Women's Contemporary Readings of Medieval (and Modern) Arabic Philosophy (Logic, Argumentation & Reasoning, 28) 3031056280, 9783031056284

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Logic, Argumentation & Reasoning 28

Saloua Chatti   Editor

Women's Contemporary Readings of Medieval (and Modern) Arabic Philosophy

Logic, Argumentation & Reasoning Interdisciplinary Perspectives from the Humanities and Social Sciences Volume 28

Series Editor Shahid Rahman, University of Lille, CNRS-UMR 8163: STL, France Editorial Board Members Frans H. van Eemeren, Amsterdam, Noord-Holland, The Netherlands Zoe McConaughey, Lille, UMR 8163, Lille, France Tony Street, Faculty of Divinity, Cambridge, UK John Woods, Dept of Philosophy, Buchanan Bldg, University of British Columbia, Vancouver, BC, Canada Gabriel Galvez-Behar, Lille, UMR 8529, Lille, France Leone Gazziero, Lille, France André Laks, Princeton/Panamericana, Paris, France Ruth Webb, University of Lille, CNRS-UMR 8163: STL, France Jacques Dubucs, Paris Cedex 05, France Karine Chemla, CNRS, Lab Sphere UMR 7219, Case 7093, Université Paris Diderot, Paris Cedex 13, France Sven Ove Hansson, Division of Philosophy, Royal Institute of Technology (KTH), Stockholm, Stockholms Län, Sweden Yann Coello, Lille, France Eric Gregoire, Lille, France Henry Prakken, Dept of Information & Computing Sci, Utrecht University, Utrecht, Utrecht, The Netherlands François Recanati, Institut Jean-Nicord, Ecole Normale Superieur, Paris, France Gerhard Heinzmann, Laboratoire de Philosophie et d’Histoire, Universite de Lorraine, Nancy Cedex, France Sonja Smets, ILLC, Amsterdam, The Netherlands Göran Sundholm, ’S-Gravenhage, Zuid-Holland, The Netherlands Michel Crubellier, University of Lille, CNRS-UMR 8163: STL, France Dov Gabbay, Dept. of Informatics, King’s College London, London, UK Tero Tulenheimo, Turku, Finland Jean-Gabriel Contamin, Lille, France Franck Fischer, Newark, USA Josh Ober, Dept of Pol Sci, West Encina Hall 100, Stanford University, Stanford, CA, USA Marc Pichard, Lille, France Managing Editor Juan Redmond, Instituto de Filosofia, University of Valparaíso, Valparaíso, Chile

Logic, Argumentation & Reasoning (LAR) explores links between the Humanities and Social Sciences, with theories (including decision and action theory) drawn from the cognitive sciences, economics, sociology, law, logic, and the philosophy of science. Its main ambitions are to develop a theoretical framework that will encourage and enable interaction between disciplines, and to integrate the Humanities and Social Sciences around their main contributions to public life, using informed debate, lucid decision-making, and action based on reflection. • • • •

Argumentation models and studies Communication, language and techniques of argumentation Reception of arguments, persuasion and the impact of power Diachronic transformations of argumentative practices

LAR is developed in partnership with the Maison Européenne des Sciences de l’Homme et de la Société (MESHS) at Nord - Pas de Calais and the UMR-STL: 8163 (CNRS). This book series is indexed in SCOPUS. Proposals should include: • • • •

A short synopsis of the work, or the introduction chapter The proposed Table of Contents The CV of the lead author(s) If available: one sample chapter

We aim to make a first decision within 1 month of submission. In case of a positive first decision, the work will be provisionally contracted—the final decision about publication will depend upon the result of an anonymous peer review of the complete manuscript. The complete work is usually peer-reviewed within 3 months of submission. LAR discourages the submission of manuscripts containing reprints of previously published material, and/or manuscripts that are less than 150 pages / 85,000 words. For inquiries and proposal submissions, authors may contact the editor-in-chief, Shahid Rahman at: [email protected], or the managing editor, Juan Redmond, at: [email protected]

Saloua Chatti Editor

Women’s Contemporary Readings of Medieval (and Modern) Arabic Philosophy

Editor Saloua Chatti Retired from the University of Tunis Tunis, Tunisia

ISSN 2214-9120 ISSN 2214-9139 (electronic) Logic, Argumentation & Reasoning ISBN 978-3-031-05628-4 ISBN 978-3-031-05629-1 (eBook) https://doi.org/10.1007/978-3-031-05629-1 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 This work is subject to copyright. All rights are solely and exclusively licensed 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 Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

To the memory of my mother Téfida To the memory of my sister Nadia Saloua Chatti

Acknowledgments

The idea of this contributed book, where the contributing authors are exclusively female contemporary researchers in the field of Arabic philosophy, comes from Professor Shahid Rahman, the editor of the series Logic, Argumentation and Reasoning, who wanted to highlight the researches of female scholars in that specific tradition. I totally accepted this idea, which seemed to me a good way to encourage these researchers, especially the young ones, and we started this adventure. I thank him for having chosen me as an editor of that book and for his constant support throughout its preparation. Our purpose was to bring attention to the work of women scholars in Arabic philosophy, whether renowned professors and researchers or young scholars at the very beginning of their careers. It is one of the ways to encourage women’s publications in that field, especially for those authors who come from underrepresented countries or are still at the beginning of their careers. Our authors come from very different geographic areas, namely North Africa, the USA, and some European countries such as France and Germany. They pertain to different traditions, for some of them are Iranian while others are German, French, or North African. This diversity enriches the book, due to the various cultural and professional backgrounds of the authors and the variety of their interests. This project, however, should not be thought of as a defense of any kind of feminist approach or trend. Rather it simply reflects the will to make the research of female scholars known by their colleagues in the same field. It makes it clear that these scholars are active and produce original research in all branches of Arabic philosophy. The authors specialize in various fields of Arabic philosophy, such as logic, mathematics, metaphysics, ethics, and aesthetics, which explains the variety of the topics studied in the book. I wish to thank all of them for having accepted to collaborate on this project, and for their patience and all the work they have done to make the project succeed. All chapters without exception have been submitted to a double-blind evaluation by reviewers specialized in the specific topic studied by the chapter. vii

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Acknowledgments

So, I would like to thank all the reviewers, who kindly accepted to evaluate these chapters and whose remarks and suggestions helped improve each of them. I greatly appreciate their valuable work and their insightful and helpful remarks which made all authors benefit from their expertise. I am also very grateful to some friends and colleagues, such as Professors Wilfrid Hodges, Thérèse-Anne Druart, and Ahmed Hasnaoui, whose advice and fruitful suggestions helped enrich and improve this contributed book. Thanks also to the team of Springer Nature for their patience and copyediting of the volume. Finally, I warmly thank my sisters Lilia and Samia for their constant support and for the very welcome help in the preparation of this volume for press.

Contents

1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Saloua Chatti

Part I

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Logic and Mathematics

2

The Hypothetical Logic in the Arabic Tradition . . . . . . . . . . . . . . . . Saloua Chatti

13

3

Mullā Ṣadrā Shīrāzī and the Meta-Theory of Logic . . . . . . . . . . . . . Sayeh Meisami

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4

Algorithms in Takmilat al-‘Uyūn of al-Iṣfahāni: Sources and Validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Nacéra Bensaou

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Part II

Metaphysics, Ethics and Aesthetics

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Some Observations on Prudence (gr. φρóνησις, ar. taʿaqqul) in Book VI of Averroes’ Middle Commentary on the Nicomachean Ethics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 Frédérique Woerther

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Ethics and Aesthetics: Theorizing Simile in Ibn Sīnā’s Risālat al-Ṭayr and Ḥayy ibn Yaqẓān . . . . . . . . . . . . . . . . . . . . . . . . . 127 Nora Jacobsen Ben Hammed

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Toward Another Understanding of the Notion of fiṭra in the Avicennian Ontology of the Rational Soul . . . . . . . . . . . . . . . . 147 Meryem Sebti

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Primacy of Existence Versus Primacy of Essence – What Is the Debate About? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163 Mansooreh Khalilizand

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“Superstition” or “Crown of Science”? Zaki Naguib Mahmoud, Youssef Karam, and Yumna Tarief El-Kholy on Metaphysics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Kata Moser

Contributors

Nacéra Bensaou is Associate Professor in Computer Science at the University of Sciences and Technology Houari Boumédiene (USTHB), Algiers (Algeria). Saloua Chatti is Professor of Philosophy and Logic. She is now retired from the University of Tunis. Nora Jacobsen Ben Hammed is Assistant Professor in the Interdisciplinary Study of Religious Program, Bard College, USA. Mansooreh Khalilizand is research fellow at the Center for Islamic Theology, University of Münster, Germany. Sayeh Meisami is Associate Professor of Philosophy at the University of Dayton, USA. Kata Moser is Junior Professor at the University of Göttingen, Germany. Meryem Sebti is Senior Researcher at the National Center for Scientific Research (CNRS) and Associate Professor at EPHE, Paris, France. Frédérique Woerther is Director of Research at the National Center for Scientific Research, (CNRS) Paris, France.

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Chapter 1

Introduction Saloua Chatti

Abstract In this first part, we provide a summary of all the articles of the book. In doing so, we follow the table of contents which contains two main parts (1) Logic and mathematics, and (2) Metaphysics, ethics and aesthetics. Within the first part, we find three chapters, two of which are about logic in the Arabic tradition (Chaps. 2 and 3), while the third one (Chap. 4) is about the history of Arabic mathematics. The second part contains five chapters, two of which are about ethics and/or aesthetics studied from the perspective of different Arabic philosophers (Chaps. 5 and 6), while the three others are about metaphysics within the Arabic tradition, whether in Medieval times (Chaps. 7 and 8), or nowadays (Chap. 9).

This contributed book aims at showing some of the researches made by well known as well as young scholars in the various sub-disciplines of Arabic philosophy. It does not claim exhaustiveness but presents some samples of original research in different fields of Arabic philosophy, both Medieval and contemporary. As I already said, the idea of focusing on the research of female scholars comes from Professor Shahid Rahman (University of Lille, France) and aims at highlighting this research, in particular that of young scholars. I myself accepted the project and Professor Rahman’s idea and tried to bring together many scholars who did not come from the same area or the same tradition. I suggested the project to many female scholars, whom I had heard about. Some accepted it, some others failed to take part to the project for various reasons, either personal or because they had previous engagements; some others joined us at the very end. Unfortunately, we had also some rejections. The project started in 2019, before the Covid crisis. It went through many steps, and changed several times during this period, following the changes of authors, who would leave it or join it. These changes affected the structure of the book and its length. In the beginning, the book was divided into three parts (1. Logic and S. Chatti (*) Retired from the University of Tunis, Tunis, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Chatti (ed.), Women’s Contemporary Readings of Medieval (and Modern) Arabic Philosophy, Logic, Argumentation & Reasoning 28, https://doi.org/10.1007/978-3-031-05629-1_1

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S. Chatti

language, 2. Science and mathematics, and 3. Metaphysics and Ethics), but due to the changes of contributors, I had to modify this whole structure and to divide the book into two main parts: (1) Logic and Mathematics and (2) Metaphysics, Ethics and Aesthetics. As a result, the book has now a various and rich content, covering different subdisciplines inside philosophy plus some sciences such as mathematics, and also numerous periods of time, starting from early Medieval Arabic philosophy, illustrated by al-Fārābī and Avicenna (ninth and tenth Centuries) and the mathematician al-Khawarizmī (ninth Century), to recent Arabic philosophers from the twentieth Century, including Arabic writing philosophers from intermediary times, such as Averroes (twelfth century), the mathematician Umar al-Khayyam (eleventh century) and Avicenna’s followers, both in Persia, such as al-Suhrawardī (twelfth Century), Fakhr al-Dīn al-Rāzī (twelfth century), Afḍal al-Dīn al-Khūnajī (thirteenth Century) and Mullā Sadrā Shirāzī (seventeenth Century), and in North Africa, such as Ibn ʽArafa (fourteenth Century), and al-Sanūsī (fifteenth Century). These authors are not all philosophers; some of them are logicians and theologians, while others are mathematicians. Within Part I (Logic and Mathematics), we count three articles: the first two articles are about logic (Saloua Chatti and Sayeh Meisami), the third one is about the history of mathematics within the Arabic tradition (Nacéra Bensaou). The first article, written by Saloua Chatti, is entitled “The hypothetical Logic in the Arabic Tradition”. It analyses the logical systems pertaining to hypothetical logic, whose inferences use conditional propositions (containing ‘if . . .then’) and disjunctive propositions (containing ‘either . . .or’) in the Arabic tradition, and assesses their novelty and their evolution. It shows that this particular logical system, rarely studied by scholars interested in Arabic logic, has been significantly modified and developed, as it largely extends the Stoic system, in particular in Avicenna’s frame. For Avicenna built several hypothetical systems by introducing quantification on the hypothetical propositions and mixing in interesting ways between hypothetical quantified propositions, both conditional and disjunctive, and predicative ones. He thus presents a first system containing only hypothetical conditional quantified propositions and exactly parallel to categorical syllogistic, then a second system where hypothetical conditional quantified propositions are mixed with disjunctive quantified ones, then a third system mixing between hypothetical conditional and disjunctive quantified propositions and categorical ones, which contains several new kinds of mixed inferences. By doing so, he considerably widens the domain of traditional syllogistic. He also presents a more traditional hypothetical system containing unquantified hypothetical propositions, where the moods are similar to the Stoic ones and are called istithnā’ī moods. In this system, the main novelty is the importance given to the inclusive disjunction (¼ P or Q or both), which is clearly distinguished from the exclusive one (¼ P or Q but not both) and from the negated conjunction (¼ not both P and Q). These changes have been endorsed and developed by his followers, in particular in Iran (with Fakhr al-Dīn al-Rāzī (d. 1210, Iran) and Afḍal al-Dīn al-Khūnajī

1 Introduction

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(d. 1248, Iran), among many others), and later in North Africa, with Ibn ʻArafa (d. 1401, Tunisia), who commented on al-Khūnajī’s treatise al-Jumal, and al-Sanūsī (d. 1490, Algeria) who commented on Ibn ʻArafa’s one. All these authors are clearly influenced by Avicenna, although they did criticize him on some points. Nevertheless, they also have their own contribution both in the quantified hypothetical system and in the unquantified or propositional one. Al-Khūnajī developed in his treatise al-Jumal the istithnā’ī moods and ‘discovered’ new laws and inferences pertaining to propositional logic, while in his book Kashf al-Asrār ʻan ghawāmid al-afkār, he focused much more on the quantified system and presented much more complex mixed inferences than Avicenna’s ones. As to Ibn ʻArafa and after him al-Sanūsī, they distinguished themselves by focusing mainly on the istithnā-ī system, i.e. the propositional system, and presenting some new kinds of inferences and above all by almost ‘extensionalizing’ the definitions of the main operators, i.e. the disjunction but also and especially the conditional, which had always been defined intensionally by their predecessors. The second article is about the logic of Mullā Ṣadrā (d. 1635). It is written by Sayeh Meisami and is entitled “Mullā Ṣadrā Shīrāzī and the Meta-Theory of Logic”. After a brief introduction of Mullā Ṣadrā, his life and his oeuvre, the author analyses his conception of logic, and its links with metaphysics, and compares this conception to Avicenna’s one, especially as it is characterized in Avicenna’s treatise on metaphysics entitled al-Ilāhiyāt. Avicenna had defined the subject matter of logic in that treatise as being the study of ‘secondary intelligible concepts’, i.e. those concepts such as the concepts of subject and predicate that apply to other concepts, not to objects. This specific conception has been attributed to Avicenna by many contemporary scholars starting from Sabra, who distinguishes it from another “rival” conception, endorsed by al-Khūnajī, according to which logic is the study of “conceptions and assents”, which became the standard conception in the Arabic tradition. However, some other scholars challenge this attribution of the first conception to Avicenna and credit him with a more extensive view, according to which logic is at the same time “a science and a tool for other sciences”, while stressing the fact that the “rival” conception is also present in Avicenna’s text. As to Mullā Ṣadrā, as the author shows in her article, he discusses the issue of the subject matter of logic by relying on Avicenna’s more general view, in particular as it is expressed in al-ishārāt wa’l-tanbīhāt, and endorses this same conception of logic as being a “tool”, “which protects from errors”. He expresses this position with “just some minor changes in wording”, but he also endorses in his Taʻlīqāt, the position defended by Avicenna in al-Ilāhiyāt, as stressed by the author (Sect. 3.2), against Abharī’s criticisms. Yet the author shows that Mullā Sadrā does acknowledge the distinction between conceptions and assents, and applies it to all “knowledge by acquisition” (Sect. 3.3). According to the author, Mullā Ṣadrā develops this distinction and the nature of both assents and conceptions by discussing and criticizing some of his predecessors’ views such as those of Quṭb al-Dīn al-Rāzī and Fakhr al-Dīn al-Rāzī. In Sect. 3.4, the author analyses Mullā Sadrā’s conception of modalities and stresses its novel character, its relation to his metaphysics, and the influence of

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al-Suhrawardī in this respect. According to Mullā Ṣadrā, all modalities must be attached to necessity as illustrated by sentences like “Necessarily all humans are possibly literate, necessarily animals, or impossibly stones” (Mullā Ṣadrā, in Sect. 3.4). He thus introduces iterated and multiple modalities inside modal logic. This position is ultimately related to his “own existence-centered metaphysics”, as the author claims. The third article of this first part is about the history of Arabic mathematics. It is written by Nacéra Bensaou and is entitled “Algorithms in Takmilat al-‛uyūn of Al-Iṣfahānī : sources and validation”. In her paper, the author analyses the mathematical algorithms used in the Arabic tradition, starting from al-Khawarizmī and his followers, by considering the paradigmatic case of solving equations’ algorithms, of different degrees (1, 2 and 3). She then focuses on the algorithms which solve the cubic equations. After presenting al-Khawarizmī’s algorithms and solutions of the different kinds of equations, and considering some of his followers, such as al-Yazdī, al-Kāshī and Sharaf al-Dīn al-Tūsī, among others, who all introduce some novel features in their solutions, the author examines in particular the solutions of the cubic equations provided by Alī Muḥammad ibn Muḥammad Ƥusayn al-Iṣfahānī, a mathematician who lived in Iran in a relatively recent period (nineteenth century), but who used the traditional methods inspired by his predecessors in his solutions of these equations, i.e. without any use of the mathematical symbolism which we find in modern mathematics. By studying these different scholars, the author aims at showing that, despite the lack of formalisation, “the several types of algorithms for solving equations, . . . ., from the 9th to the 19th centuryˮ (abstract) do indeed exhibit some kind of “reasoning” which validate these algorithms, even if this reasoning is “not necessarily [expressed] in a formal language” (abstract). She shows that by analysing the notion of validation applied to the solving equations algorithms, including those which provide an approximate solution. She does so by showing first “why the solution given by the algorithm is correct even if it is an approximate solution” (section D), second “that the iterations of the calculation can be stopped as soon as we note that between two successive approximate solutions there is very little difference” (section D), i.e. that the algorithm has a finite number of steps. She also distinguishes between the geometrical solutions which were first used by the early authors and the algebraic ones, which were most used by the later authors. In the second part of the volume (Ethics, Aesthetics and Metaphysics), we have five articles, which are either about Ethics (eventually associated with Aesthetics) (Frédérique Woerther and Nora Jacobsen Ben Hammed) or about Metaphysics (Meryem Sebti, Mansooreh Khalilizand and Kata Moser). The first article, written by Frédérique Woerther, is entitled “Some Observations on Prudence (gr. Φρóνησις, ar. taʿaqqul) in Book VI of Averroes’ Middle Commentary on the Nicomachean Ethics”. The author considers Averroes’ commentary on Book VI of Aristotle’s Nichomachean Ethics, a commentary which is available only in Latin and in Hebrew, given that its Arabic version is lost. She studies the concept of prudence called ‘Φρóνησις’ in Greek and ‘taʿaqqul’ in Arabic, and compares between Averroes’ views about this concept and Aristotle’s ones. After

1 Introduction

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presenting the main contents of Aristotle’s own Book VI of the Nichomachean Ethics, to provide and explain the general background of her study, she turns to Averroes’ text and comments on the sections devoted to prudence in Averroes’ Middle commentary on Aristotle’s Nichomachean Ethics. For this purpose, she provides the first critical edition, made from the two main Latin witnesses of the text, of the passages devoted to prudence on Averroes’ commentary on Book VI of the Nichomachean Ethics and the very first original English translation of these passages, and claims that her aim is first “to give a brief overview” of the way prudence is conceived by Averroes, secondly to complement S. Harvey’s remarks on the same topic, especially in his own analysis of Averroes’ commentary and thirdly to assess the main departures of Averroes’ Commentary toward Aristotle’s original text, which the Latin version considered shows and their eventual reasons. While analysing Averroes’ views and their justifications, she notes that the Latin text reflects in general a clear closeness between Averroes’ views on prudence and Aristotle’s ones. But some passages show, on the contrary, a deviated interpretation of the Aristotelian text, which the author attributes to a possible misunderstanding of the Aristotelian Greek text, probably due to an “obscure” Arabic translation of some passages (Sect. 5.2). In Sect. 5.3 of the article the author detects another “problem of interpretation”, probably due to “a problem of translation from Greek to Arabic” illustrated by the alleged erroneous confusion of the notion of “architectonic” with the notion of architecture by Averroes. Apart from these alleged “misinterpretations”, the author notes some differences in the way prudence is viewed by both authors, for, she claims, while Aristotle considers prudence as a “practical virtue”, Averroes sees it rather as an “intellectual virtue”, which is reflected by the very word he uses to name it (i.e. taʿaqqul), whose root is “ʿaql”, namely, the Arabic translation of reason. The second article in this Part II is written by Nora Jacobsen Ben Hammed and is entitled “Ethics and Aesthetics: Theorizing Simile in Ibn Sīnā’s Risālat al-ṭayr and Ƥayy ibn Yaqẓān”. In this article the author analyses the ‘allegorical tales’ which Avicenna presents in his two treatises Ƥayy Ibn Yaqzān and Risālat al-ṭayr, from an ethical and an aesthetic viewpoint. More precisely, she analyses the similes introduced by Avicenna in these two treatises to extract their ethical as well as their aesthetic significance and considers that Avicenna is using them in order to “produce imaginative assent in the listener and encourage the practical soul’s control of lower faculties to aid in the process of the theoretical intellect’s self-perfection” (abstract). Her analysis shows that the bizarreness of some animals portrayed by Avicenna has an impact on the human imagination, and influences the human intellect by introducing a dialectical relation between imagination and intellect. These similes are analysed by the author in the light of Avicenna’s own philosophy in particular in relation with the ideas defended in the Poetics. The author pays attention both to the content of the tales and to their form in order to exhibit Avicenna’s aims and motivations in producing such allegories and the way he uses them. She shows that in his analysis of these tales, Avicenna demonstrates that literature can be used to “convey moral content” (Sect. 6.1) to literature and the metaphors used in its context. His aim is thus

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to convince the readers of some moral concepts and practices by means of imagination, which according to the author, is an important faculty of the soul in Avicenna’s philosophy. His use of these similes to make the soul reach the realm of intelligibles shows the closeness of Avicenna’s opinion with the Neo-platonic tradition as well as the “Islamic tradition of the miˡrāj” (Sect. 6.2). The very use of strange or even “repellent” animals is meant by Avicenna to provoke the readers and “produce surprised attentiveness” in their minds, according to the author. It is a way to bring their attention to these stories, and stimulate reflection to arrive at the morals to be deduced from such tales. This analysis has also an aesthetic dimension, as the author shows, when she talks about the disgust that some animals and descriptions produce in the minds of the listeners. This disgust and the comparable negative impressions produced by these descriptions aim at pushing away the listener from these beings and actions, hence, they lead to rejection, which introduces a relation between the aesthetic and the ethical dimensions, as can be seen in particular in Ƥayy Ibn Yaqzān. This aesthetic dimension is also conveyed by poetic and imaginative syllogisms, which differ from the demonstrative ones in that they don’t produce an assent to a true conclusion, but rather either attraction or repulsion. For this reason, the imagination, which has the power of moving the soul towards good and beauty or against evil and ugliness, can play a significant role by complementing the intellect. The third article in this Part II is entitled “Toward another understanding of the notion of fiṭra in the Avicennian ontology of the rational soul” and written by Meryem Sebti. In this paper, the author raises a problem related to the Avicennian notion of fiṭra (innateness), which she analyses by focusing on its ontological dimension and situating it in the context of Avicenna’s theory of the soul. The problem raised by the author can be summarized as follows: what is fiṭra in Avicenna’s theory and how can one define it in the light of Avicenna’s ontological doctrine of the soul? After a brief review of the main studies devoted to the notion of fiṭra in the literature on Avicenna’s philosophy, which generally focuse on its relation with the most fundamental and primitive logical principles called “awwaliyāt”, in order to determine whether or not they are innate ( fiṭrī), the author distances herself from these studies, and raises the problem in a more general and fundamental way by taking into account not only the epistemological and logical aspects of this notion, but rather and more specifically its ontological dimension. According to her, the notion of fiṭra must be understood in the light of what she calls the “Avicennian ontology of the rational soul” (introduction). For that purpose, she starts by analyzing Avicenna’s doctrine of the “rational soul” which holds the following fundamental difficulty: the soul is an immaterial substance “essentially distinct from its powers” but at the same time it “acts only through the mediation of its powers” (Sect. 7.2). This difficulty is at the heart of Avicenna’s doctrine of the soul and affects his conception of fiṭra. On the other hand, she distinguishes between the ontological problem – what the soul is in itself – and the epistemological problem – how the subject perceives

1 Introduction

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himself – while analyzing the nature of the soul as essentially spiritual and distinct from the body and from its “operative powers” (Sect. 7.2.1). As a consequence, the notion of fiṭra, i.e. “the original primary nature” is what “the soul is endowed with” (Sect. 7.3). This notion is used in several contexts and is seen as the origin of some kinds of knowledge – the wahmiyyāt (fictions) and the ʻaqliyyāt (rational knowledge) -which is not acquired but rather innate. However, the innate origin of this knowledge does not warrant its truth, as witnessed by the falsity of the wahmiyyāt, even if the ʻaqliyyāt are “always true” (Sect. 7.3). So, as the author stresses, “fiṭra is ambivalent” from an epistemological point of view, given that it can lead to truth as well as to falsity. The same ambivalence affects the fiṭra from an ontological viewpoint, for as the author claims, fiṭra “is oriented towards both the intelligible world and the body” (Sect. 7.3). The fourth article in this second part is entitled “Primacy of Existence versus Primacy of Essence – What is the Debate About?” and is written by Mansooreh Khalilizand. In that article the author is interested by the very old and fundamental distinction in the Arabic tradition between essence and existence. She raises the problem from both the metaphysical and the theological viewpoints. Starting from the debate about what is more fundamental (or “authentic”) with regard to reality existence or essence - she raises the questions of “what underlies” (introduction) that debate and “what precisely this debate is about” (abstract). As she expresses it, the question raised by the various authors is the following: Is it “being and existence which fill up the texture of reality” or rather is it “the varied essences which actually are”? She then analyses the different answers provided by some authors, essentially in the Persian area, starting from Avicenna but focusing much more on post Avicennian authors, such as al-Suhrawardī and Mullā Sadrā, among others. After briefly distinguishing between the authors who defend the primacy of existence (Avicenna, according to some of his followers, and Mullā Sadrā) and those who defend the primacy of essence such as Mīr Dāmād, she claims that this debate is closely connected to two main issues, namely “the theological problem of jaʿl in the Islamic-Neoplatonic account of creation and secondly, the highly controversial question concerning the meaning of existence as predicate in a proposition” (Sect. 8.2). Having said that, she devotes the next section of the article (Sect. 8.3) to the analysis of the first issue – the theological problem of ja‛l – and she examines the second issue – the definition of existence as a predicate – in Sect. 8.4 of the article. In Sect. 8.3, she starts by a view attributed to Avicenna but not explicitly present in his numerous writings, according to which “God did not make the apricot apricot, he rather brought it into existence” and uses this quotation to introduce and define the notion of ja‛l, i.e. “instauration”, by putting forward its theological (the notion of creation and the question of God’s Knowledge) as well as its metaphysical (the distinction existence / essence) connotations. This view is the general background for a number of Persian authors who discussed the issue, which as claimed by the author, has for them “the status of a specific philosophical question, designated by the title jaʿl” (Sect. 8.3). Thus, she quotes Ibn Sīnā’s al-Ilāhiyyāt to show that, for

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him, this notion of instauration would be closely related to existence in Avicenna’s theory, since the “object of the act of instauration” is “existence”, but then she focuses on Mullā Sadrā’s view, which appears to be more elaborate, given the distinction between simple and compound instaurations introduced and examined with great detail by this author. This view is contrary to that endorsed by al-Suhrawardī, Mīr Dāmād and Jalāl al-Dīn al-Dawānī who all defend the primacy of essence, in one way or another. This latter view is in turn criticized by Mullā Sadrā. In Sect. 8.4, she examines the thesis according to which existence is a predicate in a proposition as any other predicate, and denotes an accident (‛araḍ). This view is, according to her, defended by Ibn Sīnā, against some Muslim theologians (mutakallimūn), and is grounded on the grammatical structures of the propositions in Arabic and the absence of the copula (the verb ‘to be’) in that language. However, Mullā Sadrā points out to the fact that existence is not an accident like all others; rather it is fundamentally different in that it moves the subject “beyond the realm of consciousness and puts it in the very fabric of reality” (Sect. 8.4) and elaborates his own view about existence on this basis. The fifth and last article in this second part, is written by Kata Moser and is entitled ““Superstition” or “Crown of Science”? Zaki Naguib Mahmoud, Youssef Karam, and Yumna Tarief El-Kholy on Metaphysics”. Unlike the preceding papers, which talk about Medieval authors, this one is about contemporary Arabic authors, and analyses their views about metaphysics. The author’s aim is “to demonstrate the range of solutions that Arab philosophers offer to the crisis of metaphysics and where they locate the controversial issues” (introduction), rather than assessing the originality of these views or “discuss the reception history” (introduction). In order to do so, she analyses the theories about metaphysics defended by three Egyptian authors and the discussions and interactions or “interconnectedness” (introduction) between these three authors who defend opposed views but come nevertheless from the same cultural and geographic area. The first section is devoted to Zaki Naguib Mahmoud, a philosopher who represents the neopositivist trend in contemporary Arabic philosophy. The author shows that this author follows both Carnap and Wittgenstein and rejects metaphysics on the same basis as these two authors. For him as for Carnap, metaphysics is unable to give any adequate knowledge of reality, which only science is able to do. She analyses his justifications of this rejection, which according to the author are not free of originality, despite the faithfulness of Mahmoud to Carnap’s views, and also how his theories have been criticized by his colleagues in modern Egypt. The second section is devoted to the analysis of Youssef Karam’s views and his defense of a more classical view on metaphysics, a view influenced mainly by Medieval and classical authors. The author interprets his views as Neo-thomist and says that “he posits that metaphysics is the most abstract and most comprehensive science that deals with being as such, including God, and it is necessary in order to gain a correct knowledge of reality” (abstract). His position is thus completely contrary to that of his colleague and is defended on another, more traditional, basis. Finally, the author examines Yumna Tarief El-Kholy’s views on metaphysics, which are influenced by those of

1 Introduction

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Karl Popper, the Austrian contemporary philosopher of science. This scholar is opposed to her two Egyptian colleagues in that she admits that metaphysics is necessary because it is needed by science itself. As the author characterizes her view, “El-Kholy also holds metaphysics to be necessary by the sheer fact that all scientific theories include a metaphysical theory” (abstract). She thus seems to ground science on metaphysics, which makes her defend an authentic contemporary theory, but compatible with the acceptance of classical metaphysics, and distinguishes her from her two colleagues. For unlike Mahmoud, she does not reject metaphysics, but says that it is indispensable to science itself, and unlike Karam, she does not mix between metaphysics and theology and does not evoke God in her analysis. Nevertheless, despite the differences between the views of these three philosophers, the author shows that they do interact with each other by commenting on their respective positions and exchanging their respective arguments. In this respect, the author concludes by saying that “The Arab discourse on metaphysics thus reveals a dynamic that does not merely replicate the European dynamic, but has its own traits that emerge when Arab philosophers react to positions of their colleagues and reinforce or even develop their argument on this basis as has happened in particular with regard to existentialism” (conclusion).

Part I

Logic and Mathematics

Chapter 2

The Hypothetical Logic in the Arabic Tradition Saloua Chatti

Abstract In this paper, I will present the hypothetical logics of some authors pertaining to the Arabic tradition, namely al-Fārābī, Avicenna and some later logicians, in particular in North Africa, but not Averroes, as he gives that system very little importance. The main problems that I raise are the following: what is the hypothetical logic about? How is it viewed by the Arabic logicians, whether in early Medieval times (eighth–twelfth centuries) or in later ones (fourteenth–fifteenth centuries)? What are the main theories and results provided by these authors? The study of the hypothetical logic, which analyses the propositions containing ‘if . . .then’ and those containing ‘either . . .or’ shows that in al-Fārābī’s frame, this system is given some importance and developed in various ways. But it is Avicenna who studies these propositions at the greatest length and gives them the most importance, since he introduces quantification in this field and presents several systems of hypothetical logic, having distinct features. In later times, the hypothetical logic is given much importance in particular in the Western (i.e. North-African) Arabic areas with the writings of Ibn ‛Arafa and al-Sanūsi, among others, who seem influenced by Avicenna and especially his follower al-Khūnajī. I will thus present the basic features and rules of these systems, and will show the main differences between these various systems as well as the evolution of this field in the Arabic tradition. I will also analyse the rules provided in the light of the modern classical formalism of propositional and predicate logics.

2.1

Introduction

The hypothetical (sharṭīyya) logic is a kind of propositional logic, containing conditional plus disjunctive propositions. In the Arabic tradition, this logic is more or less developed, depending on the authors. Al-Fārābī, for instance, presents the five S. Chatti (*) Retired from the University of Tunis, Tunis, Tunisia © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Chatti (ed.), Women’s Contemporary Readings of Medieval (and Modern) Arabic Philosophy, Logic, Argumentation & Reasoning 28, https://doi.org/10.1007/978-3-031-05629-1_2

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Stoic indemonstrables and adds some variants, while Averroes seems to consider this kind of logic as really secondary. By contrast, Avicenna gives it much importance and constructs a very original system of his own, which mixes between hypothetical propositions and categorical ones. Avicenna’s improvements seem to have influenced his successors, such as Fakhr-al-Dīn al-Rāzī, Afḍal-al-Dīn al-Khūnajī and other scholars in later times, in both Asian and North African areas. In this paper, I wish to assess the contribution of Arabic authors in the field of hypothetical logic. Thus, the main problems that I raise are the following: what is the hypothetical logic about? How is it viewed by Arabic logicians? What are the main results gathered by these logicians? What are the main advances in this field? How did it develop in the Arabic writings? In what follows, I will show that this specific system has given rise to many advances both in propositional logic and in predicate logic, mainly with Avicenna, but also with some later authors.

2.2

The Different Kinds of Hypothetical Propositions

The hypothetical logic studies the propositions called ‘hypothetical’ (sharṭīyya). These propositions are either conditional (muttaṣila) or disjunctive (munfaṣila). Unlike the categorical ones, they contain the logical connectives ‘if. . .then’, when they are conditional and ‘either. . .or’, when they are disjunctive. These connectives relate two elementary propositions, which are considered as the simplest entities in the system, unlike the usual Aristotelian syllogistic, which analyses the propositions into subjects and predicates (plus quantifiers and sometimes modal operators) and studies the relations between them. In this respect, it is a kind of propositional logic, unlike syllogistic, which is a predicate logic containing quantified propositions. However, as we will see below, Avicenna introduces quantification in some of his hypothetical systems and gives it much importance. So, there is no radical separation between syllogistic and hypothetical logic in his frame, since he considers categorical syllogistic as a basis for all his subsequent systems, apart from the exceptive (istithnā’ī) system where the propositions are not quantified. Among the systems containing quantified propositions, we find first the one which contains only hypothetical conditional propositions, where he applies in full all the definitions and rules of traditional syllogistic, second all other ones, namely, the one containing conditional and disjunctive propositions, and the one containing predicative plus all kinds of hypothetical propositions. All these systems are constructed in the fashion of categorical syllogistic and contain the same three figures and many moods comparable to the categorical ones, although some of their rules and moods are new and original.1

1

See Chatti, Saloua, Arabic Logic from al-Fārābī to Averroes, section 5.1.3, Springer Nature 2019.

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The hypothetical propositions are of two kinds: (1) The conditional ones, which contain ‘if. . .then’ and relate between an antecedent and a consequent, e.g. If A is B, then C is D, and (2) The disjunctive ones, which contain ‘either. . .or’, e.g. Either A is B or C is D. So, the first problem that arises is the following: how can we interpret these two kinds of hypothetical propositions? In other words, how the conditional and the disjunction are they defined by the different Arabic logicians? In answering these questions, we will consider both the definitions provided in the systems considered and the rules and moods held in these systems. Let us start by al-Fārābī’s system.

2.3

The Basic Moods and Their Variants in al-Fārābī’s Frame

In al-Fārābī’s frame, the hypothetical moods presented are first: (1) (2) (3) (4)

The Modus Ponendo Ponens [If p then q, but (¼ lākin) p; therefore q] The Modus Tollendo Tollens [If p then q, but not q; therefore not p] The Modus Ponendo Tollens [Either p or q, but p; therefore not q] The Modus Tollendo Ponens [Either p or q, but not p; therefore q] (my formalization)

Note that al-Fārābī often expresses these moods by using concrete propositions. For instance, he expresses the first mood (Modus Ponendo Ponens) as follows in his treatise Kitāb al-Qiyās: If this visible thing is a man, then it is an animal But (lākin) it is a man Therefore, it is an animal (al-Qiyās, p. 32)

But this use of concrete propositions may be seen as a pedagogical device rather than a lack of formalism in his system. For he pays attention to the structure of these syllogisms and presents them sometimes by naming their components and stating explicitly the relations between the premises and the conclusion, without any recourse to a concrete example, for instance, when he says: “the first hypothetical syllogism contains as a second premise the antecedent itself (bi ‛aynihi), thus deduces the consequent itself (bi ʻaynihi)” (al-Qiyās al-Ṣaghīr, p. 167). The conditional is not material, unlike the Stoic’s conditional, for the antecedent and the consequent must be related in one way or another. This link can be either accidental (‛aradī) as in the sentence “If Zayd comes, ʻAmr leaves” (Kitāb al-Maqūlāt, p. 127), when there is no necessity in that link, or essential, when there is a strong or at least a regular link between the antecedent and the consequent, in which case the consequent follows the antecedent either “most of the time” as in “If Sirius passes the zenith in the morning, the heat will be severe and the rains will cease” (Kitāb al-Maqūlāt, p. 127) or it follows it at all times, as in the classical

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sentence ‘If the sun rises, it is daytime’ or else in the following one ‘If this is a human, it is an animal’ (but not conversely) (al-Maqūlāt, p. 78). In the last two sentences, the link is either causal or conceptual, since by definition, humans are necessarily animals. This essential conditional may also be a biconditional, when it is convertible. This biconditional is illustrated by the classical sentence ‘If the sun rises, it is daytime’, in al-Fārābī’s frame. However, the conditional involving a regular link may raise some problems, since the link is not really necessary. So one may ask: when such a regular conditional is used in Modus Ponens, for instance, are we allowed to deduce the consequent ‘q’ from the truth of the conditional ‘if p then q’ and that of the antecedent ‘p’, since the following occurs in most times, but not in all of them? If we consider ‘if p then q’ true (on a statistical or a probabilistic basis), and we verify the truth of the antecedent (empirically), nothing warrants the truth of the consequent, which could, in certain circumstances, be false, precisely in those circumstances where there is an exception to the general (empirical) ‘law’ expressed by the conditional sentence “If Sirius passes the zenith in the morning, the heat will be severe and the rains will cease”, given as an example. This problem is raised by Kamran Karimullah, for instance, when he says, speaking of the conditional exemplified by that sentence: “when both (P1) ‘if Sirius passes the zenith in the morning, then it will be hot’. . . and (P2) ‘but Sirius passes the zenith in the morning’ are true”, the consequent could be false, since “the respondent will still not give his assent to the conclusion that it is a hot day, for the simple reason that it is not. So modus ponens with conditional premises read ‘for-the-most part’ (and, a fortiori, per accidens) is classically invalid” (Karimullah, 2014, p. 252). He uses this example and other ones to show that al-Fārābī’s conception of the conditional is not unified, and that the truth-conditions of the different conditionals are context-dependent. However, with regard to the conditional occurring at regular times, or even accidentally, one could still warrant the validity of Modus Ponens and the other hypothetical moods by adding the relevant conditions. For instance, instead of simply saying ‘if Sirius passes the zenith in the morning, then it will be hot’, one could say “if Sirius passes the zenith in the morning, then it will probably be hot’ (since it is a ‘for-the-most part’ conditional, as Karimullah calls it), in which case the conclusion of Modus Ponens will be ‘it will probably be hot’ and will follow necessarily from the two premises. Likewise with the accidental conditional illustrated by “If Zayd comes, then ʻAmr leaves”. This conditional is accidental, it may be true only at some times, not regularly nor necessarily; so we could add the condition ‘on occasion’ and say “If Zayd comes, then ʻAmr leaves (on occasion)”, meaning that it can happen to ʻAmr to leave when Zayd comes. This would allow the deduction of the consequent ‘ʻAmr leaves (on occasion)’ from the two premises in all cases. So there is a way to complement al-Fārābī’s conception by adding the required and relevant conditions that would warrant the validity of the hypothetical moods. Al-Fārābī also adds some variants, which are the following moods:

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I/ If not P then not Q, But not P Therefore not Q (al-Qiyās assaghīr, p. 167, my formalization) II/ If not P then Q But not P Therefore Q (al-Qiyās assaghīr, p. 167, my formalization) III/ If not P then not Q But Q Therefore P (al-Qiyās assaghīr, p. 167, my formalization) IV/ Not (P and Q) But P (or but Q) Therefore not Q (or therefore not P; which is the third Stoic indemonstrable) (al-Qiyās, p. 139, my formalization) VI/ P or Q or R But P Therefore not Q and not R (al-Qiyās, p. 139, my formalization) VII/ P or Q or R But not q Therefore P or R But not P Therefore R (al Qiyās, p. 139, my formalization) However, he says that from the following two premises: “Not (P and Q) But not P we cannot deduce: Therefore Q” (al Qiyās, p. 139, symbols added) Because Q could be false, since the disjunction is not exhaustive. The exhaustiveness of the disjunction means that the disjuncts are stated in full, so that there are no more alternatives that could be added. For instance, in the sentence ‘This number is either even or odd’, the disjunction is exhaustive because there is no other alternative that could be added to ‘even’ and ‘odd’. It is also exclusive, of course, because the two elements exclude each other, being contradictory. But in ‘not (P and Q)’, there is no exhaustiveness, because P and Q do not exhaust the whole set of alternatives, so that if ‘not P’ is true, this does not mean that Q can be deduced or considered true on that basis. For instance, when one says: ‘not (Zayd is in Iraq and Zayd is in Shām)’, from this and the sentence ‘Zayd is not in Iraq’, one cannot deduce that Zayd is in Shām,

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because he could be in Khurāsān or anywhere else, the alternatives being multiple and not exhausted by this kind of disjunction. As to mood VI, we can say that the disjunction which it contains is exclusive, which means ‘P or Q but not both’, for if it were inclusive (if it means ‘P or Q or both’), we could not deduce ‘not Q’ from ‘P’, because an inclusive disjunction can be true when both its elements are true, so that if one of them is known to be true, the other one is not necessarily false, and consequently its negation is not necessarily true; so this negative element cannot be deduced from the disjunctive premise and the truth of the first element. It should also be exhaustive, because it should not admit more alternatives than it actually does, otherwise, the deduction would not be valid. Mood VII contains a disjunction that can be exclusive (meaning ‘p or q but not both’) or inclusive (meaning ‘p or q or both’), but should also be exhaustive; otherwise it could admit more alternatives and the conclusion would not follow from the premises stated. For instance, we could say ‘This number is either equal or inferior or superior to Zero’, in which case the conclusion really follows from the premises; but if we say ‘This thing is either white or black or yellow’, since the whole set of colours is not exhausted, and since the thing could be blue or red, the conclusion does not necessarily follow, unless there is some evidence related to the nature of that thing, that the other colours are not pertinent anyway, for instance if we are talking about the colours of human beings or that of some animals, which are necessarily restricted to some colours, and do not extend to the whole set of colours. So, exhaustiveness is related to the matter of the propositions rather than to the definition of the word ‘or’; while exclusiveness and inclusiveness are both related to the definitions of ‘or’, which can be determined by the truth conditions of the disjunctions. As to the complete conditional, al-Fārābī says that it is the “convertible” one (al-Maqūlāt, p. 79) among the “essential” conditionals, which gives rise to the following moods: ‘If p then q; but q; therefore p’ ‘If p then q; but not p; therefore, not q’ (al-Maqūlāt, p. 79)

In these two moods, ‘if. . .then’ means ‘if and only if’. But although there is no specific group of words by which this meaning is expressed, in al-Fārābī’s frame, two words seem to clarify the meaning of this double conditional: the word ‘convertible’ which qualifies this connective, and the word ‘complete’ (tāmm), by which al-Fārābī calls this double implication. The ‘complete implication’ is the one where the two elements are convertible and symmetrical, as witnessed by the following quotation: And those expressing a complete implication (alladhāni luzūmuhumā tāmm) are those where if whatever (ayyuhumā) component holds, the other one necessarily holds too by means of it (bi-wujūdihi), for if the first one holds, the second one necessarily holds, and if the second one holds, the first one necessarily holds too (al-Maqūlāt, p. 78.16-18, my emphasis).

One might think that this double implication is truth-functional, since its truth values seem to be quite clear, but al-Fārābī does not seem to consider it in this way, since he uses the modal word ‘necessarily’ in his definition, which clearly indicates that the

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two elements must be semantically related so that the truth of the whole does not only depend on the truth values of the elements: one must know the meanings of these elements in order to determine the truth value of the whole biconditional. This biconditional could be seen as close to a strict biconditional of C. I. Lewis’ kind, given the presence of the world ‘necessarily’, but Lewis’ motivations are not shared by the traditional logicians, who do not have a clear idea about the so-called ‘paradoxes of material implication’,2 which were at the heart of Lewis’ concerns and gave rise to his specific modal logic, although according to Łukasiewicz, there were some discussions about the meanings of the implication between the Philonian school and the Diodorian one.3 So one can simply say that the conditional (and the biconditional), in al-Fārābī’s frame, express the ordinary uses of these connectives, which take into account the meanings of the antecedent and of the consequent; but they do not reflect Lewis’ concerns or problems, despite the closeness between the luzūm and the strict implication. Generally speaking, al-Fārābī’s opinion about the conditional is different from that of the Stoics. For the Stoics consider it as a material conditional or “a Philonian implication” as Łukasiewicz says in his article about the Stoic logic, whose truth conditions are determined: it is false only when the antecedent is true and the consequent is false, and true in all other cases. While al-Fārābī says that the antecedent and the consequent must be related semantically so that the conditional is not really truth-functional. According to him, the case of falsity is settled: it is the one where the antecedent is true while the consequent is false. The case where the two elements are true is true when they are semantically related. But the truth values of the two cases where the antecedent is false are not warranted: in these two cases, the conditional can be true and it can also be false depending on the meanings of its elements. So the conditional seems to be intensional in his frame. It can hardly be truth-functional. As to the disjunction, it can be exclusive and exhaustive, as when we say ‘either p or q but not both’. In this case it is complete and validates both Modus Ponendo Tollens and Modus Tollendo Ponens. But it can also be only exclusive, when it validates ‘Not (p and q), but p; therefore, not q’. In this case, it is incomplete, because it does not validate the following: ‘Not (p and q) But not p Therefore q’.

In the disjunction expressed by ‘not (p and q)’, the two disjuncts are incompatible with each other, because they cannot be true together. But although incompatible (¼ contrary) they are not necessarily contradictory, which means that they can be false together, if another alternative is true. For instance, if one says ‘Not (this thing is blue

2

See, for instance, Susan Haack, Philosophy of Logics, p. 176. See Łukasiewicz “Contribution à l’histoire de la logique des propositions” in Largeault, Jean, Logique mathématique, textes, Armand Colin, Paris, 1972, in particular, p. 15.

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and it is red)’, the sentence can be true if the thing is only red but not blue, or conversely, but also if it is neither red nor blue, e.g. white or black. The incomplete disjunction may have an undetermined number of disjuncts, so it is not exhaustive. But al-Fārābī does not admit the inclusive meaning of the disjunction (i.e. ‘either p or q or both’), since according to him, the disjunction always involves some kind of conflict (‛inād). In all cases, the disjunction is not truthfunctional, because its truth value depends mainly on the meanings of its elements, not only on their truth values. These elements are always incompatible in the examples provided. For instance, he often uses propositions containing contradictory words, such as ‘even’ and ‘odd’ when he talks about the exclusive disjunction. In other cases, he uses propositions containing contrary words such as ‘hot’, ‘middling’ and ‘cold’ or ‘white’, ‘black’ and ‘yellow’ for instance, when the disjunction admits more than two elements. In all cases, he does not use propositions containing words whose meanings are compatible with each other, such as ‘coloured’ and ‘white’, for instance. So, the disjunction in his frame seems to always involve some kind of incompatibility, and also to be non-truth-functional, since the meanings of the elements are crucial to determine the truth value of the disjunctive propositions. In this latter respect, the disjunction is close to that of the Stoics, which is also non-truthfunctional, as claimed by Susanne Bobzien (SEP, 2006, section 5.2). Let us now turn to Avicenna’s analysis of the hypothetical moods and propositions.

2.4

The Systems of Avicenna

Avicenna admits several kinds of hypothetical systems. The first one is comparable to that of al-Fārābī and contains what Avicenna calls ‘the istithnā’ī (exceptive) syllogisms’, namely the syllogisms of the Stoic kind. Its basic moods are presented in the very end of al-Qiyās. The second one contains what Avicenna calls ‘iqtirānī’ (conjunctive)4 syllogisms, that is, syllogisms that are close to the Aristotelian kind.

4

Alternative translations are available in the literature. For instance, Tony Street translates ‘istithnā’ī’ by ‘repetitive’ and ‘iqtirānī’ by ‘connective’ in his article ‘Arabic and Islamic philosophy of language and logic’ (SEP, 2008, revised 2013). However, as explained in Chatti and Hodges’ translation of al-Fārābī’s book al-Qiyās, (2020) the meaning of the word ‘istithnā’ is much closer to that of ‘exception’ in classical Arabic than it is to ‘repetition’ (see the introduction to al-Fārābī, Syllogism, p. 26 + p. 56), for ‘repetitive’ is a very indirect way to translate ‘istithnā’ī’. Besides that, if the notion of repetition occurs in this kind of moods, it occurs only in some of them, namely Modus Ponens, while in Modus Tollens, on the contrary, there is no repetition, strictly speaking of one part of the first premise, since the second premise is the contradictory of that part, not that very part. As to the word ‘connective’ used by T. Street as a translation of ‘iqtirānī’, it is more pertinent to translate ‘ittiṣālī’, which carries approximately the same vague meaning. ‘iqtirānī’ is better translated as ‘conjunctive’, since in categorical as well as in hypothetical syllogistics, the two premises, which are different from each other, are related by a conjunction in all such iqtirānī moods.

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The difference between these two kinds of syllogisms is that in the former, the second premise contains the word ‘but’ (lākin) which is the exceptive word, and repeats either one part of the first premise or the contradictory of one part of that premise, and the conclusion is either the other part or the contradictory of the other part, while in the iqtirānī syllogisms, both premises are different, although they share some terms, and the conclusion is new and different from them both. The originality of Avicenna is that he introduces this second kind of syllogisms inside the hypothetical logic, which usually contains only the first kind of syllogisms. By doing so, he creates a whole system, which mixes between all kinds of hypothetical propositions and categorical ones. This system contains three sub-systems, which are the following: A/ A system which deals with quantified hypothetical conditional propositions, parallel to the categorical ones. B/ A system which deals with quantified conditional as well as disjunctive hypothetical propositions. C/ A system which mixes between quantified conditional and disjunctive propositions and categorical ones.

2.4.1

The Istithnā’ī (Exceptive) Syllogisms

The istithnā’ī system is comparable to al-Fārābī’s one, although there are some differences in the moods held and in the definitions of the connectives. For instance, according to Avicenna, the conditional is never convertible, i.e. is not a biconditional, unlike what al-Fārābī says, although he does state in the end of al-Qiyās the moods containing a convertible implication, which he attributes to the usual (al-mashhūr) doctrine (see al-Qiyās, p. 390.6-7, for instance). The reason of this rejection of the biconditional has to do with the structure of the conditional proposition. According to Avicenna, the order of the antecedent and the consequent should not change in the conditional proposition: the antecedent is always what comes first and expresses the condition, and the consequent is always what comes next and follows the word ‘then’. If one changes this order, one changes the structure and the meaning of that connective, and introduces in his definitions matter considerations (mawādd al-muqaddima) (al-Qiyās, p. 392.2), which should not be taken into account, for this would be like considering the copula in categorical propositions as expressing a kind of identity in some cases, which would lead to different moods (for instance, to something like Darapta, instead of Darapti), in categorical logic (al-Qiyās, p. 392.4-15). This is why one has to stick to the initial form of the conditional proposition and define the conditional on that basis. Thus, by definition, the conditional is not convertible; otherwise, it becomes something else. But like al-Fārābī, Avicenna admits the following syllogisms in his istithnā’ī hypothetical system:

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1. 2. 3. 4.

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Modus Ponendo Ponens Modus Tollendo Tollens Modus Ponendo Tollens Modus Tollendo Ponens (al-Qiyās, pp. 391–401)

However, unlike the Stoics, who consider the conditional used in the first premise of these moods as a material (i.e. a truth-functional) conditional, Avicenna says that the Modus Ponens, for instance, is valid only when there is a real semantic or causal link between the antecedent and the consequent of the first conditional premise. So the conditional in the first premise of the Modus Ponens (and also the Modus Tollens), should not be an ittifāqīyya relation, where the consequent does not really follow from the antecedent (al-Qiyās, p. 390), nor is it a material implication of the Stoic kind. Rather it is a strong implicative (luzūmīyya) relation (al-Qiyās, p. 390-16-17), where the consequent must really follow from the antecedent, either semantically or causally. Note that he does not evoke al-Fārābī’s ‘most-of-the-time’ conditionals nor the accidental ones, for the conditional giving rise to the moods above should always involve a strong semantic or causal relation, otherwise, the moods are not valid. Like al-Fārābī, he also states the syllogism whose first premise is a negated conjunction, i.e. the third indemonstrable of the Stoics: 5. Not (p and q), but p; therefore, not q (al-Qiyās, p. 452) He also states the following syllogism: 6. Either not p or not q, but p; therefore, not q (al-Qiyās, p. 451) Here, ‘not (p and q)’ the first premise of (5) is replaced by ‘Either not p or not q’ the first premise of (6). This replacement might suggest that he equates ‘Either not p or not q’ with ‘not (p and q)’, which could be a reason to say that he holds implicitly the following De Morgan’s Law: ‘~ (p ^ q)  (~ p _ ~ q)’, since everything else in moods (5) and (6) is the same. But the law is not stated nor explicitly analysed in Avicenna’s frame, although Avicenna did use it implicitly in his analysis of the modal propositions, when he stated the contradictories of the bilateral possible propositions and of the quantified hypothetical ones, when analysing the contradictories of the disjunctive hypothetical A propositions.5 Nevertheless, even if we could not attribute to him the ‘discovery’ of the De Morgan’s laws, these moods show at least that he admits the inclusive meaning of the disjunction, which was not mentioned by al-Fārābī. The disjunction is itself considered either as exclusive and exhaustive, when both elements are contradictory to each other or as only exclusive when these elements are incompatible with each other, i.e. contrary, or else as inclusive as in mood (6) above. This distinction between three different meanings of the disjunctions is claimed explicitly in al-Ishārāt wa at-Tanbīhāt, where Avicenna provides three distinct meanings of the disjunction. 5 See S. Chatti, Arabic Logic from al-Fārābī to Averroes, 2019, sections 4.1.2 and 5.1.3.2 respectively.

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These meanings are explained in the following way by Naṣīr al-Dīn at-Tūsī in his comment of al-Ishārāt: According to that author, in its first meaning, the disjunction “prevents the conjunction (tamna‛u al-jam‛) and the vacuity (wa al-khuluw)” (al-Ishārāt, p. 250) [i.e. it does not admit the case where P and Q are both true, nor the case where they are both false], while in the second meaning, it prevents only the conjunction [i.e. it does not admit the case where P and Q are both true], and in the third meaning, it prevents only the vacuity [i.e. it does not admit the case where P and Q are both false]. So the first meaning seems to be the exclusive disjunction, that is, ‘P or Q but not both’, while the second meaning seems to be the negation of a conjunction, that is, ‘not both P and Q’ and the third meaning seems to be the inclusive disjunction, namely ‘P or Q or both’, since the only case which is false in that kind of disjunction is the one where both propositions are false. Does this mean that the disjunction is truth-functional in Avicenna’s view? This question is legitimate, since the truth conditions of the three disjunctions seem to be clarified by Tūsī’s explanations. But things are not so simple, for in Avicenna’s view, the truth value of the disjunction depends on the meanings of the two disjuncts, even when there is no incompatibility between these disjuncts, as in the third case. This being so, the disjunction is not really truth-functional in his frame. However, as we will see below, these definitions will become really truthfunctional in the systems of some followers of Avicenna. As to the conditional, it is either a luzūmī one (where the consequent really follows from the antecedent) or an ittifāqī one (where the consequent does not really follow from the antecedent). This distinction will also be admitted by Avicenna’s followers such as al-Khūnajī and the scholars who are influenced by him. He also admits the following equivalences between the conditional and the disjunctive propositions: (1) (P ! Q)  ~(P ^ ~ Q) (al-Qiyās, p. 280) (2) (~P ! Q)  (P _ Q) (al-Qiyās, p. 244.16) (3) (P ! Q)  (~P _ Q) (al-Qiyās, p. 251) The last equivalence, for instance, is expressed explicitly in the following passage of al-Qiyās, where Avicenna claims: When they say: ʻNot (A is B) or C is Dʼ . . ., it is undoubtedly (lā maḥāla) a hypothetical (shartiyya), [. . .], it thus resembles the following conditional: ʻIf A is B, then C is Dʼ. . . (al-Qiyās, p. 251).

All these equivalences will be used by Avicenna in his quantified hypothetical system. Avicenna also admits the principle of contraposition, which is called “‘aks al-naqīḍ’ and relates between a conditional where both elements are affirmative and another one “where the antecedent is the contradictory of the first one’s consequent and the consequent is the contradictory of the first one’s antecedent” (al-Qiyās, p. 3856). Formally, this relation is expressed as follows: (P ! Q)  (~Q ! ~P). There is an error in the text, where ‘contradictory of the antecedent’ is put in place of ‘contradictory of the consequent’.

6

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Apart from contraposition, Avicenna admits a straight conversion which is expressed as follows: “The straight conversion is where the antecedent becomes a consequent, and the consequent becomes an antecedent, while the quality remains the same and the truth is preserved” (al-Qiyās, p. 385). It is illustrated by the following example: “‘Never when every A is B, then every C is D’, it is clear that this implies ‘Never when every C is D then every A is B’” (al-Qiyās, p. 385). The conversion given in this example is the counterpart of E-conversion in categorical logic and can be formalized as follows: (8s)(Ps ! ~Qs)  (8s)(Qs ! ~Ps), which is the quantified counterpart of the following equivalence: ‘(P ! ~Q)  (Q ! ~P)’, itself deducible from the principle of contraposition stated above. We will return to this formalization of the universal negative proposition below. All these principles and equivalences will be used and discussed by his followers in eastern and western areas as we will see below. Let us now turn to the second kind of systems.

2.4.2

The Iqtirānī (Connected) Syllogisms

In this kind of hypothetical systems, Avicenna uses quantified propositions which contain either a conditional or a disjunction. These propositions are parallel to the categorical A, E, I and O propositions, except that they contain whole propositions instead of a subject and a predicate. The quantifiers range over states. The conditional hypothetical propositions are expressed as follows: – Ac: Whenever (kullamā) A is B then H is Z (al-Qiyās, p. 265) – Ec: Never when A is B then H is Z (p. 280) – Ic: Sometimes (qad yakūn) when every A is B then every H is Z (al-Qiyās, p. 278) – Oc: Not whenever A is B then C is D. While the disjunctive hypothetical propositions are expressed as follows: – – – –

AD: Always either A is B or C is D ID: Sometimes either A is B or C is D ED: Never either A is B or C is D OD: Not always either A is B or C is D

The first natural formalization of the conditional ones is the following (where ‘!’ stands for the conditional and ‘&’ stands for the conjunction in Z. Movahed’s symbolisations):

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– – – –

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Ac: (8s)(Ps ! Qs) Ic: (∃s)(Ps & Qs) Ec ¼ (8s)(Ps ! ~Qs) Oc ¼ (∃s)(Ps & ~ Qs)7 (Movahed, 2009, pp. 8–9, notation of Z. Movahed)

This formalization validates the principle of contraposition for quantified propositions, which we could express as follows: – ‘(8s)(Ps ! Qs)  (8s)(~Qs ! ~Ps)’ It validates Ec conversion and Ic conversion, which Avicenna holds too (with ‘!’ for the conditional and ‘^’ for the conjunction): – Ec: (8s)(Ps ! ~Qs)  (8s)(Qs ! ~Ps) – Ic: (∃s)(Ps ^ Qs)  (∃s)(Qs ^ Ps) However, it does not validate Ac conversion, for from ‘(8s)(Ps ! Qs)’ one cannot deduce ‘(∃s)(Ps ^ Qs)’, nor does it validate subalternation, contrariety and subcontrariety which Avicenna holds in this system too. It does not validate Darapti and Felapton either, which Avicenna does admit in this system too. To validate these rules and moods, one has to add an existential complementary clause ‘(∃s)Ps’ to the Ac propositions, which should thus be formalized as follows: Ac: (∃s)Ps ^ (8s)(Ps ! Qs) (consequently Oc must be formalized accordingly) This validates Ac-conversion, Darapti and Felapton, plus all the relations of the square. So, we must say that Ac admits two formalizations, which can be used in different occasions8: (1) Ac with the existential clause. (2) Ac without the existential clause. We will see below that the second Ac is used in the moods containing both conditional and disjunctive propositions, while the first Ac is needed mainly in the moods of the third figure containing only conditional propositions. These formalizations are not clearly distinguished but they can both be supported by some evidence in the text. (1) is supported by the following quotation: When we say: ʻIf A is B, then H is Zʼ, we assume from this (nūjibu min hāḍha) that at each time (ayyi marra min al-marrāt) where 'A is B' is the case (kāna AB) and when (wa matā kāna) A is B then H is Z, as if the fact that H is Z follows the fact that A is B, in so far as in effect A is B (min hayṭhu huwa kāʼinun A [huwa] B) (al-Qiyās, p. 263.8-9, my emphasis).

As to (2), it can be justified by the admission of contraposition and the equivalences between some conditional and disjunctive propositions. For we can say and prove

In Zia Movahed’s formalizations, ‘s’ stands for ‘situation’, but since Avicenna uses the word ‘ḥāl’, a better interpretation of ‘s’ is ‘state’, which is used by Strobino in his article ‘Ibn Sīnā’s Logic’ (SEP 2018). 8 See Chatti, 2022 for more details. 7

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that Ac (without the existential clause), i.e. ‘(8s)(Ps ! Qs)’ equals ‘(8s)(~Ps _ Qs)’, that is, ED. Below, we will see that both Ac’s are needed and are used in different sub-systems. As to the disjunctive propositions, they are formalized as follows by Nicholas Rescher (where ‘t’ refers to time, in Rescher’s interpretation): AD: (8t) (Pt _ Qt) ED: (8t) ~ (Pt _ Qt) ID: (∃t) (Pt _ Qt) OD: (∃t) ~ (Pt _ Qt) (1963, p. 233, Rescher’s notation, names of the propositions added) where the disjunction has a univocal meaning, namely, the exclusive one. Unfortunately, these formalizations, in particular those of ED and ID, do not validate Avicenna’s moods. Alternative formalizations have been suggested by Pr. Wilfrid Hodges, which can be rendered as follows (where, again, ‘t’ refers to ‘time’), namely: ED: either as 1. (8t)(Pt _ ~ Qt) or as 2. (8t)(~Pt _ Qt) (¼ (8t)(Pt ! Qt)) (or as Pr. Hodges writes “At all times t, if p is true at t then q is true at tˮ) (Hodges, 2016, p. 263). ID: either as 1. (∃t)(Pt ^ ~ Qt) (or as Pr. Hodges writes “There is a time at which p is true and q is not trueˮ (Hodges, 2016, p. 263)) or as 2. (∃t)(~Pt ^ Qt), which is required to validate some other moods, as noted by Pr. Hodges. As to AD, it has explicitly two distinct meanings in Avicenna’s texts. The first one corresponds to an inclusive disjunction, while the second one (called “strict disjunction”) corresponds to an exclusive disjunction. They can be formalized as follows (where I have replaced ‘t’ by ‘s’ (standing for ‘states’), since Avicenna uses the word ‘ḥāl’ in his text): AD1: (8s)(Ps _ Qs) AD2: (8s)(Ps _ Qs) (strict disjunction) Consequently, their contradictories are the following: OD1: ~(8s)(Ps _ Qs) (the contradictory negation of AD1) OD2: ~(8s)(Ps _ Qs) (the contradictory negation of AD2) [where ‘_’ means: P or Q but not both, ‘_’ means: P or Q or both] In all cases, however, Avicenna holds the contradictions AD / OD and ED / ID. Let us now turn to the first of these systems, which is also the simplest one.

2.4.2.1

The Iqtirānī Syllogisms with Conditional Propositions

This first system is exactly parallel to categorical syllogistic, since it deals only with conditional propositions. In this system, Ac should be the one with the existential

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clause, otherwise, Ac-conversion does not hold and Darapti and Felapton do not hold too. As an example of the moods held, we can state the following hypothetical Barbara: “Whenever A is B, then C is D Whenever C is D, then H is Z Therefore, Whenever A is B then H is Z” (al-Qiyās, p. 296.3-4) Likewise, for the other first figure moods and the second figure ones. In the third figure, Darapti, for instance, is stated as follows: “Whenever C is D, then H is Z Whenever C is D, then A is B Therefore, Sometimes, when H is Z, A is B00 (al-Qiyās, p. 302) As we can easily show, this mood is valid only when Ac contains the existential complementary clause. Otherwise, it is not valid, given that the conditional does not imply the conjunction. So, this first system should use these specific Ac and Oc. Only in that case, it would be consistent and would admit the whole number of conclusive moods, namely 4 conclusive moods in the first figure, 4 conclusive moods in the second figure and 6 conclusive moods in the third figure, as in categorical logic, which contains only three figures in Avicenna’s system, the fourth figure being rejected because of its ‘unnaturalness’. Note that even in categorical logic, the third figure moods are valid only when the categorical A contains an additional existential clause, and O is its exact contradictory. In categorical logic, this is justified by Avicenna’s opinion about the import of the different propositions, according to which affirmative propositions have an import, while negative ones do not have an import. Let us now turn to the second system, which mixes between conditional and disjunctive hypothetical propositions.

2.4.2.2

The Iqtirānī Syllogisms with Conditional and Disjunctive Propositions

In this second system, the disjunctive propositions are added to the conditional ones, which gives rise to new moods. These moods are different from the ones above, although Avicenna tends to reduce the disjunctive propositions to the conditional ones. This system is presented in section VI.2 of al-Qiyās, which is subdivided into 10 groups (governed by specific conditions), each of which containing 3 or more sub-groups.9 For instance, in Group I (sub-group I-a), we find the following mixed Barbara:

9

See Chatti, 2019, section 5.1.3.2 (B), for more details.

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“Whenever H is Z, then C is D (AC) Always either C is D or A is B (AD) Therefore, Whenever H is Z, then not A is B (AC)” (al-Qiyās, 305.8-10) Avicenna proves this mood by replacing the disjunctive proposition by a conditional one, given his claim that the disjunctive proposition is equivalent to the following conditional proposition: “Whenever C is D, then not A is B” (al-Qiyās, p. 305.10). This allows him to deduce the conclusion by Barbara ADACAC and explains why the conclusion contains a negative element. Despite this negative consequent in the conclusion, the conclusion is considered as affirmative by Avicenna because the negative (or the affirmative) character of the conditional proposition is not related to its consequent or to its antecedent. Rather it is related to the conditional itself. However, this mood is valid only when the disjunction is exclusive. It is not valid if the disjunction is inclusive, as Avicenna rightly says. Other moods are totally new such as the following AEE of the first figure: “Never if H is Z then C is D (EC) Always either C is D or A is B (AD) Therefore, Never either H is Z or A is B (ED) (al-Qiyās, p. 306.3-5) This mood is proved by reducing the conditional proposition to ‘Whenever H is Z then not (C is D)’ and the disjunctive to ‘Whenever not (C is D) then A is B’. From these premises what follows is ‘Whenever H is Z then A is B’ (al-Qiyās, p. 306). If we write down these two premises, and the conclusion that follows from them, we get the following mood: Whenever H is Z then not (C is D) (Ac) Whenever not (C is D) then A is B (Ac) Therefore, Whenever H is Z then A is B (Ac) (by Barbara AcAcAc) This conclusion, in turn, implies ‘Never either H is Z or A is B’, since ‘P ! Q’ implies ‘~P _ Q’, and ED is rendered as ‘(8s)(~Ps _ Qs)’, as we saw above. This mood I is not Celarent. Rather it is an AEE mood of the first figure, which does not correspond to any known mood of categorical logic or of the hypothetical system containing only conditionals. So it is a new mood. Another AEE mood of the first figure, pertaining to Group II (Sub-Group II-b) is stated as follows: “Never if H is Z then C is D (EC) Always either C is D or not (A is B) (AD) Therefore, Never if H is Z then A is B (EC) (al-Qiyās, 307.15-17) This mood is also valid as we can show very easily by formalizing the propositions and as Avicenna himself shows when proving it. Here too, he proves the mood by reducing the Ec proposition to the following Ac ‘Whenever H is Z, then not (C is D)’ and the disjunctive proposition to ‘Whenever not (C is D) then not (A is B)’,

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from which the conclusion ‘Whenever H is Z then not (A is B)’ is deduced. This can be expressed as follows: Whenever H is Z, then not (C is D) (Ac) Whenever not (C is D) then not (A is B) (Ac) Therefore, Whenever H is Z then not (A is B) (Ac) (by Barbara AcAcAc) This conclusion, in turn, implies the following Ec proposition ‘Never if H is Z then A is B’, which is the conclusion of the initial mood, since both Ac propositions ‘(∃s)Ps ^ (8s)(Ps ! ~Qs)’ (with the existential clause) and ‘(8s)(Ps ! ~Qs)’ (without the existential clause) entail the following Ec proposition: ‘~(∃s)(Ps ^ Qs)’ [which is equivalent to ‘(8s)(Ps ! ~Qs)’], given the formalizations above, which render Avicenna’s intuitions as closely as possible, and given Avicenna’s analyses and proofs of the different hypothetical moods (see, for instance, al-Qiyās, p. 367 and elsewhere in the same section). For since Ic is rendered as a conjunction, Ec, which is its contradictory negation, negates this conjunction, according to Avicenna’s view that the negative propositions negate the connecting (ittiṣāl) itself, not the consequent of the connected (or conditional) proposition, and this is exactly what the proposition Ec does. In these moods, as we can see, the Ac propositions used are mainly those that do not contain the existential clause, for the proofs used by Avicenna are often made by reducing the disjunctive propositions to conditional ones and these reductions are valid only when the conditional propositions do not contain the existential clause. However, the Ac propositions containing the existential clauses are also needed in particular in the moods of the third figure such as Darapti and Felapton. But in that case, the disjunctive proposition ED can no more be reduced to Ac, for Ac implies ED in all cases (that is, with or without the existential clause) but ED does not imply Ac if Ac contains the existential clause; it implies it only when it does not contain it. This can be shown by the following three implications and one non-implication: 1. [(∃s)Ps ^ (8s)(Ps ! Qs)] ⟹ (8s)(~Ps _ Qs) 2. (8s)(Ps ! Qs) ⟹ (8s)(~Ps _ Qs) 3. (8s)(~Ps _ Qs) ⟹ (8s)(Ps ! Qs) But 4. (8s)(~Ps _ Qs) ⇏ (∃s)Ps ^ (8s)(Ps ! Qs) While in the three first implications, the antecedents imply their respective consequents, in the fourth case, the antecedent does not imply its consequent, as one can easily show. Some moods presented by Avicenna contain or presuppose both kinds of Ac propositions, for instance, the following mixed Darapti ADAcIc mood, which he considers as valid: “Whenever H is Z then C is D (AC) Always either H is Z or A is B (AD) Therefore, Sometimes when C is D then not (A is B) (IC)” (al-Qiyās, p. 309.10-12) This mood is proved by reducing the disjunctive premise to a conditional one, namely the following ‘Whenever H is Z then not (A is B)’ (al-Qiyās, p. 309.13-14),

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which leads to the conclusion of the initial mood by Darapti AcAcIc. So, it is clear that the second Ac to which AD has been reduced cannot contain the existential clause, otherwise the reduction is not valid. But on the other hand, the first Ac (the minor premise of the mood) should contain the existential clause, otherwise the mood is no more valid, since all Darapti moods are valid only if at least one of the A premises contains the existential clause. So here, the Ac proposition which the mood contains must contain the existential clause, but the AD proposition must be reduced to an Ac without the existential clause, in order for the reduction to be valid. We can also add that the disjunction in AD must be real in order for the whole mood to be valid.10 All this can be clarified by formalizing the mood, but Avicenna himself does not explicitly make these precisions.

2.4.2.3

The Iqtirānī Syllogisms with Conditional, Disjunctive and Categorical Propositions

In this third system, Avicenna considers the moods where the hypothetical conditional and disjunctive propositions are combined with categorical ones. These syllogisms are presented in section VI-4 of al-Qiyās. In these moods, the categorical premise “takes the place of the minor or of the major term” (al-Qiyās, p. 325.5). The moods are structured as the usual categorical ones but their premises may share just one part of the whole proposition. The first mood considered is the following: “Whenever H is Z, then Every C is D Every D is A Whenever H is Z, then Every C is A” (al-Qiyās, 326.3-4) [in bold: Barbara, C: minor) Here, the term shared by the major premise and the consequent of the minor one is the term D. The whole mood is valid as it is stated, provided the second premise is interpreted as ‘Every D is A in every situation (or state)’,11 i.e. if we add ‘8s’ in front of it. For, thus stated, the first premise ‘Whenever H is Z, then Every C is D’ could be formalized as follows, where ‘H is Z’: P, and the whole premise: (8s)[Ps ! (8x) (Cxs ! Dxs)]. This formalization is equivalent to ‘(8s)Ps ! (8s)(8x)(Cxs ! Dxs)’, by the following rule of modal logic: □(P ! Q)  (□P ! □Q). This being so, the whole Barbara mood within this syllogism will have a doubly quantified first premise and a doubly quantified conclusion, which would also be formalized thus: ‘(8s)Ps ! (8s)(8x)(Cxs ! Axs)’, in which case, if the second premise is rendered with only the quantifier ‘(8x)’, the syllogism would not be valid. For it would have the following structure:

10

See Chatti, 2019, section 5.1.3.2, p. 310. See W. Hodges, ‘Ibn Sīnā on reductio ad absurdum’, Review of Symbolic Logic, 2017, section 4, where Prof. W. Hodges gives another example studied by Avicenna, similar to the one above, and says that the second premise should be stated as “Always no D is an A”.

11

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(8s)Ps ! (8s)(8x)(Cxs ! Dxs) And (8x)(Dx ! Ax) If follows (?)(8s)Ps ! (8s)(8x)(Cxs ! Axs) Unfortunately, thus stated, the syllogism is not valid for there are cases where the premises are true while the conclusion is false. This is presumably so because the underlying Barbara, which this syllogism contains, namely the following: (8s)(8x)(Cxs ! Dxs) (L) (8x)(Dx ! Ax) (X) Then (?) (8s)(8x)(Cxs ! Axs) (L) [D: middle, C: minor] is itself invalid, since it is an XLL Barbara with de dicto necessity. And this particular Barbara is invalid in modal logic. So, to validate this mood, one has to write it as follows: (8s)Ps ! (8s)(8x)(Cxs ! Dxs) And (8s)(8x)(Dx ! Ax) Therefore (8s) Ps ! (8s)(8x)(Cxs ! Axs) Thus written, the mood is valid, for if its premises are true, then the conclusion is true. And the underlying Barbara mood would be an LLL mood.12 In other cases, the presupposed syllogism is Celarent, or Cesare, as appears in the following examples, stated in the next sections. For instance, in section VI-5, he considers the case where the predicative unquantified proposition ‘H is Z’ is the consequent of [one of] the premises. One of the moods is stated as follows: “Every C is B Whenever No B is A then H is Z If follows: Sometimes when No C is A then H is Z” (al-Qiyās, 338.3-4; in bold, Celarent) As it is stated, however, this ‘mood’ is not valid, if the first premise has no import. For then, it would be formalized as follows (where ‘H is Z’: P) (8x)(Cx ! Bx) [(8s)[(8x)(Bxs ! ~Axs) ! Ps] ⊬ (∃s)[(8x)(Cxs ! ~Axs) ^ Ps] One of the cases of falsity occurs when both Cx1s1 and Bx1s1 are true, while ~Ax1s1 and Ps1 are false, which makes the first premise true, the first implication of the second premise false, and consequently the whole premise true, while the conjunction of the conclusion is false since its first implication and P are both false.

12

The validity of this mood (or its invalidity) has nothing to do with the existential clause as we have shown above. This mood is not validated by the addition of the underlying Barbara as a further ‘premise’. I thank K. El-Rouayheb who drew my attention to that point. However, unlike what he seems to suggest, I never said that the existential clause is needed to validate this mood.

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Even when the first premise has an import, i.e. if it is formalized as (∃x)Cx ^ (8x) (Cx ! Bx), the mood would be stated as follows, but would remain invalid: (∃x)Cx ^ (8x)(Cx ! Bx) [(8s)(8x)(Bxs ! ~Axs) ! Ps] ⊬ (∃s)[(8x)(Cxs ! ~Axs) ^ Ps] For even in that case, the two premises can be true while the conclusion is false, for instance, when ~Ax1s1 and Ps1 are false, while Cx1s1 and Bx1s1 are true. To validate the mood, one has, not only to consider the categorical premise to be necessary but also to add Celarent in full either by stating its corresponding implication ‘If Every C is B and No B is A, then No C is A’13 as a further premise, or alternatively by just adding the whole Celarent mood as an inference (stated vertically as usual). In the first case, it should be stated as follows: “[Always] Every C is B Whenever No B is A then H is Z If Every C is B and No B is A, then No C is A If follows: Sometimes when No C is A then H is Z” In the second case, it should be stated as follows: “1. [Always] Every C is B 2. Whenever No B is A then H is Z 3. Every C is B (from 1, by UI) 4. No B is A (assumption) Therefore 5. No C is A (by Celarent, from 3, 4) [6. Sometimes when No C is A then No B is A (¼ ‘No C is A and No B is A’) (from 4, 5 by the Conjunction rule14)]

13

Now it should be clear that a syllogism in traditional (Greek, Arabic or Latin) syllogistic, is an inference, and I consider it as such, as most modern scholars. So what I am saying is not that a syllogistic mood, which is an inference deducing a conclusion from two premises, is also a proposition. Rather what I am saying is that a syllogistic mood, as any inference, corresponds to (but is not identical with) a proposition whose main connective is a logically valid implication (¼ a tautological implication), in which case, the entailment that leads to the conclusion is verified semantically by showing that there is no case where the antecedent of the implication (the corresponding conjunction of the two premises) is true while its consequent (the conclusion of the corresponding inference) is false. This correspondence is also admitted by modern logicians who use both proof-theoretic methods (deductive methods) and model-theoretic methods (semantic methods based on calculus with truth tables) (see, for instance, Irving Copi, who says in Symbolic Logic: “Thus to the valid argument form ‘p _ q; ~p ∴ q’ corresponds the tautologous statement form [(p _ q) . ~ p] ⊃ q” (p. 31, my emphasis) (where the dot stands for the conjunction and ‘ ⊃ ’ stands for the valid implication). Other logicians such as Stephen Kleene in Mathematical Logic, use both proof-theoretic methods and model-theoretic ones (see, for instance, chapter 1, §§ 1 to 8 and §§ 9 to 13, in that book) and admit the correspondences between these methods. This correspondence makes the verification of the validity of the moods easier and faster. 14 See Irving Copi, Symbolic Logic, p. 36

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If follows: Sometimes when No C is A then H is Z (from 2, 6 by Darii, No B is A: middle). In section VI-6, he considers the following mood, which contains disjunctions: “Every B is either C or H or Z Every C and [every] H and [every] Z are A Therefore, Every B is A” (al-Qiyās, 350.3-4) This is called the “divided syllogism”. We can formalize it as follows: (x)[Bx ! (Cx _ Hx _ Zx)] (first (¼ minor) premise) (x)[(Cx _ Hx _ Zx) ! Ax] (second (¼ major) premise) ‘ (x)(Bx ! Ax) (conclusion) This formalization shows that Avicenna equates between ‘(x)[(Cx _ Hx _ Zx) ! Ax’ on the one hand and ‘(x)(Cx ! Ax) ^ (x)(Hx ! Ax) ^ (x)(Zx ! Ax)’ on the other hand. This is remarkable, since this equivalence is the quantified counterpart of the following law of Russell & Whitehead’s Principia Mathematica: ‘*4.77 ‘:. q ⊃ p . r ⊃ p. : q _ r . ⊃ . p (section A, p. 121).15 But it also shows that in this mood, the complex A proposition (i.e. the second premise), which is categorical, should not contain the existential clause. So, in this kind of moods, the categorical A propositions used can be without import in some cases, as in their modern interpretation. But Avicenna does not distinguish clearly and explicitly in this part of his text between the two interpretations of A and does not say which interpretation should be given to the first premise and to the second one. It is by verifying the validity of the mood with modern tools that one can give the right interpretations to each of these two A premises. As to the disjunction, it should be inclusive; otherwise, the equivalence does not hold, as can easily be shown. However, he also presents the following mood of the third figure, where the Ac proposition should contain the existential clause, because otherwise, the mood would not be valid: “Always either C is B or D is B Every C and every D is H Therefore, Some B is H” (al-Qiyās, 351.10-11)

In more usual symbols, this proposition would be written as follows: [(q ⊃ p) ^ (r ⊃ p)]  [(q_ r) ⊃ p]. Of course, since it is an equivalence, we could also start by the second part and end by the first one. 15

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So, here too, there is a lack of clarity with regard to the different interpretations of the A propositions. As a matter of fact, this mood seems to be structured in the model of a Darapti mood, since its two premises are universal while its conclusion is particular. This is why it requires the existential clause in the universal conditional premise as its counterparts in categorical syllogistic and in the first hypothetical system.

2.5

Further Developments, al-Khūnajī, Ibn ʿArafa and al-Sanūsī

Al-Khūnajī (d.1286, 685 of Hegira) was one of the followers of Avicenna, although, as stressed by El-Rouayheb (2019, p. 44) “he was critical of Avicenna” as well as “of al-Rāzī”. However, his brief ‘handbook’ entitled al-Jumal seems clearly influenced by Avicenna’s thought, even if it departs from it on some points. As claimed by El-Rouayheb, this author had a great influence on North African scholars, for he says: Rather than Kātibī’s Shamsiyya and Urmawī’s Maṭāliʿ, Khūnajī’s Jumal became the standard handbook on advanced logic in the Maghreb, and elicited numerous commentaries by fourteenth- and fifteenth-century North African scholars (El-Rouayheb, 2019, p. 121).

Furthermore, according to El-Rouayheb, these scholars were more interested by hypothetical logic than their Eastern colleagues. This is why I will present the views of two of them, namely, Ibn ‛Arafa (d.1401, 803 of Hegira) and Muhammed b. Yusuf al-Sanūsī (d. 1490, 895 of Hegira). Let us first start by al-Khūnajī, before analyzing the views of his two North African followers.

2.5.1

Further Developments: Afḍal al-Dīn al-Khūnajī (d. 1286)

In hypothetical logic, al-Khūnajī, who appears to be one of Avicenna’s followers, clarifies Avicenna’s distinctions. According to this author, the hypothetical proposition is either connected (muttasila) or disconnected (munfasila) [¼ disjunctive]. The connected one can be either (1) implicative (luzūmīyya) or (2) concordant (ittifāqīyya). In the luzūmīyya (1), the consequent really follows from the antecedent. In the ittifāqīyya (2), there is no such relation of following from, for “there is just a concordance in truth (mujarrad ittifāqun fī al-sidqi)’ (al-Jumal, p. 6). This is explained in the following quotation: . . . .the connected is the one where one of the two propositions – which is called the antecedent (muqaddim) – accompanies the other one (musṭaḥiba li-l-ukhrā) – called the consequent (tālī) – due to a relation between them both (li-‛alāqatin baynahumā) which

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requires that (taqtaḍī dhālika), [in which case] it is called implicative (luzūmīyya) or simply due to their agreement in truth (li-mujarrad ittifāqihima fi al-ṣidqi), and this is called concordant (ittifāqīyya) (al-Jumal, p. 6, my emphasis).

It seems thus clear that the implicative proposition expresses a relation of following from, since the consequent is required by the antecedent, while the concordant proposition just expresses a kind of agreement in truth, being thus very much like a conjunction. If so, the ittifāqīyya seems to be a conjunction, and its truth conditions seem to be settled: this ittifāqīyya proposition is true when both its components are true, false otherwise. This definition is clearer than the one provided by Avicenna, since it is explained in terms of the truth conditions of the proposition, which appears to be true only in the unique case where their two components agree in truth, i.e. are both true, hence false otherwise. These truth conditions are not stated as clearly in Avicenna’s various texts, as we have seen above. However, as we will see below, the implicative (luzūmīyya) remains as intensional as it was in Avicenna’s texts. As to the disjunctive proposition, it is of three kinds, as in Avicenna’s al-Ishārāt: 1. Real (ḥaqīqīyya): where the components are opposed both in truth and in falsity (ta‛ānud fī al-sidqi wa al-kadhibi ma‛an) 2. Preventive of the conjunction (māni‛atu al-jam‛): where the components are opposed only in truth (ta‛ānud fī al-ṣidqi faqaṭ) 3. Preventive of the vacuity (māni‛atu al-khuluw): the components are opposed only in falsity (ta‛ānud fī al-kadhibi faqaṭ) (al-Jumal, p. 6) However, he adds that (1) is true “when each component is contradictory to the other or equivalent to the contradictory of the other” (al-Jumal, p. 6), for instance, in the disjunctive proposition ‘this number is either even or not even’ the disjunction is real, because the two components are contradictory (never true nor false together). Likewise, when the second component is equivalent to the contradictory of the first one, as when we replace ‘not even’ by ‘odd’ to get ‘this number is either even or odd’, both components are also neither true nor false together. As to (2), it is true “when each component is more specific (akhaṣṣ) than the contradictory of the other” (al-Jumal, p. 6). (3) is true “when each component is more general (a‛amm) than the contradictory of the other” (al-Jumal, p. 6). The two expressions ‘akhass’ (more specific) and ‘a‛amm’ (more general) can already be found in Fakhr al-Dīn al-Rāzī’s treatise entitled ‘Manṭiq al-Mulakhkhaṣṣ’, where they are illustrated by the following example: ‘either this is a stone or it is a tree’. The word ‘stone’ is more specific that the word ‘not-stone’, which is more general than it. Likewise ‘not-tree’ is more general than ‘tree’, since it applies to more objects. Both ‘stone’ and ‘tree’ are contrary, for if something is a stone, it is not a tree and vice versa. But their two contradictories are subcontrary instead, for if something is not a tree (or is a not-tree), it can also not be a stone (or be a not-stone), if it is a horse, for instance. So, the second kind of disjunction, which is called the preventive of conjunction contains two contrary components, namely ‘stone’ and

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‘tree’, where each of them is more specific that ‘the contradictory of the other one’. The contradictory of ‘stone’ is ‘not stone’, and ‘tree’ is more specific than ‘not stone’, since it is one of the things that are not stones. In this case, the ‘preventive of conjunction’, is the sentence ‘either this is a stone or it is a tree’ and means ‘it is not the case that this thing is both a stone and a tree’ [¼ not (P and Q)]. As to the third kind of disjunction, which is called the ‘preventive of vacuity’, it contains two components which are both more general that the ‘contradictory of the other one’. Recall that ‘not stone’ is the contradictory of ‘stone’ and ‘not tree’ is the contradictory of ‘tree’, the elements of the disjunction will be the more general expressions, and the disjunctive sentence will be the following: ‘either this is not a tree or it is not a stone’. In this case, each component (e.g. ‘not a tree’) is more general that the contradictory of the other one (e.g. ‘stone’), and the sentence will be true when both ‘this is not a tree’ and ‘this is not a stone’ are true, or one of them is true, but it will be false only when both components are false, namely when something is both a tree and a stone. In other words, it will have the following truth conditions: Either ðthis is not a stoneÞ 1

or 1

ðitis not a treeÞ 1

1

1

0

0 0

1 0

1 0

These truth conditions are exactly those of the inclusive disjunction. So, the ‘preventive of vacuity’ is clearly an inclusive disjunction, whose truth conditions are settled. We can thus say that it is truth functional, although the definition above is explained by means of a concrete example, which makes it more intuitive and concrete that the modern definition of the inclusive disjunction, which uses only variables. So although the use of the words ‘more general’ and ‘more specific’ in the definitions provided in both al-Rāzī’s and al-Khūnajī’s theories and their illustrations by concrete examples may suggest that these disjunctions are not truth-functional, because of the precise relation between these components, this does not seem to be the case, since al-Khūnajī adds in the sequel the whole set of truth conditions of the disjunctions. Thus, according to him, the real disjunction “is true when only one of its components is true, it is false when both are true and when both are false” (al-Jumal, p. 6). So, its truth conditions are settled quite clearly, which means that it is indeed truth-functional. The second disjunction (māni‛atu al-jam‛) “is true when both components are false or when one of them is false, and it is false when both are true” (al-Jumal, p. 6). So, it is also truth-functional, since its truth conditions are given in full. When stating the truth conditions of the third disjunction, he just says: “the truth conditions of the preventive of vacuity are the inverse [of the above] (bi-al-‛aksi)” (idem, p. 6). But if we understand that as meaning that this third disjunction is true

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when both its components are true or when one of them is true, and it is false when both are false, then the third disjunction is also truth-functional. As a result of these definitions, the truth conditions of the three disjunctions are the following (where ‘_’ stand for the exclusive disjunction, ‘~’ for the negation, ‘_’ for the inclusive disjunction and ‘^’ for the conjunction; in bold the truth values of the main connectives): ð 1Þ

ð2Þ

ð3Þ

P 1

_ 0

Q 1

 0

ðP 1

^ 1

QÞ 1

P 1

_ 1

Q 1

1 0

1 1

0 1

1 1

1 0

0 0

0 1

1 0

1 1

0 1

0

0

0

1

0

0

0

0

0

0

These tables, which just reflect al-Khūnajī’s explanations above, show clearly that all the disjunctions, as defined by that author, are truth-functional. This truthfunctional character is not at all obvious in Avicenna’s text, since the exclusive disjunction, for instance, remains clearly intensional in his frame. This is a significant move in hypothetical logic, since it becomes more formal than it was in al-Fārābī’s and Avicenna’s frames. This move is confirmed even more clearly in al-Khūnajī’s treatise Kashf al-asrār ‛an ghawāmiḍ al-afkār, where he explicitly gives all the truth conditions of the three kinds of disjunction as follows: As to the disjunctive (al-munfaṣila), its true affirmative is composed only of a true [component] and a false [one] if it is real, and of two true [components] or one false [component] and one true [component], if it is preventive of the vacuity, and of two false [components] or a false one and a true one, if it is preventive of conjunction. And the false real disjunction is composed of two true [components] or of two false ones, while the preventive of the conjunction [is false] when its components are both true, and the preventive of vacuity [is false] when its components are both false. (Kashf al-asrar, edited by K. El-Rouayheb, p. 199)

As we can see, the truth conditions of the three disjunctions are stated quite clearly and confirm the three truth tables above. So, we can say that in al-Khūnajī’s system, the three disjunctions are truth-functional, unlike what we found in Avicenna’s one. As to the truth conditions of the connected propositions, they are stated as follows: The connected is true when both clauses are true, or when the consequent alone is true or when both are false, and it is false when both are false or one of them is false or both are true if it is implicative (al-Jumal, p. 6, emphasis added).

Here there is some ambiguity in the last part of the quotation (in italics), for the conditions stated seem incoherent, since they do not correspond to the meanings of the ittifāqīyya and of the luzūmīyya. We can try to summarize and clarify these truth conditions as follows, by considering what is written in italics as corresponding to

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those of the ittifāqīyya, and what is written in Roman letters as being those of the luzūmīyya. The conditions stated are the following: True when: 1. Both clauses are true (Both clauses are true) or 2. The consequent is true, or 3. Both clauses are false. False when: 1. Both clauses are false or 2. One of them is false or 3. Both are true if it is implicative. If we try to separate the two meanings of the connected propositions, we can say that the ittifāqīyya is true when both its clauses are true, and that it is false when both its clauses are false or when one of them is false. If so, its truth conditions correspond to those of a conjunction, which means that it is truth functional. This is confirmed by the fact that the ittifāqīyya proposition had been said to express concordance in truth. As to the luzūmīyya proposition, it is true when its consequent is true, or when both its clauses are true (but in that case, it can also be false), or when both its clauses are false. If so, its truth conditions resemble those of the material conditional for most of them, but not for the first line of the table. So, it does not seem to be truth functional. This being so, the tables corresponding to these two meanings seem to be the following: P

!

Q luzumiyya

1 1

1=0 0

0 0

1 1




P

^

Q ittif aqiyya

1 0

1 1

1 0

1 0

1 0

0 0

0 0

1 0




In the first table, the luzūmīyya (implicative) proposition is said to be true when its clauses are true, but also false in that specific case (second half of the quotation). So, we could consider that its truth in that specific case is conditioned by the specific (semantic) link of dependency between its two clauses. As to the second table, it is justified by the hypothesis that the condition ‘when one of them is false’ corresponding to the case of falsity applies only to the ittifāqī meaning of the connected, since when the antecedent is false, the luzūmīyya should be true, provided the consequent is true. With these truth conditions, the ittifāqīyya appears to be a conjunction, so it is extensional (i.e. truth-functional). But although the truth conditions of the luzūmīyya are close to that of the material conditional, they are not strictly determined, since the first case of the table admits two values and the consequent is said to follow from the antecedent. This luzūmī meaning seems thus to be not truth–functional; it remains intensional as in Avicenna’s theory.

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In Kashf al-Asrār ʿan ghawāmiḍ al-afkār, al-Khūnajī clarifies these truth conditions as follows: And the true affirmative implicative (luzūmīyya) is composed of two true [clauses] (ṣādiqayni) [or] two false [clauses] (wa kādhibayni), [or] a true consequent and a false antecedent, for the consequent can be more general than the antecedent, but the inverse (wa ‛aksuhu) is impossible, for it is impossible that a true [proposition] implies a false one.

This is quite clear as is the case with the disjunctive propositions, for the cases of truth explicitly stated are the following: true-true, false-false and false-true. However, when determining the truth values of the other kind of connective, namely the so-called ittifāqīyya, what he says remains somewhat confused, for he claims: As for the ittifāqīyya, since it cannot possibly be true with two false (kādhibayni) [clauses], the two remaining cases are determined (Kashf al-asrār, p. 199)

This quotation is puzzling because of the mention of the ‘two cases’ which are determined. Which are these two cases? And how are they determined? The text remains unclear. However, the editor added in the notes, some other possibilities corresponding to the cases of falsity of this connective, where the cases ‘true-false’ and ‘false-true’ are mentioned (see note 15, p. 198). Now, when talking about the case(s) of falsity, the text seems clearer since al-Khūnajī says what follows: And the implicative can be false in each of the four cases, while the ittifāqīyya, because it cannot possibly be false with true components, can be [false] in the three remaining ones (Kashf al-asrār, p. 199, emphasis added)

This clarifies the truth conditions of the ittifāqīyya, which appears to have the truth conditions of a conjunction (i.e. true when both components are true, false otherwise), but it seems to say that the luzūmīyya is not truth-functional, since it can be false in all cases (maybe, for instance, if there is no dependence between the antecedent and the consequent). So, in the light of this quotation, we can assume that the implicative is intensional as it is in Avicenna’s frame, while the ittifāqīyya has the truth conditions of a conjunction and is therefore truth-functional. Thus, we can say that the truth conditions of the ittifāqī meaning are clearer than they are in Avicenna’s theory, since Avicenna does not give the truth conditions of a conjunction to the ittifāqīyya propositions. In his theory, these truth conditions seem to be equivalent to those of the consequent (see Chatti, 2019, section 5.1.2.2). But what is really new in al-Khūnajī’s theory (if we compare it to those of his predecessors) is expressed in the following quotation: And the connected multiplies (tata‛addadu) by the multiplication of the elements of the consequent (bi-ta‛addudi ajzā’ al-tālī), not those of the antecedent (dūna al-muqaddim), due to the necessity of the implication of the part by the whole (li-mā yalzamu al-kull), but not conversely (dūna al-‛aks). And the disjunctive multiplies by the multiplication of its elements, when it prevents the vacuity but not when it prevents the conjunction (al-Jumal, p. 6).

By this unique phrase, al-Khūnajī expresses the following laws and non-implications:

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– [P ! (Q ^ R)] ⟹ [(P ! Q) ^ (P ! R)]16 This is the distributivity of conditional over conjunction. – [P ^ (Q ^ R)] ⟹ [(P ^ Q) ^ (P ^ R)] – [P _ (Q ^ R)] ⟹ [(P _ Q) ^ (P _ R)] This last line expresses the distributivity of disjunction over conjunction. But – [(P ^ Q) ! R] ⇏ [(P ! R) ^ (Q ! R)] – ~[P ^ (Q ^ R)] ⇏ [~(P ^ Q) ^ ~ (P ^ R)] This is a remarkable advance, since the laws stated can be found in modern propositional logic and are not present in any of his predecessors’ systems, although al-Khūnajī leaves the valid converses of some of these implications unstated. For the following implications: [(P _ Q) ^ (P _ R)] ⟹ [P _ (Q ^ R)] [(P ! Q) ^ (P ! R)] ⟹ [P ! (Q ^ R)] [(P ^ Q) ^ (P ^ R)] ⟹ [P ^ (Q ^ R)] are not stated by him. Nevertheless, he also states in some way (parts of) the two De Morgan’s laws by saying what follows: And each of the two non-real disjunctives implies the other one where the components composing [the latter] are the contradictories of those [of the former] (murakkaba min naqīḍay juz’ayhā), but not conversely (min ghairi ‛aksin) (al-Jumal, p. 7, my emphasis)

Given the definitions provided above, according to which the two non-real disjunctions are expressed by ‘~(P ^ Q)’ and ‘P _ Q’ respectively (see tables (2) and (3) above), this sentence expresses the relations stated by the following two implications, where the propositions in the second disjunction are the contradictories of those of the first one: (1) ~(P ^ Q) ⟹ (~P _ ~ Q)17 (a part of the first De Morgan’s law) (2) (P _ Q) ⟹ ~ (~P ^ ~ Q) (a variant of one part of the second De Morgan’s law)

Here, we have used the symbol ‘⟹’ to distinguish the main implication from those expressed by ‘!’, since this main implication is tautological anyway (even if these traditional authors do not know nor use the word ‘tautological’), while the other ones are not tautological. The symbol ‘⟹’ is also used by El-Rouayheb in his book The development of Arabic Logic, 1200–1800 (pp. 131–132). We could also have used the sign ‘‘’ of entailment, which allows the deductibility of the second proposition from the first one. 17 In this law, the sign of the implication is more commonly used than the sign of entailment, in modern logic, since the whole implication is now called a law rather than a rule, although, it can also be used as a rule to deduce the second proposition from the first one. In modern logic, the De Morgan’s laws are rather equivalences (¼ double implications). Irving Copi calls them ‘theorems’ (p. 30) and Russell & Whitehead call them ‘formulae’ in Principia Mathematica (p. 119), despite the fact that they use proof-theoretic methods in their seminal book (published in 1910), the model16

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However, the expression ‘not conversely’ shows that he is not aware of the fact that these laws are not simply implications but equivalences. This unawareness may be due to the lack of a clear formalism and of a method of calculus in his system.18 He does not either state some other basic laws, such as the law of distributivity of conjunction over disjunction, namely, the following valid implication: ‘[P ^ (Q _ R)] ⟹ [(P ^ Q) _ (P ^ R)]’, although he seems to hold the following laws related to the conjunction: ‘(P ^ Q) ⟹ P’ and ‘(P ^ Q) ⟹ Q’ given that, in this context too “the part follows the whole but not conversely” (al-Jumal, p. 6), which means that the conjunction ‘P ^ Q’ implies its components (or parts) but is not implied by either of them. He also developed a whole theory involving the quantified hypothetical propositions and the moods that can be composed of these different propositions in a way which is comparable to that of Avicenna, for the moods he presents contain conditional as well as disjunctive quantified propositions in addition to the predicative ones. Thus, like Avicenna, al-Khūnajī quantifies over hypothetical propositions, whether connected or disjunctive, and expresses the quantifiers by ‘kullamā’ (whenever) for AC, ‘laysa al-battata’ (never) for EC, ‘qad yakūn’ (Sometimes) for IC, and ‘laysa kullamā’ (not whenever) for OC. In the disjunctive propositions, the words ‘dā’iman’ (always) and ‘laysa dā’iman’ in AD and OD respectively are used, instead of ‘kullamā’ and ‘laysa kullamā’ (Kashf al-asrār, p. 206). His theory is very briefly summarized in al-Jumal, but much more developed in his book Kashf al-asrār ʿan ghawāmiḍ al-afkār. In that book, he devotes the last section (Section 10) to the study of several kinds of moods, containing all kinds of hypothetical quantified propositions.

theoretic methods having been introduced later. They prove them in the same book, where these laws are expressed as follows: “*4.51. ‘: ~(p . q) .  . ~p _ ~ q” and “*4.56 ‘: ~p . ~q .  . ~(p _ q)” (Principia Mathematica, p. 120), where the dot stands for the conjunction, and ‘’ stands for the equivalence. 18 K. El-Rouayheb says in his introduction of Kashf al-Asrār that “Khūnajī appears to have been the first to explicitly formulate what has come to be known as De Morgan’s law: A disjunction (P or Q) implies the negation of the conjunction of the contradictories of the disjuncts (Not both ~P and ~Q), and vice-versa” (p. xxxi), and he cites the following passage of Khūnajī’s book to confirm that view: “The two non-ḥaqīqī disjunctions, if they agree in quantity and quality and have contradictory disjuncts, then one implies the other and vice versa” (introduction, p. xxxii, my emphasis), which translates a phrase in page 219 of the book. However, this passage evokes explicitly the quantity and the quality of the disjunctions; so, it is about quantified disjunctions, while the De Morgan’s laws are both propositional and do not involve quantification at all. Consequently, what al-Khūnajī is talking about, in that passage, largely extends the [variant of] the De Morgan’s law cited by El-Rouayheb, and deserves to be studied more precisely, since he evokes at least four quantified disjunctions and several implications between them.

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It is not our aim here to analyse in detail these moods, since it would extend the limits of this paper, but we can evoke some of them to show the differences between his theory and Avicenna’s one. To begin with, unlike Avicenna, he admits four figures, not only three. Second, he criticizes some of Avicenna’s claims about the relations between some quantified connected propositions, for instance between the Ac and the Ec propositions whose antecedent is the same but whose consequents are contradictory. For according to Avicenna, “‛Never when every A is B then every C is D’ (laysa al-battata idhā kāna kull A B fa-kull C D” is equivalent to ( fī quwwati) the following Ac proposition: “‘Whenever every A is B, then not every C is D’ (Kullamā kāna kull A B, fa-laysa kull C D)” (al-Qiyās, p. 366). So, for him, this EC implies that AC and conversely. But al-Khūnajī says that this cannot be so because the proposition “‘Never if (idhā) two is even then it follows that (yalzamu) Zayd is at home’ is true, while the proposition ‘Whenever two is even then Zayd is not at home’ is false” (Kashf al-Asrār, p. 209), when the connection is understood as a kind of luzūm (implication). This is so because there is no entailment between the antecedent and the consequent in these two propositions, and the Ec one precisely denies the presence of the entailment, while the Ac one seems to assume it. For this reason, Ec is true while Ac is false. So, this EC does not imply that AC, unlike what Avicenna says, since the example given by al-Khūnajī shows intuitively that EC is true, while AC is false. Since Avicenna uses this equivalence in many of his proofs, one can presume from the start that al-Khūnajī’s moods will be different from those of Avicenna. Now this criticism is not necessarily correct, because it is based on a deviated interpretation of Ec. But Avicenna’s equivalence can also be shown not to hold, for one can show that Ac implies Ec but not conversely (see Chatti, ASP, 2022). As to the disjunctive propositions used in the different moods, their relations with each other and with the connected ones are determined precisely, when they have the same components or contradictory ones. For instance, “if two quantified [real] disjunctive propositions have the same quantity and the same quality, but contain contradictory components, they are equivalent and mutually convertible (talāzamatā wa ta‛ākasatā)” (Kashf al-asrār, p. 214). So, these two AD propositions with contradictory components imply each other and convert with each other. Formally, this could be expressed as follows: (8s)(Ps _ Qs) ⟹ (8s)(~Ps _ ~ Qs); (8s)(~Ps _ ~ Qs) ⟹ (8s)(Ps _ Qs) (mutual implication) (8s)(Ps _ Qs) ⟹ (8s)(~Qs _ ~ Ps); (8s)(~Ps _ ~ Qs) ⟹ (8s)(Qs _ Ps) (mutual conversion) These implications and conversions are correct, provided the disjunction is real, i.e. exclusive, as one can easily show. They are proved by reductio ad absurdum by al-Khūnajī. He provides similar relations for the other kinds of disjunctive propositions, such as the one evoked in note 18 above. These relations are used in the proofs of the moods and condition the nature of these moods.

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As to the moods admitted, as we said above, they are classified by means of the kinds of propositions used in them. For instance, the following mood is the very first mood containing connected propositions and where the premises share only one part of their antecedents: “Whenever every A is B then C is D Whenever some B is H then W is Z It follows: Sometimes (Whenever every A is H then C is D) then (Whenever every A is H then W is Z) (Kashf al-asrār, p. 331). This mood is proved by reductio ad absurdum. The proof runs as follows: If the conclusion is false, then the following proposition is true, being its contradictory: (1) Never if (whenever every A is H then C is D) then (Whenever every A is H then W is Z) To this proposition, the following connected proposition is added, as it is said to be true too: (2) Whenever if (whenever every A is H then every A is B) then (whenever every A is H then W is Z) (Kashf al-asrar, p. 331) This proposition (2) is true according to al-Khūnajī, because of the following deductions: If – Whenever every A is H then every A is H and – [whenever every A is H] then every A is B Then – whenever every A is H then some B is H (in bold Darapti) Therefore – (whenever every A is H then W is Z) (the consequent of (2)) The latter is deduced by the following Barbara mood: Whenever some B is H then W is Z (major premise of the initial mood) Whenever every A is H then some B is H (conclusion of the preceding argument) Therefore, whenever every A is H then W is Z (by Barbara) From (1) and (2), he arrives at the desired conclusion (3) by Camestres from the second figure as follows: (1) Never if (whenever every A is H then C is D) then (whenever every A is H then W is Z) (2) Whenever if (whenever every A is H then every A is B) then (whenever every A is H then W is Z) It follows: (3) Never if (whenever every A is H then every A is B) then (whenever every A is H then C is D) (by Camestres) Where the proposition in italics is the middle. This proposition (3) is false, he claims, because its contrary, namely, the following:

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‘(4) Whenever if (whenever every A is H then every A is B) then (whenever every A is H then C is D)’ is true. (4) is true because its antecedent and the minor of the initial mood imply its consequent by the following Barbara: Whenever every A is H then every A is B Whenever every A is B then C is D Therefore, whenever every A is H then C is D (by Barbara, ‘every A is B’ ¼ middle). This mood and its proof show that al-Khūnajī’s theory extends Avicenna’s one and holds much more complex moods, which use explicit and implicit deductions. The rules used are not foreign from Avicenna’s theory since Avicenna did examine the logical relations between the hypothetical connected propositions whose clauses are themselves quantified and stated the implications between these propositions in al-Qiyās. It seems then that al-Khūnajī develops further this theory by providing the syllogistic moods containing these propositions. But the validity of these inferences is not obvious, and requires a more detailed examination. Let us now turn to his followers Ibn ‛Arafa al-Warghamī al-Tūnisī (d. 1401) (usually called Ibn ‛Arafa) and Muhammed b. Yūsuf al-Sanūssi (d. 1490).

2.5.2

Further Developments, Ibn ‛Arafa and Muhammed b. Yūsuf al-Sanūsi

Let us first present these two authors. Unlike Avicenna, these authors are not really philosophers. They are much more known as theologians and jurists, especially Ibn ‛Arafa, who was one of the scholars of the famous Tunisian al-Zaytūna Mosque. Apart from his theological writings, which we won’t evoke here, Ibn ‛Arafa wrote a treatise on logic, namely a commentary on al-Khūnajī’s al-Jumal, called ‘alMukhtaṣar’, which was seen as highly “esteemed in later centuries” (El-Rouayheb, 2019, p. 127) despite its brevity and its difficulty, especially by his North African followers and colleagues. As to Muhammed b. Yusuf al-Sanūsī, he wrote a commentary on Ibn ‛Arafa’s al-Mukhtaṣar, where he developed some of the ideas stressed by Ibn ‛Arafa in that treatise, leaving aside some other ones, which he considered as being of less interest, despite their real importance (see El-Rouayheb, 2019, p. 130). Both authors were clearly influenced by al-Khūnajī, although in an indirect way for the latter, and ultimately by Avicenna, if we consider that al-Khūnajī himself was one of Avicenna’s followers. Let us start by Ibn ‛Arafa. This author makes the same distinctions as al-Khūnaji between the different disjunctive and connected operators. But unlike al-Khūnajī, who tended to view the implicative (luzūmīyya) proposition as intensional, Ibn ‛Arafa clarifies this connective and provides in some way its whole truth conditions

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by saying that “it is true in all cases other than when the antecedent is true while the consequent is false (wa tata’allafu ṣādiqatu al-muttaṣila luzūmīyya min ghairi muqaddamin ṣādiqin ma‛a tālin kādhibin)” (p. 27), despite the fact that he also says that the luzūmīyya differs from the ittifāqīyya “because of a relation (li-‛alāqatin) [between its antecedent and its consequent]” (idem, p. 26), which is not present (“aw dūnaha” (p. 26)) in the ittifāqīyya. So, for him, the luzūmīyya has the truth values of a conditional and can consequently be considered as truthfunctional in his frame, although this expression is not used by him, of course. As to the ittifāqīyya, he says that “it is true when its clauses are true and false otherwise (wa [tata’allafu ṣādiqatu] ittifāqīyya min ṣādiqatayni wa kādhibatuhumā mimmā siwā mā humā ‛anhu ṣādiqatāni)”. So, it seems to be a truth-functional conjunction. As to the disjunction, we can say that the real disjunction is true when its components have different values, while the second disjunction is true when its components are both false or one of them is true while the other one is false. They are false otherwise. For he says “the true real disjunction [contains] a true [component] and a false [one], while the preventive of the conjunction [contains] two false components or a true one and a false one, and both are false in all the cases where they are not true (wa ṣādiqatu al-munfaṣila al-ḥaqīqīyya [tata’allafu] min ṣādiqin wa kādhibin wa māni‛atu al-jam‛i min kādhibayni aw ṣādiqin wa kādhibin wa kādhibatu kullin minhumā khaṣṣaha19 al-ḥamlu bimā siwā mā hiya minhu ṣādiqa)” (al-Mukhtaṣar, p. 27, Gherab, 1980, p. 94). So, he confirms the truth-functionality of at least two kinds of disjunctions. Note that this paragraph starts by the word ‘alAthīr’ followed by a colon. So maybe he is following here Athīr al-Dīn al-Abharī. As the editor of his al-Mukhtaṣar says in his introduction, Ibn ‛Arafa has the merit of citing all his sources, for “whenever he evokes an opinion, he never omits to refer to its original author.” (Introduction to Ibn Arafa’s al-Mukhtaṣar, edited by S. Gherab, 1980, p. 53). As a matter of fact, throughout the treatise, many names, apart from al-Khūnajī whom he is commenting on, are mentioned by Ibn ‛Arafa, among which we can evoke al-Sarrāj, al-Rāzi, al-Athīr, and even Avicenna, since he sometimes evokes explicitly al-Shifā. As to the third disjunction, it is defined as it is in al-Khūnajī’s treatise, for it is said to contain “one component and another [component, which is] more general than its contradictory” (idem, p. 26). Thus it seems to have the same truth conditions as in al-Khūnajī’s system. Besides that, the real disjunction “does not contain three components” (al-Mukhtaṣar, p. 26), due to the fact that its components are contradictory to each other, while the preventive of conjunction can “contain more than two components” (idem, p. 26). In his al-mukhtaṣar fī al-Manṭiq, he states the same implications as al-Khūnajī and some of the non-implications, for he says “The multiplication (ta‛addud) of the implicative (luzūmīyya) holds when the consequent is multiple (bi-ta‛addud al-tālī),

19 This word seems ‘bizarre’ to one reviewer, who asked to ‘double-check’ it. But it is present in the two editions of the book (al-mostafa.com + the Tunisian edition by Saad Gherab), so I have kept it.

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not when the antecedent is multiple (lā muqaddimahā), due to the fact that the part follows what is followed by its whole, while the whole does not follow what is followed by its part (li-luzūmi al-juz’i malzūma kullihi wa ‛adami malzūmati al-juz’i lāzima kullihi)” (al-Mukhtaṣar, p. 27). So, like al-Khūnajī, he admits the following implication and non-implication: [P ! (Q ^ R)] ⟹ [(P ! Q) ^ (P ! R)] [(P ^ Q) ! R] ⇏ [(P ! R) ^ (Q ! R)] He adds that their negations behave in the inverse way (wa sawālibuhā ‛alā al-‛aksi fī dhālika) (idem, p. 27). Thus, the following implication is admitted and added to those claimed by his predecessor: ~ [(P ^ Q) ! R] ⟹ [~(P ! R) ^ ~ (Q ! R)] This implication is indeed valid, for there is no case where the first proposition is true while the second one is false. This is a further advance in hypothetical logic, which we don’t find in al-Khūnajī’s treatise al-Jumal and tends to show the progressive evolution of this logical field from author to author in the Arabic tradition. As to al-Sanūsī, he says that the connected propositions may involve the notion of cause but not in the same way for: 1. The antecedent is the cause of the consequent rationally (‛aqlī), e.g. ‘If this is a human, it is an animal’: here the link is necessary, given the very meanings of the antecedent and the consequent. It is called luzūmīyya (al-mukhtasar, p. 25). We might consider it as a kind of analytic relation between the antecedent and the consequent. 2. The antecedent and the consequent are related according to some lawful (or law-like) (shar‛ī), e.g. ‘If it is daytime, then the planets are hidden’: here the link is just usual (‛ādī), not really necessary. It might express some kind of empirical link between the antecedent and the consequent. 3. There is no causal relation at all between the two clauses, e.g. ‘If the sun rises, men are speaking’: this is an ittifāqiyya. In this kind of sentences, there is no link at all between the meanings of the antecedent and the consequent. In this third kind, the clauses only happen to be true together. In other examples such as ‘[even] if (law) this man presents his excuses, I would not forgive him’ (p. 24), the word ‘if’ seems to mean ‘even if’. It has thus the truth conditions of a conjunction. Al-Sanūsī states a great number of implications in his ‘Sharh Mukhtaṣar fī al-Mantiq’. For apart from the implications and non-implications (expressed here by entailments and non-entailments) already stated by al-Khūnajī and Ibn ‛Arafa, he adds the following:

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1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

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[(P ^ Q) ^ R)] ⟹ [(P ^ R) ^ (Q ^ R)] (idem, p. 73, El-Rouayheb, p. 131).20 [(P ^ Q) _ R)] ⟹ [(P _ R) ^ (Q _ R)] (idem, p. 73, El-Rouayheb, p. 131) ~[P ! (Q ^ R)] ⇏ [~(P ! Q) ^ ~ (P ! R)] (idem, p. 73, El-Rouayheb, p. 131) ~[P ^ (Q ^ R)] ⇏ [~(P ^ Q) ^ ~ (P ^ R)] (idem, p. 73, El-Rouayheb, p. 131) ~[(P ^ Q) _ R)] ⇏ [~(P _ R) ^ ~ (Q _ R)] (idem, p. 73, El-Rouayheb, p. 132) ~[P _ (Q ^ R)] ⇏ [~(P _ Q) ^ ~ (P _ R)] (idem, p. 73) (P _ Q) ⟹ [(~P ! Q) ^ (~Q ! P)] (idem, p. 75, El-Rouayheb, p. 132) ~ (P ^ Q)] ⟹ [(P ! ~Q) ^ (Q ! ~P)] (idem, p. 75, El-Rouayheb, p. 132) [(P _ Q) ^ ~ (P ^ Q)] ⟹ [(~P ! Q) ^ (~Q ! P) ^ (P ! ~Q) ^ (Q ! ~P)] (idem, p. 75, El-Rouayheb, p. 132). (P ! Q) ⟹ ~ (P ^ ~ Q) (idem, p. 75, 3.1 El-Rouayheb, p. 132) (P ! Q) ⟹ (~P _ Q) (idem, p. 75, El-Rouayheb, p. 132) ~(P ^ Q) ⟹ (~P _ ~ Q) (¼ A part of the first De Morgan’s law, idem, p. 76) (P _ Q) ⟹ ~ (~P ^ ~ Q) (¼ A variant of the second De Morgan’s Law, idem, p. 76) ‘~~P ⟹ P’ (¼ one part of the law of double negation (idem, p. 72)).

The two implications 12 and 13 are expressed in the following quotation: and each of the preventive of conjunction and the preventive of vacuity implies the other one, provided the components of the latter are the contradictories of those of the former (idem, p. 76, El-Rouayheb, p. 132, emphasis added).

So since the preventive of conjunction is ‘~(P ^ Q)’, it implies the preventive of vacuity [which had been expressed by ‘P _ Q’] only in case this preventive of vacuity involves the contradictories of the components of the first disjunctive proposition, namely ‘~P’ and ‘~Q’. As a result, we get one part of the De Morgan’s Law, i.e. ‘~(P ^ Q) ! (~P _ ~Q)’, which is the implication stated above. The real De Morgan’s Law, however, is an equivalence; it is stated as follows by modern logicians: ‘~(P ^ Q)  (~P _ ~Q)’. This equivalence is the same as a double implication, which can be stated as follows: ‘[~(P ^ Q) ⟹ (~P _ ~ Q)] ^ [(~P _ ~ Q) ⟹ ~ (P ^ Q)]’. But what al-Sanūsī is stating is only one part of the De Morgan’s law, namely the first implication; it does not include nor involve the second one. As to 13, it is one variant of [one part] of the second De Morgan’s Law, which is the following equivalence: ~(P _ Q)  (~P ^ ~ Q). This equivalence is also the same as a double implication, which would be expressed as follows: [~(P _ Q) ⟹ (~P ^ ~ Q)] ^ [(~P ^ ~ Q)] ⟹ ~ (P _ Q)]. 20

For consistency, I have slightly modified El-Rouayheb’s notation, who expresses the conditional by ‘if. . .then’, uses ‘&’ for the conjunction between P and Q, and puts the sign ‘;’ between (P&R) and (Q&R), but he also uses ‘⟹’ for the main implication. As to the text of al-Sanūsī himself, it does not use any modern symbol, of course. The sign ‘⟹’ stands for the valid (i.e. tautological) implication. The implications could be stated as entailments (allowing the deduction of the second proposition from the first one), in which case the sign ‘⟹’ would be replaced by ‘‘’.

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While the formula stated by al-Sanūsī, is a variant of the second implication of the De Morgan’s Law, since it is equivalent to it by contraposition. For if we apply the principle of contraposition to ‘(~P ^ ~Q) ⟹ ~(P _ Q)’, we get ‘~~(P _ Q) ⟹ ~(~P ^ ~Q)’, which in turn is equivalent to ‘(P _ Q) ⟹ ~(~P ^ ~Q)’, by the law of double negation, that is, to the implication stated by al-Sanūsī above (and al-Khūnajī, before him). So this implication is the exact equivalent of the second part of the second De Morgan’s law. We can thus say that both De Morgan’s laws are at least partially stated by this author, and previously by al-Khūnajī, as we saw above. Note that even in Avicenna’s texts, they were presupposed implicitly, as appears in his demonstrations of some equivalences in modal and hypothetical logics, for instance, when he expresses the negation of the bilateral possible. Proposition 14 states one part of the law of double negation quite clearly, namely ‘~~ P ⟹ P’. Here too, the converse of this law, namely ‘P ⟹ ~~P’, although much more obvious and less problematic than the first one, is not evoked. This may be due to the fact that it is the first formula that is most often used, for instance, in the proofs by reductio ad absurdum. Some more implications involving quantified hypothetical propositions are added, for instance, the following ones, which are evoked by El-Rouayheb in the same volume: 15. Always if Some A is B, then Q ⟹ Always If every A is B then Q (El-Rouayheb, 132) 16. Always If P then Every A is B ⟹ Always If P then Some A is B (El-Rouayheb, 132) In both cases, ‘Every A is B’ should have an import in order for the implications to be valid, although this specific condition is not added by El-Rouayheb or by al-Sanūsī. But it is required for otherwise, there is a case of falsity under the main implications, which means that the former proposition does not imply the latter.

2.6

Conclusion

In what precedes we have shown that the hypothetical logic has evolved in a significant way in the Arabic tradition. For while al-Fārābī presents just the Stoic indemonstrables and some of their variants, Avicenna builds several systems containing different kinds of hypothetical moods and inferences. These systems had a great influence on his followers who introduced inside the same kind of frame some changes in the definitions and the inferences. Al-Khūnajī presents hypothetical systems where elementary propositions as well as quantified ones are used. He states some laws which prefigure some basic modern propositional laws. But he also develops a hypothetical system containing quantified hypothetical propositions which extends Avicenna’s system and contains some complex moods that we don’t find in Avicenna’s hypothetical logic.

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We can also say that with al-Khūnajī first and afterwards Ibn ‛Arafa and al-Sanūsī, Avicenna’s intensional definitions of the conditional and the disjunction were clarified and became more formal and extensional, in particular with regard to the disjunction in al-Khūnajī’s frame, but also to some extent, with regard to the conditional as well, in Ibn ‛Arafa and al-Sanūsī’s frame. These extensional definitions are given in parallel to the intensional ones, and both are said to define distinct relations. This clarification and ‘extensionalization’ gave rise to some complex inferences that are not explicit in Avicenna’s frame. Some laws of propositional logic are thus stated explicitly by these logicians: one half of each of the De Morgan’s laws, the law of double negation, the law of distributivity of disjunction over conjunction etc.. . .plus implications with quantified propositions. There is thus a clear departure from intensionality. But this departure remains partial, since some definitions still have semantic features, in particular those involving the conditional propositions.

Bibliography Al-Fārābī, A. N. (1986a). Kitāb al-Maqūlāt. In R. Al Ajam (Ed.), al-Manṭiq ‘inda al-Fārābī (Vol. 1, pp. 89–132). Dar el Machriq. Al-Fārābī, A. N. (1986b). Kitāb al-Qiyās. In R. Al Ajam (Ed.), al-Manṭiq ‘inda al-Fārābī (Vol. 2, pp. 11–64). Dar el Machriq. Al-Fārābī. (1988a). al-Qawl fī al-‛Ibāra. In al-Manṭiqiyāt li-al-Farābi, volume 1, texts published by Mohamed Teki Danesh Pazuh, Edition Qom, 1409 of Hegira. Al-Fārābī. (1988b). al-Maqūlat. In al-Manṭiqiyāt li-al-Fārābi (Vol. 1, pp. 41–82). Mohamed Teki Danesh Pazuh, Edition Qom. Al-Fārābī. (1988c). al-Qiyās. In al-Manṭiqiyāt li-al-Fārābi (Vol. 1, pp. 115–151). Mohamed Teki Danesh Pazuh, Edition Qom. Al-Fārābī. (1988d). al-Qiyās al-Saghīr, In al-Manṭiqiyāt li-al-Fārābi (Vol. 1, pp. 152–194). Mohamed Teki Danesh Pazuh, Edition Qom. Al-Fārābī. (2020). Syllogism: An Abridgement of Aristotle’s Prior Analytics, in the series Ancient Commentators on Aristotle (Richard Sorabji, Ed., introduced by Wilfrid Hodges, and translated by Chatti, Saloua and Wilfrid Hodges). Bloomsbury Academic. Al-Khūnajī, Afḍal al-Dīn. al-Jumal. Available online in the site www.al-mostafa.com. Al-Khūnajī, Afḍal al-Dīn. (2010). Kashf al-Asrār ‛an Ghawāmiḍ al-Afkār (edited and introduced by Khaled El-Rouayheb). Institute for Islamic Studies & Iranian Institute of Islamic Philosophy. Al-Rāzī, Fakhr al-Dīn. (2003). Manṭiq al-Mulakhkhass (A. F. Karamaleki & A. Asgharinezhad, Eds.). ISU Press. Al-Sanūsī, Muḥammad b. Yūsuf. Sharḥ Mukhtaṣar al-manṭiq. Cairo: 1292/1875. Manuscript available at www.al-mostafa.com Avicenna. (1964). al-Shifā’, al-Mantiq 4: al-Qiyās (S. Zayed, Ed., rev. and intr. by I. Madkour). Cairo. Chatti, S. (2019). Arabic logic, from al-Fārābī to Averroes. Birkhaüser, Springer. Chatti, S. (2022). On some ambiguities in Avicenna’s analysis of the hypothetical quantified propositions. Arabic Sciences and Philosophy, 32(1). Copi, I. M. (1967). Symbolic logic (third edition of XIX 282). The Macmillan Company, CollierMacmillan Limited. El-Rouayheb, K. (2019). The development of Arabic logic (1200–1800). Schwabe Verlag. Haack, S. (2006). Philosophy of logics (15th printing). Cambridge University Press.

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Hodges, W. (2017). Ibn Sīnā on reductio ad absurdum. Review of Symbolic Logic, 10(3), 583–601. Hodges, W. (2016). Mathematical background to the logic of Avicenna. In preparation. Available online at http://wilfridhodges.co.uk/arabic44.pdf Ibn ‛Arafa. (1980). al-Mukhtaṣar fī al-Manṭiq. In Saad Gherāb, al-Maṭbaʿa al-ʿaṣriyya (Ed.), Risālatān fī al-Manṭiq (pp. 59–123). Tunis. Ibn ʽArafa. al-Mukhtaṣar fī al-Manṭiq. Manuscript available online in www.al-mostafa.com Karimullah, K. (2014). Al-Fārābī on conditionals. Arabic Sciences and Philosophy, 24, 211–267. Kleene, S. (1967). Mathematical logic French translation by Jean Largeault, Logique mathématique. Librairie Armand Colin. 1971. Łukasiewicz, J. (1972). Contribution à l’histoire de la logique des propositions. In J. Largeault (Ed.), Logique mathématique, textes. Armand Colin. Movahed, Z. (2009). A critical examination of Ibn Sīnā’s theory of the conditional syllogism. Sophia Perennis, 1, 1. Available online at: www.ensani.ir/storage/Files/20120507101758-90 55-5 Rescher, N. (2006). Studies in the history of logic. Ontos Verlag. Russell, B., & Whitehead, A. N. (1973). Principia mathematica (Paperback edition to *56). Cambridge University Press. Street, T. (2008). Arabic and Islamic philosophy of language and logic. In E. N. Zalta (Ed.), The Stanford Encyclopedia of philosophy (New Edition (2013) ed.). Stanford University. http:// plato.stanford.edu/entries/arabic-islamic-language/. Strobino, R. (2018). Ibn Sīnā’s logic. In E. N. Zalta (Ed.), The Stanford Encyclopedia of philosophy. Available online in https://plato.stanford.edu/archives/fall2018/entries/ibn-sina-logic

Chapter 3

Mullā Ṣadrā Shīrāzī and the Meta-Theory of Logic Sayeh Meisami

Abstract Introduced into Western scholarship briefly for the first time in the late nineteenth century and later more elaborately in the fourth quarter of the twentieth century, the Persian philosopher of the Safavid period Mullā Ṣadrā Shīrāzī (d. 1045/ 1635-6) has so far been studied for his ontology, epistemology, eschatology, political philosophy, and commentaries on the Qurʾan and Tradition. One area of his work that has not been studied as much is logical theory and meta-theory. This chapter focuses on Mullā Ṣadrā’s meta-theory of logic with respect to three subjects that he discussed in his logical and philosophical writings. First, the place of logic among sciences that also includes the question of the subject matter of logic. Second, the relation between logic and ontology of knowledge that shapes his position on the nature of assent. Third, his confirmation of Shihāb al-Dīn Suhrawardī’s modification of modal logic, which Mullā Ṣadrā defends based on his own existence-centered metaphysics.

3.1

Introduction to the Philosopher and His Works on Logic

Ṣadr al-Dīn Muḥammad b. Ibrāhīm b.Yaḥyā Qawāmī Shīrāzī, known as Mullā Ṣadrā (d. ca.1045/1635-6) was born into an influential family in the city of Shiraz in the Safavid Persia (907–1135/1500–1736) and contributed greatly to the intellectual milieu of his time. He studied with major scholars of his day such as Muḥammad Bāqir Mīr Dāmād (d. 1040/1631), Shaykh Bahāʾ al-Dīn al-ʿĀmilī (d. 1030/1620-1), and Abu’l-Qāsim Mīr Findiriskī (d. 1050/1640-1). He also trained great thinkers including Muḥsin Fayḍ Kāshānī (1090/1680-1) and ʿAbd al-Razzāq Lāhījī (d.1072/ I am grateful to the anonymous reviewers as well as the volume editor for their insightful comments. S. Meisami (*) University of Dayton, Dayton, OH, USA e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Chatti (ed.), Women’s Contemporary Readings of Medieval (and Modern) Arabic Philosophy, Logic, Argumentation & Reasoning 28, https://doi.org/10.1007/978-3-031-05629-1_3

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1661-2). There is no question that Mullā Ṣadrā enjoyed the opportunity to both teach and write during his lifetime despite certain disturbances (Khamenei, 2000), a fact that is evidenced by the large quantity of texts he has produced1 and his opportunity to train a good number of scholars. Yet the attraction of Mullā Ṣadrā’s philosophy in his homeland, Iran, as well as in India and beyond becomes noticeable mainly through the Qajar Period (1796–1925) and after. More recently, Asian, Middle Eastern, and Western scholars have produced texts in European languages on various aspects of Mullā Ṣadrā’s philosophy including the ontological doctrines of the graded unity of existence (Bonmariage, 2007; Kamal, 2006; Rizvi, 2009), and the unification of the subject and object of knowledge (Kalin, 2010; Safavi, 2002), philosophy of mind (Parildar, 2020), eschatology (al-Kutubi, 2015), hermeneutics (Peerwani, 2004; Rustom, 2012), spiritual methodology (Jambet, 2006; Moris, 2003), and political philosophy (Meisami, 2018; Toussi, 2020). There are also a few volumes that introduce Mullā Ṣadrā’s philosophy in its totality (Akbarian, 2009; Ha’iri Yazdi, 1992; Kalin, 2014; Meisami, 2013; Morris, 1981; Nasr, 1997; Rahman, 1975) and translations into English of several of his works. Mullā Ṣadrā’s magnum opus, al-Ḥikma al-mutaʿāliya fī’l-asfār al-ʿaqliyya al-arbaʿa, henceforth referred to as al-Asfār, is a summa of his philosophy including elaborate responses to his intellectual and spiritual predecessors. This voluminous treatise alone proves Mullā Ṣadrā to be a systematic philosopher who both builds on the findings of his predecessors and goes beyond them to offer his own philosophical views. The structure of al-Asfār is an innovation among the philosophical compendiums of similar stature. Unlike his main philosophical influences, Avicenna (d. 428/ 1037) and Shihāb al-Dīn Suhrawardī (d. 587/1191), Mullā Ṣadrā does not divide his compendium into logic, physics, and metaphysics. Rather, inspired by the symbolism of “journey” (sg. safar. pl. asfār) that is prevalent in Ṣūfī prose and poetry, Mullā Ṣadrā’s treatise presents a path toward wisdom with the philosopher as a seeker of truth on an intellectual quest that begins with an inquiry into the concept and reality of existence and ends with grasping the meaning of life after death. The non-traditional division of al-Asfār must have been one of the reasons why in Western scholarship little attention has been paid to Mullā Ṣadrā’s contribution to logical theory.2 Mullā Ṣadrā’s engagement with logical theory and meta-theory especially in the context of his responses to his predecessors deserves to be studied and discussed more diligently especially in relation to his philosophy as a whole. To this end, the present study highlights Mullā Ṣadrā’s meta-theory of logic with respect to three subjects. First, the place of logic among sciences and the subject matter of

Kalin (2003) provides an excellent annotated bibliography of Mullā Ṣadrā’s writings and Rizvi (2007) includes important historical notes on Mullā Ṣadrā’s life and works. 2 One exception is El-Rouayheb (2010, pp. 133–137) who briefly introduces Mullā Ṣadrā’s views on syllogisms without middle terms based on his Addenda on the Commentary on the Philosophy of Illumination (Shīrāzī, 2010). 1

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logic. Second, the relation between logic and ontology of knowledge that explains his position on the nature of assent. Third, his treatment and metaphysical justification of Shihāb al-Dīn Suhrawardī’s position on modalities. Mullā Ṣadrā’s writings on logic as listed by El-Rouayheb (2019, pp. 152–155) consist of (1) al-Taʿlīqāt on the well-known Sharḥ (commentary) by Quṭb al-Dīn Shīrāzī (d. 710/1311) on Suhrawardī’s Ḥikmat al-ishrāq (Philosophy of Illumination); (2) Risāla fī’l-ṭaṣawwur wa’l-taṣdīq (Treatise on Conception and Assent); (3) Al-Tanqīḥ fī’l-manṭiq;3 (4) Ḥall shubhat al-jadhr al-aṣamm that offers a solution for what is now known as the liar paradox. The last treatise is believed to have been attributed to Mullā Ṣadrā by mistake (El-Rouayheb, 2019, p. 155; Rizvi, 2007, p. 113) due to the similarity of his name to the real author, Ṣadr al-Dīn Shīrāzī Dashtakī (d. 903/1498). In addition to the first three of the sources mentioned above, I cite from Mullā Ṣadrā’s al-Taʿlīqāt ʿalā ilāhiyyāt al-shifā where he comments on Avicenna’s definition of logic and defends him against some of his critics. I also draw on his Sharḥ al-Ḥidāya al-athīriyya, which is a commentary on Athīr al-Dīn Abharī’s famous philosophical volume, al-Hidāya. Last but not least among my sources is al-Asfār which provides the philosophical background for Mullā Ṣadrā’s logical meta-theory. Further studies need to be pursued on Mullā Ṣadrā’s logical debates with his predecessors such as Avicenna, Suhrawardī, Fakhr al-Dīn Rāzī (d. 606/1210), Naṣīr al-Dīn Ṭūsī (d. 672/1274), Quṭb al-Dīn Shīrāzī, Ghyāth al-Dīn Dashtakī (d. 949/1542), Athīr al-Dīn Abharī (d. 664/1265), and Jalāl al-Dīn Dawānī (d. 908/1502) among others. A preliminary question about Mullā Ṣadrā’s place in the history of logic is whether he belongs to the Avicennan or Post-Avicennan schools. This is particularly important given his engagement with people from both camps, for instance, the post-Avicennan logician, Athīr al-Dīn Abharī, and the Avicennan Naṣīr al-Dīn Ṭūsī.

3.2

The Place of Logic Among Sciences and Its Subject Matter

Since the publication of an article by Sabra (1980) about Avicenna’s view on the subject matter of logic there has appeared a line of fruitful discussions by logicians and historians of Arabic logic which raise important questions, two of which I briefly report here to set up the background for my discussion of Mullā Ṣadrā.4 The first question is raised about attributing to Avicenna a clear-cut position on the subject

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This treatise is also known as al-Lumaʿāt al-mashriqiyya that is translated into Farsi and annotated by Mishkāt al-Dīnī (1981). 4 Since in this article I discuss Avicenna’s meta-theory of logic only as a background for understanding Mullā Ṣadrā’s position, in discussing the former, I rely on secondary sources by experts in the field and references to primary texts are through those secondary sources. The question of the subject matter of logic in Avicenna is still an ongoing conversation among Avicenna scholars.

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matter of logic as Sabra’s article seems to endorse based on a passage from Ilāhiyyāt al-shifāʾ: As you have known, the object of logic was the secondary intelligible concepts (al-maʿānī al-maʿqūla al-thāniya)-those that depend upon (tastanid ilā) the primary intelligible concepts-insofar as they may be of use in arriving at the unknown from the known, and not insofar as they are thoughts having an intellectual existence that is not attached to matter at all or attached to non-corporeal matter. (Sabra, 1980, p. 753)

Based on the above passage, Sabra concludes that for Avicenna the subject matter of logic or “secondary intelligibles” consists in “thoughts of second order” or “second order concepts. (pp. 763–64).” McGinnis (2010) elucidates this concept by describing its referents as “those accidental features that accrue to essences considered inasmuch as they are conceptualized,” and he provides examples for them: These features involve: the logical classification of things into subjects and predicates, as well as into genera, species, and the like; the various logical modes that hold of propositions formed from these predicables, as, for example, being universal or particular, essential or accidental, necessary or possible, always or sometimes, and the like; and likewise the valid inferential structures that hold between propositions so as to lead to necessary conclusions. (McGinnis, 2010, p. 35)

This has been the dominant understanding of Avicenna’s position on the subject matter of logic to date. However, Hodges (2016) calls for further analysis of different works by Avicenna such as Taʿlīqāt and Ḥikmat al-mashriqīyyīn, as well as the way Avicenna performs logic in his logical writings (p. 2). Hodges raises several issues in the contemporary attempts to understand Avicenna on the subject matter of logic due to the “confusion” that is caused by Avicenna’s “parallel discussions at corresponding points in different books (8).” In his analysis, Hodges uses the distinction between the “subject term” and the “subject individuals” of a theoretical science, for example, we can discuss “number” as the subject term of arithmetic while different “numbers” are the “subject individuals.” After comparing parallel passages where Avicenna describes logic, Hodges concludes that “with marginal exceptions, any well-defined meaning can be a subject individual of logic (p. 11)” and continues to argue that the subject term or “the main business of logic is to tell us the forms of formally valid inferences (p. 13).”5 To sum up, Hodges’ alternative reading of Avicenna widens the scope of the subject matter of logic to include any

5

In his forthcoming book, Hodges provides a more formal treatment of this topic and concludes the discussion as follows: “In a nutshell, Ibn Sīna’s characterization of the subject term of logic answers the question ‘What does it mean for logic to be formal?’, and his answer bears close comparison with the views of Bolzano. His characterization of the ‘features’ of the subject individuals answers a different question, namely ‘What are the logical constants?’ The reader should be warned that over the centuries, Ibn Sīna’s characterization of the subject term of logic has captured the interest of quite a number of people whose logical knowledge didn’t reach to distinguishing between these two questions. As a result one often sees a confusion between the subject individuals of logic, which are arbitrary well-fined meanings, and their ‘features’, which are a small group of higher-order concepts.” (Hodges, forthcoming, p. 69)

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clear and distinct notions plus the features that accrue to them only in the mind, i.e. second-order concepts, and the relations among them. This brings us to the second question that is raised with respect to Sabra’s argument regarding the subject matter of logic in Avicenna. In objection to Sabra’s argument that Avicenna’s description of the subject matter of logic as the secondary intelligibles became “henceforward the standard doctrine to which later Arabic logicians turned” (p. 756), it has been argued that certain logicians through thirteenth century and after claimed a different position from Avicenna, according to which the subject matter of logic are the objects of conceptions and assents (El-Rouayheb, 2012, 2016; Street, 2008). The allegedly divergent position is traced in the logical writings of Afḍal al-Dīn Khūnajī (d. 646/1248) who influenced a host of later logicians including Athīr al-Dīn Abharī (d. 663/1265), Najm al-Dīn Kātibī (d. 675/ 1276), Sirāj al-Dīn Urmawī (d. 682/1283), and Ibn Wāṣil al-Ḥamawī (697/1298). In his Kashf al-asrār, Khūnajī (2010) describes the subject matter of logic as “the objects of conception and assent (al-maʿlūmāt al-taṣawwuriyya wa l-taṣdīqiyya),” a position that was later highlighted and celebrated by his followers (El-Rouayheb, 2012, p. 71).6 One evidence for the influence of Khūnajī’s position is that major textbooks on logic have since been divided into two sections of conception (taṣawwur) and assent (taṣdīq). In her recent work, Chatti (2019) provides a clear exposition of the so called shift from Avicennan to post-Avicennan position on the subject matter of logic. Taking into consideration the above mentioned critique of Sabra’s understanding of Avicenna based on the passage from Ilāhiyyāt al-Shifā Chatti (2019) also consults Avicenna’s al-Najāt and Manṭiq al-mashriqiyyīn. After explaining the meaning of secondary intelligibles, or “second intentions”, she argues that for Avicenna, logic goes beyond second intentions when it is used as a tool or “Organon” for other sciences.7 Furthermore, Avicenna divides knowledge or science into conception and assent, which shows his familiarity with this division (pp. 21–22). Yet, Chatti agrees with Street (2008) and El-Rouayheb (2012) that after Khūnajī, major logicians emphatically considered conception and assent, which include both first intentions and second intentions, as the subject matter of logic, a position that rivalled Avicenna’s and became popular (p. 22).8 With this sketch of the topic’s background, I turn to Mullā Ṣadrā’s definition of logic. The first question that comes to mind in this respect is whether Mullā Ṣadrā opted for the popular post-Avicennan trend. Since nowhere in his works does he engage in an assessment of the Avicennan versus post-Avicennan positions on the

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This movement was resisted by a group of philosophers and logicians, most prominently, Naṣīr al-Dīn Ṭūsī (d. 672/1274), Shams al-Dīn Samarqandī (d. 722/1322) and Quṭb al-Dīn Rāzī Tahtānī (d. 766/1365) who advocated for the Avicennan position on the subject matter of logic and critiqued the other trend (El-Rouayheb, 2012). 7 In this regard, Chatti (2019, p. 21) quotes Ṭūsī’s commentary on al-Ishārāt where he expounds on Avicenna’s definition of logic by saying that logic is “a science by itself (ʿilmun bi nafsihi) and a tool with regard to other sciences. . .” 8 For the further shift through Khūnajī’s students from “conception and assent” to “the objects of conception and assent,” see El-Rouayheb (2012, p. 72).

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subject matter of logic, I look into scattered passages from his philosophical and logical works including his commentary on Abharī, which can indirectly point us to his preference. In his al-Tanqīḥ fī’l-manṭiq which follows the Avicennan nine-part division of logic, Mullā Ṣadrā defines logic in the Introduction (Isagoge) section as follows: Logic is a cognitive scale (qisṭās al-idrākī) by which thoughts (al-afkār) are assessed as true (al-ṣaḥīḥ) or false (al-fāsid). And thought (al-fikr) is the methodical movement of the mind through the forms of objects that are present to it, which is knowledge (al-ʿilm), toward what is not present to it with respect to that aspect which is not present, namely, ignorance (al-jahl). The sum of these movements is not based on intuition (ḥads), and if there is no [proper] order in the method of performing (naḥw al-taʾdiya) it or what results from it, it would be erroneous. Therefore, this protector (al-ʿāṣim) [i.e. logic] is indispensable.9 (Shīrāzī, 2006, p. 197)

With minor changes in wording, the content of Mullā Ṣadrā’s descriptive definition of logic is the same as Avicenna’s in Manṭiq al-ishārāt wa’l-tanbīhāt: Logic is intended to give the human being a canonical tool [(al-ʾāla al-qānūniyya)] which, if attended to, protects him from error in his thought. I mean by “thought” here that which a human being has, at the point of resolving [(ijmāʿ)], to move from things present in his mindincluding what is conceived (mutaṣawwara) or assented to (muṣaddaq bihā) . . . to things not present in it. This movement inevitably has order and format in the elements dealt with. . . Thus logic is a science by means of which one learns the kinds of movements from elements realized in the human mind to those whose realization is sought. (Avicenna, 1984, pp. 47–48).10

Both philosophers accept the cognitive nature of logic, which is necessary for enumerating it among sciences. In pre-modern times, often sciences (al-ʿulūm) were subsumed under either theoretical philosophy (al-falsafa al-naḍariyya) or practical philosophy (al-falsafa al-ʿamaliyya) with philosophy aiming at “understanding the realities of all things (ḥaqāʾiq kull al-ashyā) as much as it is possible for the human being” (Avicenna, 1952, p. 12). Obviously, if logic is supposed to guarantee that scientific discourses consist only of statements that correspond to reality,11 it cannot be itself less than a science (ʿilm). It is true that Avicenna plays down the dispute over whether logic is a tool for philosophy or part of philosophy as “futile” (Avicenna, 1952, p. 16; Sabra, 1980, p. 752), yet this is due to the fact that he believes we can consider logic both as a tool of philosophy and as a part of it depending on how we define “philosophy.” For him, science is a part of philosophy if the latter is defined as “all theoretical inquiries (kull al-baḥth al-naḍarī) in every way;” and it is a tool of philosophy if by philosophy we mean “inquiries into things in terms of being existent” (baḥth ʿan al-ashyā min ḥaythu hiya mawjūda) (Avicenna, 1952, p. 16). Moreover, in the following pages from al-Manṭiq al-shifā where he discusses the purpose of logic, Avicenna refers to logic as both the “science All quotations from Mullā Ṣadrā’s texts are my translation unless noted otherwise. I have made slight changes in Inati’s translation (1984) of this passage. 11 For Avicenna on the relation between logic and science, see Gutas (1988, pp. 280–285) and McGinnis (2010, pp. 28–35). 9

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of logic” (ʿilm al-manṭiq) and “the art of logic” (ṣināʿat al-manṭiq) (pp. 18–20). This is the starting point for Mullā Ṣadrā’s comments on the place of logic among sciences in his comments on Avicenna’s text. But, Mullā Ṣadrā pushes the discussion toward a new direction by associating logic with both theoretical and practical sciences, which I explain below. Mullā Ṣadrā’s long but unfinished commentary on Avicenna’s Ilāhiyyāt al-shifā is often referred to as al-Taʿlīqāt ʿalā ilāhiyyāt al-shifā. There is no conclusive evidence as to when exactly this commentary was written but therein Mullā Ṣadrā makes references to some of his own major philosophical treatises including al-Asfār. Although in this book Mullā Ṣadrā, as a commentator on Avicenna, limits himself to elucidating and defending him against his critics, he also delicately interjects some of his own ideas and critical points, very much with the same approach as that of Ṭūsī’s in his famous commentary on Avicenna’s al-Ishārāt wa’l-tanbīhāt. However, Mullā Ṣadrā also interprets Avicenna to his own favor in several places. One of the places where he does so to the effect of reconciling Avicenna’s position with his own is the place of logic among sciences. In this work (Shīrāzī, 2003, pp. 22–23), after explaining Avicenna’s criteria for the division of philosophy into theoretical and practical, Mullā Ṣadrā explains that Avicenna categorizes logic under theoretical philosophy not practical philosophy despite the fact that logic is concerned with the way of action (kayfiyyat al-ʿamal), i.e. the manner of our mental act. For Mullā Ṣadrā, it is possible for a science to be at the same time theoretical and also relate to the way of action. He believes this view would also be acceptable to Avicenna as the latter suggests in his major writings that for a science, relating to the way of action is not the same as having action as its subject matter (p. 7). On the other hand, Mullā Ṣadrā emphasizes, for Avicenna, the goal (ghāya) of logic is not the content of the sciences as such but protecting thought process from errors (p. 22). Thus, Mullā Ṣadrā concludes, for Avicenna, logic and other theoretical sciences have their subject matter (mawḍūʿ) in common while departing in their goals. Conversely, logic and other practical sciences differ over their subject matter, but agree in their goal that is the action as such (nafs al-ʿamal), respectively in the mental and the extra-mental domains. The difference between logic and other practical sciences over their subject matters is due to the fact that the subject matter of logic is “secondary intelligibles,” over which we have no voluntary control, and that of other practical sciences consists in actions that are subject to voluntary control (p. 23). Thus, throughout al-Taʿlīqāt, Mullā Ṣadrā attributes to Avicenna the categorization of logic under theoretical philosophy and endorses this position. Yet, he also intends to highlight the connection between logic and practical sciences. To do so, earlier in the text he suggests three ways of understanding “theoretical” and “practical” depending on the context in which they are used. Borrowing from Quṭb al-Dīn Shīrāzī (d. 710/1311), Mullā Ṣadrā argues that the terms “theoretical” and “practical” have different senses depending on whether we use them in dividing (1) all sciences in general (al-ʿulūm muṭlaqan), (2) philosophy (al-ḥikma), or (3) the arts (al-ṣināʿāt). In the first division, “theoretical” refers to those sciences that do not relate to the way of action (kayfiyyat al-ʿamal), and vice versa for “practical.” In the second division,

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“theoretical” applies to that type of philosophy with a subject matter which consists in things whose existence is not subject to our choice (ikhtiyār) such as “secondary intelligibles,” and vice versa for “practical” philosophy. In the third division, “theoretical” applies to those arts such as jurisprudence or medicine whose existence does not depend on continual practice (Shīrāzī, 2003, pp. 6–7). With the consideration of different usages for “theoretical” and “practical” in each division, Mullā Ṣadrā categorizes logic under theoretical philosophy and practical science at the same time: Logic falls under theoretical philosophy if we use the term “theoretical” from the second division since the subject matter of logic, i.e. secondary intelligibles, is not subject to our voluntary control; but, logic is a practical science when we use the term “practical” from the first division because logic is related to the way of action, i.e. mental acts. As we can see, in the above discussion from al-Taʿlīqāt, Mullā Ṣadrā has taken for granted Avicenna’s position on the subject matter of logic being the secondary intelligibles without a second doubt. Another evidence for this preference is found in his commentary on Athīr al-Dīn Abharī’s al-Hidāya. Although in this work the citation from Abharī, which relates to the subject matter of logic, and Mullā Ṣadrā’s comments are very concise, the passage is important because of Abharī’s place among the so called post-Avicennan logicians. In this short passage, after explaining Avicenna on the types of theoretical philosophy, Abharī is quoted by Mullā Ṣadrā to claim that there is an inconsistency in Avicenna’s categorization of logic under theoretical philosophy due to the latter’s position on the subject matter of logic as secondary intelligibles: Now, these are the divisions of the principle philosophy (al-ḥikmat al-aṣliyya), [i.e. theoretical philosophy]. But, how is it possible to include logic in philosophy and categorize it under the theoretical type, as the Shaykh [Avicenna] has done, while the subject matter of philosophy (al-ḥikma) is real existents (al-mawjūdāt al-ʿayniyya) with existence (al-wujūd) belonging to the general matters? (al-umūr al-ʿāmma) (Shīrāzī, n.d., p. 11)

Clearly, Abharī’s objection here does not aim at rejecting the categorization of logic under theoretical philosophy, but to prove that the subject matter of logic cannot be secondary intelligibles, a position that he defends as a post-Avicennan logician.12 Before relating Mullā Ṣadrā’s response to this objection, I need to provide a quick survey of the usage of the term “general matters” (al-ʾumūr al-ʿāmma) in Islamic philosophy because Mullā Ṣadrā’s response to Abharī relies on his understanding of this term as “philosophical secondary intelligibles.” Avicenna follows Aristotle in counting “general matters” as the subject matter of the first philosophy in the sense of those features which belong to “existent qua existent” (mawjūd bi mā huwa mawjūd). For Aristotle in his Metaphysics (IV31005a 22–24), “these truths hold good for everything that is, and not for some special genus apart from others” (Aristotle, 2001, p. 736). In the same vein, Avicenna says that “the existent

In his Tanzīl al-afkār, Abharī tries to prove that the subject matter of logic is “conceptions” and “assents,” against which Ṭūsī has a detailed argument in his Taʿdīl al-miʿyār fī naqd tanzīl al-afkār (El-Rouayheb, 2012, pp. 78–80).

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(al-mawjūd) in accepting these accidents, [i.e. features] and in being prepared for them, does not need to become specified as natural, mathematical, moral, or some other thing,” the two most obvious examples of which are “existence” and “oneness” (Avicenna, 2005, pp. 10–11). As for the ontological status of the general matters, in al-Mubāḥathāt he states that “existence is one of the attributes (al-ṣifāt) of a thing, and its being the effect (al-maʿlūl) of another is one of the fixed attributes (al-ṣifāt al-mutaqarrara) of it rather than a consideration of the mind (min al-iʿtibār), and the same is true of its being possible rather than necessary” (Avicenna, 1993 p. 131). There are different interpretations of what Avicenna really means by calling existence an attribute or accident, which is also related to the problem regarding his position on the relation between thingness (shayʾiyya) and existence (wujūd).13 For Mullā Ṣadrā whose philosophy revolves around the metaphysical primacy of existence over essence in both mental and extra-mental domains, only one interpretation holds water: Existence is not “added” to essence and we speak of the relation between the two only “figuratively” because the ‘relation’ is one of unity (ittiḥād) and “it is not like the relation between subject and its accident, or between the subject qualified and the attribute that qualifies it (Shīrāzī, 2014, p. 34).14 The question concerning the referent of “existence” is believed to be one of Suhrawardī’s main points of departure from Avicenna’s metaphysics. Suhrawardī rejects the real origin of the general matters and characterizes them as “purely intellectual.” (Suhrawardī, 1999, p. 45; Amin Razavi, 1997, pp. 33–35). Mullā Ṣadrā has a long response to Suhrawardī in his al-Taʿlīqāt and other writings to prove that general matters such as existence are not just in the mind (Shīrāzī, 2010, pp. 288–316; 1999, I: p. 139–140). Rather, as he argues in al-Asfār, secondary intelligibles such as “existence” have a real origin in the extra-mental world (al-Asfār, I: 332–33).15 It is in this context that he suggests a two-pronged typology of secondary intelligibles which was further developed and labeled by his followers as “logical secondary intelligibles” (al-maʿqūlāt al-thānīya al-manṭiqiyya) and “philosophical secondary intelligibles” (al-maʿqūlāt al-thānīya al-falsafiyya) (Sabziwārī, 2000, II: 163). One could argue that this distinction had its roots in Avicenna depending on how his position on the essence-existence relation is interpreted.16 For Mullā Ṣadrā, logical secondary 13

Robert Wisnovsky (2000) has a profound discussion on this topic and considers the difference of opinions among Avicenna scholars to have been caused by Avicenna’s developing ideas across his different writings. Wisnovsky (2000, pp. 199–200, footnote 36) correctly mentions that Catholic interpreters of Avicenna have for the most part understood him to imply that existence is “attached” to essence as an accident. 14 In Mullā Ṣadrā’s metaphysics, existence is the reality with essences being determinations of it in the mind as he says in al-Asfār “the quiddity (al-māhiyya) itself is not one of the things (shay’un min al-ashyā) unless it becomes existent because its very quiddity depends on the realization of its existence (taḥaqquq wujūdihā).” (1999, I, p. 75; 2014, p. 15) 15 The debate over the status of philosophical secondary intelligibles dates back to Naṣīr al-Dīn Ṭūsī and the ambiguity in his treatment of the topic. For a history of the topic and Ṭūsī’s influence on later thinkers, see Sharif and Javadi (2009). 16 For example, in his al-Taʿlīqāt, Avicenna (1972, p. 167) discusses secondary intelligibles in the context of describing the subject matter of logic. There he argues that “secondary intelligibles” are

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intelligibles include concepts such as genus ( jins) and differentia ( faṣl) which are only the products of the mental process, or as Suhrawardī calls it “mental consideration,” with no origin whatsoever in the extra-mental world. In contrast, philosophical secondary intelligibles, while occurring to primary intelligibles in the mind (ʿurūḍ fī’l-dhihn), are at the same time fixed attributes of extra-mental realities (ittiṣāf fī’l-khārij). For example, “oneness” occurs to the primary intelligible of “human being” in the mind and at the same time an existent human being in the extra-mental world is “one” for real and that is why we can attribute oneness to her. In al-Shawāhid al-rubūbiyya, Mullā Ṣadrā equates the latter category with the general matters and describes them as the “special accidents” (al-ʿawāriḍ al-khāṣṣa) such as “one” or “many” which are predicated of existent things (Shīrāzī, 1967, p. 15). With the above in mind, I return to Mullā Ṣadrā’s response to Abharī in Sharḥ al-hidāya. Mullā Ṣadrā briefly argues that in the context of theoretical philosophy, the general matters, i.e. philosophical secondary intelligibles are predicates (maḥmūlāt) of things in the world rather than independent subjects (mawḍūʿāt); and derivatives (mushtaqqāt) rather than origins (mabādī) “since there is no [real] distinction between existent qua existent (al- mawjūd bi mā huwa mawjūd) and existence (al-wujūd), just like there is no [real] distinction between possible qua possible (al-mumkin bi mā huwa mumkin) and possibility (al-imkān) as the Shaykh [Avicenna] says in al-Shifā” (Shīrāzī, n.d., p. 11). Mullā Ṣadrā is too concise in his argument, but, based on my elucidation of his divisions of “the secondary intelligibles,” I interpret him as follows: There is no inconsistency in Avicenna’s position because the general matters are attributes of the extra-mental existents and derivatives of them rather than independent objects in their own right. Therefore, there is no inconsistency between the categorization of logic under theoretical philosophy and considering the subject matter of theoretical philosophy as the secondary intelligibles. Mullā Ṣadrā could simply dismiss Abharī’s objection by saying that the general matters as philosophical secondary intelligibles are the subject matter of metaphysics while logic deals with the logical secondary intelligibles. All in all, Mullā Ṣadrā’s response to Abhārī is a strong evidence that with respect to the subject matter of logic, he considers himself an advocate for the Avicennan position. Nevertheless, the question remains as to why Mullā Ṣadrā did not address the so called shift among the post-Avicennan logicians in a more direct and detailed manner despite the popularity of their position? Can we speculate that he did not regard it as a drastic change with implications for how logic is actually done, at least in the context of his own philosophy? In other words, did he see the polemic simply a result of different terminological confusions as in the case of his response to Abharī? Obviously, there is room for more investigation on this matter in the future.

first proved in “metaphysics” and then they become the subject matter of logic only by virtue of directing the mind from the known to the unknown. However, this is debatable and the present article is primarily concerned with Mullā Ṣadrā’s understanding of Avicenna in favor of his own existence-centered metaphysics.

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The Status of Assent: Reinterpreting Fakhr al-Dīn Rāzī and Suhrawardī

Mullā Ṣadrā’s adherence to the classical position on the subject matter of logic as secondary intelligibles in the face of the strong preference toward conceptions (al-taṣawwurāt) and assents (al-taṣdīqāt) among later logicians such as Abharī does not detract from the significance of the latter dual concepts in his logical writings. This is particularly evidenced by the fact that he devoted one whole treatise to this subject, i.e. Risāla fī‘l-ṭaṣawwur wa’l-taṣdīq, which is an important text in an area where logic overlaps with ontology of knowledge.17 Following Fārabī and Avicenna and major philosophers after them, with the exception of Suhrawardī, Mullā Ṣadrā divides all knowledge by acquisition (ʿilm al-ḥuṣūlī) into conception and assent, which in turn corresponds to the division of logic into two major parts, i.e. the logic of definitions (al-taʿrīfāt) and the logic of propositions (al-qaḍāyā). Avicenna distinguishes conception from assent based on the simplicity of the former and the compositeness of the latter. For example, the conception of “triangle” is realized in the mind as a single meaning but “the knowledge that the angles of every triangle are equal to two right angles” consists of concepts that are related to each other via a judgment (ḥukm) (Avicenna, 1952, p. 17; 1984, p. 49). After Avicenna, this division gave rise to controversies over the exact components of assent and its relation to conception. In al-Tanqīḥ, Mullā Ṣadrā only reiterates the dominant view that assent has a compound nature. Here, he is mainly concerned with the relation of assent to truth. He says that, Knowledge is either assent that is a preponderant belief (al-iʿtiqād al-rājiḥ) [regarding the relation between subject and predicate] which in case it is an assertion ( jazm), if it corresponds to reality, it is certitude (yaqīn), and otherwise it is total lack of knowledge. And, in case it is an opinion (ẓann) that could be true or false. Other than assent is conception. Sometimes “conception” (al-taṣawwur) is used as a general term that applies to both [conception and assent] in which case it is synonymous with “knowledge.” (Shīrāzī, 2006, p. 197)

Regardless of the fact that Mullā Ṣadrā does not offer his own position in the above passage, the distinction he makes between the two meanings of the term “conception” is actually an important component of his argument for the simplicity of assent in his Risāla fī‘l-ṭaṣawwur wa’l-taṣdīq. In this treatise, building on Suhrawardī’s comments on Avicenna, Mullā Ṣadrā argues that the distinction between the two meanings of “conception” was already implied by Avicenna. He quotes from Suhrawardī’s al-Muṭāraḥāt that: The carelessness in the division of knowledge into conception and assent in the opening sections of books is due to the fact that the opening is not a good place for deep inquiries. But the most inclusive delimitation (aḥwaṭ al-taqyīdāt) is given by Master AbūʿAlī [Avicenna] in

Parildar (2017) offers a detailed discussion of how the ontology of knowledge in Mullā Ṣadrā’s philosophy explains his position on conception and assent.

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S. Meisami some places according to which knowledge is either conception alone or conception accompanied with assent -both having conception in common.18 (Shīrāzī, 1894, p. 11)

Mullā Ṣadrā’s above citation of Suhrawardī is a reinforcement of what the former argues earlier in the text. Previously, Mullā Ṣadrā argued for the above distinction regarding the meaning of “conception,” where he approached the division of knowledge into conception and assent from two perspectives, existential and conceptual. At the existential level of analysis, he describes conception and assent as “simple” (al-basīṭ) and “individualized” (al-mutashakhkhaṣ) existences in the mind. From this perspective, the division of knowledge into conception and assent resembles the division of a genus into two different species with “natural unity” (al-wiḥda al-ṭabīʿiyya) though not of a kind enjoyed by extra-mental species such as “horses” and “men” that are composed of matter and form. Rather, conception and assent enjoy a kind of natural unity that is found in “psychic qualities” (al-kayfiyyāt al-nafsāniyya) such as “will” and “anger.” He then concludes his remarks on the existential status of conception and assent by saying that “they are forms of mental existence (al-wujūd al-dhihnī)19 through which objects of knowledge (al-maʿlūmāt) come into existence in the mind. As for the conceptual level of analysis, “conception” and “assent” are proved to be “secondary intelligibles that concern logicians.” The two concepts are subsumed under the generic concept (al-maʿnā al-jinsī) of “knowledge” in the same way that “white” and “black,” as simple concepts, are subsumed under the generic concept of “color.” Thus, by applying the generic sense of “conception,” Mullā Ṣadrā tries to demonstrate that from both the existential and conceptual perspectives, conception and assent prove to be simple rather than composite (Shīrāzī, 1894, p. 5). With the above characterization of both the existential and conceptual status of conception and assent as simple, Mullā Ṣadrā challenges different positions on assent. He starts with a view that is often attributed to Fakhr al-Dīn Rāzī. According to Mullā Ṣadrā, Rāzī is responsible for the misleading view according to which assent is composed of three or four components (Shīrāzī, 1894, p. 6). What he is referring to here is the analysis of assent into three conceptions and one psychic act: (1) conception of the subject, (2) conception of the predicate, (3) conception of the relation between subject and predicate, and (4) the judgment that could be affirmative or negative. Mullā Ṣadrā attempts to refute not only the compositionality of assent but also the responses provided by logicians in defense of this position especially those by Quṭb al-Dīn Rāzī in his Sharḥ al-maṭāliʿ and Sharīf Jurjānī (d. 816/1413) in his annotations on it (Shīrāzī, 1894, pp. 6–8). Since this debate is

18

For this treatise, I translate from the original Arabic that is available as an appendix in Ibn al-Muṭahhar al-Ḥillī’s Jawhar al-naḍīd fī sharḥ manṭiq al-tajrīd (1894). For a different English translation of the cited passages from this treatise, see Lameer (2006). 19 Mullā Ṣadrā devoted a chapter of al-Asfār (1999, vol. 1, pp. 263–326) to the topic of mental existence (al-wujūd al-dhihnī) which is an essential concept in his ontology of knowledge. On this topic, see Rizvi (2009, pp. 77–101).

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beyond the scope of the present study, I will proceed to explain another position on the nature of assent that Mullā Ṣadrā tried to refute.20 Another example of a position on assent that is mentioned and refuted by Mullā Ṣadrā in his Risāla fī ‘l-ṭaṣawwur wa’l-taṣdīq is the equation of assent with judgment (al-ḥukm). He says, “how terribly absurd is the view of those who identify assent with judgment while the latter is one of the actions of the soul (al-afʿāl min al-nafs) and the former is a kind of passive knowledge (al-ʿilm al-infiʿālī) to which the soul is related as a recipient rather than agent” (Shīrāzī, 1894, p. 6). If we state the implied third premise, the argument can be formulated as: P1. Assent is passive; P2. Judgment is active; P3. Passive and active are opposites; therefore, assent is not judgment.21 For Mullā Ṣadrā, an assent such as “Joseph is tall,” is a piece of knowledge that is acquired as a simple existence in the mind. It is only upon second intention that we analyze it into subject, predicate, relation, and judgment, as did Fakhr al-Dīn Rāzī. By distinguishing the reality of assent as a simple mental existent and its analysis upon second intention into two conceptions of subject and predicate, and the conception of their relation accompanied by an affirmative or negative judgment, Mullā Ṣadrā refutes the dominant interpretation of Fakhr al-Dīn Rāzī’s position on assent and reinterprets him in agreement with himself. As he puts it: It is possible to show the agreement between the [above] position and that of the [earlier] philosophers, and also interpret later thinkers’ view of “assent” in its favor, such as the one by the Imam [Fakhr al-Dīn Rāzī] and the school of those who define “assent” as conception to which judgement occurs (al-maʿrūḍ bi’l-ḥukm) or conception that is accompanied by judgment such as the opinion of the Commentator of al-Maṭāliʿ [Quṭb al-Dīn Rāzī] even though they might not have put into words an evidence of not understanding what we have achieved. (Shīrāzī, 1894, p. 8)

The above comments fit into Mullā Ṣadrā’s general methodology of reconciling apparently conflicting positions by offering multiple levels of analysis. Another example of this approach is his treatment of Suhrawardī’s view of the nature of assent.22 Mullā Ṣadrā interprets Suhrawardī’s position on the nature of assent and

For a detailed discussion of all the different positions on the nature of assent and Mullā Ṣadrā’s responses, see Haʾirī Yazdī (1988, pp. 28–31), Lameer (2006, p. 156–182) and Parildar (2017, pp. 147–154). 21 In their footnotes on their translation of the passage, Haʾirī Yazdī (1988, p. 33, n.1) and following him Lameer (2006, p, 112, n.5) provide examples for active and passive knowledge. Active knowledge is such as God’s knowledge of everything other than his own essence or the knowledge that a cause has of its effect. Passive knowledge is the knowledge that humans have of the extramental world via noetic forms. As we can see, these examples exclude the immediate knowledge that both God and human beings have of themselves, i.e. knowledge by presence (al-ʿilm al-ḥuḍūrī), which is central to Mullā Ṣadrā’s theory of knowledge since he argues that all knowledge is possible against the backdrop of knowledge by presence that is immediate, i.e. not in need of noetic forms. For elaborate discussions of Mullā Ṣadrā’s theory of knowledge, see Haʾirī Yazdī (1992), Kalin (2010). 22 Mullā Ṣadrā’s professed indebtedness to Suhrawardī is noticeable in many parts of his philosophy. Yet, for the most part, he adopts from Suhrawardī what reinforces his own position rather than accepting the latter’s ideas in their totality. One of the most important instances of this approach is 20

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modifies it in a manner that reinforces his own view. He cites from Suhrawardī’s al-Muṭāraḥāt that according to the latter, knowledge is not divided into conception and assent; rather, it consists only in conceptions. And assent is simply a conception that is accompanied by a judgment that is not a type of knowledge but a psychic act by which a relation is either established (īqāʿ al-nisba) between subject and predicate or extracted (intizāʿ al-nisba), with the two acts respectively amounting to affirmative judgment (al-ḥukm al-ījābī) and negative judgment (al-ḥukm al-salbī). Suhrawardī is quoted to add that the act of judgment is grasped by the mind as a conception which is different from the judgment as an action that is issued by the mind, and he concludes that: Conceptions are sometimes the conception of extra-mental things (taṣawwur al-ʾumūr al-khārijiyya) and sometimes the conception of psychic judgments that are the assents, hence the fact that all our knowledge goes back to conceptions even though in the case of judgments and assents there are psychic acts of establishing [a relation] or cutting [it]. (Shīrāzī, 1894, p. 11).

At this point, Mullā Ṣadrā ends his quotation from Suhrawardī and proceeds to offer his interpretation of it. Mullā Ṣadrā begins with reinstating Suhrawardī’s view that assent is a subdivision of absolute conception (al-taṣawwur al-muṭlaq) (Shīrāzī, 1894, p. 11), which he also identifies with “knowledge” in al-Tanqīḥ as previously quoted. He continues to reiterate the characterization of judgment as a psychic act which, qua action, cannot be a type of knowledge by acquisition since all knowledge by acquisition is passive. However, he argues here that judgment can also be regarded as active knowledge that is issued from our consciousness and as such “its existence is the same as its manifestation (ḍuḥūr) and “revelation” (inkishāf) to us. This is of course not “action” ( fiʿl) in the sense of one of the nine accidental categories because those are categories of quiddities while both conception and assent are instances of existence and “existence is not included into any of the ten categories” (p. 12) and does not have a definition as Mullā Ṣadrā extensively argues in his philosophical works such as al-Asfār (Shīrāzī, 1999, I: p. 50–53). The bottom-line of Mullā Ṣadrā’s response to Suhrawardī is that assent is a type of knowledge in its own right and cannot be reduced to conception. To understand Mullā Ṣadrā’s response better we need to go back to an earlier passage in his treatise where he presents three different usages for the term “judgment.” According to him, Judgment is a conception if we consider its general realization (muṭlaq ḥuṣūlih) in the mind, but it is also assent considering its specific mode of realization. In sum, there are three considerations here: one is the judgment itself as establishing [a relation] or extracting [it], which is a psychic act and not a type of knowledge by acquisition or mental conception; the

the adoption of Suhrawardī’s concept of gradation (tashkīk) that is applied to essence/quiddity in the Illuminationist ontology (Suhrawardī, 1999). Mullā Ṣadrā depart from Suhrawardī by applying “gradation” to existence rather than essence. According to Mullā Ṣadrā, “the instances of being are different in terms of intensity and weakness as such, priority and posteriority as such; nobility and baseness as such, although the universal concepts applicable to it and abstracted from it, named quiddities, are in contrast essentially, in terms of genus, species, or accidents (Shīrāzī, 1999, IX: 186).”

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second one is the conception of the judgment (taṣawwur al-ḥukm) that is an instance of knowledge through acquisition and conception that is of course not an assent but an example of the other kind of knowledge [i.e. conception] even though the object of this [conception] is the assent since it is not impossible for the same thing to be knowledge (ʿilm) and object of knowledge (maʿlūm) at the same time by two [different] considerations (bi iʿtibārayn). Third is a conception that is not separable from judgment but requires (yastalzim) it. This is assent as the counterpart (muqābil) of conception and a subdivision [of knowledge] next to conception. (Shīrāzī, 1894, p. 9)

It is the third consideration from the above passage that Mullā Ṣadrā has in mind when he comments on Suhrawardī about the status of assent. In calling “assent” a “counterpart” of conception, Mullā Ṣadrā refers to “assent” as a conception that is conditioned by the accompaniment of a judgment. This reading is also reinforced by a passage in his al-Taʿlīqāt ʿalā sharḥ ḥikmat al-ishrāq where he characterizes the conception that is the counterpart (al-muqābil) of assent as “knowledge that is not conditioned by judgment” (Shīrāzī, 2010, p. 21). In other words, conception and assent as counterparts are species that fall under the genus of absolute conception. With the above summary of Mullā Ṣadrā’s understanding of “judgment” as entailed by conception, his approach to Suhrawardī’s position becomes clear. The reality of assent in the mind cannot be reduced to conception because assent consists in conception and assent together as one reality. Yet, at the level of conceptual analysis one can break down assent into its genus, i.e. conception, and differentia, i.e. judgment, though as a simple mental existence, this conception is unified with the judgment and both the conception and the assent are “two kinds of knowledge” (al-nawʿān li’l-ʿilm) which are “simple” (basīṭ) (Shīrāzī, 2010, p. 22). Setting the simplicity of conception and assent aside, the situation is analogically comparable to analyzing “human being” into “animal” and “rational” conceptually while in reality they are not separable. One may object to this analogy that “human being” has a real unified existence in the extra-mental world while assent is only in the mind. The response that I can give on behalf of Mullā Ṣadrā is that in his overall ontology of knowledge, mental and extra-mental existences are different grades of the same reality with the former occupying even a higher rank in existential intensity, i.e. reality, due to being detached from the matter (Shīrāzī, 1999, I: pp. 263–266).23 In sum, for Mullā Ṣadrā, assent is a simple reality in the mind though, at second intention, it can be analyzed into conception and judgment.

For Mullā Ṣadrā, knowledge is a mode of being realized for an immaterial being such as the human soul and is issued from the soul itself since “God has created the human soul with the power to create forms of both immaterial and material objects (Shīrāzī, 1999, I: pp. 264–265).” In his theories of knowledge and truth, the correspondence between our knowledge and the extra-mental world is justified because the mind creates forms as the mental or ideal existence of the material objects. The mental or ideal existence ranks higher in its intensity than those forms which are dependent on the matter because the immaterial is always more intense in existence than the material (Kalin, 2014, pp. 121–123; Meisami, 2013, pp. 44–47).

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3.4

Mullā Ṣadrā on Modalities: More Engagement with Suhrawardī

Mullā Ṣadrā follows Avicenna in his general method and principles of logic. One of the places where he seems to have deviated from Avicenna is his position on modalities. In this section, after a brief account of Suhrawardī’s controversial reduction of the modes, i.e. necessity, possibility, impossibility (ḍarūra, imkān, imtināʿ), to necessity alone, I discuss Mullā Ṣadrā’s adoption of the reduction based on his al-Tanqīḥ and al-Taʿlīqāt ʿalā sharḥ ḥikmat al-ishrāq. Suhrawardī’s position on modalities has not been adopted by most Arabic logicians due to the impact of his reduction on the application of logical rules such as conversion (ʿaks) and negation (salb), as well as syllogism (qīyās). Among later Islamic philosophers, Mullā Ṣadrā is an exception in supporting Suhrawardī’s position on modalities, a fact which, as I will argue in this section, is mostly due to the metaphysical relevance of the latter’s logical position to Mullā Ṣadrā’s philosophical system. In Ḥikmat al-ishrāq, after explaining the three modes in the manner of his preceding logicians, Suhrawardī argues that all the modes must be reduced to “definitive necessity” (al-ḍarūra al-battāta).24 According to him: Since the possibility of the possible, the impossibility of the impossible, and the necessity of the necessary are all necessary, it is better to make the modes of necessity, contingency, and impossibility parts of the predicate (ajzāʾ al-maḥmūl) so that the proposition will become necessary at all events. You would thus say “necessarily all humans are possibly literate, necessarily animals, or impossibly stones. This is definitive necessity (al-ḍarūra al-battāta).”25 In the sciences, we investigate the contingency or impossibility of things as part of what we are seeking. Yet, we can make no definitive and final judgment except for that which we know necessarily.26 (Suhrawardī, 1999, p. 17)

The gist of Suhrawardī’s suggestion is that all scientific propositions should turn into necessary propositions because the internal mode, namely the mode of the predicate ( jihat al-maḥmūl), is attributed to it necessarily. Four points must be noted here: First, in this discussion, Suhrawardī is concerned with “scientific” propositions, by which, as his above examples illustrate, he means universal propositions about what can or cannot be the case in the world. Second, in this discussion Suhrawardī uses the term “necessity” in the sense of “substantial necessity,” namely necessity of predication with respect to the essence of the subject to the exclusion of “descriptional”

24

In Suhrawardī’s texts and all commentaries on him in both Arabic and Persian this term appears as “‫ ”ﺑﺘﺎﺗﻪ‬. 25 I have made slight changes to Ziai’s translation of the passage. According to Hossein Ziai in his explanatory note on the above passage from Ḥikmat al-ishrāq, the term “definitively necessary proposition” has no precedence before Suhrawardī and in contemporary formal logic it is comparable to a type of “iterated modal proposition” (Suhrawardī, 1999, p. 173, n. 20). 26 Other than in Ḥikmat al-ishrāq, Suhrawardī mentions this thesis in the logic part of al-Talwīhāt al-lawḥiyya wa’l-ʿarshiyya only in passing: “if the mode ( jihat) becomes a part of the predicate ( juzʾ al-maḥmūl), then the relation (rabṭ) would be necessary” (Suhrawardī, 1955, p. 90).

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or “conditional” types of necessity.27 In this regard, he states that “by necessary we only mean that which it [an object] has by virtue of its essence (dhāt) (p. 17). Third, he includes only affirmative propositions in sciences. (p. 18).” Fourth, he considers modality inseparable from scientific propositions and leaves no room for non-modal propositions therein (p. 18). In brief, for Suhrawardī, sciences must only include “definitively necessary affirmative” (al-mūjiba al-ḍarūriyya al-battāta) propositions (p. 17). Before turning to Mullā Ṣadrā, it is important to note that Suhrawardī’s argument for the reduction of modalities is preceded by his attempt to prove that particular ( juzʾiyya) propositions can be transformed into universal (kulliyya) ones (p. 15), hypothetical (sharṭiyya) propositions to predicative (ḥamliyya) (pp. 15–16), and negative propositions (sāliba) to affirmative (mūjiba) (pp. 15). Below, I explain the manner of these transformations with reference to Mullā Ṣadrā’s account in his logical work, al-Tanqīḥ, and al-Taʿlīqāt, namely the commentary on Quṭb al-Dīn Shīrāzī’s Sharḥ ḥikmat al-ishrāq, where he discusses the topic in more details. Mullā Ṣadrā’s logical treatise, al-Tanqīḥ fī’l-manṭiq, is divided into nine parts, each of which is called an “Illumination” (ishrāq) and each Illumination is divided into several sections, each called a “Sparkle” (lumʿa). Only two of these sections in the entire treatise have slightly different names for an obvious reason: in both cases, Mullā Ṣadrā presents an idea that he seems to regard as different from Avicennan logic. One of the two cases is the section which discusses the modes of propositions in the style of Suhrawardī and it is called “Illuminative Sparkle” (lumʿat ishrāqiyya).28 This section belongs to the Fourth Illumination which is on the modes of propositions ( jihāt al-qaḍāyā). All the sections in this part except for the third one follow the rules and principles of Avicennan logic. Before reaching section three, Mullā Ṣadrā provides a quick summary of the types of modal propositions in classical Arabic logic, as well as the different meanings of possibility (al-imkān). Once he reaches the third section which is on the reduction of the modes of propositions to necessity, he clearly follows Suhrawardī though without mentioning his name or using the latter’s favorite term, “definitive necessity.” The section opens with this remark that propositions do not differ from each other essentially as different species do; rather they are only accidentally different to the effect that all propositions are in principle “necessary predicative universal affirmative” (mūjiba kulliyya ḥamliyya ḍarūriyya) (Shīrāzī, 2006, 212). Mullā Ṣadrā states very laconically that negative (sāliba) propositions become affirmative (mūjiba) by making them metatheic (maʿdūla), which is done by attaching the negation to the predicate as in negative predicative propositions (sālibatu’l-maḥmul). Particular ( juzʾiyya) propositions become universal (kulliya) through “positing” (iftirāḍ) which I will explain below based on al-Taʿlīqāt since he does not provide any explanation for it here. Hypothetical (sharṭiyya) propositions become predicative (ḥamliyya) through the

27

For types of necessity in Arabic logic see (Chatti, 2014a, b; Lagerlund, 2008; Street, 2004). The other case is section five under part four that he titles as “Sparkle of Wisdom” (lumʿa ḥakamiyya). In this section Mullā Ṣadrā introduces an additional condition for negation of propositions, namely, “unity of predication” (wiḥdat al-ḥaml). (2006, p. 214). 28

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articulation (taṣrīḥ) of “implication” (luzūm) or “conflict” (ʿinād) in respectively connective (muttaṣila) and disjunctive (munfaṣila) propositions. Finally, all modal propositions turn into necessary ones if the mode ( jihat) is relocated to become part of the predicate since “possibility is necessary for the possible just like impossibility is for the impossible and necessity for the necessary (Shīrāzī, 2006, p. 212).” In the end, he reiterates Suhrawardī’s argument that true sciences (al-ʿulūm al-ḥaqīqiyya) are only concerned with what is necessarily the case despite the fact that the “form” (al-ṣūra) of some propositions may appear otherwise (Shīrāzī, 2006, p. 212). This is as far as Mullā Ṣadrā goes in al-Tanqīḥ to include Suhrawardī’s suggestions, and in the rest of the treatise one can hardly find any impact of the mentioned adjustments on the other logical rules such as negation, conversion, and syllogism. In his al-Taʿlīqāt, Mullā Ṣadrā treats the types and modes of propositions, except for hypotheticals, with more elaboration. Following the order of the discussion in Rāzī’s Sharḥ ḥikmat al-ishrāq, Mullā Ṣadrā begins with the transformation of particular propositions into universal ones. Like Suhrawardī (1999, p. 18), Mullā Ṣadrā regards particular propositions such as “some (baʿḍ) artists are sculptors”29 as “existential indefinite” (al-muhmalatu’lbaʿḍiyya) and sees the removal of the indefiniteness as the key to turning it into a definite universal proposition. To do so, he suggests that we put in the subject place a definite noun (pl. asmāʾ muʿayyana) that consists of a noun “artist” plus an adjectival clause “who sculpt” which Mullā Ṣadrā calls “the posited description” (al-ʿunwān al-iftirāḍī). And then one is logically justified to add a universal quantifier to the proposition (Shīrāzī, 2010, p. 117). This way, the previous proposition will change into the universal one that “all artists who sculpt are sculptors.” For negative propositions turning into affirmatives, Suhrawardī resorts to a type of proposition with a “negative predicate” (sālibatu’l-maḥmūl) which are referred to as “metathetic” (maʿdūla) propositions (Avicenna, 1984, p. 83). Like Avicenna, Suhrawardī’s example for such propositions is the singular proposition, “Zayd is non-literate” (Zaydun huwa lā kātib), and according to him, in such propositions “the affirmative copula (al-rābiṭ al-ījābī) remains and the negation becomes part of the predicate (Suhrawardī, 1999, p. 15).” This is the most controversial part of his discussion due to the issues related to existential import. To consider as equivalent the propositions “Zayd is not literate” and “Zayd is non-literate,” Suhrawardī must commit to the existence of Zayd in both propositions, which goes against classical Arabic logic where the real negative proposition is without existential import and the two propositions do not have the same truth value. Broadly speaking, in this regard, Avicenna is in agreement with the majority of medieval logicians (Chatti, 2016; Hodges, 2012).30 Suhrawardī too only presupposes existence for the subject of Avicenna regards such propositions as definite and particular. According to him, “if it is evident that the judgment is about some and does not extend to the rest, or it extends [to the rest] in an indirect manner, then the definite proposition is particular” (Avicenna, 1984, p. 80). 30 Chatti (2016) draws attention to the “complexity” of Avicenna’s position on existential import. According to her definition of existential import, “a categorical proposition of the form subjectpredicate, whether singular, indefinite or quantified, has an existential import if and only if it 29

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metathetic propositions, and he argues that even when the subject does not have a referent in the extra-mental world, “a mental affirmative judgment (al-ḥukm al-mūjib al-dhihnī) can only apply to something established in the mind (thābitun dhihnī) (1999, p. 15).” As for Mullā Ṣadrā’s comments on this issue, he evokes the problem of existential import by first reiterating Suhrawardī’s words that in both kinds of affirmative propositions, namely the real affirmative with a positive predicate and the metathetic proposition where negation is attached to the predicate, the existence of the subject is presupposed. Mullā Ṣadrā also emphasizes that in a real negative proposition, the existence of the subject is not required and calls this an important “semantical difference” (al-farq al-maʿnawī) between the real negative and the metathetic propositions (Shīrāzī, 2010, p. 92). There are two objections here reported by Mullā Ṣadrā to which he responds in favor of the difference between the metathetic and real negative propositions over existential import. The first objection amounts to the idea that only in the case of predicating a positive property the existence of the subject is assumed, but in predicating a negative property such as “non-literate” the subject need not to be assumed as existent since even the predicate is an empty term. To this Mullā Ṣadrā responds that “although negation is included in the concept of the predicate, either it has some trace of existence in terms of its form in the mind (ṣūratihī fī’l-dhihn) or in terms of its being subcontrary (muqābil al-malaka) which assumes the capacity of the subject for having either of the two opposite properties (pp. 92–93).” The second objection is based on the assumption that in a real negative proposition when the subject is not existent, it is not absolutely non-existent (al-maʿdūm al-muṭlaq) otherwise the latter could not even be conceived; rather in such cases the subject has a realization (taḥaqquq) in the mind, therefore even in real negative propositions, the subject is not an empty term. Mullā Ṣadrā has a long response to this objection in which he dismisses existential import for real negative propositions, such as “Zayd is not literate.” The gist of Mullā Ṣadrā’s argument is that in the metathetic affirmative propositions the existence of the subject, be it mental or extra-mental, is assumed due to affirmation as such (nafsu ījābin), whether the predicate is positive or negative. This is not the case in a real negative proposition because “in the latter the negation is applied to non-existent qua non-existent (maʿdūm min ḥaythu innahū maʿdūm), which is not the case in the former (Shīrāzī, 2010, pp. 93–94).” Now, given that Mullā Ṣadrā distinguishes between the two kinds of propositions “Zayd is not literate,” and “Zayd is non-literate” in terms of their existential import, how could he regard them as equivalent? The answer lies in Rāzī’s comment that the aforementioned “semantical difference” does not apply when we are dealing with “universal definite propositions” (al-qaḍāyā al-muḥīṭa) (Shīrāzī, 2010, p. 98). Mullā Ṣadrā supports this position by arguing that in a universal proposition, quantification is over all the members under the subject term, so a universal proposition has existential import in both affirmative

requires the existence of its subject’s referent(s) to be true,” and the corollary of this is “if such a proposition has an existential import, it implies the existence of the objects satisfying the subject term. If it can be true when the subject is empty, it does not have an import.” (p. 47)

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and negative forms (pp. 98–99). What he means is that, for example, the two universal propositions, “All humans are not winged” and “All humans are non-winged” equally presuppose the existence of the subject due to the function of the universal quantifier “all,” hence their equivalence. In view of this answer, and the fact that Suhrawardī only allows for definite universal propositions in sciences (Suhrawardī, 1999, p. 15), one can infer that Mullā Ṣadrā does not find existential import problematic with regard to Suhrawardī’s suggestion that universal negative propositions can turn into universal affirmative by including the negation in the predicate.31 As for hypothetical (sharṭiyya) propositions, Mullā Ṣadrā suffices to mention Suhrawardī’s suggested transformation only in al-Tanqīḥ and unlike the previous subject, al-Taʿlīqāt does not provide a separate discussion regarding the transformation of hypothetical propositions. Neither does Suhrawardī himself say much about the adjustment of hypothetical propositions in Ḥikmat al-ishrāq though he mentions “implication” and “conflict” in the context of discussing negative propositions. In this regard, he says that “a connective conditional (muttaṣila) proposition is negated by removing the implication (rafʿ al-luzūm), and a disjunctive conditional (mufaṣila) proposition is negated by removing the conflict (rafʿal-ʿinād)” (Suhrawardī, 1999, p. 16). As I explained before, in al-Tanqīḥ Mullā Ṣadrā suggests that by articulating the implication and conflict in respectively connective and disjunctive conditionals, we can turn them into predicative propositions. To explain what he means, I borrow from Avicenna’s examples for the two kinds of propositions (Avicenna, 1984, pp. 86–87). Mullā Ṣadrā understands Suhrawardī as suggesting that a connective conditional such as “if the sun rises, then it is daytime” should turn into the universal predicative “Every sunrise implies daytime” and a disjunctive proposition like “every number is either odd or even” should turn into something like “In every number oddness and evenness are related by a disjunction.” The above suggested adjustments are all for the purpose of excluding scientific propositions to universal predicative affirmative ones. Yet, for Suhrawardī a truly scientific proposition that corresponds to reality must also be necessary in its modality. As in the case of the rest of his logical discussions, Suhrawardī addresses modal propositions via examples from Avicenna (1984, p. 79). To show the necessary mode of possible propositions (al-qaḍāyā al-mumkina), he includes the three modes in the predicate and argues that by doing so the mode of the proposition becomes “necessary:” “Necessarily all human beings are possibly literate.” To show the necessary mode of impossible propositions (al-qaḍāyā al-mumtaniʿa) he says “necessarily all human beings are impossibly stones.” (Suhrawardī, 1999, p. 17). In other words, the mode of the relation between subject and predicate, whether it is

31

The application of this position on existential import could impact syllogistics in problematic ways, which is beyond the scope of this paper and requires further study and evaluation by logicians.

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necessity, possibility, or impossibility, is always necessarily so.32 In this discussion Suhrawardī also includes that type of propositions which are necessary at an indefinite time. According to him: Even for that which is only true sometimes, we use the definitely necessary proposition. In the case of “breathing at some time,” it would be correct to say that “All men necessarily breathe at some time.” That men necessarily breathe at some time is always an attribute of man. That they necessarily do not breathe at some time is also a necessary attribute of man at all times, even at the time when he is breathing. (1999, p. 18)

Suhrawardī’s inclusion of propositions which are necessary at an indefinite time gives Mullā Ṣadrā the opportunity to initiate a metaphysical justification for the former’s reduction of scientific propositions to definitively necessary ones. In his al-Taʿlīqāt, Mullā Ṣadrā relates the logical structure of propositions to the metaphysical structure of the world. In this context, first he approves of attributing necessity to propositions which are necessary at indefinite times (al-qaḍāyā al-dāʾima), and argues that “average minds are not capable of understanding this.” He proceeds to say that for this purpose one must refer to “the science that is nobler,” i.e. metaphysics, according to which “all things, even temporal occurrences (al-ḥādithāt) are necessary in relation to the origins (al-mabādīʾ) and in relation to the [rest of] universal propositions and the order [of the universe] in total” (Shīrāzī, 2010, p. 119). At the basic level, this metaphysical position is premised on the Avicennan axiom that “whatever is possible in its existence does not exist unless rendered necessary with respect to its cause” (Avicenna, 2005, p. 32). Yet, Mullā Ṣadrā adapts this axiom into his ontology to reinforce his own argument for the necessary connection among all things in the world in virtue of their existential unity (Mullā Ṣadrā 2003, p. 101; 1999, I: pp. 221–230). In the same context of comments on Suhrawardī’s modification of modalities, we come across another example of metaphysical interjection by Mullā Ṣadrā. According to Suhrawardī, When the proposition is necessary, the mode of the copula ( jihat al-rabṭ) alone is sufficient for us; or it is posited to be definitive (al-battāta) without including another mode in the predicate – for example, when you say “all human beings are definitively animal.” In other modal propositions, whenever they are to be rendered definitive, the mode should be included in the predicate.33 (Suhrawardī, 1999, p. 18)

32

This may be interpreted as a combination of de re and de dicto senses: That the above propositions are respectively possible and impossible de re, but both are necessary de dicto. Yet, the application of the medieval logical terms de re and dicto, in Arabic logic is debatable. As far as Avicenna’s modal logic is concerned, Bäck (1992, pp. 229–231) argues against corresponding what he calls Avicenna’s “strict necessity” and “derivative necessity” with de re and de dicto despite “the temptation” to do so. On the other hand, Chatti (2014) argues for the existence of de re/de dicto distinction in Avicenna’s modal logic though she admits that his “treatment of modal propositions remains incomplete.” Also, Thom (2008) discusses the de re/de dicto distinction in the context of investigating the metaphysical application of Avicenna’s modal theory. 33 I have made slight changes in Ziai’s translation.

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Mullā Ṣadrā agrees with the above qualification of necessary propositions but he goes beyond Suhrawardī’s text in justifying the qualification based on a logical implication of his own ontology. To do so, he likens the case of necessary propositions to existential propositions in which the predicate is existence as in the proposition “X exists,” where there is no need for a copula.34 For Mullā Ṣadrā, “existence” appears in propositions either as a syncategorematic, i.e., the copula “is” or a categorematic term with an independent nominal meaning (al-maʿnā al-ismī), which he calls “the predicative existence” (al-wujūd al maḥmūlī) (Shīrāzī, 1999, I: p. 82).35 In his comment on Suhrawardī’s above quoted remarks about necessary modal propositions, Mullā Ṣadrā has the latter sense of “existence” in mind when he says that “the case is similar to that of a simple existential proposition where there is no need for any copula (al-rābiṭa) other than the predicate, [i.e. predicative existence] while other [propositions] are in need of the copula” (Shīrāzī, 2010, p. 119). Mullā Ṣadrā’s above juxtaposition of necessity and predicative existence is grounded in his ontology. Moreover, although his definition of the three modes and his division of possibility as quoted below recalls Avicenna, there is a fundamental difference between them due to their different metaphysical approaches to modalities: For Avicenna, except in the Necessary Being,36 it is the essence that is considered as necessary, possible in any of its senses (see Chatti, 2014a; Hodges, 2010), or impossible. But for Mullā Ṣadrā, since existence is the reality, the modes are attributed to existence. This shift shows itself in Mullā Ṣadrā’s characterization of “necessity” in the following passage which does not include the term “essence:” Next to the concept of existence (wujūd) and thingness (shayʾiyya) in general, there is no other concept that is conceived by the human mind prior to the concepts of necessary (al-ḍarūrī) and non-necessary (al-lāḍarūrī). So, when necessity (ḍarūra) is attributed to existence, [the mode] becomes necessity (al-wujūb);37 when it is attributed to non-existence,

This gives rise to the question whether for Mullā Ṣadrā, existence is a real predicate. In the history of Western philosophy Emmanuel Kant is credited with raising the issue and arguing that existence is not a real predicate, hence for him the failure of the ontological arguments for the existence of God. But Rescher (1960) argues that Abū Naṣr Fārābī (d. 339/950) initiated this question long before Kant, and Avicenna too pursued the discussion. Both philosophers are said to have addressed this question mainly for its metaphysical import regarding the distinction between essence and existence (pp. 429–430). Mullā Ṣadrā departs from the Peripatetic understanding of the relation between existence and essence since for him existence is the only authentic reality and in existential propositions, existence is the subject not the predicate. For Mullā Ṣadrā, “X exists” means “this existence is X” where “X” signifies a determination of existence in the mind. In the extra-mental world, there is only existence, and quiddity is only predicated of existence in the mind (Shīrāzī, 2014; p. 32). 35 On the usage of “syncategoremata” and “categoremata,” in classical Arabic logic, see Chatti (2014b) 36 Following Avicenna (2005), Mullā Ṣadrā (1999, I, 96) holds that “the essence of the Necessary Being is His existence (inna wājib al-wujūd māhiyyatuhū inniyyatuhū), by which similar to Avicenna, he means that the Necessary Being does not have any essence at all (see McGinnis, 2010, p. 169). 37 Mullā Ṣadrā’s use of two different terms, “ḍarūra” and “wujūb” for necessity recalls Avicenna’s application of them. Bäck (2018, p. 20) argues that there are two types of “necessary” for Avicenna 34

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[the mode] becomes impossibility (al-imtināʿ); and when non-necessity is attributed to either side of [existence or non-existence] or to both sides, there will respectively be general possibility (al-imkān al-ʿāmm) and narrow-possibility (al-imkān al-khāṣṣ). (Shīrāzī, 1999, I: pp. 83–84)

The idea that “necessity” in its general sense is the base for understanding the other two modes, though not Mullā Ṣadrā’s innovation,38 can be interpreted in a new way in light of his metaphysics. The conceptual reliance of the other two modes on “necessity” is rooted in the way Mullā Ṣadrā sees realities as manifestations of the one single existence, i.e. the Necessary Being. In his view, “necessity” as a concept is as evident as “existence” and co-extensional with it because in reality all that is the case is necessarily so. For Mullā Ṣadrā, it is true that we can conceptually analyze everything other than the Necessary Being into essence and existence, yet in terms of their essence those things are “pure nothingness” (al-laysiyya al-ṣirfa) while in relation to the Necessary Being, which is “the eternal necessity” (al-ḍarūra al-azaliyya), they are existences which proceeded from the Necessary Being (Shīrāzī, 1999, I: pp. 186–187). In other words, everything is “necessary in relation to the cause that necessitates it” (Ṭabāṭabāʾī, 2007, I: p. 97) but in itself suffers from “existential possibility” (al-imkān al-wujūdī) a concept that ultimately replaces “essential possibility” (al-imkān al-māhuwī) in Mullā Ṣadrā’s system. As he says in al-Mashāʿir, nothing “constitutes in reality an ipseity (al-huwiyya) that is separated from the ipseity of its existentiating cause (al-mūjid)” (Shīrāzī, 2014, p. 57). Furthermore, “the criterion for possibility [of a possible existent] is nothing other than its existential dependence (al-imkān al-wujūdī) and need (iftiqār) for another, and the criterion for necessity is nothing other than the lack of such need (Shīrāzī, 1999, I, p. 86). To conclude this section, Mullā Ṣadrā’s tendency to Suhrawardī’s reduction of the modes of scientific propositions to definitive necessity can be best understood if studied in light of his existential monism where all that exists is a manifested grade of the one existence that is necessary by itself. The metaphysical motivation behind Mullā Ṣadrā’s approval of Suhrawardī’s position on modalities is also evidenced by the fact that, unlike the latter, Mullā Ṣadrā does not try to extend its application to the other parts of logic. which are expressed by the two different Arabic terms, “wājib” and “ḍarūrī,” with the latter having a more general sense and used in logical discussions while the former is used in metaphysical discussions which address the Necessary Being. For the different meanings of necessity in Avicenna, also see Chatti (2014a, pp. 336–337). 38 According to Avicenna, the general possibility (al-imkān al-ʿāmm) is “that which accompanies the negation of the necessity of non-existence.” And the narrow-possibility (al-imkān al-khāṣṣ) “is meant that which accompanies the negation of both the necessity of non-existence as well as the necessity of existence, attributed to a subject.” (Avicenna, 1984, p. 95) In modern logic, these two notions are referred to as “one-sided possibility (“M”) and two-sided possibility (“Q”)” which in addition to the notion of “necessity” and “impossibility” comprise “the alethic modalities” (Thom, 2008, p. 361). Hodges (2010) and Chatti (2014a) examine the different meanings of possibility and according to the latter, the “narrow-possible is the one that Avicenna considers as the genuine meaning of possibility” (p. 335).

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Conclusion

The observations and arguments in the above sections were meant to indicate the significance of logical meta-theory for Mullā Ṣadrā and his strong connection to the long tradition of Arabic logic. His philosophical and logical treatises equally point to his command of logic at both the theoretical and meta-theoretical levels, as well as his constant engagement with the logical debates of his time. While the main focus of this study was Mullā Ṣadrā’s contributions to the meta-theory of logic, the discussions also indirectly indicate his command of logical theory per se. In this respect, I hope that further studies delve into Mullā Ṣadrā’s ideas in logical theory especially in the context of responding to objections concerning “mental existence” (al-wujūd al-dhihnī), and resolving some of the paradoxes associated with it (Shīrāzī, 1999, I: pp. 312–313; 1967, pp. 28–30). Yet, the distinction between theory and meta-theory is a modern-day convention. Especially for Mullā Ṣadrā, logical discussions were for the most part informed by his ontological and epistemological doctrines. For example, as I tried to show, in this study, Mullā Ṣadrā’s views on the subject matter of logic, the nature of assent, and the typology of propositions are deeply immersed in his existence-centered metaphysics that characterizes existence as the only authentic reality and essences as epiphenomenal. Lastly, my inquiry into Mullā Ṣadrā’s contributions to logical meta-theory demonstrates the strong influence of Avicenna in addition to a selective adoption of Suhrawardī’s modification of Avicennan logic. In brief, Mullā Ṣadrā fits into the category of the Avicennan logician rather than post-Avicennan, and the influence of Suhrawardī does not detract from this fact because despite his suggested modifications, the latter too belongs to the school of Avicennan logic and he never claimed otherwise. Moreover, Mullā Ṣadrā’s appreciation of Suhrawardī’s innovation regarding modalities is primarily due to metaphysical predilections rather than logical ones.

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Bäck, A. (2018). Existence and modality in Avicenna’s syllogistic. In M. Sgarbi & M. Cosci (Eds.), The aftermath of syllogism: Aristotelian logical argument from Avicenna to Hegel (pp. 11–34). Bloomsbury Publishing. Bonmariage, C. (2007). Le réel et les réalités: Mullā Ṣadrā Shīrāzī et la struture de la réalité. Vrin. Chatti, S. (2014a). Avicenna on possibility and necessity. History and Philosophy of Logic, 35(4), 332–353. Chatti, S. (2014b). Syncategoremata in Arabic logic: Al-Fārābī and Avicenna. History and Philosophy of Logic, 35(2), 67–197. Chatti, S. (2016). Existential import in Avicenna’s modal logic. Arabic Sciences and Philosophy, 26, 45–71. Chatti, S. (2019). Arabic logic from al-Fārābī to Averroes: A study of the early Arabic categorical, modal, and hypothetical syllogistics. Springer. El-Rouayheb, K. (2010). Relational syllogism and the history of Arabic logic, 900–1900. Brill. El-Rouayheb, K. (2012). Post-Avicennan logicians on the subject matter of logic: Some thirteenthand fourteenth-century discussions. Arabic Sciences and Philosophy: A Historical Journal, 22(1), 69–90. El-Rouayheb, K. (2016). Arabic logic after Avicenna. In C. D. Novaes & S. Read (Eds.), The Cambridge companion to medieval logic (pp. 67–93). Cambridge University Press. El-Rouayheb, K. (2019). The development of Arabic logic (1200–1800). Schwabe Verlag. Gutas, D. (1988). Avicenna and the Aristotelian tradition: Introduction to reading Avicenna’s philosophical works. Brill. Ḥāʼirī Yazdī, M. (1988). Āgāhī wa guwāhī: tarjuma wa sharḥ-i intiqādī-i risālah-i taṣawwur wa taṣdīq-i Mullā Ṣadrā. Muʾassasah-i iṭṭilāʿāt wa taḥqīqāt-i farhangī. Ḥāʼirī Yazdī, M. (1992). The principles of epistemology in Islamic philosophy: Knowledge by presence. State University of New York Press. Hodges, W. (2010). “Ibn Sīnā on modes, ʿIbarah ii.4.” Available online at http://wilfridhodges.co. uk/arabic07.pdf Hodges, W. (2012). Affirmative and negative in Ibn Sīnā. In C. D. D. Novaes & O. H. Thomassen (Eds.), Insolubles and consequences, essays in honour of Stephen Read (pp. 119–134). College Publications. Hodges, W. (2016). “Ibn Sīnā on the definition of logic.” Available online at http://wilfridhodges. co.uk/arabic41.pdf Hodges, W. (Forthcoming). Mathematical background to the logic of Avicenna. http:// wilfridhodges.co.uk/arabic44.pdf Jambet, C. (2006). The act of being: The philosophy of revelation in Mullā Ṣadrā. Zone Books. Kalin, I. (2003). An annotated bibliography of the works of Mullā Ṣadrā with a brief account of his life. Islamic Studies, 42(1), 21–62. Kalin, I. (2010). Knowledge in later Islamic philosophy: Mullā Ṣadrā on existence, intellect, and intuition. Oxford University Press. Kalin, I. (2014). Mullā Ṣadrā. Oxford University Press. Kamal, M. (2006). Mulla Sadra’s transcendent philosophy. Ashgate. Khamenei, M. (2000). Mullā Ṣadrā: Zindigī, shakhṣiyyat wa maktab-i Ṣadr al-mutaʾallihīn. Intishārāt-i bunyād-i ḥikmat-i Ṣadrā. Khūnajī, A. (2010). In K. El-Rouayheb (Ed.), Kashf al-asrār fī ghawāmiḍ al-afkār. Iranian Institute of Philosophy. al-Kutubi, E. S. (2015). Mullā Sadrā and eschatology: Evolution of being. Routledge. Lameer, J. (2006). Conception and belief in Sadr al-Din Shirazi (ca. 1571–1635). Iranian Institute of Philosophy. McGinnis, J. (2010). Avicenna. Oxford University Press. Meisami, S. (2013). Mullā Ṣadrā. Oneworld. Meisami, S. (2018). Knowledge and power in the philosophies of Ḥamīd al-Dīn Kirmānī and Mullā Ṣadrā Shīrāzī. Palgrave Macmillan.

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Mishkāt al-Dīnī, A.-M. (1981). Manṭiq-i nuwīn: Mushtamil bar al-lamaʿāt al-mashriqīyya fī’lfunūn al-manṭiqiyya. Mu’assasah-i intishārāt-i āgāh. Moris, Z. (2003). Revelation, intellectual intuition and reason in the philosophy of Mullā Ṣadrā: An analysis of al-Hikmah al-ʿarshiyyah. Routledge Curzon. Morris, J. W. (1981). The wisdom of the throne: An introduction to the philosophy of Mullā Ṣadrā. Princeton University Press. Nasr, S. H. (1997). Sadr al-Din Shirazi and his transcendent theosophy: Background, life and works. Institute for Humanities and Cultural Studies. Parildar, S. (2017). Applying gradational ontology to logic: Mullā Ṣadrā on propositions. In S. N. Ahmad & S. H. Rizvi (Eds.), Philosophy and the intellectual life in Shiʿah Islam (pp. 135–157). The Shīʿah Institute Press. Parildar, S. (2020). Intentionality in Mullā Ṣadrā. Springer. Peerwani, L. (Trans.). (2004). On the hermeneutics of the light-verse of the Qur’an/Mullā Ṣadrā Shīrāzī. ICAS Press. Rahman, F. (1975). The Philosophy of Mullā Ṣadrā Shirazi. State University of New York Press. Rescher, N. (1960). A ninth-century Arabic logician on: Is existence a predicate? Journal of the History of Ideas, 21(3), 428–430. Rizvi, S. H. (2007). Mullā Ṣadrā Shīrāzī: His life and works and the sources for Safavid philosophy. Oxford University Press. Rizvi, S. H. (2009). Mullā Ṣadrā and metaphysics: Modulation of being. Routledge. Rustom, M. (2012). The triumph of mercy: Philosophy and scripture in Mullā Ṣadrā. State University of New York Press. Sabra, A. I. (1980). Avicenna on the subject matter of logic. The Journal of Philosophy, 77(11), 746–764. Sabziwārī, H. (2000). In H. H.-Z. Amuli (Ed.), Sharḥ-i manẓūma (Vol. 3). Nashr-i nāb. Safavi, G. (Ed.). (2002). Perception according to Mullā Ṣadrā. London Institute of Islamic Studies. Sharif, Z., & Javadi, M. (2009). Maʿqūl-i thānī-i falsafī dar falsafah-yi mashshāʿī-ishrāqī: Az Khwajah Naṣīr tā Mīr Dāmād. Maʿrifat-i falsafī, 7(1), 37–79. Shīrāzī, M. ibn I. (1967). In J. Āshtiyānī (Ed.), Al-Shawāhid al-rubūbiyya. Chapkhānah-i Dānishgāh-i Mashhad. Shīrāzī, M. ibn I. (1999). In M. R. Muẓaffar (Ed.), Al-Ḥikmat al-mutaʿāliya fī’l-asfār al-ʿaqliyya al-arbaʿa (Vol. 9). Dār al-iḥyaʿal-turāth al-ʿArabī. Shīrāzī, M. ibn I. (2003). In N. Ḥabībī (Ed.), al-Taʿlīqāt ʿalā ilāhīyāt al-shifā li al-shaykh al-raʿīs Abū ʿAlī Ḥusayn ibn Sīnā. Bunyād-i ḥikmat-i Ṣadra. Shīrāzī, M. ibn I. (2010). In H. Ziai (Ed.), Addenda on the commentary on the philosophy of illumination. Pt. 1: On the rules of thought. Mazda Publishers. Shīrāzī, M. ibn I. (n.d.). Sharḥ al-hidāya al-athīriya (MS 6-32040). Iranian National Library. http:// dl.nlai.ir/UI/8a7be27a-e41b-4d72-93ae-34766ccadaff/LRRView.aspx Shīrāzī, M. ibn I. (1894). “Risāla fī‘l-ṭaṣawwur wa’l-taṣdīq.” Appendix to Ibn al-Muṭahhar al-Ḥillī’s al-Jawhar al-naḍīd fī sharḥ manṭiq al-tajrīd taʿlīf-i Muḥammad ibn Muḥammad ibn al-Ḥasan al-Ṭūsī. s.n. Shīrāzī, M. ibn I. (2014). The book of metaphysical penetrations: A parallel English-Arabic text (I. Kalin, Ed. & H. Nasr. Trans.). : Brigham Young University Press. Shīrāzī. M. ibn I. (2006). al-Tanqīḥ fi’l-manṭiq. In Ḥ. Nājī Isfahānī (Ed.), Majmūʿa-yi rasāʿil-i falsafī-i Ṣadr al-mutaʿallihīn (pp. 194–236). Intishārāt-i ḥikmat. Street, T. (2008). Arabic and Islamic philosophy of language and logic. In E. N. Zalta (Ed.), The Stanford Encyclopedia of Philosophy. https://plato.stanford.edu/archives/spr2015/entries/ arabic-islamic-language/ Suhrawardī, S. Y. (1955). In A. A Fayyāḍ (Ed.), Manṭiq al-talwīḥāt. Intishārāt-i dānishgāh-i Tehran. Suhrawardī, Sh. Y. (1999). The philosophy of illumination: A new critical edition of the text of Ḥikmat al-ishrāq (J. Walbridge, Ed. & H. Ziai, Trans.). Brigham Young University Press.

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Ṭabāṭabāʾī, M. Ḥ. (2007). Nihāya al-ḥikma (4th ed.). Muʾassasat al-nashr al-islāmī. Thom, P. (2008). Logic and metaphysics in Avicenna’s modal syllogistic. In S. Rahman et al. (Eds.), The unity of science in the Arabic tradition. Springer. Toussi, K. (2020). The political philosophy of Mullā Ṣadrā. Routledge. Wisnovsky, R. (2000). Notes on Avicenna’s concept of thingness (Šhayʿiyya). Arabic Sciences and Philosophy, 10, 181–221.

Chapter 4

Algorithms in Takmilat al-‘Uyūn of al-Iṣfahāni: Sources and Validation Nacéra Bensaou

Abstract Treatises on algebra in classical Arabic mathematics are texts written in the Arabic language, without mathematical symbolism and structure. An important part of these treatises is devoted to the solution of equations, of degree less than or equal to two, at the beginning of algebra (Book title of Roshdi Rashed on al Khawarizmi [1]: The Beginnings of Algebra) by al-Khawārizmī in the 9th century, then cubic equations by al-Khayyam (in the 11th century) and Sharaf Dīn al-Ṭūsī (in the 12th century) then equations of degree greater than three by al-Yazdī (in the 17th century). In 1824, an Iranian mathematician and astronomer, Alī Muḥammad ibn Muḥammad Ƥusayn al- Iṣfahānī proposed a new theory of cubic equations in a treatise titled “Takmilat al-‘Uyūn” (This title is translated by R. Rashed, in (Sharaf Al-Dīn al-Tūsī: Oeuvres mathématiques: Algèbre et Géométrie au XIIe siècle (tome 1). Société d’édition Les Belles Lettres, Paris, 1986a; Transfer of modern science and technology to the Muslim world. Istanbul, 1992): La Complétion des Fontaines). Written in Arabic in the ancient style, without mathematical symbolism, this treatise is exclusively dedicated to solving cubic equations, for which it uses only numerical algorithms. This treatise borrows some algorithms from the mathematics of its predecessors such as al-Khawarizmī, Sharaf al-Dīn al-Ṭūsī, al-Kāshī and al-Yazdī, and completes them. Is there any reasoning, not necessarily in a formal language, intended to establish the validity of the proposed algorithms, in these texts? This article aims to show, through several types of algorithms for solving equations, from a period that covers a millennium, from the ninth to the nineteenth century, that the answer is yes.

N. Bensaou (*) Faculty of Computer Science, University of Sciences and Technology Houari Boumediene (USTHB), Algiers, Algeria e-mail: [email protected] © The Author(s), under exclusive license to Springer Nature Switzerland AG 2022 S. Chatti (ed.), Women’s Contemporary Readings of Medieval (and Modern) Arabic Philosophy, Logic, Argumentation & Reasoning 28, https://doi.org/10.1007/978-3-031-05629-1_4

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Introduction

In her article on Arabic mathematics (Bellosta, n.d., p. 40), Hélène Bellosta, points out that Algebra was characterized from its beginnings by its style; both computational and demonstrative. This statement is clearly verifiable in the numerous treatises on algebra devoted to the solution of equations, which are to our interest in this paper. Indeed, in his treatise, al-Jabr wa al-Muqābala (‫)ﺍﻟﺠﺒﺮ ﻭﺍﻟﻤﻘﺎﺑﻠﺔ‬, al-Khawarizmī proposes a process for solving equations of degree less than or equal to two and a demonstration by geometry of the validity of his process. Since this founding treatise on algebra, research activities in mathematics have evolved to fall into one of two approaches: that of the arithmetic algebraists, who have developed algebra through arithmetic, and that of the geometer algebraists who have developed algebra through geometry. In the treatises of the two traditions, new algorithms for solving equations were proposed and the question of the validation of these algorithms was raised and studied.

4.2

Algorithms for Solving Equations

Exploring the question of algorithms and their validation in classical Arabic mathematics certainly requires a more in-depth and a more developed study than what this article could propose, given the large number of algorithms and the extent of mathematical disciplines and related fields that such endeavor would require, including arithmetic calculus, or the arithmetic applications of algebra, optics, astronomy, physics and so on. Therefore, we limit ourselves in this article to the algorithms of the resolution of equations as well as the methods of proofs deployed to validate them. These treatises are texts written in the Arabic language and without mathematical symbolism. To understand their contents, in addition to the lack of mathematical symbolism, a major difficulty is added in the fact that these texts are not structured. In other words, there is no explicit separation which distinguishes the different parts of the text: for example, the praises to God, the problem statement, the aim of the treatise, the presentation of the solution, the argumentation on the method used and so on. All these parts contribute to the composition of a linear text which only allows the reader to guess and detect the different parts thanks to the linguistic vocabulary specific to each one of them. To establish the mathematical commentary on a treatise, it is necessary to extract from these texts the expressions that relate to the algorithm and those corresponding to a proof or a justification of such algorithm. To highlight the computational aspect and the demonstrative aspect of the methods, we examine several treatises on algorithmic solutions of equations, from the ninth century, with the treatise al-Jabr wa al-Muqābala written by al

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Khawarizmi until Takmilat al ‘Uyūn (‫)ﺗﻜﻤﻠﺔ ﺍﻟﻌﻴﻮﻥ‬, the treatise on the solution of cubic equations of al Isfahānī from the nineteenth century. In our opinion, choosing to limit our study to algorithms for solving equations, and to some authors separated from one another by several centuries, does not reduce its importance, for the following reasons: 1. Solving equations is an ideally algorithmic discipline and a variety of algorithms of different kinds have been found. 2. The whole tradition of classical Arabic mathematics is inaugurated by a treatise on solving equations: that of al-Khawarizmi, al-Jabr wa al-Muqabala. 3. Arithmetic algebraists from the school of al Karajī (eleventh century) and al-Samaw’al (twelfth century) have developed algorithms for algebraic calculus, the theory of polynomials and the search for roots of numbers. In his work on Indian Calculus,1 al-Khawarizmī himself proposed several algorithms for arithmetic calculus based on positional numeration system, such as addition, multiplication, and the digit-by-digit extraction of the square and cubic roots of a number. Several of these problems are expressed as problems of solving equations. 4. The geometer algebraists such as ʿUmar al-Khayyām (eleventh century) and Sharaf al-Dīn al-Ṭūsī (twelfth century) contributed decisively to the solution of equations by geometry and by numerical algorithms which extract, digit by digit, the root of an equation. This algorithm, proposed by al-Ṭūsī, is an extension of the algorithm of extraction of the roots of numbers, presented by the arithmetic algebraists known since al-Khawārizmī. 5. In the fifteenth century, to calculate the Sine of one degree, al-Kāshī, mathematician and astronomer, proposed, a fixed-point iterative algorithm2 for solving a cubic equation, based on the convergence of a numerical sequence. 6. In the seventeenth century, al-Yazdī, in his voluminous treatise ‘Uyūn al Hisab (Muhammad Bāqir Zayn al-‛Ᾱbidīn al-Yazdī, n.d.) (‫ )ﻋﻴﻮﻥ ﺍﻟﺤﺴﺎﺏ‬applies the algorithm of al-Ṭūsī to solve equations of degree greater than three. 7. In the nineteenth century, al-Iṣfahānī proposed a new theory of cubic equations, all solved by numerical algorithms applicable to the resolution of an equation of degree n
2 that combines and extends (to real roots): – the algorithm of al-Ṭūsī applied to the digit-by-digit extraction of a real root; – the algorithm of al-Kāshī (and several of its variants), to solve equations that the previous algorithm does not solve; – an algorithm for reducing the interval of the root; and, al- Ƥisāb al Hindī (‫ )ﺍﻟﺤﺴﺎﺏ ﺍﻟﻬﻨﺪﻱ‬is the title of an Arabic mathematical book on arithmetic based on position numeration written by al Khawārizmī. See Allard (1992), the book written by the historian André Allard which collects latin translations of what is known about this lost book of al Khawārizmī. 2 A fixed point algorithm puts the initial equation in the form x ¼ g (x) and from an initial root x0, it iteratively calculates x1 ¼ g (x0); x2 ¼ g (x1) and so on, xi + 1 ¼ g (xi) and stops when the difference between xi and xi + 1 is infinitely small. 1

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– an algorithm that combines an indeterminate search (Istiqra’) to find the interval and the first digit of the root, and the extraction of the root digit-bydigit to find the other digits.

4.3 4.3.1

Algorithms and Validation Quadratic Equations: Al-Khawārizmī and His Successors

The algebra proposed by al-Khawārizmī in his treatise al Jabr wa al Muqābala is a discipline which aims at the development of a theory of first- and second-degree equations (Rashed, 2007). Al-Khawārizmī restricts himself to the second degree “because of the requirement of resolution by radicals, and his know-how in this field [. . .]”.3 Within the mathematical theory that he proposes, he defines, among other things: – A vocabulary of basic terms of quadratic equations: the number (“al` adad ‫)”ﺍﻟﻌﺪﺩ‬, the thing or the root (“ al-shay` ‫ ﺍﻟﺸﻲء‬or al-jidhr ‫ – ﺍﻟﺠﺬﺭ‬plural: al-judhur ‫)”ﺍﻟﺠﺬﻭﺭ‬, the square of the thing (“al-māl ‫ ;)”ﺍﻝﻡﺍﻝ‬and the set of arithmetic operators (addition, subtraction, multiplication, division and extraction of the root) as well as the predicate of the equality between terms expressed by the verb to equal (“`ādala ‫;)”ﻋﺎﺩﻝ‬ – The language of well-formed formulas of quadratic equations: This language is reduced to a set of six canonical equations which constitute the set of all possible equations of degree at most two. It corresponds to the following formulas, classified in two categories: three equations of the form: one term equal one term, called “Al mufradāt (‫”)ﺍﻟﻤﻔﺮﺩﺍﺕ‬, and three others of the form one term equal two terms, called “Al-Muqtaranāt (‫”)ﺍﻟﻤﻘﺘﺮﻧﺎﺕ‬: a  x2 ¼ b  x a  x2 ¼ c bx¼c a  x2 ¼ b  x þ c b  x ¼ a  x2 þ c c ¼ a  x2 þ b  x – A solution algorithm whose computational complexity is minimal (it consists in a constant number of operations whatever the coefficients of the equation) since it is reduced to a formula for each type of equation considered;

3

See Rashed (1997), volume 2, p. 32, article: “L’algèbre”, R. Rashed.

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– A proto-geometric proof of the validity of the solution formulas to validate the algorithm. This proof will be improved by his successors, like Ibn Turk and especially by Thābit ibn Qurra who reconstructs it based on the ‘Elements of Euclid’.4 In his book “al-Fakhrī”, al-Karajī generalizes the process of al-Khawārizmī to the solution of the following cases:5 a  x2n ¼ b  xn þ c b  xn ¼ a  x2n þ c c ¼ a  x2n þ b  xn Ibn-al Banna’, mathematician of the end of the 13th and beginning of the fourteenth century reduces the solution of an equation of degree four: x4 + 2 x3 ¼ x + 30 to simultaneous solutions of two quadratic equations.6

4.3.2

Cubic Equations: ʿUmar al-Khayyām and Sharaf al-Dīn al-Ṭūsī

4.3.2.1

ʿUmar Al Khayyām

In the eleventh century ‘Umar Al Khayyām proposed a theory of solving cubic equations by geometry. He formalizes his theory by proposing a classification of the set of all cubic equations, into twenty-five. His general approach is based on the translation between geometric figures of conics and cubic equations and vice-versa which is a method that he indeed theoretically posed formalized and demonstrated. The approach consists in translating the equation into a geometric figure by combining two conics to determine the unknown. Al-Khayyām demonstrates, for each kind of cubic equation, the existence or not of an intersection of two conics using the equations of the curves corresponding to the conics in question. The classification of

4

See Rashed (2007) p. 33–41 and Rashed (1997) volume 2, p. 35. See Rashed (1984), p. 28. 6 He adds x2 in the two sides of the equation, which gives: x4 + 2x3 + x2 ¼ x2 + x + 30 which in turn reduces to: (x2 + x)2 ¼ x2 + x + 30. By setting y ¼ x2 + x, the previous equation is transformed into: y2 ¼ y + 30 which is an instance of the canonical forms of al-Khawārizmī, its resolution by radicals gives y ¼ 6, then the resolution of x2 + x ¼ 6 gives x ¼ 2 as the root of the initial equation. See Rashed (1997) volume 2, p. 40–41. 5

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equations is based on the number of terms involved in the equation and the type of conics7 needed to solve it.

4.3.2.2

Sharaf-al-Dīn al-Ṭūsī

In the twelfth century, Sharaf al-Dīn al-Ṭūsī modifies the classification of al-Khayyām by dividing the twenty-five equations into two classes: the set of solvable equations and the set of equations which may not have a solution, and he proposes a new theory of cubic equations: – by providing a new algorithm for a digit-by-digit extraction of the roots of polynomials; and, – by inventing new analytical concepts to study equations that may not have solutions. He associates with the geometric methods of al-Khayyām new analytical methods of curves where, to solve the equation f(x) ¼ c (with f(x) being a polynomial of degree three, and c, a number), he studies the possibility that there is or is not an intersection between y ¼ f(x) and y ¼ c, by comparing c to the maximum of f(x) which he determines by calculating the root of an expression which is none other than that of the derivative of f(x). The first step is to reduce the number of cubes to a single cube. The equation is considered in this form: f ðxÞ ¼ x3 þ ax2 þ bx þ c ¼ 0 with a, b, c E ℤ f ðxÞ ¼ gðxÞ  hðxÞ ¼ 0 such that : gðxÞ contains all the positive terms of f ðxÞ, and hðxÞ contains all the negative terms of f ðxÞ: Hence, to solve: f(x) ¼ 0 consists, for al-Ṭūsī, into solving g(x) ¼ h(x). This transformation consists in putting any equation in a canonical form which will match one of the three possible cases envisaged by the algorithm: • if c ¼ 0 then we have a quadratic equation; if c < 0 then x3 is in g(x) and c is in h(x): the equation always admits at least one positive root; • if c < 0 then x3 is in g(x) and c is in h(x): the equation always admits at least one positive root; • if c > 0 then c and x3 are in g(x) and the equation may not have a positive root (or have a double or two separate).

7

The two conics used to solve these equations are precisely chosen according to this classification. (See Rashed & Vahabzadeh (1999), p. 10)

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This reasoning leads to the following classification of equations: (A) The set of solvable equations: • the set of so-called pure equations, i.e., whose resolution is an extraction of the root of a number (or a trivial case such as x ¼ c): (1) (2) (3) (4) (5) (6)

x¼c x2 ¼ c x2 ¼ bx x3 ¼ ax2 x3 ¼ bx x3 ¼ c

• The set of quadratic equations (or which reduce to the quadratic) and which are solved by radicals (by the algorithm of al-Khawārizmī): (7) (8) (9) (10) (11) (12)

x2 x2 x2 x3 x3 x3

+ bx + c + ax2 + bx

¼ ¼ ¼ ¼ ¼ ¼

c bx + c bx bx ax2 + bx ax2

• The set of solvable cubic equations which do not reduce to the second degree. They are solved by al-Ṭūsī’s new algorithm (x3 and c are not in the same member of the equation): (13) (14) (15) (16) (17) (18) (19) (20)

x3 x3 x3 x3 x3 x3 x3 x3

¼ ¼ +ax2 ¼ ¼ +ax2 + bx ¼ ¼ +ax2 ¼ +bx ¼ +bx

c c c ax2+ c c ax2 + bx+ c bx+ c ax2+ c bx+

(B) The set of five equations that may not have a solution (x3 and c are in the same side of the equation) (21) (22) (23) (24) (25)

x3 + c x3 + c x3 + c +ax2 x3 + c +bx x3 + c

¼ ¼ ¼ ¼ ¼

ax2 bx bx ax2 ax2 + bx

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(C) The importance of the coefficients in f(x) in al-Ṭūsī’s theory The new iterative, digit-by-digit, root extraction algorithm proposed by Sharaf-alDīn al-Ṭūsī is derived from arithmetic and extended to equations (Rashed, 1986a, b). He generalizes that of the resolution of the equation x3 ¼ c of al-Khawārizmī to the resolution of the different cases of the canonical equation where x3 and c are not in the same member. This generalization induces a greater complexity of the calculations to find the digits of the root because of the existence of coefficients in the terms of the equation. This problem is taken into account and studied by al-Ṭūsī. Let f(x) ¼ x3 + a1x2 + a2x ¼ c, with a1, a2, c being positive. The determination of the first digit of the root of f(x) will depend on the decimal ranks8 of a1, a2, and c. For this, al-Ṭūsī introduces definitions which allow him to compare these decimal ranks as a function of the power i of xi to which the coefficient ani is associated. To reduce the combinatorics of cases, he introduces the notion of “dominant polynomial” (a part of f(x)), because if we can no longer deduce the first digit of the root by starting only from the digits of c, al-Ṭūsī wants to deduce it from the smallest possible polynomial contained in f(x). This problem of the complexity of calculations is explicitly posed by al-Ṭūsī, which indicates that he was aware of this problem. Obtaining the “dominant polynomial” allows him to find a formula (associated with each type of equation) to determine the number of digits of the root and its first digit; that is to say, the whole approximate root, because the remaining steps to find the other digits are mechanical calculations and finite, as in the case of the equation x3 ¼ c. The processing in this algorithm is carried out in an array that maintains all the calculations of all the iterations. (D) The posterity of al-Ṭūsī’s algorithm: al-Kāshī, al-Yazdī, al-Iṣfahānī Even if al-Ṭūsī only searches for the integer root of a cubic equation, his algorithm is applicable to obtaining the root of an equation of degree n
2 and allows calculating an approximate real root. Indeed, al-Yazdī, in the seventeenth century, applies it in his treatise ‘Uyūn al-Ƥisāb to the resolution of equations of degree four and five. Al-Iṣfahānī applies it in the nineteenth century to obtain a real root for all the cubic equations that always have a root according to the formalization of al-Ṭūsī. Finally, let us recall that al-Kāshī describes, in a detailed and educational way, in his treatise Miftāḥ al-Ƥisāb (Ghiyāth al-Dīn Jamshīd Mas’ud al-Kāshī, 1967), the algorithm allowing to extract the fifth root digit by digit of a number with fourteen digits, in different bases calculation. He announces in his book that he intends to

f(x) ¼ xn + a1  xn  1 + a2  xn  2 + . . . + ai  xn  i + . . . + an  1  x ¼ an; Let ai be the coefficient of the monomial ai  xn  i (for cubic equation, n ¼ 3). The decimal rank mi of ai is defined as the result of the Euclidean division of ai by:mi = i  pi + qi; 0  qi < i. This definition makes it possible to compare the coefficients ai according to the value of pi knowing mi and i.

8

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solve all cubic equations, while specifying not to know the work that Sharaf al-Din al Mas’ūdī (instead of al-Ṭūsī) who would have solved these equations.

4.3.3

Cubic Equation: Astronomers and Arithmetician Algebraists, al-Kāshī, al-Yazdī, al-Iṣfahānī

Mathematicians and astronomers who have been interested in the elaboration of the sine table have been confronted with problems reduced algebraically to solving cubic equations. When the resulting equation is one of the five (from al-Ṭūsī’s classification) where digit-by-digit extraction does not apply, it can be solved by an indeterminate iterative algorithm (al-istiqrā’‫ )ﺍﻻﺳﺘﻘﺮﺍﺀ‬which consists in calculating the terms of a convergent sequence. (A) The calculation of chords: the algorithms of al-Kāshī and al-Iṣfahānī One of the problems studied in classical Arabic mathematics is to find the lengths of chords resulting from the subdivision of an arc whose chord length is known. It is reduced to the problem of trisection of an angle and the calculation of the sine 1 .

Knowing the length of the AE chord, we seek to know, for example, the lengths of the chords AB, BC, AD, CE, etc.

This problem is posed in Greek mathematics by Ptolemy in the Almagest. Ptolemy’s approximate solution earned him severe criticism from Ibn al Haytham (on how to deduce an approximate value), al-Samaw’al and al-Kāshī. This problem is transformed in Arabic mathematics into a cubic equation of the form x3 + N ¼ bx and will be solved for the first time without the intervention of geometry, by an indeterminate iterative algorithm established by al-Kāshī in a “treatise on the chord and the sine” entitled Risālat al watar wa l-jayb (‫( )ﺭﺳﺎﻟﺔ ﺍﻟﻮﺗﺮ ﻭﺍﻟﺠﻴﺐ‬Asger, 1979; Savadi, 1987).

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Al-Kāshī’s Algorithm Based on al-istiqrā’ Al-Kāshī is also interested in the geometry of regular polygons and its relation with trigonometry, and more precisely with the calculation of the sine table and he also seeks to improve the precision of the calculations. Striving for great precision in his calculations, al-Kāshī considers regular polygons inscribed and circumscribed at 3  228 ¼ 805,306,368 sides, where Archimedes was satisfied with polygons of 96 sides.9 Al-Kashi’s fixed-point algorithm, based on al-istiqrā’, puts the equation x3 + N ¼ bx in the following form: x ¼ (x3 + N )/b, and starts from an initial solution x0 ¼ N/b. The following steps consist in successively carrying out the following calculations:
 x1 ¼ f(x0) ¼ x30 þ N =b x2 ¼ f(x1) ¼ (x31 þ NÞ=b ... xi + 1 ¼ f(xi) ¼ (x3i þ NÞ=b The process stops when the difference between two successive root calculations is very small. The algorithm calculates the value of sin1 with a precision of ten digits in base sixty, improving the approximations obtained in the tenth century by Abū al-Wafā’ al-Buzjānī and Ibn Yūnus.10 It is also the most precise approximation calculated up to the sixteenth century. The term al-istiqrā’ is known in classical Arabic mathematics. Al-Karajī defines it in rational Diophantine analysis and more precisely when he treats the problem of extracting square roots from polynomials that are not perfect squares. In his books al-Fakhrī and al-Badi`, he defines al-istiqrā’ as the pursuit of expression calculations until we find what we want to know.11 (B) Some of al-Iṣfahānī’s Algorithms In his treatise, Takmilat al ‘Uyūn (Ali Muhammad ibn Muhammad Husayn al-Isfahani, n.d.), on solving the twenty-five cubic equations, al-Iṣfahānī proposes a new theory of cubic equations, based exclusively on approximate numerical calculation. Like his predecessors, al-Khawārizmī, al-Khayyām and al-Ṭūsī, his treatise begins with a classification of the equations which he lists in increasing order of the number of their terms. He solves the equations of the first and second degree by the methods known from al-Khawārizmī, and the algorithm for extracting the root of a number for the

See more details in the article by B. Rosenfeld and A. P. Youschkevitch “Geometry” in Rashed (1997), volume 2, pp. 126–127. 10 Gillespie (1970) entry: al-Kāshī, p. 258. 11 See Rashed (2013) by Roshdi Rashed, p. 36–39. 9

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equations: x2 ¼ N or x3 ¼ N. Al-Iṣfahānī makes no allusion to validating these methods. They are qualified as the known methods. To the eight non-reducible cubic equations and where x3 and N are not in the same member, he exploits the digit-by-digit extraction algorithm of the root of Sharaf al-Dīn al-Ṭūsī to obtain real roots. Likewise, the application of this algorithm is neither proven nor justified. Al-Iṣfahānī exploits the idea of al-Kāshī’s fixed-point algorithm and adapts it to solve the five equations where x3 and N are in the same member. He proposes variants of this algorithm which reduce the number of lines needed by the calculations. All these algorithms are explained during the process of solving the equation x3 + N ¼ bx, then are applied to the other four types of equations where x3 and N are in the same member. Al-Iṣfahānī applied these methods called by al-istiqrā’ for the solution of some of these equations by varying the function f(x) as follows: For the equation x3 + N ¼ bx he proposed the following three algorithms (that we present by using the modern mathematical formalism as follows): pffiffiffi • a first algorithm by taking as initial solution x0 ¼ b and ffiffiffiffiffiffiffiffiffiffiffiffiffiffi p 3 knowing f ðxÞ ¼ bx  N , gives the following iteration:

f ð xÞ ¼

8 > < > :

pffiffiffi b p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xiþ1 ¼ 3 bxi  N for i ¼ 0, 1, 2, . . . while jxiþ1  xi j > ε; x0 ¼

and • an algorithm with the initial solution N/b and the function f(x) ¼ (x3 + N )/b giving the following calculation expressions 8 > > > > > > > < > > > > > > > :

x0 ¼ 0 N x1 ¼ b
 xiþ1 ¼ x3i þ N =b for i ¼ 1, 2, . . . while jxiþ1  xi j > ε ;

• another2algorithm based on3the indeterminate iteration starting from x0 ¼ N/b and !3 i k P P 4 x j  x3i1 5=b, and giving a final solution xs ¼ x j where k is f ð xÞ ¼ j¼0

the last iteration:

j¼0

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8 > > > > > > > > > > > > > > > > < > > > > > > > > > > > > > > > > :

x0 ¼ N=b
 x1 ¼ x30 =b x2 ¼ ½ðx0 þ x1 Þ3  x30 =b ... 2 3 !3 i P xiþ1 ¼ 4 x j  x3i1 5=b j¼0

    i P   while xiþ1  x j  > ε:  j¼0 

For the equation x3 + N ¼ ax2 he proposed: • an algorithm based on the following calculations:

f ð xÞ ¼

8 >
:

while jxiþ1  xi j > ε;

• and a second solution with the following procedure:

f ð xÞ ¼

8 > < > :

x0 ¼ a pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi xiþ1 ¼ 3 ax2i  N for i ¼ 0, 1, 2, . . . while jxiþ1  xi j > ε:

Al-Iṣfahānī solves the three remaining equations: x3 þ N þ ax2 ¼ bx; x3 þ N þ bx ¼ ax2 ; x3 þ N ¼ ax2 þ bx by a hybrid method, which he calls the two-table method (Dhāt al-jadwalayn ‫ )ﺫﺍﺕ ﺍﻟﺠﺪﻭﻟﻴﻦ‬after a calculation by al-istiqrā’ of the first digit.12

12

The algorithm uses two tables of calculations and exploits the method of extracting the root digit by digit. The basic idea of this algorithm is to find the first digit by indeterminate iteration (Istiqrā’) and then to improve this solution by extracting the other digits using al-Ṭūsī’s method. (See Bensaou, 2018, pp. 73–76 for the details of the algorithm.)

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(C) Example of application of one of the fixed point algorithms In the following text,13 al-Iṣfahānī explains and applies one of the fixed point algorithms he proposes to solve: x3 + 210 = 121  x formalized as: x ¼ f ðxÞ, where f ðxÞ ¼

pffiffiffi x3 þ N , x0 ¼ b and N ¼ 210, b ¼ 12 b

In the following table, al-Iṣfahānī shows the detail of the iterative calculations. This type of presentation of calculations exists in the works of al-Kāshī, in Miftāḥ al Ƥisāb.

13

See page 5v in ms and pages 248–249 in Bensaou (2018).

The free translation: Example:

One cube with the number two hundred and ten equal one hundred and twenty-one things. So we equated the cube and the 121 things and we took the square root of 121, thus getting 11; we calculated its cube, which is 1331, then we subtracted from it the number, I mean the number 210. What remains is 1121; we supposed it to be a cube and we took its cubic root, namely, ten and three tenth. We multiplied it by the number of things, getting the result 1246,3 i.e. one thousand two hundred forty-six and three tenth. We subtracted the number and we got 1036,3 i.e. one thousand thirty-six and three tenth. We supposed it to be cube and we took its cubic root, which is 10,1 i.e. ten and a tenth, which we multiplied by the number of things. This gave us 1222,1 i.e. one thousand two hundred twenty-two and one tenth; we subtracted the number, getting the remainder 1012,1 i.e. one thousand twelve and one tenth. We took its cubic root, which is 10.04 i.e. ten and four parts of the tenth of a tenth. But < this root > is superior to the requested thing by this fraction while the requested thing is the integer ten. And we described this work in the following table:

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(D) Validation analysis of al-Iṣfahānī’s Algorithms Today, validating an algorithm consists in: (a) proving that the algorithm stops after a finite number of computation steps, and (b) showing that the result it provides is correct for the set (finite or infinite) of data for which this algorithm is provided.

Integer part The number of things is 121 Then the root is 11 We took its cube 1331 We subtracted the number from it 210 We supposed the remaining to be a cube 1121 We took its cubic root 10 We multiplied it by the number of things 1246 We subtracted the number from it 210 We supposed the remaining to be a cube 1036 We took its cubic root 10 We multiplied it by the number of things 1222 We subtracted the number from it 210 We supposed the remaining to be a cube 1012 We took its cubic root 10 This number is superior to the requested thing by this fraction.

Decimal part

3 3 3 1 1 1 04

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Our purpose in this paper certainly does not concern this type of validation in classical Arabic mathematics treatises. But there certainly exists in these texts, and in an explicit way, a discourse which seeks to explain why the solution given by the algorithm is correct, even if it is an approximate solution and that the iterations of the calculation can be stopped as soon as we note that between two successive approximate solutions there is very little difference. In al Khawārizmī’s treatise, the method of calculating the solution is proven correct by geometry. This type of proof (improved by his successors) is sufficient given that the solution is expressed by an algebraic expression and not by an iterative process. Regarding the digit-by-digit extraction algorithm of the root of a number, Ibn al-Haytham (in the tenth century) devoted two texts on the problem of extracting the square root and the cubic root of a number. One of these two texts, “On the cause of the root, its doubling and its displacement”14 deals with the justification of the steps of the Indian calculus algorithm to extract the root square. It rigorously validates the steps of the algorithm by justifying the method which makes it possible to obtain the positions of the root (and therefore of the number of digits of the root), the operations of the intermediate calculation during the passage from one iteration to the next and the need for the lag at the end of an iteration. This algorithm extended by al-Ṭūsī to extract the root of a cubic equation, where x3 and c are not in the same member, preserves the same type of calculations in an iteration and the fundamental steps to go from an iteration to the next one and the overall reasoning of the algorithm that extracts the root of a number.15 The indeterminate iterative algorithms, like those proposed by al-Kashi then al-Isfahani, calculate an approximate solution by an iterative process. The calculation stops when one notices that between two successive solutions there is very little difference. There exists in the treatise of al-Iṣfahānī, and in an explicit way, a discourse which seeks to explain why the solution given by the algorithm is correct, even if it is an approximate solution and that the iterations of the computation can be stopped as soon as it is observed that between two successive approximate solutions there is very little difference.

See Rashed (1993) (p. 461–487), the book of R. Rashed “Infinitesimal Mathematics from the ninth to the eleventh centuries” Vol II. 15 The taking into account the existence of the coefficients of the equation increases the combinatory nature of the calculations to obtain the first digit (therefore the initial approximate root). This problem is studied by al-Ṭūsī, as we mentioned above (see subsection 4.3.2.2). 14

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When all the algorithms for solving the equation x3 + N ¼ bx are presented, al-Iṣfahānī makes a long remark (tanbīh 16(‫ ﺗﻨﺒﻴﻪ‬on the algorithms presented in order to: 1. explain the need for their convergence; 2. explain, in the following text the meaning and principle of calculating an irrational approximate root. In other words, this type of method (with its variants) initially calculates x0 < xs where x0 ¼ N/b is the first approximation of xs, that is, the solution of the equation, because on the one hand: bx ¼ x3 þ N ) bx > N ) x > N=b ¼ x0 Then 
 x30 þ N < x3 þ N ¼ bx
 ) x1 ¼ x30 þ N =b < x

x30 < x3 )

16




See page 9v in ms and page 270 in Bensaou (2018).

The free translation of tanbih: «And the secret in this rule, with its two parts, is that the result of the first division is necessarily inferior to the requested thing since the things are equal to that number plus the cube. So, the cube of the result is necessarily inferior to the cube of the requested thing. Hence, what results from dividing the number and the cube of the result by the number of things, the second time, is inferior to the requested thing and superior to the second thing. So if we do that several times, the thing is obtained approximately; from this we know that the thing is not obtained with exactness». See Bensaou (2018), p. 270.

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‫‪From there,‬‬ ‫‬ ‫‬ ‫

‬ ‫ ‬ ‫‪x1 ¼ N þ ðN=bÞ3 =b ¼ Nb3 þ N 3 =b3 =b‬‬ ‫
‬ ‫‬ ‫‪¼ Nb3 þ N 3 =b4‬‬ ‫
‬ ‫‬ ‫‪x1 ¼ Nb3 þ N 3 =b4 ¼ N=b þ N 3 =b4 ) x1 > N=b ¼ x0‬‬ ‫‪Therefore‬‬ ‫‪x0 < x1 < xs‬‬ ‫‪This gives an increasing sequence bounded from above by xs:‬‬ ‫‪x0 < x1 < . . . < xi < xiþ1 < . . . < xs‬‬ ‫‪Hence it converges to its upper bound.‬‬ ‫‪Then, thanks to the following example, he explains how (see, the following‬‬ ‫‪Arabic text and its partial free translation in the example) this algorithm can only‬‬ ‫‪give an approximate root:‬‬ ‫ﻻ ﺗﻘﻞ أنّ ھﺬه اﻟﻘﺎﻋﺪة ﻻ ﺗﻔﻲ ﺑﺎﻟﻤﻄﻠﻮب ﻓﺈﻧﮫ اذا ﻋﺎدل ﻣﺎﺋﺔ ﺷﻲء وﺷﻲء ﻣﻊ ﻋﺸﺮة ﻣﻦ اﻟﻌﺪد وﻛﻌﺒﺎ واﺣﺪًا واﻟﻤﻔﺮوض‬ ‫أن ﻛﻞ ﺷﻲء ﻋﺸﺮة ﻣﻦ اﻟﻌﺪد ﻓﻲ اﻟﻮاﻗﻊ ﻓﺎﻟﻜﻌﺐ اﻟﻮاﺣﺪ اﻟﻒ ﻣﻦ اﻟﻌﺪد ﺑﻨﺎء ﻋﻠﻰ ذﻟﻚ ﻣﻊ أن اﻟﺨﺎرج ﻣﻦ ﻗﺴﻤﺔ اﻟﻌﺪد ﻋﻠﻰ‬ ‫ﻋﺸﺮ أﺟﺰاء ﻣﻦ ﻣﺎﺋﺔ وﺟﺰء واﺣﺪ وھﻮ أﻗ ّﻞ ﻣﻦ اﻟﻌُﺸﺮ وﻣﻜﻌﺒﮫ ﺷﻲء ﻗﻠﯿﻞ وﻣﻦ زﯾﺎدﺗﮫ ﻋﻠﻰ‬ ‫ﻋﺪة اﻷﺷﯿﺎء ﻓﻲ اﻟﻤﺮة اﻷوﻟﻰ ة‬ ‫اﻟﻌﺪد وﺗﻘﺴﯿﻢ اﻟﻤﺠﻤﻮع ﻋﻠﻰ ﻋﺪة اﻷﺷﯿﺎء ﻻ ﯾﺤﺼﻞ ﻛﺜﯿﺮ ﺗﻔﺎوت وﻟﻮ ﻛﺮرت اﻟﻌﻤﻞ ﻣﻦ اﻟﺘﻘﺴﯿﻢ واﻟﺰﯾﺎدة ﻣﺎﺋﺔ ﻣﺮة ‪ ،‬ﻷﻧﻨﺎ‬ ‫ﻧﻘﻮل ﺻﺤﺔ ھﺬا اﻟﻔﺮض ﻻ ﯾﻨﺎﻓﻲ ﺻﺤﺔ ھﺬه اﻟﻘﺎﻋﺪة ﻷن اﻟﺸﻲء ﻓﻲ ھﺬه اﻟﻤﺴﺄﻟﺔ ﻻ ﯾﻠﺰم أن ﯾﻜﻮن ﻋﺪدًا ﻣﺨﺼﻮﺻﺎ ﺑﻞ‬ ‫ﯾﺼﺢ اﻟﺠﻮاب ﺑﻌﺪدﯾﻦ ﻣﺨﺘﻠﻔﯿﻦ ﻛ ٌﻞ ﻣﻨﮭﻤﺎ ﯾﺼﻠُﺢ أن ﯾﻜﻮن ﺷﯿﺌﺎ ﻓﻜﻤﺎ أﻧﮫ ﯾﺼﺢ أن ﯾﻜﻮن اﻟﺸﻲء ﻋﺸﺮة ﻋﺪدا ﻋﻠﻰ ﻣﺎ ھﻮ‬ ‫اﻟﻤﻔﺮوض ﻛﺎن ﯾﺼﺢ أن ﯾﻜﻮن ﻗﺮﯾﺒﺎ ﻣﻦ اﻟﻌﺸﺮة وﻟﻮ ﻟﻢ ﯾﻤﻜﻦ اﻟﺘﻌﺒﯿﺮ ﻋﻨﮫ ﺑﻜﺴﺮ ﻟﻜﻮﻧﮫ أﺻ ًﻤﺎ ‪.‬ﻓﺈن ﻣﻜﻌﺐ ﻣﺎ ﯾﻜﻮن أﻗ ّﻞ ﻣﻦ‬ ‫ﻋﺸﺮ ﻋُﺸﺮ اﻟﻌُﺸﺮ ﻓﯿﻤﻜﻦ أن ﯾﻜﻮن ﻣﺎﺋﺔ ﺟﺰء وﺟﺰء واﺣﺪ ﻣﻤﺎ ھﻮ أﻗ ّﻞ ﻣﻦ اﻟﻌُﺸﺮ أﻋﻨﻲ ﻋﺪة اﻷﺷﯿﺎء ﻋﺸﺮة‬ ‫اﻟﻌُﺸﺮ أﻗ ّﻞ ﻣﻦ ُ‬ ‫ﻋﺪدا ﻣﻊ ﻣﺎ ھﻮ أﻗ ّﻞ ﻣﻦ ﻋُﺸﺮ ﻋُﺸﺮ اﻟﻌُﺸﺮ أﻋﻨﻲ ﻣﻜﻌﺒﮫ‪.‬‬

‫‪Let the equation 101  x = 10 + x3. If we take x ¼ 10 then x3 ¼ 1000‬‬ ‫‪and x3 + 10 ¼ 1010, in the same way 101  10 ¼ 1010. However, the algorithm‬‬ ‫‪cannot find this solution but determines an approximate solution because: the first‬‬ ‫‪division N/b ¼ 10/101 gives a rational number that is not exact (its decimal part is an‬‬ ‫‪infinite sequence of digits). Indeed:‬‬ ‫‪10‬‬ ‫‪1‬‬ ‫‪1‬‬ ‫¼‬ ‫