūpita “determined by” operates on a relational abstract expression to form a term. For example, X-nirūpita-pitṛtva “fatherhood determined by X.” 5. Sentence-Forming Operator Terms such as niṣṭha “resident in” and avacchinna “delimited by” combine a relational term with another term to form a sentence. 6. Negation Functor The term abhāvaḥ “negation/absence” is termed as a negation factor. Bhattacharya (1990) describes the syntactic structure of a cognition expressed by an NN expression. The canonical form for expressing the cognition in NN is either b is a
possessing, or b has a: Thus, the canonical form for “A cat is on a mat” is “A mat has a cat (on it)” or “A mat is cat-possessing.” The stock example in NN is the expression describing the reality “a pot is on the ground,” expressed in NN canonical form as either ghatavat bhutalam, or bhutale ghatah : _

_

_

A cognition called “qualificative cognition” savikalpaka jn˜ āna, according to a Naiyāyika, is of the form aRb, where a is the “qualifier” prakāra, b is the “qualificandum” vis´eṣya, and R is the “qualification” saṁsarga of the cognition. In case of the cognition arising from b is a-possessing, b is the qualificandum and a is the qualifier. But in case of the cognition of the form b has a, a is the qualificandum and b is the qualifier. Accordingly the description of the cognition differs in both these cases. This structure aRb is a complex object, according to a Naiyāyika, where ontologically a is considered to be a property viz. super-stratum (ādheya) of b in the relation R and b is the property-possessor (substratum or locus) ādhāra of a in the

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relation R. The two relata a and b of the relation R are called the pratiyogin and anuyogin, respectively. This cognition has the following structure: R
nistha
sam_ sargata
nirupita
__

a
nistha
R1
ness
nirupita
__

(1)

R2
ness
vat
b: or more elaborately as: R
avacchinna
a
ness
avacchinna
a
nistha
R1
ness
nirupita
__

(2)

b
ness
avacchinna
R2
ness
vat
b: And the cognition is described as _ R
nistha
sa msargata
nirupita
__

a
nistha
R1
ness
nirupita __


b
nistha
R2
ness
s ali
j~ n anam: __

(3)

Similarly, in Eq. 2 R2-ness-vat-b is replaced by R2-ness-s´āli-jn˜ ānam when one refers to the cognition. This structure uses three relational terms viz. avacchinna, niṣṭha, and nirūpita, the -ness suffix (which in Sanskrit is represented by -tva), and three relational abstracts viz. R1-ness and R2-ness, and saṁsargatā. For example, the knowledge nī lo ghaṭaḥ (the pot is blue) has nī la (blue) as the qualifier (prakāra), ghaṭaḥ (pot) as the qualificandum (viśeṣya), and the relation of contact samavāya (inherence) between them. The structure of this cognition has the form: samavaya
sambandha
avacchinna
nıla
tva
avacchinna
nılarupa
nistha
prakarata
nirupita
__

ghat a
tva
avacchinna
ghat a
nistha
vis esyata
s ali
j~ n anam _

_

(4)

__

The same structure is used to describe the physical reality as well. For example, the verbal cognition arising from the phrase ghaṭavad bhūtalam “the ground with a pot” is described unambiguously as sam_ yoga
sambandha
avacchinna
ghat a
tva
avacchinna
ghat a
nistha
adheyata
nirupita
_

_

__

bhutala
tva
avacchinna
adhikaran ata
vat
bhutalam: _

(5)

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Graphical Representation Ingalls (1951), Staal (1988), Bhattacharya (1990), and many others have tried to use the Western symbolic notation to represent the NN expressions. But even to translate the NN expressions symbolically, one needs to understand these expressions fully. If the understanding goes wrong, the symbolic translation will be erroneous. Further, when one translates one formal language into another, one has to ensure that the two languages are equivalent in their expressive capability and that there is commensurability between the two. Use of diagrams solved this problem to a large extent. While representing the long linear compounds as a two-dimensional figure, the compounds were broken into its components, and the connection between various components was shown. This made it easy for a student to understand the NN Expressions. Of course, while showing the connection between components, the user had to take the context into account to rule out the various possibilities. Diagrammatic representation also does not pose any commensurability problem, since what is being represented is just a “parse” of the linear string, and there is no attempt to “translate” the original expression into another formal language. We notice the use of diagrams to represent NN expressions since the beginning of twentieth century. Patil (2014) in Vidyādharī mentions the use of diagrams by Vāmācaraṇabhaṭṭācārya in the early twentieth century. Though the need for graphical representation was felt, it was not formalized till recently. We find an extensive use of diagrams for understanding Sanskrit texts, especially the NN texts, in Japan. Wada (2007) reports that the first Japanese scholar to use diagrams for interpreting Sanskrit texts was Kitagawa and later Tachikawa, Miyasaka, and Wada (2007) enhanced these diagrams further. V. N. Jha (1987) further systematized them. Concepts, in these diagrams, are represented by rectangular boxes, and relations connecting these concepts are represented by edges. Technical terms nirūpita, avacchedaka, niṣṭha, are represented by different types of arrows. There are only minor differences among these various diagrammatic representations. The differences are more at the stylistic level such as style of the arrow heads (solid, hollow, curved, etc.) and the style of the edges (single, double, solid, dotted, etc.). Figure 1 represents the cognition ghaṭavat bhūtalam “the ground with a pot,” expressed in brief without mentioning the avacchedakas (From (Jha 1987, 6). Here the plain edges denote the relation niṣṭha and the edges with an arrow represent the relation nirūpita.). _ _ samyoga
nistha
samsargata
nirupita
__

ghat a
nistha
prakarata
nirupita
_

__

(6)

n anam bhutala
nistha
vis esyata
s ali j~ __

Conceptual Graphs for NN Expressions Conceptual graph (CG) of Sowa (1985) is a KR scheme originally designed as a semantic representation for natural language. It provides a graphical representation

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Fig. 1 Description of cognition ghaṭavad bhūtalam

viSayita

prakarata

ghata

samsargata (samyoga)

viseSyata

bhutala

Fig. 2 Conceptual graph of “A cat is on a mat”

Cat

on

Mat

that is human readable and at the same time formal for computational purpose. It can represent both the epistemic structure and the ontological structure. Further the representation scheme of conceptual graph is so general that various graphical representation methods such as a parse trees, Petri net turn out to be special cases of the conceptual graph (Sowa 1992). In a CG, the concepts are related through the conceptual relations. Concepts are represented using boxes and relations using ovals. For instance, “A cat is on a mat” is represented in CG as in Fig. 2. Here “cat” and “mat” are the concepts and are represented using boxes and the relation “on” is represented using an oval. Kulkarni (1994) proposed a scheme for representation of NN using Conceptual Graphs, in an effort towards establishing a bridge between the knowledge representation scheme of NN and the Western paradigms. Over a period of time, the representation scheme was modified further and, with the availability of

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Fig. 3 NN expression as a CG

R1_ness(7)

nistha(6)

nirupita(8)

nirupita(4)

samsargata(3) sa msargata(3)

a(5)

nistha(2)

R2_ness(9)

R(1)

vat(10)

b(11)

computational tools to parse a Sanskrit text, now efforts are on at University of Hyderabad to develop a computational tool to render the NNEs through Conceptual Graphs. Graphical representations of the normal and elaborate descriptions of the cognition aRb described in Eqs. 1 and 2, according to this scheme, are shown in Figs. 3 and 4, respectively. The numbers in parenthesis indicate the position of the component in the NNE. Thus, if we navigate the graph sequentially, following the numbers, we get the corresponding NNE. The description of the perceptual cognition ghaṭavad bhūtalam “the ground with a pot,” given by Eq. 6, is shown in Fig. 5, and the corresponding verbal cognition is described by Eq. 5 is shown in Fig. 6. Note that if we condense the graph by replacing relation nodes by edges with different styles, Fig. 5 reduces to Fig. 7 which is very close to Fig. 1. However, the basic difference is, in Fig. 7, the relation saṁyoga is shown as a concept node, while in Fig. 1 it is shown by an edge. Also, Fig. 1 encloses the complete cognition in a box with viṣayitā residing in this knowledge. No such box is shown in the CG representation.

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avacchinna(4)

a_ness(3)

R1_ness(7)

nistha(6)

nirupita(8)

a(5)

avacchinna(2)

R(1)

avacchinna(10)

b_ness(9)

R2_ness(11)

vat(12)

b(13)

Fig. 4 Elaborate NN expression as a CG

Now we give an example from śābdabodha to illustrate the structure of a verbal cognition arising out of the following sentence. Sanskrit: Rāmaḥ hastena brāhmaṇāya dhanam dadāti. Gloss: Rama{nom.} hand{instr.} Brahmin{dat.} money{acc.} give{pres., active, 3sg}. English: Rama gives money to a Brahmin with (his) hands. The verbal cognition of this sentence according to the grammarian’s school is Sanskrit: rāma-niṣṭha-kartṛtva-nirūpaka-hasta-niṣṭha-karaṇatva-nirūpakabrāhmaṇa-niṣṭha-sampra-dānatva-nirūpaka-dhana-niṣṭha-karmatva-nirūpakadānakriyā. English: An activity of giving characterized by the agent-hood in Rama, the instrument-ness in the hand, the recipient-ness in a Brahmin, and the object-hood in money.

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Fig. 5 A CG expressing the cognition

prakarata(7)

nistha(6)

samsargata(3)

nirupita(8)

ghata(5)

nistha(2)

nirupita(4)

visesyata(11)

samyoga(1)

nistha(10)

bhutala(9)

Figure 8 shows the rendering of this expression as a conceptual graph. Finally let us look at the definition of pṛthivī “Earth” expressed as a NN expression. “Earth,” according to the Indian school of ontology, is an object which has a characteristic property of having smell which differentiates it from other objects. This is precisely expressed by the NN expression gandhatva
avacchinna
gandha
nistha
adheyata
nirupita __


adhikaran ata
vatı pr thivi: _

_

(7)

The conceptual graph corresponding to this structure is shown in Fig. 9. Dotted lines show the onto-logical reality viz. that smell-ness is the inherent property of the smell and that the earth has smell as its characteristic property. Solid lines show the connection between the concepts through the conceptual relations expressed in the NNE. Note the taddhita suffix vat (possessing) is represented as a relation relating the adhikaraṇatā (substratum-ness) with the bhūtala (substratum).

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avacchinna(4)

ghatatva(3)

adharata(7)

nistha(6)

ghata(5)

nirupita(8)

avacchinna(2)

samyoga(1)

avacchinna(10)

bhutalatva(9)

adhikaranata(11)

vat(12)

bhutala(13)

Fig. 6 A CG expressing the reality

Computational Parsing of an NN Expression An NN expression is a compound. A compound, in Sanskrit, is written as a single word without any gap or hyphen in between the components. These components are joined together following euphonic changes. Compound formation also results in the loss of case markers. Euphonic changes as well as loss of case markers sometimes result in ambiguous compounds. Classical example of an ambiguous Sanskrit compound is rāmes´vara. Since the written Sanskrit forms do not mark any accent, this compound

942 Fig. 7 Condensed Conceptual Graph

A. Kulkarni

prakarata(4)

ghata(3)

samsargata(2)

visesyata(6)

samyoga(1)

bhutala

Fig. 8 conceptual graph representing the Śābdabodha

can be analyzed in three different ways: as a coordinating compound “karmadhāraya” (rāma is the īśvara), as an endocentric compound “tatpuruṣa” (the īṣvara of rāma), and as an exocentric compound “bahuvrīhi” (the one whose īṣvara is rāma). Kumar et al. (2010) describe the steps involved in processing Sanskrit compounds and also discuss the associated computational complexity. The steps are

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943

avacchinna(2)

gandhatva(1)

adheyata(5)

nistha(4)

nirupita(5)

gandha(3)

adhikaranata(7)

vat(8)

pr thivi(9)

Fig. 9 conceptual graph defining pṛthivi

1. 2. 3. 4.

Splitting a compound into components Analyzing its constituent structure Identifying relations between its components Providing a paraphrase for it

We illustrate these steps with an example. Consider an NNE which describes the cognition ghaṭavat bhūtalam: _ samyogasambandhavacchinnaghat atvavacchinnaghatanisth _

_

__

adheyatanirupitadhikaranatavatbhutalam: _

1. In order to understand this compound it is first split into its components as saṁyoga-sambandha-avacchinna-ghaṭatva-avacchinna-ghaṭa-niṣṭhaādheyatā-nirūpita-adhikaraṇatāvat-bhūtalam where the components are separated by hyphen.

(8)

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A. Kulkarni

2. Next a constituency parse of this compound is obtained. ((((((saṁyogasambandha-avacchinna)-((ghaṭatva-avacchinna)-((ghaṭa-niṣṭha)ādheyatā)))-nirūpita)-adhikaraṇatā) ˆ vat)-bhūtalam) Note vat, a taddhita suffix is separated from its stem, since it is adhikaraṇatā (substratum-ness) which is nirūpita (delimited) by the ādheyatā (superstratumness), and not adhikaraṇatāvat (super stratum). 3. A graphical rendering of this expression is generated. In addition, for the benefit of Sanskrit scholars, • The relations between its components are identified. ((((((saṁyogasambandha-avacchinna)T3-((ghaṭatva-avacchinna)T3-((ghaṭaniṣṭha)T7- ādheyatā)K)K)K-nirūpita)T3-adhikaraṇ atā)K ˆ vat)-bhūtalam)K where K, T3, T6, and T7 stand for karmadhāraya, tatpuruṣa with instrumental case suffix, tatpuruṣa with genitive case suffix and tatpuruṣ a with locative case suffix, respectively. These are all endo-centric compounds, with a requirement of nominative, instrumental, genitive, and locative case suffixes for paraphrasing. • And the paraphrase of this compound is provided. Sanskrit: ghaṭatvena avacchinnā, ghaṭe niṣṭhā yā ādheyatā, tannirūpitā yā adhikaraṇatā, tad-vat bhūtalam Gloss: By pot-ness delimited in pot residing which superstratum-ness determined by that which substratum-ness that possessing ground English: The ground which has substratum-ness which is determined by the superstratum-ness that is residing in the pot and is delimited by the pot-ness. In the following sections, we describe computational modules for (a) segmentation of a given compound into its components undoing the sandhi at the junctures, (b) a human interface for parsing such split compounds semiautomatically, and (c) a graphical renderer for such a parsed NN expression.

Segmenter for NN Expressions Sanskrit is influenced by an oral tradition, and hence, in an utterance the word boundaries undergo euphonic changes “smoothing” the process of articulation of sounds that otherwise would have required more effort on the part of vocal organs. When a word w1 is followed by a word w2 the terminal phonemes in w1 and the initial phonemes in w2 undergo a “smoothing” process, which is termed as an operation of sandhi. The sandhi rules of Pāṇini are of the form ABC ! ADC where A and C provide the context under which phonemes B change to D. Thus, we get a continuous string of phonemes. In the process, information of word boundaries is lost. And this brings in nondeterminism during segmentation. For example, the

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945

sequence of phonemes sālakānanas´obhinī (Varalakshmi 2013) can be split in two possible ways as sala-kanana-s obhī nī , and sala-alaka-anana-s obhīnī In order to segment a compound into its components, we need a sandhi splitter that splits a string into morphologically valid segments restoring the phonemes that underwent transformations during the sandhi process. Building a morphological analyzer for analyzing compound words is not trivial owing to the rich productive morphology (Kulkarni and Shukl 2009). Further each of the components of a compound can itself be a compound. Because of this recursive nature, it is simply impossible to build a dictionary of all possible compound stems. In the absence of such a lexicon, the only way to analyze a compound is to split it into possible components and analyze each of the components separately. But analysis of such components is further difficult owing to special operations they undergo during compound formation. Here are some such operations. • Deletion of the final “n” of a bare stem: For example, during the compound formation, the final “n” of rājan gets deleted as in rājapuruṣa. So in order to split it as rāja-puruṣa, morphological analyzer should recognize rāja as a compounding form of the stem rājan. • Shortening or lengthening of a vowel: In some cases, the final vowel of a bare stem is either shortened or lengthened. For example, iṣṭakā is changed to iṣṭaka in iṣṭakacitam. • Substitutes: In some cases, the bare stems are substituted by their allomorphs. For example, hṛdaya is substituted by hṛd in hṛllekhaḥ. • Bound morphemes: There are certain bound morphemes such as kāra, ja, etc., that occur only as a final component of a compound as in kumbha-kāraḥ, paṅkajam, etc. The morphological analyzer should be designed to analyze such words only when they occur as a bound morpheme as a final component of a compound. • Change in gender: In case of certain compounds such as exo-centric or avyayībhāva, the final component assumes an altogether different gender than its original one. The morphological analyzer needs to be equipped to analyze such words. • Change in number: The word anekān when split will have an-ekān as its components. Now the morphological analyzer should analyze ekān as a plural of -eka, where -eka is the component of a compound. • Change in paradigm: Words such as kin˜ cana when a part of a compound akin˜ cana do not inflect as kin˜ cana but need a separate paradigm. Thus, in order to analyze the components of a compound, we need to equip our morphological analyzers to handle various phenomena described above. There are significant efforts in this area in the past by Huet (2006), Mittal (2010), Kumar et al. (2010), Natarajan and Charniak (2011), and Huet and Goyal (2013). All these efforts

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were centered around general Sanskrit texts only. These segmenters needed some adaption to handle NN texts. One such adaptation is not to split the technical terms of Navya-Nyāya. While it is desirable to split the components and preverbs (upasargas) in compounds, in case of technical terms such as avacchinna, nirūpita, samānādhikaraṇa, vyadhikaraṇa, etc., it is not desirable to split these as avacchinna, ni-rūpita, samāna-adhi-karaṇa, and vi-adhi-karaṇa. In such cases, we would rather like to hide the derivation and see such words as a single unit. Arjuna and Huet and Arjuna and Kulkarni (2014) report the development of a segmenter for NN expressions with a special morphological analyzer for the Navya-Nyāya technical terms. Figure 10 shows the interactive user interface which facilitates the user to select the correct split among all possible splits.

Constituency Parser for NNE The segmented expression needs further analysis to get the underlying constituency structure. For example, a compound with three components a-b-c may be analyzed in two different ways viz. (a-(b-c)) and ((a-b)-c). And of these two possibilities, typically for a given compound only one will be meaningful. For example, ekapriya-dars´anaḥ can be analyzed only as (eka-(priya-dars´anaḥ)) “one who is dear to all,” whereas tapassvādhyāyaniratam can be analyzed only as ((tapas-svādhyāya)niratam) “one who is constantly engaged in penance and self-study.” The constituency parsing is similar to the problem of completely parenthesizing n + 1 factors in all possible ways. Thus, the total possible ways of parsing a compound with n + 1 constituents is equal to a Catalan number, Cn (Huet 2009), where Cn ¼

ð2nÞ! for n  0: ðn þ 1Þ!n!

Thus, as n increases, the total number of possible groupings increases exponentially. Correctness of a parse for a given compound is governed by the semantics of the components involved. It is the meaning compatibility (sāmarthya) of the components that decides the correct analysis. Kulkarni and Kumar (2011) proposed a statistical constituency parser that uses statistical properties of a tagged corpus to model the sāmarthya. Due to unavailability of a tagged corpus for NN, it was not possible to follow this approach for parsing NN expressions. However, an intelligent user-interface that takes advantage of NN syntax was developed (Arjuna and Kulkarni 2016) to help the user to select the correct parse.

Semi-Automatic Parsing Following observations related to the syntax of NNEs were crucial in designing a constituency parser for NNEs.

Fig. 10 User interface showing possible splits of a NNE

34 Later Nyāya Logic: Computational Aspects 947

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1. Concepts and Relations alternate in an NNE. For example, consider gandhatvaavacchinna-gandha-niṣṭha-ādheyatā. Here the components gandhatva, gandha, and ādheyatā denote the concepts and the components avacchinna and niṣṭha denote the relations. 2. The pratiyogin of a relation always immediately precedes the relation term. Thus, for example, in the above example, the pratiyogin for avacchinna is gandhatva, and the pratiyogin for niṣṭha is gandha. If “R” is a relation which connects two concepts “a” and “b” resulting in a compound “a-R-b,” then such a compound thus always will be parsed as ((a
R)
b), and never as (a
(R
b)). Thus, this constraint rules out almost half of the possible parses. So an NNE with 3 terms “aR-b” is unambiguous. Now consider an NNE with 5 terms viz. “a-R-b-S-c” where “a,” “b,” and “c” denote the concept terms and “R” and “S” denote the relation terms. This compound is ambiguous. The ambiguity is with respect to the anuyogin of “R,” with two possible parses being, ((a
R)
((b
S)
c)) and ((((a
R)
b)
S)
c). In the first case, the anuyogin of “R” is “c,” while in the second, it is “b.” It is the context that tells us which parse is correct. For example, in samavāyasambandha-avacchinna-gandha-niṣṭha-ādheyatā, the anuyogin of avacchinna is ādheyatā, while in gandhatva-avacchinna-gandhaniṣṭha-ādheyatā, the anuyogin of avacchinna, in one reading, can be gandha. So, if there are “n” concept nodes after a relation node “R,” the anuyogin of “R” potentially can be any of these “n” concepts. It is the context that decides which is the correct anuyogin. 3. A cue that rules out some more possibilities is the use of co-relative terms in Navya-Nyaāya. Anuyogitā and pratiyogitā are the co-relative terms, similarly, ādheyatā and adhikaraṇatā are the co-relative terms. And the relation-terms nirūpita and nirūpaka always combine two co-relatives. 4. Then there is of course, a well-nested-ness constraint. The resulting constituency structure should be well bracketed, without any crossings. In other words, if the anuyogin of a relation at kth position is at “j,” then the anuyogin of any relation lying between “k” and “j” cannot be beyond “j.” Thus, the three conditions, viz. (a) the pratiyogin is always to the immediate left of a relation node, (b) nirūpita and nirūpaka always connect two co-relative terms, and (c) the well-nested-ness condition, reduce the search space to a considerable degree. Since it is not clear what other factors are responsible for the correct choice of the anuyogin, a human being is involved who is well versed in Navya-Nyāya to mark the correct anuyogins in the cases of ambiguities. The design of the interface takes care of the above three conditions and dynamically reduces the search space with every choice. For instance, the input samavāyasambandha-avacchinna-gandhatvaavacchinna-gandha-niṣṭha-ādheyatā-nirūpita-adhikaraṇatāvat-vastu will be parsed as shown in Fig. 11. After selecting the anuyogis for all relations (see Fig 12), the constituency parse as a linear bracketed expression is

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949

Fig. 11 A screenshot of interface of NN-Parser

Fig. 12 A screenshot of interface after user-selection

ððððððsamavayasambandha
avacchinnaÞ
ðgandhatva
avacchinnaÞ

ðgandha
nisthaÞ
adheyataÞÞÞ
nirupitaÞ
adhikaranataÞ∧ vat
vastu __

Sometimes NNEs do not specify the relation between the concepts explicitly. For example, the expression ghaṭa-abhāva-vat-avṛttitvam has two concepts ghaṭa and abhāva as consecutive nodes. In such cases these are treated as a compound with an un-specified relation and is parsed as (((ghaṭa-abhāva)-vat)-avṛttitvam).

Translating NN Expressions into Conceptual Graphs Formal grammar G for such a parsed structure is defined below (Arjuna and Kulkarni 2016)

950

A. Kulkarni

Let G = (N, T, P, N N E), where N: Set of nonterminal symbols = {Compound Concept, Compound Relation, Rel term, Concept term}, T: Set of terminal symbols = {relation and concept}, NNE: The start symbol, and P: Production rules as described below. NNE

:

Compound_Concept

; :

Compound_Concept ↑.head = ↓. head`

‘(’ Compound_Relation ‘-’ Concept ↑. head = Concept term.head establish an edge between the head of the Compound Relation to the head of the Concept term

Compound_Relation

; :

‘(’ Concept term ‘-’ Rel term ‘)’ ↑. head = Rel term.position draw a relation node for Rel term. establish an edge between the head of the Concept term to the relation node.

Concept_term

; :

NNE ↑. head = ↓. head

|

CONCEPT

↑. head = ↓.position

draw a concept node Rel_term

; :

RELATION

↑. head = ↓. head

;

Concepts are the nouns, relational abstract expressions, the negation functor, and the terms derived with tva suffix from nouns. Relations are (a) the sentence forming operators niṣṭha and avacchinna, (b) the conditioning operator nirūpita, and (c) along with their inverse relations viz. vṛtti (or āśraya), avacchedaka, and nirūpaka, respectively. In NN the relations are always binary (dviṣṭaḥ sambandhaḥ.). Every relation node needs two relata. Thus, in order to draw a CG corresponding to an NN relation, (i) node labels, (ii) node types, and (iii) the two relata corresponding to the given relation are needed. With each rule of this grammar, a semantics in terms of an attribute grammar is associated which then translates an NNE into a CG. The node labels and the node types correspond to the intrinsic attributes of the terminal nodes concept and relation, which are available from the lexer. The two rules in the

34

Later Nyāya Logic: Computational Aspects

Fig. 13 CG with unspecified relation “R”

sadhya (Probandum) (1)

R

951

abhava (absence) (2)

R

adhikaranata (substratum) (3)

nirupita (determined by) (4)

vastu (object) (5)

grammar above corresponding to Compound Relation and Compound Concept provide the links between a relation and a concept term. It was observed that, in actual usage, if a compound is not ambiguous, it is not expanded with NN structure. Thus typically NNEs were found to be heterogeneous mixtures of classical Sanskrit and well-structured descriptions as stated above. In order to handle such heterogeneous structures, an unspecified relation “R” is established between the two consecutive conceptual nodes. For example, the expression sadhyabhavadhikaranatanirupitavastu

(9)

contains only one NN technical term nirūpita and the remaining part of the expression is an ordinary classical Sanskrit compound with 4 components sādhya, abhāva, adhikaraṇatā, and vastu denoting concepts. Figure 13 shows the CG for this expression. A dummy relation “R” is introduced between two consecutive concept nodes. Similarly, when two relation terms follow each other, a dummy concept node “vastu” is introduced in between in order to faithfully render such an expression with a CG. Nyāyacitradīpikā combines all the three modules described above and presents a platform for a user to help him understand an NNE (http://sanskrit.uohyd.ac.in/scl/ NN/segmenter.).

Conclusion Advances in computational linguistics and availability of computational tools for the analysis of Sanskrit texts has made it possible to produce conceptual graphs for NN expressions mechanically. The only subjectivity involved in this rendering is involvement of the user in the constituency analysis of the compound. It is also possible to generate the śābdabodha of a given sentence, using the parser for Sanskrit (Kulkarni 2013). This way of analysis has its own limitations as well. For example, the nañ-tarpuruṣa compounds such as avṛttitva or asāmānādhikaraṇya are represented in the present scheme as a single concept node. But unless such compounds are

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expanded elaborating the underlying structure, even the conceptual graphs will be as difficult as the original NN expressions. In the manual diagrammatic representations of Jha (1987) and Wada (2007), for example, we find such terms are analyzed further, and proper interpretation is provided. Next natural step is to use the technical language of NN for natural deduction. Some initial work has been carried out by Srinivas Varakhedi in his doctoral thesis, in this direction. It is necessary to take that work one step ahead and build an inference engine that understands NN expressions.

References Arjuna, S.R., and Amba Kulkarni. 2014. Segmentation of Navya-Nyāya expressions. In Proceedings of international conference on NLP, Goa. Arjuna, S.R., and Amba Kulkarni. 2016. Analysis and graphical representation of navya-nyāya expressions – Nyāyacitradī pikā. In Sanskrit and computational linguistics, ed. Amba Kulkarni, 21–52. Pragun Publication, A D K Publishers Distributors Enterprise, New Delhi. Bhattacharya, Sibajiban. 1990. Some features of the technical language of Navya-Nyaya. Philosophy East and West 40 (2): 129–149. Briggs, Rick. 1985. Knowledge representation in Sanskrit and artificial intelligence. AI Magazine 6 (1). Ganeri, Jonardon. 2008. Towards a formal regimentation of the Navya-Nyāya technical language-I. In Logic, Navya-Nyāya & applications, London:College Publications. 109–124. Huet, Gérard. 2006. Lexicon-directed segmentation and tagging of Sanskrit. In Themes and tasks in old and middle indo-Aryan linguistics, 307–325. Delhi: Motilal Banarsidass. Huet, Gérard. 2009. Formal structure of Sanskrit text: Requirements analysis for a mechanical Sanskrit processor. In Sanskrit computational linguistics 1 & 2, LNAI 5402, eds. Gérard Huet, Amba Kulkarni, and Peter Scharf. Springer, Berlin Heidelberg. Huet, Gérard and Pawan Goyal. 2013. Design of a lean interface for Sanskrit corpus annotation. In Proceedings of international conference on NLP, Noida, eds. Dipti Mishra Sharma, Rajeev Sanghal, Karunesh Kr.Arora, and B.K.Murthy, 177–186. Ingalls, Daniel H.H. 1951. Materials for the study of Navya-Nyāya logic. Cambridge, MA: Harvard University Press. Jha, V.N. 1987. Viṣayatāvāda of Harirāma Tarkālaṇkāra. Pune: University of Pune. Kulkarni, Amba. 1994. Navya-Nyāya for Scientists and Technologists: A first step. MTech dissertation, IIT, Kanpur. Kulkarni, Amba. 2013. A deterministic dependency parser with dynamic programming for Sanskrit. In Proceedings of the second international conference on dependency linguistics (DepLing 2013), 157–166, Charles University in Prague, Matfyzpress, Prague. Kulkarni, Amba and Anil Kumar. 2011. Statistical constituency parser for Sanskrit compounds. In Proceedings of international conference on NLP, Macmillan Advanced Research Series. Chennai. Macmillan Publishers India Ltd. Kulkarni, Amba, and Devanand Shukl. 2009. Sanskrit morphological analyser: Some issues. Indian Linguistics 70 (1–4): 169–177. in the Festscrift volume of Bh. Krishnamoorty. Kumar, Anil, Vipul Mittal, and Amba Kulkarni. 2010. Sanskrit compound processor. In Proceedings of the fourth international Sanskrit computational linguistics symposium, LNAI 6465, ed. Delhi. Girish Nath Jha, 57–69. Springer. Matilal, Bimal Krishna. 1968. The Navya-Nyaya doctrine of negation. Cambridge, MA: Harvard University Press. Matilal, Bimal Krishna. 1977. Nyāya-Vais´eṣika. Harrassowitz, Verlag. Mittal, Vipul. 2010. Automatic Sanskrit segmentizer using finite state transducers. In Proceedings of student research workshop, 85–90. Association for Computational Linguistics. Uppsala.

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Mohanty, Jitendra Nath. 2000. Classical Indian philosophy. Oxford: Rowman & Littlefield Publishers. Natarajan, Abhiram and Eugene Charniak. 2011. Sˆ3 – Statistical Sandhi Splitting. In Proceedings of IJCNLP, Thailand. 301–308. Patil, Devadatta. 2014. Vidyādharī. Pune: Samarth media center. Shaw, J.L. 1980. The Nyāya on cognition and negation. Journal of Indian Philosophy 8 (4): 279–302. Sowa, John F. 1985. Conceptual Structures: Information Processing in Mind and Machine. Addison-Wesley, Reading, MA Sowa, John F. 1992. Semantic networks. Encyclopedia of cognitive science. Wiley. Staal, Fritz. 1988. Chapter: Means of formalization in Indian and Western logic. In Universals: Studies in Indian logic and linguistics, 81–87. Chicago: The University of Chicago Press. Varalakshmi, K. 2013. Ś leṣālaṅkāra: A challenge for testing sanskrit analytical tools. In Recent researches in Sanskrit computational linguistics fifth international symposium proceedings, ed. Malhar Kulkarni and Chaitali Dangarikar. New Delhi: D. K. Printworld. Wada, Toshihiro. 2007. The analytical method of Navya-Nyāya. Groningen: Egbert Forsten. Woods, W. 1975. What’s in a link: Foundations for semantic networks, Representation and Understanding: Studies in Cognitive Science, 35–82. Academic Press, New York.

The Logic of Late Nyāya: A PropertyTheoretic Framework for a Formal Reconstruction

35

Eberhard Guhe

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Nyāya Theory of Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Technical Language of the Navya-Naiyāyikas and Previous Attempts to Analyze It . . . . Epistemological and Ontological Presuppositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Toward an Alternative Formal Reconstruction of the Logic of Navya-Nyāya . . . . . . . . . . . . . . . . G. Bealer’s Calculus T1 as a Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensions of T1 which Function as Alternatives to Set Theories . . . . . . . . . . . . . . . . . . . . . . . . . Summary Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

956 956 959 962 964 964 966 975 976

Abstract

Late Nyāya and Navya-Nyāya are renowned for their affinity to Western formal logic. The interest of modern interpreters has especially been sparked by the technical language of the Navya-Naiyāyikas and by their attempt to formulate theorems about properties by means of these peculiar linguistic tools. The aim of the present contribution is to provide a property-theoretic framework for a formal reconstruction of Navya-Nyāya logic. “The Logic of Late Nyāya: Problems and Issues,” published in the same book, demonstrates its utility by referring to some pertinent examples. Keywords

Indian logic · Navya-Nyāya · Intensionality · Property theories · Non-wellfoundedness

E. Guhe (*) Department of Philosophy, Fudan University, Shanghai, China e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_11

955

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Introduction The type of logic which developed in ancient India is not symbolic, but it is nevertheless formal. Its formal character becomes obvious if one looks at the theory of inference in late Nyāya and in Navya-Nyāya. Navya-Naiyāyikas were particularly concerned about the definition of pervasion (vyāpti), the relation between probans and probandum in an inference. In order to define the concept of pervasion with utmost precision they designed an ideal language, a kind of Leibnizian characteristica universalis based on a canonical form of Sanskrit, which serves to explicate the objective content of verbalized and unverbalized cognitions and to disambiguate sentences formulated in ordinary Sanskrit. Some of the most significant logical insights of the Navya-Naiyāyikas are an offshoot of their attempts to elaborate on definitions of pervasion by making extensive use of their peculiar linguistic tools.

The Nyāya Theory of Inference According to the theory of inference in late Nyāya and in Navya-Nyāya, an inference (anumāna) consists of five members, a thesis, three members which are supposed to corroborate the claim formulated in the thesis, and a conclusion which restates the thesis as a result of the inference. To be more precise, the present type of inference is the so-called “inference for others” (parārthānumāna). Naiyāyikas distinguish between this verbalized five-membered inference and the so-called “inference for oneself” (svārthānumāna), “which is only drawn in the mind, but not expressed in words” (van Bijlert 1989: 64). More information on the latter will be given below. In the case of a valid inference, the content of each of the five members must be a veridical cognition. Although Indian logicians did not symbolize the components of a correct five-membered inference, it is clear that they conceived of it as an instance of formally valid reasoning, since all the components are identified by certain technical terms and the way they are related to each other and the order in which they are supposed to appear is strictly determined. In Keśava Miśra’s Tarkabhāṣā, a work which according to Dineśchandra Bhattacharya dates from the twelfth century AD (cf. Bhattacharya 1958: 64), one can find the following stock example of a valid five-membered inference. parvato ’yam agnimān dhūmavattvāt. yo dhūmavān so ’gnimān yathā mahānasaḥ. yatrāgnir nāsti tatra dhūmo ’pi nāsti yathā mahāhrade tathā cāyam. tasmāt tathā. (TBh: 40, 5f)

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[1. Thesis (pratijn˜ ā)] [2. Reason (hetu)] [3. Example (dṛṣṭānta)] [4. Application (upanaya)] [5. Conclusion (nigamana)]

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This mountain possesses fire. For it possesses smoke. Whatever possesses smoke possesses fire like the kitchen. Wherever there is no fire there is also no smoke such as in a big lake. This [mountain] is so (i.e., it possesses smoke as a sign of fire). Therefore [it is] so (i.e., the mountain possesses fire).

In this example, the mountain functions as the subject of the inference (pakṣa). Fire is the probandum (sādhya), a property which according to the first member can be proved to be present in the subject of the inference. Smoke is the probans or prover (hetu or sādhana), a property whose presence in the subject of the inference is an established fact according to the second member of the inference (which happens to be called hetu as well). The probans functions as an indicator of the probandum. According to the third member this is possible, because there is a universal relation called “pervasion” (vyāpti) between the probans and the probandum in the sense that every locus of the probans is a locus of the probandum. The reference to the examples in the third member indicates that the cognition of a pervasion involves an inductive generalization. The cognition is warranted because of wide experience of positive correlations such as smoke and fire in a kitchen hearth and negative correlations such as absence of smoke in a lake where there is absence of fire. So, the cognition of a pervasion is the result of an extrapolation from certain known cases, which are, of course, different from the case at issue in an inferential situation. Due to an epistemic process called parāmars´a (“reflective grasping”), which is indicated in the fourth member, the prover’s presence in the subject of an inference is associated with the pervasion relation and this finally gives rise to the “inferential awareness” (anumiti), the conclusion expressed in the fifth member, i.e., the cognition that the probandum is present in the subject of the inference. Naiyāyikas were well aware that inference might lose its status as an infallible means of knowledge if it involves an inductive generalization. Hence, the cognition of a pervasion was not entrusted to an inductive generalization alone: According to the Nyāya theory of perception the perception of an individual involves “a perception that grasps the general character” (sāmānyalakṣaṇapratyakṣa), i.e., the perception of a single individual was believed to give rise to a collective perception of all individuals of the same kind and of their common characteristics. Thus, Naiyāyikas claimed that the perception of all smoke individuals and their co-occurrence with fire can be achieved by means of the perception of a particular smoke. If the presence of the probandum in the subject of the inference was supposed to be a necessary precondition for the presence of the probans, one could rely on hypothetical reasoning (tarka) as a further epistemic backup. In such a case the hypothetical assumption of the absence of the probandum in the subject of the inference would entail the absence of the probans in the same place. One might, e.g., argue that there

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can be no smoke without fire. Therefore the pervasion relation between smoke and fire should include the co-occurrence of both in the case at issue in an inferential situation. Nevertheless, the five-membered inference turns out to be a purely deductive procedure if one regards only the result of the inductive process of determining the pervasion relation as a part of the inference, not the process itself. By means of the symbolization key. . . Sx: “x is a locus of smoke” Fx: “x is a locus of fire” m: “the mountain” . . .one can formalize the above-mentioned stock example of a valid inference in the following way: 1. 2. 3. 4. 5.

Fm Sm 8x(Sx ! Fx) Sm ^ (Sm ! Fm) Fm

Many interpreters have taken great pains to interpret the five-membered inference in such a way that its apparent redundancy disappears. Claus Oetke, e.g., suggests that the last member expresses the mere deducibility of Fm from the preceding members, while the first member contains a truth claim. So, one might still symbolize the last member as Fm, whereas the first member should rather be represented as T [Fm], where T [. . .] stands for “It is true that . . .” (cf. Oetke 1994: 22f). In contrast to other interpretations the present one affirms the redundant structure of the five-membered inference in Navya-Nyāya. If it serves the perspicuity of an argument, one need not have any misgivings about redundancy. The design of a fivemembered inference is, actually, quite similar to that of mathematical proofs. In mathematics, it is common to formulate a proof by, first of all, stating the theorem and then giving a proof, where in the end the theorem reappears as a result of the proof. Sometimes mathematicians even start a proof by rewriting a theorem in an equivalent form if the equivalent formulation is easier to prove. However, nobody would think of complaining about redundancy in such cases. Unlike Oetke, who renders the fourth member merely as an implication (Sm ! Fm), one might even be inclined to say that the second member recurs in the fourth one as a conjunct (Sm ^ (Sm ! Fm)). This is at least plausible if one understands the fourth member in the sense that the mountain possesses smoke as a sign of fire. The recurrence of the second member as a conjunct in the fourth member is in fact necessary in order to secure the deducibility of the conclusion in the case of the “inference for oneself” (svārthānumāna). The latter can be understood as an unverbalized cognitive process which corresponds to the last three members of a five-membered inference: tathā hi kas´cit puruṣaḥ svayam eva bhūyodars´anena yatra dhūmas tatrāgnir iti mahānasādau vyāptiṃ gṛhī tvā parvatasamī paṃ gatas tadgate

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cāgnau saṃdihānaḥ parvate vartinī m avicchinnamūlām abhraṃlihāṃ dhūmalekhāṃ pas´yan dhūmadars´anād udbuddhasaṃskāro vyāptiṃ smarati yatra dhūmas tatrāgnir iti. tadanantaraṃ vahnivyāpyadhūmavān ayaṃ parvata iti jn˜ ānam utpadyate. ayam eva liṅgaparāmars´a ity ucyate. tasmāt parvato vahnimān iti svasya jn˜ ānam anumitir utpadyate. tad etat svārthānumānam (NKoś, s.v. svārtham). – “[3] For instance, a certain man has grasped by himself by means of repeated seeing the pervasion in a kitchen etc., [namely] ‘Where there is smoke, there is fire’, and he has gone close to a mountain, being uncertain with regard to a fire situated on it. While seeing on the mountain a column of smoke which touches the clouds and which stays [there], having [the mountain as] a root, which is not disconnected [from the column of smoke], he remembers the pervasion on account of seeing the smoke, recalling to remembrance that where there is smoke there is fire. [4] Thereupon the cognition arises that this mountain has smoke pervaded by fire. This is called ‘the reflective grasping of the inferential sign’. [5] Therefore the inferential cognition arises, [i.e.,] his cognition: ‘The mountain possesses fire.’ This is inference for oneself.” In the case of the above-mentioned stock example this three-partite inference for oneself is valid if Sm is known to be true. Otherwise one cannot infer Fm from Sm ! Fm. Actually, the explication of the last but one member of such an inference in the preceding quotation (“. . .this mountain has smoke pervaded by fire.”) suggests that one should understand it in the sense of Sm ^ (Sm ! Fm).

The Technical Language of the Navya-Naiyāyikas and Previous Attempts to Analyze It The formal language of the Navya-Naiyāyikas has been studied in detail in a Sanskrit work by Maheśa Chandra dating from the nineteenth century, which will be used here as a major work of reference (cf. BN and Guhe 2014). There are also several Western publications in which the authors have tried to render linguistic conventions in Navya-Nyāya by means of some kind of symbolic notation. D. H. H. Ingalls, e.g., borrows symbols from first-order logic and set theory and combines them with some extra notational devices invented by himself (such as └ ┘, ┌┐ , ∸ etc.). (Cf. Ingalls 1951: 84) His own supplementary formalism is confined to mere abbreviations of technical terms in Navya-Nyāya. He does not outline any comprehensive logical system for a formal reconstruction of Navya-Nyāya logic, but he successfully demonstrates how Navya-Naiyāyikas used their technical language to formulate well-known logical theorems such as de Morgan’s laws. Moreover, he highlights the importance of certain quasi-axiomatic principles in Navya-Nyāya logic, such as the “tattvavat tad eva”-rule, a kind of comprehension principle for properties. As indicated below, some late Naiyāyikas and Navya-Naiyāyikas also advocated a kind of regularity axiom for properties. Thus, the introduction of the axiomatic method, which had such a great impact on the development of Western logic, is already foreboded in Navya-Nyāya. Goekoop’s formalizations of vyāpti-definitions are almost entirely based on the language of first-order logic. Only in a few cases he applies also symbols from

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set theory (cf. Goekoop 1967: 35). The purpose of his formalizations is to give a clue to the logical content of the vyāpti-definitions discussed in Gaṅgeśa’s Tattvacintāmaṇi. Thus, he compares the scopes of these definitions by proving in a first-order calculus in what way they are related to each other. In some cases a definition can be proved to imply another one, or they even turn out to be provably equivalent. Since there is no idiosyncratic symbolism involved in Goekoop’s formalizations, they are easily accessible to a reader who is familiar with modern logic. As long as one is only interested in the logical content of vyāpti-definitions in Navya-Nyāya, regardless of the way they are formulated in Sanskrit, one might make do with firstorder representations. However, sometimes Goekoop uses quantifiers where the Sanskrit original paraphrases a quantification by means of property terms, so that his formalization cannot easily be mapped onto linguistic elements in the Sanskrit original. Moreover, logcial inquiries in Navya-Nyāya which involve the abovementioned comprehension principle for properties cannot be adequately modeled within a first-order framework. For that purpose one would need some appropriate means to express a property abstraction similar to the class abstraction in set theory. Jonardon Ganeri contents himself with capturing “some of the logical apparatus used by the Navya-Nyāya authors” (Ganeri 2008: 118) and focuses on adequate formal reconstructions of relational expressions in Navya-Nyāya. In order to avoid distorting effects which might be associated with direct first-order formalizations, he pursues a two-step strategy. First, he translates the original Sanskrit formulation into a kind of formal regimentation similar to Ingalls’s ideosyncratic formalism, but far more elaborate. The result is an expression which can easily be mapped onto the linguistic elements of the Sanskrit original. In a second step, he transforms the formal regimentation into a first-order formula according to a precise translation key. By additionally applying the first-order equivalences ∃xα $ :8x:α and 8xα $ :∃x:α and the law of double negation if appropriate, he finally obtains a negated or unnegated first-order wff of the form (i) (:)aRb, (ii) (:)(Q)xRb, (iii) (:)(Q)aRy, or (iv) (:)(Q)xRy, where (Q) stands for one bounded quantifier binding the free variable in (ii) and (iii), or for two bounded quantifiers binding the free variables in (iv). He further claims that the bounded quantifiers in (ii)–(iv) can be replaced by unbounded quantifiers. If, e.g., the quantifiers in (iv) range over objects which are F and G, respectively, it should be possible to substitute unbounded quantifiers for the bounded ones and Fx ^ Gy ^ xRy for xRy (cf. Ganeri 2008: 117f). This is actually admissible in the case of two existential quantifiers bounded by the sets {x|Fx} and {x|Gx}, respectively, because ∃x  {x|Fx}∃y  {x|Gx}xRy$∃x∃y(Fx^Gy^xRy). However, a bounded universal quantifier 8x  {x | Fx} translates into an unbounded universal quantification of an implication: 8x(Fx ! . . .) To some extent the formal reconstruction of relational expressions can be handled by means of Ganeri’s approach. However, as will be shown in the following chapter, certain definitions of pervasion in Navya-Nyāya should be formalized as conjunctions in a first-order setting. So, they cannot be represented on the model of any of

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the abovementioned types (i)–(iv). Moreover, by representing relational expressions in Navya-Nyāya as first-order wffs Ganeri misses an important detail which is still present in his formal regimentation. When Navya-Naiyāyikas want to express that the ground is a locus of a certain pot, they use formulations like “The ground possesses locushood described by pot.” One of the purposes of this kind of circumbendibus is to specify the order of the members of the relation “being a locus of”: The first member is the “possessor” of the relational abstract “locushood,” which corresponds to the relation “being a locus of,” and the second member is the “describer” of the relational abstract. Instead of rendering the Navya-Nyāya formulation just by aRb Ganeri inserts a dot in his formal regimentation in order to indicate the residence relation (a.Rb, cf. Ganeri 2011: 227). In his translation of the regimentation into a first-order wff, he ignores the dot and ends up with a formalization of type (i). The dot obviously represents some kind of predication relation (like the  -relation in set theory or the Δ-relation introduced below), which cannot be represented in a first-order setting. It might not play such an important role in examples like “The ground possesses locushood described by pot,” but it definitely has to be taken into account for an adequate formalization of the abovementioned comprehension principle. Hence, interpreting the logic of Navya-Nyāya as a fragment of first-order logic would certainly oversimplify matters. There is another feature of Navya-Nyāya logic which gets lost even in Ganeri’s regimentations: In his system, Greek letters denote sets (cf. Ganeri 2011: 229) and he regards sets as the denotata of expressions for properties in Navya-Nyāya. Thus, “potness” (ghaṭatva), e.g., which he renders as π, is supposed to refer to the set of all pots. However, by identifying properties with sets Ganeri disregards the intensional character of Navya-Nyāya logic. As noted by Matilal, Navya-Naiyāyikas did not equate properties like “being created” (kṛtatva) and “being non-eternal” (anityatva), although the set of everything which is created and the set of all non-eternal entities were regarded as identical (cf. Matilal 21990: 131). Similarly, “knowability” ( jn˜ eyatva) and “nameability” (abhidheyatva) were conceived of as both universal, but nevertheless distinct properties (cf. Ingalls 1951: 61). They had to be regarded as distinct in Navya-Nyāya, because they were supposed to function as probans and probandum in an inference, and according to the Nyāya theory of inference the relata of a pervasion relation should not be identical (cf. Oetke 2009: 35f). Especially for the purpose of explicating definitions of pervasion in Navya-Nyāya T. Wada designed his method of graphic representations (cf. Wada 2007). He captures the relational structure of these definitions by means of different types of edges for the relations referred to. The relata are not symbolized, but their English translations are entered into boxes which constitute the nodes of the graph. Wada’s method can be fruitfully applied as a means to unravel the basic structure of complex definitions of pervasion. Unfortunately, it is not very effective in cases in which quantifications are implicitly part of the logical content of a definition, as will be shown in the following chapter.

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Epistemological and Ontological Presuppositions According to B. K. Matilal, the logic of Navya-Nyāya is a kind of property locationlogic (cf. Matilal 21990: 112). Maheśa Chandra defines the concept of property (dharma) in the following way: dhriyate tiṣṭhati vartate yaḥ sa dharmaḥ.. . .yatra yo vartate sa tasya dharmaḥ. (BN: 8, 9f = Guhe 2014: 60) – “What is fixed [somewhere], depends [on something or] is resident [somewhere], that is a property. . . .What resides somewhere that is the property of that something.” Navya-Naiyāyikas regard a property and its locus (adhikaraṇa) as the ultimate constituents of a cognitive event ( jn˜ āna). “A jn˜ āna is a particular just as a color spot or tone is a particular. It can very well be viewed as an event in the sense that a particular tone or sound can be viewed as a physical event. (. . .) Furthermore, a jn˜ āna is a momentary event, being in this respect also like a tone or sound” (Matilal 1968: 7). In the same way as expressions refer to something cognitions are always directed to an object. “Being directed to an object” (viṣayatā) is a special relation in Nyāya which obtains between a cognition and its content: tatra viṣayā ghaṭapaṭādayo jn˜ ānecchādau viṣayatāsaṃbandhena vartante. viṣayitāsaṃbandhena ca jn˜ ānecchādayo ghaṭapaṭādau viṣaye tiṣṭhantī ti (BN: 12, 16f = Guhe 2014: 72) – “In that case objects, [such as] a pot, a cloth etc., occur via objecthood relation in a cognition, a wish etc. And by the relation ‘being directed to an object’ a cognition, a wish etc. depend on an object, [such as] a pot, a cloth etc.” There seems to be a functional equivalence here between this objecthood relation and the reference relation in Western logic. The former links an object to a cognition, whereas the latter links an object to an expression which designates it. Navya-Naiyāyikas do not only consider ordinary physical objects (such as a pot or a cloth), when they analyze the content of cognitions into properties and loci. The logically more interesting elementary constituents of a cognition include objects designated by means of nouns ending in abstract suffixes like -tva or -tā, which can be translated by means of English abstract suffixes like “-ness” or “-hood.” In some cases circumlocutions by means of the word “being” may also be feasible as translations. So, the Sanskrit words ghaṭatva and kṛtatva, which derive from ghaṭa (“pot”) and kṛta (“created”), can be translated by “potness” and “being created,” respectively. Some of these abstract nouns denote universals (sāmānya), one of the ontological categories which the Navya-Naiyāyikas inherited from the school of the Vaiśeṣikas. Numbers were also denoted by means of such abstract nouns, but classified as qualities. Thus, “twoness” (dvitva) was supposed to refer to the number “two” as a quality of two things conceived of as a dyad. Of course, not every abstract noun of this type can be said to be a name of some kind of real entity. Matilal refers to discussions in Navya-Nyāya concerning unlocatable “properties,” such as “being the son of a barren woman,” “being a golden mountain,” etc.: “An unlocatable property is a suspect in Navya-nyāya. It is regarded as a ficticious property which cannot be located in our universe of loci” (Matilal 1998: 147).

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It might be tempting to regard what is commonly called an “imposed property” (upādhi) in Navya-Nyāya as ficticious as well. “The meanings of a large number of general terms in our language are construed as upādhis, i.e., ‘nominal’ properties, not objective universals. Kaṇāda and Praśastapāda did not really address this issue. It was, above all, Udayana who tried to provide a set of criteria for the exclusion of invalid or counterfeit universals, such as ‘cookness’ (pācakatva) or ‘being an inhabitant of Ayodhyā’ (ayodhyāvāsitva); . . .” (Halbfass 1992: 119). Some nominal properties are like the latter example compound, whereas universals should be elementary. Udayana notes six other invalidating factors ( jātibādhaka) (cf. Halbfass 1992: 132). Although such imposed properties do not belong to the elementary constituents of the empirical world, it would be a mistake to regard them as mere fabrications of the mind: “A failed universal as an upādhi would be a cognized property that like universals is ‘repeatable,’ i.e., can occur in more than one instance, but that falls to one or another of six ‘blockers of natural-kind status’, jāti-bādhaka. Such would be then a ‘surplus property,’ a property surplus to the system of ontological analysis. In other words, this would be a ‘condition’ in a very abstract and non-committal sense, a ‘something extra’ that is not just mind generated, that is a real property of something, but a property whose taxonomical character we have not determined. Some ‘surplus properties’ do seem to be mainly due to verbal excess, to saying things non-perspicaciously. But others do not seem so, and all upādhi-s are grounded in some fashion or other in the way the world is. Otherwise, they would not become objects, cognitive objects, that is, indicated by our perceptions and conversations about the world” (Phillips 2002: 25). Finally, there is an ontological category of negative properties called “absence” or “non-being” (abhāva) in Navya-Nyāya, which is logically particularly interesting, as will be shown below. Navya-Naiyāyikas distinguish two types of absence. Both of them can be illustrated by means of the example of an empty pot. Since the pot is different from a cloth, it is a locus of the “mutual absence” (anyonyābhāva) of or the “difference” (bheda) from a cloth. Since there is no food in the pot, it is also a locus of the “absence” (abhāva) of food. This is another type of negative property. More accurately, it is called “relational absence” (saṃsargābhāva), since this property characterizes a locus as being unrelated to the absentee in terms of one of the relations enunciated in the Navya-Nyāya system of ontological categories. Since food has no “contact” (saṃyoga) with the pot, the latter is a locus of the “absence of food having contact with the pot.” (To be more precise, such an absence was said to be “limited” (avacchinna) by contact.) The specification of the relation whereby an absentee fails to reside in a locus was regarded as crucial: Although potness resides in a pot via a relation called “inherence” (samavāya), a pot is a locus of the absence of potness having contact with a pot. A difference is construed as a denial of a further type of relation, namely “identity” (tādātmya).

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Toward an Alternative Formal Reconstruction of the Logic of Navya-Nyāya G. Bealer’s Calculus T1 as a Basic Framework The present formal reconstruction of the logic of Navya-Nyāya is an elaboration of B. K. Matilal’s interpretation. Some decades ago Matilal and Bocheński already talked about intensional tendencies in the logic of Navya-Nyāya (cf. Matilal 1968: 67 and 74, 21990: 169; Bocheński 41978: 513 and 517). This impression is owing to the fact that logical inquiries in Navya-Nyāya are mostly concerned with the abovementioned property names ending in abstract suffixes like -tva or -tā. Matilal’s idea was to formalize such property names by using Quine’s notation for intensional contexts: If one writes Px for “x is a pot,” then x[Px] (which can be read as “being an x such that x is P”) is an analytical expression for “potness.” The function of the variable x in front of this term is to bind the free occurrence of x in Px. The square brackets around Px indicate an intensional context. If one substitutes an expression within the bracketed part by another one which is extensionally equivalent, one might change the reference of the property term. Such restrictions concerning the substitutability of extensionally equivalent expressions generally distinguish intensional from extensional logical systems. The property theory designed in Bealer 1982 can be used to elaborate Matilal’s formal analysis of Navya-Nyāya logic. Bealer never thought about such an application of his theory. He wants to explicate his realist notion of properties, relations, and concepts, which is supposed to open up new vistas in analytical philosophy, especially in the realm of semantics, philosophical logic, philosophy of mind, and philosophy of mathematics. For that purpose Bealer designed three calculi called “T1,” “T2,” and “T20 .” T1 is especially suited to the treatment of modal matters, whereas T2 serves to check epistemic arguments. T20 is a synthesis of T1 and T2. T1 can also be used as a basis for a formal analysis of the logic of properties in NavyaNyāya, as will be shown below. The language of T1 consists of the following primitive symbols (cf. Bealer 1982: 43): (i) (ii) (iii) (iv)

Logical operators: ^, :, ∃ Predicate letters: F11 , F12 , . . . , Fqp Variables: x, y, z,. . . Brackets: (, ), [, ]

Simultaneous inductive definition of terms and formulas: (1) (2) (3) (4)

All variables are terms. If t1,. . ., tj are terms, then Fji t1 . . . tj is a formula. If A and B are formulas and vk a variable, then (A^B), :A and ∃vkA are formulas. If A is a formula and v1,. . ., vm (0  m) are distinct variables, then ½Av1 ...vm is a term.

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The predicate letter F21 is singled out as a distinguished logical predicate. Formulas of the form F21 t1 t2 are to be written in the form t1 = t2. Moreover, the symbols 8, !, _ and $, □ and ◊, which can be defined in terms of ∃, :, ^, [ and ], are included in the language of T1. (The definition of the modal operator □ by means of the square brackets will be explained below.) Remarks • In (3) A is an arbitrary formula, in which the variable vk need not occur. Similarly, in (4) the variables v1,. . ., vm are not required to be components of A. If they do occur in A, they are bound by the index variables. Generally speaking, an occurrence of a variable vi is bound (free) if and only if it lies (does not lie) within a formula of the form ∃viA or a term of the form ½Av1 ...vi ...vm . • Occasionally, a, b, c,. . . will be used as constant symbols in the present formal representation of Navya-Nyāya expressions, although they are not part of the elementary symbols of T1.

Instead of going into the details of the semantics for the language of T1, it will suffice to explain the meaning of the “exotic” expressions: A term of the form ½Av1 ...vm denotes: (a) A proposition, if m = 0 (“that A”) (b) A property, if m = 1 (“being a v1 of which A is true”) (c) An m-ary relation, if m  2 (“the relation which holds between v1,. . ., vm iff A applies to them”) Remarks • In contrast to possible-worlds approaches, Bealer treats intensional entities as individuals and not as functions. Therefore his property logic has much in common with the reification of properties in Navya-Nyāya. “The new semantic method does not appeal to possible worlds, even as a heuristic. The heuristic used is simply that of properties, relations, and propositions, taken at face value” (Bealer 1982: 42f). • The language of T1 includes also the modal operators □ and ◊, but as defined symbols. An expression of the form □A is adopted as a convenient abbreviation of expressions such as N[A], where N is a one-place predicate expressing “. . .is necessary.” The semantic model structure for T1 (cf. Bealer 1982: 49f) contains a condition which ensures that there is only one necessary truth (cf. Bealer 1982: 52f). Since [x = x] is a trivial necessary truth for any proposition x, [A] can be identified with it if A is necessarily true. Therefore it is possible to define the modal operator □ simply by means of the square brackets: □A ∶$ [A] = [[A] = [A]] (A is necessarily true iff the proposition “that A” is identical to a trivial necessary truth.) As usual, ◊A ∶$ : □ :A.

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• The term ½Av1 ...vm can be regarded as a counterpart of the class term {〈v1,. . ., vm〉| A}.

Bealer suggests the following axiomatization of T1, which can be proved to be sound and complete (cf. Bealer 1982: 58f): A1: Truth-functional tautologies A2: 8viA(vi) ! A(t), where t is free for vi in A, i.e., no free occurrence of vi in A lies within the scope of a quantifier or a sequence of index variables in a term ½. . .v1 ...vm which would bind a variable occurring in t. A3: 8vi(A ! B) ! (A ! 8viB), where vi is not free in A. A4: vi = vi A5: vi = vj ! (A(vi, vi) $ A(vi, vj)), where A(vi, vj) is a formula that arises from A(vi, vi) by replacing some (but not necessarily all) free occurrences of vi by vj, and vj is free for the occurrences of vi that it replaces. A6: ½Au1 ...up 6¼ ½Bv1 ...vq , where p 6¼ q. 
 
 A7: A u1 , . . . , up u1 ...up ¼ A v1 , . . . , vp v1 ...vp , where these two terms are alphabetic variants. A8: ½Au1 ...up ¼ ½Bu1 ...up $ □8u1 . . . 8up ðA $ BÞ A9: □A ! A A10: □(A ! B) ! (□A ! □B) A11: ◊A ! □ ◊ A R1: If ‘ A and ‘ (A ! B), then ‘ B. R2: If ‘ A, then ‘ 8viA. R3: If ‘ A, then ‘ □A. A1–A5 along with R1 and R2 constitute an axiomatization of first-order predicate logic including identity. A6–A8 determine how to deal with the intensional abstracts in T1. It is important to note that A8 furnishes a criterion for the identification of intensional abstracts. In this sense, it has the same function as the axiom of extensionality in set theory. A9–A11 and R3 are the modal part of the axiomatic system S5 of propositional modal logic. A11 is the S5-axiom E (cf. Hughes and Cresswell 1996: 58), which is misquoted by Bealer: “□A  □ ◊ A” (Bealer 1982: 59).

Extensions of T1 which Function as Alternatives to Set Theories The Naive Property Abstraction in Navya-Nyāya It is also necessary to have some methods of formalization to express the abovementioned comprehension principle: tattvavat tad eva. – “Anything which possesses the property ‘being that’ is that” (cf. Ingalls 1951: 36). In order to see how this rule works one might replace the Sanskrit word tat (“that”), which has the same function as a schematic variable here, by words

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like ghaṭa (“pot”). ghaṭatvavān ghaṭa eva means: “Anything which possesses the property ‘potness’ is a pot.” Thus, the tattvavat tad eva-rule can be regarded as a kind of counterpart of the naive class abstraction in set theory: a  {x|A(x)} $ A(a), where a is free for x in A and vice versa. This equivalence can be transformed into a formal version of the naive property abstraction rule in Navya-Nyāya by replacing {x|A(x)} by the corresponding property term in T1. In order to express that something possesses or is a locus of a class-like property one can use Bealer’s Δ-relation, which functions as a counterpart of the  -relation in set theory (cf. Bealer 1982: 96). Thus, if one understands the tattvavat tad eva-rule in the sense of “a possesses (or is a locus of) the property ‘being an x such that A is true of x’ iff A is true of a,” one can formalize it in the following way: (*) aΔ [A(x)]x $ A(a), where a is free for x in A and vice versa. Remarks • The present interpretation of the tattvavat tad eva-rule as an equivalence is confirmed by Matilal, who characterizes the specific style of Navya-Nyāya texts in the following way: “Simple predicate formulations, such as ‘x is F’ are noted, but only to be rephrased as ‘x has F-ness’ (where ‘F-ness’ stands for the property derived from ‘F’).” (Matilal 21990: 115) • Apart from property abstraction relational abstraction also plays an important role in Navya-Nyāya logic (cf. Ingalls 1951: 44f; Matilal 21990: 170f). Only dyadic relations with different relata are considered by Maheśa Chandra: saṃbandhaḥ saṃnikarṣaḥ sa ca vibhinnayor vastunor vis´eṣaṇavis´ eṣyabhāvaprayojakaḥ. (BN: 9, 13 = Guhe 2014: 63) – “A relation is a connection and it establishes the state of being qualificand and qualifier of two different objects.” It is important to note that Maheśa Chandra conceives of the pair of relata as an ordered pair. The following remark about “indirect relations” (paraṃparāsaṃbandha) is certainly applicable to Maheśa Chandra’s concept of relation in general: . . . yasya paraṃparāsaṃbandhasyārambho yasmiṃs´ ca paryavasānaṃ tat tena paraṃparāsaṃbandhena tasmiṃs tiṣṭhati (BN: 10, 12 = Guhe 2014: 66) – “. . ., from which an indirect relation starts (i.e., which is the initial point of the relation) and in which [the relation has its] final end (i.e., which is the terminal point of the relation) that depends on that on account of that indirect relation.” In order to formalize relational abstracts one can use the abovementioned T1-terms of the form ½Av1 ...vm , which are similar to Matilal’s square bracket notation with prefixed binding variables (cf. Matilal 21990: 170, where he suggests the following semi-formalization of samaniyatatva, i.e., “equi-locatability”: xy[x is samaniyata with y]). The idea that a relational abstract [xRy]xy applies to an ordered pair can be expressed as Δ [xRy]xy. According to Bealer one may understand as a property term if one replaces the class

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term {{x}, {x, y}}, which corresponds to (according to the WienerKuratowski definition), by applying the following context definition: . . .{x, y}. . . iffdf ∃z(8w(w Δ z $ (w = x _ w = y)) ^. . . z. . . ), where z is a new variable not occurring in. . . (Cf. Bealer 1982: 83) Since the Navya-Naiyāyikas had no Wiener-Kuratowski definition, they had to follow a different strategy to express that a pair of individuals is an instance of a relational abstract, as will be shown below.

A Property-Theoretic Variant of Zermelo-Russell’s Antinomy and Its Sanskrit Equivalent Now, it is possible to replace the word tat (“that”) in the tattvavat tad eva-rule by asvavṛttitva (“being not resident in itself”). This property can easily be formalized. If one admits xΔx as a formal equivalent of “x resides in itself,” “being not resident in itself” can be expressed by [:x Δ x]x. Let r be an abbreviation of this property. (Navya-)Naiyāyikas were aware that defining properties randomly can result in logical fallacies. Udayana, e.g., noticed that a contradiction in terms derives from the assumption of a universal for ultimate particularities and he included this defect in his list of criteria for identifying counterfeit universals ( jātibādhaka). Nevertheless, there is no textual evidence that Navya-Nyāya logicians were aware of the possibility to use r to derive a variant of Zermelo-Russell’s antinomy from the tattvavat tad eva-rule (cf. Guhe 1999: 22, 2000: 109, 2008: 144f): (a) If r is resident in itself (i.e., if it is svavṛtti), then the property “being not resident in itself” (asvavṛttitva) resides in r. Therefore (according to the tattvavat tad eva-rule) r is not resident in itself (i.e., it is asvavṛtti). (Contradiction!) This is the formal counterpart of the argument: rΔr ) rΔ½:xΔxx ) :rΔr |fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}

can be substituted for aΔ½AðxÞx in ðÞ

(b) If r is not resident in itself (i.e., if it is asvavṛtti), then (according to the tattvavat tad eva-rule) the property “being not resident in itself” (asvavṛttitva) resides in r. Therefore r is resident in itself (i.e., it is svavṛtti). (Contradiction!) This is the formal counterpart of the argument: :rΔr ) rΔ½:xΔxx ) rΔr |fflffl{zfflffl}

can be substituted for AðaÞ in ðÞ

(a) and (b) together yield the following variant of Zermelo-Russell’s antinomy: rΔr $ :rΔr

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An ST2-Style Extension of T1 (“T1+”) as an Appropriate Framework for a Formal Reconstruction of Navya-Nyāya Logic In order to modify (*) in such a way that its paradoxical consequence disappears one can try to imitate the strategies which were pursued by the founders of set theories in order to safeguard the naive class abstraction rule against Zermelo-Russell’s antinomy. Certain restrictions in standard systems of set theory would, however, interfere with ontological commitments in Navya-Nyāya. In ZF (Zermelo-Fraenkel set theory), e.g., sets are the only objects in the domain of models of this system. However, since Navya-Naiyāyikas also talk about non-class-like objects, one needs a system which is similar to set theories with urelements. Moreover, some logical arguments in Navya-Nyāya involve universal properties such as nameability, which can be regarded as the analog of a proper class in set theory. Talking about proper classes like, e.g., {x | x = x} (“the universal class”) is admissible in NBG (NeumannBernays-Gödel set theory), but not in ZF. Therefore a property adaptation of NBG with urelements is preferable as a system which may serve to model logical inquiries concerning properties in Navya-Nyāya. Mendelson incorporates urelements into the framework of NBG (cf. Mendelson 4 1997: 297f). He uses lower-case Latin letters (x, y, z) as restricted variables for sets, capital Latin letters (X, Y, Z) as restricted variables for classes (i.e., for sets and proper classes), and lower-case boldface Latin letters (x, y, z) as variables for classes and urelements alike (cf. Mendelson 41997: 297). In the present property adaptation of set theory, the same kinds of variables are used for set-like properties, class-like properties (i.e., set-like and properly class-like properties), and urelements, respectively. A property version of the NBG comprehension axiom seems to be still too restrictive, because it does not include impredicative instantiations, which a Navya-Naiyāyika might not want to rule out (cf. the example given below). Since impredicative comprehension is admissible in QM (Quine-Morse set theory, also known as “Morse-Kelley set theory”), but not in NBG, (*) should rather be modified according to the following QM-style comprehension axiom: (C) 8x(Ex ! (xΔ[A(y)]y $ A(x))), where x is free for y in A and vice versa. As a remedy for the property variant of Zermelo-Russell’s antinomy, the property abstraction in (*) has been relativized to “element-like” individuals. The monadic predicate E (read: “. . .is element-like”) is true of x iff x is an urelement or a set-like property. (C) can be used to formalize impredicative substitution instances of the tattvavat tad eva-rule, such as: “An element-like x is a locus of the property ‘being a locus of some property which is equi-locatable with nameability’ (abhidheyatvasamaniyatakiṃciddharmādhikaraṇatva) iff x is a locus of some property which is equi-locatable with nameability.” The symbolization key. . . Nx: “x is nameable”

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x ¼ y: “x is equi-locatable with y,” i.e., 8z(zΔx $ zΔy) . . .yields the following instantiation of (C):  8x Ex !



 

  xΔ ∃z xΔz ^ z ¼ ½Nyy $ ∃z xΔz ^ z ¼ ½Nyy x

There is still another constraint in standard systems of set theory which should not be reproduced in a formal reconstruction of Navya-Nyāya logic: It is commonly assumend that proper classes can never be elements of classes, i.e., (even finite) collections of proper classes do not exist. In Navya-Nyāya, however, it is possible to apply the -tva-abstraction technique repeatedly, so that one might create an expression like abhidheyatvatva (“nameabilityness”), which denotes a property of nameability. The analog of such a property in set theory would be the singleton of the universal class, something which does not exist according to standard systems of set theory. One might call it a “hyper-class” (Fraenkel et al. 1973: 142). The concept of a hyper-class should, however, not be confounded with that of a hyperset (i.e., a non-well-founded set) in non-well-founded systems of set theory (cf. Barwise and Moss 1996: 6). An appropriate set-theoretic system on which one can model a formal reconstruction of Navya-Nyāya logic should endorse the existence of hyper-classes, hyper-hyperclasses, etc. In Fraenkel et al. 1973 (cf. 142f), the authors design such a system by combining the set theories of QM and ZF. The resulting system ST2 can serve as a set-theoretic prototype of the Navya-Nyāya logic of property and location if one additionally takes into account urelements. In ST2 with urelements Sx (read: “x is a set”) functions as a primitive monadic predicate. The system comprises the following axioms: (a) A sethood axiom: Every member of a set is a set or urelement. (b) All the axioms of QM with urelements (with due regard to the abovementioned notational convention for variables). (c) The axioms of ZF with all variables replaced by upper-case variables. This is a two-tier set theory with sets and urelements in the bottom tier and classes in the upper tier. (c) warrants the existence of hyper-classes, hyper-hyper-classes etc. in ST2. Due to the ZF-axiom of pairing with upper-case variables one can, e.g., pair the universal class V with itself in order to obtain the hyper-class {V}. However, a hyper-class which contains all proper classes does not exist in ST2. On account of the antinomy of the sets of all sets (cf. Barwise and Moss 1996: 16), there is no universal set in ZF. Hence, there is also no way to obtain a corresponding universal hyper-class by means of the axioms in (c). Proper classes can be elements in ST2, but they should still be distinguishable from sets. This is achieved by adding (a), which ensures that proper classes cannot be elements of sets. A property-theoretic counterpart of ST2 with urelements can be obtained by transforming (a), (b), and (c) into the corresponding property versions. Only the

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variants of the axiom of extensionality in (b) and (c) have to be exlcuded, because there is already a criterion for the identification of intensional abstracts in T1, namely A8. The extension of T1 which includes the abovementioned axioms of a property adaptation of (a), (b), and (c) (exluding the variants of the axiom of extensionality in (b) and (c)) will be called “T1+” hereafter. In T1+ the notion of a set-like property is taken as primitive. Unlike set-like properties and urelements properly class-like properties cannot be loci of set-like properties in the sense of the Δ-relation. So, if xΔy, then for some set-like property x or urelement a, x = x or a = x. Although in T1+ properly class-like properties are admitted as members of the domain of the Δ-relation, Ex (“x is element-like”) in (C) is still to be understood in a QM-like sense, i.e., in the sense that x is an urelement or a set-like property. Thus, (C) warrants only the existence of class-like properties residing in urelements and set-like properties. Nevertheless, according to the property versions of the ZF-axioms with uppercase variables in (c) of ST2 terms of the form [A(X)]X as denotations of hyper-classlike properties, i.e., of properties residing in properly class-like properties, are admitted in T1+. If, e.g., A(X) is of the form XΔY ^ B(X), the property version of the ZF-axiom of separation in (c) warrants the existence of the hyper-class-like property [A(X)]X. If A(X) is the formula X = [N(x)]x _ X = [N(x)]x (where [N(x)]x stands for “nameability”), the property version of the axiom of pairing in (c) warrants the existence of the hyper-class-like property [X = [N(x)]x]X (“nameabilityness”). However, a hyper-class-like property which resides in all properly class-like properties does not exist in T1+. Since there is no universal set in ZF, there is also no way to obtain a corresponding universal hyper-class-like property by means of the property versions of the axioms in (c). A term of the form [A(x)]x might also be regarded as a denotation of a hyperclass-like property, since the variable x refers to urelements and all kinds of class-like individuals alike. This is, however, inadmissible if by rewriting A(x) with upper-case variables one obtains a formula which is not equivalent to any of the formulas whose property abstracts occur in the property version of the axioms in (c). In such a case, [A(x)]x has to be regarded as an abbreviation of [Ex ^ A(x)]x. [x = x]x, e.g., should be taken to denote a universal property of all urelements and all properties, excluding the properly class-like ones, so that a property version of the antinomy of the set of all sets will not strike here. It might be tempting to choose NFU (Quine’s New Foundation with urelements) instead of ST2 as a model for a formal reconstruction of Navya-Nyāya logic, because NFU has only two axioms and one type of variables. The NFU axiom of extensionality is obsolete in the present context. So, one might just reformulate the NFU comprehension axiom: ∃y8x(x  y $ A(x)), where y is not free in A and A is stratified, i.e., it is possible to index the variables in A such that  occurs only between variables with consecutive indices. (Cf. Fraenkel et al. 1973: 161f; Quine 1973: 210f)

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Since x = x is stratified, one can prove in NFU that V  V. Similarly, one can prove the self-residence of a universal property by means of the following property adaptation of the NFU comprehension axiom: 8x(xΔ[A( y)]y $ A(x)), where A(x) is “stratified” with respect to the Δ-relation and x is free for y in A and vice versa. Some Navya-Naiyāyikas regard the self-residence (ātmās´raya) of a property as a kind of absurdity. If one wants to model their intuitions about properties, an NFU-style extension of T1 is not an appropriate framework. It is, however, congenial with respect to Navya-Naiyāyikas who affirm the self-residence of certain properties. Since nameability, e.g., is a property which is also nameable, it would make sense to say that nameability resides in itself.

Well-Foundedness Versus Non-Well-Foundedness The Position of Varadarāja and Maheśa Chandra Some late Naiyāyikas and Navya-Naiyāyikas such as, e.g., Varadarāja and Maheśa Chandra would call for a well-foundedness condition on properties, similar to the axiom of regularity (or “foundation”) in set theory. In standard systems of set theory, this axiom excludes the existence of sets which are elements of themselves or – generally speaking – the existence of (potentially looping) infinite sequences (an)n  ℕ such that ai+1  ai for all i  ℕ: 8x(∃y(y  x) ! ∃y(y  x ^ :∃z(z  x ^ z  y))) (“Every non-empty set x contains an element y which is disjoint from x.”) Theorem (in ZF) :∃(an)n  ℕ8i  ℕ(ai+1  ai) Proof If an infinite series (an)n  ℕ such that ai+1  ai for all i  ℕ did exist, then there would be a set x = {a1, a2, a3,. . .} and 8ai(ai  x ! ai+1  ai \ x), i.e., no element of x would be disjoint from x. But according to the axiom of regularity there is no such set x. ▪ A property analog of the axiom of foundation is contained in the early Nyāya work Tārkikarakṣā (cf. TR) by Varadarāja: ātmās´rayas tathānyonyasaṃs´rayas´ cakrakās´rayaḥ/ anavasthety amī tarkāḥ svarūpāsiddhihetavaḥ// (TR: 234f) “Self-dependence, mutual dependence, circularity, the regressus in infinitum, these inferential blockers are the causes of the [probans’s] being essentially unestablished.”

In this verse, Varadarāja expresses his misgivings about looping chains of dependence relations involving one member (ātmās´raya – “self-dependence”), two

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members (anyonyasaṃs´raya – “mutual dependence”), or more than two members (cakrakās´raya – “circularity,” also: “arguing in a circular way”). Moreover, he refers to the regressus in infinitum (anavasthā – literally: “ungroundedness”) as a further type of “inferential blocker” (tarka). In contrast to the abovementioned “hypothetical reasoning” (cf. section “The Nyāya Theory of Inference”), which is also named tarka, an inferential blocker is a so-called “unfavorable tarka” (pratikūlatarka – cf. Phillips 2002: 94). An inference is blocked if the presence of the probans in the inferential subject entails an impossible circular chain of qualification or causation or a regressus in infinitum. In such a case the probans is said to be “essentially unestablished” (svarūpāsiddha), i.e., its presence in the inferential subject is in doubt (cf. Matilal 21990: 51). While Varadarāja is talking here about loops and infinitely descending chains in a more general sense, an opponent in the anonymous Navya-Nyāya treatise Upādhidarpaṇa (cf. UD) cites this verse (cf. UD: fol. 2a, 10) to support his view that in particular the existence of looping or infinitely descending chains of residence relations should be excluded. Similarly, Maheśa Chandra argues that the residence relation is irreflexive and asymmetric. So, there can be no loops involving one or two members: saṃbandho yadyapy ubhayaniṣṭho yathā kuṇḍabadarayoḥ saṃbandhaḥ kuṇḍe badare cāsti tathāpi kenacit saṃbandhena kas´cid eva kutracid eva tiṣṭhati. yathā saṃyogena saṃbandhena kuṇḍa eva badaraṃ tiṣṭhati na tu badare kuṇḍam. evaṃ bhūtala eva ghaṭo vartate na tu ghaṭe bhūtalam iti. atra kāraṇam etat. saṃbandhasyaikaṃ pratiyogi. aparaṃ cānuyogi bhavati. yasya saṃbandhasya yat pratiyogi bhavati tena saṃbandhena tad eva tiṣṭhati. yac ca yasya saṃbandhasyānuyogi bhavati tena saṃbandhena tatra pratiyogi tiṣṭhati. yathā kuṇḍabadarayoḥ saṃyoge badaraṃ pratiyogi kuṇḍaṃ cānuyogīti kuṇḍe badaraṃ vartate. dharmadharmiṇoḥ saṃbandhasya dharmaḥ pratiyogī dharmi cānuyogi bhavati. ata eva dharma eva dharmiṇi vartate na tu dharmi dharme. (BN: 12, 19f = Guhe 2014: 72) – “Although a relation is situated in both [things], such as the relation between pot and dried ginger in a pot and in dried ginger, something nevertheless depends on something via a certain relation. Dried ginger, e.g., is in the pot on account of the relation contact, but the pot is not in dried ginger. In the same way the pot occurs on the ground, but not the ground on the pot. Here is this the reason: One is the adjunct of the relation and the other is the subjunct. That is the adjunct of that relation which depends [on something] via that relation. And that is the subjunct of that relation on which the adjunct depends via that relation. In the case of the contact of the pot and the dried ginger, e.g., the dried ginger is in the pot, because the dried ginger is the adjunct and the pot is the subjunct. In the case of property and property bearer the property is the adjunct of the relation and the property bearer is the subjunct. Therefore the property occurs on the property bearer, but not the property bearer on the property.” The existence of self-resident properties is explicitly denied in the following passage: ata eva pratiyogyanuyoginor abhede ’pi ghaṭe ghaṭo nāstī ti saṃsargābhāvapratītiḥ (BN: 17, 16f = Guhe 2014: 86) – “Therefore, when there is no difference between adjunct and subjunct, there is the cognition of the relational absence ‘A pot is not in a (i.e., in the same) pot’.” This statement should be understood in a more general sense, since the word “pot” (ghaṭa) is used in

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Navya-Nyāya as a kind of dummy singular term and in some contexts it can have the function of a variable (cf. Matilal 1968: 23). In order to exclude the existence of infinitely descending or looping chains of dependence relations for properties (in accordance with Varadarāja’s and Maheśa Chandra’s misgivings about such phenomena) one can postulate the following property-theoretic version of the QM-axiom of regularity (cf. Mendelson 41997: 302): (R) 8X(∃y(yΔX) ! ∃y(yΔX ^ 8z(zΔX ! :zΔy))) Theorem :∃(an)n  ℕ8i  ℕ(ai+1Δai) Proof Assume indirectly: There is a sequence (an)n  ℕ such that 8i  ℕ(ai+1Δai). Let X be a property whose loci are the members of (an)n  ℕ, i.e., X = [∃n  ℕ(x = an)]x. (For the definiton of natural numbers as properties cf. Bealer 1982: 121 and section “A Quasi-Fregean Account of the Reference of Number Words” of the ▶ Chap. 20, “The Logic of Late Nyāya: Problems and Issues.”) Then 8ai(aiΔX ! ai+1Δai ^ ai+1ΔX). So, 8ai(aiΔX ! ∃z(zΔai ^ zΔX)) – in contradiction to (R). ▪ The Position of the UD The author of the UD does not approve of any such restriction (cf. Guhe 2015). He affirms the existence of non-well-founded properties “by assenting to the occurrence of something in itself” (svasmin svavṛttitvābhyupagamena – UD: fol. 4a, 4f). The axiom of regularity, which captures the idea of well-foundedness in standard systems of set theory, is relatively independent, i.e., it is possible to construct models for the other axioms of these systems in which it fails. So, the assumption of such an axiom is optional from a logical point of view. Some set theorists just leave it out: “It therefore seems prudent (. . .), not to assume the axiom of foundation. In practice that is no great concession, however, since we shall focus exclusively on grounded collections (i.e., sets) in everything that we do from now on. Readers who believe there are ungrounded collections as well will thus find nothing here with which they can reasonably disagree: the most they are entitled to is a mounting sense of frustration that I am silent about them” (Potter 2004: 53). By contrast, set theorists like Peter Aczel replace the axiom of regularity by an anti-foundation axiom, which “expresses, in a particular way, that every possible non-well-founded set exists” (Aczel 1988: xviii). It is especially designed to provide solutions to certain equations which cannot be solved in standard systems of set theory. Thus, in Aczel’s non-well-founded set theory the reflexive set Ω is the solution of the equation x = {x}. It would surely be hazardous to incorporate a similar anti-foundation axiom referring to properties into a formal reconstruction of the logical framework of the UD. In Navya-Nyāya, there is no solution to the equation x = xtva. However, in order to model non-well-founded intuitions about properties, such as in the UD, one might want to adopt the idea of a stratified comprehension embodied in the NFU comprehension axiom. Stratified comprehension can be used to prove the selfresidence of the property which according to the UD defines a so-called “associate

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condition” (upādhi). Although both are referred to as upādhis in Sanskrit, the concept of “associate condition” has to be distinguished from the concept of “imposed property,” which was introduced in section “Epistemological and Ontological Presuppositions.” The function of an associate condition is to refute or undercut putative inferences. The stock example is the assumed inference of smoke (= the probandum) from fire (= the probans). In this case (which is just the reversal of the correct inference of fire from smoke) wet fuel serves as an associate condition, because it was supposed to be a necessary precondition for the production of smoke. With regard to a locus like molten metal, where the associate condition is missing, smoke cannot be inferred from fire. It is important to note here that (i) an associate condition has to be absent somewhere in order to function as an undercutter. According to the UD an associate condition can also have the function of a corrector of an assumed inference in the sense that in an inferential subject exhibiting the associate condition together with the probans the probandum can be secured. Concerning the aforementioned example we can say that smoke is inferable from fire wherever the latter occurs in combination with wet fuel. So, the author of the UD also calls the associate condition an “enabling [condition]” (prayojaka) promoting an inferential knowledge. What is important here, is that (ii) the associate condition must be present somewhere in order to function as a corrector. On account of (i) and (ii), the author of the UD defines an upādhi as something which is absent somewhere and also present somewhere. This defining characteristic (“being present somewhere and absent somewhere”) applies to urelements and properties alike. However, if an urelement x (such as wet fuel) functions as an associate condition with respect to an assumed inference, we can always replace x by the equi-locatable property x-vattva (such as the property “being a locus of wet fuel”), which may equally well serve as an associate condition with respect to the same inference. So, it would be sufficient to take into account only class-like properties as associate conditions and then one can use the Δ-relation in order to state the defining characteristic of an associate condition as a property u¼df ½∃yðyΔxÞ ^ ∃yðy ΔxÞx. It resides in locatable properties which are not universal (excluding, e.g., the property “nameability”). Hence, ∃yðyΔuÞ ^ ∃yðy ΔuÞ. Since this formula is stratified with respect to the Δ-relation, the author of the UD can reasonably claim that uΔu.

Summary Points • In their attempt to develop a kind of “logic of property and location” (Matilal), Navya-Naiyāyikas came up with interesting new ideas concerning the interaction between logic and ontology. • Some logical inquiries about properties in Navya-Nyāya are related to similar problems in set theory, although the Navya-Nyāya concept of property should not be confounded with the Western concept of set.

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• In the present exposition of Navya-Nyāya logic, the author outlines a propertytheoretic framework for a formal reconstruction. • Moreover, he discusses the methods of formalization suggested by several other interpreters who have tried to render the technical language of the NavyaNaiyāyikas by means of some kind of symbolic notation.

References Aczel, P. 1988. Non-well-founded sets. Vol. 14, CSLI lecture notes. Stanford: Stanford University, Center for the Study of Language and Information. Barwise, J., and L. Moss. 1996. Vicious circles. On the mathematics of non-wellfounded phenomena. Vol. 60, CSLI lecture notes. Stanford: Stanford University, Center for the Study of Language and Information. Bealer, G. 1982. Quality and concept. Oxford: Clarendon Press. Bhattacharya, D. 1958. History of Navya-Nyāya in Mithilā. Vol. 2, Mithila Institute series. Darbhanga: Mithila Institute of Post-Graduate Studies and Research in Sanskrit Learning. BN, see Maheśa Chandra 1891. BN. 1891. Brief notes on the modern Nyāya system of philosophy and its technical terms. By Mahāmahopādhyay Maheśa Chandra Nyāyaratna. Calcutta: Hare Press. Bocheński, J.M. 41978. Formale Logik. Orbis Academicus III, 2. Freiburg/Munich: Karl Alber. Fraenkel, A., Y. Bar-Hillel, and A. Levy. 1973. Foundations of set theory. Vol. 67, Studies in logic and foundations of mathematics. Amsterdam/London/New York/Oxford/Paris/Shannon/Tokyo: North Holland. Ganeri, J. 2008. Towards a formal regimentation of the Navya-Nyāya technical language I. In Logic, Navya-Nyāya & applications. Vol. 15, Studies in logic, ed. M.K. Chakraborti et al., 105–121. London: College Publications. Ganeri, J. 2011. The lost age of reason. Philosophy in early modern India 1450–1700. London: Oxford University Press. Goekoop, C. 1967. The logic of invariable concomitance in the Tattvacintāmaṇi. Dordrecht: D. Reidel Publishing Company. Guhe, E. 1999. Die Lehre von der zusätzlichen Bestimmung (upādhi) im Upādhidarpaṇa. Unpublished Dissertation, Vienna. Guhe, E. 2000. Intensionale Aspekte der indischen Logik. Berliner Indologische Studien 13/14 (2000): 105–116. Guhe, E. 2008. George Bealer’s property theories and their relevance to the study of Navya-Nyāya logic. In Logic, Navya-Nyāya & applications. Vol. 15, Studies in logic, ed. M.K. Chakraborti et al., 139–153. London: College Publications. Guhe, E. 2014. Mahes´a Chandra Nyāyaratna’s “Brief notes on the modern Nyāya system of philosophy and its technical terms” – introduction, edition and annotated translation. Shanghai: Fudan University Press. Guhe, E. 2015. The problem of foundation in early Nyāya and in Navya-Nyāya. History and Philosophy of Logic 36/2 (2015): 97–113. Halbfass, W. 1992. On being and what there is. Classical Vais´eṣika and the history of Indian ontology. Albany: State University of New York Press. Hughes, G.E., and M.J. Cresswell. 1996. A new introduction to modal logic. London/New York: Routledge. Ingalls, D.H.H. 1951. Materials for the study of Navya-Nyāya logic. Vol. 40, Harvard oriental series. Cambridge, MA: Harvard University Press. Jhalakīkar, B. 1928. Nyāyakos´a or dictionary of technical terms of Indian philosophy. Revised by Vasudev Shastri Abhyankar. Poona: Bhandarkar Oriental Research Institute.

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Keśava Miśra. 1937. Tarkabhāṣā of Kes´ava Mis´ra. With the commentary Tarkabhāṣāprakās´ikā of Cinnaṃbhaṭṭa. Edited by D.R. Bhandarkar and K. Sāhityabhūṣaṇa. Bombay Sanskrit and Prakrit Series. Poona: Bhandarkar Oriental Research Institute. Maheśa Chandra. 1891. Brief notes on the modern Nyāya system of philosophy and its technical terms. Calcutta: Hare Press. Matilal, B.K. 1968. The Navya-Nyāya doctrine of negation. Vol. 46, Harvard oriental series. Cambridge, MA: Harvard University Press. Matilal, B.K. 21990. Logic, language and reality. Delhi: Motilal Banarsidass. Matilal, B.K. 1998. The character of logic in India. Albany: State University of New York Press. Mendelson, E. 41997. Introduction to mathematical logic. London: Chapman & Hall. NKoś, see Jhalakīkar, B. 1928. NKoś. 1928. Nyāyakos´a or dictionary of technical terms of Indian philosophy. By Bhīmācārya Jhalakīkar, revised by Vasudev Shastri Abhyankar. Poona: Bhandarkar Oriental Research Institute. Oetke, C. 1994. Vier Studien zum altindischen Syllogismus. Reinbek: Dr. Inge Wezler Verlag für orientalistische Fachpublikationen. Oetke, C. 2009. Some aspects of Vyāpti & Upādhi in the Nyāyalī lāvatī . Vol. 19, Stockholm oriental studies. Stockholm: Stockholm University Press. Phillips, S. 2002. Gaṅges´a on the Upādhi, the “inferential undercutting condition”. Introduction, translation and explanation (with N.S. Ramanuja Tatacharya). Delhi: Indian Council of Philosophical Research. Potter, M. 2004. Set theory and its philosophy. Oxford: Oxford University Press. Quine, W. v. O. 1973. Mengenlehre und ihre Logik. Transl. by Anneliese Oberschelp of Set theory and its logic. Braunschweig: Friedr. Vieweg + Sohn. TBh, see Keśava Miśra 1937. TBh. 1937. Tarkabhāṣā of Kes´ava Mis´ra. With the commentary Tarkabhāṣāprakās´ikā of Cinnaṃbhaṭṭa. Ed. by D. R. Bhandarkar and K. Sāhityabhūṣaṇa. Bombay Sanskrit and Prakrit series. Poona: Bhandarkar Oriental Research Institute. TR, see Varadarāja 1903. TR. 1903. tārkikarakṣā. s´rī madācāryavaradarājaviracitā. tatkṛtasārasaṅgrahābhidhavyākhyāsahitā. mahopādhyāyakolācas´rī mallināthasūriviracitayā niṣkaṇṭakākhyayā vyākhyayā jn˜ ānapūrṇanirmitayā laghudī pikākhyayā ṭī kayā samanvitā. Varanasi: Pandit Reprint. UD, see Upādhidarpaṇa. UD: Upādhidarpaṇa. BORI-Ms. No. 6. 1898–99. van Bijlert, V.A. 1989. Epistemology and spiritual authority. The development of epistemology and logic in the old Nyāya and the Buddhist school of epistemology with an annotated translation of Dharmakī rti’s Pramāṇavārttika II (Pramāṇasiddhi), VV. 1–7. Vol. 20, Wiener Studien zur Tibetologie und Buddhismuskunde. Wien: Arbeitskreis für Tibetische und Buddhistische Studien/Universität Wien. Varadarāja. 1903. tārkikarakṣā. s´rī madācāryavaradarājaviracitā. tatkṛtasārasaṅgrahābhidhavyākhyāsahitā. mahopādhyāyakolācas´rī mallināthasūriviracitayā niṣkaṇṭakākhyayā vyākhyayā jn˜ ānapūrṇanirmitayā laghudī pikākhyayā ṭī kayā samanvitā. Varanasi: Pandit Reprint. Wada, T. 2007. The analytical method of Navya-Nyāya. Vol. XIV, Gonda indological studies. Groningen: Egbert Forsten.

A Few Historical Glimpses into the Interplay Between Algebra and Logic and Investigations into Gautama Algebras

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Dedicated to the memory of The Founders of Indian Logic: MEDHATITHI GAUTAMA (cerca de 550 BCE) The author of Anwikshiki and Nyaya-Shastra and AKSHAPADA GAUTAMA (cerca de 150 CE) The author of Nyaya-Sutra

Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART I: A Few Historical Glimpses into the Interplay Between Logic and Algebra . . . . . . . . PART I.A: Symbolic Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART I.B: Algebra Becomes Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART I.C: The Evolution of Algebraic Logic, Mathematical Logic and Non-classical Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART I.D: Modern Algebraic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . PART II: Investigations into Gautama Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Introduction and Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A New Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Variety of Gautama Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A Logic for Gautama Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Epilogue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

This chapter consists of two parts. PART I presents a few historical glimpses into the fascinating interplay between algebra and logic that essentially started in the 2010 Mathematics Subject Classification: Primary: 03G10, 03G27, 03B50, 03G25, 06D20, 06D15; Secondary: 08B26, 08B15, 06D30 H. P. Sankappanavar (*) Department of Mathematics, State University of New York, New Paltz, NY, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_54

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middle of the nineteenth century with Boole’s work. It is mostly non-technical and highlights a few significant examples of this interplay. PART I includes a brief discussion of Boole’s algebra of logic, Frege’s mathematical logic, and the eventual meeting of Frege’s tradition with Boole’s tradition in Tarski’s papers (in the early 1930s). It also discusses Rasiowa’s implicative logics and Blok and Pigozzi’s algebraizable logics, and provides examples – some known and some new. It concludes with a discussion of the impact, on logic, of algebra arising from rough set theory. PART II further illustrates the interplay between algebra and logic. More precisely, we define a new equational class of algebras called “Gautama algebras.” They are named in honor and memory of the two founders of Indian Logic – Medhatithi Gautama and Akshapada Gautama. The variety  of Gautama algebras is a common generalization of the varieties of regular double Stone algebras and regular Kleene Stone algebras, both of which grew out of Boolean algebras, which, in turn, arose from Boole’s algebra of logic. We provide an explicit description of subdirectly irreducible Gautama algebras and the lattice of subvarieties of the variety of Gautama algebras. We also introduce another variety  of Gautama Heyting algebras and show that it is term-equivalent to the variety of Gautama algebras. We then define new propositional logics called GAUTAMA, RDBLSt, and RKLSt and show them to be algebraizable (in the sense of Blok and Pigozzi), with the varieties of Gautama algebras, regular double Stone algebras and regular Kleene Stone algebras, respectively, as their equivalent algebraic semantics. The chapter concludes with some open problems for further research and also with an extensive bibliography. Keywords

Implicative logic · Equivalent algebraic semantics · Algebraizable logic · Regular double Stone algebra · Regular Kleene Stone algebra · Gautama algebra · Subdirectly irreducible algebra · Simple algebra · Dually hemimorphic semiHeyting logic · Dually quasi-De Morgan semi-Heyting logic · Dually pseudocomplemented semi-Heyting logic · Logic GAUTAMA · Logic RDBLSt · Logic RKLSt

Introduction This chapter consists of two parts. PART I presents a few historical glimpses into the fascinating interplay between algebra and logic that essentially started in the middle of the nineteenth century with Boole’s work. We have also highlighted the names of a few mathematicians and logicians who made those discoveries or were highly influential in the propagation of the new ideas presented here. Due to the limitation of space, the choice of names has been, of necessity, very subjective. In the nineteenth century, after a long period of stagnation, two major discoveries took place in western logic that freed it from the limitations of Aristotalian logic:

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Boole’s Algebra of Logic in 1847 and Frege’s “Begriffsschrift” in 1879. It is remarkable that algebra already played a significant role in the very creation of Boole’s logic. Following these two seminal dicoveries, the westeren logic developed along two paths: “algebraic logic,” following Boole, and “mathematical logic” (or “formal logic”), following Frege. In the first two decades of the twentieth century, several non-classical logics were discovered. In the early 1930s, Tarski developed the so-called “Lindenbaum-Tarski method,” which brought algebra and logic together again. Ever since Tarski’s discoveries, a fruitful and symbiotic relationship has developed between algebra and logic. The successful adaptation of this method to several different logics over decades led Rasiowa, in 1974, to introduce the notion of “implicative logics.” Influenced by Rasiowa’s work, Blok and Pigozzi, in 1989, defined “algebraizable logics” and their “equivalent algebraic semantics,” thus heralding a new area of algebraic logic called “Abstract Algebraic Logic.” It is our view that the roots of all the discoveries where algebra and logic have mutual influence can be traced back to the systematic introduction of symbols into algebra and logic, which occurred in the middle of the sixteenth century. We, therefore, start our presentation of PART I from that period. It would be helpful, but not necessary, for the reader to have some familiarity with universal algebra and mathematical logic. We would also like to mention here that important areas of logic, like first-order logic and model theory, have not been included here due to the limitation of space; however, an extensive bibliography is included. PART II of this chapter1 further illustrates the interplay between algebra and logic. More precisely, we define a new equational class of algebras called “Gautama algebras.” They are named in honor and memory of the two founders of Indian Logic – Medhatithi Gautama and Akshapada Gautama. The variety of Gautama algebras is a common generalization of the varieties of regular double Stone algebras and regular Kleene Stone algebras, both of which grew out of Boolean algebras, which, in turn, arose from Boole’s algebra of logic. We provide an explicit description of subdirectly irreducible Gautama algebras and the lattice of subvarieties of the variety of Gautama algebras. We also introduce a related variety  of “Gautama Heyting algebras” and show that it is term-equivalent to the variety of Gautama algebras. We then turn to the problem of logicizing the variety  of Gautama algebras, via the variety GH of Gautama Heyting algebras. In fact, we define new propositional logics called GAUTAMA, RDBLSt, and RKLSt and show them to be algebraizable (in the sense of Blok and Pigozzi), with the varieties of Gautama algebras, regular double Stone algebras, and regular Kleene Stone algebras, respectively, as their equivalent algebraic semantics. PART II concludes with some open problems. PART II is directed toward advanced graduate students and researchers in lattice theory, algebraic logic, universal algebra, and computer science. The chapter

1

PART I is an expanded version of my lectures given at the silver jubilee celebration of Calcutta Logic Circle at Kolkata, India, on October 13, 2013 and at IIT, Guwahati, India, on April 28, 2017. PART II is an expanded version of my talk at IIT, Kanpur, India, on April 9, 2021.

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concludes with some open problems for further research and also with an extensive bibliography. I would like to express my appreciation to Professor Chakraborty for inviting me to contribute this chapter to the Handbook of Logical Thought in India. I am also grateful to Professor J.M. Cornejo for reading an earlier version of this chapter and making some valuable suggestions. I wish to take this opportunity to express my love and appreciation to my son Sanjay and to my daughter Geeta for their assistance in proofreading earlier versions of this chapter from the linguistic point of view, and to my wife Nalinaxi for her love and support, through thick and thin, for over half a century.

PART I: A Few Historical Glimpses into the Interplay Between Logic and Algebra PART I begins with the Introduction of symbols into mathematics and logic. We believe that it is this symbolization which set the stage for later discoveries. Historically speaking, algebraic notation had three stages of development. The first stage, Rhetorical Algebra, used only words but no symbols. The second stage, Syncopated Algebra, still used only words (no symbols), but incorporated some abbreviations. The development of algebraic notation is currently in its third stage, Symbolic Algebra, in which all algebraic expressions are completely symbolic. During the rhetorical and syncopated periods, even without the advantage of having symbols, Indian, Arabic and Chinese algebraists, writing in prose and in verses, have made significant discoveries in algebra and logic. To honor their contributions and to keep their memory alive, we mention the names of a few of those algebraists and logicians below: Algebraists: Aryabhatta, Varahamihira, Brahmagupta, Bhaskara I, Mahavira, Sridhar, Narayan Pandit, Bhaskara II, Madhava, Virasena, Nilakantha Somayaji, Al-Khwarizmi, al-Mahani, Al-Karaji, Omar Khayyám, Sharaf al-Din al-Tusi, Jamshīd al-Kāshī, Wang Xiaotong, Jia Xian, and Chu Shih-chieh. Logicians: Medhatithi Gautama, Aksapada Gautama, Vatsyayana, Vasubandhu, Divkara, Nagarjuna, Dignaga, Dharmakirti, Udayanacharya, Gangesa, Raghunatha, Gadadhara, Bhartrihari, Prashastapada, Shridhara, Amalananda, Akalanka, Alfarabi, Avicenna (Ibn Sina), Averroes, Ghazâlî, Tûsî, Râzî, Khûnajî, Hui Shi, Gongsun Long, Mozi, Xunzi.

PART I.A: Symbolic Mathematics The Begining of the Systematic Use of Symbols in Mathematics Before the sixteenth century, the language of mathematics was rhetorical with sporadic use of abbreviations.

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Francois Viète (1540–1603) The landmark advance in the use of symbols in mathematics was made by Viète. He used letters to represent known constants (parameters), in addition to the already existing use of letters for variables. This advance freed algebraists from the (restricted) consideration of a single equation at a time and allowed the study of the “form” or the “scheme” under which (infinitely) many equations would fall, thus facilitating more and more generality. It opened the possibility for studying the relationship between the coefficients of an equation and the roots of that equation. Symbolic algebra is the form employed by Viéte in his work “In artem analyticem isagoge” (i.e., “Introduction to the Analytic Art”). Although German mathematicians introduced (+) as the short-hand symbol for addition and the (
) for subtraction, these symbols were not commonly used. Their use became more common after Viéte began to use them. His most well-known contribution to the growth of algebra was his consistent use of the letters of the alphabet to represent both the constants and the variables in all equations. Although letters were used before Viéte to represent quantities occurring in an equation, there was no way of identifying quantities assumed to be known (or given) and those (unknown) quantities that were yet to be found. Viéte changed that situation; indeed, he designated consonants for known quantities and vowels for unknown quantities. Viéte’s approach enabled mathematicians to solve equations in a more general form, rather than to work out each specific equation on an individual basis. This was a crucial step in the development of today’s algebra; however, Viéte’s convention did not survive much past Viéte’s death. Viète’s algebra was still not completely symbolic. References: (Burton 1985; Kneale and Kneale 1962; Varadarajan 1998; Viète 1983; van der Waerden 1985; Mazur 2014) The Use of Symbols at Full Maturity and Accelerated Growth in Mathematics René Descartes (1596–1650) Numeric algebra became fully symbolic with the publication of La Gèomètrie by Descartes in 1637 on analytic geometry. Through his cartesian coordinate system, he gave a significant application of algebra to geometry which allowed the translation of geometric problems into algebraic problems and vice versa. He made extensive use of symbols in his book. Reversing Viéte’s convention, Descarte started using letters at the beginning of the alphabet for given quantities (i.e., constants) and the letters at the end of the alphabet for unknown quantities – a practice that is still in use. During the seventeenth century the deliberate use of symbolism – as opposed to incidental and accidental use of symbols – had taken hold in mathematics. The extensive use of symbols paved the way for mathematics to grow faster. The realization of the incredible expressive power of symbolic mathematics became quite wide-spread. In essence, mathematics had become fully symbolic by the end of the seventeenth century.

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References: (Descartes 1637, 1897)

The Systematic Use of Symbols in Logic Recognizing the power of symbols used by mathematicians, logicians also became increasingly interested in using symbols in logic as well (i.e., in symbolizing logic). They also started using letters more systematically to represent propositions, concepts, classes, properties, relations, and so on. Inspired by the triumphs achieved in mathematics after it had turned to the systematic use of special symbols, logicians hoped that through the systematic application of symbols and abbreviations, great progress could be made in logic as well. Perhaps, this is the first notable influence of mathematics on logic. Indeed, logic also began to be symbolic. Gottfried Wilhelm Leibniz (1646–1716) Leibniz proposed new ideas related to logic that were ahead of his time. Leibniz dreamed of creating: 1. A universally characteristic language (lingua characteristica universalis) by displaying the most basic concepts in symbols, 2. A calculus of reasoning (calculus ratiocinator) designed to create relations among scientific concepts. With these tools, Leibniz believed reasoning could then take place mechanically or algorithmically and thus would not be subject to individual’s mistakes and failures of ingenuity. Furthermore, such derivations could even be checked by others or performed by machines. Gottlob Frege, a prominent nineteenth century logician, was influenced by Leibniz and tried to implement some of his ideas. References: (Parkinson 1966; Loemker 1969)

PART I.B: Algebra Becomes Abstract In the early decades of the nineteenth century, algebra was no longer just numerical algebra. It became more and more abstract with the discoveries of different kinds of algebras. Logic, too, got revived.

The Beginings of Modern Algebra By the beginning of the nineteenth century, questions as to whether “negative numbers” and “complex numbers” were really numbers still lingered in the minds of mathematicians in the Western world. Questions like “what are numbers?” were becoming of serious concern to mathematicians. The discoveries made during the seventeenth century, such as derivatives, integrals, infinite series, were crying for further clarity and a firm foundation. In particular, the logical foundation for numeric

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algebra was still lacking. Numerical algebra was subject to criticism by mathematicians like Bolzano and others. Bernard Bolzano (1781–1848) Bolzano was one of the first mathematicians to reject the traditional geometric approach to mathematics and to insist on rigor in mathematics. He suggested that the foundation of mathematics (including arithmetic) ought to be based solely on logical grounds, thus raising the possibility of a significant interaction of mathematics with logic (MacTutor). The work on the foundations of algebra began in the early 1800s by mathematicians, like Whately, Peacock, Gregory, De Morgan, and Boole, in England. Richard Whately (1787–1863) During several decades prior to Whately’s book “Elements of Logic,” published in 1826, there seemed to be little or no interest in the study of logic in England. Whately is credited with bringing about a resurgence of interest in logic. The clarity of presentation of deductive logic in his book may have been responsible to attract others, like De Morgan and Boole, to logic and thereby to the foundational questions of the algebra. George Peacock (1791–1858) The first major attempt among English mathematicians to clear up the foundation problems of algebra was Peacock’s in his “Treatise on Algebra” in 1830. (Its second edition appeared as two volumes in 1842/1845). He divided the algebra into two parts: The first part was called arithmetical algebra which dealt with the algebra of positive numbers only, in that subtraction resulting in negative numbers was not allowed. The second part was called symbolical algebra, which was governed only by laws and was therefore free from any specific interpretation. For example, symbolical algebra allowed formal manipulation of symbols according to a prescribed set of laws, thus allowing unrestricted use of subtraction. According to Burris (2015), Peacock was the first to separate (what are now called) the syntactic and the semantic aspects of algebra. Peacock may have been the originator of axiomatic thinking in algebra. Duncan Gregory (1813–1844) Gergory’s main contribution was his conception of algebra. He defined algebra as the study of the operations subject to certain laws. This is indeed the beginning of abstract algebra. His work in this area is described in the paper: “On the real nature of symbolical algebra” published in the Transactions of the Royal Society of Edinburgh (see (Gregory 1840)). In this work, based on Peacock’s findings, Gregory further developed the foundations of algebra. Gregory published several papers on the method of separation of symbols into operators and objects. Gregory’s work had a significant impact on Boole’s views on logic (see “Boole” later in the section “The Birth of Algebraic Logic”).

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References: (Bolzano 1978; Ewald 1996; Whately 1827; Peacock 1830; Gregory 1840) Algebra Begins to Be Non-numeric Since the beginning of the nineteenth century, following the lead of Peacock and Gregory, mathematicians, especially algebraists, began to accept “non-numeric” objects, such as permutations, on which “interesting” operations like composition could be carried out as legitimate mathematical objects. A few highlights of this kind are mentioned below. It is widely believed that, in 1814, Argand was the first to represent complex numbers as points on the plane, that is, as ordered pairs of real numbers. However, Caspar Wessel had already published a paper in 1799, providing a geometric representation of complex numbers and a definition of vector addition. Wessel’s paper was not known till Sopus Lie republished it in 1897. In 1831, Gauss also discovered this representation of complex numbers. In 1831, Galois made the fundamental observation that the solution of an equation was related to the structure of the group of permutations of the roots of the equation. Thus, Galois came to be known as the founder of group theory. In 1832, Galois discovered what are now called “normal subgroups.” Galois’s work was not known (and, in fact, was first rejected) until Liouville published Galois’s papers in 1846, which led to the new area of algebra known as “Galois Theory.” In 1833, W.R. Hamilton demonstrated that complex numbers could be expressed as a “formal algebra” with operations defined on ordered pairs of real numbers and the usual numeric operations defined on them. In 1843, Hamilton discovered quaternions and extended the arithmetic of complex numbers to quaternions. Soon after Hamilton’s discovery, Grassmann began investigating vectors. Gibbs developed the algebra of vectors in three-dimensional space. Cayley introduced and developed the algebra of matrices. Thus, vectors, matrices and transformations with various operations became the objects of investigation in algebra, expanding the scope of algebra beyond numeric. Even algebras that failed to satisfy the commutative and associative identities were accepted as legitimate algebras. Algebraists began to call these new objects “algebraic structures”, or just “algebras.” Form and Structure Algebra was no longer limited to the study of ordinary numbers. The scope of algebra was expanded to the study of form and structure of algebras. Augustus De Morgan (1806–1878) De Morgan expanded Peacock’s work to consider operations defined on abstract symbols. In 1838, he gave a clear and strict definition of the concept of mathematical induction. In his foundations paper of 1841, he proposed a set of eight rules for working with symbolical algebra (see Burris 2015). He was partly responsible for the renaissance of logical studies in the first half of nineteenth century. He was among the first to recognize the possibility that there could be algebras of a kind different from the ordinary algebra. De Morgan’s book “Formal Logic: or, The Calculus of

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Inference, Necessary and Probable” was published in 1847 which contains, among other things, the laws now known as De Morgan’s laws. De Morgan is known, besides these laws, for his algebra of relatives (i.e., relations) and his idea of the “universe of discourse.” De Morgan was the first logician to extensively study the logic of relatives. His work was continued by Peirce and Schröder. The theory of relational algebras has remained an active field of research, thanks to Tarski and his collaborators. By way of a historical remark, it needs to be noted here that according to Professor Daniel H.H. Ingalis Sr., Wales Professor of Sanskrit at Harvard, the socalled De Morgan laws were actually known to logicians of Navya Nyaya School in India, centuries before De Morgan (see: Ingalls 1951). References: (Burris 2015; De Morgan 1839, 1841, 1966; Ingalls, D.H. 1951; Peirce 1880; Schröder 1890)

PART I.C: The Evolution of Algebraic Logic, Mathematical Logic and Non-classical Logic By the middle of the 1840s, the environment was just ripe for algebra and logic to meet. For the first time, logic evolved beyond Aristotle’s, after more than 2000 years. Until 1847, algebra and logic had developed independently of each other, perhaps with one exception, when logic followed mathematics in the use of symbols. The situation was about to change. In 1847 and 1879, two remarkable and groundbreaking discoveries occurred in logic: Boole’s Algebra of Logic and Frege’s Begriffsschrift. These discoveries transformed the very nature of logic and its philosophy, and also led to logic’s branching into two tracks: Algebraic Logic (following Boole) and Mathematical Logic (following Frege).

The Birth of Algebraic Logic George Boole (1815–1864) For centuries, algebra studied mainly equations. In the hands of Boole, logic, too, was about to become equational and, thereby, algebraic. Boole was thinking of modernizing Aristotelian logic by using symbolical algebra, as he had just become familiar with the ideas and works of Whately, Peacock, Gregory, and De Morgan. In 1847, Boole published his first book on logic: Mathematical Analysis of Logic. Being an Essay towards a Calculus of Deductive Reasoning. Boole’s contributions to logic were: • To recognize the similarity of the logical connectives and, or, not with the numeric operations of multiplication, addition, and (a sort of restricted) subtraction, respectively. • To develop a scheme to translate (certain) logical statements, including the propositions used in Aristotle’s syllogisms.

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• To develop an “equational calculus” that could be used to “derive” new equations by eliminating “intermediate” variables. • To develop a scheme to translate the concluding equation back into a statement, which would be the logical conclusion one arrives at from the given hypothesis. Thus the process of verifying the validity of a logical argument translates into the process of solving a system of equations. Boole gave many examples to “prove” that his method would give a “valid” conclusion of the syllogism. In 1854, he published his second book: An Investigation of The Laws of Thought on which are founded the Mathematical Theories of Logic and Probabilities. Boole’s “theorems”, which were completed in the book, published in 1854, gave powerful algorithms for analyzing many argument forms with more expressive power than Aristotle’s categorical proposition forms. Although the 1854 presentation of Boole is mainly concerned with the justification of his algebra of logic, it does have some innovations, like the replacement of selection operators by classes (i.e., sets, in modern terminology), the Rule of 0 and 1, and the (improved) Elimination Theorem. References: (Boole 1847, 1848, 1854; Burris 2018) Boole’s Algebra of Logic Here is a very brief description of the essential idea behind Boole’s algebra of logic. He interpreted the usual operations of symbolical algebra +, ,
, 0, 1 as follows: (i) He borrowed the concept of the “universe of discourse” from De Morgan and interpreted “1” as the universe of classes (sets, in the present terminology) and “0” as the empty set, (ii) ‘+’ was defined as the “restricted” union in the sense that it can only apply to disjoint classes, (iii) ‘’ was interpreted as intersection, (iv) ‘–’ was interpreted as the class difference with the restriction that only a subclass is to be subtracted from a class. In other cases, the addition and subtraction were simply undefined or “uninterpretable” in Boole’s words. Thus, addition and subtraction were partial operations, (v) He then added one new law, the idempotent law x  x ¼ x for multiplication, in addition to the laws of symbolical algebra. The next step in Boole’s system was to translate the propositions into equations, where X  Y is written simply as XY: “All X are Y” becomes “X ¼ XY” ‘No X is Y’ becomes “XY ¼ 0” ‘Some X is Y’ becomes “V ¼ XY”,

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where V is a new symbol signifying “some.” The third step was to “solve” equations in his system to derive the conclusion equation which would contain only the variables that occur in the conclusion of the given argument by eliminating the “other” variables in the equations. The fourth step in Boole’s system was to translate back the conclusion equation into a statement which, hopefully, would be the conclusion statement of the argument, in which case the argument would be considered valid. For a thorough discussion of Boole’s theorems, see Burris and Sankappanavar (2014c). Reference: (Boole 1847, 1854; Burris 2015, 2018, 2019; Burris and Legris 2021; Burris and Sankappanavar 2014a, 2014b, 2014c) Concerns About Boole’s Algebra of Logic There were a number of concerns about Boole’s approach to the algebra of logic. Here we mention only a few: • Was it really acceptable to work with uninterpretable terms in equational derivations? • Could one handle particular propositions (i.e., propositions with “some” in them) in Boole’s system which only focuses on equations? • Can we believe in Boole’s “Theorems”? Limitations of Boole’s Algebra of Logic Boole’s algebra of logic was not perfect in several respects: 1. His operations of addition and subtraction were only partial operations. 2. There were some serious gaps in the proofs of his theorems. 3. Boole’s algebra of logic was still of limited scope from the modern point of view since it was essentially monadic, as it did not have a way to deal with relations other than the monadic ones. 4. His treatment of particular statements (i.e., existentially quantified statements) was not satisfactory. Remark 1 (a) Boole’s algebra is not what we know today as Boolean algebra. (b) Correct proofs were given for Boole’s theorems in Hailperin (1986). (c) Boole’s claim about his “rule of 0 and 1” is stated more precisely and proved to be correct only recently in Burris and Sankappanavar (2013). An Example of Reasoning in Boole’s Algebra of Logic Here is a simple example of how Boole used his algebra of logic (aka., algebra of classes) to verify if an argument is correct.

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• Consider the following syllogism: All human beings are logicians. No logicians are crazy. Therefore, no human beings are crazy. Here is how Boole would proceed: • First Step: Boole would define three classes (sets, in the current terminology): Let A ¼ the set of human beings, B ¼ the set of logicians, C ¼ the set of crazy people. • Second Step: Boole would symbolize (translate) the three statements in the above syllogism as: All A are B. No B is C. Therefore, no A is C. • Third Step: Translate these propositions into equations involving the sets A, B, C: 1. A \ B ¼ A. 2. B \ C ¼ Ø. 3. So, A \ C ¼ Ø. The problem of checking the validity of the original syllogism is now reduced to the problem, in algebra, of solving a system of equations. • Fourth Step: Derive (3) from (1) and (2) by eliminating B. (This is easy to verify.) Thus, Boole turned the logic into algebra. In so doing, he gave an “algebra” which was different from the ordinary algebra of numbers. This algebra, after some modifications by his successors, Jevons and Peirce, would become known as “Boolean algebra.” It would play a crucial role in the creation and development of new fields such as “universal algebra,” “lattice theory,” and “algebraic logic.” For more detailed presentations of Boole’s algebra of logic, for the correct proofs of Boole’s theorems and for its future impact, we recommend that the reader consult the following references: References: (Boole 1847, 1848, 1854; Rosenbloom 1950; Shannon 1948; Hailperin 1986; Brown 2009; Burris 2018; Burris and Legris 2021; Burris and Sankappanavar 2013, 2014a, 2014b, 2014c; Burris 2019)

Modifications of Boole’s Algebra of Logic Boole’s successors, especially Jevons and Peirce, would soon replace the partial operations used in Boole’s algebra by total operations and arrive essentially at what is currently known as “Boolean algebra.”

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An Algebra of Logic Based on Total Operations William Stanley Jevons (1835–1882) In 1863, Jevons wrote to Boole that his partial operations should be replaced by the corresponding total operations. But Boole did not agree with him. Jevons was one of the first logicians to use “union” as a total operation, thus making Boole’s algebra closer to what are now known as Boolean algebras. Jevons published his system of logic in the 1864 book, Pure Logic. Charles Sanders Peirce (1839–1914) Around the same time, but independently of Jevons, Peirce also decided to replace Boole’s partial operations with the corresponding total operations. In his 1880 paper, “On the Algebra of Logic,” Peirce broke with the Aristotelian semantics of classes, by introducing modern semantics in which a class symbol could denote the empty set as well as the universe. For example, Peirce said the proposition “All A is B” is true if both A and B are empty classes (sets). Peirce used the subsumption (i.e., inclusion) relation in his algebra of classes. He stated the partial order properties of subsumption and then proceeded to define the operations + and  as the least upper bound and the greatest lower bound, respectively. In 1883, he listed the key equational properties of the algebras with two binary operations that we now call lattices. It is quite remarkable that Peirce, while doing research in logic, isolated the new concept of a lattice. This has led to the creation of a whole new area, Lattice Theory, which has grown rapidly and has had connections to other areas of mathematics such as universal algebra, logic, order theory and graph theory. Peirce is viewed as the founder of the theory of lattices. Peirce claimed that the distributive law followed from his axioms for lattices, but its proof was too tedious. Schröder asked Peirce to provide his proof of the distributive law from lattice axioms; however, in 1885, Peirce let Schröder know that he was unable to provide a proof. Friedrich Wilhelm Karl Ernst Schröder (1841–1902) Inspired by Peirce’s work, Schröder wrote an encyclopedic three-volume work called “Algebra der Logik” (1890–1910). He gave three complicated counterexamples to Peirce’s claim of distributivity in an appendix to Volume I. Volume II augments the algebra of logic for classes developed in Volume I with a way to handle existential statements. It is essentially a study of the calculus of classes using both equations and negated equations, covering the same topics as in Volume I. Using modern semantics, Schröder proved that one cannot express “Some X is Y” by using equations alone. However, he noted that one can easily express it with a negated equation, namely, “XY 6¼ 0” (Burris 2018). Julius Wilhelm Richard Dedekind (1831–1916) Schröder’s work, in turn, inspired Dedekind, who, in 1897, composed an elegant (abstract) presentation of lattices. (Note that Dedekind used the term Dualgruppen

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for lattices.) He presented a five-element counter-example to Peirce’s claim that the distributive law held in lattices. He introduced the modular law. His investigations related to it have led to an extensive theory of modular lattices which, in turn, has made it possible for the formulation of congruence-modular varieties and commutator theory in universal algebra. Alfred North Whitehead (1861–1947) Boole’s algebra of logic, Hamilton’s algebra of quaternions and Grassmann’s theory of extensions were in need of a unifying concept of a “universal algebra” which motivated Whitehead to write the treatise “Universal Algebra” in 1898, borrowing the name from Sylvester who had coined it. In this treatise, he presented Boole’s algebra of logic, Grassmann’s linear algebra and Hamilton’s quaternions. However, the treatise had no general results in it. Such results were only discovered much later in the 1930s by Birkhoff, Mal’cev, Tarski and others. Interestingly, Whitehead called the algebras created by Boole, Hamilton, and Grassmann “extraordinary algebras.” From 1900 onwards, progress in algebraic logic was relatively non-existent until the arrival of Tarski in the 1920s. References: (Jevons 1864; Peirce 1880; Schröder 1890; Dedekind 1900; Whitehead 2009) Boolean Algebras As mentioned earlier, Boole’s algebra of logic went through modifications by Jevons and Peirce and received the name “Boolean algebras.” Many axiomatizations for Boolean algebras have since been published; see, for example, (Sheffer 1913) and (Huntington 1933). We will now present one of the most commonly used definitions of Boolean algebras, which also will be useful later in PART II. Definition 1 An algebra A ≔ hA, _, ^, 0 , 0, 1i, where _, ^ are binary, 0 is unary, 0 and 1 are 0-ary (i.e., constants), is a Boolean algebra if the following axioms hold in A: (1) x (2) x (3) x (4) x (5) x (6) x (7) x

^x≈x ^y≈y ^x ^ (y ^ z) ≈ (x ^ y) ^ z ^ (x _ y) ≈ x ^ (y _ z) ≈ (x ^ y) _ (x ^ z) ^0≈0 ^ x0 ≈ 0

x _ x ≈ x, x _ y ≈ y _ x, x _ (y _ z) ≈ (x _ y) _ z, x _ (x ^ y) ≈ x, x _ (y ^ z) ≈ (x _ y) ^ (x _ z), x _ 1 ≈ 1, x _ x0 ≈ 1.

Thus, a Boolean algebra A is a complemented distributive lattice. References: (Balbes and Dwinger 1974; Burris and Sankappanavar 1981)

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The Birth of Mathematical Logic (aka: Formal Logic or Symbolic Logic) About 32 years after Boole’s discovery, a second major breakthrough in logic occurred in 1879 with Frege’s publication “Begriffsschrift” (German for, roughly, “concept-script”). In this work, Frege conceived of a system of logic that was totally different from Boole’s system. With Frege’s discovery, a second trend in logic, called the mathematical logic (or formal logic) trend, began. It is quite remarkable that Aristotle’s syllogism, which had maintained its total command for over 2000 years, finally yielded to not just one but two significant modifications: Boole’s algebra of logic in 1847 and Frege’s formal logic in 1879. Each of these carved out its own independent path for over 30 years till the arrival of Tarski. Friedrich Ludwig Gottlob Frege (1848–1925) In an attempt to realize some of Libniz’s ideas for a language of thought and a rational calculus, Frege, in 1879, developed a formal notation and precise rules for a formal language of pure thought in his first book “Begriffsschrift, eine der arithmetischen nachgebildete Formelsprache des reinen Denkens” (“Concept Notation: A formal language of pure thought, modeled upon that of arithmetic”). This system was later refined in his two volume work “Grundgesetze der Arithmetik” (1893–1903). Frege was the first also to show that all truth-functional connectives could be defined in terms of negation and the material implication and to propose the modern system of classical propositional logic. Giuseppe Peano (1858–1932) Peano, influenced by Grassmann and Dedekind, used the axiomatic method to provide an early axiomatization of arithmetic. In 1889, he published the so-called Peano’s axioms which gave a formal foundation for the arithmetic of natural numbers. His book “Formulario Mathematico” (Formulation of Mathematics), first published in 1895, expresses fundamental theorems of mathematics in a symbolic language. It also contains innovative notation for logical connectives. In 1908, he published the fifth and final edition of “Formulario Mathematico” containing complete statements of 4200 formulae and theorems with proofs for most of them. References: (Frege 1879, 1884; Peano 1889, 1895)

The Evolution of Mathematical Logic and the Rise of the Axiomatic Method Mathematicians and logicians of the twentieth century were extremely productive in developing and expanding the ideas and methods discovered up to and during the nineteenth century, as well as in proposing new ideas of their own. Their work has led to many new areas of mathematics and logic in the twenty-first century. Much of the work in logic done during the period 1900–1930 pertained to particular formal systems, one of which was the system of Principia Mathematica. This time-period

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also gave birth to a large number of new logics created either to correct the perceived weaknesses of the classical system exposed in Principia or to expand the classical system of logic. For example, certain criticisms of the classical propositional logic led to the so-called non-classical propositional logics, including many-valued logics, modal logic and intuitionistic logic. In the middle of the 1930s, universal algebra, with its powerful tools, came into existence and started influencing logic. The fruitful interplay between (universal) algebra and logic continued throughout the twentieth century and beyond. 1900: Two Important International Congresses in Paris The First International Congress of Philosophy and the Second International Congress of Mathematics were held consecutively in Paris in August, 1900. Cantor, Coutourat, Dedekind, Frege, Hilbert, Klein, Ladd-Franklin, Peano, Poincaré, Russell and Schröder were some of the members who attended the meetings. Hilbert proposed his 23 most significant unsolved problems of mathematics. Several of these problems were about foundational issues in mathematics and logic that would dominate logical research during the first half of the twentieth century. Also, Peano met and gave Russell a copy of his “Formulario Mathematico.” Russell was quite impressed by the notation used by Peano in that work. Alfred North Whitehead (1861–1947) and Bertrand Russell (1872–1970) Influenced by Frege’s formal logic and Peano’s Formulario Mathematico, Russell and Whitehead decided to use Frege’s ideas on formal logic from his Begriffsschrift and Peano’s notation from Formulario Mathematico in their monumental work “Principia Mathematica” consisting of three volumes written during 1910–1913. In that work, a system of logic was given by a formal language and a deductive calculus consisting of a set of axioms and a set of inference rules. Principia brought Frege’s formal logic to the attention of logicians, mathematicians, and philosophers. Principia also isolated the modern classical propositional logic which was already present in Frege’s work. David Hilbert (1862–1943) Through his axiomatization of Euclidean geometry, Hilbert popularized the axiomatic method. Hilbert and his associates singled out First-Order Logic (FOL) from Higher-Order Logic during the 1920s. FOL first appeared in the book “Grundzüge der Theoretischen Logik” (“Principles of Theoretical Logic”) by Hilbert and Ackermann, published in 1928. This book formulated two important problems: The Decision Problem (Entscheidungsproblem) of First-Order Logic and the problem of whether that logic was complete (i.e., whether all valid formulas of FOL were derivable

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from FOL axioms and rules). The first problem was answered in the negative by Church in 1936. The second was answered in the affirmative by Gödel in 1929. Hilbert wrote another important book “Grundlagen der Mathematik,” with Bernays, which appeared in two volumes (1934 and 1939). This book prescribed a standard way, known as the Hilbert-style presentation, to define a system of logic consisting of definitions for a set of connectives (i.e., the language), a set of (wellformed) formulas in that language, a set of axioms, and a set of valid (finitary) inference rules. The importance of symbolic logic, of axiomatization, and set theory as a foundation for mathematics was gaining momentum. References: (Whitehead and Russell 1910; Hilbert 1899, 1905, 1917, 1930; Hilbert and Ackermann 1928; Hilbert and Bernays 1934; Hilbert and Ackermann 1959; Hilbert 2013; Hilbert and Bernays 1939; Church 1956; K. Gödel 1986) The Hilbert-style Axiomatization of the Classical Propositional Logic (CPL) The rest of PART I will be limited to classical propositional logic and its variations, and will not deal with FOL due to space constraints. We will now present the Classical Propositional Logic (i.e., two-valued propositional logic) as an example of a Hilbert-style formal system using modern terminology. • Language L : h_, ^, !,ci (A formal definition of a language is given later in the section “The Lindenbaum-Tarski Method: The Bridge Between Algebra and Logic”) • The set Var of variables : ¼ {p1, p2, p3, . . ., pn, . . .} • L-formulas: The set FmL of formulas in L (L-formulas for short) is defined as the smallest set such that: (i) Var  FmL, (ii) The set of constants (i.e., the 0-ary operation symbols)  FmL, (iii) If α1
, α2 FmL, then (α1 _ α2)  FmL, (α1 ^ α2)  FmL (α1 ! α2)  FmL and αc1  FmL (for convenience, the end parentheses will be deleted in the sequel.) • Axioms: 1. α ! ðβ ! αÞ, 2. ðα ! ðβ ! γ ÞÞ ! ððα ! βÞ ! ðα ! γ ÞÞ, 3. ðα ^ βÞ ! α, 4. ðα ^ βÞ ! β, 5. α ! ðβ ! ðα ^ βÞÞ, 6. α ! ðα _ βÞ,

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7. α ! ðβ _ αÞ, 8. ðα ! γ Þ ! ððβ ! γ Þ ! ððα _ βÞ ! γ ÞÞ, 9. ðα ! βc Þ ! ðβ ! αc Þ, 10. ðα ! αÞc ! β, 11. α _ αc : • Rule of Inference: From α, α ! β, deduce β (Modus Ponens).

The Discovery of Non-classical Logics The Classical Propositional Logic, as expounded in Principia Mathematica (mentioned above), became the springboard for new logical systems, either because of concerns of philosophers with the paradoxes of the material implication or because of the need to add additional connectives or because of the criticism of classical principles like the law of the excluded middle. In the early 1900s, several systems of non-classical (i.e., not necessarily classical) logics were presented by logicians, motivated by different philosophical considerations. Such logics came in several different kinds, such as, many valued logics, modal logic, intuitionistic logic, and so on. 1909: The Birth of Many-Valued Logics Interestingly, algebras were used to define logical matrices which were, in turn, utilized to present many-valued logics. A logical matrix is a pair hA, Fi, where A is an algebra and F  A. Peirce’s 3-Valued Logic According to (Lane 2001), contrary to the established belief, it was Peirce who was the first to extend the truth-table method in 1885. Peirce’s work on 3-valued logic was brought to light in (Fisch and Turquette 1966). In February of 1909, Peirce defined his triadic logics, using “V” (for “verum” or “true”), “F” (for “falsum” or “false”), and “L” (for “the limit” or “the boundary”) as truth values, with F < L < V, in four (three-valued) unary operations (negations) and six binary connectives, using the truth-table method. In the brief description given below of Peirce’s triadic logics, we essentially follow (Lane 2001), with some notational changes. Peirce defined four different unary connectives (negations) and six different binary connectives. Peirce’s threevalued negations, with a slight change in notation, are:

(1)

– V L F

F

L V

o

(2)

V L F L

L L

(3)

¬ V L

F

F V L

(4)

‘

V L F L F V

Peirce’s three-valued binary connectives Θ, Z, Y, Ω, Φ, and Ψ are as follows:

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Remark 2 Jan Łukasiewicz and Emil Post were also pioneers of many-valued logics. Jan Łukasiewicz (1878–1956) In the 1920s, Łukasiewicz defined many-valued logics to allow “truth values” that would reflect various degrees of (un)certainty. In 1920 (see Łukasiewicz (1967)), he presented a three-valued logic in the paper “On the Concept of Possibility,” which was based on a talk given on June 5, 1920 in Lwów. Soon thereafter, he presented nvalued logics for n ⩾ 2 and an infinitely many-valued logic. Łukasiewicz’s three-valued logic is given by the matrix: hL3, {1}i, where L3 ¼ h{0, 1/2, 1}, _, ^, !,0, 0, 1i, where 0 < 1/2 < 1, is a bounded lattice (chain) and the operations ! and 0 are defined as follows: x ! y ≔ min f1, 1
x þ yg, x0 ≔1
x:

Emil Post (1897–1954) In 1921, Post developed his n-valued logics for n ⩾ 2 using the matrix method. For example, Post’s three-valued logic is given by the following matrix: hP3, {1}i, where P3 ¼ h{0, 1/2, 1}, _, ^,0, 0, 1i, where 0 < 1/2 < 1, is a bounded lattice (chain), with the unary operation 0 is defined as follows: 10 ¼ 1=2, ð1=2Þ0 ¼ 0, 00 ¼ 1: (The axiomatization of Post’s logics was given later by Rosenbloom (1942).)

1918: Modal Logic Clarence Irving Lewis (1883–1964) In 1912, Lewis (1912) criticized the use of material implication in Principia Mathematica which yielded the “paradoxes of material implication” such as α ! (β ! α) and :α ! (α ! γ). His criticism laid the foundation for the modern Modal Logic. In fact, in 1918, Lewis defined the first modal system, SSL, using an axiomatic system to express the notion of strict implication as a better substitute than the material implication for the “intuitive” notion of implication. Lewis received some criticism for this from other philosophers. In 1932, Lewis and Langford responded to the criticism of SSL by providing five systems of modal logic, S1–S5, with S3 being the earlier system SSL. They used the symbol ◊ to denote the “possibility” operation. They defined the strict implication ) (our notation, not theirs) by p ) q ≔ : ◊ (p ^ : q), and the strict equivalence , (our notation, not theirs) by p , q ≔ (q ) q) ^ q ) p.

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The System S1: Let p, q, r denote propositional variables. Axioms: (a) ( p ^ q) ) (q ^ p); (b) ( p ^ q) ) p; (c) p ) (p ^ p); (d) (( p ^ q) ^ r) ) (p ^ (q ^ r)); (e) (( p ! q) ^ (q ! r)) ) ( p ! r); (f) ( p ^ (p ! q)) ) q; (g) p ) : : p. (McKinsey (1934) proved that the axiom: p ) : : p was redundant.) Rules of Inference for S1-S5: Let α, β be propositional formulas. (a) Uniform Substitution Rule: Let α1 denote the formula obtained from the formula α by substituting another formula β (uniformly) for a propositional variable. Then From α, infer α1. (b) Substitution of Strict Equivalents: From α , β and γ, infer any formula obtained from γ by substituting β for some occurrence(s) of α. (c) Adjunction Rule: From α and β, infer α ^ β. (d) Strict Detachment Rule: From α and α ) β, infer β. The System S2: Axioms: The axioms for S1 PLUS the following axiom: (i) ◊(p ^ q) ) ◊ p. The System S3: Axioms: The axioms for S1 PLUS (ii) (p ) q) ) (: ◊ q ) : ◊ p). The System S4: Axioms: Axioms of S1 PLUS (iii) ◊ ◊ p ) ◊ p. We note that (Lewis and Langford 1932) didn’t use the symbol □ for : ◊ : . The System S5: Axioms of S5: The axioms of S1 PLUS (iv) ◊p ) ◊ ◊ p.

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Intuitionistic Logic In 1907, Brouwer raised objections to the validity of certain principles of classical mathematics such as the laws of the excluded middle: ϕ _ : ϕ and the double negation. Several logicians, including Kolmogorov, Glivenko and Heyting, proposed axiomatic systems to formalize the intuitionistic propositional logic underlying the Brouwer’s ideas. Andrey Nikolaevich Kolmogorov (1903–1987) In 1925, Kolmogorov formalized his System B, the first formal system for intuitionistic propositional logic as follows (in the modern notation): Language: {!, :}, Variables: {p1, p2, p3, . . ., pn, . . .}, Formulas: Defined similarly to those in CPL, Axioms (presented as schemas): α ! ðβ ! αÞ, ðα ! ðα ! βÞÞ ! ðα ! βÞ, ðα ! ðβ ! γ ÞÞ ! ðβ ! ðα ! γ ÞÞ, ðβ ! γ Þ ! ððα ! βÞ ! ðα ! γ ÞÞ, ðα ! βÞ ! ððα ! :βÞ ! :αÞ, where α, β, and γ denote arbitrary formulas. Inference Rule: From α, α ! β, deduce β (Modus Ponens). System B is not equivalent to Heyting’s system given below. However, it is historically the first system that attempted to capture the principles of Brouwer’s intuitionistic philosophy. System B has come to be known as “minimal calculus” as it is equivalent to the minimal calculus (see Johansson 1937). Valery Ivanovich Glivenko (1896–1940) In 1928, Glivenko gave the following axiomatization for intuitionistic propositional logic: Language: {_, ^, !, :}, Variables: {p1, p2, p3, . . ., pn, . . .}, Formulas: Defined similarly to those in CPL, Axioms: α ! α,

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ðα ! βÞ ! ððβ ! γ Þ ! ðα ! γ ÞÞ, ðα ^ βÞ ! α, ðα ^ βÞ ! β, ðγ ! αÞ ! ððγ ! βÞ ! ðγ ! ðα ^ βÞÞÞ, α ! ðα _ βÞ, β ! ðα _ βÞ, ðα ! γ Þ ! ððβ ! γ Þ ! ððα _ βÞ ! γ ÞÞ, ðα ! βÞ ! ððα ! :βÞ ! :αÞ, where α, β, and γ denote arbitrary formulas. Inference Rule: From α, α ! β, deduce β (Modus Ponens). The above system was not expressive enough to yield intuitionistic propositional logic; so, Glivenko, in 1929, added the following four axioms: ðα ! ðβ ! γ ÞÞ ! ðβ ! ðα ! γ ÞÞ, ðα ! ðα ! γ ÞÞ ! ðα ! γ Þ, α ! ðβ ! αÞ, :β ! ðβ ! αÞ: He proves an interesting theorem in this new system which says: If α is provable in the classical propositional logic, then ::α is provable in this new (stronger) system of intuitionistic propositional logic. Arend Heyting (1898–1980) In 1930, Heyting gave his axiomatization for intuitionistic propositional logic: Language: {_, ^, !, :}, Variables: {p1, p2, p3, . . ., pn, . . .}, Formulas: Defined similarly to those in CPL, Axioms: α ! ðα ^ αÞ, ðα ^ βÞ ! ðβ ^ αÞ, ðα ! βÞ ! ððα ^ γ Þ ! ðβ ^ γ ÞÞ,

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ððα ! βÞ ^ ðβ ! γ ÞÞ ! ðα ! γ Þ, β ! ðα ! βÞ, ðα ^ ðα ! βÞÞ ! β, α ! ðα _ βÞ, ðα _ βÞ ! ðβ _ αÞ, ððα ! γ Þ ^ ðβ ! γ ÞÞ ! ððα _ βÞ ! γ Þ, :α ! ðα ! βÞ, ððα ! βÞ ^ ðα ! :βÞÞ ! :α: Inference Rule: From α, α ! β, deduce β (Modus Ponens). It should be remarked that the discoveries of Peirce, Lukasiewicz, Post, Lewis, Kolmogoov, Glivenko and Heyting, mentioned above, have led to new areas of logic: Many-valued logic, modal logic, and intuitionistic logic. All these areas are, collectively, being called “non-classical logic.” It should also be mentioned that the non-classical logic now includes the study of “paraconsistent logics” and “substructural logics” and many more, the details of which are beyond the scope of this chapter. References: (Lane 2001; Fisch and Turquette 1966; Łukasiewicz 1967; Post 1921; Lewis 1912, 1914, 1920; Brouwer 1913; Kolmogorov 1925, 1932; Glivenko 1928, 1929; Heyting 1930; Rosenbloom 1942; Ballarin 2021; Goldblatt 2006)

PART I.D: Modern Algebraic Logic As mentioned earlier, by the end of second decade of the twentieth century, several non-classical logics had been discovered. Thus, by 1930, the very nature of logic had changed. The existence of different systems of logic raised the most fundamental question: What is a logic?

Logics as Deductive Systems In connection with the question raised above, Tarski made an important observation: “what follows from what” is more significant than “what is provable”. In other words, he emphasized that the concept of “logical consequence” is more characteristic of a logic than that of “proof.” Alfred Tarski (1901–1983) In the 1930s, Tarski introduced the concept of a logical consequence and the formal notion of a “deductive system,” as an abstract consequence operator (see (Tarski

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1930a, 1930b, 1936a, 1936b)). He also laid out the principles of the “deductive method.” Building on an idea of Lindenbaum, Tarski, introduced the algebra of propositional formulas and the Lindenbaum-Tarski algebra. The process of constructing Lindenbaum-Tarski algebra has come to be known as “Lindenbaum-Tarski method.” Using this method, he showed that the class of Boolean algebras is an “algebraic semantics” for the Classical Propositional Calculus. This seminal result led to the successful application of the Lindenbaum-Tarski method to several other logics. This was also the beginning of a much stronger interplay between logic and algebra. Thus, finding an algebraic semantics for logics has become an important research problem in the study of logics. For several decades, the phrase “algebraic semantics” was used informally. This notion was made precise by (Blok and Pigozzi 1989a) (see the section “Abstract Algebraic Logic (AAL)” of this chapter) with the help of powerful concepts and tools of universal algebra that were developed by Birkhoff, Maltsev, Tarski, Jónsson, Łoś, and others (see the section “The Emergence of Universal Algebra” of this chapter). Tarski defined logical consequence in terms of consequence operators, not in terms of consequence relations; however, Scott, in (Scott 1974a), pointed out that the two concepts were interdefinable and, so, can be used interchangeably. Some details of the Lindenbaum-Tarski method will be given in the section “The Lindenbaum-Tarski Method: The Bridge Between Algebra and Logic”. References: (Tarski 1930a, 1930b, 1936a, 1936b, 1956, 1994; Blok and Pigozzi 1988) Jerzy Łoś (1920–1998) and Roman Suszko (1919–1979) An important property, namely, “structurality,” was missing in Tarski’s definition of logical consequence. Later, in 1958, the concept of a “structural consequence operator” was introduced in Łoś and Suszko (1958) (see the section “The Lindenbaum-Tarski Method: The Bridge Between Algebra and Logic” for its definition). Structural deductive systems are frequently called logics.

The Emergence of Universal Algebra By 1930, several classes of algebraic structures were discovered and investigated. Those investigations revealed many similarities in concepts, results, and methods. Birkhoff, Mal’cev, and Tarski were the pioneering giants in the development of universal algebra. Garrett Birkhoff (1911–1996) Birkhoff saw many of these similarities and arrived at the unifying concept of an “(abstract) algebra” and provided some fundamental theorems about algebras and classes of algebras. (See the references given at the end of the section “A Quick Tour of Universal Algebraic Concepts and Tools”.) Borrowing Whitehead’s terminology, he called this new field “universal algebra.” Birkhoff also formulated the axioms and

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rules of inference for Equational Logic and proved the completeness theorem for that logic. His books on lattice theory have been quite influential in the universal algebraic community. Anatoly Ivanovich Mal’cev (1909–1967) He proved several fundamental theorems in universal algebra, especially in the theory of quasivarieties. One of his theorems is given a little later. Malcev’s early publications were on logic and model theory. For example, he showed the undecidability of the elementary theory of finite groups. Alfred Tarski (1901–1983) We have already seen Tarski’s contributions on the interconnections between algebra and logic. He has also made significant contributions to universal algebra, as well. He proved, among others, the ℍℙ-Theorem which will be mentioned in the next subsection. References: (Birkhoff 1933, 1935, 1944; Mal’cev 1971, 1973; A. Tarski 1968) A Quick Tour of Universal Algebraic Concepts and Tools Since Birkhoff’s fundamental discoveries, additional powerful techniques have been developed in universal algebra. Some of these techniques have become quite useful in the investigation of algebras related to logic. We will briefly present below some of these concepts and tools and refer the reader to textbooks for more details (see references at the end of this section). • Languages: A language (or type or signature) of algebras is a set L of operation symbols such that a non-negative integer n is assigned to each member f of L. This integer is called the arity (or rank) of f, and f is called an n-ary function symbol. f is finitary if f is n-ary for some n. • Algebras: Given a language L of algebras, an algebra A in the language L (or L-algebra) is an ordered pair hA, Fi where A is a nonempty set and F is a family of finitary operations indexed by the language L such that corresponding to each n-ary function symbol f there is an n-ary operation f A on A. • Subalgebras: If A and B are L-algebras, then B is a subalgebra of A if B  A and, for every f  L, the operation f B is the restriction of the corresponding operation f A. • Congruences: Let A be an algebra of type L and let θ be an equivalence relation on A. Then θ is a congruence on A if it satisfies the following compatibility property: (CP): For each n-ary operation symbol f  L and elements ai, bi  A, if ai θ bi holds for 1 O i O n then

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f A ð a1 , . . . , an Þ θ f A ð b1 , . . . , bn Þ holds. Con A and Con A denote respectively the set of congruences and the congruence lattice of A. • Simple Algebras: An algebra is simple if Con A ¼ {△,▽} where △ is the trivial congruence and ▽ = AA. • Quotient Algebras: If θ is a congruence on an L-algebra A, the quotient algebra of A by θ, denoted by A/θ, is the algebra whose universe is A/θ and whose fundamental operations satisfy: f A=θ ða1 =θ, . . . , an =θÞ ¼ f A ða1 , . . . , an Þ=θ, where a1 , . . . , an  A and f is an n-ary function symbol in L: • Homomorphisms: If A and B are L-algebras, then α : A ! B is a homomorphism from A into B if α( f A(a1, . . ., an)) ¼ f B(α(a1), . . ., α(an)). If a homomorphism is one-to-one, then it is called an embedding, and if a homomorphism is one-to-one and onto, it is called an isomorphism. • Direct Products of Algebras: Given an indexed family (Ai)i  I of L-algebras, the direct product A ¼ Πi  I Ai is an algebra with universe Πi  I Ai and such that if f is n-ary and a1, . . ., a n  Πi  I A i , f A ða1 , . . . , an ÞðiÞ ¼ f Ai ða1 ðiÞ, . . . , an ðiÞÞ,

for every i  I:

• Subdirect Products of Algebras: An L-algebra A is a subdirect product of an indexed family (A i ) i  I of L-algebras if (i) A is a subalgebra of Πi  I Ai, and (ii) πi(A) ¼ Ai for each i  I, where πi is the ith projection of A. • Subdirect Embedding: An embedding α : A ! Πi  I Ai is subdirect if α(A) is a subdirect product of the Ai’s. The following criterion is useful in practice to find subdirect embeddings. Proposition 1 Let A be an L-algebra and let θi  Con A for i  I such that


\ θi ¼ 0:

iI

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Then the natural homomorphism ν:A!

Q iI

A=θi

defined by ν(a)(i) = a/θi is a subdirect embedding. • Subdirectly Irreducible Algebras: An L-algebra A is subdirectly irreducible if for every subdirect embedding α:A!

Y Ai iI

there is an i  I such that

π i ∘α : A ! Ai

is an isomorphism (πi being the ith projection). In practice, the following Birkhoff’s characterization of subdirectly irreducible algebras is more convenient to use to verify if an algebra is subdirectly irreducible. Theorem 1 An algebra A is subdirectly irreducible iff A is trivial or A has a smallest nontrivial congruence. Birkhoff’s Subdirect Product Theorem Every algebra A is isomorphic to a subdirect product of subdirectly irreducible algebras which are homomorphic images of A. • Class Operators , , ℍ, ℙ, ℙS Class Operators: The following are useful operators mapping classes of L-algebras to classes of L-algebras: Let  be a class of L-algebras. Then A  ðÞ iff A is isomorphic to some member of , A  ðÞ iff A is a subalgebra of some member of , A  ℍðÞ iff A is a homomorphic image of some member of , A  ℙðÞ iff A is a direct product of a nonempty family of members of , A  ℙS ðÞ iff A is a subdirect product of a nonempty family of members of . • Varieties of L-algebras: A nonempty class of L-algebras is called a variety if it is closed under ℍ, , and ℙ.

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Let ðÞ denote the variety generated by a class  of L-algebras (i.e., the smallest variety containing ). Here is another fundamental theorem in universal algebra, due to Tarski. Tarski’s ℍℙ-Theorem: If  is a class of L-algebras, then ðÞ ¼ ℍℙðÞ: The following version of Birkhoff’s Subdirect Decomposition Theorem is useful in studying varieties. Proposition 2 If  is a variety, then every algebra A   is isomorphic to a subdirect product of subdirectly irreducible algebras in . • Free Algebras: Let  be a class of L-algebras and let U(X) be an L-algebra generated by X. If for every algebra A   and for every map α : X ! A, there is a homomorphism β : UðXÞ ! A, which extends α (i.e., β(x) ¼ α(x) for x  X), then we say U(X) has the universal mapping property for  over X. X is called a set of free generators of U(X), and U(X) is said to be a free algebra freely generated by X over . • Terms, Term functions, Term Algebras, and -Free Algebras: Terms: Let X be a set of (distinct) symbols called variables. Let L be a language for algebras. The set T(X ) of L-terms over X is the smallest set such that (1) X  Τ(X ), (2) If c  L is nullary, then c  T(X ), (3) If f  L is n-ary and p1, . . ., pn  T(X ), then the “string” f(p1, . . ., pn)  T(X ). If p is a term such that the variables occurring in p are among x1, . . ., xn, then p is also denoted by p(x1, . . ., xn). Term Functions: Given a term p(x1, . . ., xn) of type L over some set X and given an L-algebra A, we define a mapping pA: An ! A as follows: 1. If p is a variable xi, then pA(a1, . . ., an) ≔ ai, for a1,. . ., an  A, i.e., pA is the ith projection map;

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2. If p is of the form f ( p1(x1, . . ., xn),. . ., pk(xi, . . ., xn)), where f  L and is k-ary, then
 A pA ða1 , . . . , an Þ≔ f A pA 1 ða1 , . . . , an Þ, . . . , pk ða1 , . . . , an Þ :




Term Algebra: Let L be a language of algebras and T(X) 6¼ 0. Then the term algebra of type L over X, written T(X), has as its universe T(X) and the fundamental operations satisfy: f TðXÞ : hp1 , . . . , pn i 7! f ðp1 , . . . , pn Þ, where f  L and is n-ary and pi  T(X), 1  i  n. -free algebras: Let  be a family of L-algebras. Given a set X of variables, let Φ ðXÞ ¼ fϕ  Con TðXÞ : TðXÞ=ϕ  ðÞg: Define the congruence θ ðXÞ on T(X) by θ ðXÞ≔ \ Φ ðXÞ:
 Define F X , the -free algebra over X, by
 F X ¼ TðXÞ=θ ðXÞ, where X ¼ X=θ ðXÞ. Proposition 3 (Birkhoff 1935) Suppose T(X) exists. Then





 1. F X has the universal mapping property for  over X.
 2. For  6¼ 0, F X  ℙðÞ. • Identities, Valuations, Satisfaction, and Equational Classes: Identities: Let L be a language and T(X) the set of L-terms over X. An L-identity is an expression of the form pq where p, q  T(X ). Let {x1, . . ., xn} be the set of variables containing those that occur in the terms p and q so that the identity p ≈ q can also be written as pðx1 , . . . , xn Þ  qðx1 , . . . , xn Þ: Valuations and substitutions: Let A be an L-algebra. Then it is easy to see that any function ν : X ! A can be extended to a homomorphism from T(X ) to A denoted by the same symbol ν. Hom

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(T(X ), A) denotes the set of all homomorphisms from T(X ) into A, and these homomorphisms are also called valuations (or interpretations) in A. An endomorphism δ: T(X ) ! T(X ) is called a substitution. Satisfaction: A valuation h in A (i.e., h  Hom(T(X ), A)) satisfies an Ḻ-identity p ≈ q if hp = hq. We denote this satisfaction relation by the notation: A |=h p ≈ q. An algebra A satisfies the identity p ≈ q if all the valuations in A satisfy it; in symbols, A ⊧ p ≈ q if and only if A ⊧h p ≈ q, for all h  Hom(FmL, A). Equivalently, A ⊧ p ≈ q if and only if for every choice of a1,. . ., an  A we have pA(a1, . . ., an) = qA(a1, . . ., an). A class  of L-algebras satisfies p ≈ q when all the algebras in  satisfy it; i.e., ⊧p  q if and only if A⊧p  q, for all A  : Let Σ be a set of L-identities. We define ðΣÞ to be the class of L-algebras A that satisfy Σ. Here is a crucial connection between the semantical notion of -free algebra and the syntactical notion of an identity. Proposition 4 Given a class  of L-algebras and terms p, q  T(X ) of type L, we have ⊧p  q
 , F X ⊧p  q
 , p ¼ q in F X , hp, qi  θ ðXÞ: Equational classes: A nonempty class  of L-algebras is called an equational class if there is a set of L-identities Σ such that  ¼ ðΣÞ: In this case, we say that  is defined, or axiomatized, by Σ. • Connection Between Equational Classes and Varieties of Algebras Here is another celebrated theorem of Birkhoff-an important illustration of the interplay between (equational) logic and (universal) algebra. Birkhoff’s theorem: For a class  of L-algebras,  is an equational class iff  is a variety. • Ultraproducts: Let (Ai)i  I be an indexed family of L-algebras and let U an ultrafilter over the set I.

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Define a relation θU on Πi  I Ai by ha, bi  θU iff ½½a ¼ b  U, where [[a ¼ b]] ¼ {i  I : a(i) ¼ b(i)}. Then θU is a congruence. So we define the ultraproduct of (Ai)i  I to be Y Ai =θU : iI

It will be denoted simply by Y Ai =U: iI

The elements of

Q iI

Ai =U are denoted by a/U, where a 

Q iI

Ai .

We can now define a new class operator ℙU. A  ℙU ðÞ iff A is an ultraproduct of members of . (In the rest of this section, it is assumed that the reader is familiar with the definition of first-order formulas.) Łoś’s Theorem Let Ai, i  I be a nonempty indexed family of L-algebras. If U is an ultrafilter over I and ϕ is any first-order L-formula, then Y Ai =U ‘ ϕða1 =U, . . . , an =UÞ iI

if and only if fi  I : Ai ‘ ϕða1 ðiÞ, . . . , an ðiÞÞg  U: • Jónsson’s Theorem for Congruence Distributive Varieties: Let ðÞ be a congruence-distributive variety (i.e., the congruence lattice of every algebra in ðÞ is a distributive lattice). If A is a subdirectly irreducible algebra in V(K ), then A  ℍℙ ðÞ; hence, ðÞ ¼ ℙS ℍℙU ðÞ: The following cosequence of Jónsson’s Theorem is useful.

A Few Historical Glimpses into the Interplay Between Algebra and Logic. . .

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Corollary 1. (Jónsson) If  is a finite set of finite algebras and ðÞ is congruence-distributive, then the subdirectly irreducible algebras of ðÞ are in ℍðÞ, and ðÞ ¼ S ℍðÞ: • Quasivarieties of Algebras: A quasi-identity is either an identity or a first-order formula of the form ððp1  q1 Þ ^    ^ ðpn  qn ÞÞ ! ðp  qÞ: A quasivariety is a nonempty class of algebras defined by a set of quasi-identities. • Mal’cev’s Theorem: Let  be a class of L-algebras. Then the following are equivalent: (a)  can be axiomatized by quasi-identities. (b)  is a quasivariety. (c)  is closed under , , ℙ, and ℙU and cotains a trivial algebra. (d)  is closed under ℙℙU and cotains a trivial algebra. References: (Bergman 2011; Birkhoff 1935, 1944, 1948; Burris and Sankappanavar 1981; Chang and Keisler 1990; Gorbunov 1998; Grätzer 2008; Jónsson 1967; Mal’cev 1971, 1973; McKenzie et al. 1987; Tarski 1968)

The Lindenbaum-Tarski Method: The Bridge Between Algebra and Logic We now wish to describe the Lindenbaum-Tarski method which was first used by Tarski as a device for the algebraization of the classical propositional logic (CPL, for short). In order to describe the Lindenbaum-Tarski method, the following definitions are needed: (Propositional) Languages A language (or type or signature) of propositional logics is a set L of connectives such that a non-negative integer n is assigned to each member f of L. This integer is called the arity (or rank) of f, and f is called an n-ary connective. • Variables: Let Var denote the set of propositional variables {p0, p1,. . .}. • (Propositional) Formulas: The set of (propositional) L-formulas, denoted by FmL, is defined recursively as the smallest set of strings such that

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(i) If p is a variable then p  FmL, (ii) If c is a constant symbol (i.e., of arity 0), then c  FmL, (iii) If f  L with arity n ⩾ 1 and ϕ1, . . ., ϕn  FmL, then fϕ1. . .ϕn  FmL (prefix notation). Algebra of Formulas The algebra of L-formulas, denoted by FmL is the algebra which has the universe FmL and the fundamental operations are defined by: f FmL ðϕ1 , . . . , ϕn Þ ≔ f ðϕ1 , . . . , ϕn Þ, where f  L and is n-ary and ϕ1, . . ., ϕn  FmL. Valuations and Substitutions Let A be an L-algebra. Then it is easy to see that any function ν: Var ! A can be extended to a homomorphism from FmL into A, which will be denoted by the same symbol ν. Hom(FmL, A) denotes the set of all homomorphisms from FmL into A, and these homomorphisms are also called valuations (or interpretations) on A. An endomorphism δ : FmL ! FmL is called a substitution. Consequence Relations A consequence relation on FmL is a binary relation ‘ between sets of formulas and formulas that satisfies the following conditions for all Γ, Δ  FmL and ϕ  FmL: (i) ϕ  Γ implies Γ ‘ ϕ, (ii) Γ ‘ ϕ and Γ  Δ imply Δ ‘ ϕ, (iii) Γ ‘ ϕ and Δ ‘ β for every β  Γ imply Δ ‘ ϕ. A consequence relation ‘ is finitary if Γ ‘ ϕ implies Γ ‘ ϕ for some finite Γ0  Γ. Tarski’s axiomatization of logical consequence did not include the property of “structurality” (invariance under substitutions). The concept of a structural consequence operator was introduced in (Łoś and Suszko 1958). Structural Consequence Relations A consequence relation ‘ is structural if Γ ‘ ϕ implies σ(Γ) ‘ σ(ϕ) for every substitution σ, where σ(Γ) :¼ {σα : α  Γ}. Logics A logic (or deductive system) is a pair S ≔hL, ‘S i, where L is a propositional language and ‘S is a finitary and structural consequence relation on FmL. Let S be a logic and let Γ, Δ  FmL be finite. If Γ ‘S δ, for every δ  Δ, then we write Γ ‘S Δ. We also write Γ a‘S Δ if Γ ‘S Δ and Δ ‘S Γ. A rule of inference is a pair hΓ, ϕi, where Γ is a finite set of formulas (the premises of the rule) and ϕ is a formula (the conclusion).

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One way to present a logic S is by displaying it in Hilbert-style; that is, giving its axioms and rules of inference which induce a consequence relation ‘S as follows: Γ ‘S ϕ if there is a sequence of formulas ϕ1,. . ., ϕn, n  ℕ, such that ϕn ¼ ϕ, and for every i O n, one of the following conditions holds:




(i) ϕi  Γ (ii) There is an axiom ψ and a substitution σ such that ϕi ¼ σψ (iii) There is a rule hΔ, ψi and a substitution σ such that ϕi ¼ σψ and σ(Δ)  {ϕj : j < i}. If Γ ¼ 0, then we simply write ‘ ϕ. The sequence ϕ1,. . ., ϕn is then called a derivation of ϕ from Γ. S-theories If S is a logic (deductive system), a set T of formulas is called an S-theory if ϕ  T whenever T ‘ ϕ, for every ϕ  FmL. If Γ is a set of formulas, hΓiS ¼ fϕ : Γ ‘S ϕg is the smallest S-theory containing Γ. Extensions and Fragments A logic S 0 ¼ hL, ‘S0 i is an extension of the logic S ¼ hL, ‘S i if Γ ‘S0 ϕ whenever Γ ‘S ϕ, for all Γ [ {ϕ}  FmL or, equivalently, if Th S  Th S 0. Given a presentation of a logic S by axioms and rule of inference, S 0 is an axiomatic extension of S if S 0 can be axiomatized by adding axioms and keeping the existing rules of S. If S ¼ hL, ‘Si and L0  L, the L0 -fragment of S is the deductive system S0 ¼ hL0 , ‘s0 i defined by Γ ‘S 0 ϕ iff Γ ‘S ϕ, for all Γ [ fϕg  FmL0 . The Lindenbaum-Tarski Method Applied to Classical Propositional Logic The Lindenbaum-Tarski method was introduced by Tarski as a device for the algebraization of the classical propositional logic CPL. Let the language L ≔ h_, ^, !,ci. Recall that the logic CPL was defined in the section “The Hilbert-style Axiomatization of the Classical Propositional Logic (CPL)”. We define the relation ΘCPL (written simply as Θ) on the algebra FmL by: α Θ β if and only if ‘CPL α ! β and ‘CPL β ! α: Observe first that ‘CPL α ! α. Then it can be proved that Θ is an equivalence relation on FmL. Using the axioms of CPL, one can prove that Θ is a congruence on the algebra FmL. So, we can form the quotient algebra FmL/Θ ¼ hFmL/Θ, _, ^, !,ci, where the operations _, ^, !,c on FmL/Θ are as follows:

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Let α  Fm, β  Fm. Then α=Θ _ β=Θ ≔ðα _ βÞ=Θ, α=Θ ^ β=Θ ≔ðα ^ βÞ=Θ, α=Θ ! β=Θ ≔ðα ! βÞ=Θ, ðα=ΘÞc ≔ðαc Þ=Θ: Then the algebra FmL/Θ is called the Lindenbaum-Tarski algebra of CPL. Since the language of Boolean algebras as defined in the section “Boolean Algebras” contains the constants 0 and 1, we can define: 1 ≔ðα _ αc Þ=Θ, 0 ≔ðα ^ αc Þ=Θ: Note also, in a Boolean algebra, x ! y ¼ xc _ y. Again, using the axioms of CPL and modus ponens (see the section “The Hilbert-style Axiomatization of the Classical Propositional Logic (CPL)”), one can prove the following theorem: Theorem 2 (Lindenbaum-Tarski) FmL/ΘCPL is a Boolean algebra. There is a much stronger version of this theorem that applies to theories in CPL. The following corollary which follows from the above theorem gives the algebraic completeness of CPL. Corollary 2 Let α be a formula in L. Then BA ‘ α implies ‘CPL α. In fact, the stronger version of the above theorem (mentioned earlier) can be used to prove the strong algebraic completeness theorem for CPL. The above theorem acts as a bridge between CPL and the variety of Boolean algebras. The ideas used in the proof of this theorem became useful in many other logics. Lindenbaum-Tarski method, perhaps, inspired Rasiowa to arrive at “implicative logics” and Blok and Pigozzi to arrive at the precise notion of “algebraizability” and “equivalent algebraic semantics” (to be discussed later). References: (Tarski 1930a, 1930b, 1935, 1936a, 1936b, 1968, 1994, 1983; Blok and Pigozzi 1988; Citkin and Muravitsky 2021; Jansana 2016; Font 2016)

Implicative Logics Implicative logics were first introduced by Rasiowa in Rasiowa (1974). Helena Rasiowa (1917–1994) Since 1947, Rasiowa began investigating several logics by using an algebraic approach. She published her research in a series of papers and two famous books: Mathematics of Metamathematics, with Sikorski, first published in 1963 and An Algebraic Approach to Nonclassical Logics, published in 1974. In the latter, she provided a general method for constructing an algebraic semantics which associated

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a (suitable) class Alg*S of algebras with a logic S. She applied this method to a class of logics called “standard systems of implicative extensional propositional calculus” by generalizing the Lindenbaum-Tarski method. These logics will be referred to as “implicative logics.” They have played a pivotal role in the development of Abstract Algebraic Logic. Rasiowa’s work inspired many mathematicians and logicians including Czelakowski, Blok, and Pigozzi to extend and strengthen her method. We recommend that the reader consult (Fitting 1999; Bartol et al. 1995; Font 1999, 2006, 2016) for a more complete account of the impact of Rasiowa and her work. Rasiowa defined an implicative logic and its associated class of algebras as follows: Definition 2 Let S be a logic in a language L that includes a binary connective !, either primitive or defined by a term in exactly two variables. Then S is called an implicative logic with respect to the binary connective ! if the following conditions are satisfied: (IL1) ‘S α ! α, (IL2) α ! β, β ! γ ‘S α ! γ, (IL3) For each symbol f  L of arity n ⩾ 1,   α1 ! β 1 , . . . , αn ! β n , ‘S f ðα1 , . . . , αn Þ ! f ðβ1 , . . . , βn Þ, β1 ! α1 , . . . , βn ! αn (IL4) α, α ! β ‘S β, (IL5) α ‘S β ! α: Definition 3 Let S be an implicative logic in the language L with an implication connective !. An S-algebra is an algebra A in the language L that has an element 1 with the following properties: (LALG1) For all Γ [ {ϕ}  Fm and all h  H om(FmL, A), if Γ ‘S ϕ and hΓ  {1} then hϕ ¼ 1, (LALG2) For all a, b  A, if a ! b ¼ 1 and b ! a ¼ 1 then a ¼ b. The class of S-algebras is denoted by Alg*S. The usefulness of this class of algebras is shown by the following theorem: Theorem 3 ((Rasiowa 1974), (Font 2016, Theorem 2.9)) If S is an implicative logic, then S is complete with respect to the class Alg*S in the following sense: For all Γ [ {ϕ}  FmL, Γ ‘S ϕ iff hΓ  f1g implies hϕ ¼ 1,

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for all h  Hom (FmL, A) and for all A  Alg*S. References: (Bartol et al. 1995; Rasiowa and Sikorski 1963; Rasiowa 1974; Fitting 1999; Font 1999, 2006, 2016) Examples of Implicative Logics (Some Known and Some New) Let α!H β ≔ α ! (α ^ β). Num. (1) (2) (3) (4)

L {_, ^, {_, ^, {_, ^, {_, ^,

(5) (6) (7) (8)

{!} {!} {_, ^, !, ⊥, ⊤} {_, ^, !, 0 , ⊥, ⊤}

(9)

{_, ^, !, 0 , ⊥, ⊤}

(10)

{_, ^, !, 0 , ⊥, ⊤}

(11)

{_, ^, !, 0 , ⊥, ⊤}

(12) (13)

{_, ^, !, 0 , ⊥, ⊤} {_, ^, !, 0 , ⊥, ⊤}

(14)

{_, ^, !, 0 , ⊥, ⊤}

(15) (16) (17) (18)

{_, ^, {_, ^, {_, ^, {_, ^,

Alg L Boolean algebras Heyting algebras Generalized Heyting algebras Contrapositionally complemented lattices Positive implicational logic Hilbert algebras Implicational fragment of CPL Implication algebras Semi-intuitionistic logic Semi-Heyting algebras Dually hemimorphic Dually hemimorphic Heyting intuitionistic logic algebras Dually quasi-De Morgan semi- Dually quasi-De Morgan semiintuitionistic logic Heyting algebras Dually quasi-De Morgan Dually quasi-De Morgan intuitionistic logic Heyting algebras De Morgan semi-intuitionistic De Morgan semi-Heyting logic algebras De Morgan intuitionistic logic De Morgan Heyting algebras Dually pseudocomplemented Dually pseudocomplemented semi-intuitionistic logic Semi-Heyting algebras Dually pseudocomplemented Dually pseudocomplemented intuitionistic logic Heyting algebras Dually Stone intuitionistic logic Dually Stone Heyting algebras GAUTAMA Gautama algebras RDBLSt Regular double Stone algebras RKLSt Dually Kleene Stone algebras

Name of the logic :, ⊥, ⊤} Classical Propositional Logic !, :, ⊥, ⊤} Intuitionistic Propos. Logic !} Positive logic !, :} Johansson’s minimal logic

!, 0 , ⊥, ⊤} , 0 , ⊥, ⊤} , 0 , ⊥, ⊤} , 0 , ⊥, ⊤}

Remark 3 For definitions of logics given in (1)–(6), see Rasiowa (1974) or Font (2016), for the one in (7) see Cornejo and Viglizzo (2015), for those in (8)–(15), see Cornejo and Sankappanavar (2022), and for the ones in (16)–(18), see PART II of this chapter. References: (Rasiowa and Sikorski 1963; Rasiowa 1974; Font 2006, 2016; Cornejo and Viglizzo 2015; Cornejo and Sankappanavar 2022) A New Example of an Implicative Logic: Dually Hemimorphic SemiIntuitionistic Logic (Cornejo and Sankappanavar 2022) introduced a new example of an implicative logic which arose in an attempt to “logicize” the variety of dually hemimorphic semi-

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Heyting algebras defined in the section “The Variety GH of Gautama Heyting Algebras” of PART II. This logic will also play a crucial role in section “A Logic for Gautama Algebras” of PART II. (Cornejo and Sankappanavar 2022) defined the dually hemimorphic semiHeyting logic, DHMSH, as follows: The Logic DHMSH Let α!H β ≔ α ! (α ^ β) and let α$H β ≔ (α!H β) ^ (β!H α). Language: h_, ^, !, 0, ⊥, ⊤i Axioms: 1. α!H ðα _ βÞ, 2. β!H ðα _ βÞ, 3. ðα!H γ Þ!H ½ðβ!H γ Þ!H ðα _ βÞ!H γ Þ, 4. ðα ^ βÞ!H α, 5. ðγ !H αÞ !H ½ðγ !H βÞ !H ðγ !H ðα ^ βÞÞ, 6. ⊤, 7. ⊥!H α, 8. ½ðα ^ βÞ!H γ !H ½α!H ðβ!H γ Þ, 9. ½α!H ðβ!H γ Þ!H ½ðα ^ βÞ!H γ , 10. ðα!H βÞ!H ½ðβ!H αÞ!H ððα ! γ Þ!H ðβ ! γ ÞÞ, 11. ðα!H βÞ!H ½ðβ!H αÞ!H ððγ ! βÞ!H ðγ ! αÞÞ, 12. ⊤!H ⊥0 , 13. ⊤0 !H ⊥, 14. ðα ^ βÞ0 $H ðα0 _ β0 Þ: Rules of Inference (SMP) From ϕ and ϕ!H γ, deduce γ (semi-Modus Ponens), (SCP) From ϕ!H γ, deduce γ0!H ϕ0 (Contraposition). Theorem 4 (Cornejo and Sankappanavar 2022) The logic DHMSH is implicative with respect to the connective !H. For more examples of implicative logics, see the section “A Logic for Gautama Algebras” of PART II of this chapter as well as (Cornejo and Sankappanavar 2022).

Abstract Algebraic Logic (AAL) Willem Johannes Blok (1947–2003) and Don Pigozzi The discovery of many-valued logics, modal logics, intuitionistic logic, etc. created the need for a unifying context in which common generalizations of similar results found in “concrete” logics could be stated and proved. In 1989, Blok and Pigozzi introduced one such unifying context called Abstract Algebraic Logic (AAL, for short). AAL is also concerned with the process of algebraization itself. The motivation for this work was to find an appropriate context for investigating the “Deduction Theorem” from an algebraic point of view (see Font et al. (2000)).

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Informally speaking, a class  of algebras is an algebraic semantics, in the sense of Blok and Pigozzi, of a logic S if the consequence relation ‘S can be interpreted in ‘ by means of an “equational translation.” The interpretation will then yield a strong completeness theorem for the logic S relative to the class . Blok and Pigozzi were actually interested in a stronger notion of an “equivalent algebraic semantics” of S; that is, a notion that not only requires S to be an algebraic semantics, but also requires that there be an “inverse translation” of ‘K into ‘S. A logic is algebraizable if it possesses an equivalent algebraic semantics. With this stronger notion, Blok and Pigozzi were able to prove theorems saying that a metalogical property P holds in an algebraizable logic S iff an algebraic property P0 holds in the equivalent algebraic semantics of S – the so-called bridge theorems. For example, the “Deduction theorem” is one such property of algebraizable logics. A precise description of these notions will be given below in the section “Algebraic Semantics, Equivalent Algebraic Semantics, and Algebraizability”. References: (Blok and Pigozzi 1989a; Blok and Rebagliato 2003; Blok and Pigozzi 1991a, 1991b; Font 2016; Jansana 2016; Font et al. 2000; Czelakowski 1981) Algebraic Semantics, Equivalent Algebraic Semantics, and Algebraizability The notions of an equivalent algebraic semantics and an algebraizable logic are among the central notions of AAL. These were first introduced and studied in Blok and Pigozzi (1989a). In order to define these notions of equivalent algebraic semantics and algebraizable logics, we need the following definitions: Equational Consequence Relation For a class  of L-algebras, we define the relation ‘ that holds between a set Δ of identities and a single identity α ≈ β as follows: Δ ‘ α  β if and only if for every A   and every interpretation ā of the variables of Δ [ {α ≈ β} in A, if ϕA(ā) ≈ ψ A(ā) for every ϕ ≈ ψ  Δ, then αA(ā) ¼ βA(ā). (ā denotes the sequence ha1, a2, . . . , ani.) In this case, we say that α ≈ β is a -consequence of Δ. The relation ‘ from sets of identities to identities is called the semantic equational consequence relation determined by . (in other words, fϕi  ψ i : i  I g ‘ ϕ  ψ if and only if for every A   and every homomorphism h : FmL ! A, we have: h(ϕi) ¼ h(ψ i) for every i  I implies h(ϕ) ¼ h(ψ)).

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If  and 0 are sets of equations, then we write  ‘K 0 for the conjunction of the statements “ ‘K ϕ ≈ ψ”, where ϕ ≈ ψ  0. We also write  ‘K 0, if  ‘K 0 and 0 ‘K . The relation ‘K is finitary if {ϕi ≈ ψ i : i  I} ‘ K ϕ ≈ ψ implies {ϕi ≈ ψ i : i  J} ‘ K ϕ ≈ ψ for some finite J  I. The rest of this chapter will be concerned only with relations ‘K which are finitary. ‘

Algebraic Semantics The following definition describes the precise notion of an algebraic semantics which was introduced in Blok and Pigozzi (1989a). Definition 4 Let S be a logic and  a class of algebras.  is called an algebraic semantics (in the sense of Blok and Pigozzi) of S if there exists a finite set {δi ( p) ≈ ϵ i ( p) : i O n} of identities in a single variable p satisfying the condition: For all Γ [ {ϕ}  FmL and each j O n, Γ ‘S ϕ if and only if fδi ðγ Þ  ϵ i ðγ Þ : γ  Γ, iOng ‘K δ j ðϕÞ  ϵ j ðϕÞ: The equations δi( p) ≈ ϵ i( p), i O n, are called the defining equations for the algebraic semantics. Proposition 5 (Blok and Pigozzi 1989a) If  is an algebraic semantics of a logic S, then so is the quasivariety QðÞ, with the defining equations being the same as those for . Let L ¼ h_, ^, !, :, ⊥, ⊤i and let CPL ¼ hL, ‘CPLi denote the classical propositional   A  calculus and let  denote the variety of Boolean algebras. As : A   g is a matrix semantics of CPL, we have that  is an A, ⊤ algebraic semantics of CPL with the defining equation p ≈ ⊤. Similarly, the variety ℍ of Heyting algebras is an algebraic semantics for the intuitionistic propositional calculus CPL ¼ hL, ‘CPLi, with the same defining equation. Equivalent Algebraic Semantics and Algebraizable Logic: Definition 5 Let S be a logic over a language L and  an algebraic semantics of S with defining equations δi( p) ≈ ϵ i( p), i O n. Then, K is an equivalent algebraic semantics of S if there exists a finite set {Δj( p, q) : j O m} of formulas in two variables satisfying the condition: For every ϕ ≈ ψ  EqL, =||=K

The set {Δj( p, q) : j O m} is called an equivalence system. A logic is algebraizable (in the sense of Blok and Pigozzi) if it has an equivalent algebraic semantics.

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Both the classical propositional logic CPL and the intuitionistic propositional calculus IPL are algebraizable, with the varieties of Boolean algebras and Heyting algebras as their equivalent algebraic semantics, respectively, both with defining equation {p ≈ ⊤} and equivalence system {p ! q, q ! p}. Proposition 6 (Blok and Pigozzi 1989a) If  is an equivalent algebraic semantics of a logic S, then so is ðÞ, the quasivariety generated by . Proposition 7 Let  be an algebraic semantics for a logic S. Then  is equivalent to S if and only if ðÞ is. Proposition 8 (Blok and Pigozzi 1989a) An equivalent quasivariety semantics of a logic, if it exists, is unique.  and ℍ are, in fact, equivalent algebraic semantics of CPL and IPL, respectively. The following theorem shows that Rasiowa’s implicative logics provide a class of examples of algebraizable logics and was proved in Blok and Pigozzi (1989a), Theorem 5 (Blok and Pigozzi 1989a), (Font 2016) Every implicative logic S is algebraizable with respect to the class Alg*S and the algebraizability is witnessed by the defining identity p ≈ p ! p and the equivalence formulas Δ ¼ {p ! q, q ! p}. References: (Blok and Pigozzi 1989a; Blok and Rebagliato 2003; Font 2016; Font et al. 2003) Axiomatic Extensions of Algebraizable Logics Let S and S0 be logics in the language L. The logic S 0 is an axiomatic extension of S if S0 is obtained by adjoining new axioms but keeping the rules of inference the same as in S. Let ExtðS Þ denote the lattice of axiomatic extensions of a logic S and LV ðÞ denote the lattice of subvarieties of a variety  of algebras. The following important theorems were first proved by Blok and Pigozzi (1989a). Theorem 6 (Blok and Pigozzi 1989a), (Font 2016, Theorem 3.33) Let S be an algebraizable logic with the variety  as its equivalent algebraic semantics. Then ExtðS Þ is dually isomorphic to LV ðÞ. Theorem 7 (Blok and Pigozzi 1989a) Let S be an algebraizable logic and S 0 be an axiomatic extension of S . Then S 0 is also algbraizable with the same defining identities and equivalence formulas as those of S. Theorem 5, Theorem 6, and Theorem 7 will be useful in PART II. References: (Blok and Pigozzi 1989a; Blok and Rebagliato 2003; Font 2016) Examples of Logics and Their Equivalent Algebraic Semantics

We will now give some logics which are known to be algebraizable along with their equivalent algebraic semantics. The beauty of knowing the variety of algebras

A Few Historical Glimpses into the Interplay Between Algebra and Logic. . .

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corresponding to an algebraizable logic is that it provides a new tool in that the problems in logic could be solved using algebraic methods and vice-versa. First we note that all the implicatve logics listed in the section “Implicative Logics” are algebraizable logics. Below, we give several algebraizable logics and their equivalent algebraic semantics. Logic Modal logic K Modal logic S4 Any super-intuitionistic logic Gödel Logic n-valued Łukasiewicz logic Linear logics Moisil’s Logique modale Nelson logic Paraconsistent Nelson logic N4 BCK-logic

Equivalent Algebraic Semantics Modal algebras Interior algebras (The corresponding) subvariety of Heyting algebras Linear Heyting algebras n-valued Łukasiewicz-Moisil algebras Commutative residuated lattices De Morgan Heyting algebras Nelson algebras N4-lattices The quasi-variety of ℂ-algebras

Note: On the right column, all, except the last one, are varieties. Nelson logic is also known as “Constructive logic with strong negation.”

Examples of Logics That Are Not Algebraizable

Here are some logics that are not algebraizable: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

Quasi-classical modal logics, The system E of Entailment, da Costa’s paraconsistent logic C1, The local consequence of the normal modal logic K, The logic R!, the implication fragment of relevance logic, The fragment of classical logic with conjunction only as the connective, The fragment of the classical logic with only conjunction and disjunction, The Belnap-Dunn logic, The implicational fragment of intuitionistic linear logic, Logics with no theorems, BCI-logic.

References: (Font 2016; Blok and Pigozzi 1989a; Font et al. 2003) Bridge Theorems Bridge theorems are the results that connect metalogical properties of logics with algebraic properties of their corresponding (quasi)varieties. For an algebraizable logic S with a quasivariety ℚ of algebras as its equivalent semantics, a bridge theorem says: S satisfies a property P if and only if ℚ has the property P0. Bridge theorems certainly indicate that AAL is an area worth exploring. They clearly express the interplay between logic and algebra.

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Examples of Bridge Theorems (a) P: ¼ “Deduction theorem” and P0 : ¼ “uniformly equationally definable relative principal congruences,” (b) P: ¼ “Craig interpolation property” and P0 : ¼ “the amalgamation property,” (c) P: ¼ “Beth definability property” and P0 : ¼ “epimorphisms are surjective.” References: (Font 2016; Blok and Pigozzi 1989a; Font et al. 2003) Logicization of Algebras As mentioned earlier, in the early 1920s, Peirce, Lukasivicz, and Post introduced many-valued logics by means of logical matrices. The axiomatizations for these logics were discovered later. The problem of logicization of algebras is clearly the opposite of that of algebraizing logics. Given a quasivariety (or a variety)  of algebras, logicizing V is the problem of finding a logic S such that S is algebraizable with  as its algebraic semantics. Thus, the logic S is the “corresponding logic” to the quasivariety . To the best of our knowledge, the problem of logicization doesn’t seem to have received much interest. Nevertheless, some examples of logicizable varieties are given below. Of these, the last three are taken from PART II of the present chapter. Number (1) (2) (3) (4) (5) (6) (7) (8) (9)

Variety of algebras Semi-Heyting algebras De Morgan semi-Heyting algebras Dually pseudocomplemented semi-Heyting algebras Dually pseudocomplemented Heyting algebras Dually hemimorphic semi-Heyting algebras Semi-Nelson algebras Gautama algebras Regular double-Stone algebras Regular Kleene-Stone algebras

The corresponding logic Semi-intuitionistic logic De Morgan semi-intuitionistic logic Dually pseudocomplemented semiintuitionistic logic Dually pseudocomplemented intuitionistic logic Dually hemimorphic semi-intuitionistic logic Semi-Nelson logic GAUTAMA RDBLSt RKLSt

References: (Cornejo 2011; Cornejo and Viglizzo 2018a; Cornejo and Sankappanavar 2022; Font 2016) Further Progress in AAL The class of algebraizable logics (in the sense of Blok and Pigozzi) have been further generalized to several new classes of logics that include equivalential logics, protoalgebraic logics, algebraizable logics (in a more general sense), assertional logics, truth-equational logics, selfextensional logics, Fregean logics and so on. These classes have been classified into two hierarchies, called Leibniz Hierachy and Frege Hierachy. The discussion of these hierarchies, their mutual relation and related ideas is beyond the scope of the present chapter.

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References: (Czelakowski 1981; Czelakowski and Jansana 2000; Czelakowski 2001, 2003; Czelakowski and Pigozzi 2004a, 2004b; Czelakowski 2018; Czelakowski and Pigozzi 1999; Blok and Pigozzi 1986; Czelakowski 2001 and Font; Jansana and Pigozzi 2003; Cintula and Noguera 2011; Font and Jansana 1996; Jansana 2005, 2006; Jansana and Palmigiano 2006; Font 2016)

The Interaction Between Logics and Algebras Arising from Rough Set Theory In 1982, Pawlak introduced rough set theory to deal with vague or incomplete information. Following Pawlak’s work, there has been a considerable amount of research activity in this area by the members of Rasiowa-Pawlak School, as well as researchers from elsewhere. Right from the inception of the theory, researchers began investigating both logical and algebraic aspects of rough sets. All that research has resulted in a large body of publications and several books. Some of the early contributions to this area were made by Rasiowa, Iwiniski, Pomykala, Skowron, Orlowska, Cattaneo, Pagliani, Rauszer, Polkowski, and Obtulowicz. A group of Indian logicians and mathematicians, led by M.K. Chakraborty and M. Banerjee, has also been pursuing vigorously the research into the interconnections between algebras arising from rough sets and logics for the last 25+ years. The chapter (Banerjee 2021) in this Handbook gives an overview of the progress achieved so far, worldwide, in this area. References: (Banerjee 2021; Mani et al. 2018; Orlowska 1998; Pawlak 1982, 1991; Pagliani and Chakraborty 2008; Skowron and Suraj 2013)

PART II: Investigations into Gautama Algebras This part presents new examples of the interplay between algebra and logic. Firstly, a new equational class of algebras, called “Gautama algebras” is defined and named in honor and memory of the two founders of Indian Logic– Medhatithi Gautama and Akshapada Gautama. The variety  of Gautama algebras is a common generalization of the variety of regular double Stone algebras and that of regular De Morgan Stone algebras, both of which are generalizations of Boolean algebras. Secondly, we give an explicit description of subdirectly irreducible Gautama algebras and the lattice of subvarieties of the variety of Gautama algebras. We also introduce another variety ℍ of Gautama Heyting algebras and show that it is term-equivalent to the variety of Gautama algebras. Thirdly, a new propositional logic, called GAUTAMA, is defined and shown to be algebraizable with the variety  as its equivalent algebraic semantics. Finally, the axiomatic extensions RDBLSt and RKLSt of the logic GAUTAMA are defied and shown to have ℝt and ℝt , respectively, as their equivalent algebraic semantics. PART II concludes with some open problems for further investigation.

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Introduction and Motivation For PART II, it is assumed that the reader has had some familiarity with lattice theory and universal algebra (see (Balbes and Dwinger 1974; Burris and Sankappanavar 1981), for example). As such, for notions, notations and results assumed here, the reader can refer to these or other relevant books. Some familiarity with the first-order logic would also be helpful. It is clear from PART I that Boolean algebras have been the springboard for many new classes of algebras. Recall that an algebra A ¼ hA, _, ^, c, 0, 1i is a Boolean algebra if A is a complemented distributive lattice. Let us look a little further into Boolean algebras to motivate some of the concepts needed in what follows. Let  denote the variety of Boolean algebras. The following theorem is wellknown. Theorem 8 (a) Let A be a Boolean algebra. Then, the following statements are equivalent: 1. A is simple, 2. A is subdirectly irreducible, 3. A is finitely subdirectly irreducible, 4. A is directly indecomposable, 5. A ffi 2. (b)  ¼ ð2Þ (i.e., the variety generated by {2}). The following 2-element algebra with universe {0, 1}, denoted by 2, is the “simplest” Boolean algebra (Fig. 1).

Some (Known) Weakenings of the Boolean Complement The following observation is well-known and easy to prove: Proposition 9 Let A be a Boolean algebra with a  A, and let ac denote the complement of a. Then (i) ac is the largest element x such that a ^ x ¼ 0 (i.e., a ^ x ¼ 0 iff x O ac). (ii) ac is also the smallest element x  A such that a _ x ¼ 1 (i.e., a _ x ¼ 1 iff x ⩾ ac). (iii) The operation c satisfies: (a) (x _ y)c ≈ xc ^ yc and (x ^ y)c ≈ xc _ yc (De Morgan laws), and (b) xcc ≈ x (involution, or double negation).

2 : 0

Fig. 1

^

1

0

1

^

0

1

0

0

1

0

0

0

1

1

1

1

0

1

c

0

1

1

0

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The above proposition has led to the following three well-known generalizations of the Boolean complement: Item (i) of Proposition 9 has led to the notion of pseudocomplement and, thereby, to pseudocomplemented lattices; let denote the pseudocomplement. Item (ii) of Proposition 9 has led to the notion of dual pseudocomplement, and to dually pseudocomplemented lattices; let + denote the dual pseudocomplement. Item (iii) of Proposition 9 has led to the notion of De Morgan complement, and to De Morgan algebra; let 0 denote the De Morgan complement.

Algebras Based on The 3-element Chain Since the lattice reduct of the Boolean algebra 2 is the 2-element chain and 2 generates the variety of Boolean algebras, it is only natural to consider the above-mentioned operations on a 3-element chain (viewed as a bounded distributive lattice) denoted by 3. The three-element chain and the three operations mentioned above are shown in Fig. 2. Algebras on the 3-Element Chain with One Additional Unary Operation It is natural to consider the expansion of the language h_, ^, 0, 1i of bounded distributive lattices by adding just one of these unary operations. This leads to the following three well-known algebras on the chain 3: 1. 3st ≔ h3, _, ^, , 0, 1i-this is known as a Stone algebra. 2. 3dst ≔ h3, _, ^, +, 0, 1i-this is a dual Stone algebra. 3. 3kl ¼ h3, _, ^, 0, 0, 1i-this is a Kleene algebra. The varieties generated by 3st, 3dst, and 3kl are known, respectively, as those of Stone algebras, dual Stone algebras and Kleene algebras. We will denote these varieties by t, t and , respectively. t and  have been researched well. Consequently, there is a fair amount of literature on them. According to Problem 70 (on page 149) of Birkhoff (1948), the following problem was proposed by M.H. Stone: What is the most general pseudocomplemented distributive lattice in which a _ a ¼ 1 identically? In 1957, Stone algebras were introduced in Grätzer and Schmidt (1957) to answer the above problem of Stone. In order to define Stone algebras, we need the notion of a pseudocomplemented lattice. It is clear that the definition of pseudocomplement as given earlier is not 1

Fig. 2 3 :

a 0

*: 0 a 1 0

1 0

+

0 a 1 1

1 0

0 a 1 a

1 0

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equational. However, in 1949, Ribenboim (1949) proved that the class of pseudocomplemented lattices is a variety. The following axiomatization is a slight modification of the one given in Balbes and Dwinger (1974). An algebra A ¼ hA, _, ^, , 0, 1i is a distributive pseudocomplemented lattice ( p-algebra for short) if 1. hA, _, ^, 0, 1i is a bounded distributive lattice. 2. The operation satisfies the identities: (a) 0 ≈ 1, (b) 1 ≈ 0, (c) x ^ (x ^ y) ≈ x ^ y . A p-algebra A is a Stone algebra if A satisfies the identity: 3. x _ x ≈ 1 (Stone identity). In what follows, the symbol 2 denotes a two-element algebra whose signature, though varies, will be clear from the context where it appears. The following theorem is also well-known. Let t denote the variety of Stone algebras. Theorem 9 1. 2 and 3st are the only non-trivial subdirectly irreducible Stone algebras. 2. The variety t ¼ ð3st Þ. Dual Stone algebras are, of course, defined dually. Kleene algebras are wellknown too. They form a subvariety of De Morgan algebras, which were first introduced by Moisil in 1935 (see Moisil (1972)). They were further investigated in Bialynicki-Birula and Rasiowa (1957) and in Kalman (1958). An algebra hA, _, ^, 0, 0, 1i is a De Morgan algebra if 1. 2. 3. 4.

hA, _, ^, 0, 1i is a bounded distributive lattice. 00 ≈ 1 and 10 ≈ 0. (x ^ y)0 ≈ x0 _ y0 ( ^-De Morgan law). x00 ≈ x (Involution).

A De Morgan algebra is a Kleene algebra if it satisfies: 5. x ^ x0 O y _ y0 (Kleene identity). The following theorem is also well-known. Let  denote the variety of Kleene algebras. Theorem 10 (Bialynicki-Birula and Rasiowa 1957; Kalman 1958) 1. The only non-trivial subdirectly irreducible Kleene algebras are 2 and 3kl. 2. The variety  ¼ ð3kl Þ; 3.   .

Algebras on the 3-Element Chain with Two Additional Unary Operations The next natural step in this development was to consider the expansion of the language h_, ^, 0, 1i by adding two of the above three unary operations on the 3-element chain. This leads us to the following three algebras on the 3-element chain:

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(a) 3dblst ¼ h3,_, ^, , +, 0, 1i: This is a Double Stone algebras. Varlet (1972) and Katriňák (1973) have observed that it also satisfies an additional identity, which we shall call “regular Identity”: (R) x ^ x+ O y _ y . So, 3dblst is a “regular double Stone algebra”. (b) 3klst ¼ h3, _, ^, , 0 , 0, 1i: This is a Kleene Stone algebra (see Romanowska (1981) and Sankappanavar (1987)). This algebra also satisfies an interesting identity, also called “Regular Identity”: (R1) x ^ x0 0 O y _ y . So, it is a “regular Kleene Stone algebra”. (c) 3dstkl ¼ h3, _, ^, +, 0 , 0, 1i: This would not be of much interest to us. Thus, (a) and (b) lead us to the varieties of regular double Stone algebras and regular Kleene Stone algebras. The following definition of regular double Stone algebras is well-known. Definition 6 An algebra A ¼ hA, _, ^, , +, 0, 1i is a regular double Stone algebra if 1. hA, _, ^, , 0, 1i is a Stone algebra, 2. hA, _, ^, +, 0, 1i is a dual Stone algebra, 3. A satisfies the identity: (R) x ^ x+ O y _ y . The variety of regular double Stone algebras is denoted by ℝt. The following theorem is also well-known. Theorem 11 (Sankappanavar 1985) (i) Let A  ℝt with |A| > 1. Then, the following statements are equivalent: (a) A is simple, (b) A is subdirectly irreducible, (c) A is finitely subdirectly irreducible, (d) A is directly indecomposable, (e) A  {2, 3dblst}, up to isomorphism. (ii) The variety ℝt ¼ Vð3dblst Þ. (iii) The variety ℝt is a discriminator variety. (iv) 3dblst is quasiprimal. (v)  is the only nontrivial subvariety of ℝt. The following definition of regular Kleene Stone algebras is also well-known. Definition 7 An algebra A ¼ hA, _, ^, , 0 , 0, 1i is a regular Kleene Stone algebra if

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1. hA, _, ^, , 0, 1i is a Stone algebra, 2. hA, _, ^, 0, 0, 1i is a Kleene algebra, 3. A satisfies the identity: (R1) x ^ x0 0 O y _ y . The variety of regular Kleene Stone algebras is denoted by ℝt. The following theorem lists some of the known properties of the variety ℝt. Theorem 12 (Sankappanavar 2011) (i) Let A  ℝt with |A| > 1. Then, the following statements are equivalent: (a) A is simple, (b) A is subdirectly irreducible, (c) A is finitely subdirectly irreducible, (d) A is directly indecomposable, (e)   f2, 3klst g, up to isomorphism. (ii) The variety ℝt ¼ ð3klst Þ. (iii) The variety ℝt is a discriminator variety. (iv) 3klst is quasiprimal. (v)  is the only nontrivial subvariety of ℝt. Remark 4 It is easy to verify that the algebra 3dblst also satisfies the identity (R1) and hence the variety ℝt also satisfies (R1). In the rest of PART II, “regular” refers only to (R1).

A New Problem In view of the similarities of ℝt and ℝt, the following problem arises naturally. Problem: Find a variety  of algebras hA, _, ^, , 0 , 0, 1i such that (i) The variety of regular double Stone algebras is a subvariety of . (ii) The variety of regular Kleene Stone algebras is a subvariety of . (iii)  is “close enough” to both of these varieties. Since ℝt ¼ ð3dblst Þ and ℝt ¼ ð3klst Þ, the above problem reduces to finding “enough” identities that are true in both 3dblst and 3klst.

Toward a Solution of the Above Problem Observe: 1. Algebras 3dblst and 3klst have Stone algebra reducts. 2. It is straightforward to verify that the operation + of 3dblst and the operation 0 of 3klst satisfy the following identities:

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(a) 00 ≈ 1 and 10 ≈ 0, (b) (x ^ y)0 ≈ x0 _ y0, (c) (x _ y)00 ≈ x00 _ y00, (d) x00 O x. 3. In view of Remark 4, both algebras satisfy the following identity: (R1): x ^ x0 0 O y _ y . 4. It is easy to verify that both algebras 3dblst and 3klst also satisfy the identity: ð Þ

x 0  x :

Now, we note that the identities mentioned in item (2) above are actually duals of the identities that define the variety of (upper) quasi-De Morgan algebras first defined in (Sankappanavar 1987b) as a subvariety of the variety of “Semi-De Morgan algebras.” Thus, the condition (2) above can be replaced by: (20 ) the h_, ^, 0 , 0, 1i-reduct of each of 3dblst and 3klst is a dually quasi-De Morgan algebra.

The Variety of Gautama Algebras In this section we define and investigate a new variety of algebras, which we will call “Gautama algebras” in honor and memory of the two Gautamas, the founders of Indian Logic. Let us start with the definition of a dually quasi-De Morgan algebras (Sankappanavar 1987b). An algebra hA, _, ^, 0 , 0, 1i is a dually quasi-De Morgan algebra if the following conditions hold: (i) (ii) (iii) (iv)

00 ≈ 1 and 10 ≈ 0, (x ^ y)0 ≈ x0 _ y0, (x _ y)00 ≈ x00 _ y00, x00 O x.

Definition 8 An algebra A ¼ hA, _, ^, , 0 , 0, 1i is a Gautama algebra if the following conditions hold: (a) hA, _, ^, , 0, 1i is a Stone algebra, (b) hA, _, ^, 0 , 0, 1i is a dually quasi-De Morgan algebra, (c) A is regular; i.e., A satisfies the identity: (R1) x ^ x0 0 O y _ y , (d) A is star-regular; i.e., A satisfies the identity: ( ) x 0 ≈ x . Let  denote the variety of Gautama algebras. Clearly, 2, 3dblst, 3klst are examples of algebras in ; and so, the varieties , ℝt, and ℝt are subvarieties of the variety .

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The following lemmas will be useful in proving Theorem 13. Lemma 1 Let A   and let a  A such that a ^ a0 ¼ 0. Then (i) (a] ¼ h(a], _, ^, a, 0 a, 0, ai  G, where x a ≔ x ^ a and x0a ≔ x0 ^ a for x  (a]. (ii) The function x 7! x ^ a is a homomorphism of A onto (a]. Proof Let x  (a]. Then x ^ (x ^ y) a ¼ x ^ (x ^ y) ^ a ¼ x ^ y ^ a ¼ x ^ y a. Also, a a ¼ a ^ a ¼ 0. 0 a ¼ 0 ^ a ¼ 1 ^ a ¼ a. So, x a is the pseudocomplement of x  (a]. Since satisfies (St), we have x a _ x a a ¼ (x ^ a) _ ((x ^ a) ^ a) ¼ (x ^ a) _ ((x _ a ) ^ a) ¼ (x ^ a) _ (x ^ a) ¼ (x _ x ) ^ a ¼ 1 ^ a ¼ a. So, Stone identity holds in (a]. The verification of dually quasi-De Mogan identities, (R1), ( ) and (ii) are left to the reader to verify. □ Lemma 2 Let a, b  A such that 0 < a < b < 1. Let A   satisfy: (S) For every x  A, x _ x ¼ 1 implies x ¼ 0 or x ¼ 1. Then, b0 ¼ 0 and a ¼ 0. Proof If there is an x  G such that x 6¼ 1 and x 6¼ 0, then by (S) we would have x _ x 6¼ 1, implying 1 6¼ 1, in view of Stone identity, leading to a contradiction. So, we get (A)

For every x  A,

x ¼ 1 or x ¼ 0.

So b0 ¼ 1 or b0 ¼ 0. Hence b0 ^ b0 ¼ b0 or b0 ¼ 0; that is, b0 ¼ 0 or b0 ¼ 0, i.e., b00 ¼ 1 or b0 ¼ 0, implying b ¼ 1 or b0 ¼ 0, But b 6¼ 1, so, b0 ¼ 0, proving the first half of the conclusion. Now, from (A), we have a ¼ 1 or a ¼ 0. So, a ^ a ¼ a or a ¼ 0, implying a ¼ 0 or a ¼ 0. Since, a 6¼ 0 by the hypothesis, we obtain a ¼ 0, proving the second half of the conclusion. □ Lemma 3 Let A  , and a  A such that a _ a ¼ 1. Then a ¼ a, and a0 ¼ a . Proof From a ^ a ¼ 0, we get (a _ a ) ^ (a _ a ) ¼ a, which, together with the hypothesis, implies a O a, whence a ¼ a, Hence, from the axiom ( ), we get a0 ¼ a 0 ¼ a ¼ a . Thus, a0 ¼ a . □ The following theorem gives a concrete description of the subdirectly irreducible algebras in the variety . Theorem 13 Let A  . Then the following are equivalent: 1. A is simple. 2. A is subdirectly irreducible.

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3. A is directly indecomposable. 4. For every x  A, x _ x ¼ 1 implies x ¼ 0 or x ¼ 1. 5. A  {2, 3dblst, 3klst}, up to isomorphism. Proof It is well-known that (1) ) (2) ) (3). To prove (3) ) (4), suppose (4) is false. Then there exists an a  A such that a _ a ¼ 1, a 6¼ 0 and a 6¼ 1. From Lemma 3, we have a0 ¼ a . Hence a ^ a0 ¼ 0. So, by Lemma 1, we have (a]  , and (a ]  , since a ^ a 0 ¼ 0 by ( ). Then the map x 7! hx ^ a, x ^ a i is easily seen to be an isomorphism. Hence A ffi (a]  (a ], implying (3) is false, completing the proof of (3) ) (4). Next, to prove (4) ) (5), we assume (4). First, we claim that the height of the lattice reduct of A O 2. Suppose the claim is false. Then there exist a, b  A such that 0 < a < b < 1. In view of the axioms ( ) and (R1), we get (b ^ b0 ) _ (a _ a ) ¼ (b ^ b0 0) _ (a _ a ) ¼ (a _ a ), hence b ^ b0 O a _ a . But we know that a ¼ 0 and b0 ¼ 0 from Lemma 2. Hence, b ^ 1 O a _ 0, that is, b O a, which is a contradiction, as a < b. This proves the claim. Thus the height of the lattice reduct of A is O 2. Now it is easy to see that the only nontrivial algebras in  of height O 2, up to isomorphism, are: the 2, 3dblst, 3klst and 22. But the algebra 22 does not satisfy (4). Thus (5) holds. Finally, it is routine to check that 2, 3dblst, 3klst are indeed simple. Thus the proof of the theorem is complete. □ The following corollaries are immediate from Theorem 13. Corollary 3  ¼ ð3dblst , 3klst Þ ¼ ð3dblst Þ _ ð3klst Þ. The above corollary says that the variety  is as close to 3dblst and 3klst as it can possibly be. (This completely answers the problem raised in the section “A New Problem.”) Corollary 4 If A   and A is finite, then A is a finite direct product of copies of 2, 3dblst, and 3klst. Corollary 5 The equational theory of  is decidable. Corollary 6 The lattice of nontrivial subvarieties of  is isomorphic to the 4-element Boolean lattice.

Equational Bases for Subvarieties of  It is easy to see that 2 is the only subdirectly irreducible Gautama algebra that satisfies x ≈ x0 . Hence the following theorem is immediate from Theorem 13. Theorem 14 The variety ð2Þð¼ Þ is defined, modulo , by the identity: x  x0 :

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Since 3dblst satisfies the identities (i), (ii), (iii) listed in Theorem 15, while 3klst does not satisfy them, the following theorem follows from Theorem 13. Theorem 15 The variety ð3dblst Þ is defined, modulo , by EACH of the following identities: (i) x _ x0 ≈ 1, (ii) x0 ^ x00 ≈ 0, (iii) x0 0 ≈ x0. Observe that 3klst satisfies the identity x00 ≈ x, while 3dblst does not satisfy it. Consequently, the following theorem follows from Theorem 13. Theorem 16 The variety ð3klst Þ is defined, modulo , by the identity: x00  x:

The Variety ℍ of Gautama Heyting Algebras Note that the language of the variety  of Gautama algebras lacks an implication operation. We will now define a variety called Gautama Heyting algebras (ℍ, for short) and show that it is term-equivalent to the variety  of Gautama algebras–a fact that will play a crucial role a little later. Actually, ℍ is not a new variety; it has already appeared, without a name, as a subvariety of the variety ℍℍ of dually hemimorphic semi-Heyting algebras studied in our earlier papers (Sankappanavar (2011, 2014a, 2014b, 2016) and Cornejo and Sankappanavar (2022)). Therefore, we need to recall relevant definitions and results from those papers. Definition 9 An algebra A ¼ hA, _, ^, !, 0, 0, 1i is a dually hemimorphic semiHeyting algebra if A satisfies the following conditions: (a) hA, _, ^, !, 0, 1i is a semi-Heyting algebra; i.e., the following conditions hold: (i) hA, _, ^, 0, 1i is a bounded distributive lattice, (ii) x ^ (x ! y) ≈ x ^ y, (iii) x ^ (y ! z) ≈ x ^ [(x ^ y) ! (x ^ z)], (iv) x ! x ≈ 1. (b) The operation 0 is a dual hemimorphism; i.e., the following conditions hold: (v) 00 ≈ 1, (vi) 10 ≈ 0, (vii) (x ^ y)0 ≈ x0 _ y0 ( ^-De Morgan law). The variety of dually hemimorphic semi-Heyting algebras will be denoted by ℍℍ. A  ℍℍ is a dually quasi-De Morgan Heyting algebra if it satisfies:

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(c) (x _ y)00 ≈ x00 _ y00, (d) x00 O x, (e) (x ^ y) ! x ≈ 1 (H). The variety of dually quasi-De Morgan Heyting algebras is denoted by ℚℍ. A  ℚℍ is regular if A satisfies: (R1) x ^ x+ O y _ y , where x+ ≔ x0 0. The variety of regular dually quasi-De Morgan Heyting algebras is denoted by ℝℚℍ. A  ℝℚℍ is a regular dually quasi-De Morgan Stone Heyting algebra if A satisfies: (St) x _ x ≈ 1, where x ≔ x ! 0. Let ℝℚtℍ denote the variety of regular dually quasi De Morgan Stone Heyting algebras. Remark 5 The reader is cautioned here not to confuse the notion of regularity given in the above definition with the one given in Sankappanavar (2011). The varieties ℍℍ , ℚℍ , ℝℚℍ , ℝℚtℍ and many of their subvarieties are examined, in detail, in Sankappanavar (2011, 2014a, 2014b, 2016). The logics associated with those subvarieties of the variety ℍℍ are investigated in (Cornejo and Sankappanavar 2022). We are ready to define the variety ℍ of Gautama Heyting algebras. Definition 10 Let ℍ be the subvariety of ℍℍ defined by the following axioms: 1. 2. 3. 4. 5. 6.

(x ^ y) ! x ≈ 1 (H), x _ x ≈ 1 (St), where x ≔ x ! 0, (x _ y)00 ≈ x00 _ y00, x00 O x, x ^ x0 0 O y _ y (R1), x 0 ≈ x ( ).

Proposition 10 ℍ  ℝℚtℍ  ℚℍ  ℍℍ: Proof Axioms (1)–(5) imply that ℍ  ℝℚtℍ. The rest of the inclusions are immediate from definitions. □

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Remark 6 1. Let 3dsth ≔ h3, _, ^, !, +, 0, 1i be the algebra, where 3 is the 3-element chain (0 < a < 1) and the operation + is as given in Fig. 2, and ! is defined as follows:

o 0 a 1 0 a 1

1 1 1 0 1 1 0 a 1

2. Let 3klh ≔ h3, _, ^, !, 0, 0, 1i be the algebra, where 3 is the 3-element chain (0 < a < 1) and the operation 0 is as given in Fig. 2, and ! is defined as above. It is clear that 3dsth and 3klh are algebras in ℍ. We wish to present an explicit description of the subdirectly irreducible algebras in ℍ: To achieve this goal, we need some definitions and results from (Sankappanavar 2011, 2014a). The notion of “level n” has played an important role in the classification of subvarieties of ℍ in (Sankappanavar 2011), although this name was not explicitly used there. We only need the definition of “level 1” here. Definition 11 An algebra A  ℍℍ is of level 1 if it satisfies the identity: (L1) x ^ x0 ≈ x ^ x0 ^ x0 0 . Let ℍℍ1 denote the subvariety of ℍℍ of level 1. For a subvariety  of ℍℍ, we let 1 ≔ \ ℍℍ1 . Lemma 4 ℍ is of level 1. Proof Let A  ℍ and let a  A. Then, using the axiom (6), we get a ^ a0 ^ a0 0 ¼ a ^ a0 ^ a0 ¼ a ^ a0 . □ Corollary 7 ℍ  ℝℚtℍ1 : Proof Observe that the variety ℍ is of level 1 by Lemma 4 and ℍ  ℝℚtℍ by Proposition 10, whence ℍ  ℝℚtℍ1 : □ Remark 7 It is clear that ℍ  ℚℍ: Hence it follows from Lemma 2.4 (5) of Sankappanavar (2014a) that the identity (L1) is equivalent to the following identity in ℍ : (x ^ x0 )0 ≈ x ^ x0 . (It can also be verified directly.)

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Let ℝℚtℍ1 denote the subvariety of ℝℚtℍ of level 1. The following lemma is a special case of Lemma 4.8 of Sankappanavar (2014a), when the underlying semi-Heyting algebra is actually a Heyting algebra. Lemma 5 Let A  ℝℚtℍ1 satisfy the simplicity condition: (SC) For every x  L, if x 6¼ 1, then x ^ x0 = 0. Then A is of height at most 2. Corollary 8 Let A  ℍ: If A satisfies (SC), then A is of height at most 2. Proof We know ℍ  ℝℚtℍ1 by Corollary 7. Now apply Lemma 5.

□

The following lemma is a special case of Corollary 4.1 of Sankappanavar (2014a). Lemma 6 Let A  ℝℚtℍ1 with |A| 2. Then TFAE: (1) A is simple, (2) A is subdirectly irreducible, (3) For every x  A, if x 6¼ 1, then x ^ x0 = 0. We are now ready to describe the subdirectly irreducible algebras in ℍ: Theorem 17 Let A  ℍ with |A| 2. Then the following statements are equivalent: 1. 2. 3. 4.

A is simple, A is subdirectly irreducible, For every x  A, if x 6¼ 1, then x ^ x0 = 0, A  {2, 3dsth, 3klh}, up to isomorphism.

Proof (1) , (2) , (3) by Lemma 6. Suppose (3) holds. Then A is of height at most 2 by Corollary 8. Then it is easy to see that the algebras of height at most 2 in ℍ are, up to isomorphism, precisely 2, 3dsth, 3klh, and 2  2. It is also clear that the algebra 2  2 does not satisfy the hypothesis (3), implying that (4) holds. Thus (3) implies (4), while it is routine to verify that (4) implies (3), proving the theorem. □  Corollary 9 Let A≔ A, _, ^, !A , 0 , 0, 1  ℍ: Define !k on A by x !k y ≔ (x _ y ) ^ [(x _ x )0 0 _ x _ y _ y ], where x ≔ x !A 0. Then, !A = !k. Proof It suffices to show that the equality holds on the (non-trivial) subdirectly irreducible algebras in ℍ, which are 2, 3dsth, and 3klh, up to isomorphism, in

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view of Theorem 17. Now it is routine to verify the equality of the two implications on these three algebras. □ The following corollary will play a crucial role later. Corollary 10  and ℍ are term-equivalent. More explicitly, (a) For A ¼ hA, _ , ^ , , 0 , 0, 1i  , let A! ≔ hA, _, ^, !k, 0, 0, 1i, where !k is defined by:

 x !k y ≔ ðx _ y Þ ^ ðx _ x Þ0 0 _ x _ y _ y : Then A!  ℍ: (b) For A ¼ A, _, ^, !A , 0 , 0, 1  ℍ, let A ≔ hA, _, ^, , 0 , 0, 1i, where is defined by x ≔ x !A 0. (c) If A  , then (A!) = A. (d) If A  ℍ, then (A )! = A. Proof (a) Observe that it suffices to verify that (a) holds for nontrivial subdirectly irreducible members of : So, let A be a nontrivial subdirectly irreducible algebra in : Then A  {2, 3dblst, 3klst} by Theorem 13. It is now routine to verify that A!  {2, 3dsth, 3klh}, and hence, A!  ℍ, whence (a) is proved. (b) The proof of (b) is similar to that of (a), in view of Theorem 17. (c) is easy to verify, and (d) is also easily proved using Corollary 9. □

A Logic for Gautama Algebras The purpose of this section is to define a propositional logic called GAUTAMA in Hilbert-style and show that it is algebraizable (in the sense of Blok and Pigozzi), with the variety  of Gautama algebras as its equivalent algebraic semantics. For this purpose, we first need to recall some definitions and results from Cornejo and Sankappanavar (2022).

Algebraizability and Axiomatic Extensions of DHMSH Recall that the logic DHMSH was introduced in the section “A New Example of an Implicative Logic: Dually Hemimorphic Semi-Intuitionistic Logic” of PART I. We will now derive important connections between the logic DHMSH and the variety ℍℍ of dually hemimorphic semi-Heyting algebras defined in the section “The Variety  of Gautama Heyting Algebras”. The following Lemma is proved in Lemma 4.4 of Cornejo and Sankappanavar (2022).

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Lemma 7 Alg ðDHMSH Þ ¼ ℍℍ: As an immediate consequence of Theorems 4, 5 and Lemma 7, we have the following corollary: Corollary 11 The logic DHMSH is algebraizable, and the variety ℍℍ is the equivalent algebraic semantics for DHMSH with the defining identity p ≈ p!H p (equivalently, p ≈ 1) and the equivalence formulas Δ ¼ {p ! H q, q ! H p}. Axiomatic Extensions of DHMSH: Recall that a logic S0 is an axiomatic extension of S if S 0 is obtained by adjoining new axioms but keeping the rules of inference the same as in S. Recall also that Ext(S) denotes the lattice of axiomatic extensions of a logic S and LV ðÞ denotes the lattice of subvarieties of a variety  of algebras. The following theorem is a consequence of Theorem 6 and Corollary 11: Theorem 18 (Isomorphism Theorem for DHMSH) The lattice Ext(DHMSH) of axiomatic extensions of DHMSH is dually isomorphic to the lattice LV ðℍℍÞ of subvarieties of the variety ℍℍ. We will now present a new logic, called GAUTAMA, along with its axiomatic extensions RDBLSt and RKLSt. Recall that α!H β ≔ α ! (α ^ β). Definition 12 The logic GAUTAMA is defined as follows: • Language: h_, ^, !, 0 , ⊥, ⊤i, where _, ^ are binary, , 0 are unary, and ⊥, ⊤ are constants. Let $H be the connective defined by: α$H β ≔ (α!H β) ^ (β!H α). • Axioms: (1), (2), . . ., (14) of the logic DHMSH, together with the following additional axioms: (15) (α ^ β) ! α, (16) α _ α , where α ≔ α ! 0, (17) (α _ β)00$H (α00 _ β00), (18) (α _ α00)$H α, (19) (α ^ α+) _ (β _ β )$H (β _ β ), where α+ ≔ α0 0, (20) α 0$H α . • Rules of Inference (a) (SMP) From ϕ and ϕ ! H γ, deduce γ (semi-Modus Ponens). (b) (SCP) From ϕ ! H γ, deduce γ 0 ! H ϕ0 (contraposition rule).

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Definition 13 (a) The logic RDBLSt is the axiomatic extension of GAUTAMA defined by the following axiom: (DSt) (α0 ^ α00) $H ⊥. (b) The logic RKLSt is the axiomatic extension of GAUTAMA defined by the following axiom: (KL) ((α ^ α0) _ (β _ β0))$H (β _ β0). In view of the axiom (15), the connective !H can be replaced by ! in the axioms and rules of inference for the logics GAUTAMA, RDBLSt and RKLSt. Lemma 8 The logic GAUTAMA is an axiomatic extension of DHMSH. Proof It suffices to observe that the axioms of GAUTAMA include the axioms of DHMSH and the inference rules are the same for the two logics. □ We remark here that there is no loss of generality in extending the definition of equivalent algebraic semantics to include the varieties that are term-equivalent. Let S be an algebraizable logic with  as its equivalent algebraic semantics and let 0 be a variety term-equivalent to . Then 0 can be considered as an equivalent algebraic semantics for the logic S. Corollary 12 The logic GAUTAMA is algebraizable with the variety  as the equivalent algebraic semantics. Proof By Lemma 8, the logic GAUTAMA is an axiomatic extension of the logic DHMSH, which, by Corollary 11, is algebraizable with ℍℍ as its equivalent algebraic semantics. Hence we can conclude by Theorem 7 that GAUTAMA is algebraizable with  as its equivalent algebraic semantics, in view of Theorem 18. Noting that the variety  is term-equivalent to the variety  by Corollary 10, the corollary is proved. □ Since the logics RDBLSt and RKLSt are axiomatic extensions of the logic GAUTAMA, we have the following corollaries. Corollary 13 The logic RDBLSt is algebraizable with the variety ℝt as its equivalent algebraic semantics. Corollary 14 The logic RKLSt is algebraizable with the variety ℝt as its equivalent algebraic semantics. In view of Corollary 12 and Theorem 13 and the fact that the logics GAUTAMA, RDBLSt and RKLSt are finitely axiomatized, we have the following Corollary:

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Corollary 15 The logics GAUTAMA, RDBLSt and RKLSt are decidable.

Open Problems Problem 1 There is a representation of regular double Stone algebras in terms of rough sets. Is there a similar representation for the variety ? Problem 2 Find a Priestley-type duality for the variety . Problem 3 Katriňák has given a construction for regular double Stone algebras using triples. Is there a similar construction for the variety ? Problem 4 Do the logics GAUTAMA, RDBLSt, and RKLSt have the Disjunction Property? Problem 5 Do the logics GAUTAMA, RDBLSt, and RKLSt have the Interpolation Property? Problem 6 Are the logics GAUTAMA, RDBLSt, and RKLSt structurally complete?

Epilogue In PART I, we have given a few glimpses into the interplay between logic and algebra since Boole’s and Frege’s works, with emphasis only on propositional logics. In PART II, we have introduced and investigated the variety of Gautama algebras and the associated propositional logic GAUTAMA, as well as its axiomatic extensions RDBLSt and RKLSt. We have also provided an extensive bibliography for the reader to delve further into the topics discussed in this chapter.

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van der Waerden, B.L. 1985. A history of algebra. From al-Khwarizmi to Emmy Noether. Berlin: Springer-Verlag. van Evra, J.W. 1977. A reassessment of George Boole’s theory of logic. Notre Dame Journal of Formal Logic 18 (3): 363–377. van Heijenoort, J. 1967. From Frege to Gödel. A source book in mathematical logic, 1879–1931. Harvard University Press, Cambridge, Mass. Varadarajan, V.S. 1998. Algebra in ancient and modern times. Vol. 12. Mathematical World. American Mathematical Society, Providence, RI; Hindustan Book Agency, Delhi. Varlet, J. 1972. A regular variety of type ¡2, 2, 1, 1, 0, 0, 0 ¿. Algebra universalis 2: 218–223. Viète, F. 1983. The analytic art. Nine studies in algebra, geometry and trigonometry from the it Opus restitutae mathematicae analyseos, seu algebrâ novâ, translated from the Latin and with an introduction by T. Richard Witmer. Kent State University Press, Kent, OH. Wang, H. 1963. A survey of mathematical logic. Science Press, Peking; North-Holland Publishing Co., Amsterdam. Weyl, H. 1944. David Hilbert and his mathematical work. Bulletin of the American Mathematical Society 50: 612–654. Whately, R. 1827. Elements of logic. London: J. Mawman. Whitehead, A. N. 2009. A treatise on universal algebra. Cambridge Library Collection. With applications, Reprint of the 1898 original. Cambridge University Press, Cambridge. Whitehead, A. N., and B. Russell. 1910. Principia mathematica. 3 volume work: Vol. 1 in 1910, Vol. 2 in 1912, Vol. 3 in 1914. Cambridge: Cambridge University Press. Wójcicki, R. 1973. Matrix approach in methodology of sentential calculi. Studia Logica 32: 7–39. Wójcicki, R. 1988. Theory of logical calculi, Synthese library. Basic theory of consequence operations. Vol. 199. Dordrecht: Kluwer Academic Publishers Group. Wolenski, J. 1995. Mathematical logic in Poland 1900–1939: People, circles, institutions, ideas. Modern Logic 5 (4): 363–405. Woods, John, ed. 2009. Handbook of the history of logic. Logic from Russell to Church, Handbook of the history of logic. Vol. 5. Amsterdam: Elsevier/North-Holland. Wronski, A. 1973. Intermediate logics and the disjunction property. Reports on Mathematical Logic 1: 39–51. Wronski, A. 1976. Remarks on Hallden completeness of modal and intermediate logics. Polish Academy of Sciences Institute of Philosophy and Sociology: Bulletin of the Section of Logic 5 (4): 126–129. Zach, R. 1999. Completeness before Post: Bernays, Hilbert, and the development of propositional logic. The Bulletin of Symbolic Logic 5 (3): 331–366. Zadeh, L.A. 1965. Fuzzy sets. Information and Control 8: 338–353. Zakharyaschev, M., F. Wolter, and A. Chagrov. 2001. Advanced modal logic. In Handbook of philosophical logic, vol. 3, 83–266. Dordrecht: Kluwer Acad. Publ.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Quasi-Boolean and Kleene Algebra of Rough Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Quasi-Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kleene Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Topological Quasi-Boolean Algebras and Extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Sequent Calculi for tqBa and tqBa5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical Linear Algebras and tqBas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pre-rough Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logics for Pre-rough Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Pre-rough and Three-Valued Łukasiewicz Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intermediate Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebraic Structures Related to Pre-rough Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . System Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Implicative Quasi-Boolean Algebras with Modal Axioms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rough Lattices and Heyting Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rough Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Rough Heyting Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Contrapositionally Complemented Pseudo-Boolean Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Intuitionistic Logic with Minimal Negation (ILM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebras from Rough Concept Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Logics DBL for Contextual dBas and PDBL for Pure dBas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Information System Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebra for Deterministic Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebra for Incomplete Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Algebra for Nondeterministic Information Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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M. Banerjee (*) Department of Mathematics and Statistics, Indian Institute of Technology, Kanpur, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_35

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Abstract

The chapter attempts to give an overview of work done in India in the area of algebraic logic related to rough set theory. Algebraic structures and associated propositional logics, proposed and studied over the past twenty-five years or so, are presented here with a sketch of the main results associated with the systems.

Introduction The story of “algebraic logic,” as we know, begins with Boole – when logic was first given a flavor of an algebraic theory (Boole 2009). Later, with Tarski’s work (Tarski 1983) on logical consequence, the connection between algebra and logic became clearer. On the one hand, there were formal logical systems with their “theorems” and on the other, varied semantics for the system with “validities.” The goal: theorems would be validities, and vice versa. This link between theorems and validities was established using syntactic and semantic logical consequence relations. A particular semantics that a logical system could be endowed with was the algebraic one. A prime example is classical propositional logic and its algebraic semantics given by the class of Boolean algebras: theorems of classical logic are exactly the statements that are valid across all Boolean algebras. Similar is the connection between Intuitionistic logic and the class of Heyting or pseudo-Boolean algebras. Algebraic logic deals with algebras and logics associated in the above manner; a hope is that properties of one would give insights about the other. The area also involves studies of logical matrices and other algebraic structures that could be associated to formal systems. However, that is not dealt with in this chapter. The primary reference for algebraization of logics that is discussed here is the seminal book by Rasiowa (1974). Rough set theory was introduced in 1982 by Z. Pawlak (1982, 1991), to deal with partial or inadequate information about a domain of discourse, or more precisely, about objects of a domain of discourse. Information about the objects may be available in terms of the properties describing them. Due to inadequate information about the objects, there may be some objects that are not distinguishable from others. For instance, contrast a situation where we know only about the color and size of a set of toys with another where we know about their shape as well. Two objects that are indistinguishable in the former may become distinguishable in the latter. Pawlak defined an approximation space as a pair (U, R), where R is an equivalence (i.e., reflexive, symmetric, and transitive) relation on U representing indiscernibility in the domain. The role of R may then be understood as follows. Given a set P of properties (say, color and size), a block in the partition of U induced by R collects all objects of U indistinguishable from one another with respect to P (say, a block collects all red and small toys). A goal in rough set theory is to obtain methods to handle the inadequacy of information, leading to approximate reasoning and decision-making with “rough” concepts.

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Given an approximation space (U, R), a few of the basic notions of rough set theory are as follows. If X
U, the lower approximation XR of X is the collection of those objects of the domain U that definitely belong to X. In other words, an object x belongs to the lower approximation, if all objects that from
cannot be distinguished  x also belong to X. Mathematically expressed, XR :¼ x  U : ½xR
X , where [x]R denotes the equivalence class of x with respect to the relation R. Alternately, XR ¼
 [ ½xR  U=R : ½xR
X , where U/R denotes the quotient set of U formed by R. The upper approximation XR of X in (U, R) contains those objects x of U that possibly belong to X – there is at least from x which
one object indiscernible  belongs to X. Mathematically, X :¼ x  U : ½ x  \ X 6 ¼ 0 ; alternately, XR ¼ R
R 
 [ ½xR  U=R : ½xR \ X 6¼ 0 ¼ [ ½xR  U=R : x  X . We shall drop the suffix R when there is no confusion about the indiscernibility relation in question. Observe that the definitions of the approximations could equivalently be stated in terms of operators L, U on the power set P(U ) of U where L, U:P(U )!P(U ) such that LðXÞ :¼ X, U ðXÞ :¼ X for any X
U. The boundary BnX of X consists of elements possibly, but not definitely, in X it is given by X∖X. X is definable in (U, R) if it has an empty boundary: X ¼ X ¼ X. For X, Y (
U ), X is said to be roughly included in Y provided X
Y and X
Y . X, Y are roughly equal, denoted X ≈ Y, if X ¼ Y as well as X ¼ Y . Rough equality induces a conceptlevel equivalence – formally, an equivalence relation on the power set P(U ) that relates the roughly equal subsets of U. Pawlak (1991) defined a subset X of U as a rough set in the approximation space (U, R), provided BnX 6¼ 0. In (Banerjee and Chakraborty 1993a, 2003), the restriction was removed and the context was included to define the triple (U, R, X) as a rough set. Most commonly   however, a rough set in the approximation space (U, R) is defined as the pair X, X , for any X
U. A more general form of this definition has also been considered (Banerjee and Chakraborty 1996, 2004; Iwiński 1987): the ordered pair (D1, D2), where D1
D2 and D1, D2 are definable sets, is termed a generalized rough set in (U, R). Other definitions of a rough set are also found in literature – these are discussed along with their equivalence and relevance to the foundations of the theory, in (Banerjee and Chakraborty 2004; Chakraborty and Banerjee 2013). Complete/Deterministic information systems (DIS) constitute a “practical” source of Pawlakian approximation spaces. These are of the form S :¼ (U, A, V , f ), where U is a set of objects, A a set of attributes, V :¼ [a  A V a a set of values for the attributes, and f a function from UA to V , assigning values  from Va to an object of U for an attribute a in A. A Pawlakian approximation space U, Ind SB is then induced for each subset B of attributes from A, where x Ind SB y in U, if and only if f ðx, aÞ ¼ f ðy, aÞ, for all a  B: “Incomplete” and “nondeterministic” information systems are also defined in literature, where (respectively) the value of f (x, a) may not be known for some cases, or there may be multiple possible values for f (x, a). The definitions are as

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follows. A tuple S :¼ (U, A, [a  A V a [{}, f ) is called an information system (IS), where U, A, V a are as for DIS, and f :UA![a  A V a [{}. An information system which satisfies f (x, a) ¼  for some x  U and a  A is called an incomplete information system (IIS). The absence of attribute values for an object is indicated by “” – so rather than just a value,  may be said to represent an epistemic state of information. In this context, a similarity relation (Kryszkiewicz 1998, 1999) is defined: for any B
A, (x, y)  SimSB if and only if f (x, a)¼f (y, a) or f (x, a)¼, or f (y, a)¼  for all a  B. Note that a DIS is an IS where f (x, a) 6¼ for all x  U and a  A. When an object is allowed to take a set of values for an attribute, one gets a nondeterministic information system (NIS). A tuple S :¼ (U, A, [a  A V a, f ) is called an NIS, where U, A, V a are as for DIS and f :UA!P ([a  A V a ) such that f (x, a)
V a. Now several relations can be defined on U, apart from the indiscernibility relation Ind SB (that can be defined identically as for DISs). In (Düntsch et al. 2001, 2007), there are examples of different interpretations for “f (x, a) ¼ V.” For instance, if a is the attribute “speaking a language,” then f (x, a) ¼ {German, English} can be interpreted as (i) x speaks German and English and no other languages, (ii) x speaks German and English and possibly other languages, (iii) x speaks German or English but not both, or (iv) x speaks German or English or both. These interpretations lead to different relations (see e.g. Düntsch et al. 2007; Orłowska and Pawlak 1984; Vakarelov 1991), for instance: 

Similarity:ðx, yÞ  SimSB if and only if f ðx, aÞ \ f ðy, aÞ 6¼ 0 for all a  B, and Inclusion:ðx, yÞ  InSB if and only if f (x, a)
f (y, a) for all a  B. Since the inception of rough set theory, researchers have investigated logical and algebraic aspects of rough sets. As approximate reasoning with rough concepts was a goal, Pawlak himself proposed the first “logical system” for rough sets, called decision logic (Pawlak 1991), which was motivated by information systems. It was also clear that logics for rough sets or their generalizations would invariably involve modalities to express the concepts of lower and upper approximations and as a consequence, a number of modal systems of logic have been proposed and studied (see, e.g. Banerjee and Khan 2007). In particular, the two approximations considered as operators obey all the laws of the modal system S5. For the definition of S5 and its Kripkesemantics, we refer to (Hughes and Cresswell 1996). According to the Kripke semantics for S5, a formula α is interpreted by a “valuation” function v as a subset in a nonempty domain U, the subset representing the extension of the formula – i.e., the collection of situations/objects/worlds where the formula holds. Moreover, in an S5model M:¼ (U, R, v) (say), the accessibility relation R is an equivalence relation on U. Now the formal connection between the syntax of S5 and its semantics in terms of rough sets is given as follows. If □, ◇ denote the necessity and possibility operators, respectively, in the language of S5, then for any formula α, it may be observed that vð□αÞ ¼ vðαÞ and vð◇αÞ ¼ vðαÞ , the lower and upper approximations (respectively) of the extension v(α) of the formula α. This is in consonance with the

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interpretations of the approximation operators in rough set theory as mentioned earlier: the lower approximation of the concept represented by the formula α consists of all objects of the domain that definitely or necessarily belong to the extension of α, while the upper approximation collects all objects of the domain that possibly belong to the extension of α. Classically, a formula α is true in M, if v(α) ¼ U. Now it can easily be seen that all the S5 theorems involving □ and ◇ translate into valid properties of lower and upper approximations in the approximation space (U, R). To address the notion of rough equality (≈), a second implication ) (apart from the S5 “!”), called rough implication, is defined in the language of S5 as: α ) β :¼ ð□α ! □βÞ ^ ð◇α ! ◇βÞ: So we get a notion corresponding to rough equality in the logic, called rough equivalence: α  β :¼ ðα ) βÞ ^ ðβ ) αÞ: The following are a few theorems (Bunder et al. 2008) involving rough implication and rough equivalence that are not so obviously derivable in S5. ~ denotes the negation operator in S5. (i) ‘ α ) (α!□α). (ii) ‘ (◇α!α) ≈ (α!□α). (iii) ‘ α ≈ □α _ ( ◇α^ ~ α). Algebraically, the structure corresponding to the logic S5, the S5-algebra, is a Boolean algebra with operators (Blackburn et al. 2001) satisfying the algebraic versions of S5 axioms. However, it was observed in (Banerjee and Chakraborty 1996) that considering the Lindenbaum-Tarski algebra construction in S5 based on the rough implication and equivalence, one obtains a non-Boolean algebra, viz. a quasi-Boolean algebra (Banerjee and Chakraborty 1993b) which has an interior operator – call it the rough S5-algebra. Details of the construction are given in section “The Quasi-Boolean and Kleene Algebra of Rough Sets.” The new algebraic structure was termed a topological quasi-Boolean algebra (tqBa). This class of algebras gave rise to a whole direction of work (see, e.g., Lin and Chakraborty 2019; Lin et al. 2018; Saha et al. 2014, 2016; Sen and Chakraborty 2002). It was observed by Kumar and Banerjee in (Kumar and Banerjee 2017) that this quasi-Boolean algebra, in fact, enjoys the Kleene property a ^ ~ a  b _ ~ b, where ~ is the quasi-Boolean negation. Moreover, the class of all Kleene algebras can be represented in terms of the class of these Kleene algebras of rough sets. This observation is shown to lead to a rough set semantics for the Kleene logic. Note that if A denotes a class of abstract algebras (such as Kleene algebras) and R the class of “corresponding” set algebras (such as the Kleene algebras of rough sets), a representation result for A that we refer to is of the following kind. The result

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establishes a correspondence c :A!R such that any element A  A is isomorphic to a subalgebra of cðAÞ – in other words, A is embedded in cðAÞ. Such results will be stated for almost all the algebras presented in this chapter. Abstraction of further properties that the rough S5-algebra satisfies gave rise to two new algebraic structures based on tqBas, namely the pre-rough and rough algebras (Banerjee and Chakraborty 1996). Logics corresponding to these classes of algebras can also be imparted a rough set semantics, due to representation results proved for the algebra classes in terms of the classes of corresponding algebras of rough sets. Furthermore, well-known classes of algebras such as 3-valued Łukasiewicz algebras and regular double Stone algebras are shown to be equivalent to pre-rough algebras (Banerjee 1997; Banerjee and Chakraborty 2004). Thus logics for these classes of algebras also get a rough set semantics. In sections “The Quasi-Boolean and Kleene Algebra of Rough Sets,” “Topological Quasi-Boolean Algebras and Extensions,” and “Pre-rough Algebras,” we present a summary of the above-mentioned work. Section “Algebraic Structures Related to Pre-rough Algebras” gives a sketch of extensive investigations (Saha et al. 2014, 2016) done on structures related to pre-rough algebras, with focus on the implication operator. We then take a look at some other directions of work related to rough set theory where algebra and logic have interacted with each other. Pawlak’s rough set theory has been generalized in many ways, for instance by considering a tolerance (i.e., reflexive and symmetric) relation in place of an equivalence relation on the domain of discourse, to obtain a “generalized approximation space.” In such an altered scenario, investigations have resulted in a variety of definitions for appropriate “lower” and “upper” approximations (see, e.g., Samanta and Chakraborty 2011 for a summary), which have, in turn, led to new algebraic structures and logics. In section “Rough Lattices and Heyting Algebras,” we summarize the work done in this regard by Kumar and Banerjee (Kumar 2016; Kumar and Banerjee 2015). Categories of rough sets were introduced by Banerjee and Chakraborty (1993a, 2003). The study was followed up by More and Banerjee (2016), where it is shown that strong subobjects of objects in a category of rough sets form a new algebraic structure, when rough relative complementation is incorporated along with the classical categorical operations. The work gives rise to logics (More and Banerjee 2017) with two negation operators which are a novel amalgamation of the wellknown minimal and intuitionistic logics. This is sketched in section “Contrapositionally Complemented Pseudo-Boolean Algebras.” Formal concept analysis (see Ganter and Wille 2012) was related to rough set theory by Düntsch and Gediga (2002), and Yao (2004b) with their study of propertyoriented and object-oriented concepts. Recently, negation was introduced in the study of object-oriented concepts, and object-oriented protoconcepts and semiconcepts were defined by Howlader and Banerjee (2018, 2020). It is shown that

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the algebraic structures of contextual double Boolean and pure double Boolean algebras introduced by Wille can be represented by object-oriented protoconcept and semiconcept algebras, respectively. Corresponding logics are investigated and are shown to get an object-oriented protoconcept and semiconcept semantics. This is briefly presented in section “Algebras from Rough Concept Analysis.” A string of logics with semantics based on information systems followed the proposal of decision logic by Pawlak, a bulk of it being work by Orlowska and Balbiani (e.g., some early papers: (Balbiani 1998; Balbiani and Orlowska 1999; Orłowska 1982), also see (Banerjee and Khan 2007)). Khan and Banerjee have also worked in this area (Khan 2015; Khan and Banerjee 2009, 2011) and have introduced (Khan and Banerjee 2013) algebraic counterparts of logics for deterministic, incomplete, and nondeterministic information systems. These structures are presented in section “Information System Algebras.” Section “Conclusions” concludes the chapter. In the chapter, we follow Notation 1 Given an approximation space (U, R), – D denotes
 the collectionof definable sets, – RS :¼ X, X : X
U , the collection of rough sets, and – R :¼ {(D1, D2): D1
D2, D1, D2 definable sets in U}, the collection of generalized rough sets. (Clearly, RS
R.)

The Quasi-Boolean and Kleene Algebra of Rough Sets Initially, the endeavor in (Banerjee and Chakraborty 1993b) was to look for a suitable logic, models of which would be rough sets as defined in section “Introduction.” As mentioned earlier, the modal system S5 seemed to be an appropriate candidate, because a Kripke model (U,R) along with the interpretation v(α)(
U ) of a well-formed formula α of S5 is indeed a rough set of the form (U, R, v(α)). However, a Lindenbaum-Tarski-like construction was then carried out on the set of formulae using the rough implication ) and rough equivalence ≈ to obtain the rough S5algebra as follows. Two formulae α, β are defined to be equivalent if and only if ‘S5α ≈ β, i.e. ‘S5α ) β and ‘S5 β)α. This equivalence relation gives a quotient set, and the logical operations are extended to the set as: ½α u ½β :¼ ½ðα ^ βÞ _ ðα ^ ◇β^ ð◇ðα ^ βÞÞÞ, ½α t ½β :¼ ½ðα _ βÞ ^ ðα _ □β_ ð□ðα _ βÞÞÞ, ½α :¼ ½ α, for any [α], [β] in the quotient set. The resulting structure turns out to be a quasi-Boolean algebra (Rasiowa 1974), in which the negation operator, in fact, obeys the Kleene property – thus yielding a Kleene algebra. Let us discuss these algebras and the corresponding logics.

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Quasi-Boolean Algebras Let us recall the definition of quasi-Boolean algebras (Rasiowa 1974), also called De Morgan algebras. Definition 1. An algebra hA, , ^, _, ~, 0, 1i is a quasi-Boolean algebra (qBa) if and only if 1. hA, , ^, _, 0, 1i is a bounded distributive lattice. 2. ~~ a ¼ a, for all a  A. 3. ~ (a_b) ¼ ~a^~b, for all a, b  A. It can be shown that an “implication” (!) operation that satisfies the property ðÞ a  b if and only if a ! b ¼ 1, cannot be defined in a qBa with the help of the other operations of ^, _, ~ (Bhuvneshwar 2012). Consider the four-element qBa that is not a Boolean algebra, given in Fig. 1. If ! satisfying the property () were definable in this four-element qBa with the help of ^, _, ~, the element a!a would have to evaluate to a (6¼ 1), as each of the three operations fix a. But that would violate property (), because aa. So a Hilbert-style axiomatization for the logic of qBa is not immediately clear. However, a sequent calculus qBl was developed in (Sen and Chakraborty 2002) for the class. The language of qBl consists of propositional variables pi, constants ⊥, ⊤, and logical symbols ~, _, ^. Formulae are formed as usual, denoted by α, β, . . . A sequent is of the form Γ : Δ, where Γ and Δ are finite (possibly empty) multisets of formulae.

Fig. 1 Four-element qBa

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qBl Postulates: Ax1 α : α

Ax2 α : α 0

Γ : Δ Δ : Γ Γ : Δ RW Γ : α, Δ Γ : α, α : Δ RC Γ : α, Δ Γ : α, β, Δ R_ Γ : α _ β, Δ

0

Γ : α, Δ Γ , α : Δ Γ, Γ 0 : Δ, Δ0 Γ : Δ LW Γ, α : Δ Γ, α, α : Δ LC Γ, α : Δ Γ , α : Δ Γ 0 , β : Δ0 L_ Γ, Γ 0 : α _ β : Δ, Δ0 Γ , α, β : Δ L^ Γ, α ^ β : Δ

Rule

Cut

Γ : α, Δ Γ 0 : β, Δ0 Γ, Γ 0 : α ^ β, Δ, Δ0 R⊥ Γ : ⊤ , Δ

R^

L⊥ Γ, ⊥ : Δ

Derivability of a sequent in the logic is defined in the usual manner. A model of qBl is a quasi-Boolean algebra B together with a valuation v that is a function from the set of propositional variables of qBl to B. v is extended recursively to the set of all formulae of qBl. A sequent Γ : Δ is said to be valid in a model (B, v) if any only if v(α1) ^ . . . ^v(αm) v(β1) _. . ._v(βn). Theorem 1. The system qBl is sound and complete with respect to the class of quasi-Boolean algebras: any sequent Γ : Δ is derivable in qBl, if and only if it is valid in all models of qBl.

Kleene Algebras A Kleene algebra is a quasi-Boolean algebra hA, , ^, _, ~, 0, 1i in which the Kleene property a^~ ab_~ b holds for the negation ~. The rough S5-algebra is an instance of a Kleene algebra. The collection R is also observed to form a Kleene algebra. Note that D, the collection of all definable sets in an approximation space (U, R), forms a Boolean subalgebra of the power set Boolean algebra over U. That R forms a Kleene algebra is a consequence of the fact that it is of the form B[2] :¼ {(a, b) : ab, a, b  B}, where B is a Boolean algebra (taking B¼D ), and the latter forms a Kleene algebra with operations defined as follows (Boicescu et al. 1991). For (a, b), (c, d)  B[2], ða, bÞ _ ðc, d Þ :¼ ða _ c, b _ dÞ, ða, bÞ ^ ðc, d Þ :¼ ða ^ c, b ^ dÞ, ða, bÞ :¼ ðbc , ac Þ:

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It may be remarked here that the set B[2] was first studied by Moisil in the context of three-valued Łukasiewicz (Moisil) algebras (see Cignoli 2007). As for the collection RS of rough sets, it can be shown that it is closed with respect to the operations _, ^, ~ (Banerjee and Chakraborty 2004), that is, for any (a, b), (c, d)  RS , the pairs on the right-hand side of the above definitions of the operations also belong to RS . The collection forms a subalgebra of the Kleene algebra formed by R. The following representation result is shown to hold for Kleene algebras (Kumar and Banerjee 2017). Theorem 2. Given a Kleene algebra K , there exists an approximation space (U, R) such that K can be embedded in RS . In other words, every Kleene algebra is isomorphic to an algebra of rough sets in a Pawlak approximation space. The logic L K(K for Kalman and Kleene) is the Kalman consequence system studied by Dunn (Dunn 1999, 2000) with slight modifications. The language consists of propositional variables p,q,r,. . ., propositional constants ⊤, ⊥, and logical connectives _, ^, ~. The set F of well-formed formulae of the logic are defined through the following scheme: ⊤j⊥j p jα _ βjα ^ βj α: The following postulates and rules, taken from (Dunn 1999, 2005), give the consequence relation ‘L K . The postulates are expressed with the help of consequence pairs α ‘L K β, α, β  F . Definition 2. LK Postulates: 1. α ‘α (Reflexivity). α ‘ β, β ‘ γ (Transitivity). 2. α ‘γ 3. α^β ‘α, α ^ β ‘β (Conjunction Elimination). α ‘ β, α ‘ γ 4. (Conjunction Introduction). α ‘β^γ α ‘ γ, β ‘ γ 5. (Disjunction Elimination). α_β ‘γ 6. α ‘α_β, β ‘α _ β (Disjunction Introduction). 7. α ^ (β _ γ) ‘ (α _ β) ^ (α _ γ) (Distributivity). 8. α ‘⊤ (Top). 9. ⊥ ‘α (Bottom). α ‘β 10. (Contraposition). β ‘ α 11. ~ α ^~ β ‘~(α_β) (_-linearity). 12. ⊤ ‘ ~ ⊥ (Nor). 13. α ‘ ~~ α.

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14. ~~ α ‘α. 15. α ^ ~ α ‘ β _ ~ β (Kalman/Kleene). It may be remarked that postulates 1–9 of L K define the logic for bounded distributive lattices. A valuation is again a map v from the set of propositional variables to a Kleene algebra K that is extended to the set of all formulae of L K in the usual manner. A consequent α ‘β is valid in K under the valuation v, if v(α)  v(β). If the consequent is valid under all valuations on K, then it is valid in K. Let A K denote the class of all Kleene algebras. We write α A K β, if α ‘β is valid in each Kleene algebra. In the classical manner, one obtains Theorem 3. α ‘L K β if and only if α A K β, for any α, β  F . If A KRS denotes the class containing the collections RS of rough sets over all possible approximation spaces (U,R), we get, as a consequence of Representation Theorem 2 and Theorem 3, a rough set semantics for logic LK, that is, one in which a formula is interpreted as a rough set in some approximation space. Theorem 4. Kumar and Banerjee (2017) For any α, β  F , (i) α A K β if and only if α A KRS β, (ii) α‘L K β if and only if α A KRS β.

Topological Quasi-Boolean Algebras and Extensions In the rough S5-algebra, expectedly, the necessity operator □ of S5 induces an “interior” operation (also denoted □, by abuse of notation): □[α] :¼ [□α]. □ satisfies the 0-dimensionality property. Similarly, in the quasi-Boolean algebra formed by R , an interior operation I can be defined as: I(a,b) :¼ (a,a). The motivation of defining the operator I in the context of rough set theory is that when  R ¼ RS , I extracts the lower approximation of the set concerned: I X, X ¼ ðX, XÞ, X
U. These observations led to the notion of a topological quasi-Boolean algebra (tqBa) (Banerjee and Chakraborty 1993b). Formally, we have the following. Definition 3. An algebra hA, ^, _, ~, I, 0, 1i, where I is a unary operation, is said to be a topological quasi-Boolean algebra (tqBa) if and only if 1. hA, ^, _, ~, 0, 1) is a qBa. 2. I1 ¼ 1. 3. Iaa, for all a  A. 4. I(a^b) ¼ Ia^Ib, for all a,b  A. 5. IIa ¼ Ia, for all a  A.

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Notation 2 The symbol C is used for ~ I ~ in a tqBa, that is, Ca :¼~ I ~ a. So C is a unary operation on a tqBa acting like a closure operation. Observe that in the rough S5-algebra, C[α]¼ [◇α], while in the tqBa formed by R, C(a, b)¼ (b, b). The 0-dimensionality property is satisfied, and these algebras become instances of the following. Definition 4. A tqBa hA, ^, _, ~, Ι, 0, 1i, where I is a unary operation, is said to be a topological quasi-Boolean algebra 5 (tqBa5) if and only if 6. CIa¼Ia, for all a  A.



Recall that the relation of rough equality ≈ induces an equivalence relation on the power set P ðU Þ for any approximation space (U, R) (see section “Introduction”). As   another example of a tqBa5, we have P ðU Þ= , u , t , , L, 0 , ½U defined on the quotient set P ðU Þ=, where for any ½S, ½T   P ðU Þ= ,   c  ½S u ½T :¼ ðS \ T Þ [ S \ T \ S \ T , ½S t ½T :¼½ðS [ T Þ \ ðS [ T [ S [ T c Þ, ½S:¼½Sc , L½S:¼½S:

Sequent Calculi for tqBa and tqBa5 These systems were studied in (Sen and Chakraborty 2002). tqBl: To form the sequent calculus tqBl corresponding to the class of tqBas, a unary operator l is added to the language of qBl (see section “Quasi-Boolean Algebras”) and along with the rules of qBl, two more rules for l are taken: Ll

Γ, α : Δ Γ, lα : Δ

Rl

lΓ : α : lΓ : lα

Here lΓ means lα1, . . ., lαm, where Γ is α1, . . ., αm. tqB15: To get the logic tqB15 for the class of tqBa5, the unary operator m dual to l is defined as mα :¼~ l ~ α. One more axiom is then taken: Ax3 mlα : lα The definitions of models and validity remain the same as given for qBl. One obtains Theorem 5. The system tqBl (tqB15) is sound and complete with respect to the class of all topological quasi-Boolean algebras (topological quasi-Boolean algebras 5).

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Classical Linear Algebras and tqBas A connection between tqB15 and distributive multiplicative additive linear logic dMALL, the latter being the logic for distributive classical linear (CL-) algebras, was established through work in (Sen and Chakraborty 2002). We briefly present it below. First let us define CL-algebras. Definition 5. An algebraic structure of the form < X, \, [, ⊥,  , , 1, 0 > is a CL-algebra, where 1. < X, \, [, ⊥> is a bounded lattice with least element ⊥, 2. < X, , 1 > is a commutative monoid with the identity element 1, 3. If x  y then x  z y  z, z  x  z  y and y  z  x  z, for all x, y, z  X and 4. ~~ x ¼ x for all x  X, where ~ x:¼x  0. The language of the sequent calculus MALL for multiplicative additive linear logic corresponding to CL-algebras consists of propositional variables pi, propositional constants 0,1,⊥,⊤, and logical symbols ~, , ^. A sequent, as before, is of the form Γ : Δ, where Γ and Δ are finite (possibly empty) multisets of formulae. The axioms and rules of MALL are given in terms of sequents Γ : Δ (as defined for qBl), and are as follows. Ax α : α Γ : α, ΔΓ0 , α : Δ0 Γ, Γ 0 : Δ, Δ0 Γ : α; Δ L~ Γ; α : Δ Γ : Δ L1 Γ, 1 : Δ Γ, α : Δ L^ Γ, α ^ β : Δ Γ, α, β : Δ L Γ, α  β : Δ Cut

Γ,α : Δ Γ : α, Δ Γ : Δ R0 Γ : 0, Δ Γ : α, ΔΓ : β, Δ R^ Γ : α ^ β, Δ Γ : α, ΔΓ 0 : β, Δ0 R Γ, Γ 0 : α  β, Δ, Δ0 R⊤ Γ : ⊤, Δ R~

L⊥ Γ, ⊥ : Δ

Adding an extra axiom for distributivity, viz.: α ^ ðβ _ γ Þ : ðα ^ β Þ _ ðα ^ γ Þ gives the sequent calculus dMALL corresponding to distributive CL-algebras. Let tqBa–d denote a tqBa5 where the lattice is not necessarily distributive. The following relationship is observed in (Sen and Chakraborty 2002). Theorem 6. If < X, \, [, ⊥,  , , 1, 0 > is a CL-algebra (distributive CL-algebra), then < X, \, [, ~, I, ⊥, ⊤> is a tqBa–d (tqBa5), where ~ a :¼ a  0, Ia :¼~ (~ a ~ ⊥) and ⊤ ¼~ ⊥ is the greatest element of the lattice.

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The translation ° from the set of well-formed formulae of tqB15 to those of dMALL is given as follows: P° :¼ P for atomic P (~ α)° :¼~ α° (α^β)° :¼ α°^β° (α_β)° :¼ α°_β° (lα)° :¼ α° + ⊥. Note that (mα)° ¼ α° ⊤.





Theorem 7. If ‘Γ : Δ in tqBl5 then ‘(Γ °) : [Δ°] in dMALL, where (Γ°) means the conjuncts of the formulae of Γ ° (for Γ 6¼ 0) and [Δ°] means the disjuncts of the formulae of Δ° (for Δ 6¼ 0). It is assumed that (Γ°) is ⊤ if Γ is empty, and [Δ°] is ⊥ if Δ is empty. Replacing the Cut, R^, and L_ rules of tqBl5 appropriately, a sequent calculus of tqBa–d is obtained, and a similar embedding as in Theorem 7 results between the logic and MALL.

Pre-rough Algebras The tqBa5 formed by P ðUÞ=  satisfies a few more properties. On abstraction in two stages, one gets the notions of pre-rough and rough algebras (Banerjee and Chakraborty 1996). Definition 6. An algebra hA, , ^, _, ~, I, 0, 1i is said to be a pre-rough algebra if and only if it is a tqBa5 with the following additional axioms 7, 8 and 9. 7. ~ Ia_Ia ¼ 1, for all a  A. 8. I(a_b) ¼ Ia _ Ib, for all a, b  A. 9. CaCb, IaIb imply ab, for all a, b  A. Moreover, an operator ) is defined as follows: a ) b:¼ð:Ia _ IbÞ ^ ð:Ca _ CbÞ: A pre-rough algebra hA, , ^, _, ~, I, 0, 1i is found to be an instance of Antonio Monteiro’s well-studied tetravalent modal algebra (see Font and Rius 2000), the latter being defined as a qBa hA, ^, _, ~, 0, 1i with a unary operator I satisfying for all a  A, ~ Ia_a ¼ 1, and Ia_ ~ a ¼ a_ ~ a. The converse is not true, as shown in (Saha et al. 2014): the 4 element qBa of Fig. 1 is considered with the operators I,C defined as I (0) ¼ I (a) ¼ I (b) ¼ 0, I (1) ¼ 1, C (0) ¼ 0, C (a) ¼ C (b) ¼ C (1) ¼ 1. This forms a tetravalent modal algebra. But it is not a pre-rough algebra: a ≰ b, even though CaCb, IaIb.

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Axioms 5, 6, and 8 defining a pre-rough algebra (see Definitions 3, 4, and 6) have subsequently been shown to be derivable from the rest, and a simplified definition of pre-rough algebra is considered in (Saha et al. 2014): Definition 7. (modified form) An algebra hA, , ^, _, ~, I, 0, 1i is said to be a pre-rough algebra if and only if 10 . hA, , ^, _, ~, 0, 1i is a qBa. 20 . I1 ¼ 1. 30 . Ia  a, for all a  A. 40 . I(a^b) ¼ Ia^Ib, for all a, b  A. 50 . ~ Ia_Ia ¼1, for all a  A. 60 . CaCb, IaIb imply ab, for all a, b  A. A few “intermediate” algebras that are tqBas but not pre-rough algebras are obtained when different proper subsets of the set of axioms 7–9 are taken. These have been elaborated upon in section “Intermediate Algebras.” Pre-rough algebras have also been investigated in (Banerjee 1997). In particular, the relationship with three-valued Łukasiewicz (Moisil) algebras was studied. The latter are qBas of the form hA, , ^, _, ~, 0, 1i with a unary operator M satisfying the following for all a, b  A: L1. M(a^b) ¼ Ma^Mb, L2. M(a_b) ¼ Ma_Mb, L3. Ma^ ~ Ma ¼ 0, L4. MMa ¼ Ma, L5. M ~ Ma ¼~ Ma, L6. ~ M ~ a  Ma, L7. Ma ¼ Mb, M ~ a ¼ M ~ b imply a ¼ b, It is shown that Proposition 1. A pre-rough algebra is equivalent to a three-valued Łukasiewicz (Moisil) algebra, in the sense that the defining axioms of one are deducible from those of the other. It is also noted in (Banerjee and Chakraborty 1996) that the implication ! in a three-valued (Moisil) Łukasiewicz algebra, defined as a!b :¼ ((~ ~ M ~ a)_b)^ ((~ a)_Mb), and the implication ) in a pre-rough algebra differ. The tqBa5 formed by P ðUÞ=  not only satisfies the pre-rough algebra axioms, but more. On abstraction of the additional properties, one obtains a rough algebra: it is a pre-rough algebra P :¼ (A, , ^, _, ~, I, 0, 1) such that the subalgebra (I(A), , ^, _, ~, 0, 1) of P, where I(A) :¼ {Ia : a  A} is complete and completely distributive, i.e., _i  L ^ j  J ai,j ¼ ^ f  J L _i  L ai,f ðiÞ , for any index sets L, J and elements ai, j, i  L, j  J, of I(A), JL being the set of maps from L into J.

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Representation theorems for both pre-rough and rough algebras are established in (Banerjee and Chakraborty 1996): any pre-rough algebra (rough algebra) is embeddable in the pre-rough algebra (rough algebra) R over some approximation space.

Logics for Pre-rough Algebras Let us first present the sequent calculus PRSC for the class as proposed in (Sen and Chakraborty 2002). The language of the system is the same as that of tqBl. PRSC Postulates: All axioms and rules of tqBl5 are considered, only Rl is replaced by a generalized version Rl0 : Rl0

lΓ : α, Δ : lΓ : lα, Δ

Two rules are added: ðR Þr

lα : lβ : lα,lβ

lmR

lα : lβ mα : mβ : α:β

It can be shown that the following rules are derivable in PRSC: L

Γ : lα, Δ Γ, lα : Δ R : Γ, lα, Δ Γ : lα : Δ

Ax3 of tqBl5 (see section “Sequent Calculi for tqBa and tqBa5”) is also derivable in PRSC. Taking definitions of model and validity as done for qBl, one obtains Theorem 8. PRSC is sound as well as complete with respect to the class of all pre-rough algebras. On the other hand, a logic PRL with Hilbert-style axiomatization was proposed in (Banerjee and Chakraborty 1996) corresponding to the class of pre-rough algebras, and observed to be sound and complete also with respect to a semantics based on rough sets. ~, u, □ are the primitive logical symbols in the language of PRL. t, ◇ are duals of u, □, while ) is defined as α ) β :¼ ð □α t □βÞ u ð ◇α t ◇βÞ, for any formulae α, β of PRL. PRL Postulates: 1. α)α 2a. ~ ~ α)α

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2b. α ) ~ ~α 3. αuβ)α 4. αuβ)βuα 5a. αu (βtγ) ) (αuβ) t (αuγ) 5b. (αuβ) t (αuγ) )αu(βtγ) 6. □α)α 7a. □(αuβ))□(α)u□(β) 7b. □(α) u□(β) )□(αuβ) 8. □α)□□α 9. ◇□α)□α 10a. □(αtβ) )□αt□β 10b. □αt□β)□(αtβ) Rules of inference: 1.

3.

D DŸE E

DŸE E ŸJ D ŸJ

modus ponens

hypothetical syllogism

D

2.

4.

E ŸD 5.

DŸE D ŸJ D Ÿ E J

6.

7.

DŸE ƶD ŸƶE

8.

9.

DŸE ~ E Ÿ~D D Ÿ E, E ŸD J ŸG, G ŸJ D Ÿ J Ÿ  E Ÿ G

D ƶD

ƶD ŸƶE ƺD Ÿƺ E

DŸE

Derivability is defined in the usual manner. Given a pre-rough algebra P:¼ hA, , ^, _, ~, I, 0, 1i, a valuation v is defined to be a function from propositional variables to A that is recursively extended to the set of all PRL-formulae, where note that for the extension, v(αuβ) :¼ v(α) ^v(β) and v(αtβ) :¼ v(α) _v(β). A PRL-model M is a pair of the form (P, v). A formula α is valid in M, if v(α) ¼ 1, while α is valid in the class of all pre-rough algebras, if it is valid in all models. It can then be shown, following standard techniques, that a PRL-formula is a PRLtheorem if and only if it is valid in the class of all pre-rough algebras. In the following sections, we may not repeat these definitions of validity for the classes of algebras, and simply state that the logics being considered are sound and complete with respect to the classes.

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As mentioned earlier, PRL also gets a rough set semantics: just as for S5, a model for PRL is of the form M :¼(U, R, v), where the departure from the S5-semantics lies in the definition of the meaning function v with respect to the connectives of conjunction u and implication ). These are based on the respective operations defined on P ðU Þ=  earlier. So for instance, for any α, β in PRL, S,T
U, v(αuβ) :¼ v(α) u v(β), and v(α)β) :¼ v((~ □αt□β) u (~ ◇αt◇β)), where   c  S u T :¼ ðS \ T Þ [ S \ T \ S \ T : Definition of truth of a formula α in M remains the same: this is if and only if v(α) ¼ U. It may then be noticed that ) reflects rough inclusion: a formula α)β is true in (U, R, v) provided v(α) is roughly included in v(β). Further, u/t are operations that reduce to ordinary set intersection/union only when working on definable sets. α is valid (written ¼ RS α), if and only if α is true in every PRL-model. One can then prove, for any PRL-formula α, Theorem 9. ‘PRL α if and only if RS α. In (Banerjee 1997), a translation * of wffs of PRL into S5 is defined that assigns: (i) the operations and □ in PRL the same operations of S5, (ii) u is translated in terms of the conjunction ^ and disjunction _ of S5 as: ðα u βÞ :¼ ðα ^ β Þ _ ðα ^ Mβ ^ :Mðα ^ β ÞÞ: Then it is shown that PRL is embeddable in S5, that is ‘PRL α if and only if ‘α, for any wff α of PRL.

Pre-rough and Three-Valued Łukasiewicz Logic By virtue of Proposition 1 (see section “Pre-rough Algebras”), the logic PRL also becomes equivalent to three-valued Łukasiewicz logic L 3. A formal link between the two is established as follows. Recall Wajsberg’s axiomatization of L 3 (see Boicescu et al. 1991). The language has primitive logical symbols ~, !, and the postulates are: 1. 2. 3. 4.

α! (β!α). (α!β)! ((β!γ) ! (α!γ)). ((α!~α) !α) !α. (~α!~β) ! (β!α).

The only rule of inference is Modus Ponens. L3 is sound and complete with respect to the class of three-valued Łukasiewicz (Moisil) algebras. It is also sound and complete with respect to the semantics on

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3 :¼ {0,1/2,1} with Łukasiewicz negation and implication (Boicescu et al. 1991). In (Banerjee 1997), it is shown that there are translations ° from L3 into PRL and * from PRL into L3 given by (~ α)° :¼~ α°, (α!β)° :¼ (◇ ~ α° t β°) u(◇β° t α°); (~ α) :¼~ α, (αtβ) :¼ (α!β) !β, (αuβ) :¼ ~ (~ αt β), (◇α) :¼ ~ α!α. The translations ensure that L3 is equivalent to PRL, that is we have Theorem 10. (a) ‘L3 α, if and only if RS α°, for an L3-formula α and (b) ‘L3 α , if and only if RS α, for a PRL-formula α. Similar relationships are established between the sequent calculi corresponding to the classes of pre-rough and three-valued (Moisil) algebras in (Sen and Chakraborty 2002). Three-valued Łukasiewicz (Moisil) algebras have been shown in literature to be equivalent to other algebraic structures such as regular double Stone algebras and semisimple Nelson algebras. Hence logics that are sound and complete with respect to these classes of algebras also get a rough set semantics. A summary of existing logics for some such classes and relationships are given in (Banerjee and Khan 2007).

Intermediate Algebras Three intermediate algebras IA1-IA3 lying between tqBa5 and pre-rough algebra are defined in (Sen and Chakraborty 2002): Definition 8. Consider a tqBa5 hA, , ^, _, ~, I, 0, 1i. It is termed an 1. Intermediate algebra 1 (IA1) if and only if it satisfies (IA1.) ~Ia_Ia ¼1, for all a  A. 2. Intermediate algebra 2 (IA2) if and only if it satisfies (IA2.) I(a_b) ¼ Ia_Ib, for all a, b  A. 3. Intermediate algebra 3 (IA3) if and only if it satisfies (IA3.) CaCb, IaIb imply ab, for all a, b  A. Sequent calculi for the intermediate algebras: Sequent calculi Il1–Il3 for the intermediate algebras IA1–IA3, respectively, have been proposed and studied in (Sen 2001). The calculi are obtained by adding to/generalizing rules of tqBl5 (see section “Sequent Calculi for tqBa and tqBa5”) as follows:

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Ill: Add to tqBl5 the rule ðR Þr

lα : lβ : lα, lβ

Il2: Replace rule Rl of tqBl5 by the more general rule Rl0

lΓ : α, Δ lΓ : lα, Δ

Il3: Add to tqBl5 the rule lmR

lα : lβ mα : mβ α:β

Theorem 11. The sequent calculi are sound and complete for the respective classes of intermediate algebras. Various combinations of the algebras IA1, IA2, and IA3, such as (i) IA1 + IA2, (ii) IA1 + IA3, and (iii) IA2 + IA3, are also studied (Saha et al. 2016). Following is a summary of observations. It is found that an algebra satisfying (ii) reduces to the pre-rough algebra, as IA1 and IA3 imply IA2. In the algebraic structure (iii), no implication operator satisfying property () (see section “Quasi-Boolean Algebras”) is available, just as in the case of qBas. In case (i), it is still an open question whether such an implication operator can be found. However, there is an instance of this algebra, viz. the rough membership function algebra (RMF-algebra) (Chakraborty 2014), which has such an implication operator. This structure is obtained from another fundamental aspect of rough set theory, and it may be worth presenting the notions involved briefly. RMF-algebras: The notion of a rough membership function is a fundamental one in rough set theory. It was formally defined by Pawlak and Skowron in (Pawlak and Skowron 1994) for an approximation space (U, R) with finite domain U as follows. Definition 9. Given a subset A of U, a rough membership function fA is a mapping from U to Ra[0, 1], the set of rational numbers in [0, 1], defined by   Card ½xR \ A f A ðxÞ :¼ , Card ½xR for all x  U. As the equivalence class [x]R of the element x consists of all elements of U not distinguishable from it, to measure the membership of x to the set A, one essentially considers how much of the whole equivalence class lies within A. So it is clear that for y  [x]R, fA(y)¼fA(x).

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In (Chakraborty 2014), a rough membership function (RMF-) algebra is introduced for a fixed approximation space (U, R) with equivalence classes having finite cardinality. Consider { fA}A
U, the set of distinct rough membership functions, and operations of ^, _, ~, I on it: (fA^fB) (x) :¼ min(fA(x), fB(x)), (fA_fB) (x) :¼ max(fA(x), fB(x)), ~ fA (x) :¼ 1 – fA(x) and I fA (x) :¼ fA (x) for any x  U. It can be shown that {fA}A
U is closed with respect to all the operations ^, _, ~, I defined above, that is, in each case one can find C
U such that the expression on the right-hand side equals fC (x). A relation  can also be induced on {fA}A
U by the  relation on Ra[0,1]. An RMF-algebra is then the structure  f f A gA
U ,  , ^ , _ , , I, f 0 , f U . As mentioned in the previous section, the algebra is a tqBa5 that satisfies IA1 and IA2 axioms (Saha et al. 2016). On the other hand, it is observed in (Chakraborty 2014) that there can be RMF-algebras that are not pre-rough. However, an implication operator satisfying () can be defined on the collection {fA}A
U. This is the basis for a proposal of a Hilbert-style many-valued propositional logic that is sound for a semantics based on rough membership functions.

Algebraic Structures Related to Pre-rough Algebras There have been extensive studies in (Saha et al. 2014, 2016) in two directions: one focussing on the implication operator ) in the pre-rough algebra, another imposing an implication operator satisfying property () (see section “Quasi-Boolean Algebras”) on a qBa. Let us give a sketch of the work done.

System Algebras Recall the remark made in section “Quasi-Boolean Algebras” that an implication operator satisfying the property () is not available in a qBa, and hence in a tqBa5 also. A goal in (Saha et al. 2014) was to look for algebraic systems weaker than the pre-rough algebra, in which an implication operator may be available. The presence of such an operator would lead to Hilbert-style axiomatizations for logics corresponding to these algebras. In the following, we list the algebras and mention the corresponding logics. Each of these is based on a qBa hA, , ^, _, ~, 0, 1i with an additional unary operator I satisfying certain properties. The operator C is defined as in Notation 2. System0 algebra hA, , ^, _, ~, I, 0, 1i Properties of I: 1. 11 ¼ 1, 2. a  b implies Ia  Ib, for all a, b  A.

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M. Banerjee

Sequent calculus sq0 for system0 algebras: All axioms and rules of qBl are taken, along with two more rules: Rule l

α:β :α ðRl Þr lα : lβ : lα

SystemI algebra hA, , ^, _, ~, I, 0, 1i Properties of I, C: 1. 2. 3. 4.

I1 ¼ 1. ~ Ia_Ia ¼ 1, for all a  A. CaCb, IaIb imply ab, for all a, b  A. ab implies IaIb, for all a, b  A.

The Hilbert system LIfor systemI algebras: To give LI, a reduced axiom system for PRL is considered that is based on the modified definition of pre-rough algebra (see Definition 7) and given as follows: 1. 2. 3. 4. 5. 6. 7. 8.

α)~~α ~~ α)α αuβ)β αuβ)βuα αu (βtγ) )(αuβ)t(αuγ) (αuβ)t(αuγ) )αu (βtγ) □α)α □αu□β)□(αuβ)

All the PRL-rules of inference remain. The language of LI is the same as that of PRL. The axiomatization is derived directly from that of the reduced PRL system above: the first six axioms and all rules of PRL are taken. Sequent calculus sqI for systemI algebras: All the axioms and rules of sq0 are taken. The rules L~, R~ and lmR for the sequent calculus PRSC for pre-rough algebras are added. SystemII algebra hA, , ^, _, ~, I, 0, 1i Properties of I,C: 1. 2. 3. 4.

I1 ¼ 1. I (a^b) ¼ Ia^Ib, for all a, b  A. ~ Ia_Ia ¼ 1, for all a  A. Ca  Cb, Ia  Ib imply ab, for all a, b  A.

Proposition 2. Any systemII algebra is a systemI algebra. The converse of Proposition 2 is an open question. The Hilbert system LII for systemll algebras:

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The language of LII is the same as that of LI. All axioms and rules of LI are taken, along with the axiom: □(αuβ) )□(α) u□(β). Sequent calculus sqII for systemII algebras: It is sqI with added axiom: Ax4 lα, lβ : lðα ^ βÞ System algebras with modal axioms T, B, S4, S5: It is observed that none of the following modal properties hold in systemII algebras in general, and hence these do not hold in systemI algebras either. Note that all hold in tqBa5. (T) : Iaa. (B) : CIaa. (S4): IaIIa. (S4): CIaIa. It is established that if T is added to the set of systemI or II algebra axioms, one obtains a pre-rough algebra. A series of algebras are now defined by adding axioms B, S4 and S5 to the axioms of systemI and II algebras separately (as their equivalence is still an open question). We thus get systemIB, systemI4, and systemI5 algebras and similar for the case of systemII algebras (the nomenclature is self-explanatory). In addition, systemI4E and systemI5E algebras are defined (and others are defined similarly) where the  relation in the modal axiom is replaced by equality. A summary of the relationships between these algebras is given in Fig. 2. If A and B are two algebraic structures in the figure, A ) B indicates that one extra operator and some axioms for the new operator are added to A to obtain B. A ! B indicates that A and B have the same operators and if an algebra is of kind B then it is of the kind A. A ≈ B indicates that A and B are equivalent algebras. A. . .B means that A and B are independent. The Hilbert systems LI4, LI4E, LI5, LI5E, LIB, LII4: These correspond to systemI4, systemI4E, systemI5, systemI5E, systemIB, and systemII4 algebras, respectively; the language in each is the same as that of LI. All axioms and rules of LI are taken, together with the modal axiom(s) defining the algebras. Sequent calculi sqI4, sqI5, sqI4E, sqII4, sqII5, sqII4E, sqI5E, sqII5E, sqIB: sqI4 for systemI4 algebra: rule (Rl)r of sqI is replaced by ðRl0 Þ

r

lβ : α lβ : lα

sqI5 for systemI5 algebra: sql with Ax3 sqI4E for systemI4E algebra: sqI4 with the rule Ll00

β : lα lβ : lα

sqII4 for systemII4 algebra: sqII with rule (Rl)r replaced by rule Rl.

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qBa System 0 System I

System I4

System II System I5 System IB

System II4

System I4E

~ ~ System II4E ~ ~ System II5 ~ ~ System II5E

System I5E

Pre−rough Fig. 2 The System algebras

sqII5 for systemII5 algebra: sqII with Ax3 sqII4E for systemII4E algebra: sqII4 with the rule Ll0

Γ : lα lΓ : lα

sqI5E for systemI5E algebra and sqII5E for systemII5E algebra: The following axiom is added to sqI5 and sqII5, respectively Ax5 lα : mlα sqIB for systemIB algebra: sqI with the following axiom Ax6 mlα : α:

Implicative Quasi-Boolean Algebras with Modal Axioms In this direction of work (Saha et al. 2016), the qBa is considered along with an implication operator satisfying property (). Modal axioms are then added – these algebras and corresponding logics are listed below.

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Implicative quasi-Boolean algebra (IqBa) It is an algebra of the form hA, ^, _, ), ~, 0, 1i where 1. hA, ^, _, ~, 0, 1i is a qBa. 2. a)b ¼ 1 if and only if ab. for all a, b  A. The other algebras are based on an IqBa hA, ^, _, ), ~, 0, 1i, with the addition of a unary operator I satisfying certain properties. Implicative quasi-Boolean algebra with operator (IqBaO) Properties of I: 1. I1 ¼ 1. 2. I(a^b) ¼ Ia^Ib for all a, b  A. IqBaO with modal axiom T (IqBaT) Property of I: Iaa for all a  A. IqBaT with modal axiom S4 (IqBa4) Property of I: IaIIa for all a  A. IqBa4 with modal axiom S5 (IqBa5) Property of I: CIaIa for all a  A, where C :¼~ I ~. Hilbert systems for the IqBas: The language of LI, the logic for IqBa consists of propositional variables p, q, r, ...., a unary logical connective ~ and two binary connectives ^, !. LI Postulates: 1. 2. 3. 4. 5. 6.

α!~~ α ~~ α!α α^β!β α^β!β^α α^(β_γ) ! (α^β)_(α^γ) (α^β) _ (α^γ) !α^(β_γ) Rules of inference: 1.

D D oE E

2.

hypothetical syllogism

modus ponens

3. 5.

D E oD D oE D oJ D o E ġJ

D oE E oJ D oJ

4.

D oE ~ E o~ D

6.

D o E, E oD J oG, G oJ D  o J o E o G

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Hilbert systems LO, LT, L4, L5: These correspond to IqBaO, IqBaT, IqBa4, and IqBa5 algebras, respectively. The language of all the systems is the same as that of LI, along with the unary modal connective □. Added to the LI axioms and rules are two rules: α)β α (Necessitation) , □α ) □β □α The following axiom(s) are added according to the algebra being referred to: □αu□β)□(αuβ) for LO □αu□β)□(αuβ) and □α)α for LT □αu□β)□(αuβ), □α)α and □α)□□α for L4 □αu□β)□(αuβ), □α)α and ~ □ ~ □α)□α for L5. Expectedly, one gets the soundness and completeness results for these Hilbert systems with respect to the classes of corresponding algebras.

Rough Lattices and Heyting Algebras Equivalence classes in a Pawlakian approximation space may be regarded as granules with the help of which any subset of the domain is approximated. In a general situation however, the indiscernibility relation defining the approximation space may not induce a partition, and granules may not be disjoint. A lot of work has been done on generalized approximation spaces, and a number of different notions of approximations of sets have been proposed over the years. A comprehensive comparative study may be found in (Samanta and Chakraborty 2011). In (Kumar and Banerjee 2015), given a collection of granules {Oi : i  Λ} on a set U such that [i  ΛOi ¼ U, a set of conditions for a set X (
U ) to be approximated in the (generalized) approximation space (U,{Oi : i  Λ}) is given. A pair of operators L, U: P (U) !P (U ) that are called (respectively) the lower and upper approximation operators in (U, {Oi : i  Λ}) must satisfy the following. 1. For X
U, LX
X
UX. 2. If O is a granule, then the lower and upper approximations of O are O itself, i.e., LO ¼ O and UO ¼ O. 3. For X
U, further approximations of its lower and upper approximations do not lead to anything new, i.e., LLX ¼ LX, UUX ¼ UX, and ULX ¼ LX, LUX ¼ UX: In (Kumar and Banerjee 2015), generalized approximation spaces are considered where the equivalence relation is replaced by a covering relation C on the domain U, that is a nonempty subset of P (U ) such that [C ¼ U. More specifically, coverings generated by quasiorders (i.e., reflexive, transitive relations) are considered in the work: any quasiorder R on U yields a covering on U, viz. the collection {R(x) :

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x  U} where R(x) :¼ {y  U : xRy}, the R-neighbourhood of x. Such covering-based approximation spaces are called quasi order-generated covering-based approximation spaces (QOCAS), and for simplicity, (U, R) abbreviates (U, {R(x) : x  U}). It may be noted that any covering-based approximation space (U, C) also yields a QOCAS. Recall the mathematical expressions of the lower and upper approximations in a Pawlakian approximation space (see section “Introduction”). Considering granules to be R-neighbourhoods, definitions of the lower approximation LX and upper approximation UX for any X
U in a QOCAS (U, R) are as follows. Definition 10. For any X
U, L, U: P (U) !P (U) are such that LX :¼ [fRðxÞ : RðxÞ
Xg, UX :¼ [fRðxÞ : x  Xg: L, U of Definition 10 satisfy all the properties 1–3. In the special case of a Pawlakian approximation space where R is an equivalence relation (so that the R-neighbourhood R(x) of x is just the equivalence class [x]), L, U reduce to the classical lower and upper approximation operators. However, it may be noted that in this general scenario, the operators are not dual to each other. Interestingly, none of the various non-Pawlakian lower and upper approximation pairs studied in (Samanta and Chakraborty 2011) satisfy 1–3 together. L, U have a topological representation, from the well-known one-one correspondence between Alexandrov topologies and quasiorders on any set. The family {R(x) : x  U} forms a minimal basis for an Alexandrov topology (denoted T R) on U. The inverse R–1 of the quasiorder R is a quasiorder, whence T R1 also generates an Alexandrov topology. It can then be seen that L is the interior operator in the topology T R, while U is the closure operator in T R1 . The collection D of all definable sets in the QOCAS (U, R) is just the collection of all open sets in the topological space (U, T R) (or closed sets in the topological space ðU, T R1 Þ). In other words, D ¼ T R. D forms a Heyting algebra, and not a Boolean algebra as in the Pawlakian case; it is also a completely distributive lattice. Note that if R is an equivalence relation on U, R ¼ R–1 so that T R ¼ T R1 . L and U then reduce to the Pawlakian lower and upper approximation operators. Following Notation 1 for the Pawlakian case, RS :¼ {(LX, UX) : X
U} gives the collection of rough sets in the QOCAS (U, R), while R :¼ {(D1, D2) : D1
D2, D1, D2  D} is the collection of generalized rough sets in the QOCAS (U, R). Again, RS
R. It is shown in (Kumar and Banerjee 2012) that, as for the Pawlakian case, R¼ RS, if and only if |D2\D1| 6¼ 1 for every (D1, D2)  R.  Consider   the  natural order  on RS and R: for any (D1, D2), (D3, D4)  R and X, X , Y, Y  RS, (D1, D2),     (D3,D4) if and only if D1
D3 and D2
D4, X, X  Y, Y if and only if X
Y and X
Y.

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It is established that for every QOCAS (U, R), there exists an approximation space (U0 ,R0 ) such that (i) R0 ¼ RS 0 , and (ii) R and R0 are order isomorphic. Here, R0 and RS 0 denote the collections of generalized rough sets and rough sets, respectively, in the QOCAS (U0 , R0 ). Investigating algebraic properties of the two collections R and RS, it is observed that R forms a completely distributive lattice, and therefore forms a Heyting algebra. On the other hand, RS may not even form a lattice with respect to . Recall that a Heyting algebra (Rasiowa 1974) (A, 1, 0, _, ^, !, :), also called a pseudo-Boolean algebra, is a bounded lattice (A, 1, 0, _, ^) with the “relative pseudocomplement !” that satisfies the property a ^ x  b , x  a ! b for any a, b, x  A, and “:” defined as :a :¼ a! 0 for all a  A.

Rough Lattices Now, as done to obtain the tqBa structure (see section “Topological Quasi-Boolean Algebras and Extensions”) on the quasi-Boolean algebra R in the Pawlakian case, one introduces operators I and C on the set R for a QOCAS (U, R): for any a, b  D, define I(a, b) :¼ (a, a) and C(a,b) :¼ (b, b). Let RIC denote the lattice R enhanced with the operators I, C. It is observed that in this case too, for the QOCAS (U, R), there exists a QOCAS (U0 , R0 ) such that R0 ¼ RS 0, and RIC corresponding to (U, R) is isomorphic to R0 IC corresponding to (U0 , R0 ). On abstraction of the properties enjoyed by I and C, a new algebraic structure is defined in (Kumar 2016): Definition 11. A rough lattice L:¼ (L, _, ^, I, C, 0, 1) is a bounded distributive lattice (L, _, ^, 0, 1) such that I,C are unary operators on L satisfying the following properties. 1. 2. 3. 4. 5. 6. 7.

I(a^b) ¼ Ia^Ib,C (a_b) ¼ Ca_Cb. I(a_b) ¼ Ia_Ib, C(a^b) ¼ Ca^Cb. Iaa, aCa. I1 ¼ 1, C0 ¼ 0. IIa ¼ Ia and CCa ¼ Ca. ICa ¼ Ca and CIa ¼ Ia. IaIb and CaCb imply ab.

RIC is a rough lattice, and any pre-rough algebra is also a rough lattice. A representation result is then expectedly obtained: any rough lattice is embeddable in RIC over some QOCAS. This leads to a rough set semantics for the logic of rough lattices. Logic L RL for rough lattices: The set F of formulae is built from a set P of propositional variables p, q, r, . . . and propositional constants ⊤, ⊥, by the following scheme:¼ ⊤j⊥j p jϕ _ ψ jϕ ^ ψ j□ϕ j◇ϕ:

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Let us now define the postulates for the logic which are based on the consequence pair notion as used for Kleene logic L K in section “Topological Quasi-Boolean Algebras and Extensions.” These will give the consequence relation ‘L RL . LRL Postulates: (a) Postulates 1–9 of Definition 2 giving logic for bounded distributive lattices. (b) □α ‘ α, α ‘ ◇α. (c) □α ^ □β ‘ □(α ^ β), ◇(α _ β) ‘ ◇α _ ◇ β. (d) □(α ^ β) ‘ □α ^ □β, ◇α _ ◇β ‘ ◇(α _ β). (e) □α _ □β ‘ □(α _ β), ◇(α ^ β) ‘◇α ^ ◇β. (f) □(α _ β) ‘ □α _ □β, ◇α ^ ◇β ‘ ◇(α ^ β). (g) ⊤ ‘ □⊤, ◇⊥ ‘ ⊥. (h) □α ‘ □□α, ◇◇α ‘ ◇α. (i) ◇α ‘ □◇α, ◇□α ‘ □α. α ‘ β α ‘ β , (j) . □ α ‘ □β ◇α ‘ ◇β □α ‘ □β, ◇α ‘ ◇β . (k) α ‘ β LRL is sound and complete for the class A RL of all rough lattices. Let A RL R denote the class of all rough lattices R IC considering all possible QOCAS. Then, using similar definitions and notations for validity as given for LK, we obtain the following due to the representation result for rough lattices. Let α, β  F . Theorem 12. (Rough Set Semantics) 1. α A RL β if and only if α A RL β. R 2. α ‘L RL β if and only if α A RL β. R

Rough Heyting Algebras As R is a Heyting algebra, (Kumar 2016) investigates how the operators I and C interact with the relative pseudocomplement ! of R, and yet another new algebraic structure is obtained. Note that any Heyting algebra is a bounded distributive lattice, hence a rough lattice may also be defined on any Heyting algebra as base. Definition 12. A rough Heyting algebra H :¼ (H, _, ^, !, I, C, 0, 1) is a rough lattice on a Heyting algebra (H, _, ^, !, 0,1) in which I, C satisfy the following additional properties. 1. I(a!b) ¼ (Ia!Ib)^(Ca!Cb). 2. C(a!b) ¼ Ca!Cb. RIC is a rough Heyting algebra, and one has a representation result here as well: any rough Heyting algebra is embeddable in RIC over some QOCAS. It is observed

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in (Kumar 2016) that any rough Heyting algebra is an instance of a monadic Heyting algebra, an algebraic structure first studied by Monteiro and Varsavsky (see Bezhanishvili 1998). The structure has also been studied under the name of “bitopological pseudo-Boolean algebra” by Ono (1977) and Suzuki (1989). It is a Heyting algebra (H, _, ^, !, 0, 1) with unary operators I,C which satisfy the following properties: for all a, b  H, 1. 2. 3. 4. 5.

Iaa, aCa I(a^b) ¼ Ia^Ib,C(a_b) ¼ Ca_Cb I1 ¼ 1, C0 ¼ 0 ICa ¼ Ca, CIa ¼ Ia C(Ca^b) ¼ C(a^b).

Note that the converse of the earlier statement does not hold: there are monadic Heyting algebras that are not rough. Logic LroughInt for rough Heyting algebras: Prior in 1957 proposed a modal version of Intuitionistic logic (IPC), i.e., he added modalities to IPC calling the new logic MIPC. Monadic Heyting algebras mentioned above constitute the algebraic models for MIPC. As rough Heyting algebras are essentially Heyting algebras with operators, the corresponding logic L roughInt is also based on IPC, with added modalities. It is also a consequence that if α is a theorem of MIPC, it is a theorem of L roughInt. We refer to (Bezhanishvili 1998) for the presentation of IPC postulates (and of MIPC). The language of L roughInt consists of a set P of propositional variables p, q, r, . . ., propositional constants ⊤, ⊥, and logical connectives _, ^, !, □, ◇. The set F of formulae is given by the scheme:¼ ⊤j⊥jpjα _ βjα ^ βjα ! βj□αj◇α: The postulates of L roughInt are as follows. L roughInt Postulates: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

α!(β!α). (α!(β!γ) ! ((α!β) ! (α!γ)). (α^β) !α. (α^β) !β. α! (β! (α^β)). α!α_β. β!α_β. (α!β) ! ((β!γ) ! (α_γ!β)). ⊥!α. □α!α. α!◇α. ⊤!□⊤. ◇⊥!⊥.

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□(α_β) !□α_□β. □α^□β!□(α^β). ◇(α_β) !◇α_◇β. ◇α^◇β!◇(α^β). ◇(α!β) $ (◇α!◇β). □(α!β) $ ((□α!□β) ^ (◇α!◇β)). □α!□□α, ◇◇α!◇α. ◇α!□◇α, ◇□α!□α. Modus Ponens α!β α!β , 23. . □α ! □β ◇α ! ◇β □α ! □β, ◇α ! ◇β 24. . α!β 14. 15. 16. 17. 18. 19. 20. 21. 22.

Note that axioms 1–9 are the axioms of IPC. ℒroughInt is sound and complete with respect to the class ℝ of all rough Heyting algebras. Let A RH R denote the class of all rough Heyting algebras RIC considering all possible QOCAS. Due to the representation result for rough Heyting algebras, we obtain Theorem 13. (Rough Set Semantics) 1. ℝ α if and only if A RH α. R 2. ‘L roughInt α if and only if A RH α. R

Contrapositionally Complemented Pseudo-Boolean Algebras Motivated by the category-theoretic studies on rough sets initiated by Banerjee and Chakraborty in (1993a, 2003), More and Banerjee followed up with further investigations (More and Banerjee 2016) of the categories proposed in (Banerjee and Chakraborty 1993a), in particular of the category ROUGH of rough sets. Objects of ROUGH are triples of the form hU, R, Xi, where (U, R) is a Pawlakian approximation space, and X is a subset of U. The upper and lower approximations of X in (U, R) are used to define morphisms in the category, such that the morphisms map upper approximations in the domain approximation space to those in the codomain space, preserving information present in the lower approximations. Formally, we have the following. First a notation: let X and X denote the collections of equivalence classes in U contained in the upper approximation and lower approximation of set X (
U ), respectively, in the approximation space (U, R): 




 X :¼ ½xR ½xR \ X 6¼ 0 , and


 X :¼ ½x ½x
X : R

R

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So X ¼ [X , X ¼ [X, union being taken over elements of equivalence classes. A morphism in ROUGH with domain hU, R, Xi and codomain hV, S, Yi is then defined as a map f : X ! Y such that f ðX Þ
Y . In category theory, an elementary topos is a generalization of SET, the category of sets. It is shown in (Banerjee and Chakraborty 1993a) that ROUGH does not form a topos. In (More and Banerjee 2016), it is established that the category forms a quasitopos, which is a structure weaker than topos. Further, a study of the algebra of strong subobjects of objects in ROUGH is presented in (More and Banerjee 2016). The standard categorical constructions of the operations of intersection, union, complement, and implication on the collection of strong subobjects of any ROUGH-object leads to a Boolean algebra, and it is observed that the negation does not capture relative rough complementation – which is what is required, as one wants the complement of a subobject with respect to a ROUGH-object. A different definition of negation is then considered, in line with Iwiński’s rough difference operator. With respect to this new negation, strong ROUGH-subobjects of an object are shown to form contrapositionally complemented lattices (cc lattices) (Rasiowa 1974) with the excluded middle property, called c_c lattices (Geisler and Nowak 1994; Nowak 1995). A category RSC of rough sets is also presented in (Li and Yuan 2008; Wyler 1991), based on Iwiński’s approach in (Iwiński 1987). Iwiński considers an interpretation of rough sets based on a Boolean algebra. A generalization of the approximation space, called a rough universe, is taken in (Iwiński 1987): it is a pair (U, B), where U is a set and B is a subalgebra of the power set Boolean algebra P(U ). A pair of the form (A1, A2), where A1, A2  B and A1
A2, is called an I-rough set of (U, B). Objects of the category RSC range over I-rough sets of all rough universes. So these are of the form (X1, X2), where X1, X2  B and X1
X2 for the rough universe (U,B). A morphism in RSC with domain (X1, X2) and codomain (Y1, Y2) is a map f : X2!Y2 such that f (X1)
Y1. Note that any pair (A1, A2), where A1
A2, can be considered as an RSC-object. This is because A1
A2
A2 so that (A1, A2) is an I-rough set of the rough universe (A2, P (A2)). It is shown in (More and Banerjee 2016) that RSC is (categorially) equivalent to ROUGH. Now ROUGH and RSC are defined on the category SET as base. Utilizing the equivalence of the two categories, the format of RSC is generalized in (More and Banerjee 2016) to give a category RSC(C ) based on an arbitrary topos C . C is assumed to be nondegenerate – that is, not all C objects are isomorphic. Objects of RSC(C) are of the form (A, B), where A and B are objects in C such that there exists a monic morphism m : A ! B in C. A morphism in RSC(C ) with domain (X1, X2) and codomain (Y1, Y2) is a pair of morphisms ( f 0 , f), where f 0 : X1!Y1 and f : X2!Y2 in C such that the following diagram commutes in C:

X1

f′

m

X2

Y1 m′

f

Y2

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Algebraic Logic and Rough Set Theory

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Here m and m0 are monic arrows corresponding to the objects (X1, X2) and (Y1, Y2) in RSC(C). The following are observed. (i) In the special case when C is the topos SET, RSC(SET) is just the category RSC. From the perspective of foundations of rough sets, this exercise is therefore significant: just as a topos is a category-theoretic abstraction of SET, the quasitopos RSC(C) is an abstraction of the category ROUGH of rough sets. (ii) RSC(C) is a full subcategory of the topos C !, consisting of all the monics in C as its objects. It is subsequently proved in (More and Banerjee 2016) that RSC(C) is not a topos, but a quasitopos. Furthermore, a study of strong subobjects of an RSC(C)-object shows that these form a pseudo-Boolean algebra (aka Heyting algebra) with respect to the standard categorial constructions of intersection, union, complement, and implication. For the same reason as mentioned earlier, Iwiński’s rough difference operator is incorporated and it is found that the strong subobjects form a new algebraic structure with two negations. This structure is termed a contrapositionally complemented pseudoBoolean algebra (More and Banerjee 2016). A subclass of the class of these algebras having the excluded middle property also becomes relevant – these are termed contrapositionally _ complemented pseudo-Boolean algebras. As the names suggest, the structures are an amalgamation of cc lattices and pseudo-Boolean algebras. Definition 13. A :¼(A, 1, 0, _, ^, !, :, ~) is called a contrapositionally complemented pseudo-Boolean algebra (ccpBa), if (A, 1, 0, ^, _, !, :) forms a Heyting algebra, and for all a  A, the following condition holds: a ¼ a ! ð:: 1Þ: If, in addition, for all a  A, a_ ~a ¼ 1, we call A a contrapositionally _ complemented pseudo-Boolean algebra (c_cpBa). Substantiating through examples, it is shown that the two negation operators are independent. Furthermore, a class of examples of the algebras can be produced from Heyting or Boolean algebras as follows. For any Heyting algebra H :¼(H, 1, 0, _, ^, !, :), the set H ½2 :¼ {(a, b) : ab, a, b  H} and u :¼ (u1, u2)  H ½2 , one may define the following set Au and operators: n o Au ≔ ða1 , a2 Þ  H ½2 : a2  u2 and a1 ¼ a2 ^ u1 , t : ða1 , a2 Þ t ðb1 , b2 Þ≔ða1 _ b1 , a2 _ b2 Þ, u : ða1 , a2 Þ u ðb1 , b2 Þ≔ða1 ^ b1 , a2 ^ b2 Þ, !: ða1 , a2 Þ ! ðb1 , b2 Þ≔ðða1 ! b1 Þ ^ u1 , ða2 ! b2 Þ ^ u2 Þ, : ða1 , a2 Þ≔ðu1 ^ :a1 , u2 ^ :a1 Þ, and : : :ða1 , a2 Þ≔ða1 , a2 Þ ! ð0, 0Þ: Then one obtains

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Proposition 3. A u :¼ (Au, (u1, u2), (0, 0), u, t, !, :, ~) forms a ccpBa. Moreover, if H is a Boolean algebra, then A u forms a c_cpBa. A representation result for ccpBas is obtained: every ccpBa is embeddable in the ccpBa formed by all open subsets of some topological space. Duality results for the algebras are also proved, using ordered topological (Esakia) spaces.

Intuitionistic Logic with Minimal Negation (ILM) A detailed study of the logics for ccpBas and c_cpBas is conducted in (More 2019). Let us take a brief look at the systems. The alphabet of the language L of ILM is that of IL, consisting of propositional constants ⊤ and ⊥, a set PV of propositional variables, and logical connectives ! (implication), _ (disjunction), ^ (conjunction), and ~ (negation). Additionally, there is a unary connective :. The formulae are given by the scheme: ⊤j⊥jpjα ^ βjα _ βjα ! βj:αj α Where p  PV. The set of all formulae is denoted by F. ILM Postulates: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11.

α!(β!α) (α! (β!γ)) ! ((α!β) ! (α!γ)) α! (α_β) and β! (α_β) (α!γ) !((β!γ)! ((α_β) !γ)) (α^β) !α and (α^β) !β (α!β) ! ((α!γ)! (α! (β^γ))) α!⊤ ⊥!α (α!β) ! ((α!:β) !:α) :α! (α!β) ~α$ (α!::~⊤)

Modus ponens (MP) is the only rule of inference in ILM. Deduction procedure is defined in the usual manner. Addition of the axiom α_~α gives the axiomatic system for ILM-_. ILM and ILM-_ are sound and complete with respect to the class of ccpBas and c_cpBas, respectively. Note that minimal logic (ML) corresponds to the class of cc lattices, while intuitionistic logic (IPC) is the logic for Heyting algebras. Expectedly, ML and IPC are both embedded in ILM. It is also shown in (More 2019) that ILM is a strict extension of an extension of ML defined by Segerberg incorporating Peirce’s law. Apart from the algebraic semantics, relational semantics

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are defined for the two logics, in the line of work by Segerberg (1968), Došen (1986), and Vakarelov (1989).

Algebras from Rough Concept Analysis Seminal work in the direction of linking formal concept analysis (FCA) and rough set theory has been done by Düntsch, Gediga, and Yao. Citing limitations of FCA, Düntsch and Gediga introduced property-oriented concepts (Düntsch and Gediga 2002), and Yao proposed object-oriented concepts (Yao 2004b). The new concepts are motivated by rough sets; examples of contexts  :¼ (G, M, R) were given where the relation R (
GM ) associates properties in M to objects in G either necessarily or possibly. For any context  :¼ (G, M, R) (R
G  M ), A
G and B
M, modal style operators were introduced:






 □ B◇ R ≔ x  GjRðxÞ \ B 6¼ 0 and BR ≔fx  GjRðxÞ
Bg

 1 □ 1 A◇ R1 ≔ y  MjR ðyÞ \ A 6¼ 0 and AR1 ≔ y  MjR ðyÞ
A We omit the subscripts R or R–1 when there is no confusion. It is observed that □◇ is an interior operator on G while ◇□ is a closure operator on M. Objectoriented concepts are then defined as follows: Definition 14. Yao (2004a) (A, B) is an object-oriented concept of the context  if it satisfies the condition A□ ¼ B and B◇ ¼ A. An order relation  can be defined on the set of such pairs. For any objectoriented concepts (A1, B1), (A2, B2), (A1, B1)  (A2, B2) if and only if A1
A2 (equivalently, B1
B2). Algebraic structures resulting from the object-oriented concepts have been studied by many. In (Howlader and Banerjee 2018, 2020), negation is brought into the studies, and notions of object-oriented semiconcepts and protoconcepts are introduced in the line of Wille’s work (Wille 2000). Definition 15. – Howlader and Banerjee (2018) (A, B) is an object-oriented semiconcept of  if A□ ¼ B or B◇ ¼ A. – Howlader and Banerjee (2020) (A, B) is an object-oriented protoconcept of  if A□◇ ¼ B◇. It is observed that object-oriented protoconcepts form a double Boolean algebra that is contextual, while the object-oriented semiconcepts form a pure double Boolean algebra. These algebraic structures were introduced by Wille (2000) and are defined as follows.

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Definition 16. A double Boolean algebra (dBa) is an abstract algebra D :¼ (D, t, u, :, ⌟, ⊤, ⊥) which satisfies the following properties, for any x, y, z  D. (1a) (xux)uy¼xuy (1b) (xtx)ty¼xty (2a) xuy¼yux (2b) xty¼ytx (3a) xu(yuz)¼(xuy)uz (3b) xt(ytz)¼(xty)tz (4a) :(xux)¼:x (4b) ⌟(xtx)¼⌟x (5a) xu(xty)¼xux (5b) xt(xuy)¼xtx (6a) xu(y_z)¼(xuy)_(xuz) (6b) xt(y^z)¼(xty)^(xtz) (7a) xu(x_y)¼xux (7b) xt(x^y)¼xtx (8a) :: (xuy)¼xuy (8b) ⌟⌟(xty)¼xty (9a) xu:x¼⊥ (9b) xt⌟x¼⊤ (10a) :⊥¼ ⊤ u ⊤ (10b) ⌟⊤ ¼ ⊥ t ⊥ (11a) :⊤¼ ⊥ (11b) ⌟⊥ ¼ ⊤ (12) (xux)t(xux)¼(xtx)u(xtx), where x_y :¼ : (:xu:y) and x^y :¼⌟(⌟xt⌟y). On D, a quasiorder
is given by x v y , x u y ¼ x u x and x t y ¼ y t y, for any x, y  D: A dBa D is contextual, if the quasiorder is a partial order. A dBa D is called pure, if for all x  D either xux¼x or xtx¼ x. Representation results are established in (Howlader and Banerjee 2020): any dBa (contextual dBa) is quasiembeddable (embeddable) in the dBa formed by objectoriented protoconcepts of some context, where a quasiembedding is a dBa-homomorphism that preserves the quasiorder in the dBas. Further, any pure dBa is embeddable in the pure dBa formed by object-oriented semiconcepts of some context. We next present the logic DBL for the class of contextual dBas, which is extended to PDBL to get a hypersequent calculus for pure dBas. Due to the repesentation results, the logics get semantics based on object-oriented protoconcepts and semiconcepts.

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Logics DBL for Contextual dBas and PDBL for Pure dBas The language of DBL consists of propositional constants ⊥, ⊤, a set VAR of propositional variables, and logical connectives t, u, :, ⌟. The set F of all formulae of DBL is given by the scheme: ⊥j⊤j p jα t βjα u βj:αj⌟α, where p  VAR. _ and ^ are definable connectives: for α, β  F , α _ β≔:ð:α u :βÞ and α ^ β≔⌟ð⌟α t ⌟βÞ: The derivation procedure of DBL is given by consequence pairs α‘β with formulae α, β  F . αa‘β abbreviates (α‘β and β‘α). DBL Postulates: 1a ⊤ ‘ α 1b α ‘ ⊤ 2a α u β ‘ α 2b α ‘ α t β 3a αuβ ‘ β 3b β ‘ αtβ 4a α u β ‘ (α u β) u (α u β) 4b (α t β) t (α t β) ‘ α t β 5a α u α ‘ α u (α t β) 5b α t (α u β) ‘ α t α 6a : (α u α) ‘ :α 6b ⌟α ‘ ⌟(α t α) 7a α u :α ‘⊥ 7b ⊤ ‘α t ⌟α 8a : ⊥ a‘ ⊤ u ⊤ 8b ⌟ ⊤ a‘ ⊥ t ⊥ 9a α u α ‘α u (α_β) 9b α t (α ^ β) ‘ α t α 10a α u (β _ γ)a‘(α u β) _ (α u γ) 10b α t (β^γ)a‘(α t β) ^ (α t γ) 11a ::(α u β)a‘ (α u β) 11b ⌟⌟(α t β) a‘(α t β) 12a : ⊤ ‘ ⊥ 12b ⊤ ‘ ⌟ ⊥ 13 α ‘ α 14 (α t α) u (α t α) ‘ (α u α) t (α u α)

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Rules of inference: α‘β α‘β (R10 ) . (R1) αuγ ‘ βuγ γuα‘γuβ α‘β α‘β (R20 ) . (R2) αtγ ‘ βtγ γtα ‘ γtβ α‘β α ‘ β, β ‘ γ (R4) . (R3) :β ‘ :α α‘γ α‘β α u β a‘ α u α α t β a‘ β t β (R6) . (R5) ⌟β ‘ ⌟α α‘β The logic DBL is sound and complete with respect to the class Dℬ of contextual dBas. The logic PDBL for the class of pure dBas is obtained as an extension of DBL in the form of a hypersequent calculus. Let P ðÞ denote the collection of all objectoriented protoconcepts of a concept . Using definitions and notations given earlier, we get the following due to the representation result for contextual dBas. Theorem 14. For any α and β in F , α ‘ β is provable in DBL if and only if it is valid in P ðÞ.

Information System Algebras Algebraic structures generated from deterministic, incomplete as well as nondeterministic information systems are studied in (Khan and Banerjee 2013) and corresponding abstract algebras are defined. Representation theorems for these classes of abstract algebras are also proved. The logics corresponding to these classes of algebras have been independently studied in (Khan 2015; Khan and Banerjee 2009, 2011). We briefly present the algebras and logical systems in this section. Slight modifications in notations from the cited work have been made here, to keep this presentation uniform. Throughout this section, we consider finite sets of attributes and attribute values.

Algebra for Deterministic Information Systems Consider fixed finite sets A of attributes and V :¼[a  A V a of attribute values, and the set D of all descriptors (Pawlak 1991), viz. pairs (a, v), for each a  A, v  V a. As mentioned in section “Introduction,” a deterministic information system S :¼ (U, A, V , f ) induces a Pawlakian approximation space. It was also mentioned that lower and upper approximations of sets in an approximation space can be defined through unary operators on the power set. In the context of information systems, the upper and lower approximations with respect to the indiscernibility relations Ind SB , B
A determine unary operators Ind SB and Ind SB , respectively on the power set P (U ) of U such that

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Ind SB ðXÞ :¼ XInd SB , Ind SB ðXÞ≔XInd SB , X
U: Each descriptor (a, v) determines a nullary operation (constant) cSða,vÞ on P (U ): cSða,vÞ :¼ fx  U : f ðx, aÞ ¼ vg: We then get the following definition. Definition 17. Let S :¼ (U, A, V , f ) be a deterministic information system. A deterministic information system algebra (in brief, DIS-algebra) generated by the deterministic information system S is the structure  n o
 P ðU Þ, \ , , 0, Ind SB B
A , cSγ 

S  :¼

γD

:

It is observed in (Khan and Banerjee 2013) that this structure is an instance of a knowledge approximation algebra derived from S (Comer 1991) extended with a collection of nullary operations. On abstraction of properties of DIS-algebras generated by DISs, the following notion of abstract DIS-algebra is introduced. Definition 18. Given fixed
finite  sets A and D, an abstract DIS-algebra is a tuple A :¼ (U, ^, :, 0, fLB gB
A , d γ γ  D ), where (U, ^, :, 0) is a Boolean algebra and LB and dγ are respectively unary and nullary (constant) operations on U satisfying the following for any x,y  U. “!” is the Boolean implication defined as x!y :¼ :x_y. (C0) (C1) (C2) (C3) (C4) (C5) (C6) (C7)

L WB (x^y) ¼ LB (x) ^ LB ( y); v  V a d ða,vÞ ¼ 1; d(a,v)^d(a,u) ¼ 0 when v6¼u; LC (x) LB (x) for C
B
A; d(a,v)L{a} (d(a,v)); d(b,v)^LBU{b} (x) LB(d(b,v) ! x); L; ðxÞ ¼ 0 for x 6¼ 1; L; ð1Þ ¼ 1.

It can be proved that the reduct A :¼ (U, ^, :, 0, fLB gB
A ) is a topological Boolean algebra (Rasiowa 1974). A representation theorem is established for the class of abstract DIS-algebras with respect to that of DIS-algebras generated by DISs: every abstract DIS-algebra is isomorphic to a subalgebra of S corresponding to some DIS S. Logic LDIS for abstract DIS-algebras: Abstract DIS-algebras are the algebraic counterpart of the logic LDIS of deterministic information systems (Khan and Banerjee 2009, 2013). (The logic was denoted as LIS in (Khan and Banerjee 2009).)

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The language has a nonempty finite set AC of attribute constants, a nonempty finite set VC a of attribute value constants for each a  AC, and a nonempty countable set PV of propositional variables. The set L of LDIS-formulae is given by the following scheme: ða, vÞ  D jp  PV j:αjα ^ βj½I ðBÞα, B
AC













The operator dual to [] is defined as hI(B)iα :¼ :[I(B)]:α. When B is a singleton {a} (say), we simply write [I(a)] or hI(a)i. LDIS Postulates: Ax1. All axioms of classical propositional logic. Ax2. [I(B)](α!β) ! ([I(B)]α ! [I(B)]β).    Ax3. I 0 α ! α.      Ax4. α ! I 0 I 0 α.

   

  Ax5. I 0 I 0 α ! I 0 α. Ax6. [I(C)]α! [I(B)]α for C
B
AC . Ax7. (a, v) !:(a, v0 ), for v 6¼ v0. Ax8. _v  VC a ða, vÞ Ax9. (a, v) ! [I(a)](a,v). Ax10. ((b, v) ^ [I(B[{b})]α) ! [I(B)]((b, v) !α). B, C are arbitrary subsets of AC . Rules of inference: N:

α ½I ðBÞα where B
AC

MP:

α α!β β

Derivability in LDIS is defined in the usual manner. Let AD I S denote the class of all abstract DIS-algebras, while RD I S denotes the class of all DIS-algebras generated by DISs. LDIS can be shown to be sound and complete with respect to AD I S. Moreover, due to the representation theorem for abstract DIS-algebras, one finds Theorem 15. For any α  L, 1. α is valid in AD I S if and only if it is valid in RD I S. 2. ‘LDIS α if and only if α is valid in RD I S.

Apart from the algebraic semantics, LDIS gets an information system semantics, as shown in (Khan and Banerjee 2009). Note that from any deterministic information system S :¼ (U, AC , [a  AC VC a , f ), one gets a structure (U, fInd S ðBÞgB
AC , mS ), where mS : D !P (U ) gives the extension of the descriptors, that is mS (a, v) :¼

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{w  U: f (w, a) ¼ v}. It is called the standard DIS-structure generated
 by S (Vakarelov 1987). Abstraction of this structure gives a tuple F :¼ (U, RIðBÞ B
AC , m), where U is a nonempty set, RI(B)
UU, m : D!P(U ) is a meaning function for the descriptors, and the following conditions are satisfied:



(DIS1) For each a  AC , [{m(a, v) : v  VC a } ¼ U. (DIS2) For each a  AC , mða, vÞ \ mða, v0 Þ ¼ 0, for v 6¼ v0 . (DIS3) RIð;Þ ¼ U  U. (DIS4) RIðBÞ
RIðCÞ for C
B
AC . (DIS5) For B
AC and b  AC , RI(B)\RI(b)
RI(B[{b}). (DIS6) For b  AC , (w, w0 )  RI(b) if and only if there exists v  VC b such that w, w 0  m(b, v). Such a tuple is called a DIS-structure, and one can show that it is a standard DIS-structure generated by some information system. Together with a meaning function V : PV!P (U ) for the propositional variables, a DIS-structure is called a DIS-model. This forms the basis for the LDIS information system semantics, but expectedly (as we observe below), it will be equivalent to considering a semantics based on standard DIS-structures. In fact, it will also be equivalent to a semantics based directly on deterministic information systems – as is done for nondeterministic information systems in section “Algebra for Nondeterministic Information Systems.”
 Satisfiability of a formula α in an IS-model M :¼ (U, RIðBÞ B
AC , m, V ) at an object w of the domain U, denoted (as usual) as M, w α, is defined as below for the non-Boolean cases: M, w (a, v) if and only if w  m(a, v), for (a, v)  D. M, w p, if and only if w  V ( p), for p  PV. M, w [I(B)]α, if and only if for all w0 in U with (w, w0 )  RI(B), M, w0 α. 

Notice that when B ¼ 0, [I(B)] is the global modal operator (Blackburn et al. 2001). For any formula α in L and DIS-model M, let M(α) :¼ {w  U : M, w α}. α is valid in M, denoted M α, if and only if M(α) ¼ U. α is said to be valid ( DIS α), if M α for every DIS-model M. DIS-models based on standard DIS-structures are called standard DIS-models. SDIS α denotes that α is valid in all standard DIS-models. It then follows Proposition 4. DIS α if and only if SDIS α for all α  L. Note also that there is a one-one correspondence between the class RD I S of DIS-algebras generated by DISs and the class of standard DIS-structures. This is

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because, for any (a, v)  D , the nullary operation cSða,vÞ on P (U ) can be identified with the function mS on U defined above. The axiomatic system for LDIS is shown to give soundness and completeness with respect to the DIS-models and therefore the standard DIS-models as well. It is observed in (Khan and Banerjee 2009) that the n-agent epistemic logic S5D n (Fagin et al. 1995) with knowledge operators Ki (i ¼ 1, . . ., n) and distributed knowledge operators DG for groups G of agents is embeddable in LDIS with jAC j n. LDIS is also more expressive than S5D n because it has the descriptors as part of its language – this extra feature is essential for it to be given a semantics based on information systems.

Algebra for Incomplete Information Systems Recall the similarity relation SimSB , B
A , defined for any IS S in section “Introduction.” SimSB also determines the unary operation SimSB on P (U ) as SimSB ðXÞ≔XSimSB , X
U: Both the indiscernibility and similarity relations are incorporated in the algebraic structure defined in this context. The IS-algebra generated by the information system S is the structure 

 n o S ≔ P ðU Þ, \ , , 0, Ind SB 

B
A

n o , SimSB

B
A

n o , cSγ

γD

:

On abstraction of properties satisfied by this algebra, one obtains the definition of an abstract IS-algebra: it is a tuple 
 A≔ U, ^ , :, 0, fLB gB
A , fSB gB
A , dγ γ  D , 
 where U, ^ , :, 0, fLB gB
A , dγ γ  D is an abstract DIS-algebra and S B , B
A, are unary operations on U such that (C8) SC(x) SB (x) for C
B
A; (C9) d(a, v)S{a} (d(a,v))_d(a,)), where v 6¼ ; (C10) d(b,v)^SBU{b} (x) SB ((d(b,v)_d(b,))!x) where v 6¼ ; (C11) d(b,)^SBU{b} (x) SB(x). As in the case of DIS-algebras, a representation theorem is proved here as well: every abstract IS-algebra is embeddable in the IS-algebra generated by some information system. Logic LIS for IS-algebras: Logics for information systems were proposed in (Khan and Banerjee 2011). It can be shown that abstract IS-algebras are the algebraic counterpart for one of the systems that we denote as LIS here. Using the

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representation result for IS-algebras, LIS is sound and complete with respect to the class of IS-algebras generated by ISs as well. The language of LIS extends that of LDIS by adding unary modal connectives [S(B)] for each B
AC . The collection of postulates also extends that of LDIS, by incorporating those for [S(B)]; the N rule now holds for [S(B)] also. To accommodate the -value for attributes that is allowed here, axiom 8 and the [S(B)]-versions of axioms 9–10 of LDIS are modified to give axiom 4 and axioms 5–6 below (respectively). Axiom 7 below is also added. LIS Postulates:



[S(B)](α!β)! ([S(B)]α! [S(B)]β).       I 0 α $ S 0 α. [S(C)]α! [S(B)]α for C
B
AC . W v  VC a [fg ða, vÞ. (a, u) ! [S(a)]((a, u) _ (a, )). ((a, u) ^ [S(B[{a})]α) ! [S(B)](((a, u) _ (a, )) !α). ((a, ) ^ [S(B[{a})]α) ! [S(B)]α. 

1. 2. 3. 4. 5. 6. 7.

The information system semantics of LIS is based on IS-structures that are DIS structures defined in section “Algebra
for Deterministic Information Systems,”  extended with a collection of relations RSðBÞ B
AC on U satisfying the following properties. (1S1) For each a  AC , [{m(a, v) : v  VC a [{}} ¼ U. (1S2) RIð;Þ ¼ RSð;Þ ¼ U  U. (1S3) RS(B)
RS(C) for C
B
AC . (1S4) For B
AC and b  AC , RS(B) \ RS(b)
RSðB[fbgÞ . (1S5) For b  AC , (w, w0 )  RS(b) if and only if any of the following holds. (i) w, w 0  m(b, v) for some v  VC b. (ii) w  m(b, ). (iii) w0  m(b, ). One can also define standard IS-structures generated by information systems, and observe similar relationships as in the case of DIS-structures. With the definitions of satisfiability and validity given in section “Algebra for Deterministic Information Systems” extended to this case, one can show that LIS is sound and complete with respect to IS-models and to standard IS-models as well. An equivalent semantics based directly on information systems may also be defined.

Algebra for Nondeterministic Information Systems Recall that objects in an NIS S :¼ (U, A, V , f ) are allowed to take a set of attribute values. Consider the set D a:¼ {(a, V ) : V
V a} for each a  A,

any (a, v) in Q whereby D may be identified with (a, {v}) of D a . Observe that (i) a  A D a is finite, and

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Q

(ii) U=Ind SA  a  A D a . The main idea of the work in (Khan 2015) is to consider

Q the finite set Θ :¼ {i, j, . . .} of nominals with |Θ| ¼ a  A D a . An NIS-algebra generated by an NIS is defined as follows. Definition 19. An NIS-algebra generated by the NIS S :¼ (U, A, V , f ) is a structure  n o S ≔ P ðU Þ, \ , , 0, Ind SB 



B
A

n o , SimSB

n B
A

,

InSB

o B
A

n o , cSγ

γD

n o , cSi;

iΘ

,





where Ind SB , SimSB , InSB are operators on P (U ) mapping a set X to XInd SB , XSimSB , and XInSB , respectively, for γ :¼ (a, v)  D, cSγ is the nullary operation (constant) given by the subset {x  U : f (x, a) ¼ v} of U, and cSi are nullary operations on P (U ) satisfying the following.
 (N1) U=Ind SA
cSi : i  Θ . (N2) cSi \ cSj ¼ 0, for i 6¼ j.
 (N3) cSi  U=Ind SA [ 0 . (N4) If ðx, yÞ 2 = Ind SA , ½xInd SA ¼ cSi and ½yInd SA ¼ cSj , then i 6¼ j. (N1)–(N4) ensure that cSi name the equivalence classes of Ind SA such that different equivalence classes are provided with different names. Equivalently, elements of the set U have been named such that elements belonging to the same equivalence class of the relation Ind SA are provided with the same name and elements belonging to different equivalence classes of Ind SA have different names. Note that (i) Θ is large enough to give nominals to cover all the equivalence classes, (ii) any NIS S can generate two distinct NIS-algebras only upto difference with respect to the nullary operators corresponding to the elements from Θ. Hybrid logics (see (Blackburn et al. 2001)) use the idea of nominals. Konikowska (1987, 1997) also has made use of naming objects. As pointed out in (Khan and Banerjee 2013), the difference lies in providing names to the equivalence classes of Ind SA instead of individual elements of U. On abstraction of the properties in these structures, the following abstract algebra is defined. Definition 20. An abstract NIS-algebra is a tuple 
 A≔ U, ^ , :, 0, fI B gB
A , fSB gB
A , fN B gB
A , dγ γ  D , fdi gi  Θ , where (U, ^, :, 0) is a Boolean algebra, IB, SB, NB are unary operations, and dγ, di are nullary (constant) operations on U satisfying the following. (N1) LC(x)  LB(x) for C
B
A, L  {I, S, N}. (N2) d(a,v)  L{a}(d(a,v)), L  {I, N}.

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(N3) :d(a,v)  I{a}(:d(a,v)).    W (N4) di  Sfag v  V a dða,vÞ ^ I ; di ! d ða,vÞ .      (N5) di ^ I B[fbg ðxÞ  I B ^v  V b dðb,vÞ $ I ; d i ! dðb,vÞ ! x . (N6) d(a, v) ^ SB[{b} (x)SB (d(b, v)!x).     V (N7) di ^ N B[fbg ðxÞ  N B v  V b I ; di ! dðb,vÞ ! d ðb,vÞ ! x . (N8) I ; ðxÞ ¼ S; ðxÞ ¼ N ; ðxÞ. (N9) I ; ðxÞ 6¼ 0 implies  x ¼ 1.  (N10) di ^ dða,vÞ  I ; di ! d ða,vÞ .  (N11) di ^ :dða,vÞ  I ; d i ! :d ða,vÞ . W (N12) i  Θ di ¼ 1. (N13) :di _ :dj ¼ 1 for i 6¼ j. (N14) di  I A di . (N15) I ; ð1Þ ¼ 1. (N16) LB(x ^ y) ¼ LB(x) ^ LB( y) for L  {I, S, N}. A representation theorem is proved: every abstract NIS-algebra is embeddable in the NIS-algebra generated by some NIS. Logic for NIS-algebras: A logic LNIS, proposed in (Khan 2015), has the abstract NIS-algebras and NIS-algebras generated by NISs as the algebraic counterparts – the latter due to the representation result mentioned above. The language of LNIS is similar to LDIS, except that it considers three modal operators □1B, □2B, and □3B for each B
A, to correspond to indiscernibility, similarity and inclusion relations in NIS (see section “Introduction”). To give the axioms, it also considers the set Θ of formulae which are a conjunction of literals from the set [α  D {α, :α}, and which contain precisely one of (a, v), or :(a, v) for each (a, v)  D. LNIS Postulates: 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13.

All axioms of classical propositional logic. □B(α!β) !(□Bα!□Bβ). □1; α ! α. α ! □1; ◇1; α. ◇1; ◇1; α ! ◇1; α. n □m ; α $ □; α, where m, n  f1, 2, 3g. □Cα!□Bα for C
B. ða, vÞ ! □ka ða, vÞ for k  f1, 3g. :ða, vÞ ! □1a :ða, vÞ. W  i ! □2a v  V a ða, vÞ ^ □1; ði ! ða, vÞÞ .  V  i ^ □1B[fbg α ! □1B v  V b ðb, vÞ $ □1; ði ! ðb, vÞÞ ! α .  ðb, vÞ ^ □2B[fbg α ! □2B ððb, vÞ ! αÞ.  V  i^□3B[fbg α ! □3B v  V b □1; ði ! ðb, vÞÞ ! ðb, vÞ ! α .

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ði ^ ða, vÞÞ ! □1; ði ! ða, vÞÞ. ði ^ :ða, vÞÞ ! □1; ði ! :ða, vÞÞ. W i  Θi. :i_:j for distinct elements i and j of Θ. i ! □1A i.

Rules of inference: N:

α MP: □B α

α α!β β

LNIS gets an information system semantics with definition of models based directly on NISs. An LNIS-model is a tuple M :¼ (S, V ), where S :¼ (U, A, [a  A V a, f ) is an NIS, and V : PV!P(U ) is a valuation function. The satisfiability of a formula α in M at an object w  U, denoted as M , w α, is defined in the standard way for the Boolean connectives. For the other cases, we have Definition 21. M, w (a, v), if and only if v  f (w, a). M, w □1B α, if and only if for all w0  W withðw, w0 Þ  Ind SB , M, w0 α. M, w □2B α, if and only if for all w0  W withðw, w0 Þ  SimSB , M, w0 α. M, w □3B α, if and only if for all w0  W withðw, w0 Þ  InSB , M, w0 α. The extension ½½α M of a formula relative to a model M is given by the set {w  U : M, w α}. Validity is defined in the usual manner. Then it can be verified that, as desired, the operators □1B , □2B , and □3B , for each B
A , give the lower approximations relative to the attribute set B in an LNIS-model, with respect to the indiscernibility, similarity, and inclusion relations, respectively. Moreover, we have





Proposition 5.
 (V1) U=Ind SA
figM : i  Θ .

 (V2) figM : i  Θ
U=Ind SA [ 0 . (V3) figM \ fjgM ¼ 0, for i, j  Θ, and i 6¼ j. (V4) For i, j  Θ, if ðx, yÞ= 2Ind SA , ½xInd SA ¼ figM and ½yInd SA ¼ fjgM , then i 6¼ j. This proposition verifies that the elements of Θ are being used for the logic just as in the case of NIS-algebras (hence the notation Θ has been retained): to name the equivalence classes of Ind SA such that different equivalence classes are provided with different names. It is established in (Khan 2015) that the above axiomatization is sound and complete with respect to LNIS-models.

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Conclusions In this chapter, we have attempted to provide a glimpse of existing work in the country on interactions between logics and algebras related to rough set theory. Presented are logics for quasi-Boolean algebras and extensions, rough lattices and Heyting algebras from covering-based rough set theory, contrapositionally complemented pseudoBoolean algebras from category-theoretic studies of rough sets, double Boolean algebras from rough concept analysis, and information system algebras. New logics have emerged which have been imparted semantics other than the algebraic ones; known logics have been imparted rough set semantics due to representation results. An ongoing study is the logic of C-algebras due to McCarthy which have also been represented in terms of rough sets (Panicker and Banerjee 2019). Another direction of work that continues to be explored is related to topological quasiBoolean algebra and its extensions. These studies range from establishing finite embeddability properties of tqBa5 (Lin and Chakraborty 2019), to settling decidability questions about the equational theories of the algebras (Lin et al. 2018), and introducing multitype display calculi for the algebras (Greco et al. 2019b). Yet another current area of work is that on logics from rough concept analysis, where apart from work on interlinking algebras and logics by Howlader and Banerjee (as presented in section “Algebras from Rough Concept Analysis”), there is work on logics only in (Greco et al. 2019a). The story thus continues to evolve and fascinate.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Daniel Dennett’s three levels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Umwelts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Davidson on Animals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Do Animals Have Beliefs? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Zebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Is language Really Necessary? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Sofa and the Air . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1104 1105 1106 1107 1107 1108 1109 1110

Outline This chapter will concentrate on two themes. 1. To what extent is the reasoning of animals and children logical? What do they think? 2. To what extent can one regard groups – corporations, or political parties, etc. – as individuals to whom one can assign goals and beliefs? In other words, how far can one extend the notion of an individual? • The first class will refer to experiments in animal behavior and what known about the thinking of animals and children. • The second will refer to issues in game theory and in states of knowledge, and subsequent coordinated action, arising from communication. 3. The work of Dennett, Uexküll, Davidson, McCarthy, and Hayes and work on animal behavior will be drawn upon. R. Parikh (*) CS, Math, Philosophy, Brooklyn College and CUNY Graduate Center, City University of New York, New York, NY, USA e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_40

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Agent? Or Machine? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Two Computer Scientists Respond . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Umwelts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Turning to Mathematics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Uninformed Agent . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Learning More . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symbiosis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Animal Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . A General Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Relevance of Language . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1110 1111 1111 1112 1113 1114 1115 1115 1115 1116 1116 1117

Abstract

This chapter addresses the question of how one can extend the notion of person to sentient creatures who are not adult humans but do show signs of motivation and intelligence.

Introduction How did the world look to him? Baby Shiva at the age of four months.

And a year later?

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Daniel Dennett’s three levels • The physical stance • The design stance • The intentional stance The most concrete is the physical stance, the domain of physics and chemistry, which makes predictions from knowledge of the physical constitution of the system and the physical laws that govern its operation; and thus, given a particular set of physical laws and initial conditions, and a particular configuration, a specific future state is predicted (this could also be called the “structure stance”). At this level, we are concerned with such things as mass, energy, velocity, and chemical composition.

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When we predict where a ball is going to land based on its current trajectory, we are taking the physical stance. Somewhat more abstract is the design stance, the domain of biology and engineering, which requires no knowledge of the physical constitution or the physical laws that govern a system’s operation. Based on an implicit assumption that there is no malfunction in the system, predictions are made from knowledge of the purpose of the system’s design (this could also be called the “teleological stance”). At this level, we are concerned with such things as purpose, function, and design. When we predict that a bird will fly when it flaps its wings on the basis that wings are made for flying, we are taking the design stance. Likewise, we can understand the bimetallic strip as a particular type of thermometer, not concerning ourselves with the details of how this type of thermometer happens to work. Most abstract is the intentional stance, the domain of software and minds, which requires no knowledge of either structure or design, and “[clarifies] the logic of mentalistic explanations of behaviour, their predictive power, and their relation to other forms of explanation.” Predictions are made on the basis of explanations expressed in terms of meaningful mental states; and, given the task of predicting or explaining the behavior of a specific agent (a person, animal, corporation, artifact, nation, etc.), it is implicitly assumed that the agent will always act on the basis of its beliefs and desires in order to get precisely what it wants (this could also be called the “folk psychology stance.”) At this level, we are concerned with such things as belief, thinking, and intent. When we predict that the bird will fly away because it knows the cat is coming and is afraid of getting eaten, we are taking the intentional stance. Another example would be when we predict that Mary will leave the theater and drive to the restaurant because she sees that the movie is over and is hungry. Note that Dennett left out a crucial condition, that Mary has a car. Computer scientists are going to notice this lack because an analysis of algorithms stance implies being aware of the condition that Mary has a car is crucial. Lacking a car she might take a bus, or perhaps walk to a restaurant close to the movie theatre. What she wants and what she believes are not enough. We also need to refer to her capabilities. See Sen (2004: 77–80).

Umwelts But long before Dennett’s The Intentional Stance, and Nagel’s “What is it like to be a bat?” Jakob von Uexküll carried out a detailed investigation (in the early twentieth century) of how animals, children, and we adult humans see the world. The way we see the world as contrasted with how the world is is called the umwelt by Uexküll. It is a notion heavily influenced by Immanuel Kant. This little monograph does not claim to point the way to a new science. Perhaps it should be called strolls into unfamiliar worlds, worlds strange to us but known to other creatures manifold and varied as the animals themselves. The best time to set out on such an adventure

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is on a sunny day. The place, a flower strewn meadow humming with insects, fluttering with butterflies. Here we may glimpse the worlds of the lowly dwellers of the meadow. To do so we must first blow in fancy a soap bubble around each creature to represent its own world filled with the perceptions of which it alone knows. When we ourselves step into one of these bubbles, the familiar meadow is transformed. Many of its colorful features disappear, others no longer belong together but appear in new relationships. A new world comes into being. Through the bubble we see the world of the burrowing worm, of the butterfly, or of the field mouse; the world as it appears to the animals themselves, not as it appears to us. This we may call the phenomenal world or the self-world of the animal. Jakob von Uexküll, Forays into the worlds of animals and children, 1934

Davidson on Animals Neither an infant one week old nor a snail is a rational creature. If the infant survives long enough he will probably become rational while this is not true of the snail. . .. The difference consists, it is argued, in the having of propositional attitudes such as belief, desire, intention, and shame. This raises the question of how to tell when a creature has propositional attitudes. Snails, we may agree, do not but how about dogs or chimpanzees?. . . It is next contended that language is a necessary concommitant of any of the propositional attitudes. This idea is not new, but there seem to be few arguments in its favor in the literature. One is attempted here. Rational Animals, Dialectica, 1982

Do Animals Have Beliefs? Norman Malcolm tells this story which is intended to show that dogs think. Suppose our dog is chasing the neighbor’s cat. The latter runs full tilt towards the oak tree but suddenly swerves at the last moment and disappears up a nearby maple. The dog doesn’t see this maneuver and arriving at the oak tree he rears up on his hind feet, paws at the trunk as if trying to scale it and barks excitedly into the branches above. We who observe this whole episode from a window say “he thinks that the cat went up the oak tree” Davidson, loc cit. But how about the dog’s supposed belief that the cat went up that oak tree? That oak tree as it happens is the oldest tree in sight. Does the dog think that the cat went up the oldest tree in sight or that the cat went up the same tree it went up the last time the dog chased it? It is hard to make sense of the questions but then it does not seem possible to distinguish between quite different things the dog might be said to believe? Davidson, loc cit

Davidson’s claim is that a dog chasing a cat up a tree could not have the belief that there was a cat in the tree. The dog might just have had the belief that a furry animal, or even a funny object, was in the tree. . . But the argument proves too much. For by the same token, a child who has not had sex education cannot know that it has a mother. . . Surely we do not want to go there.

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Perhaps it makes more sense to say that the dog’s understanding of the concept cat is a little looser than ours and corresponds to a larger equivalence class in the dog’s partition of the world. It includes cats of course but also furry animals and perhaps even furry things which move in a purposeful way. We will address this issue a bit later. In the section of A Treatise of Human Nature entitled, “Of the Reason of Animals,” Hume argued by analogy that since animals behave in ways that closely resemble the behaviors of human beings that we know to be caused by associations among ideas, animals also behave as a result of forming similar associations among ideas in their minds. Given Hume’s definitions of “thought” and “reason,” he took this analogical argument to give “incontestable” proof that animals have thought and reason. Robert Lurz (2009) in Animal Minds, The Internet Encyclopedia of Philosophy Wittgenstein famously said that if a lion could talk we would not understand him (Wittgenstein 1953: 223). But while this captures some truth, it may overestimate the differences. Lions are mammals, and as such have much in common with humans. Emotionally, cognitively, anatomically, and physiologically, lions are not as remote from humans as fish, or insects. Even bats the philosophers’ current archetype of a subjective perspective that is unimaginable to humans (Nagel 1974) are not as alien to us as sharks. At the same time, it is easy to overestimate the extent to which other people are like ourselves. We tend to assume that others will enjoy what we enjoy, reject what we reject, and seek what we seek. But a bit of experience in a foreign culture soon reveals differences that are hard to imagine without that experience, and even ones friends and neighbors prove occasionally hard to fathom. Of course there are similarities too, but the point is that one cannot predict a priori where the commonalities lie (Colin Allen 2014). Dogs for instance have a much better sense of smell and much better hearing than we do. But they are partially color blind and their vision is poorer. Their umwelts are different from ours and they have beliefs and desires and plans for action within their umwelts. Similarly, the blind character Wally and the deaf character Dave in the movie See no Evil hear no Evil have different umwelts from each other and from us. Thus the umwelt is the semantics (or semiotics) of the agent. If we see this agent as having beliefs and desires (in the BDI sense) then we need to understand what world these beliefs and desires are about. Logics for action and belief need to use the real semantics of such agents. We will offer a path towards formalizing such logics. And then we can understand what actions will come about from these beliefs and desires.

Zebras Zebra stripes are dazzling particularly to flies. That’s the conclusion of scientists from the University of Bristol and the University of California at Davis who dressed horses in black-and-white striped coats to help determine why zebras have stripes.

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The researchers found that fewer horseflies landed on the bedecked horses than on those without the striped coats, suggesting that zebra stripes may offer protection from bloodsucking insects that can spread disease.

“This reduced ability to land on the zebra’s coat may be due to stripes disrupting the visual system of the horse flies during their final moments of approach,” said Martin How, a research fellow at the University of Bristol. “Stripes may dazzle flies in some way once they are close enough to see them with their low-resolution eyes.” From a distance, the flies were equally attracted to both horses and zebras, with the same number of insects hovering around both types of animals. But when the flies got closer, things get dicey. The flies landed less frequently on the zebras and the horses covered in striped coats. “Once they get close to the zebras, however, they tend to fly past or bump into them,” said Tim Caro, a professor in the U.C. Davis Department of Wildlife, Fish and Conservation Biology. “This indicates that stripes may disrupt the flies’ abilities to have a controlled landing.”

Is language Really Necessary? Shiva in His Father’s Garden Last November Shiva (then 16 m.o.) and I were together in his father’s garden and Shiva wanted to go on a swing. But the steps to the swing are a bit steep and I did not think I could keep him safe. So I refused to take him. . . On a previous occasion he had cried when I would not do what he wanted. But this time he did not cry. Instead he said, “Daddy!”. I remembered then that his father Vikram had taken him on the swing on the previous day and I called Vikram. Vikram came and took Shiva on the swing. No tears!

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Instead of using tears, Shiva used a bit of logic. But he had (then) hardly any language beyond maybe a dozen words.

The Sofa and the Air Suppose that Aruna has a sofa in her living room. If you ask her if she knows that she has a sofa in her living room she will say, “Are you crazy? Of course I know.” But if you say to her “How many pounds of air are in your apartment?” She would have no idea. (It could be about 750 pounds in a typical apartment.) The sofa is in her apartment and so is the air so why does she know about the one but not the other? Aren’t they both part of her world? But the sofa is part of her umwelt and the weight of the air is not. Here is another example. A dog sees his master from a distance but does not recognize him. But when the master comes closer the dog is very happy, wags his tail, and licks the master’s hand. What is the difference? Dogs orient themselves in the world by smell more than by sight and a distant master is not recognized. Uexküll is interested in such questions not only for Aruna and for the dog but also for various creatures like a tick or a fly. Why does the fly get caught in the spider’s web? Because a thread in the web is too fine for the fly’s vision. So it does not know that the web is there. Once caught, it knows quite well because it is no longer using its eyes but its sense of touch. There are certain things that we are all supposed to know like whether there is a sofa in our living room but we do not usually know about the weight of the air, even though it too is in our living room. Following Kant, Uexküll distinguishes between the actual world and the phenomenal world which varies from creature to creature. The phenomenal world is the umwelt. Now for us humans, our individual umwelt is supplemented by the community umwelts which include information from the umwelts of others, and also from science. The sun and the moon look to us as if they are at the same distance but science tells us that the sun is much further. And we certainly did not send a man to the moon using just the phenomenal world. But animals and young humans tend to act primarily or entirely in terms of their phenomenal worlds.

Agent? Or Machine? A Quote The mechanists have pieced together the sensory and motor organs of animals, like so many parts of a machine, ignoring their real functions of perceiving and acting, and have gone on to mechanize man himself. According to the behaviorists, man’s own sensations and will are mere appearance, to be considered, if at all, only as disturbing static. But we who still hold

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that our sense organs serve our perceptions, and our motor organs our actions, see in animals as well not only the mechanical structure, but also the operator, who is built into their organs as we are into our bodies.(Uexküll 1957)

Two Computer Scientists Respond On this basis we shall say that an entity is intelligent if it has an adequate model of the world (including the intellectual world of mathematics, understanding of its own goals and other mental processes), if it is clever enough to answer a wide variety of questions on the basis of this model, if it can get additional information from the external world when required, and can perform such tasks in the external world as its goals demand and its physical abilities permit.(McCarthy and Hayes, Some philosophical problems from the point of view of AI, 1969)

Note that McCarthy and Hayes explicitly refer to abilities which were left out of Dennett’s account (Dennett 1989). If the organism carries a ‘small-scale model’ of external reality and of its own possible actions within its head, it is able to try out various alternatives, conclude which is the best of them, react to future situations before they arise, utilize the knowledge of past events in dealing with the present and future, and in every way to react in a much fuller, safer, and more competent manner to the emergencies which face it.(Kenneth Craik, The Nature of Explanation, 1943: 61)

Umwelts We can think of the umwelt as a homomorphic image of the real world. And that means that some information is missing. In view of this missing information, the best action is not always the same as the apparent best action. Now the expected value of the apparent best action increases when more information is received. But in order to receive more information, the animal needs to develop tools for that, and they incur a cost, so unless the cost is less than the gain the improvement will not be sought. The fly could have had better eyesight and be caught less often, but that more sophisticated eye would be expensive to maintain. A question raised by Alfred Russel Wallace was why primitive men had brains nearly as large as ours when they did not have to do complex things like file tax returns. But Steve Pinker (2010) suggests that even hunters in primitive tribes use very complex procedures to hunt animals. Having a large brain enables one to make thought experiments and discover the best action on the spot. Animals and plants may have to go through thousands of years of evolution to make the corresponding discovery. This human advantage has had an unfortunate consequence. Certain species of animals were wiped out when the clever humans entered their domain. In this context, reconsider Uexküll’s account of the life story of a tick. A tick has three perceptions and three effectors (or actions).

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The typical tick climbs on a grass blade or something similar and waits (Apparently a tick can wait for several years without starving to death.) When a mammal passes under the grass blade, its skin releases butyric acid which the tick detects and it drops onto the mammal. It knows it is a mammal because of the warmth. Then it moves around in the mammal’s skin until it finds a bald spot. It sucks blood and then drops to the ground where it lays its eggs and dies. It needs to know 1. the sunlight to know which direction is up and so to rise 2. the smell of butyric acid which tells it when to drop and 3. the feeling of warmth which enables it to know that precious blood is available It also has three actions: (1) rising, (2) dropping, and (3) sucking blood, and then (again) (2) dropping. The tick can be easily represented by a transducer finite automaton. It also uses default reasoning because it does not (bother to) distinguish between blood and some other warm liquid supplied to it by an experimenter. Under normal circumstances, it is blood and the tick does not need expensive equipment to distinguish blood from fake blood. Uexküll has lots of examples of creatures being fooled in this way when the best action in their umwelt is not the best action in the real world Default reasoning is a rational strategy when we would incur too high a cost to deviate from it. It’s cheaper to assume that what you see is what you expect to see. This idea is reminscent of Kant – we perceive the world based on what is presented to us. Also related is the notion of indriya (sense) in Jainism. Each indriya (like smell or sight) is a homomorphism from the real world onto the phenomenal world. According to Jain doctrine, it is a greater sin to kill a creature with more indriyas. Uexküll is skeptical of the idea that there is the “real world.” We shall not follow him or ask the reader to. Rather our representation will assume that there is a real world which is perceived imperfectly by every creature, whether a bat or a dog or a child. Thus each creature sees a homomorphism from real world to its personal world.

Turning to Mathematics Definition 3.1 An umwelt U consists of two parts. A homomorphism H (many one mapping) from the actual world to the perceived world. And a set A of possible actions. Thus U ¼ (H, A). In addition, each creature has a utility function u, so that u(a, w) ¼ x is the utility of action a performed when the world is w. We will assume that x is a real number. (In actuality it could be some level of satisfaction for us humans, or the expected number of progeny for animals).

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H(w) ¼ H(w') means that w, w' result in the same perceptions for a particular creature. Given a world w, the best action b(w) for the creature is that a which maximizes the expected utility u(a, w') over the set {w'|H(w') ¼ H(w)}. (There is an implicit probability distribution here which we will not specify.) The expected value E(U ) of the umwelt U is the expected value of the random variable b. Definition 3.2 Umwelt U' ¼ (H', A') refines umwelt U ¼ (H, A) if a) H'(w) ¼ H'(w') ! H(w) ¼ H(w') and b) A  A'. Thus H' has more information and more abiliities than H. Theorem 3.3 If U' refines U then E(U )  E(U'). The more you know and the more you can do, the better off you are (with some caveats). Here is the intuitive idea. Suppose I am driving to New Jersey and can take either the tunnel or the bridge. Normally, the tunnel is better as it is closer. But it might be closed. The procedure if the tunnel is open then take the tunnel else take the bridge has a higher utility than either just take the tunnel or just take the bridge. But that if then else procedure can only be carried out in the refined umwelt where the question about the tunnel has been answered. Thus, it pays to know more and it also pays to have more options for action. Note: It is of course well known that ordinary human beings become confused when they have too many options. It is easier to choose one brand of milk from three or four but not from a hundred. And this is because the computational or psychological task of choosing itself become onerous. In the theorem above, we did not consider that cost whether computational or psychological.

Uninformed Agent Here A and B are incompatible conditions which might obtain. X and Y are possible actions of the agent.

Action X Action Y

A –100, 25 6

B 10 –50, 15

This is a decision theoretic matrix. In condition A, the agent does not know whether the payoff will be 25 or –100 if action X is performed. Thus, if A is true,

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then the best action is Y and if B is true then the best action is X. The utility of the umwelt is (6 + 10)/2 ¼ 8. In this more detailed table P is an additional condition about which the agent could find out.

Better informed agent

Action X Action Y

A and P –100 6

A and –P 25 6

B and P 10 15

B and -P 10 –50

So the best action is if A and P, then Y; if A and –P, then X; if B and P, then Y, and if B and –P, then X. The utility of the umwelt is now 56/4 ¼ 14. Knowing about P has pushed the utility up by 6 and so one could say that the knowledge of P is worth 6 units.

Learning More Why then does a creature not have a maximal U where H is the identity function and A is enormous? Because acquiring more information and acquiring more possible actions has a cost and the benefit may not justify the cost. And for Darwinian creatures which rely on evolution to “learn,” the entire species has to have the extra sensory ability so that one creature may benefit. The cost summed up over the entire species may not be justified by the benefit to one member or a few members of the species. If I have an umwelt U and I ask a question Q then the H becomes refined to a finer H'. The utility of the new umwelt will be greater but the question will have a cost. To ask the question requires me to make sure that the cost is less than the utility gain. If I am at a train station and ask the agent what time my train is leaving, I will benefit from the answer. But if I ask how many dishes are available in the dining car, the agent’s rudeness will be too high a price to pay for any answer. Similarly for an increase in actions: If I am going mountain climbing, then it makes sense for me to undergo training so that I have more actions available while on the mountain. But if I am not going mountain climbing then the effort gains me nothing. Suppose that two different creatures have two different umwelts. For example, a man with eyesight but no legs riding another man without vision but with legs. Note: Something like this happens in one of the Sinbad stories. Or it could be a dog leading a man who is blind. In that case, the combined umwelt would be to the benefit of both. What is essential in that case that the umwelts supplement each other and that their utilities align.

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Symbiosis Consider two creatures with umwelts U ands U' and a common utility note: The utility need not be common but the two utilities can be compatible (See, e.g., John Nash’s work on The Bargaining Problem, Econometrica 1950). Todd Stambaugh, CUNY, has done an interesting sequel to Nash (see Economics Letters 1917). Then the two together have joint umwelt S U” whose H” is the least upper bound of H and H' and whose action set A” is A A'. I.e., H”(w) ¼ H”(w') iff H(w) ¼ H(w') and H'(w) ¼ H'(w'). Then the joint umwelt refines both the individual umwelts and (with a reasonable bargain) yields a higher utility for both creatures. This explains why we have cases of symbiosis among animals and massive cooperation among humans. (There is also the issue of compatible utilities. A leopard and a deer do not have compatible utilities unless we think of the leopard as having the job of keeping the deer herd under control.) Here is an example: In the ocean, certain species, like shrimps and gobies, will clean fish. They remove parasites, dead tissue, and mucous. Another example: The relationship between goby fish and shrimp. The shrimp digs a burrow into the sand and both organisms live there. Because the shrimp is almost blind, the goby fish will touch the shrimp when a predator is near.

Animal Logic A tiger watches a deer going toward a bush from the left. Then the deer is not seen any more. And it has not emerged on the other side. So the tiger knows and believes that the deer is behind the bush. The tiger is inferring the presence of the deer behind the bush, which it does not see, from the previous appearance of the deer to the left of the bush, and from the nonappearance of the deer to the right of the bush. The tiger is inferring some variable-free sentences which it does not experience, from other variable-free sentences which it has experienced.

A General Framework Suppose we are given a first order theory T with plenty of constants and variable free terms. T defines a relation R between finite sets X of variable free sentences and other sets Y of variable free sentences as follows: S R(X, Y ) iff T X |¼ ϕ for all ϕ  Y. Clearly R is monotonic in X, in T, and anti-monotonic in Y. Suppose the tiger’s behavior shows awareness of Y on the basis of X.

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Does the tiger know such a theory T? Not necessarily. There are many such theories which will work. And the tiger may be using some other means to infer Y. But it can be harmless if we attribute to the tiger such a theory T as long as we are aware that this is merely a fac¸on de parler. Thus it is fine for us to say, “the tiger acts as if it believes T.” Question: For which relations R does there exist a finite first order theory which “explains” R?. One could also ask which R are computable in polytime or even in linear time.

The Relevance of Language Two kinds of agents considered by AI are as follows: 1. Stimulus response creatures: These are creatures whose reactions are fixed given what they perceive. In AI they are represented by means of a table. And indeed the head of a Turing machine is just that. It sees something on a square and it acts. 2. Creatures with a “knowledge base”: These are creatures who have some cache of ‘facts’ which they revise and which they use to infer other facts. But do these facts need to be expressed in language? David Lewis (1976) showed convincingly that probabilistic conditionals are not propositions and cannot be. But they are something. Can we represent them using some mechanism other than adding to or subtracting from a knowledge base consisting of propositions? Hanoch Ben-Yami (1997) suggests that we can. Gilbert Ryle, preceding him by several decades, suggested something similar. Perhaps this is a very fruitful direction to go in in but not in this chapter.

Conclusion We have made a start toward formalizing some ideas implicit in Uexküll, Dennett and Nagel as well as others. Such a preliminary effort must leave many loose ends untied. Here are two examples. Darwin was puzzled by the long and beautiful tail of the peacock. The tail is expensive and makes the bird easier to catch. So why bother? One explanation is that hens like it and the poor peacock has to fall in line. But the tail does not contribute to the peacock’s own utility, only to the expected number of progeny. So there are two utilities involved here: the peacock’s own utility and that of its DNA. (And the utility of the hens who like to see something pretty.) The two can conflict and then the DNA will probably win. But how do these two utilities bargain? A second technical point is that the actions in the set A not only have a utility but also change the world in some way. It could well be that a sequence of actions a1 and a2 is what is actually useful. In that case, the only benefit of AI is to change the world so that a2 becomes useful. But these are also issues for a sequel.

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References Allen, Colin. 2014. Umwelt or Umwelten? How should shared representation be understood given such diversity? Semiotica 198: 137–158. Ben-Yami, Hanoch. 1997. Against characterizing mental states as propositional attitudes. The Philosophical Quarterly 47 (186): 84–89. Craik, Kenneth James Williams. 1952. The nature of explanation. Vol. 445. Cambridge: CUP Archive. Davidson, Donald. 1982. Rational animals. Dialectica 36 (4): 317–327. Dennett, Daniel Clement. 1989. The intentional stance. Cambridge, MA: MIT Press. Lewis, David. 1976. Probabilities of conditionals and conditional probabilities. In Ifs, 129–147. Dordrecht: Springer. Lurz, Robert. 2009. Animal minds. In Internet encyclopedia of philosophy. McCarthy, John, and Patrick J. Hayes. 1969. Some philosophical problems from the standpoint of artificial intelligence. Readings in artificial intelligence. Morgan Kaufmann, Burlington, Massachusetts,1981. 431–450. Nagel, Thomas. 1974. What is it like to be a bat? The Philosophical Review 83 (4): 435–450. Nash, John F., Jr. 1950. The bargaining problem. Econometrica: Journal of the Econometric Society 18: 155–162. Pinker, Steven. 2010. The cognitive niche: Coevolution of intelligence, sociality, and language. Proceedings of the National Academy of Sciences 107 (Suppl 2): 8993–8999. Ryle, Gilbert. 1949. The concept of mind. Sen, Amartya. 2004. Capabilities, lists, and public reason: continuing the conversation. Feminist Economics 10 (3): 77–80. Sen, Amartya. 2005. Human rights and capabilities. Journal of Human Development 6 (2): 151–166. Skyrms, Brian. 2004. The stag hunt and the evolution of social structure. Cambridge: Cambridge University Press. Stambaugh, Todd. 2017. Coincidence of two solutions to Nashs bargaining problem. Economics Letters 157: 148–151. Von Uexküll, Jakob. 1992. A stroll through the worlds of animals and men: A picture book of invisible worlds. Semiotica 89 (4): 319–391. Wittgenstein, Ludwig. 1953. Philosophical investigations. Oxford: JBasil Blackwell.

Philosophical Aspects of Constructivism in Logic

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Meaning: Truth-Theoretic and Proof-Theoretic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructivism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructivism Through the Truth-Theoretic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bivalence and the Law of Excluded Middle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Constructivism Through the Proof-Theoretic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Final Words . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Proofs are distinguished by identifying some – in fact many – of them as classical and the rest as constructive. This chapter attempts to understand, from a philosophical perspective, the difference between the classical and the constructive. The constructivism latent in constructive proofs is sought to be seen as grounded in a variety of a philosophical position called anti-realism. The ties of the assumptions, or the lack of them, in this variety, with the resulting presentations of logic are explored in the context of truth as well as in that of proofs. Keywords

Deductive argument · Validity · Logical words · Meaning · Recognition transcendent notion of truth · Recognition sensitive notion of truth · Principle of bivalence · Law of excluded middle · Classical logic · Intuitionist logic · Proof · Introduction rule · Elimination rule · Cut rule · Indirect proof · Direct proof · Cut elimination · Harmony · Principle of inversion · Stability · Justification of logic

R. Mukhopadhyay (*) Visva-Bharati University, Santiniketan, West Bengal, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_53

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Introduction Modern deductive logic as it is presently understood in its usual normal sense believes that the validity of a deductive argument depends on the meanings of what are called logical words (particles/constants), and also on the positions where such words occur within a given deductive argument. The meanings of logical words are thought to be given, at least at the primal level, by the truth-conditions of sentences – which have to be descriptive in nature to function as premises or conclusion of an argument – containing them. It is taken that when the conditions of truth of the premises containing any of the logical words, as per such truthconditional meanings of only the logical words, are also the conditions of truth of the conclusion, the argument under consideration is valid. This basic understanding of validity has developed into very sophisticated presentations in relevantly diverse ways for different areas of deductive argumentation – and the sophistications are becoming more and more complex day by day. The complexities get revealed by the mathematical structures discoverable in the presentation of explanations of validity in the area of logic under consideration. The benefits of such an evolution of logic, since the days of Augustus De-Morgan, George Boole, and Gottlob Frege, respectively, have been immense in that the properties of the mathematical structures used in the explanations, did and do, have the potential to illuminate many of our philosophical curiosities related to logical reasoning. Though not entirely unrelated essentially to this basic insight about validity, an apparently alternative way of explaining validity has been explored, mainly, following the works of Gerhard Gentzen (1969). In this approach, the meanings of logical words lie not in the explicitly stateable truth-conditions of sentences containing them, but in the (linguistic) uses they are put to within the context of argumentation: the requirements for bringing in (introducing) a given logical word in the course of a proof, and the same for extracting the consequences of using the given logical word (eliminating the logical word) within the proof – all these presented only as syntactic relationships between sentences occurring as premises and as conclusion comprise whatever meaning the logical word has. If the earlier approach is seen to be semantic then this approach can be said to be proof-theoretic – which is one variety of syntactic approach. The other variety of the syntactic approach is found employed in course of axiomatic investigations into different areas of logic: the ‘facts’ taken to be unquestionably axiomatic about the logical words are stated, in this approach, in terms of syntactic relationships between expressions, given which, other facts about the words can be seen to be derivable – purely syntactically again – as theorems of the area under investigation. The intuitive understanding of the syntactic relationships between expressions as expressed through the axioms and theorems may get manifested by a proposed semantics of the language syntactically laid down for the axiomatic investigation – helping for setting up possible later major results about soundness and completeness of the axiomatization.

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Though the above are not the only ways in which logic presently has found its appearance, yet these are the main and widely used presentations for making accessible whatever is thought to be important about logic. Whichever way the logical words are investigated there remains a few questions which are as yet difficult to answer. Not because they have not been so far investigated well, but because the attempted answers have always been laced with differing standpoints while dealing with the questions: the same question can be answered from differing standpoints leading to different parallel understandings of any given area of logic. This phenomenon of having a distinct standpoint while proceeding to answer such a question actually brings in different philosophies while investigations into logic are carried out. The perceived distinction between “classical logic” and “constructivist logic” ’is really a fall out of this phenomenon.

Meaning: Truth-Theoretic and Proof-Theoretic Apart from the fundamental question about what it is that distinguishes a logical word from a nonlogical word – which may be taken to be the question of demarcating the province of logic – even though there may be a full agreement about some instances of logical words and some others of nonlogical ones, there remaining a grey area in-between calling for a very sharp criterion of distinction, another very crucial question pertains to our understanding of the very notion of ‘meaning’ itself. This is primarily a philosophical question. That the notion of meaning plays a crucial role in logic cannot be overemphasized given the above narrative. But then the question “what are meanings of expressions?” has to be given an answer. Even here, most thinkers have, after considerable attempts and questions thereof, veered round to the view that a better formulation of the relevant question should be “what do we know when we know the meaning of an expression?”. Cutting a longish story short, the overall consensus reached so far has been that to understand the meaning of an expression is to understand the conditions of truth (truth-conditions) of the sentences containing the expression. An example illuminating this idea will be helpful at this stage. Let us take the sentence “Coal is green.” We, surely, do know the meaning of this English sentence, even though it is a false one in this world. Then, what is it that we know when we know the meaning of this sentence? The most naturally suggestive answer would be that we know what has to happen in the world so that the sentence can be true of this world: that coal has to be green indeed! This answer manifests the insight that to know the meaning of a sentence is to know the conditions of truth of the sentence. The philosophical journey culminating into holding this belief now relates to what we were encountering in what was called the semantic way of investigating validity in our opening passages of this article. But one has to note also that in this semantic approach the specification of truth-conditions of sentences needs to specify the respective denotations of the subject and the predicate expressions of the given sentence to arrive at the specification of the condition of truth for the whole sentence in question – which again, may give rise to further philosophical debates which we need not bring in here.

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Is the above answer to the question of meaning the only answer to it? There has, as mentioned earlier, been an alternative answer to this question. According to this answer, roughly, the meaning of an expression lies in all possible uses of it within given contexts – contexts not being limited to just linguistic ones, but also getting extended to the surroundings comprising the speaker, hearer, the place, the time, and may be, more. This reminds us, however, of the later Ludwig Wittgenstein (1953) and the slogan “meaning is use” – a standpoint, though independent, echoing earlier ideas of Gentzen but realized to be applicable to all the ingredients of language, not just logical words. The notion of truth as presupposed in the earlier understanding of meaning, along with the prerequisite notions of denotations of subject and predicate expressions, almost evaporates in this understanding. If we restrict our interest to the meanings of only the logical words, then, it transpires that meanings of logical words can be seen to be found in their uses within just the linguistic contexts of formal proofs (formal counterparts of deductive argumentations). And such uses are of two varieties, one being that of appearing in a segment of the proof (getting introduced into the segment), and the other being that of disappearing from a segment of the proof (getting eliminated from the segment). The general idea of having such an understanding of the notion of meaning can be illustrated through a somewhat mundane and crude example of a nonlogical word. A set of sense-experiences gives rise to the occasion of using the word “water” in an utterance of the sentence “Here is water,” and upon hearing the sentence we may proceed to bring about another set of sense-experiences which is describable by the utterance of the sentence “The floor is being cleaned.” The first set of sense-experiences gives us the grounds for the use of the word “water” in the sentence “Here is water,” and the second set of sense-experiences gives us the use of another sentence describing one of the consequences of using the earlier one. Such interconnections between uses of sentences, for that matter, expressions, in – starting from the sense-experiential, and, then reaching gradually the higher levels of cognitive – contexts give the meanings of expressions. Applying such an understanding of the notion of meaning in the case of logical words as used in the well-circumscribed contexts of proofs, we get two kinds of uses of them: one giving us the grounds of their use in a segment of proof, and the other, likewise, giving us the consequences of their use. This philosophical understanding of meaning (especially of logical words) relates to what was said earlier about the proof-theoretic way of presenting logic. We are already finding that the two differing ways of investigating logic, discussed so far, are really laced with differing philosophical standpoints – at various layers – regarding how to understand the notion of meaning. We shall see that even more such differences are lurking for us.

Constructivism Constructivism in logic, as opposed to classicism, can be characterized roughly in terms of a kind of requirement on certain proofs. For certain proofs the constructive ones provide some additional information, in comparison to the classical proofs, in

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that they also provide effective means, at least in principle, to prove a case – which is not apparently demanded by the conclusion to be proved, so far as the classical understanding goes – relative to the conclusion. This can be illustrated with the help of the following. When we are supposed to prove that there exists at least one thing, say x, such that x is F, on the basis of certain other given premises, the classical approach would require only that it proves from those premises that there exists some such x – not also requiring that the proof comes up with an instance, as well, of an x which indeed is F – at least by providing, in principle, an effective means which when followed throws up the relevant instance. This additional requirement is precisely the requirement of a constructivist in comparison to that of the classicist. Such a demand of the constructivist, however, does not necessarily clash with Platonic realism – which is the natural philosophical (metaphysical) companion, or even an underpinning, of the classical logician. Platonic realism believes in the existence of entities irrespective of our abilities to know them, access them. Hence, of such an entity it is obvious that either the entity has a given property or else does not have the property (unbeknownst to us) – which is a manifestation of what is called the Law of Excluded Middle in logic. Despite being a Platonic realist a classical logician may still want, for some specific purpose of hers, to come up with an instance of a thing which is F, when she proves that there exists a thing which is F. In such a case the classical logician prefers a constructive proof of her claim within the classical framework. Of course, such constructive proofs may not be available for many of such claims – in which case one has to remain satisfied with nonconstructive ones. Now, if constructivism in logic is just this much then there is nothing much to create a stir over it. It becomes serious if there are people who would not accept a nonconstructive proof for any claim whatsoever, as they would have a far stronger belief that nonconstructive ones do not count as proofs at all. Such a position is manifest in Intuitionist Logic. Hence, the serious significance of constructivism can be best investigated in intuitionist logic. Intuitionism would want an ascription of meanings to the logical words which is different in a consistently deviant way from the classical meanings of them. This so-called consistent deviation is powered by some philosophical considerations – which will now be discussed, not in their entire gamut, though, because of lack of that scope, but rather in reasonable brevity.

Constructivism Through the Truth-Theoretic There are, as expected, at least two directions from which the philosophical background of Intuitionist Logic can be looked at: the semantic and the proof-theoretic. The semantic one relates to how the notion of meaning, and consequently, the notion of truth is understood. We have remarked earlier that the relevant question presently has become “what do we know when we know the meaning of an expression?”. Presently, again, the consensus, after deliberations, is that the minimum units of meanings are sentences, rather than words (the atoms) which constitute the sentence

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(the molecule). The atoms – the words occurring in the sentences – can be thought of having meanings only in terms of the role they play in the unitary meaning of the entire sentence. So, from such a molecularist perspective, the relevant question has become “what do we know when we know the meaning of a sentence?”. And, the answer, largely accepted, has been that to know the meaning of a sentence is to know the conditions of truth of the sentence. The focus then shifts from meanings – may be – as Platonic entities to meanings as knowable, accessible something. This shift, though looks like a trivial one, makes way for bringing in questions about acceptance of a particular theory of knowledge. For a Platonist realist, the Platonic entities are indeed best accessible by what they call the faculty of intuition. But now, since we are supposed to know here conditions of truth (instead of some Platonic entity) – which are again expressible by sentences – we need to employ ways of knowing which are apt for knowing sentences expressing all sorts of states of affairs – sense-experiential, or, those involving non-sensual entities, or, even a mix of these two. The importance that this question about knowledge gains is reflective of the fact that meaning is a phenomenon relating to the users of language; without users of language, there is no language, and no meaning to convey either. So, meaning is inextricably connected with users’ access to meanings. In suitable conditions, of course, we can ignore this question of accessibility to meanings, and treat meanings as just given – accessible or not – but surely, a general theory of meaning has to make room for this accessibility. Now, since knowledge of meaning of a sentence, presently, is being understood in terms of knowledge of truth-conditions of the sentence, the understanding of truth itself has to be such that it makes room for accessibility to the conditions identified for the truth of a sentence. Let us consider the following situation: I am given a sentence of a language which I claim to have known. I am asked whether I know the meaning of the sentence. I claim that yes I know its meaning. Since, it is taken that to know the meaning is to know the truth-conditions, it is presumed, by my answer, that I know the truth-conditions really. So, I am further asked whether I can tell what the sentence is – true or false? I cannot under the above narrative reply that recognizing whether the sentence is true or false, that is, whether the conditions of truth, of which I am supposedly knowledgeable, are indeed obtaining, is beyond me, or even, beyond my capacity – that I know of no method or means which when followed would tell me whether the sentence is true or not. For, this reply would tantamount to saying that I do not at all know the meaning of the sentence. That is, the crucial question becomes: can I say, simultaneously, that I know what has to obtain in the world so that the sentence be true, and yet, even if I am suitably placed, do not know whether the conditions are indeed obtaining? Can I, in such a case, really claim to have known the conditions of truth? A notion of truth which allows such a situation, so far as truth is thought to be so intimately connected with meaning which is taken to be something to be known, must be a very unusual and strange notion. We then face the prospect of having two different notions of truth: a recognition transcendent notion of truth and a recognition sensitive notion of truth. Let us note that this recognition transcendent notion of truth is typically the semantic cognate of the philosophical position called “realism.” Realism, generally,

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is thought to be a doctrine which holds that there are objects and entities existing independently of our knowledge (in its usual strict sense of the term: sensual and deductive) of them – and consequently, sentences about them are true or false independently of any availability of a proof or a disproof of the sentence. Realism does not need any further mechanism to provide additional information as to which one is the specific case/instance relating to the sentence in question: not even a means (of recognition) which when followed leads to the case, is required by the realist. Many would have us believe that matter exists, that I as a self-exist, that other minds exist, imaginary numbers exist, that God exists, that universals exist, moral rightness exists, and so on, and take the claims as just obvious even if such claims are not provable (in the strict sense of the term). Hence, sentences about such entities are either true, or else, false irrespective of availability of any proof or disproof for them. Realism, though, has currently been scaled down to a piecemeal characterization rather than a global one. One may be a realist with respect to self and other minds but not with respect to moral rightness – and many such other combinations, throwing up different metaphysical positions. Then, a very insightful characterization of realism of any sort is that to assume for a given class of sentences about an entity that the sentences of the class possess exactly one of the two truth-values – truth and falsity – is to hold realism about the entity. This characterization uses a principle which is known as the principle of bivalence: that there are two truth-values, and exactly one of the two values is possessed by the sentences belonging to the class. Once we accept the assumption of the principle of bivalence, we not only are embracing realism, we are also embracing classical logic. One of the hall-marks of classical logic is the validity of what is known as the Law of Excluded Middle – which says that a sentence of the form ‘p or not-p’ is logically true or valid. The difference between the principle of bivalence and the Law of Excluded Middle lies precisely in the fact that whereas the principle of bivalence is a principle which can be assumed or not assumed about a class of sentences, the Law of Excluded Middle is a sentence, of a particular form, within the class itself such that it is true under any interpretation, keeping the interpretations (meanings) of the words ‘or’ and ‘not’ same, in a particular way, across the different interpretations. The interpretations of the words ‘or’ and ‘not’ can be changed even under the assumption of the principle of bivalence in such a way that sentences of the form ‘p or not-p’ no longer come out true under every interpretation of the other expressions. But of course, if we stick to the usual interpretations of ‘or’ and ‘not’, then the Law of Excluded Middle is an obvious consequence within the class, of the principle of bivalence. So, classical logic, by accepting the usual meanings of ‘or’ and ‘not’, accepts the validity of any sentence of the form ‘p or not-p’ without ever bothering about the need for a proof either of ‘p’ or of ‘not-p’. That is, without bothering to know whether the conditions of truth of ‘p’ or of ‘not-p’ has indeed obtained in the world, classical logic is ready to pronounce the sentence of the form ‘p or not-p’ as a true, in fact a logically true, i.e., valid, sentence. It does not bother to check whether the conditions of truth of the whole sentence of the form ‘p or not-p’ obtains or not. This is typically the realist streak getting manifested in classical logic. The assumption of the principle of bivalence for a given class of sentences makes the Law of

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Excluded middle valid within the class – and as a result the given class of sentences reflect the philosophical position of realism regarding the entities that the sentences of the class talk about, and classical logic gets applicable to them. And, this is happening in the case of classical logic because the recognition transcendent notion of truth is at work in the background. Let us now have a look at the recognition sensitive notion of truth. Since the notion of truth is supposed to be intimately connected with the notion of meaning – which is something to be known, the appropriate notion of truth has to cater to such a notion of knowable meaning. Specification of the conditions of truth of a sentence will, surely, include what has to happen in the world so that the sentence be true – just as it is done in the case of the recognition transcendent notion of truth – but, additionally, in this case, the specification must also include ways and means to recognize whether the condition obtains. Without this additional input one does not really get any idea of what it is for a given condition to obtain – that is, one does not really have the means to recognize whether the condition obtains. For the sentence “Coal is green” to be true the condition that has to obtain in the world is that coal has to be green. Now, what is it for coal to be green – where ‘green’ is an experiential predicate? Obviously, experiential ways of recognizing whether something is green come into play here. So, specification of experiential ways of recognizing whether an experiential predicate applies to something will be the additional input that will be needed in the specification of the conditions of recognition sensitive truth of a sentence involving an experiential predicate. Generally speaking, specification of recognition sensitive truth-conditions of atomic sentences will lay down, if not actual ways of recognizing that the conditions of truth obtain, at least the methods or means which when followed lead to recognizing that the conditions obtain. Unless such means of recognition are made available to the speakers of a language it becomes too difficult to explain how one learns and gains mastery over the language. When this spirit of specification gets applied in logic, the specification of recognition sensitive truth of sentences containing a logical word involves specification of procedures of proofs of sentences containing the logical word in question. This is the prooftheoretic treatment of logical words.

Bivalence and the Law of Excluded Middle Before we enter into an investigation into this proof-theoretic treatment of logical words, we need to note certain consequences of embracing this philosophical position of understanding meanings through recognition sensitive notion of truth. When we accept such a recognition sensitive notion of truth the most prominent casualty, from the widely practised domain of classical logic, turns out to be the validity of the Law of Excluded Middle – which we have already pointed out earlier as the hallmark of classical logic. This paves way for Intuitionist Logic which is a robust manifestation of constructivism in logic. This gets reflected in current presentations of logic through derivability of the Law of Excluded Middle in classical propositional logic on the one hand, and the non-derivability of it in intuitionist

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propositional logic on the other. As a result of nonacceptance of the Law of Excluded Middle in intuitionist logic, the assumption of the principle of bivalence also becomes unacceptable, since the Law of Excluded Middle has been seen to be following from the assumption of the principle of bivalence. The upshot of this is that any doctrine having an underlying intuitionist logic no longer remains a realist doctrine – it falls within the class of what are called antirealist doctrines – denying the principle of bivalence. In fact, within intuitionist framework there are sentences which are neither true nor false. Again, because of the unacceptability of the Law of Excluded Middle in intuitionist logic one direction of the classical Law of Double Negation too becomes unacceptable: one cannot hold in intuitionist logic that notnot-p implies p – which again leads to rejection of other classical laws of logic dependent on this direction of the implication between not-not-p and p. One other very important noticeable change that comes into intuitionist logic in contrast to the classical one concerns claims about existence. Existential claims cannot be established within the framework of intuitionist logic except by producing (through proof) an instance or a case of the kind that is claimed to be existing, or, at least by specifying ways which when followed leads to such instances. All these changes from classical logic together with resultant ramifications define intuitionist logic. Intuitionist logic, from the above perspective, is a proper subpart of classical logic, or, alternatively, classical logic is a nontrivial extension of intuitionist logic. This may give the impression that just presence or absence in logical systems, of relevant formulas or linguistic representations of the logical laws mentioned and discussed above demarcate the classical from the intuitionist logic. This, however, would be a very misleading way of differentiation between the two logics. The difference, rather, lies in how meanings of expressions – especially logical words – are understood. Two different notions of truth carrying two different philosophies underlying them help us decipher meanings in the two cases. One notion, the realist, classical, takes the universe, even if infinite, as a completed universe existing on its own right as given, independent of anybody’s knowledge of it; and the other, the antirealist intuitionist, takes the universe only in so far as it is encounterable in its ever unfolding variegation through the senses and intellect that are possessed by humans for understanding and knowing their surroundings – nothing is taken as given for it, and only legitimate “constructions” give us what can be accepted about it. So, the logical words, the sentences which contain them, carry two different meanings under the two different philosophies. The characteristic mark of the latter philosophy get revealed in starker ways in the proof-theoretic treatment of the logical words.

Constructivism Through the Proof-Theoretic As we have noted earlier, proof-theoretically, whatever meaning a logical word has, is specifiable in terms of the role the word plays in the context of proofs. The roles are twofold and specified in terms of the introduction and the elimination rules (intelim rules) for the word – the rules containing no words other than the one for which the rules are given. If the roles are specified in the context of proofs, the

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context too has to be defined well. The minimum that the context can be for having logical words, will be the context of deducibility as such – deducibility involving no logical word. Rules which determine such a context of deducibility as such are called structural rules. The structural rules will, for example, in simple language, accommodate the fact that sequence in the order of premises in the set of premises does not matter for deduction as such: any sequential order of premises would facilitate the same deduction. One other such fact will be that repetition of a premise within the set of premises does not have any effect for deduction. A similar third one will be that if a deduction is effected from a given set of premises then the deduction remains intact even if the premise set is expanded. The first one is called “exchange,” the second is called “contraction” and the third “weakening.” There is one other structural rule which will be seen later to play a significant role. It is called “cut” and its somewhat formal presentation is the following. A, A j-- B, B, B j-- C A, B, A j-- C [where bold capital letters stand for sets of formulas, normal capital letters stand for single formulas each, the turnstile stands for “yields” (or, in reverse order “deducible from”) and the bold horizontal line signifies which deducibility relation follow from which others]. The cut rule says, in effect, that if the formula B is deducible from the formula A along with the set of formulas A, and the formula C is deducible from the formula B along with the set of formulas B, then the formula C is deducible from A along with the sets of formulas A and B. What is really acknowledged through this rule is that instead of a direct proof of C from A, proof of C from A by a detour through B when other supporting premises are present in the sets of premises in A and B, is possible. In such a case the formula B is called the cut-formula. These four structural rules and the fact that any formula is deducible from itself (i.e., “A |– A” taken as an axiom) characterize one notion of deducibility as such. Of course, other notions, characterized by changes in the above structural rules and axioms, may be suitable for specific purposes, but this notion of deducibility as such serves as a perfect background context for both classical and intuitionist logic. Both these differing logics, then, can be seen, separately, as extensions from this logic of deducibility as such. So, logical words with their respective meanings in the two logics can be seen as added in the extensions through the specific pairs of introduction and elimination rules for the relevant words. A serious question that arises at this point is whether any – as one can wish – proposed pair of introduction and elimination rules can be taken to lay down the meaning of a word in question. There are two apparently different answers which seem ultimately to converge into the same insight. The net effect of the answers is a clear no. By one answer, the extension has to be unique and also conservative (Belnap 1962). The extension has to be unique in that same configuration of syntactic relations between premises and conclusion as used in the intelim rules will be taken as defining one and the same logical word even if they differ in terms of

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the symbols chosen. More crucially, the extension by adding intelim rules for a given word has to be conservative in that all the new deducibility relations emerging through the extension must involve the new logical word: that is, it will conserve all the deducibility relations present before the extension and that no new deducibility relations will emerge as a result of the extension which does not contain the new word. So, only such intelim rules for words are taken to be permissible which satisfy the constraints of uniqueness and conservative extension. The other answer looks at the relationship between the introduction rules and the elimination rules for a given word. If it is thought that the introduction rules give the ways or the requirements for proving a sentence containing a word then the corresponding elimination rules for the same word must, while stating what can be proved without using the word from a sentence which uses the word, be drawing upon elements which were present in the ways of proving the sentence containing the word. That is, ways of extracting the consequences of having a sentence with a logical word cannot go beyond what was provided by the grounds for having such a sentence. One understands what can be said given a sentence containing a logical word, only in so far as she was made to understand how the sentence with the logical word was established or asserted in the first place. In simpler terms, the consequences (the elimination rule) of an assertion of a sentence with a logical word must lie within the grounds of the assertion of the sentence. This is the principle of inversion, according to Prawitz (1965), which must be present in a pair of intelim rules for a logical word. The formal presentation of the idea contained in the principle of inversion can be stated as the requirement that an application of an elimination rule for a word immediately after an application of an introduction rule for the same word must be superfluous, or, result in going back within the premises required for the application of the introduction rule. Dummett (1991) called this same requirement as harmony between the intelim rules. So, by this answer again, any pair of intelim rules, as one may wish, cannot be taken to be defining or giving the meaning of a logical word; some constraints are there: the pair must satisfy the principle of inversion, or, what is the same thing the requirement of harmony. It can be shown that a language extended by adding logical words through intelim rules satisfying the requirement of conservative extension are just those where the intelim rules for the logical words satisfy the principle of inversion or the requirement of harmony. One curiosity remains however: how is the extension done for a set of logical words (say, those needed for first order predicate logic)? In one go or separately, one by one, checking at each stage of extension whether the extension is conservative, and that too for any order of addition? The answer, at least for first order predicate logic, is that it does not matter whether the words are added in one go or separately, so far as the intelim rules for each word satisfies the principle of inversion or the requirement of harmony. So far as the insight of specifying whatever meanings the logical words have through intelim rules goes, one very important question is about whether the elimination rules harmonious with the corresponding introduction rules for a logical word capture the full import of the logical word as given by the introduction rules. If it does not happen for a given word then it cannot be claimed that the full meaning

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has been given through the intelim rules – by proof-theoretic means. The resulting logic in such a case will not be able to explain and contain all that was expected of it. Dummett (1991) pointed out that a further requirement then must be there: that of stability: the elimination rules for a word must be such that they are in harmony with such introduction rules for the word which are themselves such that any application of them immediately after an application of an elimination rule is superfluous, or, gives back the premises for the application of the elimination rule. This additional requirement of stability apart from harmony for the intelim rules really looks for complete specification of the (proof-theoretic) meaning of a logical word. In such proof-theoretic presentations of an area of logic comprising the structural rules, the relevant logical rules (the intelim rules for the logical words) and the axioms needed, it is an important and, also interesting, curiosity whether the structural rule “cut” is eliminable. If cut is eliminable in the case of any such presentation, then certain immediate corollaries follow for that presentation which are of great interest to the logicians. But apart from these there is a serious philosophical implication of eliminability of cut as well. As we have noted earlier cut really accommodates proofs of sentences through detours from given premises along with possible additional premises via a mediatory sentence called the cut-formula. Now, if it can be shown that every proof involving the cut rule can be converted into a proof without the use of the cut rule – that is, cut is eliminable – then it amounts to saying that every proof possible in the area of logic concerned has a proof where the conclusion is provable directly from the premises without any detour – without the help of any intermediary sentences and other possible supportive sentences. We, at this stage, keep in record that the meta-theoretic proof that cut is eliminable for a given presentation needs to exploit the principle of inversion, or the requirement of harmony – the intimate relationship between the introduction and the elimination rules for the logical words. That is, until and unless the principle of inversion is operative, or the requirement of harmony is fulfilled the elimination of cut cannot be achieved: any study of cut-elimination theorem would reveal that. Now, In the case of such a direct proof what happens in this proof-theoretic setting is the following. The logical words that occur in their relative strengths of dominance over each other show which introduction rule is to be applied at which stage of the proof. For example, if the most dominant such word in the conclusion is that for ‘and’, then of course the last step of the cut-free proof will see an application of the introduction rule for ‘and’. Of course, previous to this application will be a presence of the two components needed for the application of the introduction rule for ‘and’. The two components will either be given as premises, or, as extractable from earlier stages of the proof by elimination rules or introduction rules (or a combination of them) for relevant other logical words. What is being hinted here is that if the presentation of logic of a particular area is shown to be cut-free then the conclusions themselves there show how to go about proving the sentence. From the philosophical point of view this amounts to saying that such a cut-free presentation tells us the conditions of assertion of the sentence being proved, or what is the same thing, from the semantic perspective, the means of recognizing the truth of the sentence.

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The above brings us almost, but not quite, to constructivism in logic. This is so because presentations of logic through such proof-theoretic means also allow for use of axioms. Acceptance of the Law of Excluded Middle as an additional axiom with matching intelim logical rules for negation in a proof-theoretic presentation of first order logic gives us the classical understanding of this logic. On the other hand, absence of the Law of Excluded Middle with matching changes in the rules for negation gives us intuitionist logic. Now, cut can be shown to be eliminable in both the presentations. But, eliminability of cut in the classical case is facilitated only by the use of the axiom stating the Law of Excluded Middle – which helps secure principle of inversion or requirement of harmony within the intelim rules for classical negation. The rules for classical negation, as they are, in the absence of this axiom fail to satisfy the principle of inversion or the requirement of harmony. Though there are variations in presenting classical logic in regard to intelim rules for negation and accompanying axiom of excluded middle, the crux ultimately shows up in not satisfying the principle of inversion or the requirement of harmony. Given, from the above, that any and every proposed pair of intelim rules for a symbol cannot be taken as defining a logical word: they have to satisfy certain constraints without which they do not count as proper logical words, we can, again, have a look at the kind of philosophical position that is embraced in such prooftheoretic understanding of logical words. Plainly, conditions of assertion of sentences containing logical words are in focus. But what can be taken as legitimate conditions of assertions is being sought to be identified by weighing in the internal relationship between the two sides of assertion: the grounds of assertion and the consequences of assertion. If none of these grounds and consequences go beyond the other, and further, if each of them exploits fully what the other provides, then it is a very reliable indication that the conditions of assertion as specified are legitimate. Seeking legitimacy of the conditions of assertion within some other domain would have just shifted the burden of the question of legitimacy to this other domain. Hence, the best strategy to stop this shifting of burden of legitimacy infinitely or circularly into other domains is to look for legitimacy within the domain itself. This, precisely, is what is happening here when the two sides of the conditions of assertion are sought to be checked as whether they fulfill the conditions of harmony and stability. If they do, it is atleast, if not anything more, satisfactory to the extent that the practice of using these two sides of conditions of assertion would never lead to any inconsistency, or even to accept anything not envisaged by the practice – both of which are dreaded by any serious enquiry. This demand of finding legitimacy of use of a logical word within the balance between the two sides of the use is compatible as well with how we understand the meanings of nonlogical words. In case of use of nonlogical words too some such balance is seen to be present – even if consideration of sensual experiences come in, and that helps explaining learnability of language, largely with an empiricist outlook. Argumentation within a framework of Intuitionist Logic, which satisfies the internal constraints of, namely, harmony and stability, then always keeps track – proof-theoretically, or even truth-theoretically – of the grounds of assertion and the consequences of assertion telling us how to go about proving – constructing – the

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intended claim of the conclusion. Hence, in the truth-theoretic perspective, means of recognizing the truth of the claim of the conclusion also get manifested in such argumentation.

Final Words Intuitionist logic, then, as a constructivist logic is really offering itself as an answer to the question of justification of logic: which logic seems to be justifiable, i.e., acceptable as a reliable tool for drawing logical conclusions? Justified not from some other perspective or within some other domain – because, that would really postpone the question of justification to some other stage or domain – but justified from within! Apart from claiming its stature as a justified logic Intuitionist Logic also manifests a particular philosophy of language – hence that of mind as well – dealing with meaning and truth wedded to an empiricist epistemology. One must note that the intuitionist outlook as presented here clearly hints that any “logic” – be it first order, modal propositional, modal quantificational, many-valued, fuzzy, etc., can have two presentations – the classically flavored, without requiring that means of recognizing whether a given sentence possesses the designated value in the logic be provided also, and, on the other, the intuitionistically flavored with precisely this requirement. We need to touch upon, very briefly, some other kinds of treatment of Intuitionist Logic, and see whether the treatments in those kinds illuminate the constructivist character of Intuitionist Logic. First, there are axiomatizations of Intuitionist Logic – its propositional part, its quantificational part, and meta-theoretic investigations of them. By axiomatization of a discipline we get a systematization of sentences – axioms and theorems – claims that are held true in the discipline. By that kind of an activity we get a clear disparity between an axiomatization of Classical Logic and an axiomatization of Intuitionist Logic: in the intuitionist case some claims get missing which are present in the classical case – of course, notably of them being that which expresses the law of excluded middle, and some more such. Even then one can compare and study the two systems for various purposes. But one thing has to be borne in mind that in the two cases the intended meanings of the logical words are qualitatively different. In the classical case the meanings are such that the provable theorems of the system from the axioms are sentences for which it is assumed that they are true or else false irrespective of their provability. But in the intuitionist case the sentences must be amenable to proofs or disproofs, with the possibility that sentences which are not axioms or theorems may neither be provable nor disprovable. This happens, because the meanings of logical words are understood in this case by provability of sentences containing them. So, syntactically speaking the two axiomatizations differ only by presence or absence of sentences as syntactic items, but semantically speaking they differ hugely by how they are interpreted in the two cases. Now, second, because of the difference in interpretation, the semantics of Intuitionist Logic as presented through axiomatization becomes essentially modal – as pointed out by van Dalen (1986) – in character bringing in possible worlds and

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knowledge or proofs in them as is found in Kripke’s (1965) work. Consistency and completeness of Intuitionist Predicate Logic can be shown with respect to such semantics. Third, in the domain of algebraic treatments, Boolean algebra fits for classical logic, whereas Heyting algebra fits for the intuitionist variety. Such algebras really bring out the properties of the truth-values, truth and falsity, in so far as they behave with respect to the operators like meet, join (conjunction and disjunction) and the like. But, a little reflection reveals that none of these three kinds of treatment focuses on the demand for ‘means of recognition’ factor, and, ‘conditions of assertion’ factor which have been highlighted above. As, presently the last comment we can just have a look, from the perspective of philosophical underpinnings, at the possibilities of combinations of nonlogical theories with alternative base logics. If we stick to just this so-called balance between the two sides of assertion of a sentence containing a logical word – that requirements of harmony and stability must be fulfilled – and require nothing else, apart from the initial context of deducibility as such, then we are left with intuitionist logic – which is obviously constructivist. Addition of the Law of Excluded Middle with accompanying necessary changes, if any, would give us classical logic. If we base a nonlogical theory on intuitionist logic and provide an understanding of the primitive concepts of the non-logical theory in a way which too embraces the philosophy behind intuitionist logic then the resultant theory would be an example of a theory which is thoroughly following what we may call an anti-realist philosophy. Likewise, if we base a nonlogical theory on classical logic and provide an understanding of the primitive concepts of the nonlogical theory in a way which embraces realism about the entities involved, then the resulting theory would be thoroughly following a realist philosophy. The other combinations of the base logic following one philosophy and the nonlogical theory based on it following a contrary one, would be a difficult one to comprehend philosophically.

References Belnap, N.D. 1962. Tonk, plonk and plink. Analysis 22: 130–134. Bendall, K. 1978. Natural deduction, separation and the meaning of logical constants. Journal of Philosophical Logic 7: 245–276. Dicher, B. 2016. Weak disharmony: Some lessons for proof-theoretic semantics. The Review of Symbolic Logic 9: 583–602. Dummett, M. 1977. Elements of intuitionism. Oxford: Oxford University Press. ———. 1978a. The justification of deduction. In Truth and other enigmas, 290–318. London: Duckworth. ———. 1978b. The philosophical basis of intuitionistic logic. In Truth and other enigmas, 215– 247. London: Duckworth. ———. 1991. The logical basis of metaphysics. London: Duckworth. Gentzen, G. 1969. Investigations into logical deductions. In The collected papers of Gerhard Gentzen, ed. M.E. Szabo, 68–131. Amsterdam: North-Holland Pub. Heyting, A. 1956. Intuitionism: An introduction. Amsterdam: North-Holland Pub. Kripke, S.A. 1965. Semantical analysis of intuitionistic logic. In Formal systems and recursive functions, ed. J. Crossley and M. Dummett, 92–130. Amsterdam: North-Holland Pub.

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Mukhopadhyay, R. 1990. Self-justifying rules. In Foundations of logic and language, Jadavpur studies in philosophy, second series, ed. Pranab Kumar Sen, 82–106. New Delhi: Allied Publishers Limited. ———. 2010. Dummett on realism and antirealism. In Materialism and immaterialism in India and the West: Varying vistas, ed. Partha Ghose, 481–497. New Delhi: Munshiram Manoharlal Pub. (for Center for Studies in Civilizations). Prawitz, D. 1965. Natural deduction: A proof theoretical study. Stockholm: Almqvist and Wiksell. Prior, A.N. 1960. The runabout inference-ticket. Analysis 21 (2): 38–39. van Dalen, D. 1986. Intuitionistic logic. In Handbook of philosophical logic, vol III: Alternatives in classical logic, ed. D. Gabbay and F. Guenthner, 225–339. Dordrecht: D. Reidel. Wittgenstein, L. 1953. Philosophical investigations. Oxford: Basil Blackwell.

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Contents Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . On Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Related Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Games as Models of Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Game Logics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Extensive-Form Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategy Specifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Semantics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Compositional Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategy Switching and Stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Player Types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Large Games . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Interaction About Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Aggregation of Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Expressing Deliberation on Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Strategic Reasoning . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Preferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Abstract

Reasoning about games involves elucidating the rational structure of preference that players have over outcomes, as well as the strategies they employ to achieve their preferred outcomes. This involves mutual intersubjectivity of epistemic S. Ghosh (*) Indian Statistical Institute, Chennai, India e-mail: [email protected] R. Ramanujam Instititute of Mathematical Sciences, Chennai, India e-mail: [email protected] © Springer Nature India Private Limited 2022 S. Sarukkai, M. K. Chakraborty (eds.), Handbook of Logical Thought in India, https://doi.org/10.1007/978-81-322-2577-5_42

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attitudes. We discuss some propositional modal logics of strategic interaction and point to interesting questions for further research.

Introduction Games are models of interaction where individual players (or agents) make choices, and obtain outcomes based on what everyone chooses. They have preferences over the outcomes, so each player would make a choice anticipating what others might do, so that they can yet obtain the best possible outcome. Such a simple description already leads to interesting notions. A player might want to get the best that she can, no matter what others do, simply choose the best outcome possible for herself. She might actually do better if others might choose appropriately, but she might not want to make assumptions about others’ behavior. If everyone behaves like this, we have a situation where they all choose conservatively, and we would have a dominant strategy equilibrium. This is when everyone acts unilaterally, but still the outcome is stable: none of the players would choose differently, even after being informed of the others’ choices. Alternatively, an agent may not settle for this; he can reason that everyone will want the best, and hence will assume others to do the same. If it does turn out that everyone does this, we have a Nash equilibrium. Again, such an outcome would be stable, now in a slightly different sense: none of the players would have any incentive to unilaterally deviate from this choice. Note that the latter behavior involves mutual intersubjectivity: a player’s decision depends on what she believes that others would do, but in turn what they do depends on what they believe she would do, and so on. This iteration of epistemic attitudes constitutes the foundational justification of the equilibrium notion. See (Osborne and Rubinstein 1994; Perea 2012) for a detailed discussion of the basic notions of game theory and their epistemic foundations. An underlying assumption in such reasoning is that players are rational: their awareness of their own preferences, ability to introspect on their actions, beliefs about others’ preferences and capacity to act, are all based on logically consistent reasoning. Indeed, not only are players rational, but believe that everyone is rational, that everyone believes this, and so on. This brief discussion suggests that the logical foundations of game theory involve indexical epistemic attitudes, preferences, strategization towards achievement of outcomes, and ability of players. Logicians would like to pin down the logical resources necessary for the description of such interaction and for the reasoning involved. For instance, are strategies first-class objects of a logical language, or are they composite entities built from other first-class objects? If the former, what are the characteristic properties of quantification over strategies? Fixed-point operators are natural in the description of equilibria; what is their expressiveness in the presence of epistemic modal operators? And so on. This chapter is situated in such a discourse. We take up a very small fragment of the logical study of strategic interaction, highlighting some of our own work in this

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arena. This involves an exploration of the rational structure of preferences, algebraic compositional structure in games, and a similar structure in strategies. Indeed we can see a pleasing duality between studying strategization in composite games, and studying the composition of strategies in the course of large games. These are elements of theories of play, and the notion of player types in such theories raises interesting questions for study. We also refer to how similar theories may be built for games with a large number of players.

On Preferences While strategic reasoning focusses on the relationship between individual choices and social outcomes, reasoning about preferences focusses on the interaction between individual preferences and group preferences. This study puts the notions of aggregation and deliberation in the spotlight, and these are the two main approaches in collective decision-making. Aggregation is mostly achieved by voting where the origins of the individual preferences do not come under consideration, only the actual preferences do. There is a plethora of work on aggregation of preferences together with critical formal studies on the advantages and disadvantages of different aggregation processes (e.g., see Arrow et al. 2002; Grüne-Yanoff and Hansson 2009; Endriss 2011). However, the effectivity of the aggregation process has been questioned by philosophers like Elster (1986); Habermas (1996) and others, who point to the merits of the deliberative process which make people reflect on their preferences thus influencing possible changes. Thus, the process of deliberation is also an important aspect of group decision-making – in case unanimity is reached via deliberation, there is no need to consider some (possibly artificial) aggregation process. Sometimes, even though deliberation may not lead to unanimity in preferences, the ensuing discussion may lead to certain form of “preference uniformity” (see how deliberation can help in bypassing social choice theory’s impossibility results in Dryzek and List 2003), which might facilitate their eventual aggregation. In addition, a combination of both aggregation and deliberation processes may provide a more realistic model for decision-making scenarios (Perote-Pena and Piggins 2015). In light of this status quo, more focus can be given on the formal study of achieving such preference uniformities, e.g., single-peaked, single-caved, single-crossing, value-restricted, best-restricted, worst-restricted, medium-restricted, or groupseparable profiles (see Bredereck et al. 2013) and references there in for relevant results and discussion). What we are essentially analyzing are the compositions of individual structured preferences which bring about the group preferences, and as mentioned earlier, logic can play an important role in such composition analyses. In this part of the chapter, we provide a comparative analysis of the processes of aggregation and deliberation from a logical perspective. The main objective here is to motivate a combined formal analysis of these aspects towards determining the subtle commonalities as well differences.

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Related Work A substantial part of the work discussed here originates from the seminal work of Parikh (1985) on Propositional Game Logic, which suggested algebraic game composition as a tool for logical study. Goranko (2003) looks at an algebraic characterization of games and presents a complete axiomatization of identities of the basic game algebra. Pauly (2001) has built on this to provide interesting relationships between programs and games, and to describe coalitions to achieve desired goals. Goranko (2001) relates Pauly’s coalition logics with alternating temporal logic (Alur et al. 1998), which combines temporal logic with a form of quantification over strategies. The logics presented here can be seen as a process logic extension within a branching time framework (Ramanujam and Simon 2009). Another major influence on the work presented here is that of Johan van Benthem (and co-authors). van Benthem (2001) considered both perfect and imperfect information games and analyzed them at the local action level as well as the global outcome level. From a purely logic perspective, van Benthem (2003b) established a connection between the algebra of game operations considered in the game logics (Parikh 1985; Parikh and Pauly 2003) and that of logical evaluation games, showing the latter to be quite general. van Benthem (2012) highlights the importance of having logics expressing strategies explicitly which could in turn give more pragmatic models for interaction. van Benthem et al. (2011) provided a distinct dynamic perspective into the continuing studies on logic and games. Different notions of information and interaction related to agency were brought in the foray to propose a “theory of play” interlacing the notions of logic and games. Finally, van Benthem (2014) gave a detailed description of his research agenda at the interface of logic and games. He provided a host of perspectives into the ever-continuing studies on logic and games, bringing out a number of open research problems. In his words, “this book is meant to open up an area, not to close it.” We discuss other related work in context, in the course of developing the notions. Parts of the work presented here on games arose from joint work with Soumya Paul and Sunil Simon.

Games as Models of Interaction Consider the classic situation of two children wanting to divide a piece of cake among themselves. The solution is to let one child divide the cake and let the other choose which piece she wants. Each child wants to maximize the size of his piece, and therefore this process ensures fair division. The first child cannot complain that the cake was divided unevenly and the second child cannot object since she has the piece of her choice. This is a very simple example of a game where two players have conflicting interests and each player is trying to maximize his payoff. The final outcome of the division depends on how well each child can anticipate the reaction of the other and this makes the situation game-theoretic.

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A game can be presented by specifying the players, the strategies available to each player and the payoff for each player. In the case of a two-person game, this can be presented efficiently in a matrix form. For instance the cake cutting game can be represented using the matrix shown in Fig. 1. We will refer to the players as cutter and chooser. Here both players have two strategies, each can choose to cut the cake evenly or to make one piece bigger than the other, which corresponds to picking one of the rows of the matrix. Chooser can choose the bigger piece or the smaller piece, which corresponds to picking one of the columns of the matrix. The outcome for the cutter, after both the players choose their strategies, is the corresponding entry in the matrix. For instance, if the cutter chooses to make one piece bigger and the chooser picks the bigger piece, then the outcome will be that the smaller piece goes to the cutter (bottom left cell). The chooser’s outcome is the complement of the cutter’s. An equivalent representation of the game can be obtained by replacing the outcomes with numbers representing payoffs as shown in Fig. 2. The cake cutting game captures the situation of pure conflict, where cutter’s gain is chooser’s loss and vice-versa. Such games are called zero-sum or win-loss games. If the cutter had the option of choosing any of the four available outcomes, he would prefer to have the big piece. However, he realizes that expecting this outcome is highly unrealistic. He knows that if he were to make one piece bigger, then the chooser will pick the bigger piece leaving him with the remaining smaller one. If he divides evenly, then he will end up with half of the cake. The cutter’s choice is really between the smaller piece and half of the cake. Therefore he will choose to take half of the cake (top left cell) by making an even split of the cake. This amount is the maximum row minimum and is referred to as the maximin. Now consider a variation of this game where the chooser is required to announce her choice (big or small piece) before the cake is cut. This does not change the Fig. 1 Cutting a cake: Instinctive payoffs

Choose bigger piece Cut the cake Half of the evenly cake Make one piece bigger

Fig. 2 Cutting a cake: Cardinal payoffs

Small piece

Choose bigger piece 0 Cut the cake evenly Make one piece bigger

-1

Choose smaller piece Half of the cake Big piece

Choose smaller piece 0

1

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situation: The chooser would still choose a bigger piece irrespective of how the cutter divides the cake. That is, the chooser looks for the minimum column maximum (minimax) value, which happens to be the top left cell. In this example, the maximin value and the minimax value both happen to be the top left cell. For a game when the maximin value and the minimax value is identical, the outcome is called the saddle point. When a game has a saddle point it is the expected rational play since either player cannot unilaterally improve his payoff. A win-loss game is said to be determined if a saddle point exists. Formally a two-player zero-sum game where player 1 has m strategies and player 2 has n can be represented by an m
n array A, where the (i, j)th entry ai, j represents the payoff of player 1 when he chooses the strategy i and player 2 picks strategy j. The payoff for player 2 for the corresponding entry is –1
ai, j. Note that non-zerosum payoffs can easily be represented by replacing each matrix entry by a tuple of payoffs for each player. Unfortunately not all games have saddle points. One of the simplest examples is the game of “Matching pennies” depicted in Fig. 3. In this game, two players simultaneously place a penny (heads or tails up). When both the pennies match, player 1 gets to keep both. If the pennies do not match, then player 2 gets to keep both. It’s easy to see from the payoff matrix that maximin is 1 whereas minimax is 1. It is well known that the best way of playing matching pennies is to play heads with probability half and tails with probability half. This amounts to a mixed strategy rather than the pure strategy of picking an action with absolute certainty. The minimax theorem (von Neumann and Morgenstern 1947) asserts that for all two player zero-sum games, there is a rational outcome in mixed strategies. The theory can be extended from zero-sum objectives to non-zero-sum objectives with more than just two players. In this case, the outcome of the game will specify a payoff for each of the players. The commonly used solution concept in this context is that of Nash equilibrium which corresponds to a profile of strategies, one for each player which satisfies the property that no player gains by unilaterally deviating from his equilibrium strategy. Nash (1950) formulated this notion of equilibrium for multiplayer non zero-sum games and proved the analogue of the min-max theorem for such games. The result states that for all finite multiplayer games, there exists a mixed strategy (Nash) equilibrium profile. Much of the mathematical theory developed for games talks about existence of equilibrium and does not shed light on how the players should go about playing the game. For two-person zero-sum games, one can show that the maximin theorem is equivalent to the LP (linear programming) duality problem. Therefore construction of optimal strategies is possible using linear programming techniques

Fig. 3 Matching pennies

Heads Tails

Heads Tails 1 -1 -1

1

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(von Stengel 2002). For two person non zero-sum games, optimal strategies can be constructed using techniques for solving the linear complementarity problem as shown in (Lemke and Howson 1964). For a multi-player game, Nash’s theorem talks of existence of equilibrium but it is not known how to actually construct the equilibrium strategy. Strategic form games give a highly abstracted presentation of a game. The representation typically assumes “small” games where the structure of the strategy (individual moves which build up to form the strategy) is absent (or abstracted away). The existence theorems suggest which strategy a player would employ in the game. However, we also need to analyze larger games where the players’ actions are part of the representation. We now address reasoning in such a context.

Game Logics One natural way is to consider a large game as being built up structurally from small atomic games by means of composition. This suggests an algebraic structure in games, and one line of work in game logics proceeds by imposing a program-like compositional structure on games. Program logics like the propositional dynamic logic (PDL) (Harel et al. 2000) have been developed to reason about programs. The idea here is to model programs as being constructed using operations like sequential composition, iteration, etc. on simple atomic programs. This compositional approach in program reasoning has been successful in the analysis and verification of programs, especially in giving us insights into the expressive power of various programming constructs. The natural extension to this methodology is to come up with a dynamic logic to reason about multi-agent programs and protocols. Game logic (Parikh 1985) that we briefly introduced in section “Related Work” addresses this issue. Game logic (GL) is a generalization of PDL for reasoning about determined two person games. Let the two players be denoted as player 1 and player 2. Like PDL, the language of GL consists of two sorts, games and propositions. Let Γ0 be a set of atomic games and P a set of atomic propositions. The set of GL-games Γ and the set of GL-formulas Φ is built from the following syntax: Γ :¼ gjγ 1 ; γ 2 jγ 1 [ γ 2 jγ  jγ d Φ :¼ pj:φjφ1 ^ φ2 jhγ iφ


d where p  P and g  Γ0. Let [γ]φ :¼ : hγi : φ and γ 1 \ γ 2 :¼ γ d1 [ γ d2 . The formula hγiφ asserts that player 1 has a strategy in game γ to ensure φ and [γ]φ expresses that player 1 does not have a strategy to ensure :φ, which by determinacy is equivalent to the fact that player 2 has a strategy to ensure φ. The intuitive definitions of the games are as follows: γ1; γ2 is the game where γ1 is played first followed by γ2, γ1 [ γ2 is the game where player 1 moves first and decides whether to play γ1 or γ2 and then the chosen game is played. In the iterated game γ,

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player 1 can choose how often to play γ (possibly zero times). He need not declare in advance how many times γ needs to be played, but is required to eventually stop. The dual game γd is the same as playing the game γ with the roles interchanged. The formal semantics is given below. A game model M ¼ ((S, {Eg|g  Γ0}), V ) where S is a set of states, V : P ! 2S is S the valuation function and Eg : S ! 22 is a collection of effectivity functions which are monotonic, i.e., X  Eg(s) and X  X 0 imply X 0  Eg(s). The idea is that X  Eg(s) holds whenever player 1 has a strategy in game g to achieve X. The truth of a formula φ in a model M at a state s (denoted M, s  φ) is defined as follows: M, sp

iff s  V ðpÞ

M, s:φ M, sφ1 ^ φ2

iff M, s⊭φ iff M, sφ1 or M, sφ2

M, shγ iφ

iff φM  Eγ ðsÞ

where φM ¼ {s  S|M, s  φ}. The effectivity function Eγ is defined inductively for non-atomic games as follows. Let Eγ(Y) ¼ {s  S|Y  Eγ(s)}. Then
 E γ1 ;γ2 ðY Þ ¼ Eγ1 E γ2 ðY Þ Eγ1 [γ2 ðY Þ ¼ Eγ1 ðY Þ [ E γ2 ðY Þ E γd ðY Þ ¼ Eγ ðYÞ E γ ðY Þ ¼ μX :Y [ E γ ðX Þ where μ denotes the least fixpoint operator. It can be shown that the monotonicity of Eg is preserved under the game operations and therefore the least fixpoint μX. Y [ Eγ(X) always exists. Since game logic was designed to reason about multi-agent programs, the modelling approach is quite different from traditional game theoretic notions. Pauly (2000) presents a semantics for Game logic which is closer to the standard game-theoretic approach. Theorem 1 (Parikh 1985) The satisfiability problem for Game Logic is in EXPTIME. Theorem 2 Given a Game Logic  formula φ and  a finite game model M, model ad ðφÞþ1 checking can be done in time O jMj  jφj where ad(φ) is the alternation depth of φ. A proof of this theorem can be found in Pauly (2001), Theorem 6.21 (page 122). As shown in (Parikh 1985), it is possible to interpret Game logic over Kripke structures. Over Kripke structures, Game logic can be embedded into μ-calculus

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(Kozen 1983). Whether Game logic is a proper fragment of the μ-calculus is not known. It is quite conceivable that model checking for Game logic is easier than model checking for the full μ-calculus. However, Berwanger (2003) shows that this is not the case. A long-standing open problem in Game logic, to give a complete axiomatization of valid formulas of the logic, was settled recently (Enqvist et al. 2019). Parikh (1985) proposed an axiom system and conjectured that it is complete, which has now been proven. For the dual free fragment of Game logic, a complete axiomatization is presented in (Parikh 1985). In Game logic, starting with simple atomic games, one can construct large complex games using operators like composition and union. Due to the presence of the Box-Diamond duality hγiφ  :[γ] :φ, it is easy to see that the games constructed remain determined. The compositional syntax of Game logic presents an algebra for game construction. Rather than look at arbitrarily large games, this approach gives us a way of systematically studying complex games in a structured manner and to also look at their algebraic properties. One should however note that the emphasis in this approach is to reason about games, to study the structure of games with interesting properties and definability conditions.

Extensive-Form Games When games are finite, strategies are complete plans and each player has only finitely many strategies to choose from, normal form games abstract strategies and sets of choices and study the effect of each player making a choice simultaneously. Extensive-form games retain the structure of games and we study a game as a tree of possible sequences of player moves. Then a backward induction procedure (BI procedure) can be employed to effectively compute optimal strategies for players, leading to predictions of stable play by rational players. Questions of how players may arrive at selecting such strategies and playing them, and their expectations of other players symmetrically choosing such strategies are (rightly) glossed over. However, when we consider players as being decisive and active agents but who are limited in their computational and reasoning ability, the situation changes entirely. To see this, note that the BI-procedure works bottom up on the game tree, whereas strategizing follows the flow of time and hence works top down. Hence, unless a player has access to the entire subtree issuing at a node, she cannot compute optimal strategies, however well she is assured of their existence. It is in fact for this reason that though the determinacy of chess was established by Zermelo (1913) the game remains fascinating to play as well as study even today. (The notion of players whose rationality is also limited in some way is interesting but more complex to formalize; for our considerations perfectly rational but resource bounded players suffice.) Indeed resource-limited players working top down are forced to strategize locally, by selecting what part of the past history they choose to carry in their memory, and

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how much they can look ahead in their analysis. In combinatorial games, complexity considerations dictate such economizing in strategy selection. Predicting rational play by resource-limited players is then quite interesting. Knowing that other players are also similarly limited does not trivialize the problem in any way, it only leads to more interesting epistemic situations. Since each player is (symmetrically) strategizing top down, not only is the strategy of each player partial, so is her expectation of the strategies being followed by other players. Such a dependence is recursive and leads to considerable complexity in epistemic attitudes. When game situations involve uncertainty, as inevitably happens in the case of games with large structure or large number of players, such top-down strategizing is further necessitated by players having only a partial view of not only the past and the future but also the present as well. Once again, we are led to the notion of a strategy as something different from a complete plan, something analogous to a heuristic, whose applicability is dictated by local observations of game situations, for achieving local outcomes, based on expectations of other players’ locally observed behavior. The notion of locality in this description is imprecise, and pinning it down becomes an interesting challenge for a formal theory. As an example, consider a heuristic in chess such as pawn promotion. This is generic advice to any player in any chess game, but it is local in the sense that it fulfils only a short-term goal, it is not an advice for winning the game. A more interesting example is the heuristic employed by the computer Deep Blue against Gary Kasparov (on February 10, 1996) threatening Kasparov’s queen with a knight (in response to Kasparov’s 11th move). The move famously slowed down Kasparov for 27 min, and was later hailed as an important strategy (http://www.research.ibm. com/deepblue/meet/html/d.2.html). The point is that such strategizing involves more than “look-ahead.” The foregoing discussion motivates a formal study of strategies in extensive-form games, where we go beyond looking for existence of strategies for players to ensure desired outcomes, but take into account strategy structure as well. This can be carried out in two ways: one way is to consider strategies to be local partial plans on a tree. A dual approach is to keep the notion of strategy simple, but consider the game tree to be structured, and composed of many simple subgames. In this view, a strategy would be seen as a complete plan to ensure a local outcome (which may be the initiation of a desired subgame). Thus we are led to the notion of strategy composition and game composition. We present two contrasting propositional modal logics embodying the two approaches and present complete axiom systems for the logics. The former can be seen to be in the spirit of a process logic (Harel et al. 1982) and the latter in that of dynamic logic (Harel et al. 2000). Indeed, the latter logic is based on Parikh’s game logic discussed earlier. This work is placed in the context of logical studies on games (cf. see van Benthem 2003a, 2012; Harrenstein et al. 2003; Bonanno 2001). Before we proceed with the formal development, we observe that the study of strategy structure (rather than only the existence of strategies) may be relevant not

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only in games that “people play” but also in abstract game situations. For instance, consider the class of evaluation games studied by logicians (Ebbinghaus et al. 1996). Indeed, such games have inspired the idea that semantics of logics may be provided using games, thus offering an “operational” or “constructive” view of logical truth (Hintikka 1968). Here again, the truth of a formula in a structure is equated with the existence of a strategy for an associated player. However, a computational or process interpretation of such a player (representing perhaps the evaluation or model checking algorithm) would be able to determine the truth or falsity of the formula only by considering the entire evaluation tree, unless the semantics of the logic somehow can cause truth in the structure to be determined by truth in substructures. When the tree is finite and large or infinite (as in the case of fixpoint logics), bottom up constructions may not be easy or even available. Here again strategizing can be viewed top down or by bottom up composition of subgames. The work presented here is based on earlier work in games and strategies dealt with in (Ghosh 2008) and (Ramanujam and Simon 2008a, b). Ghosh and Ramanujam (2011) and Paul et al. (2015) present automata theoretic accounts of the logics discussed here.

Strategy Specifications Game logic asserts the existence of strategies that achieve specified outcomes. The question remains how a player selects a strategy, or indeed, constructs a strategy. Reasoning about strategies can be carried out in a compositional manner, much as we spoke of composing games. We conceive of strategy specifications as being built up from atomic ones using some grammar. The atomic case specifies, for a player, what conditions she tests for before making a move. These constitute positional strategies and the pre-condition for the move depends on observables that hold at the current game position and some finite look-ahead that each player can perform in terms of the structure of the game tree. One elegant method is to state these preconditions as future time formulas of a simple action indexed tense logic over the observables. The structured strategy specifications are then built from atomic ones using connectives.   Let N denote a finite non-empty set of players. Let Pi ¼ pi0 , pi1 , . . . be a countable set of observables for i  N and P ¼ Ui  N Pi. The syntax of strategy specifications is given by:
 Strat i Pi :¼ ½ψ 7! a i jσ 1 þ σ 2 jσ 1  σ 2 where ψ  BF(Pi), the Boolean closure of P. The idea is to use the above constructs to specify properties of strategies. For instance the interpretation of a player i specification [p 7! a]i where p  Pi is to choose move “a” at every player i game position where p holds. At positions where p does not hold, the strategy is allowed to choose any move that is possible at that

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node, σ 1 + σ 2 says that the strategy of player i conforms to the specification σ 1 or σ 2. The construct σ 1  σ 2 says that the strategy conforms to specifications σ 1 and σ 2. Let  ¼ {a1, . . . , am}, we also make use of the following abbreviation. – nulli ¼ [⊤ 7! a1] +    + [⊤ 7! am]. It will be clear from the semantics (which is defined shortly) that any strategy of player i conforms to nulli, or in other words this is an empty specification. The empty specification is particularly useful for assertions of the form “there exists a strategy” where the property of the strategy is not of any relevance.

Semantics
 An extensive form game tree is a tuple T ¼ S, ), s0 , ^λ where  ¼ ðS, ), s0 Þ is a tree. The set S denotes the set of game positions with s0 being the initial game position. The edge function ) specifies the moves enabled at a game position and the turn function ^λ : S ! N associates each game position with a player. (Technically, we need player labelling only at the non-leaf nodes. However, for the sake of uniform presentation, we do not distinguish between leaf nodes and non-leaf nodes as far as player labelling is concerned.)
 Let M ¼ (T, V ) where T ¼ S, ), s0 , ^λ is an extensive form game tree and V : S ! 2P a valuation function. The truth of a formula ψ  BF(P) at the state s, denoted M, s  ψ is defined as follows: – M, s  p iff p  V(s). – M, s  : ψ iff M, s ⊭ ψ. – M, s  ψ 1 _ ψ 2 iff M, s  ψ 1 or M, s  ψ 2. Strategy specifications are interpreted on strategy trees of T. We assume the presence of two special propositions turn1 and turn2 that specifies which player’s turn it is to move, i.e., the valuation function satisfies the property – for all i  N, turni  V(w) iff λ(w) ¼ i. Recall that a strategy μ of player i is a subtree of T. For a strategy specification σ  Strati(Pi), we define the notion of μ conforming to σ (denoted μi σ) as follows: – μi σ iff for all player i nodes s  μ, we have μ, si σ. where we define μ, si σ as, – μ, si [ψ 7! a]i iff M, s  ψ implies outμ(s) ¼ a.

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– μ, si σ 1 + σ 2 iff μ, si σ 1 or μ, si σ 2. – μ, si σ 1  σ 2 iff μ, si σ 1 and μ, si σ 2. Above, ψ  BF(Pi) and outμ(s) is the unique outgoing edge in μ at s. Recall that s is a player i node and therefore by definition there is a unique outgoing edge at s. A strategy logic: We now discuss how we may embed structured strategies in a formal logic. Formulas of the logic are built up using structured strategy specifications. The formulas describe the game arena in a standard modal logic, and in addition specify the result of a player following a particular strategy at a game position, to choose a specific move a. Using these formulas one can specify how a strategy helps to eventually win (ensure) an outcome β. The syntax of the logic is given by:   П :¼ p  Pðσ Þi : c :α j α1 _ α2 j hαi α j σ ⇝i β: where a, c  Σ, σ  Strati(P i), β W Bool(Pi). The derived connectives ^, and [a]α are defined as usual. Let ○α ¼ a  Σ haiα and ☉α ¼ :○:α. The formula (σ)i : c asserts, at any game position, that the strategy specification σ for player i suggests that the move c can be played at that position. The formula σ⇝i β says that from this position, following the strategy σ for player i ensures the outcome β. These two modalities constitute the main constructs of our logic. Model: As
mentioned  earlier, models of the logic are of the form M ¼ (T, V ) where T ¼ S, ), s0 , ^λ is an extensive-form game tree and V : S ! 2P is a valuation function that satisfies the condition: – For all s  S and i  N, turni  V ðsÞ iff ^λðsÞ ¼ i: For the purpose of defining the logic it is convenient to define the notion of the set of moves enabled by a strategy specification σ at a game position s (denoted σ(s)). These moves can also
be thought  of as those which conform to σ at s. For a tree T ¼ S, ), s0 , ^λ , a node s  S and a strategy specification σ  Strati(Pi) we define σ(s) as follows:



8 < fag if λ^ðsÞ ¼ i, T , sψ and a  movesðsÞ: – ½ψ 7! a i ðsÞ ¼ 0 if λ^ðsÞ ¼ i, T , sψ and a 2 = movesðsÞ: : Σ otherwise: – (σ 1 + σ 2)(s) ¼ σ 1(s) [ σ 2(s). – (σ 1  σ 2)(s) ¼ σ 1(s) \ σ 2(s). 0

a1

am1

We say that a path ρss : s ¼ s1 ) s2    ) sm ¼ s0 in T conforms to σ if 8j: 1  j < m, aj  σ(sj). When the path constitutes a proper play, i.e., when s ¼ s0, we say that the play conforms to σ. The following proposition is easy to see from the definition.

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 Proposition 1 Given a strategy μ ¼ Sμ , )μ , s0 , ^λμ for player i along with a specification σ, μi σ iff for all s  Sμ such that ^λμ ðsÞ ¼ i we have outμ(s)  σ(s).

+

For a game tree T and a node s  S, let Ts denote the tree which consists of the unique path ρss0 and the subtree rooted at s. For a strategy specification σ  Strati(Pi),
 we define T s σ ¼ Sσ , )σ , s0 , ^λσ to be the least subtree of Ts which contains the unique path from s0 to s and satisfies the property: for every s1  Sσ , a a – If ^λσ ðs1 Þ ¼ i then for all s2 with s1 ) s2 and a  σ(s1) we have s1 )σ s2 and ^λσ ðs2 Þ ¼ ^λðs2 Þ. a a – If ^λσ ðs1 Þ ¼ ι then for all s2 with s1 ) s2 we have s1 )σ s2 and ^λσ ðs2 Þ ¼ ^λðs2 Þ.

The truth of a formula α  Π in a model M and position s (denoted M, s  α) is defined by induction on the structure of α, as usual. – M, s  p iff p  V(s). – M, s  : α iff M, s ⊭ α. – M, s  α1 _ α2 iff M, s  α1 or M, s  α2. a

+

– M, s haiα iff there exists s0 such that s ) s0 and M, s0 α: – M, s  (σ)i : c iff c  σ(s). – M, s σ⇝i β iff for all s0 such that s)σ s0 in T s σ, we have M, s0 β ^ ðturni enabled σ Þ: where enabled σ ¼

 W
hai⊤ ^ ðσ Þi : a and )σ denotes the reflexive, transitive

aΣ

closure of )σ . Figure 4 illustrates the semantics of σ⇝1 β. It says, for any 1 node β is ensured by playing according to σ; for a 2 node, all actions should ensure β. The notions of satisfiablility and validity can be defined in the standard way. A formula α is satisfiable iff there exists a model M and s such that M, s  α. A formula α is said to be valid iff for all models M and for all nodes s, we have M, s  α. Axiom system: We now present our axiomatization of the valid formulas of the logic. We find the following abbreviations useful: Fig. 4 Interpretation of σ⇝iβ

1

s σ(s)a

β x

β

y

x

¬β

2 z

y

β

β

β

β

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– δσi ðaÞ ¼ turni ^ ðσ Þi : a denotes that move “a” is enabled by σ at an i node.
 – invσi ða, βÞ ¼ turni ^ ðσ Þi : a ½a ðσ ⇝i βÞ denotes the fact that after an “a” move by player i which J conforms to σ, σ ⇝i β continues to hold. – invσι ðβÞ ¼ turnι ðσ ⇝i βÞ says that after any move of ı̄ , σ ⇝i β continues to hold. The Axiom Schemes (A0) All the substitutional instances of the tautologies of propositional calculus. 1. 2. 3.

4. 5.

(a) [a](α1 α2) ([a]α1 [a]α2) (a) haiα [a]α (a) hai ⊤ ([ψ 7! a]i)i : a for all a  Σ (b) (turni ^ ψ ^ ([ψ 7! a]i)i : a) hai⊤ (c) turni ^ ([ψ 7! a]i)i : c  : ψ for all a 6¼ c (a) (σ 1 + σ 2)i : c  (σ 1)i : c _ (σ 2)i : c (b) (σ 1  σ
2)i : c  (σ 1)i : c ^ (σ 2)i : c  σ ⇝i β β ^ invσi ða, βÞ ^ invσι ðβÞ ^ enabled σ Inference rules α, α β α ðNGÞ β ½a α J α ^ δσi ðaÞ ½a α, α ^ turnι

α, α β ^ enabled σ ðInd⇝Þ α σ ⇝i β

ðMPÞ

The axioms are mostly standard, (A3) and (A4) describe the semantics of strategy specifications. The rule Ind ⇝ illustrates the new kind of reasoning in the logic. It says that to infer that the formula σ ⇝i β holds in all reachable states, β must hold at the asserted state and – For a player i node after every move which conforms to σ, β continues to hold. – For a player ı̄ node after every enabled move, β continues to hold. – Player i does not get stuck by playing σ. Note that this notion of strategy composition extends to infinite game trees as well. Such a consideration naturally leads us to think of temporal logics on game trees and incorporate strategic reasoning in them, which is the line of work initiated by Alternating Temporal Logic (ATL). For a detailed analysis, see Bulling et al. (2015). The notion of strategy composition can be seen as constituting a theory of play whereby rational players observe and respond to play, updating their strategies. These considerations are discussed extensively in papers like Bonanno (2015) and Pacuit (2015). Our point of departure is in the use of structural composition to represent such rational deliberation.

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From strategy composition to subgame composition: We have talked of local partial strategies applicable on a game tree and composing them to build a complete strategy for the entire game. A dual approach is to consider the game to be composed of subgames and combine complete strategies from each to build a strategy for the original game. Consider the game tree given in Fig. 5. The nodes are labelled with turns of players and edges with actions. For propositions p and q let the valuation function be as shown in the figure. Let the valuation at the leaf nodes satisfy the following constraints: – Proposition r1 holds at states u1, u2, u3, u4 and w1, w2, w3, w4. – Proposition r2 holds at states u2, u4, u6, u8 and w2, w4, w6, w8. – Proposition r3 holds at states u1, w1, u3, w3. Consider the following strategy specifications: – σ 1 ¼ [p ! 7 a]1. – σ 2 ¼ [q ! 7 b]1. It is easy to see that if player 1 plays action a at the root followed by action b, then she can ensure the outcome r2 no matter what player 2 does. This can be expressed in the logic as (σ 1  σ 2) ⇝ r2. This however specifies a complete strategy for player 1. Now consider the specification σ 1, this is a partial specification since it does not uniquely dictate player 1’s actions at any node other than the root node. It can be easily verified that any (functional) strategy of player 1 which conforms to σ 1 ensures the outcome r1. This can be expressed as σ 1 ⇝ r1. Subgame composition: The dual approach in strategizing is to consider games to be structured, composed of many simple subgames and to retain the functional notion of strategies. In this setting we can define when a simple atomic game h is enabled at a game position s of the game tree T. Intuitively this holds if it is possible to embed the structure h in the game tree T starting at the game position s. The formal definition is presented in section “Compositional Games.” As an example consider the atomic game tree h1 given in Fig. 6a, the game h1 is enabled at the root node of T since the structure h1 can be embedded in T starting at

Fig. 5 Game tree T

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Fig. 6 Atomic game and strategy trees

Fig. 7 Atomic game and strategy trees

the root node. There are two valid strategies μ1 and μ2 for player 1 in the game h1, these are represented in Fig. 6b, c respectively. Similarly, for the game tree h2 given in Fig. 7a, the valid strategies π1 and π2 of player 1 are given in Fig. 7b, c. Now consider the composite game h1 followed by h2 (denoted by h1; h2). This corresponds to pasting the tree structure h2 at all leaf nodes of h1. It can be easily checked that the game h1; h2 is enabled at the root node of T. The following assertion then holds: – In the composite game h1; h2, player 1 has a strategy to ensure the outcome r3. The strategy for player 1 is basically to play according to the strategy μ1 in the game h1 and π1 in the game h2. We could also consider the repetition of the game h1 twice, in which case the following assertion holds: – In the composite game h1; h1, player 1 has a strategy to ensure the outcome r2. Here player 1 needs to play according to strategy μ1 in the game h1 and π2 in h2. In other words, player 1 needs to follow the strategy which conforms to the specification σ 1  σ 2 in the game tree T.

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A logic for compositional games: The notions of composing games and reasoning about strategies in compositional games suggest a natural logical formalism. For a composite game g, let A ðgÞ denote the set of all atomic games occurring in g (the composition operators are formally defined in section “Compositional Games”). Consider the following logic: Φ :¼ p  Pj:αjα1 _ α2 j hg, η, iiα: where g is a composite game and i  N. The map η: A ðgÞ
N ! [i  N Ωi ðA ðgÞÞ specifies for each atomic game h  A ðgÞ and a player i, a functional strategy for player i in h. The construct hg, η, iiα can then interpreted as, – hg, η, iiα: In the game g, player i can ensure the outcome α by playing according to the strategies provided by the map η. Composition of game-strategy pairs: The compositional framework in itself is however more powerful and can be employed to capture the notion of strategic response of players. For instance, let h1 and h2 be the atomic games given in Figs. 6a and 7a. Let τ shown in Fig. 8 be a strategy of player 2 in the game h1 and let π2 be the strategy of player 1 (shown in Fig. 7c). Consider the following assertion: – If player 2 plays according to strategy τ in game h1 then player 1 can respond with strategy π2 in h2 to ensure the outcome r2. In terms of composition of game-strategy pairs, the above assertion can be represented as, – h(h1, τ); (h2, π2)ir2. It can be verified that the above assertion holds in the game tree T given in Fig. 5. This shows that in order to express complex strategizing notions, it is useful to compose game-strategy pairs rather than to treat game composition and strategic analysis as independent entities. We now proceed to formalize this notion of composition. In fact we work with a more general framework of game-outcome pairs. A game-outcome pair in effect defines the functional strategies which ensure the specified outcome. Fig. 8 Strategy τ for player 2

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Compositional Games For a finite set of action symbols Σ, let T ðΣ Þ be a countable set of finite extensiveform game trees over the action set Σ which is closed under subtree inclusion. That is, if T  T ðΣ Þ and T 0 is a subtree of T then T 0  T ðΣ Þ. We also assume that for each a  Σ, the tree consisting of the single edge labelled with a is in T ðΣ Þ. Let ℍ be a countable set and h, h0 range over this set. Elements of ℍ are referred to in the formulas of the logic and the idea is to use them as names for extensive-form game trees in T ðΣ Þ. Formally we have a map v: ℍ ! T ðΣ Þ which given any name h  ℍ associates a tree vðhÞ  T ðΣ Þ. The logic: Let P be a countable set of propositions; the syntax of the logic is given by: Γ :¼ ðh, βÞjg1 ; g2 jg1 [ g2 jg Φ :¼ p  Pj:αjα1 _ α2 jhg, iiα where h  ℍ, β  Bool(P) and g  Γ.
 Models of the logic are pairs M ¼ (T, V) where T ¼ S, ), s0 , ^λ is an extensiveform game tree and V : S ! 2P is a valuation function. The truth of a formula α  Φ in a model M and a position s (denoted M, s  α) is defined as follows: – – – –

M, s  p iff p  V(s). M, s  : α iff M, s ⊭ α. M, s  α1 _ α2 iff M, s  α1 or M, s  α2. M, s hg, iiα iff ∃ðs, XÞ  Rig such that 8s0  X we have M, s0 α:

A formula α is satisfiable if there exists a model M and a state s such that M, s  α. For g  Γ and i  N, we want Rig  W
2W . To define the relation formally, let us first assume that Rig is defined for the atomic case, namely, when g ¼ (h, β). The semantics for composite games is given as follows: n i – Ri g1 ; g2 ¼ ðu, X Þj ∃Y  W such that ðu, Y Þ  Rg1 and 8v  Y there existsj o Xv  X such thatðv, Xv Þ  Rig2 and [v  Y Xv ¼ X : – Rig1 [g2 ¼ Rig1 [ Rig2 :  n  n – Rig ¼ [n 0 Rig where Rig denotes the n-fold relational composition: In the atomic case when g ¼ (h, β) we want a pair (s, X) to be in Rig if the game h is enabled at state s and there is a strategy for player i to ensure the outcome β such that X is the set of leaf nodes of the strategy. We make this notion precise below. Enabling of trees: For a game position s  S, let Ts denote the subtree of T rooted at s. We say the game h is enabled at a state s if the structure ν(h) can be

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+

+

embedded in Ts with respect to the enabled actions and player labelling. Formally this can be defined as follows:
 Given a state s and h  ℍ, let T s ¼ SsM , )M , ^λM , s and νðhÞ ¼ T h ¼
 Sh , )h , ^λh , sh,0 . The restriction of Ts with respect to the game tree h (denoted Ts h) by Th. The restriction is the subtree of Ts which is generated by the
structure specified  is defined inductively as follows: T s h ¼ S, ), ^λ, s0 , f where f : S ! Sh. Initially S ¼ {s}, ^λðsÞ ¼ ^λM ðsÞ, s0 ¼ s and f(s0) ¼ sh, 0. For any s  S, let f (s) ¼ t  Sh. Let {an1, . . . , ak} be o the outgoing edges of t, i.e., aj

for all j: 1  j  k, t)h t j. For each aj, let s1j , . . . , smj be the nodes in SsM such that aj

aj

l

s ) Ms j for all l : 1  l  m. Add nodes s1j , . . . , smj to S and the edges s ) slj for all l       : 1  l  m. Also set ^λ sl ¼ ^λM sl and f sl ¼ t j . j

j

j

+

We say that a game h is enabled at s (denoted enabled(h, s)) if the tree T s h ¼
 S, ), ^λ, s0 , f satisfies the following properties: for all s  S,



– moves(s) ¼ moves( f (s)), – if movesðsÞ 6¼ 0 then ^λðsÞ ¼ ^λh ð f ðsÞÞ: For a game tree T, let Ωi(T ) denote the set of strategies of player i on the game tree T and let frontier (T ) denote the set of all leaf nodes of T. Atomic pair: For an atomic pair g ¼ (h, β) and i  N, we define Rig as follows: +

 – Riðh, βÞ ¼ ðu, XÞjenabled ðh, uÞ and ∃μ  Ωi ðT u hÞ such that frontierðμÞ ¼ X and 8s  X, s  βg: Axiom system: We present an axiomatization of the valid formulas of the logic. We find it convenient
to make use  of the following notations. We call a tree T ¼ S, ), s0 , ^λ atomic if |S| ¼ 1, i.e., the tree consists of a single node. Given an h  ℍ such that ν(h) is a non-atomic tree T and an action a  ! s0 we a denote by ha the subtree of T rooted at a node s0 with s0 ) s0 . For each a  Σ, we define trees T ia and T ιa as,
 a – T ia ¼ S, ), s0 , ^λ where S ¼ fs0 , s1 g, s0 ) s1 , ^λðs0 Þ ¼ i and ^λðs1 Þ  N: – T ιa is similar to T ia except for the player label at game position s0 where we have ^λðs0 Þ ¼ ι:
 use hia and hιa as names denoting these trees. That is, ν hia ¼ T ia and
We  ν hιa ¼ T ιa . We can then define haiα with the standard modal logic interpretation as follows:

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  – haiα ¼ turni hia , ⊤ , i α ^ turnι hιa , ⊤ , ι α :


For h  ℍ, we use the notation h√ to denote that the tree structure νðhÞ ¼  S, ) , s0 , ^λ is enabled. This can be defined as follows:

– If ν(h) is atomic then h√ ¼ ⊤. ^ – If νðhÞ is not atomic V and λðs0 Þ ¼ i then

  √ a j ⊤ ^ a j h√a j : • h ¼ turni ^ a j  movesðs0 Þ The Axiom Schemes 1. Propositional axioms: (a) All the substitutional instances of tautologies of PC. (b) turni  :turnι : 2. Axiom for single edge games: (a) hai(α1 _ α2)  haiα1 _ haiα2. (b) haiturni [a]turni. 3. Dynamic logic axioms: (a) hg1 [ g2, iiα  hg1, iiα _ hg2, iiα. (b) hg1; g2, iiα  hg1, iihg2, iiα. (c) hg, iiα  α _ hg, iihg, iiα. 4. h(h, β), iiα  h√ ^ # (h, i, β, α).
 where for any h  ℍ feel with νðhÞ ¼ T ¼ S, ), s0 , ^λ we define #(h, i, β, α) as follow:

– #ðh, i, β, αÞ ¼

8 ^α >

:V

haihðha , βÞ, iiα

a  Σ ½a hðha , β Þ, iiα

if T is an atomic game: if T is not atomic and ^λðs0 Þ ¼ i: if T is not atomic and ^λðs0 Þ ¼ ι:

Inference Rules α, α β α ðNGÞ β ½a α hg, iiα α ðINDÞ  hg , iiα α

ðMPÞ

The axioms and inference rules form an extension of the axiom system for propositional dynamic logic to trees. The difficult part is ‘pushing’ the enabling condition down into the program structure, which complicates the proof of completeness as well. The details are similar to the one in Ramanujam and Simon (2009).

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Strategy Switching and Stability We have argued that resource-limited players do not select complete strategies. Rather, they start initially with a set of possible strategies, knowledge about the game and other players’ skills. As the game progresses, they compose/switch to devise new strategies. This can be specified in a syntax for strategy specification that crucially uses a construct for players to play the game with a strategy ν1 up to some point and then switch to a strategy ν2.

Using the “test operator” ψ?Strat, a player checks whether an observable condition ψ holds and then decides on a strategy. We think of these conditions as past time formulas of a simple tense logic over an atomic set of observables. In the atomic case, ν simply denotes a partial strategy. The intuitive meaning of the operators are given as: – Strat1 [ Strat2 means that the player plays according to the strategy Strat1 or the strategy Strat2. – Strat1 \ Strat2 means that if at a history t  T, Strat1 is defined then the player plays according to Strat1; else if Strat2 is defined at t then the player plays according to Strat2. If both Strat1 and Strat2 are defined at t then the moves that Strat1 and Strat2 specify at t must be the same (we call such a pair Strat1 and Strat2, compatible). – Strat1  Strat2 means that the player plays according to the strategy Strat1 and then after some history, switches to playing according to Strat2. The position at which she makes the switch is not fixed in advance. – (Strat1 + Strat2) says that at every point, the player can choose to follow either Strat1 or Strat2. – ψ? Strat says at every history, the player tests if the property ψ holds of that history. If it does then she plays according to Strat. The following lemma relates strategy specifications to finite state transducers, which are automata that output advice. Below note that Strat is a strategy specification, a syntactic object, and μ is a (functional) strategy, defined earlier to be a subtree of the tree unfolding of the game arena. Lemma 1 Given game arena G , a player i  N and a strategy specification Strat  Ωi, where all the atomic strategies mentioned in Strat are bounded memory, we can construct a transducer A Strat such that for all μ  Ωi we have G, μStrat iff μ  Lang ðA Strat Þ. Call a strategy Strat switch-free if it does not have any of the constructs.

or the +

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Given a game arena G and strategy specifications of the players, we may ask whether there exists some sub-arena of G that the game settles down to if the players play according to their strategy specifications. (Note that a play being an infinite path in a finite graph, and by settling down, we refer to the connected component that the play is eventually confined to.) This sub-arena is in some sense the equilibrium states of the game. It is also meaningful to ask if the game settles down to such an equilibrium subarena, then whether the strategy of a particular player attains stability with respect to switching. Theorem 3 Given a game arena G ¼ ðW, !, w0 Þ a sub-arena R of G and strategy specifications Strat1, . . . , Stratn for players 1 to n, the following questions are decidable. – Do all plays conforming to these specifications eventually settle down to R? – Given strategy specifications Strat1, . . . , Stratn for players 1 to n, if all plays conforming to these specifications converge to R, does the strategy of player i become eventually stable with respect to switching? For a detailed study of strategy switching, see (Paul et al. 2009a).

Player Types For finite extensive-form games of perfect information, backward induction (BI) offers a solution that is simple and attractive as prediction of stable play. However, this critically depends on reasoning being backward, or bottom-up on the tree from the leaves to the root. In some games such as the famous example of the centipede game, this solution is somewhat counter-intuitive. In general, an extensive-form game can have several Nash equilibria apart from the one given by the backward induction solution. If this is the statement we make about the game, how does the player reason in the game? Surprise moves and forward induction: The following example (Figs. 9 and 10) is given by Perea (2010): it is a two-player extensive-form game in which the first player chooses a move a that ends the game or the move b that leads to a normal form game g1, in which the players concurrently choose between {c1, c2} and {d1, d2, d3}, respectively. The backward induction solution advises player 1 to choose a, so player 2 does not expect to have any role. But suppose player 1 chooses b and the game does reach g1. How should player 2 reason at this node? Should player 2 conclude that 1 is irrational and choose arbitrarily, or should 2 treat the subgame as a new game ab initio expecting rational play in the future? Note that player 2 can ascribe a good reason for 1 to choose b: the expectation that 2 would choose d3 in game g1. (In this case, 1 can be expected to play c2, and then player 2’s best response would be d2.)

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Fig. 9 An extensiveform game

Fig. 10 Normal form game g1

Such issues have been discussed extensively in the literature, and many resolutions have been suggested. Some of them go as follows: – Players’ actions are to be based on substantive stable common belief in future rationality (Baltag et al. 2009; Halpern 2001). – Treat the first move of player 1 as a mistake, and either ignore past information or update beliefs accordingly (Hoshi and Isaac 2011). – Players come in different types, and deviations from expected behavior are interpreted according to players’ knowledge of each other’s type (van Benthem and Liu 2004; van Benthem 2009). – Players rationalize each other’s behavior (Pearce 1984). Among these the last requires an explanation: according to this view, a player, at a game node, asks what rational strategy choices of the opponent could have led the history to this node. In such a situation, she must also ask whether the node could also have been reached by the opponent who does not only choose rationally herself, but who also believes that the other players choose rationally as well. This argument can be iterated and leads to a form of forward induction (Battigalli and Siniscalchi 1999; Perea 2010). This leads to an interesting algorithm that can be seen as an alternative to backward induction (Perea 2012). A small point is worth noting here: the way forward induction (FI) is formalized as above, both BI and FI yield the same outcome (Perea 2010) in generic extensiveform games of perfect information (where payoffs at leaves are distinct). The strategies would in general be different, and this is in itself important for a theory of play. van Benthem (2014) suggests an alternative viewpoint: rather than looking for a normal form subgame as above, he suggests that any sufficiently abstract representation of the subgame may result in FI yielding a different outcome. For instance, if the players were computationally limited, they would have only a limited view of a large subgame, and this is a very relevant consideration for a theory of play. In general, how a player reasons in the game involves not only reasoning such as the above, but also computational abilities of the player. As the game unfolds,

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players have to record their observations, and a memory-restricted player needs to select what to record (van Benthem 2011; de Bruin 2010). Aumann and Dreze (2005) make a strong case for the focus of game theory to shift from equilibrium computation to questions of how rational players should play. For zero-sum games, the value of the game is unique and rational players will play to achieve this value. However, in the case of non-zero-sum games as mentioned above, multiple Nash equilibria can exist. This implies that players cannot extract an advice as to which strategy to employ from the equilibrium values. According to Aumann and Dreze, for a game to be well defined, it is also necessary that players have an expectation on what the other players will do. In estimating how the others will play, a rational player should take into account that others are estimating how he will play. The interactive element is crucial and a rational player should then play so as to maximize his utility, given how he thinks the others will play. The strategy specifications we introduce below are in the same spirit, since such a specification will be interactive in the sense of Aumann and Dreze (2005). Players matter: van Benthem (2014) offers a masterly analysis of the many issues that distinguish reasoning about games and reasoning in games. Briefly, he points out that even if we consider BI as pre-game deliberation, there are aspects of dynamic belief revision to be considered; then there is the range of events that occur during play: players’ observations, information received about other players, etc.; then there is post-game reflection. As we move from deliberation to actual play, our interpretation of game theory requires considerable re-examination. We leave the reader to the pleasure of reading (van Benthem 2014) for more on this, but pick up one slogan from there for discussion here: the players matter. Briefly, reasoning inside games involves reasoning about actual play, and about the players involved. The standard game-theoretic approach uses uniform algorithms (such as BI and FI) to talk of reasoning during play (including the actuality of surprise moves) and type spaces encode all hypotheses that players have about each other. However, the latter is again of the pre-game deliberative kind (as in BI), and abstracts all considerations of actual play into the type space. It is in this spirit that Perea (2010) talks of completeness of type spaces for FI, whereas the van Benthem analysis is a (clarion) call for dynamics in both aspects: dynamic decision-making during play and a theory of player types that’s dynamically constructed as well. We suggest that this is a critical issue for logical foundations of game theory. A node of a game tree is a history of play, and unless all players have a logical explanation of how play got there, it is hard to see them making rational decisions at that point. The rationale that players employ then critically depends on perceived continuity in other players’ behavior, which needs to be construed during the course of play. However, while this is easily said, it raises many questions that do not seem to have obvious answers. What would be a logic in which such reasoning as proceeds during play can be expressed? What would we ask of such a logic – that it provides formulas for every possible strategy that a player might employ in every possible game? That it be expressively complete to describe the (bewildering) diversity of player types? That we may derive stable strategy profiles using an inference engine

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underlying the logic? That we discover new strategies from the axioms and inference rules of the logic? Several logics have been studied in the context of reasoning about strategic ability. van Benthem (2012) studies strategies in a dynamic logic, and in the context of alternating temporal logics, a variety of approaches have been studied (van der Hoek et al. 2005; Walther et al. 2007; Jamroga and van der Hoek 2004; Agotnes and Walther Ågotnes and Walther 2009). While these logics reason with the functional notion of strategy, a theory of play requires reasoning about the dynamics of player types as well. Logic and automata for player types: We suggest that the logical language attempt to describe a universe of constructible player types. Therefore, players in this framework are of definable type and considerations of other players are also restricted to definable types. Rationalizability becomes relative to the expressiveness of the underlying formalism; we can perhaps call this notion ‘extensive-form reasonability’. We are less interested in completeness of the proposed language here, than in expressing interesting patterns of reasoning such as the ones alluded to above. Our commitment is not only to simple modal logics to describe types, but also to realizing types by automata. By automata, we refer only to finite state devices here, though probablistic polynomial time Turing machines are a natural class to consider as well (Fortnow and Whang 1994). A number of reasons underlie this decision: for one, resource limitations of players critically affect course of play and selection of strategies. For another, automata present a nice tangible class of players that require rationale of the kind discussed above and yet restrict the complexity that human players bring in. Suprise move by an opponent is perhaps much harder for an automaton to digest than for a human player. Further, automata theory highlights memory structure in players, and the selective process of observation and update. Why is such an approach needed, or indeed relevant, considering that an elegant topological construction of type spaces is already provided by Battigalli and Siniscalchi (1999); Perea (2010) and others, with a completeness theorem as well? A crucial departure lies in the emphasis on constructivity and computability of types and strategies (rather than their existence). Moreover, if our attempt is not only to enrich the type space but also to provide explanations of types, logical means seem more attractive. The price to pay lies, of course, in the restrictive simplicity of the logics and automata employed, and it is very likely that such reasoning is much less expressive than the topological type spaces. Types as formulas: Let N denote the set of players, we use i to range over this set. For technical convenience, we restrict our attention to two player games, i.e., we take N ¼ {1, 2}. We often use the notation i and ι to denote the players where ι ¼ 2 when i ¼ 1 and ι ¼ 1 when i ¼ 2. Let Σ be a finite set of action symbols representing moves of players, we let a, b range over Σ. Strategizing during play involves making observations about moves, forming beliefs and revising them. Player types are constructed precisely in the same manner:

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– Patterns of the form ‘when condition p holds, player 2 chooses a’ are observations by player 1 and help to assign a basic type to player 2. – Such a process clearly involves nondeterminism to accommodate apparently contradictory behavior, so a player needs to assign a disjunction of types to the other. – The process of reasoning proceeds by case analysis: in situations such as x, the other player is seen to play conservatively whereas in other situations such as y, the type is apparently aggressive. Thus type construction is conjunctive as well. – The planning of a player also includes how he responds to perceived opponent strategies that lie within this plan. Therefore type definition includes such responses. – Rationalization: perceived behavior can be explained by actual play being part of a strategy that involves the future as well, and this is articulated as a belief by the player about the opponent. Moreover, such beliefs include the opponent’s beliefs about the player as well, and iterating the process builds a hierarchy of beliefs. Above, we have spoken of the type of a player as it is ascribed by the opponent. Note that the same reasoning works for ascribing types “from above” to a player. Such considerations lead us to a syntax for player types, which is again a two-level syntax as we had earlier: we have strategy specifications, and formulas from a simple action-indexed tense logic, enriched with a belief operator. In particular, we have formulas of the form Bi π@ι which is read as: i believes that the opponent is playing a strategy that conforms to π. The use of the @ symbol is to shift location to the opponent. The semantics of the belief operator is based on the rationalizability considerations discussed above. For details, please see (Ramanujam 2014). The construct Bi π@ι describes (a kind of) belief hierarchy, player i believes that opponent behavior corresponds to some complete plan π. Note that π, in turn could be referring to some type σ 0 of player i, and so on. In this sense, a player holds a belief about opponent’s strategy choices, about opponents’ beliefs about other agents’ choices, opponents’ beliefs about others’ beliefs etc. Since this is essentially how type spaces are defined, these specifications offer a compositional means for structuring type spaces. The semantics of a player type is given as a set of the player’s plan subtrees of the given game tree, based on observables. It is defined at every player i node, specifying player i’s beliefs about opponents’ strategies that could have resulted in play reaching that node. But since every opponent’s type specifies the opponent’s beliefs about others’ strategy choices, this results in a recursive structure and we can build a hierarchy of types. Note that the belief assertions can specify different strategic choices based on the past, and thus talk of how a player may, during the game, revise her beliefs, a form of dynamics. This further suggests that we wish to derive types during play. Thus, rather than types as being fixed for the class of games, we consider types as those start perhaps as heuristics, and grow during play.

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Once we have the notion of types, it induces a notion of local equilibrium as follows. Consider player 1’s response to player 2’s strategy τ: here, 1’s best response is not to τ, but to every type π that τ satisfies. Symmetrically, 2’s best response is not to a strategy μ of player 1 but to every type σ that μ satisfies. Thus we can speak of the type pair (σ, π) being in equilibrium. We merely remark on this induced notion here, one well worth developing further on in future. We have suggested that our definition of player types has been guided by concerns of constructability and simplicity. Yet, we need to discuss how types as we have defined relate to the topological type spaces considered by game theorists, especially since forward induction is justified by the completeness of such spaces. Let ðΣ Þ ¼ ðS, ) , s0 Þ be an extensive-form game. A type space over  is a tuple G ¼ (Ui, δi)i  N where each Ui is a compact topological space, representing the set of types for player i, and δi is a function that assigns to every type u  Ui and tree node s, a probability distribution δi ðu, sÞ  ΔðΩι ðsÞ, U ι Þ. Note that Ωι ðsÞ represents the set of opponent strategies that potentially reach node s, U ι ¼ П j6¼i U j is the set of opponents’ type combinations, and Δ(X) is the set of probability distributions on X with respect to the Borel σ-algebra. In game theory, type spaces are typically defined for games of imperfect information, and the definition above coincides with the standard one when the information set for every player is a singleton. A natural question arises whether the concept makes sense for games of perfect information. In the discussions on forward induction, as for instance in Battigalli and Siniscalchi (1999) or Perea (2010), the BI and FI solutions coincide for generic games, and the analysis differentiates games with nontrivial information sets. However, as van Benthem (2014) argues, there are other interpretations of forward induction that are relevant for a theory of play: when the game tree is large, a player at a tree node s may be able to reason only about a small initial fragment of the subtree issued at s, and the subsequent abstraction may be seen as imperfect information as well. We refer the reader to [van Benthem 2014] for a more detailed justification. Moreover, rationalizing by the player i does induce an equivalence relation ~i on the tree nodes in our analysis. Note the similarity of our definition of types to the standard notion, without the use of probability distributions. The use of equivalence relations between nodes is an implicit form of qualitative expectations and we choose the simpler formalism as it is more amenable to modal logics. With these observations, consider the type space ‘induced’ in our framework.
 Consider a model M ¼ (T, V ) where S, ), s0 , ^λ is an extensive form game tree and V : S ! 2P a valuation function. Then we define the logical type space over  to be a tuple L ¼ (Sati(T ), θi)i  N
, Sati(T ) is the set of player i type specifications satisfiable in the game, and θi : Sat i ðT Þ
S ! Sat ι ðT Þ was defined earlier. Recall that this set represents the beliefs of player i about the opponent implied by the type σ at the node s. Now one can see the close correspondence between the two definitions, as well as the differences. The type space G is globally defined, and can be seen as fixing an encoding of all possible beliefs of players about opponent behavior a priori. In

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contrast, the type space L has more local structure, and is crucially determined by the expressiveness of the logic. The topological structure of the type space in G is replaced by logical structure in L. For instance, the types in L are down-ward closed: if σ 1  σ 2 is a type, then so are σ 1 and σ 2; it is closed under entailment: if σ 1 is a type and σ 1 entails σ 2, then σ 2 is a type, and so on. There are other symmetries such as: if π  θi(σ, s) for some tree node s, then there exists a tree node t such that σ  θι ðπ, tÞ. Characterizing this logical structure by a completeness theorem is an important question, but we do not proceed further on this here. At this juncture, our claim to “growing” types can be explained. Consider the root node s0 in the tree T. Notice that the beliefs of the players about each other refer only to invariant properties in the game tree (as specified by the observables), and hence the only definite assertions are about the present, namely, the root node itself. However, as play progresses, we have definite assertions about the past, as well as about the choices thus eliminated, and we have sharper type formulas. This process may be understood as a construction of the type space that proceeds top-down, starting from the root node and enriching players’ beliefs based on observations as the game tree gets pruned by play. A formal characterization of this process as a recursive function on the tree is in progress, but there are many technical challenges. While this is a general picture, we focus on a specific question: does this ‘construction’ of a logical type require an unbounded amount of information? We now proceed to show that the required information is in fact finite state, and hence can be checked by an automaton. Further, we show that, in principle, a Turing machine can construct the type space. Once we consider types to be logical, a natural question is whether a given type is consistent: we want a player to be a reasoner whose reasoning is coherent. It is this question we address here. Note that a type corresponds to a set of plans and beliefs in our framework. We have spoken of a model in which a player records observations during play and rationalizes opponents’ behavior by considering what strategies might have led to opponents playing in a particular way. The meaning we offer for constructability of such a type is a finite state automaton that ‘plays out’ such plans and rationalizes course of play.

Large Games The main strand running through the discussion so far is that we have considered temporally large games (that have episodic structure). We now consider spatially large games (where the number of players is too large for rationality to be based on exhaustive intersubjectivity). Issues in games with a large number of players Game models of social situations typically involve large populations of players. However, common knowledge of rationality symmetrizes player behavior and allows us to predict behavior of any rational player. On the other hand, it is virtually impossible for each player to reason about the behavior of every other player in such games, since a player may

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not even know how many players are in the game, let alone how they are likely to play. What is the technical implication of the number of players being large? In our view, in large games, the payoffs are usually dependent on the ‘distribution’ of the actions played by the players rather than the action profiles themselves. Moreover, in such games the payoffs are independent of the identities of the players. An action distribution is a tuple y ¼ (y1, y2, . . ., y|A|) such that yi 0, 8i and PjAj i¼1 yi  n. Let Y be the set of all action distributions. Given an action profile a, we let y(a) be its corresponding action distribution, that is, y(a)(k) gives the number of players playing the kth action in A. Every player i has a rational valued function fi : Y ! ℚ which can be seen as the payoff of i for a particular distribution. Why are such distributions interesting? They are used in many social situations. For instance, recently Singapore decided to make the entire city Wi-Fi enabled. How is it decided that a facility be provided as infrastructure? Typically such analysis involves determining when usage crosses a threshold. But then understanding why usage of one facility increases vastly, rather than another, despite the presence of several alternatives, is tricky. But this is what strategy selection is about. However, we are not as much bothered about strategy selection by an individual player but by a significant fraction of the population. Similar situations occur in the management of the Internet. Policies for bandwidth allocation are not static. They are dynamic, based on studying both volumes of traffic and type of traffic. The popularity of an application like YouTube dramatically changes such usage, calling for changes in Internet policies. Predicting such future requirements is tricky, but much wanted by the engineers. Herd mentality and imitation are common in such situations. In large games, payoffs are associated not with strategy profiles, but with type distributions. Suppose there are k strategies used in the population. Then the outcome is specified as a map μ : П k(n) ! Pk, where П k(n) is a set of distributions: k-tuples that sum up to n, and P is a payoff. Thus every player playing the jth strategy gets the payoff given by the jth component specified by μ for a given distribution. Typically there is usually a small number t of types such that t < n where n is the number of players. Can one carry out all the analysis using only the t types and then lift the results to the entire game? Why should such an analysis be possible? When we confine our attention to finite memory players, for n players, the strategy space is the n-fold product of these memory states. What we wish to do is to map this space into a t-fold product, whereby we wish to identify two players of the same type. We can show that in the case of deterministic transducers, such a blow-up is avoidable, since the product of a type with itself is then isomorphic to the type. A population of 1000 players with only two types needs to be represented only by pairs of states and not 1000-tuples. But we need to determinize transducers, and that leads to exponential blow-up. So one might ask, when is the determinizing procedure worthwhile? Suppose we have n players, t types, and p is the maximum size of the state space of any nondeterministic type finite state transducer. It turns out

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(Paul and Ramanujam 2011a) that the construction is worthwhile when n > 0.693  t  π( p), where π( p) is the number of primes below p. As we are talking about large games, the inequality above can be expected to hold. In general, while we have spoken only of qualitative outcomes, and this is natural for a logical study, it makes sense to consider quantitative objectives as well, especially when outcomes are distribution determined. For infinite play, such outcomes may diverge, and we then need to consider limit-average payoffs (or other discounted payoffs). It is still possible to carry out the kind of analysis as we have discussed here, to show existence of equilibria in finite memory strategies, as for instance, in (Paul and Ramanujam 2011a). Neighborhood structures: In large games, it is convenient to think of players arranged in neighborhoods. A player strategizes locally, observing behavior and outcomes within her neighborhood, but may switch to an adjacent neighborhood. As an example, consider vegetable sellers in India. In Indian towns, it is still possible to see vegetable sellers who carry vegetables in baskets or pushcarts and set up shop in some neighborhood. The location of their ‘shop’ changes dynamically, based on the seller’s perception of demand for vegetables in different neighborhoods in the town, but also on who else is setting up shop near her, and on her perception of how well these (or other) sellers are doing. Indeed, when she buys a lot of vegetables in the wholesale market, the choice of her ‘product mix’ as well as her choice of location are determined by a complex rationale. While the prices she quotes do vary depending on the general market situation, the neighborhoods where she sells also influence the prices significantly: she knows that in the poorer neighborhoods, her buyers cannot afford to pay much. She can be thought of as a small player in a large game, one who is affected to some extent by play in the entire game, but whose strategizing is local where such locality is itself dynamic. In the same town, there are other, relatively better off vegetable sellers who have fixed shops. Their prices and product range are determined largely by the wholesale market situation, and relatively unaffected by the presence of the itinerant vegetable sellers. If at all, they see themselves in competition only against other fixed-shop sellers. They can be seen as big players in a large game. What is interesting in this scenario is the movement of a large number of itinerant vegetable sellers across the town, and the resultant increase and decrease in availability of specific vegetables as well as their prices. We can see the vegetable market as composed of dynamic neighborhoods that expand and contract, and the dynamics of such a structure dictates, and is in turn dictated by the strategies of itinerant players. When we model games with such neighborhood structures, the central question to study is that of stability of game configurations. When can we guarantee that game dynamics leads to a configuration that does not change from then on, or oscillated between fixed configurations? Do finite memory strategies suffice? What kind of game-theoretic tools are used for such analysis? Paul and Ramanujam (2011b) offer an instance, where a characterization is presented in terms of potential games (Monderer and Shapley 1996). However, the general question of what stable

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configurations are of interest and how to strategize to achieve them is of general interest, as well as obtaining bounds on when stability is attained. Dynamic game forms: Social situations often involve strategies that are generic, (almost) game-independent: threat and punishment; go with the winner / follow the leader; try to take the lead, and if you can’t, follow a leader; imitate someone you think well of; and so on. They have some (limited) efficacy in many interaction situations. But when a significant proportion of players use such heuristics, it may affect game dynamics significantly. In general, we can consider such dynamics as follows. An individual player has to make choices; making choices has a cost. Society provides choices, incurs cost to do so. Society revises choices and costs from time to time based on the history and prediction of the future. This affects individual strategies who switch between the available choices. Then the game arena is not static but changes dynamically. We can then ask several questions based on eventual patterns dictated by the dynamics: – Does the play finally settle down to some subset of the game? – Can a player ensure certain objectives using a strategy that doesn’t involve switching? – Given a sub-arena, is a particular strategy live? – Does an action profile eventually become part of the social infrastructure? – Do the rules of the society and the behavior of other players drive a particular player out of the game? Paul et al. (2009b) offer a formal model in which such questions are posed and it is shown that these can be checked algorithmically. Therefore, it is possible to compare between game restriction rules in terms of their imposed social cost. For a player, if the game restriction rules are known and the type of the other players are known then she can compare between her strategy specifications. The more general objective of such study is to explore the rationale of when and how should society intervene, and when such rationale is common knowledge among players, how they should strategize. In this sense, individual rationality and societal rationality are mutually recursive in each other, and the study of such interdependence offers an interesting challenge for logical models. The imitation heuristic: In a large population of players, where resources and computational abilities are asymmetrically distributed, it is natural to consider a population where the players are predominantly of two kinds: optimizers and imitators. Asymmetry in resources and abilities can then lead to different types of imitation and thus ensure that we do not end up with “herd behavior.” Mutual reasoning and strategizing process between optimizers and imitators leads to interesting questions for game dynamics in these contexts. Is imitation justified? We can say no since it does not achieve optimal in most cases. But we can also say yes, since it saves time, uses less resource and does not do much worse than optimal outcomes in most cases.

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The rationality (or otherwise) of imitation has been studied (though perhaps not extensively) in game theory. In (Paul and Ramanujam 2010), games of unbounded duration on finite graphs are studied, where players may have overlapping objectives, and are divided into players who optimize and others who imitate. In this setting, it is shown that the following questions can be answered algorithmically: – If the optimizers and the imitators play according to certain specifications, is a global outcome eventually attained? – What sort of imitative behavior (subtypes) eventually survive in the game? – How worse-off are the imitators from an equilibrium outcome? However, this is a preliminary study, and more sophisticated models would involve randomizing players as well as more nuanced player types in the population.

Interaction About Preferences From modelling interactions per se, in form of games and strategies, we now move on to study preferences which constitute a motive or incitement for interaction. Preferences are integral to any decision-making process, be it individual or collective. In general, an agent is said to prefer some option a over another option b if a is more desirable, advantageous, beneficial or choice-worthy than b for the agent. Thus, a motivational attitude like preference in decision-making scenarios is basically a comparative attitude. It can also be considered as an appraisal of matters of value in such situations leading to certain choices, and as such it is different from the more informational attitudes like knowledge which concern facts. These preferences are also subjective in nature as they are attributed to agents under consideration. In his seminal work (von Wright 1963) on preferences, von Wright distinguished between two kinds of preferences: extrinsic and intrinsic. The kind of preferences discussed above constitute extrinsic preferences – in his words, “a judgement of betterness serves as a ground or reason for preference.” On the other hand one might simply like one choice over another for no reason whatsoever, e.g., one might prefer tea over coffee simply because one likes tea more. An extensive amount of work has been done on analyzing both extrinsic and intrinsic preferences from the logic viewpoint. For this chapter, the focus is on the extrinsic preferences to provide an overview of the work done towards situating and modelling certain approaches on (combining) preferences in the collective decision-making scenarios. The growing importance of decision-making in AI has resulted in a significant increase in focus on representation of preferences and reasoning about preferences, where logic plays an important role. From the computational viewpoint, algorithmic and complexity considerations in various social procedures, e.g., voting (Lang 2004; Conitzer et al. 2007), fair allocation (Chevaleyre et al. 2006), and others have led to the development of computational social choice (Chevaleyre et al. 2008), a crucial area in current day research. Another relevant issue is automated learning of preferences, where the approaches for preference elicitation are quite varied,

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e.g., range from asking effective questions (Haddawy et al. 2003) to analyzing others’ preferences (Pazzani 1999). When we deal with preferences in decision-making scenarios, more often than not we consider the following questions: What are the reasons behind such preferences? Can these preferences be changed/modified in some way? For a detailed logical study that deals with such questions, see (Liu 2011). Often, interaction/deliberation between agents influences their respective preferences – the individual decisionmaking processes of the agents’ affect one another’s viewpoints towards collective decision-making and in such cases, we need ways to consider the individual changes in preferences towards attaining some uniform preferences in the group. Even without interaction, one can combine the individual preferences based on some specific rules of combination, e.g., majority rules, and we get the group preference. In what follows we provide a comparative study of some of the logical approaches which are built on similar frameworks. As discussed in the introduction, we focus our study on preference aggregation as well as deliberation about preferences. As a running example for this section, let us consider the following paradigmatic situation: Three friends Aniket, Brishti, and Chirayu would like to go to a restaurant together and they have shortlisted three restaurants. Out of these three, they all have their individual choices. Now, the question is how to come up with a single choice so that all of them can go together. We now provide a logical framework to ground our current discussion. Since the focus is on multiagent preferences, we consider A to be a finite non-empty set of agents, where |A| ¼ n 2. We first define simple preference structures as follows: Definition 1 (Preference Frame). A preference frame F is a tuple (W, {i}i  A) where (1) W is a finite non-empty set of worlds; (2) for all agents i, i  W
W is a preorder (i.e., a reflexive and transitive relation), agent i’s preference relation among worlds in W (uiv is read as “world v is at least as preferable as world u for agent i”). We define u 0 such that infαINCONS({α, : α}) ¼ k. INCONS(X) denotes the degree of inconsistency of the set X of wffs. Thus the metalogical notion turns out to be a graded one, not crisp, giving simply values 1 (yes) or 0 (no). INC1-INC3 gives generalizations of axiomatic definition of classical notion of negation inconsistency. The following theorems establish the connection between (or interconvertibility) the notions of extended graded consequence and graded inconsistency.

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Theorem 5.3 (a) Let j be a fuzzy relation from P ðFÞ to F, satisfying (GC1) - (GC5). Let INCONS(X) be a fuzzy subset of P ðFÞ defined by INCONS (X) ¼ infαgr(X j α). Then INCONS satisfies (INC1) – (INC3). (b) Let INCONS(X) be a fuzzy subset of P ðFÞ, satisfying (INC1) - (INC3). Let c  L be such that for all x  c, xmc ¼ x and for any X  F either INCONS(X) ¼1 or INCONS(X) c. Let a fuzzy relation j from P ðFÞ to F be defined as follows: gr(X j α) ¼ 1 if and only if α  X, otherwise ¼INCONS(X [ {:α}) if α does not start with a : sign ¼INCONS(X [ {β}) if α : β. Then j satisfies (GC1) - (GC5). Proof. Proof of (a): Let j be a graded consequence relation following Definition 5.1. (INC1): Let X  Y. Then by (GC2), gr(X j α)  gr(Y j α). Hence, inf grðX j αÞ  inf grðY j αÞ i.e., INCONS(X) INCONS(Y ). α

α

(INC2): From (GC3) we have the following: inf grðX [ Y j αÞm inf grðX j βÞ  inf grðX j αÞ . . .(i) α

α

βY

Also, from (GC5) for each β  Y, for some c > 0, we have grðX [ fβg j βÞm grðX [ f:βg j βÞm c  grðX j βÞ: That is, by (GC1), gr(X [ {:β} j β)mc  gr(X j β) for each β  Y. So, inf grðX [ f:βg j αÞm c  grðX [ f:βg j βÞm c  grðX j βÞ for any α

β  Y. That is, inf inf grðX [ f:βg j αÞm c  inf grðX j βÞ . . .(ii) βY α

βY

Hence using (i) and (ii), inf grðX [ Y j αÞm inf inf grðX [ f:βg j αÞm c α

βY α

 inf grðX [ Y j αÞm inf grðX j βÞ  inf grðX j αÞ α

βY

α

By definition of INCONS, we have the following: INCONS ðX [ Y Þm inf INCONS ðX [ f:βgÞm c  INCONS ðXÞ. βY

(INC3): By (GC4) there exists k > 0 such that infα, β gr({α, : α}j β) ¼ k. That is, infαINCONS({α, : α}) ¼ k. Proof of (b): With the definition of j , given in terms of INCONS, we see (GC1) follows immediately from the definition. (GC2): Let X  Y. If gr(Y j α) ¼ 1 we are done. Let gr(Y j α) 6¼ 1. Then α 2 = Y, and hence α 2 = X. So, irrespective of the structural form of α, we can write gr(X j α) ¼ INCONS(X [ {:α}) and gr(Y j α) ¼ INCONS(Y [ {:α}). By INC1, we know INCONS (X [ {:α})  INCONS(Y [ {:α}). So, gr(X j α)  gr(Y j α).

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(GC3): We want to show infβ  Y gr(X j β)m gr(X [ Y j α)  gr(X j α). If gr(X j α) ¼ 1, we are done. So, let gr(X j α) 6¼ 1. That is α 2 = X. Now two cases arise. (i) α  Y (ii) α 2 = Y. (i) Let α  Y. Then gr(X [ Y j α) ¼ 1. Hence, infβ  Y gr(X j β)m gr(X [ Y j α) ¼ infβ  Y gr(X j β) gr(X j α) [since α  Y]. (ii) Let α 2 = Y. So, gr(X [ Y j α) ¼ INCONS(X[Y[{:α}) and as α2 = X, grðX j αÞ ¼ INCONS ðX [ f:αgÞ. Now consider any β  Y. If for all β  Y, β  X, then infβ  Y gr(X j β) ¼ 1 and gr(X [ Y j α) ¼ gr(X j α) as X [ Y ¼ X. Hence the required inequality for (GC3) holds. Let for some β, β  Y and β 2 = X, and hence gr(X j β) ¼ INCONS(X [ {:β}). So, inf grðX j βÞ ¼ inf grðX j βÞ as gr(X j β) ¼ 1 for β  X. βY

β  Y, β 2 = X

Hence inf gr(X j β)mgr(X [ Y j α) βY

=X ¼ inf grðX j βÞm grðX [ Yj αÞ for β 2 βY

¼ inf INCONS ðX [ f:βgÞm INCONS ðX [ Y [ f:αgÞ βY

infβ  Y INCONS(X [ {:α} [ {:β})mINCONS(X [ Y [ {:α}) (by INC1) ¼ inf INCONS ðX [ f:αg [ f:βgÞm INCONS ðX [ Y [ f:αgÞm c βY

INCONS(X [ {:α}) (by INC2) ¼gr(X jα) [since α 2 = X]. So, combining the cases, inf grðX j βÞm grðX [ Y j αÞ  grðX j αÞ. βY

(GC4): For any β 6¼ α, : α, gr({α, : α}j β) ¼ INCONS({α, : α} [ {:β}) INCONS({α, : α}) (by INC1). Hence, inf grðfα, :αg j βÞ  INCONS ðfα, :αgÞ. β6¼α, :α

As for β ¼ α or β ¼ : α, we have gr({α, : α}j β) ¼ 1; combining all possibilities we can claim for any α, infβ gr({α, : α}j β)  INCONS({α, : α}). Hence, inf grðfα, :αgj βÞ  inf α INCONS ðfα, :αgÞ ¼ k > 0 (by INC3). α, β

So, there is some k0 > 0, such that infα, β gr({α, : α}j β) ¼ k0. (GC5): If β 2 = X [ fαg as well as β 2 = X [ f:αg, then grðX [ fαg j βÞm grðX [ f:αgj βÞ ¼INCONS(X [ {α} [ {:β})m INCONS(X [ {:α} [ {:β}) (by definition) INCONS(X [ {:β}) (by INC2 where Y ¼ {α})

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¼gr(X j β) [since β 2 = X [ fαg implies β 2 = X]. We have to now consider three cases: (i) β  X [ {α}, (ii) β  X [ {:α}, and (iii) β  X [ {α}, X [ {:α}. (i) Let β  X [ {α}. Then either β  X or β ¼ α. If β  X then the required inequality follows immediately as gr(X j β) ¼ 1. So let β 2 = X and β ¼ α. gr(X [ {α}j β)m gr(X [ {:α}j β) ¼ 1m gr(X [ {:β}j β) ¼INCONS(X [ {:β}) [since β 2 = X [ f:βg ¼gr(X j β) [since β 2 = X]. For case (ii), i.e., β 2 = X and β ¼ : α, we now proceed as follows. grðX [ fαgj βÞm grðX [ f:αgj βÞ ¼ grðX [ fαgj :αÞm 1 ¼ INCONS (X [ {α}) [since :α 2 = X [ fαg ¼gr(X j : α) [since :α 2 = X] ¼ grðX j βÞ: (iii) β  X [ {α}, X [ {:α} imply β  X. Hence gr(X j β) ¼ 1, and the inequality holds trivially. □ It has been mentioned before that GC4 and GC5 are many-valued generalizations of the rules “explosiveness condition” (or “ex falso qudolibet”) and “reasoning by cases,” respectively. We shall now discuss the semantic consequences of these conditions. In other words, it will be explained what are the results in the semantic constituents due to the incorporation of GC4 and GC5 in the metatheory of a logic. Conversely, it will also be investigated which conditions in the semantic items give rise to GC4 and GC5 in the logic. That means necessary and sufficient (both semantic) conditions will be presented here. The formal proofs are available in Chakraborty and Dutta (2019). Readers have to assume two points, viz.: (i) The logics are truth-functional, that is, the valuation functions Ti determine the value of a wff by operations on the values of its component wffs. (ii) They preserve the classical rules for the classical values 1 and 0. It is to be recalled that because of representation theorem, any graded consequence relation j can be considered as the semantic consequence relation j fT i g for some set {Ti} of valuations. It is necesarry to observe that GC4 and GC5 are principles which need to have negation (:) in the object language, and so in the algebraic structure, to interpret this language a corresponding unary operator :o is to be present. As mentioned earlier the base set for the interpretation of the object language is the same set L as that for the meta-level sentences. Thus the set L is endowed with two algebraic structures: (L, ^, _, m, !m, 0, 1), the residuated lattice for the metalanguage and (L, :0) for the object language with negation. There may be other operators in L corresponding to other operators in the object language. But for understanding the import of GC4 and GC5, those operators are not required.

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Theorem 5.4 (Necessary condition for GC4). Let a graded consequence relationj fT i g satisfy GC4. Then either no wff receives the value 0 under any Ti or for any non-zero Ti(α) ^:O Ti(α) there is a non-zero z  L such that (Ti(α) ^ :O Ti(α))mz ¼ 0. For a formal proof of the theorem, readers are referred to Chakraborty and Dutta (2019). However, we give an explanation here. Let there be a logic satisfying GC4. Let the algebraic structure for calculation of truth-values of the metalinguistic sentences be the residuated lattice (L, ^, _, m, !m, 0, 1). The object language has a negation : (along with other connectives in general) for which there corresponds an operator :o in L. The consequence relation is always definable in terms of a set {Ti} of valuation functions and hence may be taken as j fT i g . The theorem says that satisfaction of the condition GC4 by the consequence relation j fT i g implies either of the following alternatives. (i) Ti(α) 6¼ 0 for all valuation Ti and all wff α. (This is possible because Ti’s are arbitrary functions. If for some α, Ti(α) ¼ 1, then, of course, Ti(:α) ¼ 0 because Ti is classical value preserving. So, Ti(α) 6¼ 1 for any α and any Ti as well.) (ii) Ti(α) ^ Ti(:α) ¼ 0 for all α and Ti. (This is always true if Ti(α) ¼ 1 or 0, but not only so. Neither Ti(α) nor Ti(:α) may be 1 or 0, yet Ti(α) ^ Ti(:α) may be 0. This is because L is a general lattice and no condition is specified on : except that all the Ti’s behave classically on classical values.) (iii) Ti(α) ^ Ti(:α) 6¼ 0 for some Ti and α. Then the condition says that there exists an element z  L such that (Ti(α) ^ Ti(:α))mz ¼ 0. That is, some factor z pulls it down to 0. In case of classical logic z ¼ 1, and (iii) reduces to (ii). A consequence of the above theorem is that if for any collection {Ti} of valuations, the consequence relation j fT i g satisfies GC4, then for any element b  L either b ^ :o b ¼ 0 or if not, there exists a factor z 6¼ 0 in L such that (b ^ :ob)mz ¼ 0. Let us now look at GC5. A necessary condition for GC5 to hold is given in Chakraborty and Dutta (2019). We, however, are presenting another one here and proving the same. Theorem 5.5 (Necessary condition for GC5). Let the meta-level truth structure be a complete residuated lattice (L, ^, _, m, !m, 0, 1) and (L, :o) be the structure for the object language. Let for a set {Ti} of valuations j fT i g satisfy the condition GC5. Then for any wffs α, β the following holds: ðT i ðαÞ!m inf i T i ðβÞÞm ðT i ð:αÞ!m inf i T i ðβÞÞ  c!m inf i T i ðβÞ: Proof. Let for some j fT i g satisfy GC5, i.e., gr(X [ {α}j β)m gr(X [ {:α}jβ)mc  gr(X j β) for all X, α, β. Then for all X,

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 inf i

   inf T i ðγ Þ ^ T i ðαÞ!m T i ðβÞ m inf T i ðγ Þ ^ T i ð:αÞ!m T i ðβÞ m c

γX



 inf i

γX

 inf T i ðγ Þ!m T i ðβÞ :

γX

Now take X ¼ ∅. So, inf T i ðγ Þ ¼ 1. γX

So the above expression becomes inf ½ðT i ðαÞ!m T i ðβÞÞm ðT i ð:αÞ!m T i ðβÞÞm c  inf T i ðβÞ: i

i

inf ½ðT i ðαÞ!m T i ðβÞÞm ðT i ð:αÞ!m T i ðβÞÞ  c!m inf T i ðβÞ: i

i

So, inf ðT i ðαÞ!m T i ðβÞÞm inf ðT i ð:αÞ!m T i ðβÞÞ  c!m inf T i ðβÞ. i

i

i

Hence for any particular i,     T i ðαÞ!m inf T i ðβÞ m T i ð:αÞ!m inf T i ðβÞ  c!m inf T i ðβÞ. □ i

i

i

The sufficient conditions for GC4 and GC5 are established in Chakraborty and Dutta (2019). As before the metalinguistic algebraic structure is taken as (L, ^, _, m, !m, 0, 1), and the structure for object language is (L, :o) with other operators if required. Theorem 5.6 (Sufficient condition for GC4). Let (L, ^, _, m, !m, 0, 1) be a complete residuated lattice and :o be the operator for object language negation satisfying the following conditions: (i) There is some k0(6¼1), such that b ^ :ob  k0 for any b in L. (ii) If the above k0 6¼ 0 then there exists z(6¼0) such that k0mz ¼ 0. Then for any graded consequence relation j fT i gi  I with the above structure for the meta-language, GC4 holds. Theorem 5.7 (Sufficient condition for GC5). Let (L, ^, _, m, !m, 0, 1), a complete residuated lattice, be the meta-level structure and :o be the operator corresponding to the object-level negation : satisfying the following conditions: (i) There is an element c(6¼0) in L such that x  c or x > c for all x in L. (ii) x _ :ox  c for all x  L. (iii) cmz ¼ z for all z  c. Then for any graded consequence relation j fT i gi  I , with the above structure for the meta-language, GC5 holds. In the following theorem is stated a sufficient condition for both GC4 and GC5 to hold. Theorem 5.8 (Sufficient condition for both GC4 and GC5). Let the meta-level structure be (L, m, !m, 0, 1), a complete residuated lattice, and :o be the operator for object-level negation satisfying the following conditions:

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There is an element c(6¼0)  L such that for all x  L, x  c or x > c. cmz ¼ z for all z  c. x _ :ox  c for all x  L. There is an element k0(0  k0 < c)  L such that x ^ :ox  k0 for all x  L and if k0 6¼ 0 then k0m p ¼ 0 for some p(6¼0).

Then for any j fT i gi  I defined with respect to the structure (L, m, !m, 0, 1), GC4 and GC5 hold simultaneously. As mentioned in the introduction, almost in all logic practices, we actually travel to and fro between three levels of activity generally without proper attention to it. Level 2 requires a very small fragment of classical logic. During standard practices of classical logic, levels 1 and 0 are both taken to be classical. For instance reductio ad absurdum is a wff of level 0 viz., ((α ! β) ^ (α !  β)) ! α which has to be proven as a theorem. At the same time reduction is used to establish metatheorems. In case of many-valued logics, distinction of levels 0 and 1 becomes apparent; while wffs are generally assigned more than two values, the metalogical notions are only two-valued. However, it is possible to bring many-valued logics within the ambit of GCT. For simplicity let us consider the base truth set for a many-value logic as [0, 1]. Depending on the connectives present at the level 0 (object-level) language, operators are taken in [0, 1]. For example, for conjunction and disjunction, t-norms and s-norms are taken, respectively, similarly for negation and implication. A subset D of [0, 1] is fixed as the designated set such that 0 2 = D. In most cases D ¼ {1}. Semantic consequence relation is defined as follows: X α if and only if for all valuations whenever all the premises get the designated value, α gets a designated value also. It is to be marked that the above definition is crisp, two-valued, resulting in either “yes” or “no.” To place formally, let {Ti} be the set of all valuations in [0, 1]. Then X MLα holds if and only if for all Ti, Ti(X)  D implies Ti(α)  D, ML being a particular many-valued logic. The relation ML can be modeled in GCT framework in the following way.  of functions be constructed out of the collection {Ti} by the Let a collection T D i rule that for any wff x, T D i ðxÞ ¼ 1 if Ti(x)  D and 0 otherwise. Identifying the functions T D i with the set of wffs receiving the value 1, the above D D definition of ML reduces to “for all T D i , X  T i implies  D α  T i .” We now construct the graded consequence Ti which turns out to be  w.r.t. the collection    D D gr Xj fT D g α ¼ inf inf T i ðxÞ ! cT i ðαÞ where !c is the classical implicai

i

xX

tion operator defined by a!C b ¼ 1 if and only if a  b and 0 otherwise. One can check that j fT D g is a graded consequence relation which is, in fact, a crisp relation i   and X ML α if and only if gr X j fT D g α ¼ 1. i

by M^, ! c), Thus the meta-level structure of level 1 is ([0, 1], ^, !C) (denoted
D though the value obtained from inf i inf x  X T D i ðxÞ ! cT i ðαÞ lies in {0, 1}. One can say that according to the GCT framework, any many-valued logic ML is the pair (ML, M^, ! c). For more details see Chakraborty and Dutta (2019).

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A major difference of GCT with fuzzy logics lies in that while in the former the premise set X is crisp, in the latter it is a fuzzy set. However in Chakraborty and Dutta (2019, Chap. 5), GCT with fuzzy set of premises has been considered, and a comparative study of the two kinds of logic has been made. For some other significant observations, see the concluding remarks.

Solution of the Sorites Paradox: Graded Consequence vis-a´-vis Some Other Approaches It is held that the capability to solve the sorites paradox is an acid test of the acceptability of a theory of vagueness, because, except philosophers like Crispin Wright (1987), almost all persons admit that sorites paradox is genuine. It does not only baffle our senses, it raises serious questions regarding the validity of some classical logical rules and hence about the acceptability of classical logic in general. In this final section it will be discussed how a logic following the framework of GCT can handle sorites paradoxes, vis-á-vis the account of some other approaches. The sorites paradox or the paradox of the heap (“soros” means heap in ancient Greek) originated quite a few centuries back when Eubulides, the Megarian, observed that from obviously true claims that 100 million grains make a heap and that removal of one grain from a heap does not deter it from being a heap, one arrives at a false conclusion that 1 or 0 grain makes a heap by repeated application of classical logical rules. This is perplexing. There are several other versions of the paradox of the heap. Each such paradox is a chain of classically valid arguments involving some vague predicate. The extension of a vague predicate can be arranged in a series of non-discriminably non-identical objects. According to Dummett (1978), the nondiscriminability of non-identical objects is a non-transitive relation which is the source of the paradox. Let us present the structure of a sorites involving a vague predicate P, e.g., “heap.” Let further x1, . . ., xn be a portion of the extension of the predicate P, arranged gradually in a manner, where x1 is the most clear instance of P. Px1. If Px1 then Px2, ∴Px2; If Px2 then Px3, ∴Px3; ∴Pxn. Thus, starting from an obviously true premise “Px1” and a collection of conditional premises of the form “if Pxi then Pxi + 1,” for 1  i  n  1, one arrives at an obviously false conclusion “Pxn” for a suitably large n by Modus Ponens. Each conditional premise in the above sorites is supported by a principle governing the use of the predicate P0, namely, that if a vague predicate, particularly an observationally

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vague predicate, applies to an object in its extension, then it must also apply to that object which is indiscernible from it with respect to P in the same extension. Another form of the sorites contains a generalized conditional premise, and the chain of arguments proceeds by repeated application of the rule Universal Instantiation or UI and the rule Modus Ponens or MP. For an epistemicist like Timothy Williamson (1994), Roy Sorensen (1988, 2001), and others, sorites paradoxes do not pose any problem that cannot be solved within classical logic. According to them, vagueness is just a case of absence of knowledge. The so-called borderline instances of a vague predicate appear to lack any sharp boundary because we do not know where to draw a sharp line between the positive and the negative extension of that predicate. Vague predicates indeed have sharp boundaries. On this view, sorites paradoxes can be disposed of easily, as one of the conditional premises is simply false. Bivalence is preserved. However, those who think that the vagueness of a term lies in the meaning of the term itself adopts some non-classical approach to vagueness and sorites paradoxes.

Some Non-classical Approaches to the Sorites Before presenting a probable solution of sorites paradoxes following the framework of GCT, we present some other well-known non-classical approaches so that the merit of the solution offered in GCT can be judged in comparison to those other approaches.

Kit Fine’s Supervaluational Approach According to Kit Fine (1975), there is nothing wrong in the reasoning process behind the sorites paradox. The paradox can be dissolved by making classical logic adapt to a non-classical semantics. The basic assumption of Fine’s non-classical semantics for vague predicates is that the meaning of a vague predicate can be precisified by distributing its borderline instances between its positive extension and its negative extension. There may be more than one precisification for a single predicate. Any precisification can be improved till it becomes complete. Each precisification gives rise to a valuation or specification that is two-valued. A supervaluation, defined over a set of complete and admissible valuations in a given context, assigns “true” to a sentence involving a vague predicate if all members of that set assign the value “true” to that sentence, and, it assigns the value “false” if all the valuations of that set assign “false” to that sentence. According to Fine, in a sorites involving the vague predicate “bald,” there exists a hair splitting i, 0  i  1, such that “Bald (xi)” is true but “Bald (xi + 1)” is false for a complete and admissible valuation corresponding to a complete precisification of the predicate “bald.” The hair-splitting i may vary for different complete precisifications of “bald,” but such an i exists for each complete precisification. Thus on the supervaluation theory, one of the instances of the generalized conditional premise 8xi(B(xi) ! B(xi + 1)) in the sorites is false. One of the instances being false, the generalized conditional premise becomes false in the supervaluation. Consequently, the whole chain of arguments is valid in the classical sense.

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Fuzzy Set-Theoretic Approach: Goguen and Zadeh From the point of view of fuzzy logic, the key to understanding the sorites paradox lies in recognizing that a conditional can be so interpreted in a multivalent context that its value may be anywhere in the real interval [0, 1] and that Modus Ponens may not be valid in a vague context. As an example we present Goguen’s (1968) approach to sorites paradox. He however used the predicate “short,” which we follow here. According to Goguen, each conditional premise in a sorites involving the vague predicate “short” such as “if xi with height h(xi) is short, then xj with height h(xj) is short” should provide a way of computing the value of the consequent given the value of the antecedent. He interprets the above conditional premise as a fuzzy relation H(xi, xj) read as “the relative shortness of xj with height h(xj) with respect to the shortness of a man with height h(xi).” H(xi, xj) satisfies the equation: S(xj) ¼ H(xi, xj)  S(xi), where S is a fuzzy set corresponding to the predicate “short.” It follows that H(xi, xj) ¼ S(xj)/S(xi). As the fuzzy set representing the predicate “short” is continuous and monotone decreasing in nature, there is some k, such that H(xk  1, xk) is non-unit. Hence, in the sorites series, if S(x0) is 1, then Sx1000000 ¼ Π1000000 H ðxi1 , xi Þ, which is a i¼1 result of repetitive products of non-unit numbers that might be close to zero as the number of steps increases. This explains why the ultimate conclusion of the sorites appears to be false. Let us now discuss how a sorites paradox can be handled in Zadeh’s system of fuzzy set theory and approximate reasoning. Although Zadeh did not deal with sorites paradoxes directly, a sorites can be looked upon as a case of approximate reasoning in his system. A sentence like “a is bald” would be treated as providing an imprecise information regarding the number of hairs on a’s head. The sentence induces a possibility distribution function π number of hairsðaÞ : N ! [0, 1], N ¼ {0, 1, . . .. ., n}. Then the possibility that a has i hairs on his head, 0  i  n, for a suitably large n is equated to the value of the membership function μB for i, where B is the fuzzy set bald corresponding to the vague predicate “bald.” Thus, the above proposition “a is bald” is translated in Zadeh’s system by π number of hairsðaÞ ¼ μB (Zadeh 1975, 1977a, 1978). We can formulate a sorites-like paradox in Zadeh’s system of approximate reasoning with the help of the compositional rule. Let xi stand for “a man with i hairs on his head,” 0  i  n, for a suitably large n. Then, the paradox involving the vague predicate “bald” can be reframed in Zadeh’s system in the following way: x0 is bald, x0 and x1 are approximately equal in respect of the density of hairs on their heads, ∴x1 is (more or less) bald or m0 bald; x1 and x2 are approximately equal in respect of the density of hairs on their heads, ∴x2 is m1 bald; ∴xn is mn  1 bald. Here mi’s, 0  i  n  1 are different linguistic modifiers on the vague predicate “bald.”

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Every inference in the above derivation is drawing of a new possibility distribution equation from the given possibility distribution equations by the compositional rule. Thus, “x0 is bald” can be translated by: π number of hairsðx0 Þ ¼ μB, where B is the fuzzy set bald on the universe N ¼ {0, 1, . . ., n}. “x0 and x1 are approximately equal in respect of the density of hairs on their heads” can be translated by: π number of hairsðx0 Þ, number of hairsðx1 Þ ¼ μE, where E is the fuzzy relation of being approximately equal in N N. By the compositional rule, from these two possibility distribution equations, one can infer π number of hairsðx1 Þ ¼ μB∘E, where μB∘E ð jÞ ¼ sup½ min ðμB ðiÞ, μE ði, jÞÞ, j  N . The iN

linguistic retranslation of the above possibility distribution equation would be “x1 is m0 bald.” In this way, π number of hairsðx2 Þ ¼ μðB∘EÞ∘E can be inferred from μB∘E and μE by applying the compositional rule just as μB∘E has been computed above. The linguistic retranslation of the possibility distribution equation π number of hairsðx2 Þ ¼ μðB∘EÞ∘E will be of the form “x2 is m1 bald” (more or less (more or less bald)) or equivalently (more or less2 bald). If the compositional rule is applied n-times, then the ultimate conclusion of the above sorites would be an approximate linguistic translation of the possibility distribution equation: π number of hairsðxn Þ ¼ μ½ððBoEÞoEÞ...oE , the composition being done n-times. The values of the successive possibility distributions computed by the compositional rule in the above chain of inferences gradually increase, i.e., the function π number of hairsðxi Þ  π number of hairsðxiþ1 Þ pointwise. This is appropriately reflected in the gradually changing linguistic modifiers attached with the predicate “bald” in the retranslations of the successively inferred possibility distribution equations. The paradoxical nature of the sorites can be explained if the whole process is recognized as a series of inferences from fuzzy premises to fuzzy conclusions, with each successive conclusion becoming fuzzier than the preceding one. This means that the modified fuzzy predicate “mi bald” (more or 1essi + 1 bald) in a retranslation from the deduced possibility distribution equation at the (i + 1)-th stage would lessen the possibility of xi + 1 being considered as bald. In other words, the compatibility of the possibility distribution function π number of hairsðx j Þ , 0  j  n with the fuzzy set bald would tend to 0 as j tends to n. Thus the final stage inferring the predicate “mn  1 bald” gives the information that the baldness status of xn, the last individual in the sorites series, is virtually zero. Nonetheless, each move in the above chain of approximate inferences is a precise derivation in the sense that the premises of each such derivation semantically entail its conclusion provided a suitable operator is chosen to compute the value of the conjunction of two premises. According to Zadeh, a proposition P semantically entails another proposition Q if and only if the possibility distribution equation by which P is translated, say, πx ¼ μF and the possibility distribution equation by which Q is translated, say, πx ¼ μG are such that μF  μG pointwise in the universe of discourse.

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Nicholas Smith’s Approach to Sorites Paradoxes Nicholas Smith’s (2008) approach to the semantic indeterminacy caused by vagueness in language is to adopt a semantics that takes the real interval [0,1] as the value set on which various logical operations can be defined. Nevertheless, this non-classical semantics validates classical logical inferences and classical logical truths. According to Smith, α is a fuzzy consequence of a set X of formulas just in case there is no interpretation in which the value of x  X is greater than or equal to 0.5 while the value of α is less than 0.5. As per this definition, an argument is fuzzily valid if and only if it is classically valid, provided that all values in [0,1] greater than or equals to 0.5 are treated as 1 (true) and all values in [0, 1] less than 0.5 are treated as 0 (false) in the corresponding model for classical logic. Smith’s solution of sorites paradoxes relies on his interpretation of the conditional premises of a sorites. Usually, the conditional premises are understood as expressing the tolerance of vague predicates. The tolerance principle says that if two objects a and b are close (similar) to each other in P-relevant respects, where P is a vague predicate, then the claims “a is P” and “b is P” have the same truth value. On the other hand, Smith understands the conditional premises in a sorites as expressing a more general principle, namely, the principle of closeness. According to this principle, if two objects a and b are close to each other in P-relevant respects, where P is a vague predicate, then the claims “a is P” and “b is P” will be close in respect of truth-values also. Tolerance is a special case of closeness, when the truth-values of the claims concerned are identical under the same condition. On closeness interpretation, the truth-value of the consequent will be close to the truth-value of its antecedent, but not the same; it will be slightly lesser in degree. However, all the conditionals are taken to be true for all practical purposes. But each successive statement “ai is P” being slightly lesser in the degree of truth than the preceding one, the conclusion of the sorites must be false in the literal sense. Thus, on closeness reading of the conditional premises, the sorites is invalid classically, hence on Smith’s view also. Thus, according to Smith, the sorites paradox is mistaken and hence unconvincing, though it is compelling. It is compelling because, “   if we believe Closeness but not Tolerance we will thereby be licensed to accept Tolerance as a useful approximation of our real belief. Being thus accustomed to working with Tolerance as a useful approximation, we will be inclined to accept the second premise of the Sorites reasoning, saying that ‘heap’ conforms to Tolerance. That-together with the fact that the reasoning is valid   and the fact that the first premise is obviously true explains the force of the paradox.    However”,    as Smith observes, “the reasoning itself shows us that this is one of those contexts in which the approximation is inappropriate and needs to be replaced  ” (Smith 2008, pp. 169– 170). With the closeness reading of the second (conditional) premise, the argument is invalid. That explains why the paradox is mistaken.

Solution of the Sorites in GCT The way GCT addresses the issue of sorites paradox is different from both fuzzy set-theoretic approaches and the degree-theoretic approach of Smith. It has been

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observed that GCT has a mechanism to compute the degree of an inference rule, particularly of MP, which is not in general 1. It is evident that as per the ordinary understanding, in a sorites series involving the vague predicate “bald,” the degree of baldness of the first few individual ai’s is 1. The degree of baldness of the last few individual aj’s is 0. For some individual ak’s more or less in the middle of the series, this value is around 0.5. Further, according to the tolerance property of vague predicates, there cannot be a pair (ak, ak + 1) such that B(ak) ¼ 1 and B(ak + 1) < 1; similarly, it is implausible that B(ak) > 0 and B(ak + 1) ¼ 0. Following Tarski’s Convention T it can be said that “ai is bald” is true if and only if ai is bald. If T is a valuation function representing a state of affairs, then T ðαi Þ ¼ T ðai is baldÞ ¼ Bðai Þ, T (if αi, then αi + 1) ¼T (if ai is bald, then ai + 1 is bald) ¼B(ai)!oB(ai + 1), where “!o” is an operator for computing the value of the object language “if. . . then.” The paradox arises because T ðα1 Þ ¼ T ða1 is baldÞ ¼ Bða1 Þ ¼ 1 and T ðαi  αiþ1 Þ ¼ T ðIf ai is bald, then aiþ1 is baldÞ ¼ Bðai Þ!o Bðaiþ1 Þ ¼ 1 for 1  i  n  1, but T(αn) ¼ B(an is bald) ¼0. GCT provides an answer to this problem by admitting that the rule MP in this context is not of full strength; it is rather a weak rule with some grade jMP j < 1. Assuming that the value of the rule jMP j < 1, let us show how in GCT the chain of derivation gradually becomes weaker in strength and loses its credibility in yielding an acceptable conclusion (Chakraborty and Dutta 2019). Let us have a look at Table 1 where the first five steps of the sorites chain have been evaluated as per derivation principle in GCT (vide Chakraborty and Dutta 2019, p. 27). At step 3, the value of α3 would be (1m1m| MP| ) ¼ j MPj. This value represents the derivability degree of α3 from the two premises, and it is not necessarily the actual value of the sentence α3, i.e., “a3 is bald,” which is B(a3). As soundness holds for a logic pertaining to GCT, the derivability degree of the sentence and of any such sentence would be less than or equal to the actual truthvalue of that sentence. At the ultimate step, the value of the entire derivation would be |MP|n  1. This value gets transmitted to the concluding step as well but, because of soundness, is less than or equal to the actual value of the sentence αn, viz., B(an). Thus, the derivability degree of αn is |MP|n  1. If jMP j < 1, then for a sufficiently large n, |MP|n  1 would be close to 0. So, according to the mechanism of syntactic derivation within the framework of GCT, it is no longer necessary that starting off with true premises one has to reach a conclusion whose derivability degree is also 1, particularly when the derivation process is long. The advantage of the GCT model is that it keeps the paradoxical nature of the sorites alive, as the transmitted value of “an is bald” (e.g., “a man with 100,000 hairs on his head is bald”) is so very close to 0 that the language users would acknowledge it as false. Given any small ε(>0), it is always possible to have n such that |MP|n  1 < ε. Moreover, the sorites itself as a

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chain of derivations loses its credibility to yield any acceptable conclusion as it becomes weaker and weaker in strength when it becomes longer and longer. This explains the paradox. GCT provides some method of computing jMPj the strength of the rule MP in the following way. Let {Ti : i  I} be a collection of functions from the set of formulas/ sentences to [0,1] assigning values to atomic formulas/sentences, viz., αi or “ai is bald.” Then, following (A) in section “Basic Notions of the Theory of Graded Consequence”   j MP j¼ inf gr fα, α  βgj fT i gi  I β : α, β

¼ inf inf ½fT i ðαÞ ^ ðT i ðαÞ!O T i ðβÞÞg!m T i ðβÞ α, β

i

Since a!mb ¼ 1 if a  b, we need to consider only those α, β and Ti’s such that Ti(α) ^ (Ti(α)!OTi(β)) > Ti(β). . .(A1) Are there plenty of such α, β, and Ti’s in this context of sorites-like situation? The above inequality also implies that: T i ðαÞ  T i ðαÞ ^ ðT i ðαÞ!O T i ðβÞÞ > T i ðβÞ: So, we look for α, β such that Ti(α) > Ti(β). We focus on the atomic formulae only, i.e,. sentences appeared at the steps 1, 3, 5, (Table 1), and so on up to n. Since B is monotonic decreasing function, there are recognizable cases like ak and al with k < l, such that B(ak) > B(al). This function B is supposed to represent the actual membership function for baldness, which we assume to exist but not always knowable. It is not expected that Ti (ak is bald) ¼Ti(αk) is to be exactly B(ak) and Ti(al is bald) ¼Ti(αl) is to be exactly B(al), but these valuations {Ti}i  I should comply with the fact that ak is more bald than al, i.e., Ti(αk) > Ti(αl). There is further restriction to be followed. (A1) contains a component Ti(α)!OTi(β) in the left-hand side of >. Since we have not yet imposed any restriction on the object-level, !o, Ti(α) > Ti(β) does not automatically imply Ti(α)!oTi(β) 6¼ 1. So, there may be two possibilities, viz., Ti(α)!OTi(β) ¼ 1 and Ti(α)!OTi(β) < 1. In the first case, the first restriction Ti(α) > Ti(β) becomes compatible with (A1) automatically. In the second case, one needs to choose α, β such that (A1) holds. Generally the many-valued implications, which can be chosen for !o, obey the property that Ti(α)!oTi(β)  Ti(β). An implication that gives Ti(α)!oTi(β) ¼ Ti(β) cannot be chosen since in that case (A1) will fail. Hence taking into consideration the above constraints, α, β, and Ti are to be properly chosen, viz., Ti(α) > Ti(β) and Ti(α)!oTi(β) > Ti(β). With such choices of α, β, and Ti j MP j < 1. Mathematically, Ti’s are infinite in number, but, in a practical situation like the context of vagueness, each Ti represents a valuation by an “expert” in the domain concerned. The language users accept these valuations because they have confidence in experts’ opinion. Thus, in the case of the sorites paradox involving the vague predicate “bald,” an expert’s valuation (assessment) Ti assigns a value Ti(αk) to the

1: 2: 3: 4: 5:

α1 ð Bða1 ÞÞ α1  α2 ð Bða1 Þ  Bða2 ÞÞ α2 ð Bða2 ÞÞ α2  α3 ð Bða2 Þ  Bða3 ÞÞ α3 ð B ða3 ÞÞ ⋮

ðPrÞ ðPrÞ ðMPÞ ðPrÞ ðMPÞ

Table 1 Derivation of sorites chain in GCT Step value Bða1 Þ ¼ 1 Bða1 Þ!o Bða2 Þ ¼ 1 j MP j Bða2 Þ!o Bða3 Þ ¼ 1 j MP j ⋮

Value transmitted (value of derivation) 1 1  m1 1  m 1  m j MP j 1  m 1  m j MP j m 1 1  m 1  m j MP j m 1  m j MP j ⋮

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sentence αk, i.e., “ak is bald.” This value is the degree of truth of the sentence on the basis of his/her perception, experience, and other considerations. A set of such expert valuations constitutes the set {Ti}i  I in the case under discussion. In summing up, some observations can be made regarding the solution provided in GCT in comparison with the other approaches discussed in the previous section. Solution of the sorites in supervaluational semantics offered by Kit Fine rests on the possibility of drawing a boundary between the positive and the negative extensions of a vague predicate, however much flexible that boundary may be. This, as we have pointed out, defies the essential nature of vague predicates. According to Gougen’s fuzzy set-theoretic approach, some conditional premise in the sorites receives a non-unit value. Goguen’s solution fails to explain why the reasoning process behind the sorites appears to be correct, though the conclusion is unacceptable, because an argument with premises having non-unit value is to be treated as valid. On the other hand, the sorites as constructed in Zadeh’s system is a chain of approximate inferences in the sense that the conclusion inferred from the premises at each stage is approximate. It is an approximate linguistic retranslation of the possibility distribution equation deduced from the possibility distribution equations induced by the premises by the compositional rule at each stage. Such a retranslation like “more or lessn1 bald” seems to be artificial. However, Zadeh (1977a) did not consider the possibility of assigning degrees to the notion of validity or semantic entailment. In Zadeh (2009), he gave it a thought though he never developed that thought fully. Smith’s degreetheoretic approach to the sorites in effect does away with the paradox by admitting the sorites to be invalid. The solution proposed within GCT on the other hand explains the paradox by introducing grades to meta-level predicates like “   is a consequence of,” “   is derivable from,” and similar others. It explains why the sorites is appealing at the initial stages with derivations having higher values, but losing its credibility gradually as the derivations at the latter stages start having lesser values, until the final conclusion is derived with a degree of derivability that is very close to 0.

Concluding Remarks Besides the philosophical problem with sorites, GCT has the potentiality to address more down to earth issues like decision-making through interaction with several agents involving local and global logics and infomorphism. Interested readers may consult Chap. 7 of Chakraborty and Dutta (2019). In our opinion, there is ample scope for improvement and application in concrete cases of distributed systems. But the central theoretical aspect of the theory is that it reveals the underlying three levels in any logic discourse. GCT is not any particular logic system but offers a general scheme for generating logics having two different truth set structures for the objectlevel and meta-level languages. The idea is to generate different logic systems taking different pairs of algebraic structures L0 and Lm at different levels such as L0 Lm

Ƚukasiewicz Godel

Godel Ƚukasiewicz

Kleene Ƚukasiewicz

CL-algebra. . . Heyting algebra. . .

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It has also been observed (in the necessary and sufficient conditions for GC4 and GC5) that the two truth structures are to be related in special ways in order to have certain properties to be satisfied by the logic generated. For example, the necessary as well as sufficient conditions for Deduction Theorem (DT) to hold in a logic, there should be the relation a!m b  a!ob for all a, b belonging to the base set L of both the truth structures where !m and !o are respectively the implication operators of the meta and object languages respectively [for proof, see Chakraborty and Dutta (2019)]. Similar conditions are available for other logical properties. Lm algebras should be residuated lattices, but there should, in principle, be no restriction on Lo in general. As has been discussed before, many-valued logics may be seen as special cases of GCT. If by (OŁukasiewicz, MGodel) is meant a logic in which the object-level and meta-level algebras are taken to be Łukasiewicz and Godel, respectively, one can see that the above condition holds between !o and !m, and hence DT holds in the constructed logic. While in the reverse case, viz., (OGodel, MŁukasiewicz), DT does not hold. For other results of this kind, readers are referred to Chakraborty and Dutta (2019). A user of logic is at liberty to create her/his own logic as per necessity subject to certain localized constraints. So GCT may be considered as a sort of realization of Carnap’s principle of tolerance: In logic there are no morals. Evereyone is at liberty to build up his own logic, i.e., his own form of language, as he wishes. All that is required of him is that, if he wishes to discuss it he must state his method clearly and give syntactic rules instead of philosophical argumets (Carnap 2001, p. 52)

There are numerous issues and open questions in this area from theoretical angle. Those who are interested in the application of logic may also find GCT quite resourceful. Acknowledgments The authors are extremely thankful to Dr. Jayanta Sen for his technical help in finalizing the paper.

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Index

A Abādhitaviṣayatva, 262 Abduction, 357, 358 ābhāsas, 393 Abhāvavyavahāra, 15 Abhayadeva, 37 Abhihitānvayavāda, 688, 689, 693 Abhihitānvāyavādins, 689, 690 Abhinava, 378, 391 Abhinavabhāratī, 804 Abhinavagupta, 230–232, 238–247, 249, 250, 374–376, 382, 395, 397–400, 402, 404, 408, 410, 817 ātmajñāna, 814–816 Buddhist position, 238 nirodha, 809–810 nirveda, 810–814 non-dualistic Śaiva tradition, 232 rasa, 805–809 Tantrāloka, 230, 231 tantric sources, philosophical rationalisation of, 233–237 theory of causality, 250 thesis, 238 Abhyāsa, 381, 574, 575, 584 Absence (abhāva), 963 Absolute absence (atyantābhāva), 559, 560, 563 Absolute inconsistency, 1307 Absolutely negative mark, 834 Absolutely positive mark, 833 Absolute presuppositions, 521, 528, 530 Abstract algebraic logic (AAL), 1017 algebraic semantics, 1018–1020 axiomatic extension of algebraizable logics, 1020–1021 Bridge theorem, 1021 logicization of algebra, 1022

Adaptive logics, 1258 Adjunct (pratiyogin), 973–974 Adluri, Vishwa, 493 Advaita, 622 Aṟavaṇa aṭikaḷ, 46, 54, 62, 63 aḷavai (kaḷ), 45–47, 51, 55, 62, 63, 206, 222, 224 aḷavai vāti, 51 aḷavaiyiyal, 206, 217 aṉumāṉa, 206 defects and aḷavaikaḷ, 51 kāṭciyil aḷavaikaḷ, 47 kāṇṭal, 47 karuttaḷavai, 59 pramāṇas, 206 validative criteria, 47, 48 aḷaviyal aṟattoṭunilai, 598 aḷavai and ethicality, 601 aḷavai(kaḷ), 596, 597, 600, 613 aḷaviyal, 597, 598, 600, 601, 613 aḷaviyal marapu, 597, 612 analogy (uvamai/uvamam), 596, 600, 613 ethicality, 601 inference and land, 598, 599 kāṭṭal, 600 karutal, 598 logical skills, 599 logical techniques, 597 perception, 599 primal society, 598 scrutinizing situations and counseling, 598 speech-attribution context, 597 Ten kinds of aḷavai, 600 ulakurai, 598, 600, 613 Aḷavu, 597, 598, 600, 601 Aṇṇāturai ci. eṉ., 211

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1330 Āgama, 374–377, 381, 382, 384, 388, 390–396, 574 as āpti, 406–413 epistemological structure of, 412–417 source of validity, 398–401 as verbalising (śabdana)/intuitive reflecting (pratibhāna), 401–406 Āgamakāṇḍa, 382 Āgama-pramāṇa, 379, 385 Āgamic scriptures, 234 Agency, 491, 492 Agential autonomy (svātantrya), 492, 504–506 Ages of Jaina logic, 26–28 āhnika, 94–96, 99, 100, 106, 111 Ājīvika doctrine, 188 Akalaṅka, 25, 28, 32, 848, 856, 860, 865 Akama viruttam, 55 Akanāṉūṟu, 598, 611 ākāṅkṣā (expectancy), 689 Akṣapāda Gautama, 259, 696 Alaṇkāra, 698 Alexandrov topology, 1079 Algebraic logic, 987–988, 1054 Algebraic semantics, 1003, 1018–1020 Algebraic structures, 986 Algebra of matrices, 986 Algebra valued models, for paraconsistent set theory, 1270 algebra-valued model V(PS3), 1279–1283 construction and reasonable implication algebra, 1271–1275 deductive reasonable implication algebra PS3 and logic LPS3, 1276–1279 ordinals in V(PS3), 1283–1285 Alphabetical language, 573 Alternating temporal logic (ATL), 1149 Anaikāntika, 271, 272 Anaikāntika hetvābhāsa, 466 Analogical cognition, 315 Analogy, 48, 51, 52, 60, 209, 210, 214, 220, 222, 596, 597, 600, 606, 608, 705 analogical knowledge, 606 analogical language, 207 analogical reasoning, 596, 603 Ānandavardhana, 804, 817 ananugama (non-generalization), 685 Anekānta-vāda, 24, 27, 34, 184, 186, 192, 198, 200, 878–879 Anibaddha-prasiddhi, 388, 406, 408 Aniṣṭaprasaṅga, 275 Animal logic, 1115 Anirvacanīya, 194 Anitya, 658

Index Annihilationism, 163 Anselm, 496 Anselm of Canterbury, 496 Antarvyāpti, 272–273, 447 acoustic dimensions of, 457–458 concomitance, 449–453 and HD method in science, 456–457 Jaina concept of, 456 logical and methodological aspects, 459–461 and synthetic a-priori, 453–456 Anti-essentialist doctrine, 163 Anti-foundation axiom, 974 Anumāna, 26, 359, 362, 423, 424, 432, 438, 479, 619, 700, 829–833 avayavas, 468–472 definition, 440 and hetvābhāsa, 463 Indian theory of, 454 kevalānvayī, 450 roles and contributions, 444 svārthānumāna, 443 types of, 445 ur-linkage-based, 441 See also Inference Anumiti, 822 Anupalabdhi-hetu (reason based on non-observation), 13, 15, 175, 430 anupapatti, 686 Anu-pramāṇa, 822 Anuyogin, 670, 677 Anuyogita, 670 Anvaya, 20 anvayānupapatti, 687 Anvaya-vyāpti, 270, 833 Anvaya-vyatireki liṅga, 834 Anvitābhidhāna, 581–582 Anvitābhidhānavāda, 690 Anvitābhidhānavādins, 690 Anyatarābhāva, 644 Anyathānupapannatva, 427 Anyathānupapatti, 423, 426, 433 Aparokṣa-jñāna, 823 Aphorism, 152, 575, 576 aphorisms and pseudo-aphorisms, 154–156 Apoha (exclusion), 162, 176–180, 278–279 A posteriori, 497–502 Application, 832 Approximate reasoning, 1297, 1318 Approximation space, 1054 Apramā, 823 Aprayojakatva, 464–465 A priori, 495–499, 501, 502, 511, 513

Index Āptattva, 407–409 Āpti, 379, 406, 409, 410 Araṇmaṉaikkāri, 209 Āraṇyakas, 142 Arational hypnosis, 208, 209 Arcaṭa, 82–83 Arccuṉaṉ tapacu, 215, 216 Ardra-indhana-saṁyoga, 831 Argumentation, 1131 Aristotelian logic, 890, 980 Aristotelian model, 523 Aristotelian syllogism, 130, 603, 604 Aristotle, 697–699, 705–708, 1291 Arithmetical algebra, 985 Arjuna, 711 Asatpratipakṣatva, 262, 272 Asiddha, 272 Asiddha hetvābhāsa, 466 Assertion, 521, 524, 534 Associate condition (upādhi), 974 Āstika, 148 Atadvyāvṛtti, 69 Atharvaveda, 148 Āticinēntiraṉ, 52 Ātmajñāna, 814–816 Atomic game, 1151 Authentic being, 407–409 Authority, 50 Automata, 1226, 1229–1231, 1233–1235, 1242, 1247, 1250, 1251 Avacchedaka, 670 Avacchedakatva, 671 Avai, 209, 216, 220, 224 Avaktavya, 189–192, 194 Avayava, 425, 700 Avayava of an anumāna, 458, 482 Avinābhāva, 69, 72, 81, 267, 423, 426, 457, 459, 461, 480, 828 Axiomatic extension of algebraizable logics, 1020–1021 Axiomatic investigations, 1120 Axiomatic system, 901 Axiomatization, 966, 1132 Axiom of choice (AC), 1272 Axiom of regularity, 974 Axiom schemes, 1149, 1155 Axiom system, 1148–1149, 1154–1155 Ayathārtha-jñāna, 823 Ayurveda, 709 B B3, 886, 887 Backing, 522, 523, 525, 529, 533

1331 Backward induction (BI), 1143, 1157–1159 Bagchee, Joydeep, 493 Bahirvyāpti, 447, 449, 451, 452, 473 Bārhaspatya, 151 Basic region, 733 Bateson, Gregory, 510 Baumer, Bettina, 502 Bṛhad-āraṇyaka, 142, 146 Bṛhardāraṇyaka Upaniṣad, 500 Belief, 297–300 Belief hierarchy, 1161 Bhagavad Gita, 711 Bhamatī, 661 Bharata, 804, 808, 809, 814, 815, 818 Bhargaśikhā, 396 Bhartṛhari, 492, 507, 512, 572, 573, 578, 583, 584 epistemological reflections, 584–587 sentence-holism, 580–581 theory of anvitābhidhāna, 581–582 Bhāva, 804, 809, 811, 814, 816 Bhūtas or mahābhūtas, 147 Bhūyodarśana, 447, 448, 457, 458, 830 Binary -fuzzy relation, 1302 Birkhoff, G., 1003 Birkhoff’s subdirect product theorem, 1006 Blok, W.J., 1017 Bochvar logic, 886 Bochvar’s system, 1292 Bolzano, B., 985 Boole, G., 987 Boolean algebra (BA), 134, 990, 992, 1272–1275, 1285, 1306 Boolean circuits, 1194 Boolean functional synthesis, 1193 Boolean Skolem function synthesis algorithmic paradigms, 1188 applications, 1195 computation, 1196 computational hardness, 1200 functionally determined/implicitly defined variables, 1203 knowledge representation, 1206 maximal falsifiable sets of input clauses, 1204 unate variables, 1201 unification, 1191 Boole’s algebra of logic, 988–989 concerns about, 989 example of reasoning in, 989–990 limitations of, 989 modifications of, 990–992 Brāhmaṇas, 142

1332 Brahma-rāśi, 384 Brahmodyas, 142 Bridge theorem, 1021 Brihadāraṇyaka Upaniṣad, 710–712 Buddha, 141 Buddhism, 46, 50, 62, 235, 848, 859 buddhist truth, 62 and catuṣkoṭi (see Catuṣkoṭi) Buddhist Logic vs. Nyāya logic approaches to ontology and epistemology, 260 hetu-sādhya-relation, 262–264 history, 258–260 inference for others, 264–265 oneself and inference for others, 260–262 Buddhists, 379 Buddhist theory, 184, 199 C Caiva cittāntam, 210 Cakradhara, 153 Cālanīya, 559, 563, 565 Caraka Saṃhitā, 696, 698, 699, 705 and Aristotle’s Rhetoric, 706 categories, 703 evaluation of arguments, 703 fallacies and defects, 703–704 Cārvākas, 68 Cāttiyāviyā virutti, 59 Catuṣkoṭi, 70, 195, 872–873, 875, 1259 FDE, 873–875 Nāgārjuna, 875–877 paradox, 877–878 Catuṣkoṭi (tetralemma), 166 Causality, 233, 235, 237–239, 241, 243, 245–248, 250 Causation, 167 Cause as necessary condition, 17 as sufficient condition, 16 Cavil, 837 Ceviyaṟivuṟū, 209 Chakraborty, Mihir Kumar, 729 Chala, 703, 838 Ch’an, 502 Chāndogya, 142 Charvaka (Cārvāka) āstika and nāstika, 148 bhūtavāda, 149 Buddhists and Jains, 156–157 caste and gender discriminations, 157 concept of natural elements, 147

Index disappearance of, 157 doctrine of annihilation, 141 early materialism, 143–144 evidence of Upaniṣads, 146–147 exponent of bhūtavāda, 149 heterodox and heretical doctrines, 147 idealism and materialism, 142–143 lokasiddha and tantrasiddha, 149 Lokāyata, 150–152 Maṇimēkalai, 149 mythology to logical thinking, 141 pre-Cārvāka materialists, 141 pre-Vedic, Vedic, Puranic, and Local Indigenous Cults, 144–146 refuting and denouncing the materialists, 141 Satirical epigrams, 156 scepticism and agnosticism in Vedas, 144 svabhāva-as-causality, 149 Choudhury, Lopamudra, 729 Cilappatikāram, 210, 220, 221 Circularity, 278 Citrādvaita, 390 Citrarūpatā, 392 Cittācittam, 60 Civañāṉa cittiyār, 206, 217 Claim, 522, 523, 529, 530, 532, 534 Clarke, W. Norris, 497 Classe, 556 Classical consequence, 1296–1298, 1302, 1303 Classical linear (CL-) algebras, 1065 Classical logic, 1121, 1123, 1125–1127, 1133 Classical propositional logic (CPL), 995, 1278 Classical semantic consequence relation, 1302–1306 Cognisability, condition of, 16 Cognition, 683 Cognitive event ( jñāna), 962 Common experience (lokānubhava), 689 Comparative rhetoric, 697 Completion, 509 Complex causality of totality factor, 245–248 Complex numbers, 986 Composition, 544 Compositional games, 1152–1155 Compound imperatives, 904 Comprehension principle, 959, 960, 966 Conceptual graph (CG), 936–943, 950 Conclusion, 832 Concurrency theory definition, 1246 labelled posets, 1249–1250 Petri nets, 1247–1248

Index poset languages, bounded width and bounded layering, 1250 k-safe Petri nets, 1248–1249 Conditional imperative, 893, 898, 899, 905, 907 Conflict-driven clause learning (CDCL), 1212 Congruences, 1004 of finite index, 1234–1235 Congruity (yogyatā), 691 Conjunction, 541–542, 553, 554, 898–900, 908, 914 Conjunctive normal form (CNF), 1191, 1194 Consciousness, 248, 249 Consequence and inconsistency :-fragment, 1261–1264 {:, ˅, &, }-fragment, 1267–1270 {:, ˅, &}-fragment, 1265–1267 {:, ˅}-fragment, 1264–1265 Consequence axioms, 1264 Consequence operator, 1290 Consequence relations, 1012 Construction, 520, 531 Constructivism, 1122 proof-theoretic, 1127–1131 truth-theoretic, 1123–1126 Constructivist logic, 1121 Contact with wet fuel, 831 Content-validity, 921–924 Context-free grammar, 1232, 1235, 1236, 1250 Context-sensitivity, 480 Contingency, 497, 500, 511, 513 Continuum hypothesis, 1286 Contradiction, 635 Contrapositionally complemented pseudoBoolean algebra (ccpBa), 1085 Contrary probans, 9 Conversion, 543 Coreth, Emerich, 497–500 Corrector, 975 Counter-balanced reason, 635 Counter-factual reason, 827, 835 Counterpositiveness, 547–549 Counting formulae, 1228 Curatorial, 493 Cut rule, 1128, 1130 D Darśanas, 659 Datum, 523, 524 Debate, 46, 49, 50, 697, 701–703–705, 707 contentious debate, 61 Decision logic, 1056 Declaration, 19

1333 Dedekind, J.W.R., 991 Deduction, 358, 824, 825 Deduction theorem, 1017, 1278 Deductive argument, 1120 Deductive procedure, 958 Dṛṣṭānta, 700, 702, 709 Dṛṣṭāntābhāsa, 269, 270 Default reasoning, 1112 Degrees of truth, 1295, 1299 De Morgan, A., 986 De Morgan’s laws, 553–554, 959 Dennett, Daniel, 1106 design stance, 1106 intentional stance, 1106 physical stance, 1105 Deontic sentences, 891 Dependency (Upajīvakatva), 619 Dependent origination, 239 Derivability, 1069 Descartes, Rene, 496, 983 Describers, 545–546, 961 Deterministic information systems (DIS), 1055 Dharmakīrti, 69, 94, 103, 104, 146, 174–176, 259, 260, 265, 268, 270, 274, 495, 510 Arcaṭa, 82 Dharmottara, 83 Hetubindu, 79–80 Nyāybindu, 77–79 Prajñākaragupta 84 Pramāṇavārttika, 80–81 Tibetan commentators of, 85–86 Vādanyāya, 81–82 Vinītadeva, 84 Dharmakīrti’s logic vs. Diṅnāga logic anvaya, vyatireka interrelation, 269–270 avinābhāva nature, 267–268 general issue, 265–267 Parārthānumāna nature, 268–269 viruddhāvyabhicārī, 267–268 Dharmakīrti’s Nyāyabindu, 4 Dharmakīrti’s theory of inference, see Inferential reasoning Dharmottara, 83–84 dhvani, 683 Dhvanyāloka, 804 Dialectics, 598, 702 Dialetheism, 1265 Didacticism, 520 Didhiti, 664, 665 Diṅnāga, 54, 259–262, 267, 270 Dignāga, 69, 73–75, 230, 247, 510 Directed acyclic graph (DAG), 1206 Direct product of algebra, 1005

1334 Direct proof, 1128, 1130 Direct relation (sākṣsātsaṃbandha), 544 Discernment of universals, 856–859, 866 Discussion, 837 Disjunction, 542, 553–554, 898, 900, 914 Disjunctive syllogism (DS), 1257, 1261, 1264, 1265 Dissimilar case, 13 Divisions of Vāk, 584 Doctrine of the middle way, 163 Dominant strategy equilibrium, 1136 Double Boolean algebra (dBa), 1088 Double negation, 167, 675 Dravya, 184 Dravyakiranāvali, 663 Dualgruppen, 991 Dually hemimorphic semi-Heyting logic, 1016–1017 Dually quasi-De Morgan algebra, 1029 Dūṣaṇānumāna, 837 Durkheimian, 512 Dvaita school, 621, 624 Dvaitins, 835, 837, 842, 844 Dynamic logic (DL), 912 axioms, 1155 Dzogchen, 502 E Effectivity functions, 1142 Ēkāntika, 60 ekārthaka (having one śakti), 686 2-element algebra, 1024 Element-like, 971 Elenchus, 702 Eliade, Mircea, 491, 512 Elimination rules, 1127–1130 Elision, 638 Embedding, 1005 Encapsulation, 645–648 Enthymemes, 528, 532, 699, 706 Epistemic and ontic views of logic, 483 Epistemic/heuristic method, 606 Epistemology, 26, 28, 32, 38 Equational calculus, 988 Equational theory, 1031 Equi-locatable, 551 Equivalence theorem, 909 Equivalent algebraic semantics, 1019–1020 Eristic, 702 Erroneous cognitions, 823, 834 Established well-known usage, 399, 400, 403, 404, 406, 408, 417 testimonial authenticity, 409–413

Index Eternalism, 163 Ethos, 708 ētu (cause), 599 ētīṭu, 599 Eṭuttukkāṭṭu, 522, 524, 527, 529 Etymology, 377 Euclidean geometry, 994 Euler diagrams, 721 Eva (only, exactly, really), 174 Evidence, 849–856, 859, 860, 863, 864 Exclusion class, 178 Exclusive injunction, 898 Expansion based repair, 1217 Exponent, 521 Expression based on difference, 18 Expression based on similarity, 18 Extended graded consequence, 1309 Extensive-form games, 1143–1145

F Factuality bias, 458, 473, 482 Factum probandum, 534 Fallacies asiddha (unestablished), 308–309 bādha (absence of the probandum in the locus), 312–313 of probans, 7 of reason, 838 satpratipakṣa (existence of counter-thesis), 311–312 viruddha (opposed), 311 vyabhicāra (deviation), 309–311 Fallacious instance, 10 Fallacious thesis, 6 Fallacious validity, 467 Female Tamil logicians, 46 Fine, Kit, 1317 Finite-index congruence, 1234–1235 Finite sets, 1224 First Degree Entailment (FDE), 873–875, 881, 882, 885–886 First order logic, 894 Five avayavas, 264 Five-limbed syllogism, 832 Folk songs, 212–214 Folk drama, 214–216 Formal concept analysis (FCA), 1087 Formal regimentation, 960 Forward induction (FI), 1158, 1159 Free algebras, 1007 Frege, F.L.G., 555, 993

Index Fregean Comprehension Principle, 1271 Fullness, 509 Futile objections, 838 Fuzzy logic, 1297, 1298, 1316, 1318 Fuzzy relation, 1305, 1309, 1310, 1318, 1319 Fuzzy set of premises, 1316 Fuzzy set-theoretic approach, 1318–1319 G gaccha, 684 Gadādhara’s technique, 336 Gaṅgā, 687 gaṅgātīre ghoṣaḥ, 687 Galois theory, 986 gamana kriyā (action of going), 684, 690 Game logic (GL), 1141–1143 Games, 1136, 1138–1141 compositional, 1153–1155 extensive-form, 1143–1145 game logics, 1141–1143 large, 1163–1167 player types, 1157–1163 semantics, 1146–1152 strategy specifications, 1145–1146 strategy switching and stability, 1156–1157 Gandhi, Mohandas K., 494 Ganeri, Jonardon, 494 Gautama algebra based on 3-element chain, 1025 3-element chain with one additional unary operation, 1025–1026 3-element chain with two additional unary operation, 1026–1028 logic for, 1036–1039 variety of, 1023, 1029 weakening of Boolean complement, 1024–1025 Geach, P.T., 891 Geertz, Clifford, 512 Generalized rough set, 1055 Geometrico-mathematical model, 120 Gergory, D., 985 Ghaṭapaṭānyatarābhāvaḥ, 672 ghaṭatvam, 685 Gītā, 398 Givenness, 502–504 Glivenko, V.I., 1000 Gödel algebra, 1301 Goguen’s approach, 1318 Gotama, 60, 61

1335 Graded consequence, 1300–1302 classical consequence, generalisation of, 1296–1300 classical semantic consequence relation, 1302–1306 consistency and inconsistency, generalization of, 1306–1309 many-valued logic, fuzzy logic, 1291–1293 negation, in object language, 1309–1316 sorites paradox, solution of, 1320–1324 uncertainty and degrees of truth, 1293–1296 Graded inconsistency, 1309 grāma, 689 grāmam gacchati, 689, 690 Grammar, 572 corrupt word, detection of, 574 divisions of Vāk, 584 epistemological reflections, 584–587 flexibility of language, 579–580 justification of word, 574–578 language, role of grammar in, 573–574 meaningfulness of null-class, 578–579 sentence-holism, 580–581 sphoṭa theory, 582–583 spontaneous overflow of language, 580 theory of anvitābhidhāna, 581–582 Grammarians (Vaiyākaraṇas), 683 Graphic representations, 562–565 Grelling, Kurt, 903 Group preference model, 1172 Guṇa, 184, 701 Guess-check-repair paradigm, 1212, 1214, 1215, 1218 Gunakiranāvali, 663 H H3, 887 Halldén logic, 887 Hammer’s system, 727 Harappan Civilization, 144 Hare, R.M., 891, 892 Harmony, 1129–1131, 1133 Hartshorne, Charles, 496, 500 HD method, 456–457 Hemacandra, 36 Hermeneutics, 522, 536 Hetu, 75, 77, 79, 366, 367, 423–426, 700, 702, 704, 705, 711, 712, 714, 826, 832 Hetubindu, 4, 17, 79, 174 Hetucakra, 10, 175 Hetucakraḍamaru (Hetucakranirṇaya), 75, 174 Hetu (inferential sign), 163, 172

1336 Hetu (reason for the thesis), 171 Hetu-sādhya-relation, 261–263 Hetvābhāsa, 269, 838 nature of, 461–464 Prayojakatva and Aprayojakatva of, 464–465 Heyting, A., 1001 Heyting algebra, 1054, 1272–1275, 1285, 1308 Hilbert, D., 994 Hilbert-style, 1013 Hilbert-style axiomatizations, 1073 Hilbert systems, 1078 Hīnayāna, 141 Homomorphisms, 1005, 1112, 1229 automata, 1229–1231 nets, 1231–1232 transducers and realization of tree homomorphisms, 1238–1239 Hyper-class, 970 Hypothetical logic, 835–837 Hypothetical reasoning (tarka), 957 I Identification of cosmogony and teleology, 512 Iḷampūraṇar, 61 Illocutionary negation, 169 Illustration (E), 57, 832 aṉaṉṉuvayam, 58 aṉṉuvayam, 57 defective Es of the cātaṉmiyam Type, 58, 59 defective Es of the vaitaṉmiyam type, 59, 60 Immediate cognition, 823 Imperative logic concept of validity, 916–921 conditional prescription, 912, 913 content-validity, 921–924 in Indian philosophy, 894–900 possibility of imperative inference, 891–893 prescription, 913 satisfaction, violation and avoidance, 913 unconditional prescription, 912–914 in Western tradition, 900–912 Implicative logics, 1014–1017 dually hemimorphic semi-Heyting logic, 1016–1017 examples of, 1016 Imposed property (upādhi), 963 Incompatibility principle, 16 Incomplete information system (IIS), 1056 Inconclusive probans, 7 Inconsistency axioms, 1264, 1269 Indian Logic, 363 modern, 128–135 negation, 123–126

Index principle of non-contradiction, 127–128 structure of inference, 121–123 vs. western logic, 118–121 Indian philosophy, study of imperatives in, 894–900 Indian syllogism, 130, 533 Indian syllogistic text, 525 Indigenous non-religious logic, 46 Indirect relation (paraṃparāsaṃbandha), 544 Indiscernibility, 1054 Indology and Sanskrit studies, 492 Indraśatru, 574 Induction, 358 Inductive process, 958 Inferable entity, 13 Inference (anumāna), 26, 28, 33, 36, 303–306, 596, 600, 607, 619, 689, 826–827, 956 dependence on perception and verbal testimony, 621 dependency on perception and verbal testimony, 619 fallacies, 307–313 perception as supportive (Upajīvya) for, 621 rules, 1149, 1155 and tarka, 313–315 valid and invalid, 306–307 Inference-for-the-sake-of-others (parārthānumāna), 503–505 Inferential cognition, 235, 822 concept of, 825–826 constituents of, 827–829 Inferential power, 599 Inferential reasoning canonical form of, 849–850 Dharmakīrti’s theory of ontic foundations, 859–860 evolution of, 850–852 Jaina amendments to Dharmakīrti’s theory, 860–862 negative inferential statements, 862–865 Prabhācandra, 855, 858 Information-content (zero/non-zero), 455, 471 Information system (IS), 1056 Information system algebras, 1090 Injunction, 898 Input-output separation, 1213 Instance, 5 Instrumentalism, 165 Intelim rules, 1129 Intensional character, 961 Intentionality (tātparya), 551, 692 Intermediate algebras IA1-IA3, 1071 Internal perception, 150 Intrinsic property, 163

Index Introduction rules, 1129, 1130 Intuitionistic linear logic (ILL), 912 Intuitionistic propositional logic, 1000 Intuitionistic set theory, 1271 Intuitionist logic, 1123, 1127, 1131, 1133 Intuitive induction, 453, 455 Invalid inference, 291, 306 Invariable relation, 824 Inverse translation, 1018 Īśā Upaniṣad, 142 Isomorphism, 1005 Iterated absences, 549–553 J Jaina epistemology, 184 Jaina logic, 121, 128 Abhayadeva, 37 ages of, 26 Akalaṅka, 32–34 anekānta-vāda, 24 Hemacandra, 36 Kundakunda, 29–30 Māṇikyanandin, 36 Mallavādin, 34 Prabhācandra, 36–37 Samantabhadra, 30–31 Siddhasena Divākara, 32 syād-vāda, 24–26 Umāsvāti, 31–32 Vādidevasūri, 37 Vidyānandin, 34–36 Yaśovijaya, 37–38 Jainism, 46, 48, 50, 52, 62, 184, 848, 859, 866 and Advaita-vedānta, 194–195 and Buddhism, 195 and saptabhaṇgī (see Saptabhaṇgī) Jalpa, 702, 837 Janaka, 712 Janya-pratyaksha, 676 Jāti, 838 Jātiśaktivāda, 685 Jātiśaktivādins, 685 Jātiśaktivādis, 685 Jātivādins, 686 Jayanta, 153 Jayatilleke’s interpretation, 168–169 Jesus, 494 Jevons, W.S., 991 Jñānaśrīmitra, 86–87 Jnānātmaka, 440, 444, 446 Jónsson’s theorem, 1010 Jorgensen, Jorgen, 901 Justification, 521 Justification of logic, 1132

1337 K K3, 881, 882, 886, 887 Kāṭci, 206, 610–613 otta kāṭci, 206, 207, 209, 210, 221, 224, 596, 611 Kāṇṭikai, 47, 48, 52, 59, 60, 62, 63, 206, 216, 217, 222, 224, 596, 603, 608, 610, 613 absolute presupposition, 530 acceptability, 529 analogical reasoning, 522, 523, 528, 531 application, 220, 224, 603, 607, 609, 612 argumentative text, 601, 603 col, 217 conclusion, 216–218, 220, 224 cūttiram, 604, 605, 608, 612 didactic composition, 532 didactic texts, 526 disputations, 529 ēṟṟal, 217 eṭuttukkāṭṭu and comparison, 606 eṭuttukkāṭṭu (illustration), 603, 606, 608, 609, 612, 613 enthymemes, 528, 532 ētu (reason), 599, 600, 604, 606, 607, 609, 612, 613 five members, 605, 606, 608 form of argument, 524, 529–533 iḷampūraṇar and kāṇṭikai, 605 Illustration, 211, 224, 522, 525, 531 inductive-deductive, 525, 534 kāṇṭikai as commentary, 604 kōḷ, 602, 605, 613 linking axiom, 606 Literary and commentatorial tradition, 603 logical text, 612 muṭipu (conclusion), 603, 608, 609, 612, 613 mukkūṟṟu muṭipu, 603 naṭai (conclusion), 217, 603, 607, 609, 613 non-eternity, 525 pakkam, 604 parts, 522 philosophy and rhetoric, 596 premise, 525, 526, 531 proposition, 211, 213–215, 217, 220, 224, 602, 604, 609, 611 proverbs kāṇṭikai, 612 reason, 211, 213, 215, 217, 220, 224 rhetorical (logic), 602 structure, 523, 526, 528, 534 syllogism, 522 syllogistic text, 526, 528 terms, 525, 526 30th aphorism, 532 upanayam, 608

1338 Kāṇṭikai (cont.) validity, 528 warrant, 522, 523, 526 Kaṭha Upaniṣad, 142, 145 Kalittokai, 601, 603, 610 Kalmer’s method, 1277 Kant, Immanuel, 495–498 Kantian, 495–498, 501 Kāraṇānupalabdhi, 15 Kārāṇatā, 668 Kāraṇa-viruddha-kāryopalabdhi, 15 Kāraṇa-viruddhopalabdhi, 15 Karmā, 701, 712 karmatvam, 689, 690 Kārya-hetu (reason qua effect), 13, 175 kāryakāraṇabhāva, 233 Kāryānupalabdhi, 15 Kārya-viruddhopalabdhi, 15 Kashmir Śaivism, 403 Kautsya, 145 Kevala-anvayi liṅga, 833 Kevalānvayī anumāna, 450 Kevalānvayī hetu, 262 Kevalānvayin, 649–650 Kevala-pramāṇa, 822 Kevalavyatirekī hetu, 262 Kevala-vyatireki liṅga, 834 Kleene algebra, 1025, 1057, 1059–1061 Kleene’s system, 1292 Knowledge, 142, 823 Knowledge representation (KR), 933, 936 Kōḷ, 534, 535 Kolmogorov, A.N., 1000 Kripal, Jeffrey J., 494 Kripke model, 1059 Kripke semantics, 1056 Krishna, 711 kuṟuntokai, 596, 599, 600 Kumāra Kassapa, 156 Kundakunda, 28, 29, 31, 33 L lakṣaṇā, 686, 687 Language, 1004 recognition, 1198 relevance of, 1116 Lattice, 1272 theory, 991 Law of Double Negation, 1127 Law of excluded middle, 1132 Law of identity, 55 Law of non-contradiction, 166

Index Law of the excluded middle, 166 Lawrence, David, 238 Laws of logic, 276, 277 Laws of reasoning, 598 Lawvere theory, 1251 Left-restriction, 544–546 Leibniz, G.W., 984 Leibniz’s law of indecernibility of identicals, 1279 Lewis, C.I., 998 Lewis, David, 1116 L-fuzzy sets, 1301 Limited (avacchinna), 547 Limiting condition, 831 Limiting relation, 548 Limitors (avacchedaka), 546–549 Lindenbaum-Tarski algebra, 1057 Lindenbaum-Tarski method, 1003 application to classical propositional logic, 1013–1014 Local equilibrium, 1162 Locana, 810 Locusness, 545–547 Logic, 224, 228, 229, 598, 864, 1012 alogicality, 212 calculative logic, 206 courtship, 212 dialogically rhetorical, 225 divisive logic, 209 end of logic, 221, 223 epistemic aspect of logic, 206 epistemic (logic), 601 in India (see Indian logic) logic in tangible qualities, 206 monologically rhetorical logic, 224 reasoning praxis, 206 rhetorical aspects of logic, 206 Logical consequence, 1003 Logical discourse, 1290–1291 Logical inference, 893 Logical possibilities, 169 Logical reductionism, 60 Logical type space, 1162 Logical words, 1120–1123, 1126–1132 Logicization of algebra, 1022 Logic of diagrams Euler diagrams, 721 Hammer’s system, 727 Shin’s system, 727 spider diagram, 728 square of opposition, 788 Venn diagram, 724, 732 Venn-Peirce diagrams, 725

Index Logic of formal inconsistency (LFI), 1258 Lokaprasiddhi, 397, 408 Lokāyata, 141 Lokottara, 806, 807, 813, 816, 818 Lonergan, Bernard, 497–499, 501 Łoś’s theorem, 1010 Lower approximation, 1055 LP, 881, 882, 887 Łukasiewicz, J., 998 Lukasiewicz algebra, 1301 M Madhusūdana Sarasvatī, 148 Mādhyamika, 163, 495 Māṇikyanandin, 26, 28, 36, 38, 848, 849, 851, 853–855, 860, 863 Maṇimēkalai, 46, 51, 55, 62, 522, 524, 600, 604, 606, 608 Mahābhāşya, 574, 635–640 Mahājanaprasiddhi, 383, 397, 398, 408 Mahāvīra, 141 Mahāyāna, 141 Maitrī Upaniṣad, 142, 151 Mal’cev, A.I, 1004 Mal’cev’s theorem, 1011 Mālinīvijayottaratantra, 230, 231 Mallavādin, 34 Mantramārga traditions, 230 Mantras, 142 Many-valued logics, 874, 996–998, 1291, 1292, 1297, 1315, 1325 Matam, 605 Materialism, 143 Mathematical logic, 993 Matilal, B. K., 266, 278 Matiyasevich-Robinson-Davis-Putnam (MRDP) theorem, 1190 Maximal falsifiable subset (MFS), 1204–1206 Maximal saitsifable subset (MSS), 1204–1206 Māyā Śakti, 505, 510 māyūram vētanāyakam piḷḷai, 212, 218, 220 Mbiti, John S., 501 Medhatithi Gautama, 696, 709 Mediate cognition, 823 Mēṭait tamiḷ, 211 Mēṟkōḷ, 522, 529, 534, 535 Meiners, Christoph, 493 Menger, Karl, 902 Mīṁāmsa Inspired Representation of Actions (MIRA), 897, 900, 915 Mill, J. S., 173 Mīmāṃsā, 381

1339 Mīmāṃsakas, 379, 384, 685, 686, 692 Mīmāṁsā system, 894, 898, 912 Minimal region, 733 Modal logic, 998 Modern algebraic logic, 1002 Modern deductive logic, 1120 Modern Indian logic, 128–135 Modern semantics, 991 Modus ponens (MP), 1267, 1268 Mokṣa, 658, 701 Mokṣākaragupta, 88, 275 Monadic second-order logic, 1226, 1228, 1230, 1235 Monism, 503 Monotonic logic, 366, 367 Muṭipu, 522, 529 Mūlamadhyamakakārikā, 162 Mūlamādhyamikakārikā, 238 Multivalued logic, 45, 46, 54, 62 Mutual absence (anyonyābhāva), 540–541 Mutumoḻi, 214, 221, 222, 224, 530, 608, 610, 612, 613 enthymeme, 608, 609 implicit reason, 608, 609 nayam, 609, 610 niyāyam, 609, 610 proverb and preliterate societies, 609 N Naṭai, 52, 522–524, 526, 527, 529 Naṉṉeṟi, 521 Naṟṟiṇai, 599, 603 Naṉṉūl, 210, 604 Naṭu, 60 impartiality, 60 Nāgārjuna, 70–72, 238, 246, 259, 275–278, 495, 875–877, 878 Nāgārjuna’s prasaṅga technique, 162 Naiyāyikas, 172, 661, 683, 685–687, 689, 690, 692, 832, 837, 842 nānārthakas (polysemous), 686 Nash equilibrium, 1136, 1140 Natural language instruction, 912 Natural numbers, 554 Natural potency (Jātyā Prābalya), 623–624 Navadvipa School, 665 Nāvalar, 211, 212 cōmacuntara pāratiyār, 211, 212 na. mu. vēṅkaṭacāmi nāṭṭār, 211 Navya-Nyāya (NN), 665, 666, 698, 932, 933 advantages, 676, 677 avacchedaka, 670

1340 Navya-Nyāya (NN) (cont.) categories, 933–935 cause-effect relation, 652–654 complex absence, 644–645 computational parsing, 941 conceptual graphs, 937, 949 constituency parser for, 946–949 double negation, 675 encapsulation (anugama), 645–648 fallacies of reason, 635–640 four-fold classification, 674, 675 inherent cause and efficient cause, 640–642 interpretation of Sūtra “rucyarthānāṁ prīyamāṇaḥ”, 651–652 kevalānvayin, 649–650 kinds of absences, 671–673 kinds of relations, 673, 674 language, 129 negation of absence, 642–644 Nirūpya-Nirūpakabhāva relation, 667, 668 non-difference, 650–651 pervaded and pervading entities in invariable pervasion, 640 pratiyogin, 669 relation of identity, 648–649 segmenter for, 946 Navya-Nyāya, problems and issues composition of relation, 544 conjunction property, 541 conversation of relation, 543 De Morgan’s laws, 553 describers, 545 disjunction, 542 graphic representation, Toshihiro Wada, 562 iterated absences, 549 limitors (avacchedaka), 546 mutual absence (anyonyābhāva), 540 negation property, 540 number words reference, 554 pervasion, definition, 557 relational absence (saṃsargābhāva), 541 Sheffer stroke, 542 Navya-Nyāya, theoretic framework for formal reconstruction Bealer’s calculus, 964–966 epistemological presuppositions, 962–963 inference theory, 956–959 naive property abstraction, 966–968 ontological presuppositions, 962 ST2 style extension, 969–972 technical language, 959–961 UD position, 974–975

Index Varadarāja and Maheśa Chandra position, 972–974 Zermelo-Russell’s antinomy, 968 Nayam, 46, 49, 52, 54, 60, 63, 222–224, 606, 607, 609 analogy and nayam, 52 nayap piramāṇaṅkaḷ, 48 niyāyam, 607, 609, 610 Naya-vāda, 25, 31, 185 NBG, 969 Necessity, 853, 861, 863 Negation, 123–126, 319–323, 540, 578, 606, 849, 862–864, 908, 914 Negation-free fragment (NFF), 1271, 1275, 1283 Negative concomitance, 20 Negative pervasion, 20 Neighbourhood structures, 1165–1166 Nemec, John, 492 Neoplatonic, 496 NFU, 971 Nibaddha-prasiddhi, 406 Niḥsvabhāvatā, 275 Nigamana, 19, 171, 700, 832 Nigrahasthāna, 838 Nikaṇṭam, 51 Nikamaṉam, 52, 523, 524, 527, 531 Nīlakēci, 45, 46, 48, 50, 59, 62, 63, 604 aḷavaikaḷ, 47 debate, 49, 51 debate between picācakaṉ and nīlakēci, 49, 50 Jaina preceptor, 47 kāṇṭikai, 59 nayam, 47, 48 uttikaḷ, 49 validative criteria, 46, 47, 49 Nimittakāraṇam, 522 Nirdośopapattiḥ, 827 Nirodha, 807, 809–810 Nirūpya-Nirūpakabhāva relation, 667 Nirveda, 810–814 Niścitam (determinately, necessarily), 175 Nitya, 658 Niyama, 427 Niyati Śakti, 511 Niyāyam, 52, 60–62, 206, 221, 222 antakaja niyāyam, 48 NN expressions, see Navya-Nyāya (NN) Non-alphabetical language, 573 Non-apprehension, 14, 18 principle, 16 Non-being (abhāva), 963

Index Non-Boolean algebra, 1271 Non-classical logics, 870, 883, 996–1002, 1297 Non-cognitivism, 909 Non-contradiction, 127–128 Non-denoting term, 560–562 Non-deterministic information system (NIS), 1056 Non-dualism, 96–98, 503 Non-dual metaphysics, 232 Non-dual Śaivism, 98–111 Non-explosion, 1257, 1261–1264, 1266 Non-monistic non-duality, 60 Non-monotonic logic, 354, 363, 366, 367 Non-numeric objects, 986 Non-pervasive absence, 559 Non-qualificative cognition, 284, 285 Non-reductionist approach, 901 Non-reductionist systems, 906 Non-referring expressions, 165 Non-sequential process, 1249 Nonterminals, 1235 Non-unidirectional types, 56 Non-well-foundedness, 972 Normative sentences, 891 Number words, 557–565 Numeric algebra, 983 Nyāya, 48, 60, 222, 659, 698–699, 714 Aṣṭāvakragītā, 712 Bhagavad Gita, 711–712 Brihadāraṇyaka Upaniṣad, 710, 712 Buddhists, 666 concept of pramāṇa, 696 Gaṅgeśa, 667 Grammarians, 666 human error, 660 Indian writers, 660 method of reasoning, 699 methodology, 662 Mithilā region, 664 monoism, 666 new school, 663 non-existence/non- availability, 660 old school, 660 Sūtra, 660 Nyāya Bhāṣya, 359–361 Nyāyabindu, 13, 176 Nyāyakandalī, 662 Nyāya logic, 120 Nyāya logical thought, 347–349 ādhāra – ādheya, 340–341 analogy/comparison, 315–316 anuyogī – pratiyogī, 339–340 belief, 297–300

1341 cause, 292–297 cessation of objectionable questions, 292 cognition to objects, 289 elements of cognition, 290 inference, 303–315 justification, 291–292 memory-context, 291 negation and classification, 319–323 non-qualificative cognition, 284, 285 objects of cognition, 289, 290 pakṣa – sādhya, 344–346 perception, 301–303 principle of contradiction, 328–332 qualificative cognition, 284, 285, 288 sources of knowledge, 300–301 theory of definite descriptions, 332–339 uddeśya – vidheya, 343–344 universal quantifier, 324–328 verbal cognition (testimony), 316–319 viśeṣya – prakāra, 342–343 viśeṣya – viśeṣaṇa, 341–342 Nyāyamañjarī, 662 Nyāya method, 697, 699–702, 705, 706, 708, 709, 711, 714 Nyāyamukha, 74–75 Nyāyapraveśa, 76 Nyāyapraveśakasūtram, 4 Nyāya School, 121, 122, 127 Nyāyasūtra, 72, 171, 229, 355, 359, 526, 659, 697, 698, 704–706, 709 categories, 700, 702 evaluation of arguments, 703 fallacies and defects, 704 Nyāya syllogism, 129, 171 Nyaya-Vaisesika system, 520 Nyāya view, 683 Nyāybindu, 77 O Object-ness (karmatva), 690 Occam’s razor, 282, 349 Occurrentness, 545–546 Oetke, Claus, 355, 362–363, 367 Ontology, 223, 975 definition of entities, 223 differentiated continuity and perceptual contact, 206 ontic continuity, 222–224 ontology of differentiated continuity, 223 ontology of discontinuity, 223 Order of premises, 20 Ordinal numbers, 1283, 1284

1342 Origination, 165 Own-being (svabhāva), 147 Own-nature, 18 P pada, 683 Padārthadharmasaṃgrahaḥ, 662 Padārthatatvanirupanam, 665 Padma Purāṇa, 151 Paḻamoḻi nāṉūṟu, 608, 609 Pāṇini, 572, 573, 575, 578 Pakṣa, 261, 262, 264, 423–425 Pakṣa, 431–432 Pakṣābhāsa, 268 Pakṣadhara, 664 Pakṣadharmatā-jñāna, 825 Pakṣa (locus of inference), 172 Pakkam, 53–55, 522, 535 Pali, 713 Pāñcarātra, 393 Pañcāvayava-vākya, 832 Panentheism, 496, 497, 503 Parācakti, 218 Paraconsistent logics, 1256–1259, 1266, 1269–1271, 1276, 1279, 1286, 1293 Paraconsistent set theory, 1260 algebra valued models for, 1270–1285 Paradigma, 699 Paramarsha/Parāmarśa, 825, 835 Parameśvara, 389 Pārameśvarāgama, 399 Parapratipattīcchā, 684 Parārthānumāna, 121–123, 261, 265, 268, 384, 439–446, 832 Pariṣad, 707 Pariṇāma-viśeṣaḥ, 155 Park, Peter J., 493 Parokṣa-jñāna, 823 Parsons, Josh, 911, 912, 915, 921, 922, 924 Partial order relation, 1272 Parvato vahnimān, 683 Paryāpti, 555–556 Paryāya, 184 Paryudāsa (nominally bound negation), 169, 278 Pawlakian approximation space, 1055, 1078, 1090 Peacock, G., 985 Peano, G., 993 Peano’s axioms, 993 Peano’s fifth postulate, 557 Peirce, Charles Sanders, 355, 356, 991

Index Peirce’s 3-valued logic, 996 p-equivalence, 765 Perception, 619 dependence of inference, 621 factors of potency, 624–625 inference’s dependency on, 619–621 natural potency, 623 as supportive (Upajīvya) for inference, 621 Vācāspati Miśra’s objection on supportive nature of perception, 628 Perceptual cognition, 284, 294, 302, 303, 315, 316, 342, 343 Perfection, 496, 500, 505, 507, 509 Persuasion, 521, 697, 699, 702 Pervaded entity, 828 Pervading entity, 828 Pervasion (vyāpti), 17, 557, 827, 828, 831, 833, 844, 956, 960, 972–975 Petri net, 1231, 1243, 1247–1249, 1251 Philological and historical style of scholarship, 492 Philology and descriptive history, 493 Philosophical logic, 134 Philosophy of Recognition, 230, 232, 234, 235 Philosophy of Vibration, 231, 232 Pigozzi, D., 1017 Platonic realist, 1123 Plausibility, 532 Plenitude (pūrṇatva), 505 Pleroma, 509 Plurivalent logic, 881–883 Poṅkal, 220 Point of seizure, 838 Polynomial hierarchy, 1199 Poset language, 1249–1251 Positive concomitance, 20 Positive cum negative mark, 834 Positive pervasion, 20 Possibility distribution, 1318, 1319, 1324 Post, E., 998 Post-prediction confirmation, 457 Prabhācandra, 26, 28, 33, 36, 848, 853–856, 858–859 Pragmatism, 355–358 Praijñā, 18 Prajñākargupta, 84–85 Prakaraṇasama (counter-acted), 172 Prakāśa, 376, 413, 415 Pramā, 823 Pramāṇa, 118, 184, 374–376, 400, 405, 406, 411, 412, 414–416, 418, 422, 425, 618–620, 683

Index Pramāṇa-saṅgraha, 33 Pramāṇasamuccaya, 74, 176 Pramāṇavādins, 376 Pramāṇavārttika (PV), 81, 96 Prāmāṇya-vāda, 26 Prameya, 701 Prasaṅga, 274–278 Prasaṅga method Nāgārjuna’s use of, 275–278 Prasaṅgānumāna, 274–275 Prasaṅgānumāna, 274, 275 Prasaṅga technique (technique showing the “consequences”), 162–171 Prāsaṅgika-Mādhyamika, 277 Prasajya-pratiśedha (verbally-bound negation), 169, 278 Prasiddhi, 377, 378, 380, 382, 383, 385–390, 397, 400–403, 406, 409–411, 417 Pratibhāna, 402, 403, 406 Pratijñā (thesis), 171, 700, 702, 704, 705, 709, 711–713, 832 pratītyasamutpāda, 238, 239 Pratiyogin, 669, 677 types, 669 Pratiyogitāvacchadeka-dharmah, 670 Pratyabhijñā, 92–94, 232, 234–237, 249, 409, 413, 416, 418 Pratyabhijñā inference, transcendental argument anticolonial resistance, Western theorization, 494–495 a priori and a posteriori, 502–503 identification of cosmogony and teleology, 512–513 Niyati Śakti, 511 nondual Śaiva myth and ritual, 491–492 philolological objections to dialogical engagements, 492–494 pleromatic fragmentation and inductive noncommitance, 509–511 scriptural traditions, 511–512 vocabulary of recognition, 507–509 Pratyakṣa, 700 See also Perception Pratyaya, 163 pravṛttinimitta, 685, 686 Praxic equivalence, 207 Prayoga, 423, 424–426 Prayojakatva-aprayojakatva, 464–465 Predicates and logical formulae, 1226–1228 Preferences, 1137, 1167–1168, 1180–1181 aggregation of, 1170–1173 deliberation on, 1173–1179

1343 preference frame, 1168 preference model, 1169 satisfiability and validity, 1169 Prenex normal form, 1189 Preprocessing, 1215 Pre-rough algebra, 1066 Presburger successor constraints, 1228 Presumption, 520, 534 Presupposition, 520, 530, 532 Principal injunction, 898 Principia Mathematica, 993, 994 Principle of adjunction, 1261 Principle of bivalence, 1125, 1127 Principle of Excluded Middle (PEM), 870–872, 875, 881, 883 Principle of explosion, 480 Principle of inversion, 1129–1131 Principle of non-contradiction (PNC), 127–128, 870–872, 875, 878, 881, 883 Probandum (sādhya) , 558, 957 Probans (sādhana, hetu), 5, 558–560, 957 Probative value, 524 Product algebra, 1301 Projection rule, 464, 466 Proofs, 958 Property abstraction, 966–968 Property location-logic, 962 Property theory, 964 Proposition, 53, 54, 522, 524, 525, 527, 532, 832 aṉumāṉa viruttam, 54 ākama viruttam, 55 appiracitta campantam, 54 appiracitta upayam, 54 appiracitta vicēṭaṇam, 54 appiracitta vicēṭiyam, 54 capakkam, 53, 55, 62 pakkam, 53, 55, 62 pirattiyakka viruttam, 55 ulaka viruttam, 55 vipakkam, 53, 55, 62 Propositional dynamic logic (PDL), 912, 1141 Propositional formula, 900 Propositional languages, 1011 Proto-anumāna, 443 Proximity (āsatti), 691 Pseudo-Boolean algebras, 1054 Pseudocomplemented lattice, 1025 Pseudo-refutation, 12 Puṟam, 61 puṟanāṉūṟu, 596, 598, 603 Puṣpavanta, 686 Purandara, 150

1344 Pūrṇatā, 505, 507, 509, 510 Pure Wisdom (śuddhavidyā), 492 Puruṣāptavāda, 408 Puruṣārthās, 658 Purva-Mīmāṃsā, 623, 624 Pūrvavat, 359 Q QBF-SAT, 1196 QM (Quine-Morse set theory), 969 Qualificative cognition, 284, 288 Qualities (rūpa), 962 Quantified propositional logic (QPL), 1191–1193 Quantifiers, 557–562 Quasi-Boolean algebra, 135, 1059–1061, 1063 Quasi-hetvābhāsas, 467 Quasi order-generated covering-based approximation spaces (QOCAS), 1079 Quasivarieties of algebra, 1011 Quibble, 838 Quotient algebra, 1005 R Racial considerations, 493 Rasa, 804–809 Rasiowa, H., 1014 Ratio decidendi, 217, 220, 221 Rationality, 228, 229 Ratnakīrti, 87–88 Reach, Karl, 903 Realism, 1125 Reason (R), 55, 228, 229, 234, 235, 503, 504, 832 Contrary Rs, 57 ētu, 53, 55, 60, 62 non-unidirectional R, 56, 57 unacceptable types of R, 56 Recognition (anuvyavasāya), 507, 689 Recognition sensitive notion of truth, 1124, 1126 Recognition transcendent notion of truth, 1124, 1126 Recognitive apprehension, 507, 508, 513 Recognitive synthesis, 492, 507–509 Reduced ordered binary decision diagrams (ROBDDs), 1194, 1213 Reductio ad Absurdum, 276 Reductionist approach, 901 Reduction rules, 550, 552

Index Reflective awareness, 415 Reflective grasping (parāmarśa), 957 Refutation, 12 Regular double Stone algebras, 1027 Regular identity, 1027 Regular Kleene Stone algebra, 1028 Regular languages, 1233–1236 Reification, 965 Relational absence (saṃsargābhāva), 541, 550–552 Relational abstract, 545, 546, 967 Relation of generality, 832 Relative computation, 1199 Relative rough complementation, 1084 Religion, 144 Rescher, Nicholas, 905 Research methodology, 206, 210 Residence relation, 973 Residuated lattice, 1301–1304, 1309, 1312–1314, 1325 Restrictive injunction, 898 Ṛgveda, 145 Rhetoric, 206, 208, 212, 697, 706 Aristotelian and Indian/Nyāya, 698 comparative, 697 dialogical rhetoric, 224 global, 697 injunctive monological rhetoric, 210 monological rhetoric, 211, 224 multiplying analogies, 211 multiplying reasons, 211, 214, 219, 220 political oration as monological rhetoric, 211 rhetor, 207, 209–211 Right-restriction, 544 Robinson’s interpretation, 167–168 Rodl, Sebastian, 503 Rough equivalence, 1057 Rough Heyting algebras, 1081 Rough implication, 1057 Rough lattice, 1080 Rough membership function algebra (RMF-algebra), 1072 Rough relative complementation, 1058 Rough S5-algebra, 1057 Rough set, 1055 theory, 1023, 1054 Ruegg, Seyfor D., 278 Russell set, 1271 Russell’s theory of definite descriptions, 333–335

Index S Śabda, 683, 701 independent pramāṇa, 687–689 as pada, 683 as pramāṇa, 683 See also Verbal testimony śābdabodha definition, 683 essential factors for, 690–692 Śabdana, 376–378, 402, 404, 406, 409–411, 415–417 Śabdaśaktiprakāśikā, 665 Sacrificial ritual, 142 Śaddarśanas, 659 Sādṛśya, 832 Sādhana, 423–429 Sādhanānumāna, 837 Sādhya, 80, 172, 423–425, 432, 826 Sādhyasama (unproved), 172 Śaḍāstikadarśanas, 659 Saṃhitās, 142 Śaṁkarasvāmin, 75–76 Sāṃkhya Sūtras, 659 Sāṃkhyatattvakaumudī, 661 Śāṅkarācārya, 501 Śaṅkarasvāmin, 265 k-safe Petri nets, 1248–1249 Śaivas, 385 Śaiva Siddhānta, 230–233 Śaivism, 230, 233, 235–237, 249 Sajātīya, 378 śakti, 685 śaktijñānam, 684 Śalākā Parīksā, 664 Samantabhadra, 30 Samantāt, 377 Sāmānya, 274 Sāmānyalakṣaṇa, 260 Sāmānyalakṣaṇā-pratyāsatti, 832 samavāya, 685 Samucita-deśādau vṛttitvam of hetu, 826 Samyak āgama, 395 Sandhāya, 705 sannidhi, 691, 692 Śānta, 804, 806–809, 813, 817, 818 Sapakṣa, 261–262, 269–270 Sapakṣa (homologue or place which is known to possess the sādhya already), 172, 173 Saptabhaṇgī, 879–881, 883 See also Syād-vāda Sarvavīra, 396

1345 Śaśadhara, 563 Satkāryavāda, 236 Satpratipakṣa, 272 Savyabhicāra (deviating), 171 Scepticism, 145 Schröder, F.W.K.E., 991 Scriptural traditions (āgama), 511 Śeṣavat, 359, 361 Segment of the proof appearing, 1122 disappearing, 1122 Self-residence (ātmāśraya), 972 Self-validity, 622 Semantic competence, 1294 Semantic-conceptual linkage, 453 Semantic consequence relation, 1302–1306 Semantic equational consequence relation, 1018 Semantic interpretations, 1240–1242 Semantico-inferential exclusion (apoha), 510 Semantics, 899 Semi-automatic parsing, 946–949 Semi-De Morgan algebra, 1029 Semi-Heyting algebra, 1032 Sense-experiences, 1122 Sentence (vākya), 683 Sequence of action, 898 Series-parallel poset, 1225, 1232, 1235, 1242 Settled opinion (siddhānta), 94–96 Sheffer stroke, 542–543 Shin’s system, 726–727 siddhānta, 94–96 Siddhāntalakṣaṇa, 558, 560 Siddhāntalakṣaṇa-definition, 563 Siddhasena Divākara, 26, 32 Significative power (śakti), 684, 685 Significative power (śaktijñāna), 684 Śīlāṅka, 152 Similar case, 13 Similarity (sādṛśya), 832 Simple algebra, 1005 Śiromaṇi, Raghunātha, 551 Śivadṛṣṭi (ŚD), 92–94 Śivasūtra, 231, 232 Skinner, Quentin, 494 Skolem constant, 1190 Skolem functions, 1188, 1192 algorithmic paradigms, 1188 applications, 1195 computation, 1196 knowledge representation, 1206 unification, 1191

1346 Skolem normal form, 1189 Smith, Nicholas, 1320 Somānanda’s siddhānta, 94–96 Sorites paradox, 1316–1324 Fine’s supervaluational approach, 1317 Fuzzy set-theoretic approach, 1318–1319 GCT, 1320–1324 Smith’s approach, 1320 Soundness, 900 Sources of knowledge, 300–301 Speech-act theory, 891 Sphoṭa theory, 582–583 Spider diagram, 728 Śrī vyāsatīrtha, 621, 624, 625, 628 Śruti (Vedic texts), 628 ST2, 969, 970 Stability, 1130, 1131, 1133 Sthāpanā, 699 Sthāyī-bhāva, 804 Stone algebras, 1025 Straight-line program, 1243–1244 composition, 1245–1246 hypergraph representation, 1244–1245 labelled posets, 1245 Strategic reasoning, 1179–1180 Straying reason, 635, 636 Strong paraconsistency, 1257 Structural consequence relation, 1012 Structure of inference, 121–123 Subalgebra, 1004 Subaltern, 493 Subdirect embedding, 1005 Subdirectly irreducible algebras, 1006 Subgame composition, 1150 Subjunct (anuyogin), 974–975 Sublated reason, 635 Sublation, 628, 630 Subsumption (inclusion) relation, 991 Sufficient reason, 496, 499, 501, 502 Śūnya (empty or void), 162 śūnyatā, 275, 277 Supportive (Upajīvya), 621 Suszko, R., 1003 Sūtra-Bhāṣya tradition, 262, 266 Svabhāva-hetu, 13, 18 Svabhāva-hetu (natural reason), 175 Svabhāva (intrinsic nature), 163 Svabhāvānupalabdhi, 16 Svabhāvānupalbdh, 15 Svabhāva-pratibandha, 14 Svabhāva-viruddhopalabdhi, 15, 16 Svagata, 378 Svalakṣaṇa, 260

Index Svārthānumāna, 13, 121, 123, 261, 439–446, 832 Svātantrika- Mādhyamika, 277 Syād-vāda, 24, 26, 31 historical development of sapta-bhaṅgī, 186–189 and logic principles, 195–200 sevenfold predication, structure of, 189–195 Syllogism, 522, 526, 619, 620 Symbiosis, 1115 Symbolic algebra, 983, 985 Symbol usage in logic, 984 T T1+, 566, 969 T1 11, 965 Taber, John, 355, 362, 366–367 Tamil didactic literature absolute presuppositions, 521 arguments, 520 assumptions, 521 issue, 521 religious and philosophical traditions, 520 See also Kāṇṭikai Tantra, 375, 400 Tantrāloka, 230, 231, 237, 249 Tantric sources, philosophical rationalisation of, 232–237 Tarka, 423, 424, 426, 827, 835–837 Tarski, A., 1002, 1004 Tarski’s Convention, 1321 Tarukkam, 206, 207, 212, 216, 224, 601, 603, 609, 610 debate, 601 dissertation as monological tarukkam, 210 kāṇṭikai, 602, 603 monological logic, 602, 613 monological rhetorical logic, 206, 212 tarkkam, 212, 216 tarukkam in literature, 207, 210 wordplay, 602 Taruma uraic carukkam, 47 tātparyagrāhaka, 692 Tātparyatīkā, 662 Tattvacintāmaṇi, 663, 665, 667 Tautology, 278, 279 Template/sketch-based techniques, 1214 Temporal logics, 1138, 1149, 1160 Ten religio-philosophical exponents, 51, 61 Term rewriting, 1225 Testimonial authenticity, 406, 409–413

Index The Debate of King Milinda (Milinda Pamha), 714 The Debate of King Milinda (Milinda Panha), 713 Theism, 658 Theories of logic Anumāna, 829 cognition, 823–825 fallacies of inference, 837–842 hypothetica, 835 inference, 826 inferential cognition, 825, 827 Vyāpti, 833–835 Theory of anvitābhidhāna, 581–582 Theory of argumentation, 848, 865 Theory of causality, 233–236, 238, 250 Theory of graded consequence (GCT), 1300–1302 Carnap’s principle of tolerence, 1325 classical consequence, generalisation of, 1296–1300 classical semantic consequence relation, 1302–1306 consistency and inconsistency, generalization of, 1306–1309 many-valued logic, fuzzy logic and graded consequence, 1291–1293 negation, in object language, 1309–1316 sorites paradox, solution of, 1320–1324 uncertainty and degrees of truth, 1293–1296 Theory of inference, 162, 171 Theory of intention, 912 Theory of knowability, 234 Theory of knowledge, 857, 858, 862 Theory of pāl, 48 Thesis, 5 Three-valued logic, 170, 1292 Tiṇai, 50, 61, 207, 212, 217, 218, 220, 222–224, 596, 598, 607, 610, 611, 613 kaikkiḷai, 207–209, 211, 212, 215, 216, 224 Tirukkuṟaḷ, 52, 207, 210, 221, 223, 599, 600 Tiruppati veṅkaṭēcap perumāḷ ṇāṭakam, 216 Tolkāppiyam (tol.), 45, 48, 50, 53, 55, 60, 61, 206, 210, 212, 216, 217, 221, 224, 596, 608, 610, 611 Ontology in tol., 48 pakkam, 53 uttikaḷ, 49, 50, 62, 63 Topological Boolean algebra, 1091 Topological quasi-Boolean algebra 5 (tqBa5), 1064 Topological quasi-Boolean algebra (tqBa), 1057, 1063

1347 Totality factor, 245–248 Transcendental argument, Pratybhijñā inference, see Pratyabhijñā inference, transcendental argument Transfinite recursion, 1274 Transformational grammars, 1239–1240 Tree language, 1228, 1235, 1240 Tree transducer, 1238 Tree transduction, 1238, 1240 Trika, 230–234, 236, 237, 250 Trirūpa, 13 Truth, 206, 212, 625 anarchic truth, 217, 220, 223 desacralized truth, 217, 223 differentiated continuity, 206 epistemic anarchism, 210 fact, 221 final truth (cittāntam), 216, 218, 224 foregone conclusion, 206, 218, 224, 225 heterodox truth, 217 hierarchization of truth, 217 truth as rule, 218, 223, 225 truth as the plausible, 221 truth-claims, 209 truth-texts, 216 Truth-preserving argument, 848, 849, 853, 862, 866 Turning machine (TM), 772 Two-valued logic, 50, 54, 60, 62, 63, 169 bivalent logic, 45, 46 Type space, 1159, 1160, 1162 U Ubhayābhāva, 644 Udāharaṇa (a general rule plus an example), 171, 832 Uddālaka Āruṇi, 146 Uddyotakara, 174, 261, 661 Udyotakara, 259, 261, 262 ūha, 428 Umāsvāti, 28, 29, 31 Umwelts, 1106–1107, 1111–1112 Unacceptable types, 56 Uncertainty, 1293–1296 Undercutter (upādhi), 975 Unification rule, 752 Universal algebra, 992 Universal mapping property, 1007 Universal quantifier, 324–328 Universals (sāmānya), 962 Universe of discourse concept, 988 Unknown reason, 635, 637

1348 Upādhi, 831 Upādhinirāsa, 458 Upalabdhi-hetu, 429–430 Upalabdhilakṣaṅaprāpta (observable, that which fulfils the condition of observability), 175 Upamaya, 700 Upanaya, 19, 700, 832 Upanaya (he application of the rule to the case in question), 171 Upanayam, 52, 62, 523, 527, 531 Upper approximation, 1055 Urai, 51 Ur-linkage-based anumāna, 441 Utpala, 378 Utpaladeva, 230–233, 235, 236, 239, 250, 374, 376 Utpatti, 184 Uttara, 706 Uttikaḷ, 45, 49, 50, 62, 63, 206, 210 Utti (logical techniques), 596, 597, 610, 613 V Vācāspati Miśra, 628–629 Vāda, 72, 75, 82, 422, 696, 697, 702, 705–707, 709–714, 837 Vādanyāya, 82 Vādavidhi, 72–73 Vādidevasūri, 37 Vaialyaprakaraṇa, 71–72 Vaiśeṣika, 701 Vaiśeṣika school, 662 Vaiyākaraṇas, 683, 692 Vāk, 584 Vākai, 207, 208, 212, 221, 224, 601, 609, 610 aṉal vātam, 216 aḷavai tarukkam and vākai, 611 contestatory nature of vakai, 610 dialogical logic, 613 dialogical rhetorical logic, 206, 212, 216 folk drama, 214, 216 folk songs, 212, 214 modern court of law, 217 paṭṭimaṉṟam, 220, 221, 224 poruḷ vātam, 216 puṉal vātam, 216 puṟattiṇai, 216 religious polemics, 216, 217 screenplay, 218, 220 Song duel, 610 uṟaḻcci nūl, 611 utti and vākai, 611

Index vaḻakkāṭu maṉṟam, 220, 221, 224 vaḷḷi tirumaṇam, 214, 215 verbal boast, 208 verbal duel, 206, 212, 214 Vāko-vākya, 142 vākyārtha, 689 Vākyātmaka, 440 Valid form of argument, 17 Valid inference, 282, 291, 307, 312, 349 Validity, 916–921, 1120, 1121, 1125, 1126 3-valued Łukasiewicz logic, 1070, 1071 3-valued Łukasiewicz (Moisil) algebra, 1067, 1071 Variant, 524, 529 Vasubandhu, 72–73 Vātam, 45, 46, 50, 60–62 Vātsyāyana, 260, 263, 661 Vātu/vātam, 206, 207, 224 Vedānta, 148, 712 vṛddhavyavahāra, 684 Vedic sentences, 894 Venn diagram, 724 Venni, 732 Vennin, 771 Venni consistent diagram, 767 counterpart relation, 735 diagrammatic language, 732 elimination rules, 748 extension rules, 745–746 identity, 736 inconsistent diagram, 765 introduction rules, 739 normal form, 738 provability/syntactic consequence, 769 rule of construction, 758 rule of excluded middle, 756 rule of splitting sequences, 754 semantics, 770 unification rule, 752 well-formed diagrams, 733 Vennin, 771 diagrammatic language, 772 inconsistent diagram, 773 introduction rule, 772 provability/syntactic consequence, 774 semantics, 774 Vennio1 elimination rules, 783 extension rules, 782 inconsistent diagram, 784 introduction rules, 780 normal form, 779–780

Index primitive symbols, 777–778 semantics, 784 well-formed diagram, 779 Venn-Peirce diagrams, 725 Verbal cognition, 316–319, 653 Verbal expectancy (ākāṅkṣā), 690–691 Verbal knowledge, 648 Verbal suffix (tiṅ vibhakti), 683 Verbal testimony dependence of inference, 621–623 inference’s dependency on, 619 Vidyānandin, 25, 34 Viṣayatā, 475, 476 Viṣayatāpatti, 388 Viétei, F., 983 Vigrahavyāvartanī, 71, 163 Vijātīya, 378 Vijñānavādin, 237, 238 Vikalpa, 473–475 Vikalpabuddhi, 274 Village assembly, 611, 612 ampalam, 611 avai, 596, 610 debate in assembly, 611, 612 maṉṟam, 611 potiyil, 611 Vimarśa, 376, 402, 413–416 Vinītadeva, 84 Vipakṣa (heterologue or place which is known not to possess the sādhya), 172 Vipaksa, 261, 269 Vīrapāṇṭiya kaṭṭappommaṉ, 219, 220 Viruddha (contradictory), 171, 272 Viruddha hetvābhāsa, 466 Viruddha-kāryopalabdhi, 15 Viruddhāvyabhicārī, 267, 270, 272 Viruddha-vyāptopalabdhi, 15 Viruddhopalabdhi (observation of the contrary), 176 Virutti urai, 604 Viśiṣṭābhāva, 644, 645, 671 Viśiṣṭaśaktivāda, 685 Viśrānti, 808 Vitaṇḍā, 702, 837 Vmarśana, 378 Voegelin, Eric, 496, 501, 512 Vranas, Peter B.M., 893, 907, 916, 919–921 Vyabhicāra-adarśana, 830 Vyākaraṇa, 634, 640, 641, 643–654

1349 vyakti (ghaṭa), 685 Vyāpaka, 828 Vyāpakānupalabdhi, 15 Vyāpaka-viruddhopalabdhi, 15 Vyāpti, 119, 122, 262–265, 700, 825, 827 Vyāpti-definition, 565 Vyaptigraha/Vyāptigraha, 830–832 Vyāpti-jñāna, 825 Vyāpti-smṛti, 825 Vyāpya, 828 Vyāsa, 383 Vyatireka, 20 Vyatireka-vyāpti, 270, 833 Vyatirekī hetu, 262 Vyaya, 184 W Warrant, 522–526, 529, 530, 532, 534 Warrant-establishing, 526 Weak Kleene logic, 886 Weak paraconsistency, 1257 Weber, Max, 493, 495 Wedeking, Gary A., 893 Well-foundedness, 972 Western logic, vs. Indian Logic, 118–121 Western tradition, study of imperatives in, 900–912 Whitehead, A.N., 992, 994 Wrangling, 837 Y Yādṛcchika (accidental object), 169 Yājñyavalka, 658 Yaśovijaya, 33, 37 Yathārtha-jñāna, 823 Yelle, Robert, 493 Yoga Sūtras, 659 Yogī, 244 Yukti-śāstra/śāstra, 822 Yuktisastra, 48, 49 Z Zadeh’s system, 1318, 1324 Zermelo-Fraenkel axiomatic set theory, 1272 Zermelo-Russell’s antinomy, 968 Zimmer, Heinrich, 497