Beyond Faith and Rationality: Essays on Logic, Religion and Philosophy [1st ed.] 9783030435349, 9783030435356

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Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34

Ricardo Sousa Silvestre Benedikt Paul Göcke Jean-Yvez Béziau Purushottama Bilimoria  Editors

Beyond Faith and Rationality Essays on Logic, Religion and Philosophy

Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures Volume 34 Series Editors Editor-in-Chief Purushottama Bilimoria, The University of Melbourne, Australia University of California, Berkeley, CA, USA Co-Editor Christian Coseru, College of Charleston, Charleston, SC, USA Associate Editors Jay Garfield, The University of Melbourne, Melbourne, Australia Editorial Assistants Sherah Bloor, Harvard University, Cambridge, MA, USA Amy Rayner, The University of Melbourne, Melbourne, Australia Peter Yih Jiun Wong, The University of Melbourne, Melbourne, Australia Editorial Board Balbinder Bhogal, Hofstra University, Hempstead, USA Christopher Chapple, Loyola Marymount University, Los Angeles, USA Vrinda Dalmiya, University of Hawaii at Manoa, Honolulu, USA Gavin Flood, Oxford University, Oxford, UK Jessica Frazier, University of Kent, Canterbury, UK Kathleen Higgins, University of Texas at Austin, Austin, USA Patrick Hutchings, Deakin University, The University of Melbourne, Parkville, Australia Morny Joy, University of Calgary, Calgary, Canada Carool Kersten, King’s College London, London, UK Richard King, University of Kent, Canterbury, UK Arvind-Pal Maindair, University of Michigan, Ann Arbor, USA Rekha Nath, University of Alabama, Tuscaloosa, USA Parimal Patil, Harvard University, Cambridge, USA Laurie Patton, Duke University, Durham, USA Stephen Phillips, The University of Texas at Austin, Austin, USA Joseph Prabhu, California State University, Los Angeles, USA Annupama Rao, Columbia University, New York, USA Anand J. Vaidya, San Jose State University, San Jose, USA

The Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures focuses on the broader aspects of philosophy and traditional intellectual patterns of religion and cultures. The series encompasses global traditions, and critical treatments that draw from cognate disciplines, inclusive of feminist, postmodern, and postcolonial approaches. By global traditions we mean religions and cultures that go from Asia to the Middle East to Africa and the Americas, including indigenous traditions in places such as Oceania. Of course this does not leave out good and suitable work in Western traditions where the analytical or conceptual treatment engages Continental (European) or Cross-cultural traditions in addition to the Judeo-Christian tradition. The book series invites innovative scholarship that takes up newer challenges and makes original contributions to the field of knowledge in areas that have hitherto not received such dedicated treatment. For example, rather than rehearsing the same old Ontological Argument in the conventional way, the series would be interested in innovative ways of conceiving the erstwhile concerns while also bringing new sets of questions and responses, methodologically also from more imaginative and critical sources of thinking. Work going on in the forefront of the frontiers of science and religion beaconing a well-nuanced philosophical response that may even extend its boundaries beyond the confines of this debate in the West – e.g. from the perspective of the ‘Third World’ and the impact of this interface (or clash) on other cultures, their economy, sociality, and ecological challenges facing them – will be highly valued by readers of this series. All books to be published in this Series will be fully peer-reviewed before final acceptance.

More information about this series at http://www.springer.com/series/8880

Ricardo Sousa Silvestre • Benedikt Paul G¨ocke Jean-Yvez Béziau • Purushottama Bilimoria Editors

Beyond Faith and Rationality Essays on Logic, Religion and Philosophy

Editors Ricardo Sousa Silvestre Federal University of Campina Grande Campina Grande, Brazil

Benedikt Paul G¨ocke University of Oxford Oxford, UK

Jean-Yvez Béziau Federal University of Rio de Janeiro Rio de Janeiro, Brazil

Purushottama Bilimoria University of California at Berkeley Berkeley, CA, USA

ISSN 2211-1107 ISSN 2211-1115 (electronic) Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures ISBN 978-3-030-43534-9 ISBN 978-3-030-43535-6 (eBook) https://doi.org/10.1007/978-3-030-43535-6 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express 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

Contents

Part I Introduction 1

Beyond Faith and Rationality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Benedikt Paul Göcke, Ricardo Sousa Silvestre, Jean-Yvez Béziau, and Purushottama Bilimoria

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Part II Analytic Philosophy of Religion 2

Why Believe That There Is a God? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Richard Swinburne

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The Failure of van Inwagen’s Solution to the Problem of Evil . . . . . . . . Benedikt Paul Göcke

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Saadia Gaon on the Problem of Evil. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Eleonore Stump

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Some Problems with Miracles and a Religious Approach to Them . . . Agnaldo Cuoco Portugal

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Part III Logical Philosophy of Religion 6

An Even More Leibnizian Version of Gödel’s Ontological Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ´ etorzecka and Marcin Łyczak Kordula Swi˛

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A Tractarian Resolution to the Ontological Argument . . . . . . . . . . . . . . . . . 105 Erik Thomsen

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Some Thoughts on the Logical Aspects of the Problem of Evil . . . . . . . . 121 Ricardo Sousa Silvestre

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The Logic of the Trinity and the Filioque Question in Thomas Aquinas: A Formal Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137 Fábio Maia Bertato v

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Contents

Contradictions and Rationality: An Analysis of Two Biblical Cases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153 Susana Gómez Gutiérrez

Part IV Computational Philosophy and Religion 11

Talmudic Norms Approach to Mixtures with a Solution to the Paradox of the Heap: A Position Paper . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 Esther David, Rabbi S. David, Dov M. Gabbay and Uri J. Schild

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A Case Study on Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195 David Fuenmayor and Christoph Benzmüller

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A Mechanically Assisted Examination of Vacuity and Question Begging in Anselm’s Ontological Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 John Rushby

Part V Logic, Language and Religion 14

Logic and Religion: The Essential Connection . . . . . . . . . . . . . . . . . . . . . . . . . . 257 Benjamin Murphy

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Logic in Islam and Islamic Logic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 Musa Akrami

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Thinking Negation in Early Hinduism & Classical Indian Philosophy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301 Purushottama Bilimoria

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Is God Paraconsistent?. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 Newton C. A. da Costa and Jean-Yves Beziau

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 335

Part I

Introduction

Chapter 1

Beyond Faith and Rationality Benedikt Paul Göcke, Ricardo Sousa Silvestre, Jean-Yvez Béziau, and Purushottama Bilimoria

There is a perennial philosophical debate on the relation between faith and religion on one side, and rationality and logical thinking on the other side. Philosophers and theologians have struggled to bring into harmony these otherwise conflicting concepts, with some defending that there is place, and perhaps need, for rational thought in religious discourse and analysis. Despite the diversity of approaches about what rationality effectively means, logic remains the canon of objective and rational thought. Not unlike what happened in physics in the sixteenth and seventeenth centuries, logic was mathematized at the end of the Nineteenth Century and beginning of the Twentieth Century. This gave it an impressive power to deal with its basic issues as well as a remarkable interdisciplinary cross-fertilization. It also turned it into a powerful theory of representation, able to foster our understanding of a wide range of concepts and principles, such as the ones present in the living religious traditions. Modern logic has been crucial to the development of analytic philosophy and its ideal of conceptual precision, argumentative clarity and transparency. In a very

B. P. Göcke Ian Ramsey Centre for Science and Religion, University of Oxford, Oxford, UK e-mail: [email protected] R. S. Silvestre () Federal University of Campina Grande, Campina Grande, Brazil e-mail: [email protected] J.-Y. Béziau Federal University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] P. Bilimoria University of California at Berkeley, Berkeley, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_1

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important sense, it helped the advance of a strongly rational and logical style of philosophizing; when applied to issues of faith and religion, it is named analytic philosophy of religion, or less commonly but perhaps more precisely, logical philosophy of religion. The present collection of essays deals with matters of Faith in general as well as the faith of different religions—the volume contemplates Christianity, Islam, Judaism and Indian Traditions such as Buddhism, Jainism, M¯ım¯am . s¯a, Ny¯aya and Ved¯anta—from the rational point of view of analytic philosophy of religion and logical philosophy of religion. It offers a unique approach to the relation between faith and reason as it brings the latest developments of formal methods, including computational tools, into scene; the authors analyze several issues pertaining to the philosophy of religion and philosophical theology from the perspective of their relation to logic and the benefit they can derive from the use of logical and formal tools. We briefly sketch here, in a first step, what analytic and logical philosophy are all about, and, in a second step, clarify the particular features of analytic philosophy of religion and logical philosophy of religion. In the last section we survey the contributions of the volume.

1.1 A Brief Analysis of Analytic and Logical Philosophy In the period 1930 to 1950 the concept of analytic philosophy served both as the designation of the Cambridge School of Analysis, maintained by Russell, Moore, and Wittgenstein, and as the designation of the Logical Positivism found in the Vienna Circle. Since the 1950s, the concept of analytic philosophy has been used, beyond its original scope, to denote the world’s leading philosophical research program. As Beaney (2013a, p. 3) states: “Over the course of the Twentieth Century analytic philosophy developed into the dominant philosophical tradition in the English-speaking world, and it is now steadily growing in the non-English-speaking world.” From its very beginning quite different philosophical positions have been argued for in analytic philosophy which, from metaphysical realism to verificationism, included contrary, if not contradictory, positions. To determine the concept of analytic philosophy more precisely, two central features are distinguished which express the unity of analytic philosophy, despite its different philosophical positions, and which may be accepted as central by anyone seeing their work as within the paradigm of analytic philosophy. First, the assumption of a legitimate division between the genesis and the validity of a philosophical position, with a concomitant emphasis on the greater relevance of the validity of philosophical theses. Second, emphasis on the greatest possible conceptual and argumentative clarity and transparency in the analysis of philosophical positions.

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1.1.1 The Genesis and Validity of Philosophical Theses By the concept of the genesis of a philosophical thesis is denoted the diachronic process which led to its development and precise formulation. The analysis of the genesis of a philosophical thesis makes it possible to explain why this thesis was developed by its representatives and for what reasons it was understood in a specific way. The concept of the validity of a philosophical thesis expresses its claim to truth or rational acceptability. The analysis of the validity of philosophical theses is therefore interested in the reasons which speak systematically for or against the truth or rational acceptability of a philosophical thesis. Regarding the truth of a proposition, there is no strong logical connection between the genesis and the validity of a philosophical thesis because an understanding of the genesis of a philosophical thesis is neither sufficient nor necessary for the question of its validity, and the knowledge of the validity of a philosophical thesis does not imply any knowledge of its genesis. If, for example, there is an adequate historical explanation, tracing the developments and conditions that led Descartes to formulate substance dualism, then nothing follows from this explanation about the validity of substance dualism. And someone who investigates arguments for and against substance dualism can pursue this activity successfully, without knowing the historical development of this position. Given this background, the first characteristic of analytic philosophy consists in its being primarily concerned with the validity of philosophical theses. The analysis of the validity of a philosophical thesis has to be informed of historical developments only in so far as it is necessary to arrive at a systematically clear formulation of the thesis.

1.1.2 Conceptual and Argumentative Transparency In addition to its primary interest in the validity of philosophical theses, a second feature of analytic philosophy consists in its executing the analysis of their validity in at least three methodological stages that aim for maximal epistemic vulnerability. First In the clarification of philosophical questions, the first priority of analytic philosophy is achieving the greatest possible conceptual precision. It tries to analyze the concepts which are decisive for the topic dealt with, by providing necessary and sufficient conditions for the fulfillment of those concepts, in a clear, comprehensible manner. Therefore, in principle, any person replicating the analysis ought to be in a position to state what is meant by these concepts, or what ought to be meant beyond the quirks of natural language. In this way, analytic philosophy is open to the greatest possible dialogue with critical responses and makes possible the discovery of errors in conceptual analysis. As Parsons (2013, pp. 247–248) argues: “Analytic philosophy is a genre or style of philosophy [ . . . ] that advocates rigorous forms of logical or conceptual analysis as the central method of philosophy. [ . . . ] [Analytic

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philosophers share] a commitment to the rigorous examination of philosophical problems in the light of tools and methods drawn from formal logic, set theory, and the natural sciences.” Second Based on conceptual analysis, analytic philosophy tries to formulate as precisely as possible philosophical theses which it understands as claims to truth, and therefore as factual or normative theses about reality and our perception of reality. As Russell (1900, p. 8) says, “that all sound philosophy should begin with an analysis of propositions, is a truth too evident, perhaps, to demand a proof.” Based on the clarification of the concepts involved, it attempts to clarify as clearly as possible what a particular philosophical thesis asserts about reality or our perception of reality, which other theses are sufficient for the thesis examined, and what the truth or falsehood of this thesis logically implies. In other words, it attempts, in the course of the precise formulation of a philosophical thesis, to state the necessary and sufficient conditions of the truth or falsity of this thesis. That the genesis of the philosophical thesis to be examined may be derived from very different sources of knowledge, such as tradition, intuition, and inspiration, is a self-evident fact within the framework of analytical philosophy. But this does not imply anything about the validity of the corresponding thesis. Third After conceptual analysis, and the clarification of the presuppositions and consequences of the philosophical thesis to be examined, the work of the analytic philosopher turns to the core of analytic philosophy: the argument. The argument is the decisive instance of the work of the analytic philosopher for, in an argument, the reasons which speak for or against the truth and rational acceptability of a philosophical thesis are comprehensibly formulated, expressis verbis, in order to make possible, in addition to the desired conceptual precision, the greatest possible argumentative transparency, and therefore, again, maximal criticism in philosophical dialogue. That is, in the analysis of arguments for the truth of a particular conclusion, the logical form of the argument is examined by analyzing the logical relation between the putative truth of the premises with the truth of the conclusion. The analysis of the validity of an argument examines whether, based on the assumption of the truth of the premises, it is reasonable to proceed to the truth of the conclusion. The analysis of the soundness of an argument further asks whether the premises are indeed true, and how their truth may be justified. In the case of a valid argument, therefore, it gives the reasons which speak for the truth of the premises and thus for the truth of the conclusion.

1.1.3 Analytic and Classical Philosophy The defining characteristic of analytic philosophy consists in stressing the analysis of the validity of philosophical theses using the three-steps: conceptual analysis, thesis specification, argument analysis. The vast majority of philosophical faculties’ research and teaching, as well as the greater part of philosophical work found in

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professional journals, anthologies, and monographs is based on that assumption. Despite the numerous different positions represented, analytical philosophy is united by the assumption that philosophical questions concern reality and our perception of reality, and may be treated rationally under the regulative ideal of truth.1 Although, until the middle of the last century, analytic philosophy was empirical, materialistic, or influenced by the linguistic turn (according to which philosophical problems are merely linguistic illusions) today it is no longer de facto true that the concept of analytic philosophy is used to characterize a particular position; instead, it is used to characterize a method, and a style, for approaching genuine philosophical questions. Understood in this way, analytic philosophy essentially coincides with the philosophical work predominating in the history of philosophy, for example, in Plato, Aristotle, Thomas Aquinas, Descartes, Leibniz, Locke, Hume, or Berkeley.2 There is hardly a field of classical philosophical work which is not dealt with in analytic philosophy. Therefore, one might well ask what is to be understood under the concept of non-analytic philosophy. Because the often-mentioned distinction between analytic and continental philosophy cannot be established either historically, geographically, or systematically, in a satisfactory way, and leads only to the formation of misplaced fronts, it is more meaningful not to confront analytic philosophy with continental philosophy, but to a methodical ideal of an intellectual activity, in which either no commitment to conceptual and argumentative clarity, precision and transparency is felt, or in which it is assumed that the analysis of the validity of philosophical theses is not a primary philosophical task.3

1 See Beaney (2013a, p. 26): “The analytic philosopher might then be characterized as someone who knows how to use these tools [of conceptual and argumentative analysis], through training in modern logic and study of the work of their predecessors. Each analytic philosopher may have different aims, ambitions, backgrounds, concerns, motivations, presuppositions, and projects, and they may use these tools in different ways to make different constructions, criticisms, evaluations, and syntheses; but there is a common repertoire of analytic techniques and a rich fund of instructive examples to be draw upon; and it is these that form the methodological basis of analytic philosophy. As analytic philosophy has developed and ramified, so has its toolbox been enlarged and the examples of practice (both good and bad) expanded.” See also Löffler (2007, p. 375), our translation from the German original: “Analytical philosophizing is a style of philosophizing and not a bundle of particular positions. It is, therefore, nothing to which theologians and Christian philosophers have to have fear or excessive reserves about—but also nothing that carries an intrinsic quality guarantee.” 2 Prominent authors such as Russell and Stump count, for example, Leibniz and Thomas Aquinas as historical practitioners of analytical philosophy. See also Beaney (2013a, p. 10): “But if Leibniz so counts [as an analytic philosopher for Russell], then how far back can we go? To Descartes? To Ockham, Buridan, and other medieval logicians? To Aristotle or even Plato?” 3 Cf. Priest (2000) for more on the analysis of analytic and continental philosophy.

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1.1.4 Logical Philosophy Logical philosophy is nothing but analytic philosophy taken to its extreme. First of all, like analytic philosophy, logical philosophy also emphasizes the validity of philosophical theses (at the expense of their genesis) according to at least one of the members of the three-part schema we have delineated above (conceptual analysis, thesis specification and argument analysis). Second, as we said, logic was mathematized at the end of the Nineteenth Century and beginning of the Twentieth Century, which turned it into a powerful theory of representation almost indispensable in the pursuit of the analytic ideal of conceptual precision, argumentative clarity and transparency. Third, what we have identified as the core of analytic philosophy—the argument—is precisely the main object of study of logic; issues such as the logical form of an argument and its validity have reached the highest degree of precision in the past 100 years. The distinction between logical philosophy and analytic philosophy might be seen as somehow artificial. In a very important sense, analytic philosophy has always been logical. The views of Russell and Carnap on the role of logic and logical analysis in philosophy, for instance, are still paradigmatic. While Russell famously identified philosophy with logic—according to him, “Every philosophical problem, when it is subjected to the necessary analysis and justification, is found either to be not really philosophical at all, or else to be, in the sense in which we are using the word, logical (Russell 1914, p. 14), Carnap’s project of conceptual analysis had formal logic as an indispensable element (Carnap 1950, pp. 1–18). Besides, many developments of analytic philosophy which would perhaps not be classified as logical philosophy make strong use of logical tools and results. If we use the world “logic” in the broad sense of being in accordance with the main developments of logic in the Twentieth Century, we can safely say that analytic philosophy is logical and logical philosophy is analytic.

1.2 Analytic and Logical Philosophy of Religion Once the anti-metaphysical approach to philosophy that was prominent up until the middle of the last century was overcome, philosophers (again) became aware that the methods of analytic philosophy are particularly useful and valuable when it comes to the analysis of matters that constitute the universe of discourse of what today is known as philosophy of religion. Philosophy of religion deals with questions regarding the ultimate constitution of reality, in particular, questions regarding the existence of God, from a purely philosophical point of view. For instance, discussions of the varieties of theism and atheism, and of arguments in favor of each, belong to the philosophy of religion, as well as questions concerning the possibility and nature of life after death.

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A distinction here needs to be made between analytic philosophy of religion and analytic theology. In contrast to analytic philosophy of religion, analytic theology is not only based on purely philosophical insights, but takes, depending on its religious context, genuine religious assumptions regarding the nature of reality for granted. In the same way in which works in analytic philosophy of religion can include both arguments for and against the plausibility of a particular philosophical position, so works placed in the tradition of analytic theology can include arguments for and against the plausibility of a particular religious tradition and its assumptions regarding the nature of the existence of the universe. Philosophy of religion also deals with the rationality of theistic belief; here also arguments for and against the existence of God play a crucial role. Providing a good argument for the conclusion that God does exist, or that it is highly probable that he exists, might be a strong case for the thesis that belief in his existence is rational. Similarly, a good argument for the conclusion that God does not exist could be said to support the thesis that theistic belief is irrational. A paradigmatic example of this are the several versions of the ontological argument which appeared since Anselm’s seminal work, the Proslogion, in the Eleventh Century, as well as the several kinds of logical scrutiny which they have been subjected to. Some of the greatest pre-Twentieth Century philosophers, including Descartes, Leibniz, Spinoza and Kant, have either proposed or analyzed ontological arguments. Ontological arguments are of course just one type of theist arguments, albeit the best exemplar of a priori arguments for the existence of God. Other (kinds of) theist arguments of historical importance are cosmological arguments, moral arguments, teleological and design arguments, arguments from miracles, etc. From the side of atheist arguments, the problem of evil occupies a prominent place. Names such as Epicurus, Aquinas, Leibniz, Hume and Kant have addressed it. For instance, Hume’s Dialogues concerning Natural Religion, from the Eighteenth Century, remains one of the best pre-Twentieth Century expositions and analysis of the problem of evil, even anticipating some of the tenets which would guide the contemporary debate on evil and God. Although many times put as an argument against the existence of God, the problem of evil has been traditionally described as an inconsistency, either logical or evidential, between the existence of an omnipotent, omniscient and wholly good being and the existence of evil and suffering in our world. Even though they are in most cases equivalent, the later way of presenting the problem illustrates the real point of the problem, which is to challenge the rationality of theist belief. A more basic approach than that (concerning the rationality of theistic belief) would be to analyze the very concept of God. Can God create a stone so heavy that he cannot lift? If we say yes, then there is something God cannot do, namely to create such a stone; if we say no, there is also something he cannot do, namely to lift the stone. In either case he is not omnipotent. If really unsolvable, paradoxes like this (this is the paradox of the stone) show that the concept of God (who is,

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besides other things, omnipotent) is incoherent or contradictory. Like the concept of a squared circle, it could never be instantiated. Theists must of course refute claims like this; ideally, they must provide arguments showing that the concept of God is coherent or consistent.4 Consistency is a logical concept; as said, arguments are the main object of study of logic. It is therefore natural that logic had played a significant role in the two approaches to appraising the rationality of theistic belief sketched above (the construction and analysis of arguments for and against the existence of God and the analysis of the concept of God). Indeed, analytic philosophers of religion such as Alvin Plantinga, William Rowe, Richard Swinburne and Peter van Inwagen, just to mention a few, have in some way or other incorporated results and tools of logic in their works. There have also been heavier uses, we might say, of logic in the philosophy of religion; attempts to formally analyze the ontological arguments attributed to Anselm, for instance, are abundant.5

1.3 Beyond Faith and Rationality The essays collected in this volume are written from the point of view of analytic and logical philosophy of religion. The book is divided according to a broad and somehow artificial, we acknowledge, methodological perspective. Despite this, a great diversity of themes inside the philosophy of religion is considered: there are contributions dealing with both metaphysical and epistemological sides of philosophy of religion, as well as with the two main ways of dealing with the rationally of theist belief: proposing and analyzing arguments for and against the existence of God, and analyzing the concept of God itself. The second part of the volume, entitled simply Analytic Philosophy of Religion, contains four essays belonging to a more classical and less formal approach to analytic philosophy of religion. In Why Believe that there is a God?, Richard Swinburne argues that based on (a) the existence of a physical universe, (b) its conformity to simple natural laws, (c) those laws being such as to lead to the existence of human and animal bodies, and (d) those bodies being the bodies of reasoning humans who choose between good and evil, a sound inductive argument

4 We

are here using the terms “contradictory” and “consistent” as applied also to concepts. A concept C is consistent or coherent if and only if the set composed by “There is an object x which is C.”, “The concept of C is defined as . . . ” and whatever other sentence is needed to turn the definition into a complete one, is consistent. A non-consistent concept is called contradictory or non-coherent. Here is an example. The concept of squared cirque is contradictory, for the set {“There is an object x which is a squared circle.”, “A squared circle is defined as a figure which, as a square, has four sides and, as a circle, has no sides.”, “If a figure has no sides, then it is false that it has four sides.”} is not consistent. 5 See for instance Hartshorne (1962, pp. 49–57), Adams (1971), Oppenheimer and Zalta (1991), Klima (2000), Sobel (2004, pp. 60–65) and Maydole (2009).

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for the existence of God can be elaborated. According to Swinburne, these four phenomena are such that it is moderately probable that they will occur if there is a God, and almost certain that they will not occur if there is not a God. Swinburne therefore concludes that these phenomena provide strong probabilistic evidence for cogent arguments to the existence of God. In The Failure of van Inwagen’s Solution to the Problem of Evil Benedikt Paul Göcke first analyzes Peter van Inwagen’s concept of an ideal philosophical debate on the plausibility of the argument from evil. In this debate amongst theists, atheists, and agnostics, according to van Inwagen, the Christian worldview is likely to turn out to be sufficiently plausible to leave agnostics unconvinced of the power of the argument from evil—which, according to van Inwagen, is enough to defeat the argument from evil. Göcke then argues that van Inwagen is too optimistic and that contrary to what he expects, the agnostics will not grant that the argument from evil is a failure. They will argue that van Inwagen’s defence is situated in a methodologically inacceptable paradigm of philosophical discourse and, apart from that, not even a consistent story about the suffering of animals and humans in a world created by God. Eleonore Stump continues the analysis of the problem of evil and argues that although considerable effort has been expended on constructing theodicies which try to reconcile the suffering of unwilling innocents, such as Job, with the existence and nature of God as understood in Christian theology, the abundant reflections on the problem of evil and the story of Job in the history of Jewish thought have not been discussed much in contemporary philosophical literature. In her contribution Saadia Gaon on the Problem of Evil Stump takes a step towards remedying this defect by examining the interpretation of the story of Job and the solution to the problem of evil given by one important and influential Jewish thinker: Saadia Gaon. The part ends with a chapter by Agnaldo Cuoco, entitled Some Problems with Miracles, in which three types of problem regarding the concept of miracle are analyzed: the definitional question (what is the meaning of ‘miracle’?), the epistemological challenge (can the belief in miracles be rational?), and the ethical issue (is anything wrong in claiming that a miracle has occurred?). The aim of the chapter is to compare a skeptical and a religious point of view regarding miracles concerning these problems in order to better understand and evaluate this concept in current philosophy of religion. The third part of the book, entitled Logical Philosophy of Religion, contains more explicit uses of logic and formal methods in the philosophical analysis of ´ etorzecka and Marcin Łyczak. religious issues. It starts with a text by Kordula Swi˛ In their An even more Leibnizian version of Gödel’s ontological argument, they propose a modification of Gödel’s ontological argument for God’s existence. They operate in the framework of a Leibnizian onto-theology that is specified in some ´ etorzecka and Łyczak of Leibniz’ letters written between 1676 and 1677. Swi˛ consider two differences between Gödel’s and Leibniz’ system and argues for the superiority of Leibniz’s ideas. The first aim of their chapter is to bring the concept of

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positiveness used in Gödel’s argument closer to the idea of perfection deployed by Leibniz. The second aim is to consider the concept of something’s being necessary in terms of Leibnizian demonstrability. Based on their analysis, they elaborate an S4 version of Gödel’s argument that does not use negative predicate terms. In his A Tractarian Resolution to the Ontological Argument, Erik Thomsen deals with ontological arguments for the existence of God and highlights the problematic treatment of existential entailments of true propositions. He attempts to simultaneously resolve these problems in both their predicate-based and their argument-based versions and relate this solution to the ontological argument by replacing the notion of existential entailment with the notion of sequenced evaluation. To that end, he uses a logic consistent with the principles laid out in Wittgenstein’s Tractatus. He concludes that the recasting of ontological arguments in Tractarian terms shows a fundamental mistake made by all approaches to the ontological argument. In his paper Some Thoughts on the Logical Aspects of the Problem of Evil, Ricardo Silvestre proposes to take seriously the characterization of the problem of evil as a supposed incompatibility between the proposition that the world was created and is ruled by an omnipotent, omniscient and unlimitedly good being and one that says that there is evil and suffering in our world. Besides doing justice to much that has been said about the problem of evil, he argues that this characterization takes the problem at its face value, that is to say, as an incompatibility problem. He then tries to show how from this characterization one can satisfactorily define some key concepts for the debate on God and evil: the concept of problem of evil itself as well as the concepts of argument from evil, logical problem of evil, evidential problem of evil, theodicy and defense. Although still dealing with the broad field of arguments for and against the existence of God, Silvestre’s contribution is mainly a paper of conceptual analysis. This leaves the path open to Fabio Bertato’s The Logic of Trinity and the Filioque question in Thomas Aquinas: a Formal Approach, which is directly concerned with the conceptual analysis of the notion of God. He provides an attempt to elucidate the Christian trinitarian concept of God. More specifically, he presents a formal approach to the Filioque question based on traditional discussions, especially on Aquinas’ solution. In order to do this, he constructs and introduces a First Order Theory called Thomistic Logic of the Trinity, which is an attempt to formalize the Trinitarian Theology. Besides, he also presents a model for this theory, which allows him to verify its consistency. In the final chapter of this part—Contradictions and Rationality: An Analysis of two biblical Cases—Susana Gómez Gutiérrez looks at the story of Susanna and the Elders in the Book of Daniel, and the narrations of Jesus’s resurrection by the four Evangelists as biblical cases for examining two philosophical problems, namely, how to deal logically with contradictions and the consequences that follow from the alternatives examined regarding the notion of rationality. Taking into account the two cases mentioned and considering a perspective according to which rationality is closely connected to a certain idea of rationality and the purpose of logic, Gutiérrez argues that, on one hand, in the story of Susanna, Daniel’s belief in Susanna’s chastity can be understood as supported by a reasoning whose

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underlying logic is classical logic and, for this reason, fits perfectly into the classical model of rationality. On the other hand, the narrations of Jesus’s resurrection demand a different approach both to contradictions and to the relationship between contradictions and rationality. With regard to the latter, Gutiérrez contends that to be considered rational, the believer in the resurrection should support her belief with a reasoning that has some form of paraconsistent logic as the underlying logic. The fourth part of the book, entitled Computational Philosophy and Religion, contains approaches which in one way or other make use of computational tools in the pursuit of their goals. It begins with a text by E. David, Rabbi S. David, D. Gabbay and Uri J. Schild entitled Talmudic Norms Approach to the Paradox of the Heap: A Position Paper in which they offer a Talmudic norms solution to the paradox of the heap. The claim is that the paradox arises because philosophers use the wrong language to discuss it. On their account, the appropriate language is that of an extended blocks world language, together with the Talmudic normative theory of mixing and the principle that a property of any mixture depends on how it was constructed. They seek a correlation between Talmudic positions on mixtures and philosophical positions on Sorites. In their chapter A Case Study on Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument, David Fuenmayor and Christoph Benzmüller argue that computers may help us to understand philosophical arguments. Through the mechanization of a variant of St. Anselm’s ontological argument by E. J. Lowe, they offer an ideal showcase for a computer-assisted interpretive method. This method, which they name “computational hermeneutics”, has been specifically conceived for use in interactive proof assistants and is aimed at shedding light on concepts and beliefs implicitly presupposed in arguments—with a special emphasis on metaphysical and religious ones. Computational hermeneutics draws on both a compositional and a holistic view of meaning by working our way towards understanding of an argument by circular movements between its parts and the whole. They argue that this approach also allows us to expose the assumptions we indirectly commit ourselves to every time we opt for some particular logical formalization, and also fosters the explication and revision of our beliefs and commitments until arriving at a state of reflective equilibrium: a state where our beliefs have the highest degree of coherence and acceptability. Finally, in his A Mechanically Assisted Examination of Begging the Question in Anselm’s Ontological Argument, John Rushby defends several first and higher-order formalizations of Anselm’s Ontological Argument against the charge of begging the question. He then proposes three different criteria for a premise to beg the question in fully formal proofs and argues that one or another applies to all the formalizations examined. His purpose is to demonstrate that a mechanized verification of the Ontological Argument provides an effective and reliable technique to perform these analyses. The last part of the book, entitled Logic, Language and Religion, starts with a text by Benjamin Murphy, Logic and Religion: The Essential Connection, in which he first clarifies the standard definition of validity: in classical logic, an argument is valid if and only if it is impossible for the premises to be true and the conclusion

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false, so the premises and conclusion must be truth-bearers. Murphy then proposes that there are different types of validity and argues that in any valid argument, some form of commitment must be transmitted from premises to conclusion, but it need not be a commitment to truth. According to Murphy, it is possible for us to evaluate the validity of arguments because we are capable of recognizing that some commitments exclude others. Although petitionary prayers are not truth-bearers, Murphy continues to argue that they do express commitments of a kind that could be used to form valid arguments and that therefore it would be a mistake to treat a religion as nothing more than a set of claims about what is true. Instead, members of a religion are bound together by a set of shared commitments. Any attempt to make the core commitments of a religion explicit or to reconsider the content of the core commitments will necessarily involve a consideration of how one set of commitments excludes others. Logic is thus an essential part of any religion. In the next chapter, From Logic in Islam to Islamic Logic, Musa Akrami deals with relations between logic and religion in the Islamic world. More specifically, he analyzes seven issues: (1) the different manifestations of using logical reasoning, particularly analogy, in Qur’¯anic arguments, e.g. arguments for the existence of God and resurrection after death; (2) some contradictions or paradoxes reported by different opponents in the verses of Qur’¯an; (3) the place of logic in the classification of disciplines and the courses taught at the schools and seminaries; (4) the influence of the attitudes of different religious sects on logic; (5) the instrumental role of logic for both religious and secular reasonings; (6) the relation between reason and dogmatic religious doctrines, and, finally, (7) the reflection of this relation on progress or recession of logic in medieval Islamic world. The next chapter of the volume, Thinking Negation in Early Hinduism and Classical Indian Philosophy, by Purushottama Bilimoria, deals with the several kinds of negation and negation of negation that were developed in Indian thought, from ancient religious texts to classical philosophy. He explores the M¯ım¯am . s¯a, Ny¯aya, Jaina and Buddhist theorizing on the various forms and permutations of negation, denial, nullity, nothing and nothingness, or emptiness. The main thesis argued for is that in the broad Indic tradition, negation cannot be viewed as a mere classical operator turning the true into the false (and conversely), nor reduced to the mainstream Boolean dichotomy: 1 versus 0. Special attention is given to how contradiction is handled in Jaina and Buddhist logic. The volume ends with a paper by Newton da Costa and Jean-Yves Béziau entitled Is God Paraconsistent? where they examine the general question of the relation between God and Paraconsistency. They start by some general considerations about paraconsistent negations and paraconsistent things to then examine in which sense the “thing” called “God” is paraconsistent or not. They finish by discussing the divinity of paraconsistency.

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References Adams, Robert. 1971. The logical structure of Anselm’s arguments. The Philosophical Review 80: 28–54. Beaney, Michael. 2013a. What is analytic philosophy? In The Oxford handbook of the history of analytic philosophy, ed. Michael Beaney, 3–29. Oxford: Oxford University Press. Carnap, Rudolf. 1950. Logical foundations of probability. Chicago: University of Chicago Press. Hartshorne, Charles. 1962. The logic of perfection. LaSalle, IL: Open Court. Klima, Gyula. 2000. Saint Anselm’s proof: A problem of reference, intentional identity and mutual understanding. In Medieval philosophy and modern times, ed. Ghita Holmström-Hintikka, vol. 2000, 69–87. Dordrecht: Kluwer. Löffler, Winfried. 2007. Wer hat Angst vor analytischer Philosophie? Zu einem immer noch getrübten Verhältnis. Stimmen der Zeit 7: 375–388. Maydole, Robert. 2009. The ontological argument. In The Blackwell companion to natural theology, ed. W.L. Craig and J.P. Moreland, vol. 2009, 553–592. Oxford: Blackwell. Oppenheimer, Paul, and Edward Zalta. 1991. On the logic of the ontological argument. In Philosophical perspectives 5: The philosophy of religion, ed. James Tomblin. Atascadero: Ridgeview Press. Parsons, Keith M. 2013. Perspectives on natural theology from analytic philosophy. In The Oxford handbook of natural theology, ed. Russell Re Manning, 247–261. Oxford: Oxford University Press. Priest, Stephen. 2000. Analytical and continental. In The Oxford companion to philosophy, ed. T. Honderich. Oxford: Routledge. Russell, Bertrand. 1900. A critical exposition of the philosophy of Leibniz. Cambridge: Cambridge University Press. ———. 1914. Our knowledge of the external world. Chicago: The Open Court Publishing Company. Sobel, Jordan. 2004. Logic and theism. New York: Cambridge University Press.

Benedikt Paul Göcke is a Research Fellow at the Ian Ramsey Centre for Science and Religion and a Member of the Faculty of Theology at University of Oxford. He is also a Member of Blackfriars Hall, Oxford. Göcke is author of: A Theory of the Absolute (Macmillan, 2014), Alles in Gott? (Friedrich Pustet, 2012) and editor of After Physicalism (Notre Dame, 2012). He has published articles in journals such as The International Journal for Philosophy of Religion, Zygon, Sophia, The European Journal for Philosophy of Religion, and Theologie und Philosophie, among others. Ricardo Sousa Silvestre holds a Ph.D. in Philosophy from the University of Montreal. He has been Visiting Scholar at the Universities of Oxford (UK), Notre Dame (USA) and Québec (Canada). He is the author of several papers on Logic and Philosophy of Religion and guest-editor of a couple of special issues on the field of Logic and Religion, namely the special issues on the Concept of God (2019) and Formal Approaches to the Ontological Argument (2018) of the Journal of Applied Logics (College Publications), the special issue on Logic and Philosophy of Religion (2017) of Sophia: International Journal of Philosophy and Traditions (Springer) and the special issue on Logic and Religion (2017) of Logica Universalis (Springer). He is one of the creators and main organizers of the World Congress on Logic and Religion series. He is presently Associate Professor at Federal University of Campina Grande (Brazil).

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Jean-Yvez Béziau has a Ph.D. in Mathematics (University of Paris 7) and a Ph.D. in Philosophy (University of São Paulo, Brazil). He has done research in France, Brazil, Poland, California (UCLA, Stanford, UCSD) and Switzerland. He is presently professor of logic in Rio de Janeiro at the University of Brazil, and the President of the Brazilian Academy of Philosophy. He is the promoter of Universal Logic as a general theory of logical structures, the founder and Editor-inChief of the journal Logica Universalis and book series Studies in Universal Logic, both published by Birkhäuser/Springer, Basel. He has organized a series of events on universal logic around the world (Montreux 2005, Xi’an 2007, Lisbon 2010, Rio de Janeiro, 2013, Istanbul 2015, Vichy 2018). He has renewed the study of the square of opposition, organizing interdisciplinary world events on the topic (Montreux 2007, Corsica, 2010, Beirut 2012, Vatican 2014, Easter Island 2016, Crete 2018) and the publication of special issues of journals and books. In 2019 he launched on January 14 the 1st World Logic Day which was celebrated in 60 locations around the world and was subsequently approved as an international day of UNESCO. Purushottama Bilimoria PhD is presently a Fulbright-Nehru Distinguished Fellow in India affiliated with Ashoka University. He is also lecturer with Legal Studies in the University of California, Berkeley, and serves as a senior fellow in Indian Philosophy with the Center for Dharma Studies at Graduate Theological Union, in Berkeley. He is otherwise an Honorary Research Professor of Philosophy and Comparative Studies at Deakin University and Senior Fellow at University of Melbourne, in Australia. He is a Permanent Fellow with the Oxford Centre for Hindu Studies in Oxford University, and past visiting scholar at All Souls College, University of Oxford, Harvard University, Emory University and UC Santa Barbara, and Visiting Professor in two universities in Brazil. He is a Co-Editor-in-Chief of Sophia, international journal of philosophy & traditions and of the Journal of Dharma Studies, both published by Springer.

Part II

Analytic Philosophy of Religion

Chapter 2

Why Believe That There Is a God? Richard Swinburne

2.1 Inductive Natural Theology St. Paul famously claimed that pagans who did not worship God were ‘without excuse’, because ‘ever since the creation of the world [God’s] eternal power and divine nature, invisible though they are, have been understood and seen through the things which he has made’.1 Inspired by this text, many Christian thinkers from the second to the eighteenth centuries put forward arguments from premises ‘evident to the senses’ to the existence of God. Putting forward such arguments is called ‘natural theology’. One kind of natural theology attempts to produce deductively valid arguments. It is a not unreasonable interpretation of Aquinas’s famous ‘five ways’,2 that he was seeking to give five such arguments there. But the enterprise of producing deductive arguments is, I think, an enterprise doomed to failure. For if it could be achieved, then a proposition which was a conjunction of the evident premises together with ‘there is no God’ would be incoherent, would involve selfcontradiction. But again propositions such as ‘there is a Universe, but there is no God’, though perhaps false, seem fairly evidently coherent. So my own preference is for the ‘inductive’ form of natural theology. This begins from premises evident to the senses and claims that they make probable (though not certain) the existence of God. Thinkers were not very clear about the distinction between inductive and deductive arguments during the first 1000 years

1 Letter 2 See

to the Romans 1:20. his Summa Theologiae 1.2.3.

R. Swinburne () University of Oxford, Oxford, UK e-mail: [email protected] © The Author(s), under exclusive licence to Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_2

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of the Christian era, and not much clearer until the eighteenth century. So it would be anachronistic to say that the patristic writers were explicitly seeking to give inductive, or alternatively deductive, arguments, but I consider that some patristic and later arguments do conform to an inductive pattern. What I have sought to do in my own natural theology is to give rigorous form to inductive arguments to the existence of God from premises reporting phenomena evident to the senses; and to bring out the close similarities between such arguments and arguments to a deep theory of physics such as Quantum theory, and to arguments of historians or detectives to some particular person having done some deed. The many such arguments for the existence of God can be ordered by the generality of their premises—the phenomena from which they begin. The most general phenomenon is that there is a physical Universe; the argument from the physical Universe to God is the cosmological argument. Then there are arguments of two main kinds from the order in the Universe; these are teleological arguments. One is the argument from the universal operation of simple natural laws, which I call the argument from temporal order. The other is the argument from those laws being such as (given a particular early state of the universe) to lead to the existence of human bodies, which I call the argument from spatial order. Then there is the argument from consciousness—that humans are not merely bodily organisms, but are conscious beings (having sensations and beliefs, thoughts, desires and purposes, and the ability to reason and to choose to bring about good or evil.) Then there are arguments from particular miraculous events within history, and above all from the Resurrection of Jesus—or rather, since it is disputed whether these events happened, from the public evidence about them. And finally there are arguments from the very widespread phenomena of religious experience. Like all inductive arguments from particular phenomena to some deep physical hypothesis, or to some claim of a historian, the arguments from phenomena to God are cumulative. Each phenomenon gives some degree of probability to the hypothesis; taken together with arguments from phenomena against the existence of God, they give an overall probability to the existence of God. I have argued at length elsewhere,3 that overall these arguments make the existence of God significantly more probable than not. In this paper I hope to show in brief outline the force of the positive arguments for the existence of God from the first four phenomena listed above—the existence of a physical universe, its conformity to simple natural laws, those laws being such as to lead to the existence of human and animal bodies, and those bodies being the bodies of reasoning humans who choose between good and evil. And again for reasons of space I shall not be able to discuss the negative argument against the existence of God from the existence of pain and other suffering.

3 See

my The Existence of God, 2nd ed., Oxford University Press, 2004; and the short simplified version Is There a God? Oxford University Press, rev. ed. 2010’.

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2.2 Criteria for a Probably True Causal Explanation Theism, the claim that there is a God is an explanatory hypothesis, one which purports to explain why certain observed phenomena (that is, data or evidence) are as they are. There are two basic kinds of explanatory hypothesis—personal and inanimate (or scientific) hypotheses. A personal hypothesis explains some phenomenon in terms of it being caused by a substance, a person, acting earlier with certain powers (to bring about effects), certain beliefs (about how to do so), and a certain purpose (or intention), to bring about a particular effect, either for its own sake or as a step towards a further effect. I (a substance) cause the motion of my hand in virtue of my powers (to move my limbs), my belief (that moving my hand will attract attention) and my purpose (to attract attention). An inanimate explanation is usually represented as explaining some phenomenon in terms of it being caused by some initial state of affairs and the operation on that state of laws of nature. The present positions of the planets are explained by their earlier positions and that of the Sun, and the operation on them of Newton’s laws. But I think that this is a misleading way of analysing inanimate explanation—because ‘laws’ are not things; to say that Newton’s law of gravity is a law is simply to say that each material body in the universe has the power to attract every other material body with a force proportional to mm /r2 and the liability to exercise that power on every such body. So construed, like personal explanation, inanimate explanation of some phenomenon (e.g. the present positions of the planets) explains it in terms of it being caused by substances (e.g. the Sun and the planets) acting earlier with certain powers (to cause material bodies to move in the way codified in Newton’s laws) and the liability always to exercise those powers. So, both kinds of explanation explain phenomena in terms of the earlier actions of substances having certain powers to produce effects. But while personal explanation explains how substances exercise their powers because of their purposes and their beliefs, inanimate explanation explains how substances exercise their powers because of their liabilities to do so. I suggest that we judge a postulated hypothesis (of either kind) as probably true insofar as it satisfies four criteria. First, we must have observed many phenomena which it is quite probable would occur and no phenomena which it is quite probable would not occur, if the hypothesis is true. Secondly, it must be much less probable that the phenomena would occur in the normal course of things, that is if the hypothesis is false. Thirdly, the hypothesis must be simple. That is, it must postulate the existence and operation of few substances, few kinds of substance, with few easily describable properties behaving in mathematically simple kinds of way.4 We can always postulate many new substances with complicated properties to explain anything which we find. But our hypothesis will only be supported by the evidence if it is a simple hypothesis which leads us to expect the various phenomena that form

4 For

a full account of the nature of simplicity, see my Simplicity as Evidence of Truth, Marquette University Press, 1997; or my Epistemic Justification, Oxford University Press, 2001, chapter 4.

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the evidence. And fourthly, the hypothesis must fit in with our knowledge of how the world works in wider fields—what I shall call our ‘background evidence’. I now illustrate these criteria at work in assessing postulated explanations. I begin with a postulated personal explanation. Suppose that there has been a burglary; money has been stolen from a safe. A detective has discovered these pieces of evidence: John’s fingerprints are on the safe, someone reports having seen John near the scene of the burglary at the time it was committed, and there is in John’s house an amount of money equivalent to the amount stolen. The detective puts forward as the explanation of the burglary the hypothesis that John robbed the safe. If John did rob the safe, it would be to some modest degree probable that his fingerprints would be found on the safe, that someone would report having seen him near the scene of the crime at the time it was committed, and that money of the amount stolen would be found in his house. But these phenomena are much less to be expected with any modest degree of probability if John did not rob the safe; they therefore constitute positive evidence, evidence favouring the hypothesis. On the other hand, if John robbed the safe, it would be most unexpected (it would be most improbable) that many people would report seeing him in a foreign country at the time of the burglary. Such reports would constitute negative evidence, evidence counting strongly against the hypothesis. Let us suppose that there is no such negative evidence. The more probable it is that we would find the positive evidence if the hypothesis is true, and the more improbable it is that we would find that evidence if the hypothesis is false, the more probable the evidence makes the hypothesis. But a hypothesis is only rendered probable by evidence insofar as it is simple. Consider the following hypothesis as an explanation of the detective’s positive data: David stole the money; quite unknown to David, George dressed up to look like John at the scene of the crime; Tony planted John’s fingerprints on the safe just for fun; and, unknown to the others, Stephen hid money stolen from another robbery (coincidentally of exactly the same amount) in John’s house. If this complicated hypothesis were true, we would expect to find all the positive evidence which I described, while it remains not nearly as probable otherwise that we would find this evidence. But this evidence does not make the complicated hypothesis probable, although it does make the hypothesis that John robbed the safe probable; and that is because the latter hypothesis is simple. The detective’s original hypothesis postulates only one substance (John) doing one thing (robbing the safe) which leads us to expect the various pieces of evidence; while the rival hypothesis which I have just set out postulates many substances (many persons) doing different unconnected things. But as well as the evidence of the kind which I have illustrated, there may be ‘background evidence’, that is evidence about matters which the hypothesis does not purport to explain, but comes from an area outside the scope of that hypothesis. We may have evidence about what John has done on other occasions, for example evidence making probable a hypothesis that he has often robbed safes in the past. This latter evidence would make the hypothesis that John robbed the safe on this occasion much more probable than it would be without that evidence. Conversely, evidence that John has lived a crime-free life in the past would make it much

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less probable that he robbed the safe on this occasion. A hypothesis fits with such background evidence insofar as the background evidence makes probable a theory of wider scope (e.g. that John is a regular safe-robber) which in turn makes the hypothesis in question more probable than it would otherwise be. The same four criteria are at work in assessing postulated inanimate hypotheses. Consider again the hypothesis that the present positions of the planets are to be explained by their positions and that of the sun five hundred years ago (which we learn from reports of observers) and the operation of Newton’s laws—which I’ll rephrase in due course in my preferred way. Newton’s theory of gravitation consisted of his three laws of motion and his inverse square law of gravitational attraction. The evidence available at the end of the seventeenth century favouring this theory consisted of evidence about the paths taken (given certain initial positions) by our moon, by the planets, by the moons of planets, the velocities with which bodies fall to the earth, the motions of pendula, the occurrence of tides etc. Newton’s theory made it very probable that these phenomena would occur as observed. It would be very unlikely that they would occur if Newton’s theory were not true. There was no significant negative evidence. The theory was very simple, consisting of just four laws, the mathematical relations postulated by which were very simple (F = mm /r2 being the most complicated one). Yet innumerable other laws would have satisfied the first two criteria equally well. Within the limits of accuracy then detectable any law in which you substitute a slightly different value for the ‘2’ (e.g. ‘2.0000974’) would have satisfied the first two criteria as well as did the inverse square law. So too would a theory which postulated that the inverse square law held only until AD 2969 after which a quite different law, a cube law of attraction would operate, or a theory containing a law claiming that quite different forces operated outside the solar system. But Newton’s theory, unlike such theories, was rendered probable by the evidence because it was a very simple theory. There was no relevant background evidence, because there was no evidence outside the scope of Newton’s theory making probable any explanatory theory (e.g. a theory of electromagnetism) with which Newton’s theory needed to fit. Hence Newton’s theory was very probable on the evidence available in the seventeenth century because it satisfied our four criteria; and so therefore is the hypothesis that it together with the initial positions of Sun and planets explains the present positions of the planets. Rephrased in a more satisfactory way, that hypothesis is the hypothesis that the Sun and each of planets have simple powers and liabilities (as codified by Newton’s laws) and initial positions which explain the present positions of the planets. I stress again the importance of the criterion of simplicity. There are always an infinite number of mutually incompatible theories which could be constructed which predict all the observed data when these would not otherwise be expected, yet make different predictions from each other about what will happen tomorrow. Without the criterion of simplicity it would be impossible to predict anything beyond what we immediately observe. If the hypothesis is concerned only with a narrow field, it has to fit with any background evidence. But for many hypotheses there may be no relevant background evidence, and the wider the scope of a hypothesis (that is, the

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more it purports to tell us about the world), the less background evidence there will be. For a very large-scale theory of physics (such as Quantum Theory) there will be few physical phenomena apart from those within its scope (ones which it purports to explain), and so little—if any—background evidence.

2.3 Theism as a Simple Hypothesis Such are the criteria for the probable truth of some postulated explanatory hypothesis. I now spell out the hypothesis of theism. Theism is clearly a personal hypothesis. God is supposed to be one person who is essentially omnipotent, omniscient, perfectly free and eternal. (If you emphasize that God is a Trinity, ‘three persons of one substance’ as Christianity claims, regard these arguments as arguments to the existence of God the Father who is the cause of everything else.) A person is a being who has powers (to perform intentional actions, that is actions which he or she means to do), beliefs, and purposes (choosing among alternative actions which to perform). It is simpler to suppose that the cause of the universe has zero limits to his power (that is, is omnipotent), rather than that he can only make a universe of a certain size and duration. An omnipotent person can do any logically possible action, that is any action which can be described without contradiction; and so he cannot make me both exist and not exist at the same time. But since it makes no sense to suppose that I could both exist and not exist at the same time, a logically impossible action is not really an action at all—any more than an imaginary person is really a person. A truly omnipotent person would not be subject to irrational forces in forming his purposes, as so often are the choices of humans; he would be influenced by reason alone and so by what he believes good to do. In that sense of ‘perfectly free’, an omnipotent person is necessarily perfectly free. A truly omnipotent person would know all the possible actions open to him, and so know whether they are good or bad, and so being perfectly free would do actions only insofar as they are good. So he would be perfectly good. A truly omnipotent person would be essentially omnipotent, for otherwise his omnipotence would be precarious. It is simpler to suppose that God is unlimited in time as well as in power, and so essentially eternal. In my view that should be interpreted as God being everlasting (existing at every moment of past and future time), since I regard the Boethian view of God as outside time, yet simultaneously present at all moments of human time, as a view to which it is very difficult to give any sense, and a totally unnecessary burden on theism. So often there must be before God, as there are before us, a choice between equally good incompatible actions. And, since God is omnipotent, the range of incompatible equal best actions available to him would be so much greater than the range available to us. Further, God must often be in a situation where we cannot be, of having a choice between an infinite number of possible actions, each of which is less good than some other action he could do. For example, angels and bears and elephants are good things; they can be happy and loving. So, the more of them the

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better (given that in the case of bears and elephants they are spread out among an infinite number of planets, so that they do not crowd each other out). So, however many of these creatures God creates, God would believe it better if he had created more. (And he could still have created more, even if he created an infinite number of them.) It may be however that when there is no best or equal best action available to God, there may be a best kind of action available to God, such that it would be better to do some action of that kind than to do any number of actions of any other incompatible kind. For example, God can create creatures of many different types, including angels, humans and animals. If it were the case that it would be better to create at least some humans (even if he creates no angels or animals), than to create any number of angels and animals and no humans or to do an action of any other incompatible kind, then it would be a best kind of action for God to create some humans, although there would be no best number for him to create. In that case, I suggest, God being influenced by reason alone would inevitably create some humans. And if there are two or more equal best kinds of action available to him he will inevitably do some action of one of these kinds. So God will inevitably always do the best or equal best action, or if there is no such action, at any rate an action of the best or equal best kind, and otherwise some good action; but he would never do a bad action. Given the logical impossibility of backward causation, God will not be able to cause past events, but he will be able to cause any future event. The simplest supposition about God’s knowledge is that there will be zero limits to it compatibly with his omnipotence. Hence his knowledge will be confined to knowledge of the past and of any necessary truths, but the future will be entirely subject to his choice. Just as omnipotence is to be understood as the power to do anything logically possible, so omniscience should be understood as knowledge of everything logically possible to know. (Of course, on the Boethian account of God’s eternity, then God’ s omniscience would include knowledge of the whole (to us) future as well as of the whole past.) Insofar as moral principles (e.g. that one ought to keep one’s just promises) are necessary truths, and so independent of the will of God,5 God will have true beliefs about what they are; and so not merely do what he believes to be the best (or whatever), but what is in fact the best or equal best (or one of the best or equal best kinds of action) or otherwise any good action. In that case he will be as good as it is logically possible to be, which is to say that he will be perfectly good. I conclude that theism is a very simple hypothesis indeed. It postulates just one substance, God, having essentially the simplest degree of power, and lasting for the simplest length of time; all the other essential divine properties follow from that. God being what he is in virtue of these essential properties makes God a ‘person’ in a sense somewhat analogical to the sense in which we are persons. Theism is such a 5 For

justification of my view that the fundamental moral principles are independent of the will of God; but that—in virtue of a fundamental moral principle, that everyone has an obligation to please their benefactors—since God is the source of our existence and all that we have, humans have an obligation to obey the commands of God, see my The Coherence of Theism, 2nd ed. (New York: Oxford University Press, 2016) chapter 11.

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wide-ranging hypothesis (it purports to explain all the most general features of the universe) that there is no background evidence; all the evidence (whether positive or negative) is within its scope. So the hypothesis of theism satisfies the third criterion superbly well; and does not need to satisfy the fourth criterion. Hence whether the hypothesis of theism, that God exists, is probable on the evidence of the phenomena which I outlined earlier turns on how well that evidence satisfies the first two criteria.

2.4 It Is Quite Probable That God Will Bring About the Observed Phenomena So, first, are the phenomena such as, if there is a God, it is probable that he would bring them about? If there is a God, he will seek to bring about good things. It is good that there should be a beautiful universe. Beauty arises from order of some kind—the orderly interactions and movements of objects in accord with natural laws is beautiful indeed; and even more beautiful are the plants and animals which evolved on Earth. Animals have sensations, beliefs and desires, and that is clearly a great good. Humans have the power to reason and understand the universe, and that is an even greater good. But all these kinds of goodness are kinds of goodness which God himself possesses. God is beautiful and has beliefs and desires (and in my view, also sensations), and the power to reason and understand. But there is one kind of great goodness which God himself does not possess—the power to bring about good or evil. God can only bring about good. Yet it would be very good indeed that there should be persons who have the free will to make this all– important difference to the world, the power to benefit or harm themselves, each other and other creatures. So, if there is a God, we have very good reason to suppose that there will be persons who have, as I believe humans have, that freedom.6 But clearly there is a bad aspect to the existence of such persons; they may cause much evil. So, it cannot be a unique best action to create such persons, but in view of the unique kind of goodness which they would possess, surely it must be an equal best action to create such persons; and, if so, it is as probable as not that if there is a God, there will be such persons, that is persons like us humans. But if God is to create us, he must provide a universe in which we can exercise our choices to benefit or harm ourselves and each other. We can only do that if we have bodies, through which we can learn about the world and make a difference to it, and so places where we can get hold of each other and escape from each other.7 But only if there are comprehensible regularities which we can discover will there be ways in which my doing this or that will make a predictable difference to me or you, and so we can have a choice of

6 For

my defence of the claim that humans have libertarian free will, that is freedom to make choices, either good or evil, despite all the influences to which they are subject, see my Mind, Brain, and Free Will, Oxford University Press, 2013. 7 For proper argument in favour of this claim, see The Existence of God, pp. 123–131.

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how to treat each other. Only if humans know that by sowing certain seeds, weeding and watering them, they will get corn, can they develop an agriculture. And only if they know that by rubbing sticks together they can make fire, will they be able to burn the food supplies of others. But comprehensible observable regularities are only possible if the fundamental laws of nature are simple ones. Further, if God is to create embodied humans, the laws must be such as to allow the existence of human bodies, either brought about by an evolutionary process or created directly by God. And finally, human bodies only have a point if they are controlled by conscious persons. So the four phenomena to which I have referred are to be expected (that is, it is quite probable that they will occur) if there is a God.

2.5 The Phenomena Would Be Immensely Improbable if There Were No God But if there is no God, it is immensely improbable that these phenomena will occur. It is enormously improbable that each of the innumerably many fundamental particles, or rather chunks of compressed energy) immediately after the Big Bang, should just happen to exist. And it is even more improbable that each such chunk should behave in exactly the same fairly simple way as each other chunk (the way codified in the laws of Relativity and Quantum theory and the four forces). So while there are fairly simple laws, their instantiation in each of innumerably many chunks of matter-energy would be an enormous coincidence unless caused by some external agent. And even if such an enormous coincidence occurred by chance, it is immensely improbable that those laws should be such as together with the boundary conditions of the Universe (which are its initial conditions if the Universe had a beginning) should have given rise to human bodies. And even if this too occurred by chance, as far as any plausible scientific laws are concerned, the laws might just as easily have given rise to robots. Consciousness is totally improbable, unless there is a creator who gave it first to the higher animals and then to us. Some contemporary physicists will tell you that we live in a multiverse such that many different possible universes (with different laws of nature, and different initial conditions) will eventually occur, and so it is not surprising that there is one like ours. But we could only have reason to believe what they tell us if the most probable explanation of phenomena observable in our universe was that the most general laws of nature are such as to bring about these many universes; and to postulate that is to postulate that all the particles, not merely of our universe, but of the vastly bigger multiverse behave in accord with the same very general laws, which throw up particular variants thereof in different universes—which is to postulate an even bigger coincidence. And the laws of that multiverse would have to be such as to produce at some stage a universe like ours which in turn produces us, when almost all logically possible multiverses would not have this characteristic. So even if our universe does belong to a multiverse, it is immensely improbable (if there is no more

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ultimate explanation thereof—e.g. God) that that would be a multiverse of the kind to bring about the existence of humans. So, the possible existence of a multiverse makes little difference to the force of the arguments which I have discussed. So, these four general phenomena are such as it is moderately probable will occur if there is a God, and almost certainly will not occur if there is not a God. Theism is a very simple hypothesis indeed, and simpler—I suggest—than any inanimate hypothesis which could be constructed. I conclude that arguments from the phenomena which I have discussed are strong cogent arguments to the existence of God. Richard Swinburne is Emeritus Professor of the Philosophy of Religion at the University of Oxford, and a Fellow of the British Academy. He has written a trilogy on the meaning and justification of theism, The Coherence of Theism, The Existence of God, and Faith and Reason, the main points of which are summarized in a short book, Is there a God? He has written a tetralogy of books on the meaning and justification of central Christian doctrines including Providence and the Problem of Evil, followed subsequently by The Resurrection of God Incarnate, the main points of which are summarized in a short book Was Jesus God? He has also written in defense of substance dualism, the view that each of us consists of two parts—a body and a soul, especially in his books Mind, Brain, and Free Will , and his most recent book Are We Bodies or Souls?. His book Epistemic Justification discusses the criteria which determine how probable some evidence makes some hypothesis. He has given many lectures in many countries.

Chapter 3

The Failure of van Inwagen’s Solution to the Problem of Evil Benedikt Paul Göcke

Imagine you wonder about the existence of evil in a world created by a perfect being. Suppose someone provides the following answer: ‘Listen: The existence of evil is not only consistent with the existence of an omnipotent and morally perfect God, but is in fact part of a Christian worldview that may be true. According to this worldview, animal suffering is acceptable to God because it occurs in a regular universe and leads to a greater evolutionary good. Human suffering is acceptable to God because our biological ancestors freely turned away from God. They left us genetically corrupted in a hideous world that is penetrated by an arbitrary amount of suffering. It is only if we freely return to God that the suffering will end.’ To show the plausibility of such a response to arguments from evil, van Inwagen introduces an ideal philosophical debate amongst theists, atheists, and agnostics. He argues that the Christian worldview, if extended appropriately, is sufficiently plausible to leave agnostics unconvinced of the power of arguments from evil: If the agnostics grant that God exists, they will also accept that this story could be true, which, according to van Inwagen, is all that is needed for the argument from evil to fail. I show that van Inwagen is too optimistic about the power of his defence in this ideal philosophical debate. Contrary to what van Inwagen expects, the agnostics will not grant that the argument from evil is a failure. They will argue that van Inwagen’s imagined debate is situated in a methodologically inacceptable paradigm of philosophical discourse and, apart from that, not even a consistent story about the suffering of animals and humans in a world created by God.

B. P. Göcke () Ian Ramsey Centre for Science and Religion, University of Oxford, Oxford, UK e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_3

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3.1 Philosophical Success and Philosophical Failure in an Ideal Debate People often seem to expect that an argument for a substantial philosophical thesis is a good argument if and only if every rational person as a matter of psychology or logic has to accept its premises and has to agree that the conclusion is deductively or inductively supported by the premises of the argument. As van Inwagen, however, rightly points out, if this was an adequate definition of ‘philosophical success’, then up to now there would be no successful argument for any substantial philosophical position: There is no such argument on which all philosophers agree.1 To rescue the possibility of philosophy as a substantial discipline on its own, van Inwagen suggests lowering the criterion of argumentative success: an evaluation of the quality of a substantial philosophical argument should not ask whether every rational person is convinced of the truth of the premises and, based on this, is convinced of the truth of its conclusion. Instead, it should look at the way the argument would be treated in an ideal philosophical debate. According to van Inwagen, an ideal philosophical debate consists of contemporary proponents of the argument, contemporary opponents, and an audience that is agnostic with respect to the conclusion. Nevertheless, the agnostic audience would like to know whether the conclusion is deductively or inductively supported by the premises.2 The participants are ideal members of the debate in the following sense: [They] are of the highest possible intelligence and of the highest possible degree of philosophical and logical acumen, and they are intellectually honest in this sense: when they are considering an argument for some thesis, they do their best to understand the argument and to evaluate it dispassionately. [They] have unlimited time at their disposal and are patient to a preternatural degree [ . . . ] and if their opponents think it necessary to undertake some lengthy digression into an area whose relevance to the debate is not immediately evident, they will cooperate. (van Inwagen 2006, p. 42)

If the proponent of the argument can convince the agnostics that they should believe the conclusion of the argument if they believe the premises, then the argument is a success—even if the proponent cannot convince the opponent of the argument (cf. van Inwagen 2006, p. 47). If, however, the opponent of the argument can convince the agnostics that the conclusion of the argument is not warranted by

1 Cf.

van Inwagen (2006, p. 39): “Only one thing can be said against this standard of philosophical success: if it were adapted, almost no arguments for any substantive philosophical thesis would count as a success [ . . . ] If there were an argument, an argument for a substantial philosophical thesis, that was a success by this standard, there would be a substantive philosophical thesis such that every philosopher who rejected it was either uninformed [ . . . ] or irrational or mad. Are there any?” Cf. also Göcke (2014, pp. 19–39) for a discussion of the potential of philosophical argument. 2 Cf. also van Inwagen (2006, p. 44): “On this model, two representatives of opposed positions carry on an exchange of arguments before an audience, and their purpose is not to convert each other but rather to convert the audience—an audience whose members (in theory) bear no initial allegiance to either position, although they regard the question ‘Which of these two positions is correct?’ as an interesting and important one.”

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the premises or that there is something wrong with the premises, then the argument in question is a failure. To show the failure of an argument, van Inwagen’s model of an ideal philosophical debate enables the opponent to proceed in different ways: he can cast doubt on the logical validity of the argument in question, he can provide the agnostic with reasons to doubt the truth of the premises, and he can tell a plausible story that (a) entails the falsity of the conclusion of the argument, the falsity of at least one of its premises and (b) could be true for all the agnostic knows. In effect, all the opponent has to do to turn the corresponding argument into a philosophical failure is to find an epistemologically possible scenario in which the argument fails.

3.2 van Inwagen’s Defence of the Existence of Evil The common ground of all arguments from evil is the apparent conflict between a particular philosophical concept of the divine being as perfect and our experience of a large variety of kinds of evil in a world created ex nihilo. In the discussion, this common ground is taken as a starting point for the formulation of many versions of the argument from evil that concern both animal and human suffering. Although a distinction is often drawn between logical and evidential arguments from evil, van Inwagen prefers to distinguish a global from a local argument from of evil. The second distinction does not replace the first: The first is about the strength of the conclusion, the second about kinds of evil found in the actual world. There are, then, at least four different arguments from evil: (1) the global logical, (2) the global evidential, (3) the local logical, and (4) the local evidential. But, following van Inwagen, I speak only of the global and the local argument from evil. On the global argument from evil, the existence of God is incompatible with the huge amounts of horrendous evil we observe in the world: God, if he existed, would prevent these evils from occurring (cf. van Inwagen 2006, p. 56). On the local argument from evil, the existence of God is inconsistent with, or at least is highly improbable in the light of particular evils, that is, evils occurring at places and times: God, if he existed, would prevent them, since apparently no greater good depends on their occurrence (cf. van Inwagen 2006, p. 8). Because the existence of a particular evil entails the existence of global evil, but not vice versa, it follows that a solution to the problem of global evil is not yet a solution to the problem of local evil. Both problems have to be treated separately for both the suffering of non-rational and the suffering of rational animals. There are, in principle, several ways to tackle arguments from evil. One could (a) give up the assumption that God is omnipotent, omniscient and morally perfect, (b)

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change the assumption that God creates a world ex nihilo, (c) deny the existence of evil, or (d) provide reasons God has, or could have, for allowing evil.3 Van Inwagen chooses the last option. He tries to specify the reasons a morally perfect, omniscient, and omnipotent God, who creates the world ex nihilo, has, or could have, for allowing the occurrence of kinds of evil. This story can be told as a defence or as a theodicy. A defence is philosophically humbler than a theodicy. Where a theodicy is presented with the force of truth, a defence is just an account that shows that the existence of evil is, for all we know, consistent with the existence of an omnipotent, omniscient, and morally perfect God.4 Based on his criterion of philosophical success, and deploying his definition of an ideal philosophical debate, van Inwagen only needs to present a successful defence of animal and human suffering to show that the local and the global argument from evil fail.

3.2.1 Van Inwagen’s Concept of God Van Inwagen’s defence consists of three parts: a defence of animal suffering before the fall of mankind, a defence of global animal and human suffering after the fall of mankind, and a defence of local suffering after the fall. Before summarizing them, I look at van Inwagen’s concept of God, which the agnostic is asked to accept in the ideal debate. According to van Inwagen, God is the single greatest possible being, an eternal, omnipotent, morally perfect, and omniscient person that is able to intervene in the world on special occasions. Classical theists assume that God’s eternity entails God’s atemporality and that God’s omniscience entails knowledge of the truth value of propositions that refer to what, from our perspective, look like future states of

3 One

could also use a mixture of (a) to (d). Those who choose the first option tend to argue that God is not omnipotent, but only satispotent: powerful enough to raise us from the dead and to be our perfect moral judge, but not (yet) powerful enough to prevent evil in the world (cf. Rohs 2013). Those who reject the assumption that God creates the world ex nihilo typically argue that God is not responsible for the evil to be found in the world: he does not create the world ex nihilo but only functions like a final cause who cannot interfere in the world (cf. Griffin 2014). 4 “A ‘defense’ in the weakest sense in which the word is used is an internally consistent story according to which God and evil both exist. Sometimes the following two requirements are added: The evil in the story must be of the amounts and kinds that we observe in the actual world, and the story must contain no element that we have good scientific or historical reasons to regard false. A theodicy is a story that has the same internal features as a defense, but which the theodicist, the person telling the story, puts forward as true or at least highly plausible.” (van Inwagen 2002, p. 30). Furthermore, “A defence will ascribe to God some reason for allowing the possibility of evil in his creation (for example, creaturely free will is a very great good, a good so great that its existence justifies the risk of its possible abuse). It will go on to say that this source, whatever it may have been, produced not just some evil, but vast amounts of horrendous evil, and it will, finally, ascribe to God another reason for not simply removing from his creation by fiat the vast amounts of evil that issued from the Source of Evil, a reason for allowing the vast amounts of horrendous evil produced by the Source to continue to exist.” (van Inwagen 2001, pp. 66–67).

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affairs. Van Inwagen rejects these elements of theism. Instead, he advocates ‘open theism’: God exists in time, without end or beginning. So there is no time at which God does not exist.5 God can only know the future of the actual world insofar as it is metaphysically possible to know it for a temporal being that is itself subject to the flow of time. Since according to van Inwagen, it is metaphysically impossible to foreknow free actions, it follows that God cannot know the future free acts of human beings.6 God only knows every world insofar as it is possible, he knows what happened in our past, what happens in our present, what could happen in our future given the present state of the world, and what will of necessity happen in the future of our world. Since there is nothing that can be known but is not known by God, God is still properly referred to as an omniscient being on this account.

3.2.2 Animal Suffering and Massive Irregularity To account for the suffering of animals before the fall of mankind, van Inwagen asks us, the audience of agnostics, to consider that every world that enables the evolution of higher-level sentient creatures—those that feel pain—either entails an amount of animal suffering equivalent to the amount found in the actual world or else is massively irregular. God could only avoid the suffering of animals if he constantly intervened in this world, thereby destroying the regularity imposed on this world by the laws of nature that, although not deterministic, still are regular enough in their probabilistic determination of what will happen. Since, though, such a massive irregularity would destroy the very reliability of the divinely ordained laws of nature and therefore would, according to van Inwagen, at least be as big a defect in a world as the suffering of animals, and since the development of human beings is a greater good than the suffering of animals on which it evolutionarily depends, God is free to choose between massive irregularity and animal suffering as two morally equivalent possible ways of achieving this greater good. Apparently, he chose the latter option.7 5 Cf.

van Inwagen (2006, p. 81): “In what follows, I am going to suppose that God is everlasting but temporal, not outside time. I make this assumption for two reasons. First, I do not really know how to write coherently and in detail about a non-temporal being’s knowledge of what is to us the future. Secondly, it would seem that the problem of God’s knowledge of what is to us the future is particularly acute if this knowledge is foreknowledge.” 6 Cf. van Inwagen (2006, p. 82): “Even an omniscient being is unable to know certain things—those such that its knowing them would be an intrinsically impossible state of affairs. [ . . . ] That is, such a being must yesterday have had no beliefs about what I should do freely today.” 7 According to van Inwagen, the following four proposition are true for all anyone knows: “(1) Every world God could have made that contains higher-level sentient creatures either contains patterns of suffering morally equivalent to those of the actual world, or else is massively irregular. (2) Some important intrinsic or extrinsic good depends on the existence of higher-level sentient creatures; this good is of sufficient magnitude that it outweighs the patterns of suffering found in

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3.2.3 Global Suffering and Free Will A few hundred thousand years ago, the evolutionary process that involved enormous amounts of divinely acceptable animal suffering eventually led to the evolution of a small group of clever primates, the immediate evolutionary predecessor of Homo sapiens. They lived together in a certain region of the world and, miraculously, God bestowed the ability for rational thinking and free will on them. For some time, our predecessors, who possessed preternatural powers to avoid natural evils, lived in perfect union and free love with God. Then, however, they freely decided to follow their own hearts, and turned away from God. They abused their free will to engage in evil activities, and as a consequence became mortal and fallen. They lost their preternatural powers to avoid natural evil and found themselves suddenly in a world reigned by chance and pervaded by animal and human suffering (cf. van Inwagen 1988, p. 163). We, Homo sapiens, are the genetically depraved biological heirs of the first generation of human beings that suffer from a morally corrupted and genetically enforced link between our ability to act rationally and the egocentric fulfilment of our own desires.8 Instead of deleting us from history, though, God, in his mercy, decided to install a rescue plan to bring us, once more, into perfect union with him. Since he cannot force us to love him, because we have free will, and since we cannot unite with God on our own, we need to freely choose to love God and to cooperate with Him in order to be worthy of His grace.9 So long as we do not, we will be subject to random suffering in the world because “that is part of what being separated from God means; it means being the playthings of chance. It means living in a world in which innocent children die horribly, and it means something worse than that: it means living in a world in which innocent children die horribly for no reason at all” (van Inwagen 2006, p. 89). The suffering will end only if we accept God’s offer and cooperate with him again: “When God’s plan of Atonement comes to fruition, there will never be undeserved suffering or any other sort of evil. The

the actual world. (3) Being massively irregular is a defect in a world, a defect at least as great as the defect of containing patterns of suffering morally equivalent to those found in the actual world. (4) The world—the cosmos, the physical universe—has been created by God.” 8 Cf. van Inwagen (2006, pp. 86–88): “God not only raised these primates to rationality – not only made of them what we call human beings—but also took them into a kind of mystical union with himself, the sort of union that Christian hope for in Heaven and call the Beatific Vision. Being in union with God, these new human beings, these primates who had become human beings at a certain point in their lives [ . . . ] also possessed what theologians used to call preternatural powers. [ . . . ] There was thus no evil in the world.” 9 Cf. van Inwagen (2006, pp. 87–88):“its object is to bring it about that human beings once more love God. And, since love essentially involves free will, love is not something that can be imposed from the outside, by an act of sheer power. Human beings must choose freely to be reunited with God and to live him, and this is something they are unable to do by their own efforts. They must cooperate with God.”

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‘age of evil’ will eventually be remembered as a sort of transient ‘flicker’ at the very beginning of human history.” (van Inwagen 1988, p. 165).

3.2.4 Local Suffering and Arbitrary Suffering The abuse of free will in leaving the union with God, in which our ancestors lived, caused human suffering in the world in the first place. This, however, does not explain why there is a particular amount of evil to be found in the actual world, or why horrendous evils occur at specific times and places. The reason for this, ultimately, is that there is no reason why we live in a world that contains the particular evils we know of. Although prima facie it is plausible to suppose that for any particular evil one might pick—be it animal or human suffering—God should have created a world that does not contain that particular evil, secunda facie this assumption is implausible in a fallen world since it leads to the conclusion that God should have created a world containing no evils at all. Since, however, in this case he would frustrate his own plan of showing human beings how it is to live separated from God, God chose an arbitrary amount of suffering as the amount of suffering found in the actual world: “He had to draw an arbitrary line, and he drew it. And that’s all there is to be said.” (van Inwagen 2006, p. 108).10

3.2.5 The Agnostic’s Response to van Inwagen’s Defence (according to van Inwagen) The arguments are, except for the defence of animal suffering, firmly based on Christian traditions and a particular interpretation of the Bible. According to this interpretation, the Bible is the inspired word of God that, although not historically accurate en detail, is a true account of what happened cum grano salis.11 As van Inwagen says, Suppose that someone who had never heard of the Bible and had never so much as thought about the beginning of the world were one day to read the book of Genesis and were to

10 Cf.

also van Inwagen (2001) and (2006, pp. 14–05): “[God] he cannot remove all the horrors from the world, for that would frustrate his plan for reuniting human beings with himself. And if he prevents only some horrors, how shall he decide which ones to prevent? Where shall he draw the line? [ . . . ] I suggest that wherever he draws the line, it will be an arbitrary line.” 11 Cf. van Inwagen (1995, p. 139): “What is the purpose of the first chapter of Genesis? What is their purpose in relation to the Hebrew Bible as a whole? [ . . . ] They are intended to describe and explain the relations between God and humanity as they stood when God made a covenant with Abraham. Thus, Genesis brings with it an account of the creation of the world and of human beings, an account which displays God as the maker and sovereign of the world and the ordainer of the place of humanity in the world. [ . . . ] I say that Genesis does get it right—in essence.”

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B. P. Göcke take everything it contained in a pretty literal sense and were to believe every word of it. This person would thereby come to believe many true things and many false things. [ . . . ] The true things [, however,] are much more important than the false things. In fact, the true things are among the most important there are, and the false things are not very important at all. (van Inwagen 2010a, p. 840)

Among the true things are: [Human beings,] these divine images, the stewards of nature, have, almost from their creation, disobeyed God, and have thereby marred the primal goodness of the world and have separated themselves from God and now wander as exiles in a realm of sin and death. (van Inwagen 2010a, p. 840)

In its essence, van Inwagen’s defence of human suffering together with his defence of animal suffering is a witty elaboration of this very idea of original sin. And according to van Inwagen, it is decisive as a defence of the existence of animal and human suffering. If the agnostic accepts, for sake of argument, the coherence of the idea that an omnipotent, omniscient, and morally perfect God exists and creates the world ex nihilo, he will, according to van Inwagen, acknowledge that the defence leads the ship of Christianity on a safe route between the Scylla of global evil and the Charybdis of local evil. The agnostic, according to van Inwagen, will grant that, for all he knows, the defence could be a true story and therefore the argument from evil is a failure.12

3.3 The Failure of van Inwagen’s Solution to the Problem of Evil One is tempted to imagine all participants of the ideal debate standing up from their chairs, stretching their legs and arms, grasping their jackets and umbrellas, constantly yearning and mumbling with each other on their slow way out from the lecture theatre: “Uff, that was a tough one!”, “At least I can see clearly now why the argument from evil is a failure!”, “I really don’t believe in God, but if you do, what the theist said makes sense”. Such a reaction of the agnostic audience is unlikely. It is more likely that when everybody is tired and secretly enjoys the few seconds of silence following the usual concluding remark—“Any further questions?”—some agnostics will have further caveats to the presented defence. They will point out that although the defence prima facie looks like a consistent story that could be true, secunda facie it has to be rejected: On the one hand, with regard to the form of the debate, the defence is based on a definition of philosophical success and failure that, amongst other things, entails that all

12 Cf.

van Inwagen (2006, p. 113): “I maintain that the [irregularity defence of animal suffering], when it is conjoined with the free-will defence, will constitute a composite defence that accounts for the sufferings of both human beings and beasts, both rational or sapient animals and merely sentient animals.”

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philosophical arguments fail and therefore is situated in a philosophical paradigm that the agnostic will reject as an adequate paradigm in which to discuss the argument from evil. On the other hand, with regard to the content of the defence, the defence appears to be consistent only because it is based on the unwarranted and silent rejection of important consequences and implications of the concept of God that van Inwagen deploys. The agnostic will argue that secunda facie the defence leads to contradictions and inconsistencies concerning the omnipotence, omniscience, and moral perfection of God. Contrary to what van Inwagen wants to make us believe, therefore, the agnostic audience will not grant that the argument from evil is a failure. It is van Inwagen’s solution to the problem of evil that fails.

3.3.1 The Inadequacy of van Inwagen’s Definition of the Ideal Philosophical Debate The first type of challenge to van Inwagen’s defence concerns the plausibility of the definition of philosophical success and failure that is operative in the discussion of the quality of the problem of evil. I present two arguments showing that van Inwagen’s conception of the ideal debate is unable to host a fair discussion of the problem of evil and will even lead the agnostic to the conclusion that the argument from evil is a success.

3.3.1.1

The Impossibility of a Fair Discussion of the Problem of Evil

Van Inwagen is aware of the following challenge to his conception of an ideal philosophical debate and its definition of the success and failure of philosophical arguments. As he says, If any reasonably well-known philosophical argument for a substantial conclusion had the power to convert an unbiased ideal audience to its conclusion [ . . . ], then, to a high probability, assent to the conclusion of that argument would be more widespread among philosophers than assent to any substantive philosophical thesis actually is. (van Inwagen 2006, pp. 52–53)

Van Inwagen, however, does not seem to care much about the fact that his proposed replacement criterion for philosophical success and failure does not fare better than the one it replaces. The agnostic, though, is unlikely to share van Inwagen’s gracious attitude. Since the debater is free to discuss whatever is relevant in respect to the evaluation of the argument from evil, here is what the agnostic might say: Wait a minute: it says here in the terms of agreement specifying the structure and goals of our debate that a philosophical successful argument is an argument that leads us, the audience of agnostics, to the acceptance of its conclusion. A substantial philosophical argument fails if and only if there is an epistemically possible scenario in which the argument fails. But this implies that all philosophical arguments are bound to fail, for our agnosticism entails that for any substantial philosophical argument there is an

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B. P. Göcke epistemologically possible scenario in which this argument fails. Furthermore, it seems that as the kind of agnostic you want me to be in this debate, I am not allowed to have any substantial knowledge, which means that I cannot be sure of even the most trivial rules of logic. It is therefore futile to engage in any substantial philosophical debate of the problem of evil.

Let me explain what the agnostic has in mind: A proposition p is epistemologically possible relative to a set S of known propositions {r, s, t} in case the union of S and p does not entail a logical contradiction, and therefore may be true. The set of known propositions, the background knowledge accessible in the debate, is such that it is either the same set for each of the agnostic participants of the debate or else is such that each of them has his own set of known propositions S1 , . . . ,Sn . In the first case, two problems arise that show that van Inwagen’s conception of the ideal debate of the problem of evil either entails the impossibility of a debate of the problem of evil or that all philosophical arguments fail. Let us start with the latter point and suppose that the agnostic participants have the same background knowledge only if the set of known propositions S is restricted to the most trivial elements of knowledge you can think of. Let us say that the set of background knowledge is constituted by propositions expressing simple truths of sense perception and simple mathematical and logical truths: The agnostics are agnostics in respect to any substantial interpretation of the world. This, however, entails that, as long as the opponent of any substantial philosophical argument can tell a consistent story entailing the falsity of the conclusion of this argument, each and every substantial philosophical argument is bound to fail. And telling such a story is always possible for the simple reason that there is an infinite number of epistemologically possible stories that are consistent with the background knowledge of the agnostics. In fact, advocates of conspiracy theories could use van Inwagen’s platform to convince the agnostics that each and every argument against their theory fails. For example, there is a consistent story according to which mankind never landed on the moon: because the Cold War forced the US to degrade the Soviet Union as the less effective political system, the NASA had to forge all the known photo and video materials in a Hollywood film studio etc. Let us turn to the second problem based on the assumption that the agnostics share the same background knowledge. This problem is that it is impossible by definition for the agnostics to engage in a meaningful discussion. Let me explain: although we assumed that the background knowledge of the ideal agnostics consists only of trivial elements of knowledge about simple truths of sense perception and mathematical and logical truths, it can be shown that we already assumed too much. Not even these putatively trivial elements of knowledge can be granted to the agnostic. With regard to each of these putatively simple elements of knowledge, there is philosophical discussion of their existence. For instance, eliminative materialists think there are no qualia, John Stuart Mill thinks the sentences of arithmetic are contingent, and Quine thinks logical truths are revisable.13 Therefore, since not

13 Thanks

to Stephen Priest for pointing this out to me.

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even the apparently simplest kind of knowledge is beyond dispute, it follows, by induction, that there is no knowledge whatsoever the agnostic is allowed to rely on in his evaluation of the problem of evil. But a debate is impossible if the audience is not allowed to hold any background knowledge whatsoever. The situation does not fare better in the second case, in which the agnostics do not share the same background knowledge S. Let us suppose there are two groups of agnostics, distinguished by sets of background knowledge S1 and S2 . The problem is that, in this case, the audience cannot be said to be an audience of ideal and neutral agnostics because those different sets of background knowledge will lead to different evaluations of the opponent’s and proponent’s presentations of their arguments. For instance, it will make a difference to their evaluation of van Inwagen’s defence if some agnostics suppose that there is an immortal, immaterial soul and others assume that life after death is impossible. Since these beliefs are not of necessity related to the existence of a Christian God, the agnostic, in the case at hand, apparently is allowed to hold them in the debate. But, of course, based on their beliefs they will evaluate in different ways any defence that, for instance, entails that God atones for the suffering in this world with an infinite amount of joy in the afterlife.

3.3.1.2

The Agnostic Audience and the Success of the Problem of Evil

Van Inwagen’s concept of an ideal philosophical debate entails that every substantial philosophical argument is a failure, that the debate does not even get off the ground, or that an audience of neutral agnostics is impossible because, based on their individual background knowledge, they are biased towards the acceptance or the refutation of the problem of evil. In other words, van Inwagen’s ideal debate is not a fitting place in which to discuss the problem of evil. In the light of this conclusion, the agnostic might even conclude that the argument from evil is a success because it forces the theist to establish a criterion of philosophical success in which every substantial philosophical argument and every argument against conspiracy theories fails. The agnostic might argue as follows: If the argument from evil forces philosophers like van Inwagen to throw the baby out with the bathwater, then it seems to us that, after all, the argument from evil is a sound argument. Apparently, it can only be shown to be a failure based on a definition of success and failure on which all philosophical arguments fail, no discussion is possible, or the audience is biased towards acceptance and refutation of the problem of evil.

3.3.1.3

The Impossibility of a Reliable Debate of the Problem of Evil

For the sake of argument, I will assume that it is possible to discuss the problem of evil in front of an audience of ideal and neutral agnostics. There is still a further problem with van Inwagen’s ideal philosophical debate. The problem is based on the fact that van Inwagen’s definition of the success and failure of a philosophical

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argument is psychologistic and obfuscates the distinction between the soundness of an argument and the belief that the argument is sound. On van Inwagen’s conception of an ideal debate, an argument can be sound and nevertheless easily fail to be recognised as such in the eyes of ideal agnostics. Consider, for instance, the following type of argument. If A, then B. A. Therefore, B. Suppose that “If A, then B” and “A” are actually true for a chosen A and B. The argument is sound because the truth of the conclusion is entailed by the truth of the premises. However, to convince the audience of ideal agnostics of the failure of the argument all one has to do is to establish a possible epistemic scenario in which the second occurrence of A in the argument is replaced by A∗ , and ask the audience whether they can really exclude that it is A∗ and not A that is true. Since they are asked, in the debate, to evaluate the quality of an argument in terms of consistent epistemic scenarios based on very little background knowledge, they will, of course, not be able to exclude A∗ as an epistemic possibility because, for all they know, A∗ and not A could be true. Therefore, they will judge that the argument is a failure, although in fact it is a sound argument. Applied to the problem of evil it follows that the argument from evil could be a sound argument albeit the audience of agnostics, relying on the resources of evaluation they are restricted to, is unable to believe this. There is, in other words, something wrong with the statutes of van Inwagen’s ideal philosophical discussion if it enables an audience of agnostics to believe that a sound argument is unsound. Adopting the point of view of the agnostics, we can use this to argue against the plausibility of van Inwagen’s defence as follows: If the problem of evil, in addition to the difficulties mentioned above, forces the theist to set up normative parameters of a philosophical discussion in which a sound argument could be judged to be unsound, then we cannot exclude that the problem of evil is a sound argument although, in this debate and its rules, we would be bound to judge that it is unsound. In this case, however, we refuse participation in the debate. It is just not a reliable method to discover anything interesting about the quality of an argument.

In sum, contrary to what van Inwagen asserts, his conception of an ideal philosophical debate and its definitions of success and failure are unable to host a fair discussion of the problem of evil. Since van Inwagen’s conception of the ideal debate is an essential part of his defence of the existence of evil, it follows that van Inwagen’s solution to the problem of evil fails.

3.3.2 Inconsistencies in van Inwagen’s Defence of the Existence of Evil The second kind of argument against van Inwagen’s defence is not directed upon the problematic form of the debate but concerns the content of the actual defence. Once van Inwagen’s definitions of failure and success are granted, however, it is almost as difficult as the attempt to prove a conspiracy theory false to show that even in this case van Inwagen’s solution to the problem of evil fails. The reason is that van

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Inwagen’s defence, from a systematic point of view, demands such little intellectual commitment that it is hard to point out problems that van Inwagen could not object to by pointing out that all he is asserting is epistemic consistency. Therefore, to be successful, the method to criticise van Inwagen’s defence has to aim at the consistency of his defence story. Since at first glance van Inwagen’s defence appears to be consistent, we have to distinguish between kinds of consistency: prima facie consistency and secunda facie consistency. An epistemic scenario is prima facie consistent if and only if it can be expressed as a set of propositions that are consistent if the entailments of the propositions of the set are bracketed. An epistemic scenario is secunda facie consistent if and only it can be expressed as a set of propositions that are consistent even if the entailments of the propositions of the set are added to the set as elements. The distinction between prima facie and secunda facie consistency is an epistemological distinction that adequately mirrors how we deal with any kind of story, be it scientific, philosophical, or theological: At first glance we might judge that a story is plausible and consistent before we, in deeper reflection and analysis, realize that there are implications of the story that turn it into an inconsistent one. My claim in this section, then, is that van Inwagen’s defence is prima facie consistent, but secunda facie inconsistent. It appears to be consistent only because van Inwagen implicitly brackets implications of his concept of God. The agnostic, however, will make these explicit to show the inconsistency of van Inwagen’s defence.

3.3.2.1

Van Inwagen and the Best of All Possible Worlds

Here is how the debate might continue: “Wait,” says the agnostic to the theist, you provided a prima facie consistent story of the reasons God might have to allow the suffering of animals and human beings. And you, dear fellow agnostic audience, probably assume that this story could be true, for all we know. I ask you, though, to withhold your judgement on the success or failure of the argument from evil for a few more moments. I would like to tell you a story, intended as a reductio ad absurdum, to convince you that according to van Inwagen’s defence we are living in the best of all possible worlds. My story begins with a clarification of the growing block model of reality and then continues with a sketch of a coherent model of time-travel that allows the traveller to change the past.

Reality as a growing block: let us assume that physical reality is a growing block of events, structured by the relations later than, earlier than, or simultaneously with. This structure does not change once fixed. If, for instance, it is true to say at a time that x occurred earlier than y, then, as long as this reality exists, it will always be true that x occurred earlier than y. Some models of reality entail that future events, timelessly, exist ontologically on the same level as past and present events—on these models reality is already a ‘complete’ block with us occupying some slice of it—but the growing block model of reality assumes that only past and present events exist. Future times and events do not exist until they happen. As van Inwagen says, “there are, timelessly, events that

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have happened and there are events that are happening, but there are (timelessly) no events that have not happened yet but will happen.” (van Inwagen 2010b, p. 10). As time goes by, though, present events become past events, and new events are present until it is their turn to become past. Whenever a present moment of time becomes past, on this model, the block of reality grows: “It is growing, because, if we think of the temporal axis as a dimension, then Reality is, as we might say, growing along the temporal dimension: at every time it contains—timelessly—all the parts it contained at earlier times and other parts as well.” (van Inwagen 2010b, p. 14). Time-travel: there are, in principle, two kinds of time-travel: time-travel that does not change the past and time-travel that does. Following van Inwagen, I refer to the first as Ludovician and to the second as non-Ludovician time-travel.14 According to Ludovician time-travel, the time-traveller’s visit to the past is part of the history of the world, it is an unchangeable part of the growing block of reality. An omniscient being that looks from the outside of physical reality at the history of the world up to now would know whether Ludovician time-travel occurred in the block of reality and could tell a coherent story of the history of the world that does not involve any paradoxes or contradictions. Non-Ludovician time-travel is possible if there is a model of non-Ludovician time-travel that does not entail a contradiction. Based on the assumption that reality is a growing block, here then is how non-Ludovician time travel is possible: Suppose a time-traveller, call him Tim, succeeds after years of intense research and engineering in building a functional time-machine in his family’s barn. Excited of what lays ahead, Tim enters the time machine at a moment of time that constitutes the latest slice of the growing block of reality. After all, it’s the present moment of time. He turns some gears and aims to travel to the year 1920. He then pushes the “Go-Baby-Go”-button and, suddenly, in a flash of lightning, the machine disappears from the family’s barn and re-appears at the same place in the year 1920. Slightly dizzy, Tim opens the door of the time machine and takes a look around. Since he is a non-Ludovician time-traveller, he knows that he is free to do whatever pleases him. Greedy Tim, thus, goes to the next patent office and, taking out of his pockets numerous blueprints for personal computers, smartphones, rockets, and what have you, starts a company and makes a fortune. As soon, however, as Tim changes anything in his environment that was not yet part of the history of the world as it occurred seen from the point of view of an omniscient being that observed Tim entering the time-machine in his family’s barn—like for instance, setting his family’s barn on fire and poisoning the land— something unexpected happens: on the growing block model of reality, unbeknown to Tim, the block of reality reaching from the moment of his arrival in 1920 to

14 Cf.

van Inwagen (2010b, pp. 4–5): “Travel to the past is (like time-travel simpliciter) of two kinds. I will call them Ludovician and non-Ludovician time-travel. Ludovician time-travel does not involve changing the past. [ . . . ] My topic is non-Ludovician time-travel: time-travel that involves changing the past.”

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the moment where he entered the time-machine vanishes into mere nothingness. Everything that happened in this period as part of the original growing block of reality never happened, although, of course, only an observer independent from physical reality could remember the old block of reality in total. As van Inwagen says, “an episode of non-Ludovician time-travel annihilates the part of the block after the arrival; then the block commences re-growth, but, as happens with starfish that re-grow a severed limb, the re-grown part will not be exactly like the original.” (van Inwagen 2010b, p. 23).15 The agnostic continues as follows: Dear agnostics, for all we know, reality could be a growing block that allows for the possibility of time-travel. And, for all we know, van Inwagen’s God exists independently from the physical world. Although he exists in time, it is possible for him to exist while there is no physical time, so it is plausible to assume that God is an omniscient observer of the growing block of reality that is independent from what happens in physical reality – divine time, so to speak, is hyper-time. Now let me ask you: what would this omniscient, morally perfect and omnipotent God do in the light of the possibility of time travel in a growing block universe? Well, I can tell you: since (a) he cannot influence the free decisions of human beings, since (b) he cannot determine the results of probabilistic laws of nature, since (c) he cannot deceive human beings constantly, and since (d) he is in deep divine sorrow due to the fall of mankind, he, of course, would start the whole of physical reality anew whenever it goes astray and evil happens. Like I press the reset button whenever my computer stops working as I expect it to do, an omniscient, omnipotent and morally perfect God would reset reality until it is the best of all possible worlds. Whenever physical reality exists, it is the best reality possible in the eyes of a perfect being.

Although at first glance it might look impossible for God to do this, van Inwagen assures that God is able to start reality anew: “After all, if God exists he can create an object to any set of specifications that doesn’t involve a contradiction” (van Inwagen 2005, p. 637). God could do so either by a divine decree or by way of directly commanding a human being to create a time machine and to go back to whichever point of time God wants to use as the point of a new start. In this scenario, nobody is deceived because all those that could complain of being deceived never existed in reality, and therefore simply cannot complain. They exist in the mind of God only as parts of a possible but non-actual world. Now, this scenario is relevant to the assessment of van Inwagen’s defence because it shows that the defence is only prima facie, but not secunda facie

15 Cf.

also van Inwagen (2010b, p. 17): “The Intelligence sees the block grow beyond the 1921New Year’s point with no apparently ex nihilo appearance of any sort of machine having been included in the block. After the block has grown about a century longer, the Intelligence sees one of its inhabitants, Tim, enter an elaborate machine. At the moment t, Tim presses a button [ . . . ]— and suddenly, all in a hyper-instant, the futuremost century-long part of the block vanishes. The events that hyper-now constitute the ‘leading face’ of the block occur at a moment t0 in 1920. But those events are hyper-now a bit different from the way the t0-events hyper-were before the block lost its century-long terminal segment, for the leading face hyper-now contains the appearance of a machine-Tim composite. [ . . . ] Thereafter, the block grows at the same rate (so many seconds per hyper-second) it hyper-always has. But it does not grow in quite the same way it did following the previous hyper-occasion on which it passed the point t0.”

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consistent. That is, the defence is only consistent if based on the implicit assumption that the agnostics do not, as part of their background knowledge, believe that reality is a growing block or believe that van Inwagen’s concept of God entails that God would create reality as a growing block. Although van Inwagen could argue that the assumption that reality is a growing block is controversial and that the theist only has to point out that, for all we know, it could be false that reality is a growing block to bypass the argument, the agnostic is likely to argue as follows: The concept of God which we are asked to take for granted in our discussion of the problem of evil is the concept of an omnipotent, morally perfect, and omniscient being. On this model, God can do anything that does not involve a contradiction. Since the growing block model of reality does not entail a contradiction, given his omnipotence, God can create reality as a growing block. Furthermore, on this model, God wants to avoid suffering and evil to occur in the world, but, given his moral perfection, he is unable to simply take it away after the fall of mankind, since this would entail deceiving human beings. However, given his omniscience, God knows that there is a way in which he can achieve both aims of not deceiving human beings and of creating a world without suffering and evil: all God has to do, is to start the growing block of reality anew anytime his creation goes astray. In fact, given van Inwagen’s concept of God, by the nature of his character, God is forced to do so, once he freely decided to create a world at all, because this way of acting is the only possible way given the omnipotence, omniscience, and moral perfection of God. Therefore, on van Inwagen’s concept of God, this world in which we are living is the best of all possible worlds that a perfect being can create. Since, however, this is obviously not the best of all possible worlds that a perfect being could create, because, as the theist has yet already granted, there are arbitrary amounts of gratuitous evil to be found in the actual world, God, if he existed, could not be omniscient, morally perfect, and omnipotent. Therefore, van Inwagen’s defence is only prima facie consistent, but secunda facie inconsistent. In other words, the agnostic will assume that van Inwagen’s defence is a failure because it entails the absurd consequence that we are living in the best of all possible worlds.

3.3.2.2

Determinism, Indeterminism, and the Suffering of Animals

A second problem for van Inwagen’s defence concerns the suffering of animals before the fall of mankind. According to van Inwagen, God could only avoid this suffering of animals if he constantly intervened in the world, thereby destroying the regularity imposed on this world by the laws of nature that, although not deterministic, still are regular enough in their probabilistic determination of what will happen. Since, according to van Inwagen, this massive irregularity would at least be as big a defect in a world than the actual suffering of animals, God was free to choose which option to realize and apparently decided that the animal suffering plus the regularity of the world is acceptable from a moral point of view. Although this story is prima facie consistent, it is secunda facie inconsistent with regard to the nature of God. It is prima facie consistent only because van Inwagen

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implicitly assumes that God only has two options: either creating a probabilistic world with no interventions that entails the suffering of animals or creating a massively irregular world with interventions and no animal suffering. If the agnostic agreed that these are the only relevant options, he would be tricked to assume that van Inwagen’s defence of the suffering of animals is also secunda facie consistent. However, there is no reason to assume that the agnostic would assume that the two options van Inwagen mentions are the only relevant ones. Consider the following reply to the theist: Dear theist, you want us to accept a scenario that is consistent with what we know and therefore could be true for all we know. You want to show that consistency of the existence of an omnipotent, morally perfect, and omniscient God and the suffering of animals in a world before the fall of mankind. You say that God only has two options: creating a massively irregular world without the suffering of animals or creating a regular world entailing the suffering of animals. If this was the case, then your story might well be true. However, you implicitly bracket some important options that are relevant to our evaluation of the consistency of your defence. Contrary to what you are saying, your concept of God entails that God has to create a world that is deterministic, and therefore can easily avoid the suffering of animals, until the point of time free human beings enter the stage of creation and need to live in an indeterministic world. Since this is not what the God of your defence is doing, your defence entails a contradiction.

The agnostic has the following argument in mind: if God exists and is omnipotent, morally perfect, and omniscient, and if the evolution of animals is only a means to an end in order for Homo sapiens to evolve, then, by his omniscience, God knows, first, the following: It is false that there are only the two options of either creating a massively irregular world without animal pain or creating a regular world full of animal pain. Since God knows everything that can be known, He knows that there is a possible world that is deterministic until a particular moment of time and indeterministic afterwards—it is false that a possible world has to be either fully deterministic or fully indeterministic. God further knows that there is a possible world in which determinism holds until the point of time human beings appear, such that up to this point of time the determined history of this world entails that there is no suffering in the evolution of animals. Second, by this omnipotence, God could create a world in which this deterministic and pain-free evolution of animals occurs while it still leads to the evolution of human beings that as soon as they have free will start living in an indeterministic world. Third, by his moral perfection, since nobody is deceived in this scenario and since animals are not deprived of their libertarian free will, which they do not have, it follows that God, who wants to avoid suffering and evil in a world before the fall of mankind, has to create the world in such a way that it is at first a deterministic world without the suffering of animals and an indeterministic world open to suffering once human beings entered the stage. Therefore, despite the prima facie consistency of van Inwagen’s defence of animal suffering, the agnostic will argue that secunda facie it is a story inconsistent with the concept of God he is asked to take for granted in the discussion of the problem of evil.

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3.3.2.3

Van Inwagen and the Arbitrary Love of God

“I am not done yet”, says the agnostic, I have another story to tell you that should convince you once more that the theist’s defence itself is a failure. Remember when the theist said that the age of evil will be over once we cooperate with God in his divine rescue plan for humanity? If you reflect on this, you should see it entails that God can only arbitrarily pick a particular number of human beings the love of whom is enough in order for his plan to be fulfilled. But, as you will agree, this contradicts the moral perfection of God. In other words, the theist’s defence leads to the conclusion that God is not morally perfect. Therefore, since God is assumed to be essentially morally perfect, van Inwagen’s defence of the existence of evil is a failure.

Here is the argument that the agnostic has in mind: We know that there is a relation between, on the one hand, human and animal suffering in a fallen world and, on the other, God’s rescue plan for humanity. As long as we humans do not freely return to God by freely deciding to love God again, we will live in a world that is penetrated by an arbitrary amount of suffering, since this is what it means not to live in union with God. But who is “we”? How much love is enough love in the eyes of God for him to bring the suffering in this world to an end? Here is what we know: since God cannot know the future free decisions of human beings, he cannot know whether any human being will freely return to love him. Therefore, it is unlikely that God’s master plan demands that all human beings of all times and places freely return to love him. This would be rational if and only if God knew that the probability that all human beings freely return to love him is high enough in order for him to place a fair bet on this outcome. This, however, is prevented by God’s not knowing what human beings will do. If God’s plan demanded that all human beings have to love God before he ends the age of evil, then a single person who freely refused to love God had the power to justify, in the eyes of God, the continuance of the suffering of billions and billions of creatures. If, however, God cannot rationally demand that all human beings return to him in order for God to end the suffering in the world, he can only decide that an arbitrary number of returning human beings is enough for him to end the age of evil. The argument for this conclusion proceeds along the following lines: (1) If Tim loves God, this is not enough for God to end the age of evil. (2) If Tim and Paul love God, this is not enough for God to end the age of evil. (3) If Tim, Paul, and Sarah love God, this is not enough for God to end the age of evil . . . —you know where this is going. In the light of this argument, God has to decide that for an arbitrarily chosen number n, the love of n people is enough in order to satisfy his divine rescue plan: “If n people freely return to Me, then I will end the age of evil.” Depending on how we understand “The love of n people is sufficient for God to end the suffering in the world”, however, we obtain different problems: On the one hand, if we suppose that it is only the number that counts, then it is completely irrelevant which human beings love God. As long as n people love God, God does not care who it is that loves him. On the other hand, if God decides that the number he arbitrarily picked is the number of particular people the love of whom is sufficient for him to end the age of evil, then the rest of humanity does not matter at all for the

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divine plan. Even if they all loved God, that would not be a reason for God to end the suffering, as long as the chosen one’s do not love God. A morally perfect God, however, could not decide either way: even bracketing the fact that God does not know whether n people will freely return to love Him, a morally perfect being is unable to pick an arbitrary number of people as the ones the love of whom is sufficient to end the age of evil, or to decide that only some particular people across all times and places are soteriologically relevant. Any such decision would entail that there will be human beings that will suffer for no reason at all, that will be soteriologically irrelevant, and that do not even have the chance that their love affects God to the slightest degree. Therefore, although prima facie the theistic defence could be true, secunda facie the agnostic will come to the conclusion that the defence does not present a consistent story because it entails that God is not morally perfect: I can see no way how a being that is supposed to be morally perfect could decide that only a few arbitrarily chosen people are soteriologically important and justify the huge amounts of suffering in the world, while at the same time all other people have to suffer for no reason at all and, even if they love God, are utterly impotent to move the heart of God to the slightest degree. Van Inwagen’s defence of evil therefore fails since it entails a contradiction in respect to the moral perfection of God.

3.4 Summary: The Failure of van Inwagen’s Solution to the Problem of Evil To deal with the problem of evil van Inwagen introduces a particular conception of an ideal philosophical debate as well as corresponding definitions of the success and failure of substantial philosophical arguments. He claims that the problem of evil is a failure if in an ideal debate the audience of agnostics will suppose that, for all they know, the problem of evil could be a failure, where it is enough if they are presented with epistemic scenarios in which the argument fails. Van Inwagen claims to have presented a story on which the problem of evil fails. However, as we have seen, van Inwagen’s solution to the problem of evil fails for two kinds of reason. On the first kind of reason, van Inwagen’s conception of an ideal debate is not a proper place in which a fair discussion of the problem of evil is possible. According to van Inwagen’s ideal debate, no conspiracy theory could be rejected, all arguments fail, the audience is biased, might judge sound arguments to be unsound or, worst, is so severely limited with regard to their background knowledge that no discussion is possible. On the second kind of reason, the defence itself is only prima facie consistent, but secunda facie inconsistent. It is inconsistent because contrary to what van Inwagen asserts, the concept of God deployed in the defence entails conclusions in direct contradiction to van Inwagen’s defence story.

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Therefore, given that there are both severe problems with the form and the content of van Inwagen’s defence of the existence of evil, the agnostic will assume that the defence itself is a failure.

References Göcke, Benedikt Paul. 2014. A Theory of the Absolute. Basingstoke: Palgrave Macmillan. Griffin, David Jay. 2014. Panentheism and scientific naturalism. Rethinking evil, morality, religious experience, religious pluralism, and the academic study of religion. Claremont: Process Century Press. Rohs, Peter. 2013. Der Platz zum Glauben. Paderborn: Mentis. van Inwagen, Peter. 1988. The magnitude, duration, and distribution of evil: A theodicy. Philosophical Topics 16 (2): 161–187. ———. 1995. Genesis and evolution. In God, Knowledge, and Mystery. Essays in Philosophical Theology, 128–162. London: Cornell University Press. ———. 1997. Against middle knowledge. Midwest Studies in Philosophy 21: 225–236. ———. 2001. The Argument from Particular Horrendous Evils. American Catholic Philosophical Association, ACPA Proceedings 74: 65–80. ———. 2002. What is the problem of the hiddenness of god? In Divine hiddenness. New essays, ed. Daniel Howard-Snyder, Paul K. Moser, 24–32. Cambridge: Cambridge UP. ———. 2005. Is God an Unnecessary Hypothesis? In God and the Ethics of Belief. New Essays in Philosophy of Religion, ed. Andrew Dole and Andrew Chignell, 131–149. Cambridge: Cambridge University Press. ———. 2006. The Problem of Evil. Oxford: Oxford University Press. ———. 2008. What does an omniscient being know about the future? In Oxford Studies in Philosophy of Religion, ed. Jonathan L. Kvanvig, vol. 1, 216–230. Oxford: Oxford University Press. ———. 2010a. Darwinism and Design. In Science and Religion in Dialogue, ed. Melville Stewart, 825–834. Oxford: Wiley-Blackwell. ———. 2010b. Changing the past. In Oxford studies in metaphysics, ed. Dean W. Zimmerman, 3–28. Oxford: Oxford University Press.

Benedikt Paul Göcke, Dr. Phil, Dr. Theol., is Professor of the Philosophy of Religion and Philosophy of Science at Ruhr-University Bochum, Germany. He is also a Research Fellow at the Ian Ramsey Centre for Science and Religion and an associated Member of the Faculty of Theology at University of Oxford. Göcke is author of: A Theory of the Absolute (Macmillan, 2014), Alles in Gott? (Friedrich Pustet, 2012) and editor of After Physicalism (Nore Dame, 2012) and The Infinity of God (Notre Dame, 2018). He published articles in The International Journal for Philosophy of Religion, Zygon, Sophia, The European Journal for Philosophy of Religion, and Theologie und Philosophie, among others.

Chapter 4

Saadia Gaon on the Problem of Evil Eleonore Stump

4.1 Introduction The problem of evil is raised by the combination of traditional theistic beliefs and the acknowledgement that there is evil in the world. If there is a perfectly good, omnipotent, omniscient God who creates and governs the world, how can the world he created and governs have suffering in it? This question, of course, needs to be made more precise. It is widely accepted that human beings have free will and are able to misuse that free will in immoral ways. Some punishment of the wicked doesn’t seem incompatible with God’s existence. As far as that goes, some people freely choose to subject themselves to suffering. Odysseus chooses to join the expedition to Troy although he knows that doing so will result in all the sufferings of war. Such suffering, freely brought on himself by the sufferer, also doesn’t seem incompatible with the existence of a God with the traditional attributes. So when the existence of evil in the world looks difficult to reconcile with the existence of God, the evil in question must be the suffering of unwilling innocents, those whose suffering is neither chosen nor deserved by them. Job is the classic case in literature of an unwilling innocent whose suffering is, on the face of it, incompatible with the existence of God. It is an explicit and much emphasized part of the story that Job is entirely innocent, that his suffering in no way constitutes punishment for wrongdoing. It is equally obvious that his suffering isn’t anything that he has chosen; it is simply inflicted on him, by marauding evildoers, by nature, and, in some sense, by God. For many contemporary thinkers, suffering such as that endured by Job is evidence against the existence of God. For committed theists, such as Job himself in the story, what is called in question by such suffering

E. Stump () Saint Louis University, St. Louis, MO, USA e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_4

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is not God’s existence but one or more of the attributes traditionally assigned to him. Job himself, with some passion, questions God’s goodness. How a commentator on the story of Job deals with the apparent incompatibility of Job’s suffering with the existence or traditional attributes of God thus reveals his attitude towards the problem of evil. In philosophy of religion recently, considerable effort has been expended on constructing (or reconstructing) Christian theodicies, which try to find ways consonant with Christian belief to reconcile the suffering of unwilling innocents, such as Job, with the existence and nature of God as understood in Christian theology. Theodicies of this sort make use of such Christian doctrines as original sin, incarnation, and atonement.1 But, of course, the story of Job is part of the Hebrew Bible, and there is abundant reflection on the story of Job in Jewish philosophical theology. This material, however, has not been discussed much in contemporary philosophical literature on the problem of evil. In this chapter I want to take a step towards remedying this defect by examining the interpretation of the story of Job and the solution to the problem of evil given by one important and influential Jewish thinker, Saadia Gaon.2 As I have argued elsewhere, the sort of approach to the problem of evil of which Saadia’s theodicy is one example can be defended against many of the main objections it invites:3 But in this chapter my aim will be largely to investigate and clarify what Saadia’s theodicy is, not to defend it. Since Saadia is not as well known as some other medieval philosophers, it may also be helpful at the outset to say something very briefly about his life. Saadia ben Josef al-Fayyumi was born in Fayyum in Egypt in 892.4 Even in his youth, he was involved in intellectual and religious controversy. He was an active opponent of the Karaites, who rejected the importance of the Talmud and Midrash and argued for a solely literal interpretation of biblical texts. He was also an outspoken and effective participant in a controversy between Palestinian and Babylonian rabbis over the religious calendar. Saadia was on the side of the Babylonians and was thought to have refuted the Palestinian rabbis. In 921/2, in recognition of his success in this controversy, he was made a member of the famous and influential Academy of Sura. In 928 he was appointed Gaon or Head of the Academy, a position of considerable eminence in the medieval Jewish world. After a brief period in office, he quarreled with the Exiliarch, the political leader of the Jewish community in Babylon, and the quarrel became so fierce that it led to Saadia’s removal from the Gaonate. Saadia was in retirement for the next 5 years, a 1 To

take just two examples, see, for example, my “Aquinas on the Sufferings of Job”, in Stump (1993), pp. 328–357, and Adams (1986), pp. 248–267. 2 According to The Jewish Encyclopedia, he is the first to bear this name, which is a Hebraicized version of the Arabic name ‘Sai’id’. His name is also transliterated as ‘Saadya’ or ‘Saadiah’. 3 Stump (1985a), pp. 392–424; Stump (1985b), pp. 430–435; Stump (1986), pp. 181–198; Stump (1990), pp. 51–91; Stump (1993), pp. 328–357. 4 Although the date is commonly given as 892, it is listed as 882 in The Encyclopedia of Philosophy and the Encyclopedia Judaica in the entry under his name and in Sirat’s (1985, reprinted 1990), p. 18.

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time which proved to be very productive for him; his main philosophical work was written in this period. Eventually, he was reconciled with the Exiliarch and restored to his position as Gaon, but he did not remain Gaon long. He died in 942, only 5 years after resuming his position.5 He was an influential thinker and a prolific author, writing on topics as diverse as chronology and psychology; but his magnum opus, which contains a considerable discussion of the problem of evil, is The Book of Beliefs and Opinions.6 He also wrote a lengthy commentary on the book of Job.7 In what follows, I will be considering Saadia’s views of the problem of evil as they are presented in these two works. In order to show the nature of Saadia’s theodicy and his interpretation of the story of Job, I think it is helpful to have a medieval Christian analogue for comparison. Because medieval Jews and Christians share many philosophical and theological views, one way to elucidate a medieval Jewish theodicy such as Saadia’s is to overlay Saadia’s theodicy with that of a typical medieval Christian and to look for the differences between them. So I will first give a brief summary of Aquinas’s reading of the story of Job, and then I will present Saadia’s, as Saadia is commonly understood. I will then argue that this common understanding of Saadia’s views is inaccurate and simplistic. If we look carefully at Saadia’s theodicy in The Book of Beliefs and Opinions, which, according to contemporary scholars, contains generally the same view of the problem of evil as his commentary on Job does, we can see that his theodicy is subtler and more defensible than has been generally supposed. Finally, I will argue that, interpreted in the way I am arguing for, Saadia’s theodicy is much closer to Aquinas’s than at first appears. Seeing why their positions are as close as they are gives us some insight, I think, into the issues at stake in considerations of the problem of evil.

4.2 Aquinas on Job Aquinas approaches the book of Job with the conviction that God’s existence is not in doubt, either for the characters in the story of Job or for the readers of that story.8 On his view, those who go astray in contemplating sufferings such as Job’s do so because, like Job’s comforters, they mistakenly suppose that happiness and

5 The

major work on Saadia’s life is Malter’s (1926; reprinted 1969).

6 There is a complete translation of Saadia’s (1948), and an abridged translation by Altmann (1960). 7 Saadia’s

commentary on Job has also been translated: Goodman (1988). commentary, Expositio super Job ad litteram, is available in the Leonine edition of Aquinas’s works, vol. 26, and in an English translation: Aquinas (1989). (I will give references to this work both to the Latin and to the Damico and Yaffe translation.) The commentary was probably written while Aquinas was at Orvieto, in the period 1261/2–1264. See Weisheipl (1983), p. 153; see also Tugwell (1988), p. 223.

8 Aquinas’s

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unhappiness are functions just of things in this life. Aquinas, on the other hand, takes the book of Job to be trying to instill in us the conviction that there is another life after this one, that our ultimate happiness lies there rather than here, and that we attain to that happiness only through suffering.9 On Aquinas’s view, all human beings have a terminal cancer of soul, a proneness to evil which invariably eventuates in sin and which in the right circumstances blows up into monstrosity. On his view, even “our senses and our thoughts are prone to evil.”10 The pure and innocent among human beings are no exception to this claim. When the biblical text says that Job was righteous, Aquinas takes the text to mean that Job was pure by human standards. By the objective, uncurved standards of God, even Job was infected with the radical human tendencies toward evil.11 No human being who remains uncured of this disease can see God. On Aquinas’s view, then, the primary obstacle to union with God, in which true and ultimate human happiness consists, is the sinful character of human beings. Aquinas thinks that pain and suffering of all sorts are God’s medicine for this spiritual cancer,12 and he emphasizes this view repeatedly.13 Arguing that temporal goods such as those Job lost are given and taken away according to God’s will, Aquinas says someone’s suffering adversity would not be pleasing to God except for the sake of some good coming from the adversity. And so although adversity is in itself bitter and gives rise to sadness, it should nonetheless be agreeable [to us] when we consider its usefulness, on

9 See,

for example, Aquinas (1989), chap. 7, sect. 1, Damico and Yaffe, p. 145; and Chap. 19: 23–29, Damico and Yaffe, pp. 268–271, where Aquinas makes these points clear and maintains that Job was already among the redeemed awaiting the resurrection and union with God. Someone might wonder whether it is possible to maintain this approach to suffering when the suffering consists in madness, mental retardation, or some form of dementia. This doubt is based on the unreflective assumption that those suffering from these afflictions have lost all the mental faculties needed for moral or spiritual development. For some suggestions to the contrary, see the sensitive and insightful discussion of retarded and autistic patients in Sacks (1985). 10 Thomas Aquinas, Super ad Hebraeos, chap. 12, lec. 2. 11 Aquinas (1989), chap. 9, sects. 24–30; Damico and Yaffe, pp. 1–79. 12 I have explored and defended Aquinas’s approach to the problem of evil in different ways in the papers listed in footnote 3. (This section on Aquinas is largely taken from “Aquinas on the Sufferings of Job”.) In those papers I discuss and defend Aquinas’s claims that a good God would create a world in which human beings have such a cancer of the soul, that suffering is the best available means to cure the cancer in the soul, and that God can justifiably allow suffering even though it sometimes eventuates in the opposite of moral goodness or love of God. 13 One shouldn’t misunderstand this claim and suppose Aquinas to be claiming that human beings can earn their way to heaven by the merit badges of suffering. Aquinas is quite explicit that salvation is through Christ only. His claim here is not about what causes salvation but only about what is efficacious in the process of salvation. It would take us too far afield here to consider Aquinas’s view of the relation between Christ’s work of redemption and the role of human suffering in that process. What is important for my purposes is just to see that on Aquinas’s account suffering is an indispensable element in the course of human salvation, initiated and merited by Christ.

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account of which it is pleasing to God [ . . . ]. For in his reason a person rejoices over the taking of bitter medicine because of the hope of health, even though in his senses he is troubled.14

For Aquinas, then, what justifies the suffering of an unwilling innocent is that the suffering acts as a spiritual chemotherapeutic agent, keeping the spiritual cancer of the soul from killing the patient. Aquinas thus sets fairly strenuous standards for theodicy. The morally sufficient reason for God’s allowing unwilling innocents to suffer consists in a benefit which comes, largely or primarily, to the sufferer and which consists in warding off a greater evil for the sufferer. So, for example, commenting on a line in Job containing the complaint that God sometimes doesn’t hear a needy person’s prayers, Aquinas says, Now it sometimes happens that God hearkens not to a person’s pleas but rather to his advantage. A doctor does not hearken to the pleas of the sick person who requests that the bitter medicine be taken away (supposing that the doctor doesn’t take it away because he knows that it contributes to health); instead he hearkens to [the patient’s] advantage, because by doing so he produces health, which the sick person wants most of all. In the same way, God does not remove tribulations from the person stuck in them, even though he prays earnestly for God to do so, because God knows these tribulations help him forward to final salvation. And so although God truly does hearken, the person stuck in afflictions believes that God hasn’t hearkened to him.15

In fact, on Aquinas’s view, the better the person, the more likely it is that he will experience suffering. In explicating two metaphors of Job’s,16 comparing human beings in this life to soldiers on a military campaign and to servants, Aquinas makes the point in this way: It is plain that the general of an army does not spare [his] more active soldiers dangers or exertions, but as the plan of battle requires, he sometimes lays them open to greater dangers and greater exertions. But after the attainment of victory, he bestows greater honor on the more active soldiers. So also the head of a household assigns greater exertions to his better servants, but when it is time to reward them, he lavishes greater gifts on them. And so neither is it characteristic of divine providence that it should exempt good people more from the adversities and exertions of the present life, but rather that it reward them more at the end.17

Underlying these remarks of Aquinas’s is the thought that, just as there are degrees of bodily health, so there are also various gradations of spiritual health. Those persons who are morally or spiritually stronger are given more suffering so that they might be more thoroughly cured of their own evil and brought to a more

14 Thomas

Aquinas, Super ad Hebraeos, chap. 12, lec. 2. (1989), chap. 9, secs. 15–21; Damico and Yaffe, p. 174. 16 Only one of the two metaphors is in the Revised Standard Version, the King James, and the Anchor Bible. 17 Aquinas (1989), chap. 7, sec. 1; Damico and Yaffe, p. 146. The mention of reward may lead someone to suppose that what is at issue for Aquinas is God’s providing a greater good to justify the evil suffered. But this is a mistaken supposition, as I explain below in connection with a similarly mistaken interpretation of Saadia’s theodicy. 15 Aquinas

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robust state of spiritual wellbeing. Someone might suppose here that on Aquinas’s views he ought to say not that better people suffer more but rather that worse people, who need more suffering, suffer more. But an analogy with chemotherapy is helpful here. Sometimes the most effective kinds of chemotherapy can’t be used on those who need it most because their systems are too weak to bear the treatments, and so the strongest kinds of treatment tend to be reserved for those who aren’t too old or too advanced in the disease or too riddled with secondary complications—in other words, for those who are (apart from their cancer) strong and healthy. So, on Aquinas’s view, Job has more suffering than ordinary people not because he is morally worse than ordinary, as the comforters assume, but just because he is better. Because he is a better soldier in the war against his own evil and a better servant of God’s, God can give him more to bear here. Even the dreadful suffering Job experiences at the death of his good and virtuous children becomes transformed on this account from the unbearable awfulness of total loss to the bitter but temporary agony of separation, since in being united to God in love in heaven, a person is also united with others. The ultimate good of union with God, like any great good, is by nature shareable. Aquinas recognizes that his position will seem counter-intuitive or worse to some people, and he takes this difference in attitude to stem from a more general difference in philosophical and theological worldview. “If there were no resurrection of the dead,” he says, “people wouldn’t think it was a power and a glory to abandon all that can give pleasure and to bear the pains of death and dishonor; instead they would think it was stupid.”18 His theodicy seems as reasonable as it does to him because it is set in a whole web of Christian beliefs not only about God’s existence and attributes but also about the nature of human beings in this world, the existence and nature of the afterlife, and the means by which human beings are brought to happiness in it.

4.3 Saadia’s Account of Job Saadia accepts the same basic account of the story of Job as Aquinas and most other readers of the story do. He agrees that Job is morally innocent, that he suffers horribly, and that his suffering is in no way deserved, contrary to the position of the ill-named Comforters. Like Aquinas, he also supposes that God is omnipotent, omniscient, and perfectly good. Furthermore, by ‘goodness’ here he means goodness by some objective standard, and not just goodness constituted by whatever God wills. As one scholar commenting on Saadia’s position puts it: Princes too can be guilty of injustice, and it is not by right of ownership that God is just, as was argued by the Ash’arites, whose slogan was: “There is no injustice to a chattle.”

18 Super

I ad Corinthios, chap. 15, lec. 2.

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On the contrary, Saadiah argues . . . that one cannot ascribe to God actions which would be unjust by human standards. Saadiah rejects the theistic subjectivism of his Ash’arite contemporaries [ . . . ]19

Given these views of Saadia’s, it is obvious that from his point of view the story of Job, which stands as representative for all the suffering of unwilling innocents, requires some theodicy. Like Aquinas, Saadia has strenuous requirements for theodicy. He says, “God’s creating suffering, sickness, and injury in the world is also an act of beneficence and in the interest of humanity [ . . . ] What is true of sufferings felt without affecting the body is true also of those that do affect it—the Creator does not so afflict His servant except in his [the servant’s] own interest and for his own good.”20 So Saadia, like Aquinas, thinks that the benefits which justify God in permitting suffering must go primarily to the sufferer. There are three ways in which this can occur, on Saadia’s view. First, there is the sort of suffering which constitutes training and characterbuilding. Saadia says, Although these may be painful for human beings, hard, wearying, and troubling of mind, all this is for our own good. Of this the prophet says, the chastening of the Lord, my son, despise not . . . we know from our own experience that one who is wise does burden himself with late hours and hard work, reading books, taxing his mental powers and discernment, to understand. But this is no injustice and not wrong in the least on his part.21

Here the idea seems to be that just as it is not wrong for the scholar to afflict himself for the sake of excellence in scholarship, so it is not wrong for God to afflict a person for the sake of the excellence of that person’s character. Second, there is “purgation and punishment”. If the first case can be thought of as making a basically good person better, this second case can be thought of as keeping a person who has done something bad from getting worse and rectifying his accounts so that he is not in moral debt any more. In explaining the nature of purgative punishments, Saadia says, if a servant [of God’s] does commit an offense deserving punishment, part of the goodness of the All-Merciful . . . is in His causing some form of suffering to clear the transgressor’s guilt wholly or in part. In such a case that suffering is called purgative: although it is a punishment, its object is that of grace, for it deters the transgressor from repeating his offenses and purifies him of those already committed.22

Those familiar with John Hick’s theodicy might suppose that these two categories of Saadia’s would be enough to construct a theodicy. The second case could account for all those sufferers who aren’t innocent, and the first case could then

19 Goodman

(1988), pp. 103–104. (1988), pp. 124–125. 21 Goodman (1988), p. 125. 22 Goodman (1988), p. 125. 20 Goodman

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be made to accommodate the suffering of unwilling innocents such as Job. But, on Saadia’s view, to explain such suffering as Job’s we need yet a third category: The third case is that of trial and testing. An upright servant, whose Lord knows that he will bear sufferings loosed upon him and hold steadfast in his uprightness, is subjected to certain sufferings, so that when he steadfastly bears them, his Lord may reward and bless him. This too is a kind of bounty and beneficence, for it brings the servant to everlasting blessedness.23

That is why, Saadia maintains, one kind of goodness that God shows his creatures is recompense for tribulations with which He has afflicted us and which we have borne with fortitude [ . . . ]. For the tribulations are not on account of some past sin on the servant’s part. They are spontaneously initiated by God. Their purpose, therefore, lies in the future [ . . . ] The Allwise knows that when we are visited with sufferings they are abhorrent to our natures and harrowing to us in our struggle to surmount them. So He records all to our account in His books. If we were to read these ledgers, we would find all we have suffered made good, and we would be confirmed in our acceptance of His decree.24

According to Saadia, then, God permits suffering to come to an unwilling innocent, but apparently just for the sake of rewarding him in the afterlife for his having endured such suffering. The sufferings of Job, in Saadia’s view, fall into this third sort of case, and what Job calls into question with his complaints is only God’s recompense for the sufferings of the righteous. Supporting Elihu’s side in the dispute among Job and his comforters, Saadia says, “Elihu denies [ . . . ] Job’s claim that God has caused him to suffer [ . . . ] without affording him any recompense in the hereafter.”25 It is not entirely clear how Saadia’s first and third cases differ from each other, since one might suppose that in the third case the righteous person’s character was in fact strengthened by his enduring afflictions with fortitude and in the first case the sufferer whose character is being strengthened will ultimately reap some reward in the afterlife for having developed a better character through suffering. Saadia himself sometimes seems to conflate the first and third cases. So, for example, he says, when sufferings and calamities befall us, [ . . . ] they must be of one of two classes: either they occur on account of prior sins of ours, in which case they are to be called punishments [ . . . ] Or they are a trial from the Allwise, which we must bear steadfastly, after which He will reward us.26

When he does make some remarks aimed at distinguishing these two cases, he tends to say things of this sort:

23 Goodman

(1988), pp. 125–126. (1988), p. 127. 25 Goodman (1988), p. 357. 26 Goodman (1988), p. 130. See also p. 332: “Sufferings . . . are of two kinds: either they are on account of some past act and are called punishments, or they are unprovoked and are called trials when God’s servant endures them.” 24 Goodman

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Job held it admissible that the Allwise might cause suffering to His servant despite that servant’s being guilty of no sin. By our account, such sufferings would be called chastisements—unless they were for the sake of future recompense, in which case they would be called trials.27

Here the idea seems to be that the first and third cases are distinguished by the purpose for which God allows the suffering of the righteous. If it is largely for the sake of building character now, we have the first case; but if it is primarily for the sake of a reward later, in the afterlife, we have the third case. Sometimes it seems as if Saadia postulates the third case just because it is needed to cover the sorts of suffering of unwilling innocents which would be difficult to construe as character-building. So, for example, in describing the various species of suffering that are included in the third case, Saadia says, there are three kinds of trial: by way of property, by way of body, and by way of soul. Two of these [ . . . ] are called tests. But the third, by way of the soul, is not called a test, because when suffered to the full it results in death. Rather it is called immolation [ . . . ] This too God may inflict upon the righteous without any prior offense but with subsequent recompense—as He did with the infants at the flood, the infants of the seven (Canaanite) nations, Job’s children, and others.28

Punishing infants for sin seems morally absurd, and it is not much more plausible to suppose that infants who die in their sufferings are allowed to suffer for the sake of developing their character. So Saadia is right to suppose that the suffering of infants would be hard to assimilate to either of his first two cases. That he needs some additional explanation of the suffering of unwilling innocents is therefore clear. It is not nearly as clear, however, that his third case provides the needed explanation.

4.4 Objections to Saadia’s Theory of Suffering as Trial Although Saadia’s third case covers instances of innocent suffering which his position would otherwise be hard-pressed to account for, there is nonetheless something morally distressing about this case, as it is commonly understood. If one of the purposes of theodicy is to show that the evil in the world is compatible with the existence of a God who is good, it isn’t at all clear that Saadia’s theodicy succeeds in this regard. Maimonides, for example, rejects it with vehemence. He reads Saadia’s interpretation of Job into the views of one of Job’s comforters, Bildad the Shuhite.29

27 Goodman

(1988), p. 128. (1988), pp. 161–162. Why Job’s children are in this list is not easy to say, since they were adults, or at least old enough to be capable of serious sin, at the time of their death. 29 Although Maimonides doesn’t explicitly associate this view with Saadia, it is clear that he is aware of Saadia’s philosophical and theological positions, and he is generally taken to be opposing Saadia here and also elsewhere. See, for example, Leaman (1995), p. 77; Sirat (1985), pp. 175 and 211. 28 Goodman

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According to Maimonides, Bildad’s line to Job comes to this: If you are innocent and have not sinned, the reason for these great events [Job’s sufferings] is to make great your reward. You will receive the finest of compensations. All this is good for you, so that the good that you will obtain [will] in the end be increased.30

Maimonides thinks that this view is common, vulgar, stupid, and impious. He says, What is generally accepted among people regarding the subject of trial is this: God sends down calamities upon an individual without their having been preceded by a sin, in order that his reward be increased. However, this principle is not at all mentioned in the Torah in an explicit text [ . . . ]. The principle of the Law that runs counter to this opinion, is that contained in His dictum, may He be exalted: A God of faithfulness and without iniquity. Nor do all the Sages profess this opinion of the multitude, for they say sometimes: There is no death without sin and no sufferings without transgression. And this [the quoted view of the Sages] is the opinion that ought to be believed by every adherent of the Law who is endowed with intellect, for he should not ascribe injustice to God, may He be exalted above this, so that he believes that Zayd is innocent of sin and is perfect and that he does not deserve what befell him.31

It should perhaps be said that Maimonides’s own account of suffering, if it is represented accurately in this passage, seems considerably less palatable than the view of Saadia’s which he is attacking. It is not always easy to know what Maimonides’s own opinions are, however, given the commitment to caution and secrecy evinced in the Guide, and perhaps Maimonides here means to be presenting only religious views suitable for the unlearned. But there are certainly passages in which Maimonides appears to be arguing explicitly for the view that every sufferer deserves exactly what he suffers. So, for example, he says, It is likewise one of the fundamental principles of the Law of Moses our Master that [1] it is in no way possible that He, may He be exalted, should be unjust, and that [2] all the calamities that befall men and the good things that come to men, be it a single individual or a group, are all of them determined according to the deserts of the men concerned through equitable judgment in which there is no injustice whatever. Thus if some individual were wounded in the hand by a thorn, which he would take out immediately, this would be a punishment for him, and if he received the slightest pleasure, this would be a reward for him—all this being according to his deserts. Thus He, may He be exalted, says: For all His ways are judgment [ . . . ]32

And as a palliative for what seems to be the manifest mistakenness of his position, Maimonides adds that human judgments of the moral state of others are often wrong: “But we are ignorant of the various modes of deserts.”33

30 Maimonides

(1963; reprinted 1974), II.23, p. 493. III.24, pp. 497–498. 32 Ibid., III.17, p. 469. 33 Ibid., III.17, p. 469. David Shatz has pointed out to me the need for caution with regard to Maimonides’s position on deserts. It is complicated by his unusual account of providence, which ties divine providential protection to an individual’s intellectual development. 31 Ibid.,

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Even those commentators who think Maimonides’s own account of suffering needs some detailed, explanatory apologetic are inclined to accept his evaluation of Saadia’s position. So, for example, Oliver Leaman says, It is not just that Saaya represents God as rather like a judge, a very human judge, but also that as a judge he seems to be particularly unpleasant. He makes it all right in the end, but seems to torment people for no other reason than to test them, or for no reason at all, with the ultimate promise that compensation will be available . . . . Is this how we should view the deity?34

What gives rise to the sort of complaint made by both Maimonides and Leaman is an important difference between Aquinas’s theodicy and Saadia’s, if Leaman and Maimonides understand Saadia correctly here. Aquinas and Saadia clearly share certain theological as well as ethical views. Both of them assume, for example, that God knows and cares about individual human beings, unlike the fourteenth-century Jewish philosopher Gersonides, for example, who seems to think that God’s providence doesn’t extend to all individual human beings.35 Unlike the Ash’arites Saadia opposed, who apparently thought that God’s will constitutes morality, both Aquinas and Saadia also assume that God’s goodness isn’t simply constituted by his will; it isn’t the case, in their view, that whatever God wills is good just because he wills it. Finally, both of them suppose that God is justified in allowing some unwilling innocent to suffer only in case the benefit that justifies the suffering goes primarily to the sufferer. In trying to explain how it is that God is justified in allowing Job to suffer, both of them look only for benefits that accrue solely or primarily to Job; neither of them entertains the possibility that God might be just in allowing Job to suffer because of benefits which come to, say, Elihu or others who might learn from what happens to Job. But as Saadia is understood by Maimonides and contemporary scholars such as Leaman, Saadia and Aquinas differ on one important issue. On Aquinas’s view, suffering is medicinal for the cancer of the will innate in all post-Fall human beings. Unless that cancer is cured, human beings cannot be united to God in the afterlife, and not being ultimately united to God is the worst evil that can befall a human being. Undeserved suffering, then, is allowed by God in order to help ward off a greater evil. On Saadia’s view, however, the situation is different, at least on the

34 Leaman

(1995), p. 78. for example, Gershom (1987), Book Four, chapter iv, p. 174: “it is evident that individual providence must operate in some people but not in others [ . . . ] It is evident that what is more noble and closer to the perfection of the Agent Intellect receives the divine providence to a greater degree and is given by God the proper means for its preservation. [ . . . ] Since man exhibits different levels of proximity to and remoteness from the Agent Intellect by virtue of his individual character, those that are more strongly attached to it receive divine providence individually. And since some men never go beyond the disposition with which they are endowed as members of the human species [ . . . ] such people are obviously not within the scope of divine providence except in a general way as members of the human species, for they have no individual [perfections] that warrant [individual] providence. Accordingly, divine providence operates individually in some men [ . . . ] and in others it does not appear at all.” 35 See,

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interpretation being considered here. For Saadia, undeserved suffering is allowed by God for the sake of a greater good for the sufferer—the compensation God will give to the innocent sufferer in the afterlife—and not to ward off a greater evil. If Maimonides and Leaman are right about the nature of Saadia’s position, then, there are two problems with Saadia’s theodicy. First, it isn’t clear that, on Saadia’s view, the benefit and the suffering are connected in the right sort of way. Aquinas supposes that suffering can have an effect on the will and that without the sort of change in the will which suffering is designed to help bring about, a human being will not be in the right state to be united to God in heaven. Furthermore, even an omnipotent God cannot produce such a change in the will directly, by an exercise of his power, because human wills are free. So suffering may be the best available means in the circumstances, even for omnipotent God, to keep human beings from the state in which they can’t be united to him in the afterlife. But there doesn’t seem to be any such essential connection between suffering and the benefit it yields on Saadia’s view. If God compensates a person for undeserved suffering by giving that person some gift in the afterlife, why couldn’t God simply choose to give that person such a gift even without the suffering? And if God could give the benefit without the suffering, is it morally right of him to allow the suffering just for the sake of the benefit? If nothing about the sufferer’s circumstances or choice means that the benefit can’t come to him without the suffering, isn’t the suffering entirely gratuitous? And is God good if he allows entirely gratuitous suffering? Secondly, even if there were the requisite sort of connection between the suffering and the benefit, it isn’t clear that it is, in general, morally right to bring it about (or allow it to occur when one could readily prevent it) that an innocent person suffers unwillingly for the sake of some greater good for that person. It is, of course, not always easy to make a distinction between acting to produce a greater good and acting to ward off a greater evil.36 But we do often make a rough and intuitive distinction of this sort, and we are generally much more willing to conscience suffering induced or allowed to ward off a greater evil than suffering induced or allowed just for the sake of a greater good. Consider, for example, this case. For the sake of argument, suppose that a person who is deprived of all sensory stimulation for a long period in childhood and subjected to severe bodily hardships will in after years always react with great pleasure to things other people take for granted sunlight, fresh air, even minimal food, the presence of other human beings, and so on. And suppose also that on some reasonably plausible scale of hedonistic value, the pleasure produced as a result of a period of such hardship and deprivation is enormously greater than the pain associated with it. Even if we granted these assumptions, we certainly would not suppose that the desire to produce such subsequent pleasure for a child

36 There

is also some difference in our moral intuitions between cases of inflicting suffering and cases of allowing suffering which arises from elsewhere. In what follows, for the sake of brevity, I will give examples just concerning inflicting suffering.

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rendered good a person who kept a 3-year old isolated and half starved in a dark and airless closet for a year or two. And our negative evaluation of the child’s tormenter wouldn’t be lessened by discovering that she had some special rights with regard to the child, that her relationship to the child gave her a special responsibility for his well-being. A mother who did such things would surely lose her child to social services (or so one would hope). On the other hand, a mother who subjected her child to such misery for the sake of warding off from the child some greater evil wouldn’t meet with similar moral disapproval. If it turned out that sensory deprivation and restricted food intake for a period of a year or two kept the child from a lingering and painful death, we would approve of such treatment for the child. In fact, a mother who couldn’t bring herself to consent to such treatment of her child, who preferred her child’s slow and wretched death, would strike us as culpably weak, or worse. So we do make a rough distinction between acquiring a greater good and warding off a greater evil, and it makes a difference to our moral evaluation whether an agent who could prevent suffering of an unwilling innocent allows it to occur for the sake of a greater good for the sufferer or to ward off a greater harm. We are not in general inclined to suppose that it is morally acceptable to allow suffering just for the sake of a greater good for the sufferer. When it is also the case that the greater good can be obtained without the suffering, as it apparently can in Saadia’s case, then we are even less likely to suppose that the benefit justifies the suffering. So the difference between Aquinas’s account of suffering and Saadia’s only highlights the inadequacies of Saadia’s theodicy.

4.5 Saadia’s Theodicy Reconsidered This negative evaluation of Saadia is based, of course, on the assumption that interpreters such as Maimonides and Leaman have understood Saadia’s position correctly.37 And here, I think, there is room for dispute. To see why this is so, it is important to set Saadia’s theodicy within the context of his broader philosophical and theological views. To begin with, he supposes that a human being consists of a body and a soul and that these can be separated. At death, the body rots, but the soul persists. After a certain time, the soul and the body are reunited, and the resurrected individual lives forever. Not only does Saadia hold this belief, but, in his view, so does every Jew. He says, as far as the resurrection of the dead is concerned [ . . . ] it is a matter upon which our nation is in complete agreement. The basis of this conclusion is a premise mentioned previously in the first treatises of this book: namely, that man is the goal of all creation. The reason why he has been distinguished above all other creatures is that he might serve God, and

37 See

also, for example, Husik (1944), pp. 42–43.

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E. Stump the reward for this service is life eternal in the world of recompense. Prior to this event, whenever He sees fit to do so, God separates man’s spirit from his body until the time when the number of souls meant to be created has been fulfilled, whereupon God brings about the union of all bodies and souls again [ . . . ] We [ . . . ] do not know of any Jew who would disagree with this belief.38

Additionally, in the afterlife, human beings will be divided into two groups, those receiving reward and those receiving punishment. Rewards and punishments will be meted out to resurrected individuals, and they will be perpetual. Nonetheless, these perpetual punishments or rewards will not all be maximal or infinite in quality. Rather, they will be graded and proportional to a person’s good or bad deeds: even though the reward and the punishment [ . . . ] will be everlasting, their extent will vary according to the act. Thus, for example, the nature of a person’s reward will be dependent upon whether he presents one or ten or one hundred or one thousand good deeds, except that it will be eternal in duration [ . . . ]. Likewise will the extent of a person’s punishment vary according to whether he presents one or ten or a hundred or a thousand evil deeds, except that, whatever the intensity of the punishment may be, it will be everlasting.39

Furthermore, there is no change from one state to another in the afterlife; an individual remains forever in whatever state he was in when he entered the afterlife.40 Saadia recognizes, of course, that some people will wonder whether a good God shouldn’t simply have created people perfectly, unalterably good from the beginning, so that they would go directly from creation into the state of happiness in the afterlife. Since some people will end up in that condition, unable to do evil any more, why not just create them in that state in the first place and thus avoid all the suffering in this life and the next? Saadia’s response is twofold. First, he gives a familiar answer to the question, namely, that God can’t do so because doing so would require determining human wills, and human wills can’t be determined if (as Saadia supposes) they are free. Furthermore, he thinks that free will is of great value, so that it doesn’t occur to him to suppose human beings would be better off if God took away freedom in the interests of removing suffering; on the contrary, he explicitly values suffering as an aid to the rectitude of free will.41 Secondly, he argues that God’s not creating human beings in the state of bliss and sinlessness the righteous will have in the afterlife is for the good of human beings for another reason: according to the judgment of reason the person who achieves some good by means of the effort that he has expended for its attainment obtains double the advantage gained by him

38 Rosenblatt

(1948), p. 264. (1948), pp. 347–348. 40 Rosenblatt (1948), p. 247. 41 Rosenblatt (1948), pp. 181 and 184–185. For some explanation of how such an answer can be consistent with the position that those in heaven never do evil, see my “Intellect, Will, and the Principle of. Alternate Possibilities”, in Beaty (1990), pp. 234–285; reprinted in Fischer and Ravizza (1993), pp. 237–262. 39 Rosenblatt

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who achieves this good without any effort but merely as a result of the kindness shown him by God.42

The afterlife bestowed as a result of human choices and human effort will be more precious, Saadia thinks, than it would be if (per impossibile) people were simply born into it as a result of God’s decree. Next, although he thinks that the afterlife admits of gradations of reward and punishment, Saadia recognizes only two groups in the afterlife, those who are unendingly rewarded and those who are unendingly punished. He has a less stem notion of the requirements for being in the group of the righteous, however. The completely righteous person is someone who has always fulfilled all the commandments. About this sort of person, Saadia says, even though in the opinion of men the probability of the existence of such a person who is blameless in every respect appears to be extremely remote, I yet consider it possible. For were it not so, the All-Wise would not have prescribed such a goal.43

Saadia, in other words, feels he has to argue for the possibility of there being a completely righteous person, on the theoretical grounds that God’s commandments would otherwise be futile; but he seems to join what he presents as the general consensus in supposing that the chances of there actually being such a person is “extremely remote.” The group of the righteous in the afterlife is therefore not populated, largely or entirely, by people from this category. Instead, the righteous in the afterlife will consist of sinners who have repented their sins. By repentance, Saadia explains he means “(a) the renunciation of sin, (b) remorse, (c) the quest of forgiveness, and (d) the assumption of the obligation not to relapse into sin.”44 In Saadia’s view, most Jews fall into this category, or are very close to it: Now I have no fears, so far as the majority of our people are concerned, in regard to their being remiss in their fulfillment of any of the conditions of repentance except this fourth category—I mean that of lapsing back into sin. For I believe that at the time when they fast and pray, they sincerely mean to abandon their sinful way and experience remorse and seek God’s pardon.45

On Saadia’s view, however, real repentance is not canceled by subsequent sin: if the resolve on the part of a servant of God not to lapse into sin again is sincere, his repentance is accepted, so that if, as a result of temptation, he falls once more, his repentance is not thereby forfeited. What happens is rather that the iniquities he committed before his repentance are canceled, only those committed by him thereafter being charged against him. The same would apply even if this were to occur several times; namely, that he repent and lapse back into sin. Only the wrongs perpetrated by him after his repentance would count against him, that is, provided he has been sincere each time in his resolve not to relapse.46

42 Rosenblatt

(1948), p. 138. (1948), pp. 217–218. 44 Rosenblatt (1948), p. 220. 45 Rosenblatt (1948), pp. 221–222. 46 Rosenblatt (1948), p. 223. 43 Rosenblatt

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So the group of the righteous who are unendingly rewarded in the afterlife will consist largely of ordinary people who have repented their sins. What happens, however, if someone has repented most of his sins but not all, perhaps because he has forgotten some or perhaps because some of them are so rooted in his character that repentance for them isn’t psychologically possible for him? Saadia supposes that no one can be a member of both the rewarded and the punished in the afterlife. Will the person who hasn’t repented all his sins forgo his reward in the afterlife? Saadia’s answer to this question is that individuals of this sort receive punishment for such sins in this life precisely so that those unrepented sins don’t imperil their otherworldly reward. By the same token, unrepentant sinners who do some good receive the reward of their good acts in this life, so that their otherworldly punishment doesn’t keep them from reaping whatever little reward is due them for their few good acts.47 Saadia’s position here will seem to some philosophers to raise serious questions about his conception of God’s justice, since one might wonder whether it is fair of God to assign perpetual reward or punishment for temporally limited actions. But I bring up this feature of Saadia’s position here only to point out how much more complicated his views about suffering are than they initially appeared to be. When one of the righteous suffers in this world in order that his few unrepented sins might not imperil his salvation, in which category of suffering does his pain fall? On the face of it, his pain falls into the second of Saadia’s three categories, suffering as punishment for sin. But, on the other hand, the point of the suffering is not so much backward-looking—as we might have expected in the case of suffering in Saadia’s second category—as forward-looking: the point of the suffering is that God can then with justice give the unrepentant sinner eternal reward. And so this sort of suffering shares some features of the third category of suffering, namely, that the point of God’s allowing the suffering is the sufferer’s reward in the afterlife. On the other hand, in explaining why it is appropriate for God to punish in this life people who are generally righteous, Saadia gives an explanation which makes it look as if the suffering belongs in the first category, the category of sufferings allowed in order to build the character of the sufferer. So, for example, he says, Should someone ask, however, on what ground they [the righteous person’s unrepented evils] are pardoned, seeing that no repentance of them has taken place, we would answer: [ . . . ] when a person follows such a [generally righteous] course and most of his actions are good, retribution for these relatively minor misdeeds is exacted from him in this world, so that he departs from it cleared of all blemish [ . . . ]48

So even the relatively simple second category of suffering, punishment for sin, is in fact much more complicated in Saadia’s developed philosophical theology than at first appears. Part of the reason for the confusion is, no doubt, that we tend to think of punishment simply as retributive, but, for Saadia, punishment at least in 47 Rosenblatt 48 Rosenblatt

(1948), pp. 210–211. (1948), p. 351.

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this life is medicinal and therefore, like all successful medicine, carries with it both improvement and the rewards that improvement brings with it. That is why his second category of suffering contains some elements of the other categories as well, namely, the betterment of the sinner and the consequent reward of the morally healthy righteous. In explaining his third category of suffering, the trial of a righteous person, Saadia makes a point which at first glance looks only lamentably ridiculous. He says, the sufferings to which the virtuous are subjected in this world fall into two categories. One of these constitutes the penalties for slight [unrepented] failings, as I have explained previously. The second consists of incipient trials with which God tests them, when He knows that they are able to endure them, only in order to compensate them for these trials later on with good.49

There is something apparently absurd and unjust about God’s testing an individual only when he is sure of the outcome of the test and only for the sake of compensating the wretched and innocent victim of the test afterwards. Maimonides’s diatribe against this view seems entirely warranted. But if we look more closely, Saadia’s third category of suffering, like his second category, becomes more complicated and more interesting. The group of the righteous, according to Saadia, consists largely in those who are repentant for their sins. But sins leave a stain on the soul, on his views.50 So, he says, “obedience [to God’s commandments] increases the luminosity of the soul’s substance, whereas sin renders its substance turbid and black.”51 I’m not sure to what extent Saadia means this point literally or metaphorically, but his general idea is not hard to grasp. Any instance of moral wrongdoing carries with it two problems for the wrongdoer, one as regards the future and one as regards the past. Doing a wrong act has some influence on character; it increases the likelihood that one will do such a wrong act again in the future. The solution for this future-oriented problem lies in repentance, which to some extent unravels the twist in the character produced by doing the wrong act. But there remains a problem for the wrongdoer as regards the past: he is still a person who has done such a wrong act. If Goebbels, for example, truly repented and entirely regretted the evil he did, there would still be a problem because of what occurred in the past. What he has already done has turned him into something from which other people want to shy away; and this remains the case even if people were to know that Goebbels regretted his past evils. Because of the evil he has perpetrated, Goebbels, even in a repentant state, is turbid, as Saadia says—stained or polluted, in Saadia’s idiom; in our health-oriented idiom, psychologically sick. Someone might suppose that repentance itself is enough to remove what Saadia thinks of as the stain on the soul, but this seems to me a mistaken view. Consider the effects of wrongdoing on the body, rather than the mind. Consider, for example, 49 Rosenblatt

(1948), p. 213. (1948), pp. 205–207. 51 Rosenblatt (1948), p. 246. 50 Rosenblatt

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someone who for many years has indulged in excessive eating and has also avoided exercise. On the day on which such a person truly repents of that behavior and enters on a new life of exercise and right eating, his repentance by itself won’t be enough to give him a healthy and athletic body. The effects of his previous acts will obviously remain, and it will take a long time of dieting and exercising before those effects are no longer evident. Why suppose things are any better as regards the mind? Why suppose that all the bad mental effects of previous wrongdoing can be wiped away in a moment with an act of repentance? It seems clear, for example, that even earnest and sincere repentance can co-exist with many of the old attitudes and habits laid down by previous wrongdoing. An act of repentance by itself won’t be enough to undo those old dispositions or to make a person internally integrated and unified around the good newly willed. Repentance is thus psychologically compatible with many of the dispositions laid down by the previous wrongdoing.52 A repentant agent in such a condition will, of course, be internally divided or even irrational, but these are not uncommon human conditions. Whole-heartedness and rationality may be as hard to come by, for a newly repentant person, as a healthy and athletic body is for a person with a newly refined lifestyle.53 I do not want to claim that dispositions and attitudes left by wrongdoing exhaust all that Saadia has in mind with his notion of the stain on the soul, but they are enough, I think, to show that the backwardslooking problem of moral wrongdoing isn’t likely to be solved by repentance alone. The solution to this backwards-looking problem of moral wrongdoing, in Saadia’s view, is suffering on the part of the wrongdoer. This way of thinking about Saadia’s third category of suffering also helps explain his distinction between this category and the first one, where the suffering is for the sake of character-building. Saadia thinks of the righteous as comprised mainly of those who are struggling with their own moral wrong-doing: sinning, repenting, and then sinning again. Building character, as suffering in the first category is said to do, will be a matter of strengthening a person in this struggle, so that the suffering brings him to repentance or confirms him in his repentant resolve not to sin again in that way. Suffering that builds character thus helps to overcome the future-oriented problem of a person’s wrong-doing. But the backwards-looking problem remains. When suffering helps solve this problem, the suffering serves not so much to build character for the future as to purge the polluted state of soul the sinner has already acquired. Using suffering as an antidote to pollution takes God’s omniscient providence, since only God can see the heart and can understand what pollution is there and how suffering can cure it. So Saadia says, Now He that subjects the soul to, its trials is none other than the Master of the universe, who is, of course, acquainted with all its doings. This testing of the soul has been compared to the assaying by means of fire of [lumps of metal] that have been referred to as gold or silver.

52 A

good example of a person in such a condition is Albert Speer; see Sereny (1995). am grateful to Christopher Hughes, whose persistent questions pushed me to think through this point.

53 I

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It is thereby that the true nature of their composition is clearly established. For the original gold and silver remain, while the alloys that have been mingled with them are partly burned and partly take flight . . . The pure, clear souls that have been refined are thereupon exalted and ennobled.54

That is why God allows suffering as test only for those people he knows can endure it. If those tested lacked strength for the test, then in the process of testing they would succumb to further sin, and the test would make them worse, rather than purging the stains from the soul. Like Aquinas, then, Saadia thinks that those like Job who experience perplexing, agonizing suffering which they do not deserve do so just because they are better servants of God and more able to sustain the rigors of therapy than God’s weaker, smaller servants. For the same reason, Saadia says, if the pain to which the servant of God is subjected constitutes punishment and he asks his Master to enlighten him thereon [and explain to him why he is suffering], it is a rule with Him to do so [ . . . ]. On the other hand, if the pain to which the servant of God is subjected serves as a form of trial and he asks his Master to inform him why He has brought this trial upon him, it is a rule with Him not to inform him when Job asked: Make me know wherefore Thou contendest with me [ . . . ], no explanation was offered to him.55

If it was clear to the sufferer that the suffering was for cleansing of the soul and its subsequent rewards, then, in Saadia’s view, the suffering would lose some of its therapeutic value, since then one might endure the suffering simply for the sake of the reward. So, examined more closely, Saadia’s third category of suffering looks very different from the way it is presented by some of its interpreters. In the first place, the righteous who are suffering are, in general, not those who are entirely without moral wrongdoing, but rather are those who have repented their wrongdoings, or most of them. Furthermore, the suffering isn’t gratuitous or only accidentally related to the benefit. On Saadia’s view, the suffering is in some way instrumental in bringing about the benefit. The stain on the soul brought about by wrongdoing is removed by suffering, and nothing in Saadia’s account suggests he supposes that the stain could be removed just as well in some other way, for example, by omnipotent God’s acting directly to remove it. Finally, the feature that initially seemed to render Saadia’s theodicy philosophically more problematic than Aquinas’s is now called into question. It is no longer so clear that, for Saadia, the benefit which justifies God in allowing the suffering of unwilling innocents is a greater good for the sufferer, as distinct from the warding off of a greater evil. On Saadia’s account, the perpetual rewards in the afterlife are distributed in accordance with an individual’s state of soul. A righteous individual who enters the afterlife with certain stains on the soul will forever lose part of the reward he might have had if he had purged those stains. For all time to come, he will be less or have less than he might otherwise have been or had.

54 Rosenblatt 55 Rosenblatt

(1948), pp. 246–247. (1948), pp. 213–214.

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Perhaps, because the afterlife is supposed to be a state of bliss, the righteous person in this diminished state won’t mind it, as the souls in Dante’s Paradiso explain to him that they don’t mind not being in the top rank of those in heaven. A loss can be a loss, however, even if one doesn’t mind it. The cancer patient not accepted for bone marrow transplant, which holds out the only hope of a cure, might well not mind, considering the difficulty of the therapy and the suffering it occasions. But he would have lost something anyway. All the more so, then, if what is at issue is not some temporary state of physical health but a permanent state of spiritual wellbeing. On this way of interpreting Saadia’s account, then, the benefit which justifies the suffering in the third category, of tests and trials, is not the acquisition of a greater good for the sufferer but the warding off of a greater evil. One might suppose that there is one group of human beings whose sufferings Saadia assigns to the third category but for whom my revised interpretation of this category is bound to fail. This is the group of those children who die in their suffering. Saadia says, for example, we are confronted by the fact that God, the just, ordered the killing of the young children of the Midianites and the extermination of the young children of the generation of the deluge. We note also how He continually causes pain and even death to little babes. Logical necessity, therefore, demands that there exist after death a state in which they would obtain compensation for the pain suffered [ . . . ]56

Here, one would suppose, Saadia must adopt the line attributed to him by Maimonides. But, in fact, when Saadia elaborates on the suffering of children, he makes remarks of this sort: I will go still further and say that it is even possible for a completely guiltless individual to be subjected to trials in order to be compensated for them afterwards, for I find that children are made to suffer pain, and I have no doubt about their eventual compensation for these sufferings. The sorrows brought upon them by the All-Wise might, therefore, be compared to the discipline that their father might administer to them in the form of flogging or detention in order to keep them from harm, or to the repulsive, bitter medicines that he might make them drink in order to put an end to their illness.57

Here the two examples given to illustrate God’s purpose in allowing the suffering of children are both examples in which a father causes suffering to his children in order to ward off greater evil—“harm”, in the first example, and continuing illness, in the second. It is far from clear that Saadia’s position here is consistent. The children in the example are apparently already in trouble, since medicine is administered to those who are sick and “flogging” to those who are being punished for some wrong, and yet the point of these examples is to provide some explanation of the way in which God deals with children whom Saadia himself takes to be perfectly innocent. But what is important for my purposes here is just the fact that when Saadia considers the suffering of children in any detail, he supposes that allowing

56 Rosenblatt 57 Rosenblatt

(1948), p. 330. (1948), p. 214.

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their suffering is justified in virtue of the fact that their suffering prevents a greater harm to them. Looked at in this way, then, Saadia’s theodicy appears very different from the way it has been taken by some of its interpreters. In fact, although Saadia and Aquinas are separated by a great gulf of religion, time, and culture, and although Saadia certainly repudiates Christianity with vehemence approaching scorn,58 Saadia’s account of the suffering of unwilling innocents looks very close to that of Aquinas, for all their other differences in theological doctrine.59

4.6 Saadia’s Judaism and Leaman’s Objection I suggested near the outset that approaching Saadia’s theodicy with that of Aquinas in mind would help us see what is paradigmatically Jewish about Saadia’s, but my revised interpretation seems to imply that there is nothing of that sort to see. This is a mistaken impression, however. The main difference between Aquinas and Saadia comes not so much in the way they justify God’s allowing suffering as in the nature of the suffering they consider. Aquinas thinks largely or exclusively of the sufferings of individuals. Saadia is concerned as well with a higher level of organization; he focuses also on communal suffering, the afflictions and tribulations of a whole people. Everyone knows, Saadia says, that when there is some disaster that overtakes a whole people, the suffering of that disaster can plausibly be construed as punishment

58 In

discussing the Christian view that the Mosaic law was divinely abrogated by the advent of Christianity, Saadia considers the claim made by some Christian of Saadia’s acquaintance that the miracles associated with the spread of Christianity confirm its divine origin and its claim to be the successor to the Mosaic law. In response, Saadia says, “our reply to him [Saadia’s Christian acquaintance] should be the same as that of all of us would be to anyone who would show us miracles and marvels for the purpose of making us give up such rational convictions as that the truth is good and lying reprehensible and the like. [After hearing this reply, the Christian was] . . . compelled to take refuge in the theory that the disapproval of lying and the approval of the truth were not prompted by reason but were the result of the commandments and the prohibitions of Scripture, and the same was true for the rejection of murder, adultery, and stealing. When he had come to that, however, I felt that I needed no longer concern myself with him and that I had my fill of discussion with him.” (Rosenblatt 1948, p. 164) 59 It is, of course, always possible that Saadia meant to be espousing only the simple position Leaman and others have attributed to him and that in finding evidence of a more complicated and palatable position in his work I’ve shown only that he was inconsistent and confused. It would take more historical scholarship in ninth-century philosophical thought than I can muster to sort out with any confidence what exactly Saadia himself meant his position on the problem of evil to be. But if, as Maimonides seems to think, the simple position generally attributed to Saadia is both stupid and impious, then the principle of charity might also be invoked here in support of my interpretation of him.

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only for some of those involved; for the others, it is a trial, that is, suffering in the third category: This is, as it is well known, a rule that applies to every universal catastrophe occurring at different times, such as famine, war, and pestilence. These serve as punishment for some and as a trial for others.60

This is true also of the sufferings of the Jews, according to Saadia: God is just, doing no injustice, and He has already subjected this nation to a great and longprotracted trial, which undoubtedly serves partly as punishment and partly as a test for us [ . . . ] [such operations] cannot proceed endlessly. Once, then, the end has been reached, there must needs be a cessation of the [this-worldly] punishment of those punishable and compensation for those subjected to trial.

In fact, on Saadia’s view, the Jews have had a larger share of suffering than other peoples. Just as Job, who was one of God’s better, stronger servants and so better able to endure the rigors of the spiritual therapy of suffering, experienced much more suffering than most ordinary individuals, so the Jews have had more to bear in the way of trials than other peoples. For this reason, the resurrection of the dead will begin with the Jews, at the time of the Messianic redemption, and only subsequently will other peoples be resurrected as well. Saadia says, Now let me ask this general question: “Do not we, the congregation of monotheists, acknowledge that the Creator, magnified be His Majesty, will resurrect all the dead in the world to come for the occasion of their retribution?” But what is there in this that would contradict the view that this nation [the Jews] would enjoy an advantage in being granted an additional period during which our dead would be resurrected by God prior to the world to come, that new life of theirs being extended by Him up to the time of the life of the world to come? [ . . . ] why should it not be considered as a mere act of justice whereby whoever has been tried receives compensation in proportion to his trials, since this nation of ours has been subjected by God to great trials, as Scripture says: For Thou, O God, hast tried us; Thou hast refined us? [ . . . ] It is most fitting, therefore, that He should grant to it this additional period prior to the world to come so that it might have an advantage over all those [others] who have conducted themselves well in this world, just as its patience and its trials have exceeded those of the others.61

And in another place he says, God has made us great and liberal promises of the well-being and bliss and greatness and might and glory that He will grant us twofold [ . . . ] for the humiliation and misery that have been our lot [ . . . ] what has befallen us has been likened by Scripture to a brief twinkling of the eye, whereas the compensation God will give us in return therefor has been referred to as His great mercy. For it says: For a small moment have I forsaken thee; but with great compassion will I gather thee.62

60 Rosenblatt

(1948), p. 295. (1948), pp. 284–285. 62 Rosenblatt (1948), p. 292. 61 Rosenblatt

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If Saadia’s views on the suffering of children leave him open to moral vituperation, his views of the communal suffering of his people and its reward render him vulnerable to ridicule, and he knows it. He says, in his own defense, [you will find the Jews] patiently awaiting what God has promised us, not entertaining any doubts concerning it, nor worrying or despairing. On the contrary, our courage and tenacity increase constantly, as is expressed in Scripture: Be strong, and let your heart take courage, all ye that wait for the Lord . . . Now whoever sees us behaving in this fashion may be surprised at us or regard us as fools for the simple reason that he has not experienced what we have nor believed as we have believed. He resembles a person who has never seen how wheat is sown, wherefore, when he sees someone throw it into the cracks of the earth in order to let it grow, he thinks that that individual is a fool. It is at the time of the threshing, when every measure yields twenty or thirty measures, that he first realizes that it is he who has been the fool.63

It is instructive to compare Saadia’s position here with that of someone taking what Saadia considers the real fool’s position. So, for example, Leaman says, It is a shame that the innocent suffer [ . . . ], but it is a fact in the sort of world which we inhabit. The Book of Job represents the terrible things which happen to people as brute facts, things which just happen and which we can often do nothing to prevent. Saadya cannot accept this at face value [ . . . ] he thinks of the events of the world falling under an objective standard of justice which must regulate the balance between innocent pains and pleasures. If the innocent do suffer, then they must eventually be compensated for their suffering. If they are not thus compensated, then the situation is unjust. Of course, he has great difficulty fitting such a theory of justice onto the Book of Job, since it is precisely the message of the Book that that theory of justice is vacuous. There is no evidence of such justice in this world, and little reason to hope for it in another life.64

According to Leaman, our evidence and our reason are against Saadia’s position, that the world is ruled by a just God. Leaman begins with a fact that Saadia also grants, namely, that in this world the innocent suffer. For Leaman, to accept this fact at face value is to reject the claim that the suffering of the innocent is somehow justified or rectified. So Leaman begins with the fact of innocent suffering in the world, adds the belief that there is no morally sufficient reason which justifies such suffering, and concludes that the world is not ruled justly or by a just God. By contrast, Saadia’s position constitutes what William Rowe has labeled a ‘G.E.Moore shift’ on the sort of argument represented by this quotation from Leaman. Saadia turns Leaman’s sort of argument on its head; he begins with a firm belief that the world is ruled justly by a just God and concludes that there must be a morally sufficient reason for God to allow innocent suffering. His theodicy is an attempt to figure out, by reason, revelation, and religious tradition, what sort of benefit might plausibly constitute such a morally sufficient reason. This is no doubt why, contrary to expectations, his theodicy and Aquinas’s are so much alike in their general outlines, despite the vast differences in philosophy and theology which separate the Jewish from the Christian thinker. They each begin

63 Rosenblatt 64 Leaman

(1948), pp. 292–293. (1995), pp. 292–293.

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with a commitment to belief in a God who is omniscient and omnipotent, who is the creator of the world, who knows and cares about all his creatures, and who is good by objective standards, where by ‘good’ we mean at least roughly what we mean when we call any person ‘good’. Since neither Aquinas nor Saadia is willing to sustain belief in such a God by denying (as Maimonides in some passages appears to do) the plain fact of innocent suffering in the world, they conclude that there must be a morally sufficient reason for God to allow such suffering. There are also common moral intuitions, widely shared even across cultures and times, about what would justify any person in allowing innocent suffering when he could readily prevent it: the suffering must be necessary, or the best means available in the circumstances, for producing a benefit that goes primarily or largely to the sufferer; and, generally, the benefit must be a matter of warding off a greater evil, rather than producing a greater good. Because, however much Judaism and Christianity differ otherwise, Saadia and Aquinas share these moral intuitions and the relevant traditional theistic beliefs, their theodicies take shape in the same way.65 On Leaman’s view, of course, both Saadia and Aquinas are grossly mistaken, refusing to take facts “at face value” because of their religious commitment. But why should Leaman suppose that his version of the argument from evil is superior to or more rational than theirs? How does Leaman know that there is no morally sufficient reason for innocent suffering? Given the complexity of the ways in which Saadia thinks suffering can lead to a benefit for the sufferer, one couldn’t just look at the world and see that no such benefit obtained. Questions of this sort and the epistemological issues surrounding them have been the subject of sophisticated philosophical scrutiny in recent years, and it is no part of my aim here to survey that discussion or add to it.66 I bring it up just in order to point out one feature of Saadia’s approach which seems to me significant for the discussion. How would someone come to know or be justified in believing the claim with which Saadia starts, namely, that there is a just God with the standard divine attributes who governs the earth? Answers that have been given include reason, as when a person takes himself to have a proof for God’s existence, and religious experience, as when someone supposes that her experiences have given her the religious analogue of perception of God. Saadia, too, points to reason and religious 65 I

do not mean to suggest that these are the only reasons for the similarity in their views. Their religious and philosophical traditions influence each other, and no doubt Aquinas’s position is partly shaped by influences from Jewish sources. 66 See, for example, Rowe (1979), pp. 335–341, and “The Empirical Argument from Evil”, in Audi and Wainwright (1986). For an opposing view, see Stephen Wykstra, “The Humean Obstacle to Evidential Arguments from Suffering: On Avoiding the Evils of Appearance” and Rowe’s response, “Evil and the Theistic Hypothesis: A Response to Wykstra”, both in The International Journal for the Philosophy of Religion 16 (1984). See also Plantinga (1979), pp. 1–53. The discussion is carried further in the following essays in Howard-Snyder (1996): Alvin Plantinga, “Epistemic Probability and Evil” and “On Being Evidentially Challenged”; Stephen Wykstra, “Rowe’s Noseeum Arguments from Evil”; Peter van Inwagen, “The Problem of Evil, the Problem of Air, and the Problem of Silence”; and William Rowe, “The Evidential Argument from Evil: A Second Look”.

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experience. The person who mocks as stupid and foolish Jewish expectations of everlasting glory garnered through generations of purgative, refining suffering takes this foolish attitude, in Saadia’s view, because he differs from Jews in reason and experience. But it is worth noticing that the reason and experience Saadia points to are communal, not individual. It is the failure of his opponent to experience what ‘we’ have experienced or to believe what ‘we’ have believed that makes ‘the fool’ run the argument from evil in the way he does. The experiences Saadia discusses in this connection are not his own religious experiences but the community’s experience of God’s parting the sea for the Jews in the exodus from Egypt; and the reasoned beliefs on which Saadia relies, here and throughout his whole treatise, are those which, as he says in one phrase or another, “our nation is in complete agreement [about].”67 So Saadia supposes that epistemological excellence or virtue can be vested in a community, as well as in an individual. Justification for a belief can come, at least in part, from the experiences and epistemic commitments of a whole people. In weighing reasons and evidence for one belief or another, on Saadia’s view, it is permissible or even imperative to avail oneself of that communal experience and expertise. This is an attitude with which we are familiar from the practice of science,68 but it hasn’t been the subject of much reflection in philosophy of religion. Saadia’s continual consciousness of belonging to a people whose life over many generations has shaped a common set of religious commitments is, in my view, a salutary corrective to the individualism typically found in contemporary discussions of the problem of evil.69

References Adams, Marilyn. 1986. Redemptive Suffering: A Christian Solution to the Problem of Evil. In Rationality, Religious Belief, and Moral Commitment, ed. Robert Audi and William Wainwright, 248–267. Ithaca, NY: Cornell University Press. Altmann, Alexander. 1960. Three Jewish Philosophers. New York: Meridian Books. Aquinas, Thomas. 1989. The Literal Exposition on Job: A Scriptural Commentary Concerning Providence. Translated by Anthony Damico and Martin Yaffe, The American Academy of Religion. Classics in Religious Studies. Atlanta: Scholars Press. Audi, Robert, and William Wainwright, eds. 1986. Rationality, Religious Belief, and Moral Commitment: New Essays in the Philosophy of Religion. Ithaca, NY: Cornell University Press. Beaty, Michael, ed. 1990. Christian Theism and the Problems of Philosophy, 234–285. Notre Dame, IN: University of Notre Dame Press. Fischer, John Martin, and Mark Ravizza, eds. 1993. Moral Responsibility, 237–262. Ithaca, NY: Cornell University Press.

67 Rosenblatt

(1948), p. 264. an excellent argument to this effect as regards science, see Longino (1993), pp. 257–272. 69 I am grateful to Shalom Carmy, Christopher Hughes, Norman Kretzmann, David Shatz, and David Widerker for very helpful comments on an earlier draft of this paper. 68 For

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Gershom, Levi ben. 1987. The Wars of the Lord. Translated by Seymour Feldman, 174. New York: The Jewish Publication Society. Goodman, Len, ed. 1988. The Book of Theodicy: Translation and Commentary on the Book of Job, cr. New Haven: Yale University Press. Howard-Snyder, Daniel, ed. 1996. The Evidential Argument from Evil. Bloomington, IN: Indiana University Press. Husik, Isaac. 1944. A History of Mediaeval Jewish Philosophy, 42–43. Philadelphia: The Jewish Publication Society of America. Leaman, Oliver. 1995. Evil and Suffering in Jewish Philosophy, Cambridge Studies in Religious Traditions. Vol. 6, 77. Cambridge: Cambridge University Press. Longino, Helen. 1993. Essential Tensions—Phase Two: Feminist, Philosophical, and Social Studies of Science. In A Mind of One’s Own. Feminist Essays on Reason and Objectivity, ed. Louise Anthony and Charlotte Witt, 257–272. Boulder, CO: Westview Press. Maimonides, Moses. 1963. The Guide of the Perplexed. Translated by Shlomo Pines, 493. Chicago: University of Chicago Press; reprinted 1974. Malter, Henry. 1926. Life and Works of Saadia Gaon. New York: Hermon Press; reprinted 1969. Plantinga, Alvin. 1979. The Probabilistic Argument from Evil. Philosophical Studies 35: 1–53. Rosenblatt, Samuel. 1948. The Book of Beliefs and Opinions, tr, 264. New Haven: Yale University Press. Rowe, William. 1979. The Problem of Evil and Some Varieties of Atheism. American Philosophical Quarterly 16: 335–341. Saadia, Gaon. 1948. In The Book of Beliefs and Opinions, ed. Samuel Rosenblatt. New Haven: Yale University Press. Sacks, Oliver. 1985. The Man Who Mistook His Wife for a Hat. New York: Summit Books. Sereny, Gitta. 1995. Albert Speer: His Battle with Truth. New York: Albert Knopf. Sirat, Colette. 1985. A History of Jewish Philosophy in the Middle Ages, 18. Cambridge: Cambridge University Press; reprinted 1990. Stump, Eleonore. 1985a. The Problem of Evil. Faith and Philosophy 2: 392–424. ———. 1985b. Suffering for Redemption: A Reply to Smith. Faith and Philosophy 2: 430–435. ———. 1986. Dante’s Hell, Aquinas’s Theory of Morality, and the Love of God. The Canadian Journal of Philosophy 16: 181–198. ———. 1990. Providence and the Problem of Evil. In In: Christian Philosophy, ed. Thomas Flint, 51–91. Notre Dame, IN: University of Notre Dame Press. ———. 1993. Aquinas on the Sufferings of Job. In Reasoned Faith, ed. Eleonore Stump, 328–357. Ithaca, NY: Cornell University Press. Tugwell, Simon, ed. 1988. Albert and Thomas: Selected Writings, Classics of Western Spirituality, 223. Mahwah, N.J.: Paulist Press. Weisheipl, James. 1983. Friar Thomas D’Aquino: His Life, Thought, and Works. 2nd ed, 153. Washington, D.C.: Catholic University of America Press.

Eleonore Stump is the Robert J. Henle Professor of Philosophy at Saint Louis University. She is also Honorary Professor at Wuhan University and at the Logos Institute, St. Andrews, and a Professorial Fellow at Australian Catholic University. She has published extensively in philosophy of religion, contemporary metaphysics, and medieval philosophy. Her books include Aquinas (2003), Wandering in Darkness: Narrative and the Problem of Suffering (2010), and Atonement (2018). She has given the Gifford Lectures at Aberdeen (2003), the Wilde lectures at Oxford (2006), the Stewart lectures at Princeton (2009), and the Stanton lectures at Cambridge (2018). She is past president of the Society of Christian Philosophers, the American Catholic Philosophical Association, and the American Philosophical Association, Central Division; and she is a member of the American Academy of Arts and Sciences.

Chapter 5

Some Problems with Miracles and a Religious Approach to Them Agnaldo Cuoco Portugal

5.1 Introduction Miracles play a central part in Christian religion. The Bible has plenty of accounts reporting very unusual happenings such as the opening of the Red Sea by Moses or the resurrection of Jesus, which are very important to Christian religious doctrine. The faithful should not only believe that these occurrences took place somehow because of direct action of God upon the world, but that that kind of God’s action may still occur nowadays as a response to prayer by the worshiper, either directly or through the mediation of saints or the Virgin Mary. So, the Catholic Church takes miracles as a crucial piece of evidence in canonization processes, and if you turn your TV on in the middle of the night in Brazil you will probably see Christian churches programs in which people claim that miraculous cures happened to them in response to their strong faith and constant prayer. Small hours programs with reports of miracles you are going to see on Brazilian TV are mostly made by churches that are part of a rapidly growing movement in the Latin American religious landscape called neo-Pentecostalism. Some might say that their strong appeal to miracles is part of a strategy of attracting more adherents to their congregations. The more miracles reported in a TV program the more attractive is the church, resulting in more people contributing with money for that particular institution. These critics would say that, given the interest in having more sheep in

I am grateful to Hubert Cormier, Rodrigo Silveira, Felipe Medeiros, Marco Antonio Silva Filho, Nelson Gomes and two anonymous reviewers for useful comments on a first draft of this paper. A. C. Portugal () University of Brasilia, Brasília, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_5

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their flock—which means more cash in the church leaders’ bank accounts—these reports are more probably fraudulent. The idea is that miracles are part of a market strategy for the growth of a particular business, whose main goal is the financial health of the company, fed by the contribution of desperate people who seek for divine intervention in their difficult problems. So, miracle claims are not something from the past, but a current phenomenon, which deserves attention of philosophical reflection. Two main references are generally considered in the modern philosophical debate about miracles. On the one hand, David Hume’s analysis in section x of the An Enquiry Concerning Human Understanding (1748) is a landmark in modern treatment of miracle reports. His definition of this supposed event as an alleged violation of natural laws and his sceptic rejection of their veracity has been very influential. On the other hand, C. S. Lewis’s Miracles (1947) was also very prominent in this discussion. He objected to Hume’s scepticism and the naturalistic stances according to which the belief in miracles is not rational because there is no place for supernatural intervention in a law guided nature. In this text I am going to discuss three problems regarding this central element of Christian religion: the definitional problem (what is the meaning of ‘miracle’?), the epistemological question (is it rational to believe in miracles?), and what I am calling the ethical problem of miracles, which could be stated as ‘what is wrong in claiming that a miracle happened?’. I intend to take Hume as the main representative of the sceptical position, but will try to defend a religious interpretation of miracles in a different way from Lewis, despite sharing with him some points on how to approach the epistemological problem mainly.

5.2 Definitional Problems with Miracles Miracles are reported in the Bible and are part of the Christian and other religious traditions, but the most important definition in current analytic philosophy of religion was the one proposed by Hume. In the famous section x of the Enquiry concerning Human Understanding, he defined miracles as ‘a violation of the laws of nature’ (Hume E 10.12, SBN 114–5),1 because ‘there must, therefore, be a uniform experience against every miraculous event, otherwise the event would not merit that appellation’ (Ibidem). In other words, from the point of view of the events observable by experience, miracles are a violation of a regularity registered by science in natural laws. So a necessary condition for a happening to be called a miracle is that it is so uncommon that may be considered contrary to uniform natural order. However, despite being a violation of the laws of nature might allegedly be a necessary condition for an event to be a miracle, this is clearly not sufficient. Suppose I am teaching a regular class at the University of Brasília and the pen that I 1 Quotations

are taken from Hume Texts Online (https://davidhume.org/texts/e/full).

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accidentally drop does not fall on the floor, but strangely floats in the air and lands on the table in front of me. This would be really weird and physically unexplainable at first, but would not be a miracle by itself. Why? Something is clearly missing for it to be called miraculous: a religious meaning. If we do not have reason to think that this strange occurrence was due to a divine being acting in the world, a being to which we attribute a sacred character, it would be improper to call it miraculous. Strange things are miracles not because they are strange only. This is why Hume amends his initial definition in a footnote, calling it—‘accurately’ he says—‘a transgression of a law of nature by a particular volition of the Deity, or by the interposition of some invisible agent’ (E 10.n23, SBN 115). Putting differently, miracle is not the name of any unusual happening, but it presupposes a religious background. So, although he mentions it only marginally in his definition, even Hume admits that without this background, without claiming there would be a divine reality, which could be held responsible for the event, it should not be named a miracle. Yet, a religious definition may claim a very different perspective of miracles from the one centred in the contrast with scientific natural laws. For example, according to John Driscoll, in an article first published in 1911, but transcribed in the Catholic Encyclopedia2 recently, a miracle should be defined in view of three notions: wonder, power and sign. The first notion is in the very etymology of the word, since it comes from the Latin noun miraculum, an object of wonder. So, a primary characteristic of a miracle is that it provokes a reaction of astonishment in face of a highly extraordinary happening. This may be seen as dislocating the perspective from the comparison with the uniformity of natural laws to the reaction it triggers in the presented does not need to look this subjective. The wonderful aspect of the concept of miracle is better understood as a cognitive element, in the sense that there is a subject that learns something from what he seems to be seeing occurring in the world. So, as in all cognitive relationship, there is not only a subjective element involved in the notion of miracle, but also an objective one, which is included in the perceptual claim by the epistemic agent. The wonder the occurrence provokes in the perceiver is related to the nature of what happened in her presence. The unusual character of a miracle, which marvels the perceiver, is viewed as caused by the power of a purposeful Deity—the God of theism, for example. The circumstances of the happening are perceived as denoting not only natural forces involved, but mainly the action of a reality that is not part of nature, and has the power to act in it intentionally. In other words, there is a metaphysical assumption that nature is not the whole reality. This means that if you are an ontological naturalist, who believes reality is restricted to natural events and objects, then you will not use the term ‘miracle’ in the original sense that is being argued here—but at most with a metaphorical meaning. From a naturalistic point of view, there may be puzzles, enigmas, challenging occurrences, but never miracles. From this type of

2 Driscoll,

John T. “Miracle.” The Catholic Encyclopedia. Vol. 10. New York: Robert Appleton Company, 1911. 2 Nov. 2018. http://www.newadvent.org/cathen/10338a.htm.

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naturalistic perspective, reality is what natural sciences can explain in principle. Scientific activity investigates nature using certain background assumptions, expected patterns, hypotheses, analogies and so on. Empirical observations that do not fit in those assumptions, expectations and hypotheses are a common feature of natural sciences—they are what makes science advance. It is the effort for solving these puzzles that makes for most of normal scientific work. However, miracles are not enigmas in this sense, they are not motivations for more methodical research, but the striking manifestation of God’s power. The appropriate reaction to a miracle is wonder and discernment of a deep reality, not curiosity and questioning. Apart from wonder and power, the third element in the definition of miracle is sign, which is even a synonym for ‘miracle’ in the Bible frequently. This means that apart from an epistemological (wonder) and a metaphysical element (power), miracles have a linguistic or communicative side. They are taken to be a manner through which God conveys a message to humans. The content of this message varies according to the situation, but it generally has to do with the attestation of divine presence, or the certification that someone is God’s prophet or simply about the nature of His power to us—for our benefit in healings or our correction in punishments. This linguistic aspect of miracles is also revealed in the common interpretation of miracles as answers to our prayers, a way God relates to us in a very concrete way for our good. It also confirms the essentially religious meaning the concept of miracle has. Hence, as part of human communication with God, miracles are typically an element of the relationship between man and the sacred, which we normally call religion. So, we may use the three concepts above in a different order and say that a miracle is better understood as a sign of God’s power that causes wonder in the person who testifies it. The contention here is that a miracle is a typically religious idea, and this is why the influential definition proposed by Hume is problematic, even with the footnote amendment (‘by a particular volition of the Deity, or by the interposition of some invisible agent’). Defining a miracle in terms of violations of natural laws is disregarding its religious essence. It is analogous to say that axioms are a boring way to start telling a story. It is not wrong to say so, since to start telling a story with this kind of linguistic construct would be very dull,3 but it simply misses the point as regards what an axiom is. Axioms cannot be properly understood out of a mathematical or other formal scientific context to which it belongs. You can analyse them from a literary point of view of course, but then something will be missing of the axiom’s meaning. This is the problem with the common definition of miracle put forward originally by Hume and still very common in use in today’s analytic philosophy of religion.4 3 Perhaps

an example could be: “Let there be three pigs P1 , P2 and P3 , such that for any Pn , Pi is little, and a wolf W by definition big and bad”, which is a good way to get your children to sleep quicker probably. 4 See for example both the entry “Miracles” by McGrew (2016) in the Stanford Encyclopedia of Philosophy and Swinburne (2004), chap. 12, which also see problems with Hume’s definition, but recognize its enormous influence in philosophical discussion since the eighteenth century.

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5.3 Epistemological Problems with Miracles The epistemological problem of miracles is certainly the most discussed in the analytic philosophy of religion. I propose here to deal with it in view of three elements: the a priori contention that the belief in miracles is irrational in general; the a posteriori argument, which challenges the rationality of believing that a determinate happening was a miracle; and the problem of how it is possible to conciliate the belief in miracles with modern natural sciences. Although recent debate has been intense, Hume’s arguments are still a reference to the thesis that the belief in miracles is irrational. His argument was that to accept this kind of putative event is to disregard the correct proportion between empirical evidence and testimony. On the one hand, miracles contradict the uniform regularity of natural laws, which is strongly confirmed by observation. On the other hand, testimony is the only evidence for miracles (at least the biblical ones), and testimony is much weaker than direct observation, since the reliance on the former depends on perception. So a first step in the analysis of the epistemological problem with miracles is to think about Hume’s criticisms to the rationality of belief in them. Starting with the second part of his argument, Hume’s thesis about the relationship between testimony and perception is a classic example of what was later on called ‘reductionism’. According to it, the justification of a testimonial belief is not ultimately rooted in testimony as a source of information, but in more basic forms of belief formation, like perception or reasoning. So, for reductionists like Hume, the reliability of testimony depends ultimately on the acceptability of perception, which is the real belief source. Other people’s reports are merely a shortcut for data obtained through perception. And the very fact that we believe what someone says depends on the truthfulness of that person, which is something we acquire by means of perception. Now, as the important work by Coady (1992) shows, the interaction between testimony and perception is not that unilateral. Sometimes we correct our perception by asking someone beside us whether she saw the same we did, and it is perfectly acceptable to recognize the authority of a third party in overcoming our own direct perceptual beliefs. Some pieces of information can be obtained through testimony as the main (maybe only) source, like our own names: I was told by my parents the name I was given, and the official birth certificate I read with this detail is a written testimony of that speech act. In addition, in contrast with what Hume claims, testimony can preserve and even enhance the quality of information originally obtained by perception. This is why we can think of progressive knowledge in history, even considering that as time goes by we get further and further from the event we are describing. We can know better of the Paraguayan War (1864–1870) today than a hundred years ago because we are less emotionally involved in the event now, and because we have more information from different points of view about it, so that this happening can be better told (and understood) presently than it was in the beginning of twentieth century. This is so—in part at least—because testimony is not a passive transmission of data, but involves critical thinking, since

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speaking is an active dimension of communication. The reconstruction involved in handing over a fact we have been told does not necessarily mean distortion, but can represent an enhancement in the understanding of what happened. So, recent literature on testimony5 points to non-reductionism as a legitimate alternative to Hume’s position, arguing that it may be seen as form of knowledge on a par with perception, instead of epistemically lower and dependent on it. As to the first of Hume’s arguments for the irrationality of belief in miracles stated above, we could start analysing it from his own sceptical perspective. The argument holds that, since miracles are violations of natural laws and since these are strongly confirmed by observation, experience is a full proof against the occurrence of miracles (HUME, E10.12, SBN 114–5). So, belief in miracles is so absurd that should be taken as miraculous itself. Now, a possible response to this criticism could be based on Hume’s scepticism regarding induction as a guide to true belief. In that very same book, in the initial sections, he considers that matters of fact cannot be linked by necessity. In other words, the negation of a proposition established inductively does not imply a contradiction. So, if natural laws are inferred inductively, there is no necessity involved in them and there is no absurd—strictly speaking at least—in their negation. This means that in Hume’s own sceptical philosophy terms there is no a priori reason to condemn the belief in miracles as irrational because they do not conform to what is expected from what we know about natural laws. Still, even if there is no problem in principle in claiming the occurrence of a miracle, we could say that belief in miracles is irrational because they are so improbable that are practically impossible. However, the idea that miracles are practically impossible is admitted by the religious believer as well; this is why an event like this is worth being considered a sign of God’s power, which causes wonder in those who testify it. As Hume also argued, the very low probability of a miracle to happen is a condition for it to be considered so, i.e. normal regularity of natural occurrences is a supposition in the concept of miracle itself. One needs the action of a divine power to make it happen, which is practically impossible in a natural sense, but is still open for the supernatural action of God. This means that the a priori or de jure condemnation of belief in miracles presupposes the denial of a non-natural force acting in nature. Since the possibility of God’s existence is admitted, there is no irrationality in claiming that miracles can happen, even if they are very unlikely. Now, one may argue that what has been said above amounts only to an argument for the possibility of—and consequently, the rational acceptability of—miracles. An a posteriori challenge against the rationality of belief in miracles refers instead to the identification of individual events as resulting from this kind of divine action in the world. In other words, the a posteriori question is why we should accept that a given extraordinary happening is a miracle, instead of being explainable by science (in the future perhaps). A first answer to this challenge is that it presupposes

5 See

Jennifer Lackey (2008) for additional arguments for non-reductionism in testimony.

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the inappropriate concept of miracle as a violation of natural laws, which was criticized in the first part of this text. Although it may serve better for an argument to theism from miracles, since it assumes a notion that the sceptic could also accept, it does so at the expense of a conceptual disfiguration. Beginning with the Humean conception would be very close to begging the question against the compatibility of science and belief in miracles. A sudden and unexpected cure of cancer after an intense period of prayer may be explainable scientifically and yet, the person who endured that experience may still be right in seeing in it a sign of God’s power in a wonderful way. She may see the doctors who operated her as serving as God’s instruments for this very improbable result. And the fact that the cure has a scientific explanation does not preclude the interpretation of it as a miracle, since the fact of its opportunity—why it happened that moment in that way—may still be seen as a striking sign of God’s power.6 Since natural laws concern general causal relations, the fact that it happened in that very moment against the odds—in an extraordinary way in this sense—does not contradict a scientific interpretation of the event nor excludes a human technical intervention. The move proposed in the previous paragraph may appear to exclude the possibility of elaborating an argument to theism from miracles, since a religious interpretation of the concept of miracle would not be acceptable by a non-religious counterpart in the debate. However, not only ‘miracle’ should preserve its religious meaning, but its degradation into a happening not explained by science yet is not helpful to theism either. The reason for shunning this proposal is that it falls into the infamous ‘god of the gaps’ strategy, which runs a serious risk of backfiring. In other words, to take miracles out of their religious context and consider them as violations of natural laws that are better explained by God’s action does no good to theism. Firstly, because the ‘god of the gaps’, as a mere entity among others (or just an ad hoc hypothesis), does not correspond to the theistic concept of God. Secondly, because it reinforces the idea of competition between science and religion, which is not only merely partially true (at most), but also harmful to both. In addition, given the continuous progress of science so far, a miraculous event in this sense may eventually be seen as strictly natural. Indeed, a good reason for adopting the religious concept of miracle is that it permits to see nature as part of a much broader concept of reality, and to propose the theistic God as a personal force acting in the world not only occasionally in a striking way as a sign of His power, but permanently as the ultimate cause of the existence of the physical universe and its order. So, an argument from miracles would better be taken as a complement to a cumulative case that could establish the probability of God in non-religious terms initially, so that these highly rare events could be seen as additional signs of His action, and so of His existence as a personal reality.

6 See

Humphreys (2003) for the idea that the main characteristic of miracles (he focuses on the ones reported in the Exodus) is the precise timing of the occurrence of the events identified as such, rather than the violation of a natural law.

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So, we can say the rationality of belief in miracles is a corollary of the rationality of belief in God. If there is a God like the one postulated by theistic religions, then it is perfectly rational to believe that sometimes there may be striking manifestations of His power in the world. Thus, in a naturalistic world view like Hume’s, with no place for a non-natural reality there is no room for belief in miracles. If so, the question is more metaphysical than epistemological, or at least a dispute about what we should think about reality in ultimate terms before we had a debate on what is rational to believe. This is analogous to Plantinga’s conclusion in Warranted Christian Belief (2000) that the de jure objection to the belief in God depends on a de facto rejection of the existence of God. One may say this move does not look very favourable to the belief in miracles, since the question about the existence of God is even more complicated, and in a way depends on epistemological issues about how to assess evidence claimed in favour of theism. Yet, the point considered here still holds: arguments against the belief in miracles do not work independently, i.e. there is no problem in accepting that they can (rarely) happen if there is a God, and if the extraordinary event did happen, it is evidence of God’s action, given a religious background. Perhaps it is useful to translate the above argument in Bayesian terms in order to clarify both its answer to the a posteriori challenge and its import to a cumulative case for theism. In the former case, we are looking for the probability of the hypothesis h that a miracle occurred, given an extraordinary event e and background knowledge k, which includes scientific information and the possibility that there is a God as postulated by theism, i.e. we looking for the value of P(h/e. k). This is a function of the low prior probability of the event e given k—P(e/k) —and the likelihood it acquires in view of h—P(e/h. k). Since we admit the prior probability of the miracle hypothesis—P(h/k)—is not zero in view of k as stated above, the occurrence of the very improbable event may be said to confirm h, that is, the hypothesis that it was a miracle,7 given the following version of Bayes’s theorem: P (h/e.k) =

P (e/ h.k) P (h/k) P (e/k)

On the other hand, an argument from miracles could be an additional one in a cumulative case for theism that started from more ‘public evidence’, in the sense of information shared by both theists and non-theists, such as the existence of reality, the fact there is a physical universe, that this universe is ordered etc. The argument from miracles would come into play when (if at all of course) public

7 The Bayesian formalization can help us to see why a happening e that has no scientific explanation

confirms the miracle hypothesis in a higher degree than one which is scientifically explainable. In the formalization proposed, scientific information will be included in k. If there is a scientific explanation to e, then P(e/k) is higher than if e is scientifically unexplained. According to Bayes’s theorem, the higher is the prior probability of e, the lower is the explanatory power of h, which h.k) is the proportion PP(e/ (e/k) . As was said, however, the concept of miracle proposed here does not require the alleged miraculous happening has no scientific explanation.

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evidence had already confirmed theism (as the hypothesis h under assessment) such that the occurrence of a miracle (now as evidence e) could confirm h a little more. This additional confirmation would mean that the theistic hypothesis which gave intelligibility to all previous pieces of evidence (and whose credence was increased by them) was also able to illuminate an extraordinary happening, which we call a ‘miracle’, and to obtain further confirmation from it (maybe reinforcing the idea of a personal God, who interacts with conscious beings, instead of an impersonal ordering intelligence). Still, even if we could answer to the a priori and the a posteriori challenges as above, another facet of the epistemological problem of miracles is its compatibility with scientific method and the knowledge about nature it makes possible. If there would be any incompatibility, this would be another important de facto argument against the belief in miracles, given the value and importance of natural sciences today. Do people who believe in miracles have the right of praising scientific progress in unravelling the puzzles of the physical universe and using technology that is being developed from it? Wouldn’t they be more coherent in renouncing this kind of knowledge and the technical devices it generates? As to the use of technology even without sharing the same worldview in which it was created, this does not seem to be a problem at all. Cultures as different as many Brazilian indigenous groups resort to this type of material artefact without suffering any harm in their values, ways of life and metaphysical conceptions. Even if this alien technology may cause some cultural difference, it does not amount to incoherence and certainly does not prevent them to use it. In addition, there is no irrationality in believing in miracles and going to a doctor for treating cancer because, as stated above, according to the approach defended here, the action of supernatural agency is not viewed as cancelling natural uniformity (and the technology that is made possible from the knowledge about natural laws), but as complementing it. As said above, the doctor may be seen as an instrument of God’s action in the world, for example. Indeed, theistic religions claim that both the very existence of a physical world and the fact that it is ordered so that scientific activity is possible depend on God. This means that for theism, the ultimate conditions for the possibility of science lie on God’s activity. Although this is not a very popular conception among philosophers of science today, it is still a perfectly tenable philosophical claim, with many advantages over a naturalistic metaphysics of science according to some participants in the more recent debate on this.8 After Newton and particularly from the nineteenth century onwards, mentions to God in natural science texts have gradually disappeared. Historically many important scientists saw their activity as an alternative to a religious view about the world (Laplace is often quoted on this). In addition, there is evidence that

8 See

Plantinga (2011) and Rea (2002) for criticisms to naturalism as an alleged scientific world view, and Clark (2016) for a comprehensive display of the positions involved in this debate.

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religious belief is less frequent in scientific community than in society at large.9 In spite of all the aforementioned facts, this does not mean an incompatibility between natural sciences and belief in miracles. Many important scientists (all key pioneers of early modern scientific revolution and more recent ones such as Faraday, Maxwell and Mendel, for example) were devout Christians. Apart from this, the large ontological picture of natural world is not a scientific matter, since this question is beyond the range of scientific method in empirical sciences. Although the development of science may contribute to important changes in that picture, this discussion is mainly a philosophical one, or at least involves metaphysical debates in a very central form. As Thomas Kuhn has famously argued,10 normal scientists (in contrast with the leading ones) are not involved, not even interested, in foundational issues, rather, they take them for granted in a very dogmatic fashion. This has to do with the manner new scientists are normally trained in the scientific activity: they learn how to do science in a tacit way so that this method is not distinguished from the ontological presuppositions attached to it. This means that methodological naturalism (searching for answers about the natural world within the limits of what is empirically testable) is easily confused with ontological naturalism (all reality is restricted to what science can investigate). So, there is a sociological motive for the low relative frequency of religious belief among scientists, which does not mean an argumentative reason. Putting it differently, the fact that there are fewer scientists who accept the belief in miracles than people outside academic circles does not mean this sort of belief is incompatible with scientific activity. On the contrary, given the historical background of natural sciences, the problems of ontological naturalism and some features of the theistic hypothesis (accepted by many key natural scientists and rival to ontological naturalism), we may hold that the belief in miracles is clearly compatible with that activity. Although epistemology is a vast field, and probably its application to the analysis of miracles has other elements than the ones that have been explored here, let us move to the third and last question this text aims to discuss about this subject, namely, the ethical problem of miracles.

5.4 Ethical Problems with Miracles The relation between miracles and morality is a much less usual discussion than the other two issues analysed so far. The intention here is to elaborate a little on two main dimensions of this matter. On the one hand the question is whether miracles are a good thing so that we could expect them as a result of an infinitely good God’s action. The other problem concerns the morality of reporting a miracle, i.e. whether there is anything wrong in claiming that a miracle has happened. 9 See

Brooke (2009). (1962).

10 Kuhn

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The first side of this question is related to the famous problem of evil, more specifically to a theodicy, which in general aims to improve the comprehension about what kind of world we could expect God to create. Would a world like this have room for miracles? Understood in the way philosophical discussion has mostly been putting it, as violations of natural laws, one may doubt whether a world containing them would be a good one. The reason is that a place like this would be too unpredictable to live in and virtually impossible to understand. If God can intervene at random whenever pleases Him, then anything can happen at any time, and natural science would be an unmanageable enterprise. Miracles would be a barrier to the regularity needed for a rational understanding of the world. On the other hand, a world where we could never have a minimum of certainty about what was going to happen would be a very difficult place to live, if life were going to emerge and have continuity at all in these conditions. The fact we can count on natural regularity is a condition for our adaptation and intelligent control over what happens around us. A universe dominated by miracles would be inhospitable to science and a very rough—or even impossible—place to survive. The description above is part of the story of why philosophers of science such as Hume and others in the eighteenth century thought the belief in miracles would be irrational and antiscientific. Since we have already discussed this position in the previous section from the epistemological point of view, let us try to see it from the ethical side. The problem stated above presupposes that regularity and uniformity are good things because they allow for science and for a normal, ordained life, which are good things too. Interventions in this uniformity would be morally bad in case they contradicted those qualities. However, why should one grant the inference from the goodness involved in regularity and uniformity to an ethical rejection of miracles? The crucial step is the incompatibility between them, that is, the idea that a miracle is contradictory with uniformity and regularity. Yet, as was mentioned previously we do not need to think these properties in an absolute degree in order that science may be possible and life sufficiently expectable; and miracles are by definition a very rare phenomenon. In fact, natural science deals with irregularities all the time, since they constitute the puzzles scientists are dedicated to solve in their activity, and a too predictable existential order would be a very boring and unchallenging one. So, one might expect from a good God to act in exceptional occasions, since this would be coherent with a degree of order good enough for scientific activity and existential responsiveness to the world. On the other hand, in the definition of miracle as a wonderful sign of God’s power there is even less reason to take this as an objection, since it does not need to be seen as a violation of natural laws necessarily. In addition, these signs involve commonly a beneficial consequence, being unexpected cures the typical examples, which is obviously type of thing we could expect from a good God. Yet, why would an infinitely good God benefit with a miraculous cure only some of those who resort to Him in prayer? It seems to be a morally reprehensible discrimination to help only a few of those who suffer. To answer this question does not seem to be too different from responding to the classical problem of evil, since the latter can be rephrased in terms of “why does God allow that some people suffer

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more than others?”, which is implied in the more general question “why does God permit suffering at all?”. The free will defence, the theodicy of natural order, the soul making theodicy—they are all available here as purported solutions. Yet, if one accepts the possibility that miracles can happen, then they can play an important role in still another type of this sort of philosophical theory—an eternal life theodicy, i.e. a conception of God’s redemption in a eschatological time, when there will be no more suffering and pain, not for a while, but for eternity. Miracles would be a sign of this future reality, given in order to encourage us to endure the hardships of this world and get more prepared for the eternal life of infinite goodness and deep communion with God, when pain and suffering (even boredom) would have no place. Since they are only indications of what things will be like in the eschatological future, there is no point in censuring the fact that they are for only a few. In this theodicy with a special place for miracles, redemption is forever and for everyone who freely accepts it, and its goodness supersedes current world evil by far. So, miracles might have a role in a response to the problem of evil, and its apparent selective discrimination should be taken as just a superficial issue. The other ethical problem regarding the matter we aim to discuss here is the morality of claiming that a miracle has happened. Why would this be ethically problematic in the first place? As was said in the introduction of this text, many people see the ample time miracle reports have in some religious TV programs as a malicious strategy for attracting more adepts to the respective churches and getting more money from them. This seeming fraud is aggravated by the fact that generally the persons addressed by this strategy are economically poor and existentially desperate, which makes this appeal to miracles a coward action directed to miserable human beings in a situation of vulnerability. So, claiming that a miracle has happened in this context is morally reproachable because this assertion is intentionally misleading, and lying is one of the paradigms of immoral behaviour. In addition, it aims at people in particularly fragile situations, very willing to cling to any minimal hope that can be offered, and the possibility of a miracle serves better than anything to keep them captive to the church that offers it. Now, this accusation of maliciousness in reporting miracles contrasts with the curious discretion with which Jesus is reported to treat some of his own miracles in the New Testament. Although the recommendation attaches to only 5 out of the 35 miracles performed by Jesus,11 it is curious that this request were ever made considering the general purpose of announcing a message—the Gospel itself. If the intention was marketing the Good News the most as possible, nothing would be more contrary to it than secrecy. However, this stance regarding some of Jesus’ miracles is coherent with the general attitude commended in the Sermon on the Mount as to the way one should look when fasting, praying and giving alms (Mt 6, 2–4.5-6.16-18). The idea is that instead of public, external approval, those actions should be occasions for inner personal relationship with God. So, the secrecy surrounding some of the miracles performed by Jesus could be interpreted as an

11 See

Mk 1:40–45; Lk 8:49–56; Mt. 9:27–31; Mk 7:31–37; Mk 8:22–26.

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emphasis on the linguistic or communicative element included in the notion of a marvellous sign of God’s power. In this sense, a miracle can certainly be used for announcing this message, which as such is not problematic. But this type of communication is different from advertising a product in the market. Instead of a visible growth of a church both in number of adherents and wealth, a miraculous sign from God aims to enable an invisible relationship with the ultimate reality. It should keep its religious meaning instead of being downgraded into some financial or political advantage it may bring. In other terms, we can see in the religious sense of miracle a deeper pattern to evaluate the ethical problem formulated above. The idea is that, from the account outlined in the beginning of this section, reporting a miracle is morally problematic only if it is part of a fraud or a lie. However, suppose the reports are true. In this case, from a naturalistic point of view, there is no ethical issue involved here, not even in the result of increasing the number of church adherents and the amount of money this institution gets from them. Yet, when we view miracles from the perspective presented in the first section of this text (as an alternative to Hume’s view), then the case is morally problematic not only if there is a fraud involved, but also when the reports are true. Hence, the Christian moral assessment seems to be even more rigorous against disclosing the occurrence of a miracle, since it rejects frauds and lies as the naturalistic evaluation, and in addition holds that it should be kept as discrete as possible so not to harm its religious significance. Of course, understood as signs of God’s power manifested in a marvellous way, miracles have a communicative meaning, and perhaps this is why the vast majority of Jesus’s miracles reports do not contain the secrecy clause. Sometimes the disclosing of a miracle may even be a duty.12 However, the claim here is that a religious approach to miracles is able to show what is wrong in broadcasting a miracle for non-religious purposes, such as the increase in political power or growth in material wealth, and even the positive evaluation of announcing it is clearer from a religious point of view.

5.5 Final Remarks This paper has attempted to hold that miracles are better construed as a religious concept, which has a double meaning. On the one hand, it means that not all extraordinary happening can be properly named ‘miraculous’. On the other hand, it implies that not all God’s direct action in the world should be called a miracle. Religion is a relationship between conscious beings and a reality taken to be sacred. So, to interpret miracles in religious terms means not to take them as events simply, but as happenings which have significance for someone. In other words, in a physical universe with no conscious beings, there might be other kinds of God’s

12 I

owe this observation to an anonymous reviewer.

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activities in the world, but not miracles in the sense proposed here, because of the communicative element that is essential to this concept, which would be lacking in a world like that. Metaphysically speaking, miracles counter natural laws as human actions do. When I intentionally throw my pen up I am making it to follow a course because of a force I press, which is in the opposite direction to the natural gravitational force. From the interaction of these forces—an intentional and a natural one—we have the resulting movement of the pen. From a religious perspective, a miracle would be a special type of God’s action in the world. If there is a God, then it is not irrational to expect Him to act in the world in a way so to communicate with us. If so, a miracle could be understood as an extraordinary event where physical processes do not operate normally, because there is an external force acting in the world, so that it is a sign of a divine being’s power. Thus, the central question is the existence of a God like the one postulated by theism. From a naturalistic point of view, the causal closure of the physical—the principle according to which ‘every physical event has a sufficient physical explanation’—is not only methodological, but also ontological. In other words, a naturalist like Hume takes the physical causal closure not only as a heuristic principle to guide natural sciences research, but as a description of reality as a whole as well. Viewed this way, the principle prevents God’s action in the world and then makes the occurrence of miracles impossible (and its corresponding belief irrational) a priori. This is part of the reason why Hume’s argument against miracles is criticised even by non-theist authors, such as Michael Martin (1990) and Earman (2000). Apart from the a priori objection, this text approached the epistemological problem of miracles elaborated as an a posteriori challenge. Since the concept of miracle proposed here was not of violation of natural laws, and since the notion of extraordinary occurrence that is a sign of God’s does not preclude the possibility of a scientific explanation of that happening, the a posteriori objection has no place. In addition, although it may play only a marginal role, the notion of miracle held here is also able to be a basis for an inductive argument for the theistic hypothesis in a Bayesian cumulative case. Moreover, it was claimed that, given the poor content of naturalistic ontology, a metaphysics that allowed for miracles would be more interesting as an ultimate theoretical background for natural sciences. As regards the ethical issue, we discussed the role miracles might play in a theodicy, and sketched an argument for a positive part in a specific kind of justification of God’s permission for the occurrence of evil in the world. In addition, in view of a possible moral problem related to claiming that a miracle has happened, it was argued that the religious approach seems richer than a sceptical one. An approach compatible with the occurrence of miracles deals with the possibility of fraud as the naturalistic does, but it is even more rigorous that the sceptic in the ethical analysis by using specific Christian values in assessing some moral issues involved even in non-fraudulent miracle claims. The line attempted here has taken miracles as very special occasions of religious experiences, and has put little weight in the famous question about a supposed violation of natural laws. They are striking signs of God’s power, an extraordinary

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happening that seems inexplicable from a natural point of view at the time it happens. The fact that the way a miracle has occurred becomes explained in scientific terms eventually does not explain why the event happened that way in that exact opportunity to that person or group. Apart from this, the eventual scientific explanation does not affect the communicative purpose of a miracle—it may be interpreted as God’s action in the world by means of natural events or human interventions. I do not deny there are important points in common between the religious approach to miracles this article proposes and C. S. Lewis’s position in his famous Miracles (1947). Both are critical to Hume’s sceptical denial of any place for supernatural action in the physical world, both take divine action as another factor to feed the regular working of natural laws, and both hold that belief in miracles is not necessarily unscientific (rather, it is compatible with theism, allegedly the most amicable metaphysics to natural science). However, Lewis does not propose a definition for miracles alternative to the one held by Hume. Instead, his aim is to show that miracles conform to the definition of violations of natural laws, and its correspondent belief is not irrational as a result. The present text has contended that the Humean definition was defective for various reasons, and that it was inappropriate as defining an axiom as a boring way to start telling a children’s story and blind to essential aspects of miracle as a religious concept. The aim here was to explore a comprehension of miracles other than the modern philosophy understanding of them in view of natural laws, even a friendlier one as that put forward by Leibniz. Besides, the text argued that there are various reasons to counter Hume’s arguments against the rationality of belief in miracles, and some of them were not discussed by Lewis either. In addition, I cannot see any indication in Lewis’s defence of what I called the ethical problem of miracles, which was outlined in the third part. I hope this text has helped to clarify this fascinating religious concept. The socalled modern academic culture has many other problems with miracles probably. But I think the main philosophical ones are those dealt with here.

References Brooke, John. 2009. That modern science secularized western culture. In Galileo goes to jail and other myths about science and religion, ed. Ronald Numbers. Cambridge, MA: Harvard University Press. Clark, Kelly, ed. 2016. The Blackwell companion to naturalism. Oxford: Blackwell. Coady, C.A.J. 1992. Testimony—A philosophical study. Oxford: Oxford University Press. Driscoll, John T. 1911. Miracle. In The catholic encyclopedia, vol. 10. New York: Robert Appleton Company. http://www.newadvent.org/cathen/10338a.htm. Earman, John. 2000. Hume’s abject failure—The argument against miracles. Oxford: Oxford University Press. Hume, David. 1748. Enquiries concerning human understanding. Section X. http://davidhume.org. Accessed November 2018.

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Humphreys, Colin. 2003. The miracles of Exodus: A scientist reveals the extraordinary natural causes underlying the biblical miracles. New York: Harper Collins. Kuhn, Thomas. 1962. The structure of scientific revolutions. Chicago: University of Chicago Press. Lackey, Jennifer. 2008. Learning from words—Testimony as a source of knowledge. Oxford: Oxford University Press. Lewis, C.S. 1947. Miracles. London: Haper Collins. Martin, Michael. 1990. Atheism: A philosophical justification. Philosophy of religion. Philadelphia: Temple University Press. https://books.google.com.br/books?id=MNZqCoor4eoC McGrew, Timothy. 2016. Miracles. In The Stanford encyclopedia of philosophy, ed. Edward N. Zalta. Stanford, CA: Stanford University. https://plato.stanford.edu/archives/win2016/entries/ miracles/. Plantinga, Alvin. 2011. Where the conflict really lies—Science religion and naturalism. New York/Oxford: Oxford University Press. ———. 2000. Warranted Christian belief. New York/Oxford: Oxford University Press. Rea, Michael. 2002. World without design—The ontological consequences of naturalism. Oxford: Oxford University Press. Swinburne, Richard. 2004. The existence of god—Second edition. Oxford: Oxford University Press.

Agnaldo Cuoco Portugal earned his PhD in the philosophy of religion at King’s College London (2003), and was visiting scholar in the Center for Philosophy of Religion at the University of Notre Dame (2017–2018). He is associate professor at the University of Brasilia, and was president of the Brazilian Association for Philosophy of Religion (2010–2015). He published the book Filosofía de la Religión. Una Perspectiva Analítica (2015) in Spanish, and several articles in Portuguese and English on philosophy of religion, epistemology and philosophy of science.

Part III

Logical Philosophy of Religion

Chapter 6

An Even More Leibnizian Version of Gödel’s Ontological Argument ´ etorzecka and Marcin Łyczak Kordula Swi˛

6.1 Introduction We propose a modification of Gödel’s argument for God’s existence, originally expressed in his 1970 manuscript ‘Ontologischer Beweis’ (Gödel 1970, OB). Our proposal follows a Leibnizian onto-theology, which Gödel referred to in many of his philosophical writings. We claim that Gödel’s approach combines the main ideas of two Leibnizian writings from 1676 and 1677, respectively. The first one, titled ‘That the most perfect being exists’, was included in a letter to Spinoza (Leibniz 1987, 427, trans. in Leibniz 1989, 167–169), and is usually considered by commentators of OB as the main source for Gödel’s view. The second source comes from Leibniz’s correspondence with Eckhard (Leibniz 1987, 588 trans. in Adams 1994, 136– 137), in which Leibniz addresses the question of God’s existence in connection with the concept of God’s essence. This feature is also present and significant in Gödel’s argument. It is well known that Leibniz’s letter to Spinoza influenced Gödel’s formulation of the proof of the Cartesian lemma, which is one of two main lemmata of the key thesis of God’s necessary existence. It states that the existence of the most perfect being is possible. The second pillar of Gödel’s ontological argument is the Leibnizian lemma according to which the possible existence of God implies His necessary existence. The proof of this second lemma is based on the mentioned concept of essence, which is understood quite differently from Leibniz’s interpretation.

´ etorzecka () · M. Łyczak K. Swi˛ Institute of Philosophy, Cardinal Stefan Wyszy´nski University in Warsaw, Poland e-mail: [email protected]; [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_6

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A comparative analysis of OB and Leibniz’s texts has already been carried out by ´ Adams (1995) and Swietorzecka (2016). Here we want to consider two differences between Gödel’s and Leibniz’s approaches. We will argue for the superiority of Leibniz’s approach, while preserving the main structure of the Gödelian argument. Our first aim is to bring the key concept of positiveness used by Gödel closer to Leibniz’s idea of perfectio, which underlies the ontological argument from 1676. It is true that Leibniz considered perfections as simple, positive, and absolute qualities, whereas Gödel’s concept of positiveness comprises just properties (including complex and relational ones, which do not qualify as Leibnizian perfections). However, our aim is to impose on Gödel’s positive properties at least the condition accepted by Leibniz for positive qualities. The latter are not to be “understood through negations” (Leibniz 1989, 167–169). Following this restriction, we modify Gödel’s proof in such a way as to eliminate negative predicate terms from it. The language of the whole theory still includes term negation, but in the sense of contrariness. Our second modification is also inspired by Leibniz’s letter to Spinoza. This time we want to consider the original idea of a connection between the concept of necessity and the concept of demonstrability. Leibniz explicitly understands necessarily true propositions that are not known per se as demonstrable ones. This motivates us to use a formal basis for OB different from the one that is usually considered. It is generally assumed that Gödel’s approach falls within the framework of logic S5.1 However, the concept of demonstrability does not fulfill axiom 5; rather, it fulfills axiom 4. By combining Leibnizian modalities in their prooftheoretical meaning with S4 modalities we follow a proposal advanced by Adams (1994, 46–50).2 Our approach further employs the idea of grounding OB on the ´ etorzecka (2012). That formalism is a ‘compromise’ S4 system, introduced by Swi˛ between S4 and S5 modalities. Our formal frame is logic S4, but a specific instantiation of 5 is needed as an axiom to derive the Leibnizian lemma. As a result we do not claim that S4 exclusively describes the modalities of Leibniz’s texts. It may be worth mentioning that our proposal also addresses Gödel’s doubts about “using some principle in modal logic” in OB as being too strong—Adams supposes that this principle is the axiom 5 (Adams 1995, 391).

1 A survey of the logics used in various formalizations of OB is given by Swietorzecka ´ (2016, 25– 29). Modal frames were investigated earlier by Kovaˇc (2003). In addition to S5, the mostly used modal logics which do not require many changes in the original structure of the argument in OB are K5, KD45, and KB. 2 Interestingly, the formula (4) A → A follows from the law about the rationality of necessary statements, which is also attributed to Leibniz by Perzanowski. If we assume that contingency is described by KA =: ♦A ∧ ♦¬A, then we can express the statement “Everything that is necessary, is not contingent” as: ¬KA. Using the standard definition: ♦A ↔ ¬¬A, we obtain precisely A → A (cf. Perzanowski 1989, 99).

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We formulate a new version of OB, which also offers interesting references to some of Leibniz’s views on the ‘nature’ of God. It turns out that our theory has models in which God’s nature is not maximal, in the sense that there are properties such that God does not possess them or their contraries (negations). This corresponds with the Leibnizian concept of God, Who is neither finite nor infinite in time and space. Secondly, in our proposal God is unchanging in the sense that all of His properties are His attributes, but He is not determined by at least some relative properties, that He only possibly possesses (like “being the creator of any possible but not actual world”). We begin our discussion with D. Scott’s presentation of Gödel’s argument with an explicit description of the assumed formal system (6.2). Next, we show the possibility of eliminating negative terms from the ontological proof and highlight certain derivative dependencies between this proposal and certain axioms of Scott’s theory (6.3). Finally, we introduce the axiomatic description of our modification of OB, sketch its semantics and show a few interesting semantical observations within this new framework (6.4).

6.2 Starting Point: The Formalization of OB by D. Scott The original OB manuscript was discussed by Gödel with Scott who presented it during his seminars in 1970. As a result of these exchanges we now have the socalled Scott version of Gödel’s argument, which is considered the version closest to the original ideas of the manuscript. We accept the Scott version as an adequate supplement to Gödel’s approach. For the convenience of the reader we provide a short presentation of the Scott ´ theory, following its description by Swietorzecka (2016, 17–24). The vocabulary of the used symbolic language consists of: individual variables x, y, z, . . . ; first-order unary predicate variables ϕ, ψ, χ , . . . ; constants G (is God), and N E (is necessarily existent); first order identity predicate =; second-order predicate P (is positive); symbols: − (for term negation), ¬, →, ∀, ; and parentheses. We accept the following definitions of predicate terms and formulas: τ ::= ϕ | ϕ | G | NE A ::= τ x | x = x | P(τ ) |¬A | A → A | ∀xA | ∀ϕA | A| ♦A Symbols ∧, ∨, ↔, ∃ are defined in the standard way. We use α, β as representing individual or predicate variables. The Scott theory is based on a logic characterized by: – all classical sentential tautologies (PC) – formulas of the following shapes: (Q2)

∀αA → A(τ /α ), τ is substitutable for α

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(id1) (id2) (K) (T) (5) (♦/)

x=x A ∧ x = y → A[y /x ] (A → B) → (A → B) A → A ♦A → A ♦A ↔ ¬¬A

The primitive rules are modus ponens; rules for introducing quantifiers:

A → B ⇒ A → ∀αB, where α is not free in A; and the necessitation rule:

A ⇒ A (R). The above system is called Q2S5. We add to Q2S5,  closures of the following equivalences: (− ) (G) (NE)

τ x ↔ ¬τ x (negative property) Gx ↔ ∀ϕ(P(ϕ) → ϕx) (God) N Ex ↔ ∀ϕ(ϕEss.x → ∃yϕy) (necessary existence) where ϕEss.x means ϕx ∧ ∀ψ(ψ(x) → ∀y(ϕ(y) → ψ(y))) (Ess.)

The expression ϕEss.x reads as follows: property ϕ is an essence of x. In Q2S5 extended by a modal version of the comprehension schema: (MCS) ∃ϕ∀x(ϕx ↔ A(x)), ϕ is not in A, our  closures of − , G, NE are cases of MCS. The following are specific axioms for P: (A1) (A2) (A3) (A4) (A5)

P(ϕ) ↔ ¬P(ϕ) P(ϕ) ∧ ϕ ⊂ ψ → P(ψ), where ϕ ⊂ ψ means ∀x(ϕx → ψx) P(G) P(ϕ) → P(ϕ) P(N E)

The Scott theory is just the extension of Q2S5 by the set of axioms A1–A5 above and the set of  closures of the mentioned cases of the comprehension schema. We call it GOo = Q2S5[A1–A5, (− , G, NE)]. Gödel’s ontological argument may now be reconstructed in a simple way. We start with the theorem about the possible existence of the subject of any positive properties: Th 1 P(ϕ) → ♦∃xϕx Proof 1. P(ϕ) ∧ ϕ ⊂ ϕ → P(ϕ) [A2], 2. P(ϕ) ∧ ∀x(ϕx → ϕx) → ¬P(ϕ) [1, A1, ⊂ ], 3. P(ϕ) ∧ ∀x(ϕx → ¬ϕx) → ¬P(ϕ) [2,− ], 4. P(ϕ) → ¬∀x(ϕx → ¬ϕx) [3, PC], 5. P(ϕ) → ♦∃xϕx [4, ♦/]. Next, we derive the counterpart of the Cartesian lemma: (CL)

♦∃xGx

[Th1, A3]

The Leibnizian lemma is obtained through the thesis that the property of being God is essential to Him: Th 2 Gx → GEss.x

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Proof 1. P(ϕ) → ∀x(Gx → ϕx) [G, K], 2. P(ϕ) → ∀x(Gx → ϕx) [1, A4], 3. Gx → (ϕx → P(ϕ)) [G, A1,− ], 4. Gx → (ϕx → ∀x(Gx → ϕx)) [PC, 2, 3], 5. Gx → Gx ∧ ∀ϕ(ϕx → ∀x(Gx → ϕx)) [Q2, 4], 6. Gx → GEss.x [Ess., 5]. The Leibnizian lemma is expressed as follows: (LL)

♦∃xGx → ∃xGx.

Proof 1. Gx → N Ex [G, A5], 2. Gx → ∀ϕ(ϕEss.x → ∃xϕx) [NE, 1], 3. Gx → (GEss.x → ∃xGx) [2, Q2], 4. Gx → ∃xGx [Th2, 3], 5. ∃xGx → ∃xGx [Q2,4], 6. ♦∃xGx → ♦∃xGx [ A → B ⇒ ♦A → ♦B, 5]3, 7. ♦∃xGx → ∃xGx [5, 6]. Finally we derive the key thesis from CL and LL: Th 3 ∃xGx. Theory GOo has further interesting theses: (G, − , A4)

Th 4 Gx → ∀ϕ(ϕx → P(ϕ)) Th 5 ♦P(ϕ) → P(ϕ)

(A4, A1)

Th 6 ϕEss.x ∧ ψEss.x → (ϕ = ψ) where ϕ = ψ stands for ∀x(ϕx ↔ ψx)

(Ess.)

Th 7 ϕEss.x ∧ ϕEss.y → ∀ψ(ψx ↔ ψy)

(Ess., T)

Th 8 Gx ∧ Gy → ∀ψ(ψx ↔ ψy) We can also add to GOo all  closures of (Ix ) Ix y ↔ x = y In

GOo [I

x]

(Th2, Th7) (identity in relation to x)

we can next prove the following further theses:

Th 9 ϕEss.x ∧ ϕEss.y → x = y

(Th7, Ix , id1)

Th 10 Gx ∧ Gy → x = y

(Th8, Ix , id1)

Th 11 Gx → ∀y(Gy → x = y) Th 12 Gx → Gx Th 13 ∃xGx

(Ess., Ix , id1, Th2) (Th11, id1, Th3) (Th12, Th3, T)

Th 14 Gx → ∀ϕ(ϕx → ϕx)

(G, K, A4, Th12, Th4)

Th 15 Gx → ∀ϕ(♦ϕx → ϕx).

(Th14, − )

We will refer to some of these theses in our further considerations. rule A → B ⇒ ♦A → ♦B is derivable from R. The theory GOo is  transparent in the sense that for every A which is GOo thesis we get A → A only with modus ponens and rules for ∀ (Czermak 2002, 317).

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´ A description of an adequate semantics for GOo [Ix ] is given by Swietorzecka (2016, 23–24) (which follows Hájek 2002). We will adopt this semantics when formulating our proposal in Sect. 6.4.

6.3 Negative Terms Are Not Needed in the Argument A significant difference between Leibnizian perfections and Gödel’s positive properties, which we want to eliminate in favor of the former, concerns the use of negative predicate terms. These are introduced in GOo with the equivalence − , which also enables us to describe positive properties as complements (negations) of others. Leibnizian perfections, instead, are not describable via negations. As we shall see, the elimination of negative predicate terms does not turn Gödel’s argument into a theory weaker than GOo . Let us consider two more GOo theses: (G↔ ) (¬P)

Gx ↔ ∀ϕ(P(ϕ) ↔ ϕx), ∃ϕ¬P(ϕ).

(G, Th4) (A3, A1)

We call + Q2S5 the logic expressed in the language of Q2S5 but with no negative terms and characterized in the same way as Q2S5 without the case − of the comprehension schema. The labels + Q2♦ and + Q2♦KT are used for the classical second order fragment of + Q2S5 extended by ♦/ and ♦/, K, T, respectively. It is now to be observed that the formula concerning the possible existence of the subject of any positive properties—Th1—is derivable from ¬P, and A2, which are added to + Q2♦: Fact 1

+

Q2♦[¬P, A2] P(ϕ) → ♦∃xϕx.

Proof 1. P(ϕ) ∧ ∀x(ϕx → ψx) → P(ψ) [A2, ⊂ ], 2. P(ϕ) ∧ ¬P(ψ) → ♦∃x(ϕx ∧ ¬ψx) [1], 3. ¬P(ψ) → (P(ϕ) → ♦∃xϕx) [2], 4. ∃ψ¬P(ψ) → (P(ϕ) → ♦∃xϕx) [3], 5. P(ϕ) → ♦∃xϕx [4, ¬P]. The formula stating that the God-like property is the essence of a subject of this property—Th 2—is derivable from A4 and G↔ in + Q2♦KT: Fact 2

+

Q2♦KT[A4, ∀G↔ ] Gx → GEss.x.

Proof 1. P(ϕ) → ∀x(Gx → ϕx) [∀G↔ , K]4, 2. P(ϕ) → ∀x(Gx → ϕx) [1, A4], 3. Gx → (ϕx → P(ϕ)) [G↔ , T], 4. Gx → (ϕx → ∀x(Gx → ϕx)) [PC, 2, 3], 5. Gx → Gx ∧ ∀ϕ(ϕx → ∀x(Gx → ϕx)) [Q2, 4]. 6. Gx → GEss.x [Ess., 5].

4 We

need  closure of general closure of G↔ because we are not in S5.

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Following the derivations of CL and LL in GOo and in view of facts 1 and 2, we can see that the proof of the key thesis Th3 need not be dependent on negative terms: Fact 3 + Q2S5[¬P, A2 − A5, (G↔ , NE)] ∃xGx. Of course, a language with no negative terms gives less ability to express many theses contained in GOo . For instance, the formula A1 is not expressible in this language. Interestingly, however, the addition of (− ) to + Q2S5[¬P, A2-A5, (G↔ , NE)] changes this situation. The formula A1 is now derivable in Q2S5[¬P, A2-A5, (− , G↔ , NE)]: Fact 4 Q2S5[¬P, A2-A5, (− , G↔ , NE)] P(ϕ) ↔ ¬P(ϕ).5 Proof (←) 1. Gx → (ϕx ∨ ϕx) [− ], 2. Gx → (P (ϕ) ↔ ϕx) [G↔ , ϕ/ϕ], 3. Gx → (¬ϕx → ϕx) [1], 4. Gx → (¬ϕx → P(ϕ)) [3, 2], 5. Gx → (¬P(ϕ) → ϕx) [4], 6. Gx → (¬P(ϕ) → P(ϕ)) [5, G↔ ], 7. ∃xGx → (¬P(ϕ) → P(ϕ)) [5, G↔ , 6], 8. ¬P(ϕ) → P(ϕ) [5, G↔ , 7, ∃xGx] (→) 1. P(ϕ) ∧ P(ϕ) → (Gx → (ϕx ∧ ϕx)) [G↔ ], 2. P(ϕ) ∧ P(ϕ) → (Gx → (ϕx ∧ ¬ϕx)) [2, − ], 3. P(ϕ) ∧ P(ϕ) → ¬Gx [2], 4. P(ϕ) ∧ P(ϕ) → ∀x¬Gx [3], 5. P(ϕ) → ¬P(ϕ) [4, ¬∀x¬Gx] However, the acceptability of A1 is questioned on philosophical grounds and Gödel himself considered the possibility to understand the concept of positiveness without assuming A1.6 Our suggestion is to keep ¬P as an axiom, due to its being more intuitive than A1, without impoverishing the language of the formalism by excluding from it the possibility of using negative terms. Following this idea, we accept the weaker meaning of (− ) as contrary negation, taking as an axiom the following: ( ) ϕx → ¬ϕx. From this, only the following can be proved: (A1→ )

P(ϕ) → ¬P(ϕ).

[see 4 →]

This is generally considered an acceptable part of A1, and it is also assumed by A. Anderson in his simplification of Gödel’s argument (Anderson 1990). A falsification of A1← will be shown at the end of the next section. The weakening of (− ) to ( ) yields another interesting result. Since the formula ∀ϕ(ϕx ∨ ϕx) is not valid in our semantics, we can give models in which, for some properties, God does not possess them nor their negations.

5 The inference from ¬P(I ), where: I x ↔ x = x, and A2 to Th1 was elaborated by Czermak (2002, 316). Christian has shown that the Scott theory is deductively equivalent to the theory which comes from it by replacing A1 with ¬P(I ) (Christian 1989, 6–7). Our observations 1 and 4 use the formula ∃ϕ¬P(ϕ) which is weaker than ¬P(I ). 6 In some of his philosophical notes, Gödel considered positiveness in the sense of “purely good” or “assertions” and suggested a simplification of the ontological argument without A1 (Gödel 1995, 435).

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6.4 An S4 Version of Gödel’s Argument with Contrary Terms In his argument for the Cartesian lemma, Leibniz states that: [A]ll propositions which are necessarily true are either demonstrable or known per se. Therefore this proposition is not necessarily true, or it is not necessary that A and B should not be in the same subject. Therefore they can be in the same subject. (Leibniz 1989, 167–168)

An understanding of necessity in terms of demonstrability can be found in many Leibnizian texts, and in some contexts it may also be associated with S4 modalities (Adams 1994, 46–50).7 We want to stress this connection also with respect to our pragmatic strategy to use the weakest possible logic as a basis for the proposed reconstruction. With regard to the proof of the Leibnizian lemma, it is enough to use proposed a specific instantiation of 5 stating that the possible necessary existence of God implies His necessary existence, and not 5 in its full extent. In this way we do not endorse the S5 system of modal logic, which seemed too strong to Gödel himself, and at the same time we are faithful to the original structure of the ´ etorzecka (2012). argument. This idea comes from Swi˛ The language of our approach is the same as that of Q2S5. The logical axioms differ only for schema 5. Instead of this, we assume the weaker (4) A → A The rules are the same as in Q2S5. Now we are in Q2S4. The admissibility of the necessitation rule in a weaker logic than S5 forces us to consider  closures and ∀ closures of the next axioms. Concerning the cases of the comprehension schema, we accept ∀ closures of (G↔ ) Gx ↔ ∀ϕ(P(ϕ) ↔ ϕx), (NE) N Ex ↔ ∀ϕ(ϕEss.x → ∃yϕy), (again ϕEss.x means ϕx ∧ ∀ψ(ψ(x) → ∀y(ϕ(y) → ψ(y)))) and the ∀ closure of the implication ( ) ϕx → ¬ϕx

7 Kovaˇ c

stressed this connection with respect to the passage quoted above. He formalized it in a fragment of second order logic with a ♦ version of axiom 4: ♦♦A → ♦A (Kovaˇc 2017). According to this approach, we begin with the assumption (a1) A then (A is provable from other propositions or true per se). Compossibility of two properties is defined as follows: (Comp) Comp(λx.(Xx ∧ Y x)) ⇔ ♦∃x(λx.(Xx ∧ Y x))(x). We consider two perfections X and Y . We assume indirectly that (1) ¬Comp(λx.(Xx ∧ Y x)). Because (1) is not provable (X and Y are not analyzable) and not true per se, by (a1) we have (2) ¬¬Comp(λx.(Xx ∧ Y x)). By ♦/ and 2 we obtain (3) ♦Comp(λx.(Xx ∧ Y x)) and so by ♦ version of 4 we obtain (4) ♦∃x(λx.(Xx ∧ Y x))(x).

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The following are the next specific axioms used in our formulation: – ∀ closures of (A2) P(ϕ) ∧ ϕ ⊂ ψ → P(ψ), –  closure of (¬P) ∃ϕ¬P(ϕ), – formulas:

(A4) P(ϕ) → P(ϕ),

(A3) P(G), (A5) P(N E), (A6) ♦∃xGx → ∃xGx. Our theory is the extension of Q2S4 by the above axioms: ¬P = Q2S4[¬P, A3, A5, A6, ∀( , G↔ , NE, A2, A4)]. GO 4 In simple words, we developed an S4 version of Gödel’s ontological argument allowing contrary predicate terms. ¬P is  transparent like GOo (cf. ft. 3) The closures A3 and A5 are The GO 4 to be obtained from A4. We get A6 from ¬∃xGx ∨∃xGx [A6, → /∨, ♦/] applied to A ∨ B ↔ (A ∨ B) [S4]. It is interesting that, unlike GOo our approach blocks full P-necessitarianism as expressed by the equivalence: (NEC P) ♦P(ϕ) ↔ P(ϕ) ↔ P(ϕ). ¬P we do not get the implication ♦P(ϕ) → P(ϕ), which is equivalent In GO 4 to ¬P(ϕ) → ¬P(ϕ) added by Gödel to the axiom A4. Although, the formula ♦P(ϕ) → P(ϕ) is independently derivable in GOo via A1 or the instantiation of 5: ♦P(ϕ) → P(ϕ). However, we believe, once again that our approach is closer to a Leibnizian point of view. Let ϕ represent such a relative property as ‘being the creator of a possible world w’. God may create any one of infinitely many worlds, but He chooses only one, and this is the actual world. By definition, God possesses only positive properties, from which it follows that our property is possibly positive. But if ♦P(ϕ) → P(ϕ) is valid, then the above property relative to any possible world (including worlds that are not actual) is also positive. It follows from this that God, Who possesses all positive properties, is the creator of infinitely many possible worlds. This is obviously rejected by Leibniz. The reconstruction of Gödel’s argument is now simple. The formula Th1 is derivable in our theory in the same way as in fact 1. The Cartesian lemma CL follows directly from Th1 and A3. The proof of Th2 is formed as in fact 2. The Leibnizian lemma LL is derivable in the same way as in GOo (in step 7 of the proof we use our instantiation of 5, which we call A6). Finally the thesis Th3 is derived from CL and LL. ¬P [∀I ] At the end of our consideration let us describe a model for GO x 4 ← which falsifies the problematic implication A1 , the formula Th5 expressing ♦Pnecessitarianism, and also Th15 according to which all possible properties of God are His actual properties. We adopt a modified semantics for GOo [∀Ix ] given by Hájek (2002). A Kripke frame is K = W, R, D, P rop, P, where: W is a set of possible worlds; R is a reflexive and transitive accessibility relation in W ; D is a set of individuals (all sets are nonempty); P rop is a set of mappings D × W → {0, 1} representing properties and such that:

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(*) ∀p∈P rop ∃p ∈P rop ∀d∈D ∀w∈W (p(d, w) · p (d, w) = 0); P is a mapping that assigns value 1 to properties that are positive in possible worlds: P rop × W → {0, 1}, where (**) ∀p∈P rop ∀w∈W (P(p, w) = 1 ⇒ ∀w (wRw  ⇒ P(p, w  ) = 1)). The constant domain of individuals is not needed in case of our logic, however it ¬P [∀I ]. simplifies our interpretation of GO x 4 We do not assume that the universal property is an element of P rop. In Hájek’s semantics, the characteristics of P In not dependent on possible worlds. The valuation function is v(x) ∈ D, v(τ ) ∈ P rop; for contrary terms we have: v(τ ) = p ⇒ ∃p v(τ ) = p , where: ∀d,w (p(d, w) · p (d, w) = 0). Taking valuation v, the validity of formulas in possible world w ∈ W is described as follows: K, w K, w K, w K, w

|v |v |v |v

x=y ϕx P(ϕ) A

iff iff iff iff

v(x) = v(y) v(ϕ)(v(x), w) = 1 P(v(ϕ), w) = 1 ∀w ∈W (wRw  ⇒ K, w  |v A)

The conditions for ¬A, A → B, ∀xA, ∀ϕA and ♦A are usual. The truthconditions for G, and NE are already determined in the following way: K, w |v Gx K, w |v N Ex

iff iff

K, w |v ∀ϕ(P(ϕ) ↔ ϕx) K, w |v ∀ϕ(ϕEss.x → ∃xϕx) (cf. (Ess.))

A formula A is valid in M iff ∀w∈W K, w |v A, for all valuations v. Similarly to Hájek, we assume that our structures fulfill the following conditions: (C.I ) (C.N E) (C.G)

∀x∃ϕ∀y(ϕy ↔ y = x) is valid ∃ϕ∀x(ϕx ↔ ∀ψ(ψEss.x → ∃yψy)) is valid ∃ϕ∀x(ϕx ↔ ∀ψ(P(ψ) ↔ ψx)

and that there is a g ∈ D such that: (g.1) (g.2)

∀p∈P rop ∀w∈W (p(g, w) = 1 ⇒ ∀w ∈W (wRw  ⇒ p(g, w  ) = 1)) ∀p∈P rop (P(p, w) = 1 iff ∀w ∈W (wRw  ⇒ (p(g, w) = 1)))

¬P [I ] are valid in the structures described above. All theorems of GO 4 x K , v  where K =< W , R , D , P rop Now we consider M = K rop, P >;

• • • • • • • • • •

D = {g, a}; W = {w1 , w2 }; R = {< w1 , w2 >, < w1 , w1 >, < w2 , w2 >}; P rop = {G∗ , NE ∗ , Ia∗ , p∗ , ∅∗ , U ∗ }, where: G∗ (g, w1 ) = G∗ (g, w2 ) = 1, G∗ (a, w1 ) = G∗ (a, w2 ) = 0; N E ∗ = G∗ ; Ia∗ (a, w1 ) = Ia∗ (a, w2 ) = 1, Ia∗ (g, w1 ) = Ia∗ (g, w2 ) = 0; p∗ (g, w1 ) = p∗ (a, w1 ) = 0, p∗ (g, w2 ) = p∗ (a, w2 ) = 1; ∀d∈D ∀w∈W (∅∗ (d, w) = 0 and U ∗ (d, w) = 1); P (G∗ , w1 ) = P (G∗ , w2 ) = 1; P (Ia∗ , w1 ) = P (Ia∗ , w2 ) = 0; P (p∗ , w1 ) = 0, P (p∗ , w2 ) = 1; P (∅∗ , w1 ) = P (∅∗ , w2 ) = 0 and P (U ∗ , w1 ) = P (U ∗ , w2 ) = 1.

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Let v (x) = g, v (G) = G∗ , v (N E) = NE ∗ , v (ϕ) = p∗ , v (ϕ) = ∅∗ . Now we can see that: (*) K , w1 |v P(ϕ) ∨ P(ϕ). Thus the axiom A1← of GOo is falsified in our semantics. The proposed formalization departs from Gödel’s approach with respect to both the modalities in the strong sense of S5 and classical term negation. Despite the fact that our analysis brings Gödel’s ideas close to Leibniz’s view only to a limited extend, a few promising details can be highlighted that capture the Leibnizian concept of God. Let us note that (**) K , w1 |v ♦P(ϕ) → P(ϕ),

(***) K , w1 |v Gx → (♦ϕx → ϕx).

According to (**), our formalism rejects ♦P-necessitarianism. According to G↔ and (*), it also rules out a conception of God as possessing every property or its negation.8Lastly, it can happen that some possible properties of God are not His actual properties (***), although all of His actual properties are necessarily possessed by Him (Th14 is provable in our theory). The question of whether our theory makes a significant contribution to the whole of Leibnizian natural theology remains open. Our goal was much more modest: it was bringing a few of Gödel’s theological ideas closer to Leibnizian ones following the main structure of a formalism proposed by the former.

References Adams, R. M. 1994. Leibniz. Determinist, Theist, Idealist. Oxford: Oxford Univ. Press. ——. 1995. “Introductory note to *1970”. In K. Gödel, Collected Works, vol. 3, 388–402. Anderson, C. A. 1990. “Some emendations of Gödel’s ontological proof”. Faith and Philosophy 7: 291–303. Christian, C. 1989. “Gödel Version des Ontologischen Gottesbeweises”. Sitzungsberichte der Österreichischen Akademie der Wissenschaften, Abt. II 198: 1–26. Czermak, J. 2002. “Abriss des ontologischen Argumentes”. In Kurt Gödel. Wahrheit und Beweisbarkeit, vol. II. Kompedium zum Werk, ed. B. Buldt, E. Köhler, M. Stöltzner, P. Weibel, C. Klein, W. DePauli-Schimanowich-Göttig, 309–324. Viena: ÖBV et HPT VerlagsgmbH and Co. KG. Futch, M. 2008. Leibniz’s Metaphysics of Time and Space. Boston: Springer. Gödel, K. 1970. Ontologischer Beweis. February 10th 1970. Faksimile from Nachlaß reprinted in: Kurt Gödel. Wahrheit und Beweisbarkeit, vol. II. Kompedium zum Werk, ed. B. Buldt, E. Köhler, M. Stöltzner, P. Weibel, C. Klein, W. DePauli-Schimanowich-Göttig, 307–308. Viena: ÖBV et HPT VerlagsgmbH and Co. KG. ——. 1995. “Texts relating to the ontological proof”. In Kurt Gödel, Collected Works, ed. S. Feferman et al., vol. 3, 429–437. Oxford: Oxford University Press. Hájek, P. 2002. “Der Mathematiker und die Frage der Existenz Gottes (betreffend Gödels ontologischen Beweis)”. In Kurt Gödel. Wahrheit und Beweisbarkeit, vol. II. Kompedium zum

8 The

Leibnizian God is neither finite nor infinite in time and space (Futch 2008, 171–194).

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Werk, ed. B. Buldt, E. Köhler, M. Stöltzner, P. Weibel, C. Klein, W. DePauli-SchimanowichGöttig, 325–336. Viena: ÖBV et HPT VerlagsgmbH and Co. KG. Kovaˇc, S. 2003. “Some weakened Gödelian ontological systems”. Journal of Philosophical Logic 32: 565–588. ——. 2017. “The Concept of Possibility in Ontological Proofs”. Presentation of the contributed paper in 2nd World Congress on Logic and Religion University of Warsaw, 18.06–22.06.2017. Leibniz, G. W. 1987. Sämtliche Schriften und Briefe. Reihe II: Philosophischer Briefwechsel. Band 1. Auflage Darmstadt 1926; zweiter, unveränderter Nachdruck Berlin 1987. (Available on the Internet: http://www.uni-muenster.de/Leibniz/DatenII1/II1_B.pdf 11.09.2017). ——. 1989. Philosophical Papers and Letters, ed. II, transl. and ed. by L. E. Loemker, The New Synthese Historical Library, vol. 2. Dordrecht: Kluwer Academic. Perzanowski, J. 1989. Logiki modalne a filozofia [Modal Logic and Philosophy]. Cracow: Jagiellonian University. ´ etorzecka, K. 2012. “Ontologiczny dowód Gödla z ograniczona˛ redukcja˛ modalno´sci” Swi˛ [“Gödel’s ontological proof with limited reduction of modalities”]. Przeglad ˛ Filozoficzny Nowa Seria 3 (83): 21–34. ´ Swietorzecka, K. ed., 2016. Gödel’s Ontological Argument. History, Modifications, and Controversies. Warszawa: Semper. ´ etorzecka has a PhD in philosophy (philosophical logic) from the Cardinal Stefan Kordula Swi˛ Wyszy´nski University in Warsaw (CSWU). Presently she is professor at CSWU and head of the Department of Logic in the Institute of Philosophy of CSWU. Marcin Łyczak is a doctorate student of at the Cardinal Stefan Wyszy´nski University working ´ etorzecka in a project funded by the Polish Ministry of Higher under the supervision of Kordula Swi˛ Education.

Chapter 7

A Tractarian Resolution to the Ontological Argument Erik Thomsen

The [ontological] argument does not, to a modern mind, seem very convincing, but it is easier to feel convinced that it must be fallacious than to find out precisely where the fallacy lies.—Bertrand Russell (1946), History of Western Philosophy, p. 585

7.1 Introduction An ontological argument for the existence of God begins with one or more aspects of our common intuition intended to be so self-evident that they can be taken as analytic or necessary premises. And then attempts, using only logic as reasoning tool, to conclude that there must exist something in our shared reality that exemplifies or corresponds to the concept of God. Although there are numerous exemplars (Nagasawa 2011) and even classifications of ontological arguments (including definitional, conceptual, modal, and mereological (Oppy 1995), they all rely on some notion of existence in both premises and conclusion. For example, Anselm (1965) relies on the existence in reality of something that exemplifies a concept as being greater,1 in some sense (having added the generic qualifier ‘in some sense’ because Anselm does not

1 “And

certainly that greater than which cannot be understood cannot exist only in thought, for if it exists only in thought it could also be thought of as existing in reality as well, which is greater”. From Chapter 2 ‘That God really exists’ in Anslem https://sourcebooks.fordham.edu/ source/anselm.asp.

E. Thomsen () Blender Logic, Inc., Cambridge, MA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_7

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provide an operational definition of what it means to be greater), than the mere concept itself; Descartes (1968) relies on existence being a perfection; Leibniz (1896) distinguished existence from essence; Kant (1933) relies on existence being implicitly associated with the logical subject of an assertion; Frege (1980) equates existence with number, and Plantinga (1974) treats existence as a modal possibility. The notion of existence and how it ought to be associated with the premises and/or conclusions that make up the individual propositions belonging to an ontological argument for God is thus one major problem that must be solved to either construct a successful ontological argument or to successfully demonstrate that no such argument is possible. Either outcome may be called a resolution of the ontological argument. A second major problem that receives nowhere near the same amount of attention, but which nevertheless must be solved, is understanding how the semantics (or definitions) of key terms (e.g., ‘God’, perfections) can impact the validity, soundness (and triviality) of a logical argument. For example, the mereological argument attributed to Lewis in (Oppy 1995) rests on a very different semantics for the term ‘God’ than do most other ontological arguments (whose semantics will be discussed below). By equating God as the mereological sum of what exists, God is stripped of its divine attributes and defined simply in terms of non-sentient existence. From the fact that anything exists (e.g., you now reading this paper) it is trivial to conclude that something exists. In this sense, Lewis’ (1970) semantics trivializes the ontological argument. The focus therefore, in this paper, will be on two kinds of ontological arguments that (1) exemplify the wrestling within the logic community over how to associate existence with propositions (e.g., through explicit predication, through logical subjects, or some new way), and that (2) illustrate the importance of term semantics to the construction and characterization of logical arguments and the propositions of which they are composed. Specifically, I will discuss (1) Descartes’s reformulation in the Meditations of Anselm’s original argument where necessary existence is treated as a valid predicate; and (2) a version of the ontological argument not found in the literature outside of (Mion 2018) that I include as a means to highlight the impact of existential generalization on any ontological argument and on existential entailments more generally.2 Ontological arguments remain of current interest in large part because they highlight areas where there is still disagreement about relevant logical principles. For example, by rejecting the Kantian hypothesis where existence is entailed by the logical subject of an assertion, (which one might argue had been done in an attempt to refute Anselm’s and Descartes’s ontological argument) and by treating existence as a real predicate, Priest (2008) and Berto (2013) now seem to be returning to Descartes’s implicit treatment of existence as a real perfection (i.e., predicate) for

2 Leibniz’s

shoring up of Anselm’s argument by more precisely justifying the ability to conceive of a perfect being is interesting but outside the scope of this paper which focuses on those aspects of the Ontological argument directly impacted by logic’s approach to existence.

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which the problems in so doing were well understood by Kant. This flip flop is but one indication of a foundational problem having to do with the essence, not of God, but of the components of a proposition (i.e., logical subject and predicate) and with the relationship between logic and the world (a point of contention between Wittgenstein and his view of Russell and Frege (TLP 5.4)) whose implications run through the heart of modern logic, both classical and non-classical. In this paper, I attempt to resolve both kinds of ontological arguments. (admittedly with the less-than-shocking conclusion that they fail, but with new reasons for why they fail). I will try to do this by showing (1) the fundamental mistakes that occur in both Descartes’s argument and any argument from existential generalization; and (2) how these mistakes reflect a foundational problem that lies at the heart of both classical logic and the supposed real predicate fixes of Meinong, Priest and Berto that diverge from the classical tradition. This will be done by using a logic consistent with the principles laid out in Wittgenstein’s Tractatus (Thomsen 1990). These principles include a radical reinterpretation of the components of a proposition that defines logical subjects and functions/predicates in terms of sequenced computational processes instead of as references to general objects and properties. Towards that end, the rest of the paper is divided in three sections. In Sect. 7.2, I provide an analysis of Descartes’s argument and an illustrative argument from existential generalization and include a critical discussion of Priest’s recent claim that existential entailment is irrelevant. In Sect. 7.3, I describe relevant aspects of an alternative approach to logic; what might be called Tractarian logic. And in Sect. 7.4, I attempt to resolve the two contrasting ontological arguments by recasting them in Tractarian terms.

7.2 Problems with the Ontological Arguments The two major problems with the ontological arguments—existential implications and term semantics are now discussed. 1. Existential implications are the first problem with ontological arguments. At first sight, ontological arguments highlight logic’s reliance on some notion of existence either being implicitly entailed by the logical subject of a true proposition or being explicitly predicated. For Descartes, existence (i.e., necessary existence) was a perfection (i.e., a positive predicate).3 Consider Descartes argument from his Meditations on First Philosophy; Fifth meditation (p. 24): Whenever it happens that I think of a first and a sovereign Being, and, so to speak, derive the idea of Him from the storehouse of my mind, it is necessary that I should attribute to Him every sort of perfection, although I do not get so far as to enumerate them all, or to

3 Descartes

also thought that whatever could be clearly and distinctly perceived to be contained in the idea of something was true of that thing.

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apply my mind to each one in particular. And this necessity suffices to make me conclude (after having recognized that existence is a perfection) that this first and sovereign Being really exists.

Descartes’s argument can be put in a more explicitly logical form as follows:

Descartes’s Argument Premise (#1): I have an idea of a supreme being that has all perfections; Premise (#2): Necessary existence is a perfection. Conclusion (#3): A supreme being (i.e., God) necessarily exists. For Kant, the fault in Descartes’s argument lay in the second premise. Existence could not be allowed to be a perfection (i.e., a predicate) lest Descartes’s conclusion prove true. In his motivation to disallow the existential predicates assumed by Descartes, (and Anselm, Leibniz, and Spinoza (1955) before), Kant argued in the Critique of Pure Reason (1787) that existence is analytically presumed in the logical subject and so not a real predicate.4 Kant’s view of the existential import of logical subjects was later baked into the fabric of what is called the existential quantifier in classical predicate or first order logic ‘FOL’. However, from the standard interpretation of the existential quantifier, another kind of ontological argument can arise;5 one based directly on the classical notion of existential generalization ‘EG’ where f(a) implies ∃x f(x). (Read: f(a) implies that there exists an ‘x’ such that f is true of x.) Consider, now, an EG argument.

(2) An EG Argument Premise: God is perfect. Conclusion: There exists an X (e.g., ‘God’) such that X is perfect. Since the asserted premise is true by definition, the conclusion is not only valid, but also sound. Therefore, God exists. But something is clearly wrong. This EG argument seems like it should be trivially invalid. It shouldn’t be that easy to infer an existential claim for the logical subject of an assertion.

4 In

On Denoting (Russell 1905) p. 491, Russell also criticized Descartes’s argument. But his criticism stating that Descartes’s ontological argument fails for “want of a proof of the premise” was unfair because Descartes treated God’s perfection as analytic (coming from ‘the storehouse of my mind’). 5 Though not an ontological argument in the sense of an argument widely discussed in the literature, the EG argument is a way of exemplifying the problems that arise from associating the quantification of a variable with any kind of existential entailment which is a significant problem as described elsewhere in this essay.

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So, it is not surprising that the entailment of existence from the logical subjects of true assertions has been challenged. Beginning in the last century, such non-classical thinkers as Meinong, (in Modenato 1995) and more recently Lambert (2003), Oppenheimer and Zalta (1991) Priest (2008) and Berto (2013), found the classical, or logical subject-based FOL view of existential entailments to be problematic. There are two aspects to this more modern view of the problem. The first aspect relates to the EG argument presented above where non-classical thinkers did not wish to see existential claims implicitly attributed to the logical subject of an assertion. Rather, they wanted to be able to freely assert or negate the (real) existence of the logical subject of a true assertion. The second aspect is the classically interpreted FOL’s seeming inability to support useful reasoning over fictional objects (e.g., Sherlock Holmes) which is something these same individuals quite understandably wanted to be able to do. And in a formally supported way. Solving both aspects to the problem, it was argued, required existence to be a real predicate and, concomitantly, for the quantifier to free itself of any existential implications. What was hitherto called the existential quantifier is now referred to by some as the particular quantifier; sometimes even given a new symbol as with Lambert’s (2003) Free Logic. While the recent advocacy for existence as a real predicate seems to have solved the second aspect to the problem of existential entailments, namely to allow useful reasoning over logical subjects that admittedly have no reference, it came at a heavy cost; for it reopened the door to the problems in Descartes’s original existence-as-a-predicated-perfection ontological argument. Thus, in the sense of whether existence is associated with the predicate or logical subject of a proposition, we are back to the view initially espoused by Descartes that existence is a real perfection or predicate.6 The logical approach to existential entailments has come full circle. 6 Some

authors who follow in the wake of Meinong (e.g., Routley (1980), Paoletti (2013), Priest (2016), and Bacigalupo (2017)) debate the issue of existential entailments using the term characterization instead of ‘predications assumed to be true of, or that are part of the identity of, an object’. Priest (2016), for example, asserts on p xviii (and then in chapter 4 beginning p. 83 under the heading ‘characterization principle’ or CP) that “an object has those properties that it is characterized as having”. On closer examination, however, this characterization principle reduces to the circular and trivial assertion that an object for which certain predicates are assumed true (i.e., the properties the object is characterized as having) may be assumed to be truthfully predicated with those predicates (i.e., the object has those properties it is characterized as having). Priest himself criticizes CP as being too general, but when it comes to discussing non-existent objects, for example on page 13, he uses the more traditional language of predicates and refers to existence as a special predicate. Regardless whether he eventually recasts the notion of predicatable existence in suitably restricted CP terms, it is clear that he treats existence as something that may or may not be predicated of an object. (E.g., Sherlock Holmes is characterized as a detective which is to say that it is assumed that the predicate ‘is a detective’ is true of the object Sherlock Holmes. But this characterization is independent of whether Sherlock Holmes exists). So, although Priest and others use, at times, different terminology, they adhere to the notion that existence is not entailed by the logical subject of a true proposition but rather that existence is associated with an object through predication. Their arguments thus fit into the existence-as-predicate group described in this paper and so are a part of the train of logical approaches to existence that have come full circle.

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2. Term semantics are the second problem with ontological arguments. Though it is common to see analyses of Descartes’s argument (Sobel 2004), (Priest 2017), omit the propositional attitude ‘I have an idea that x’ in order to focus on the existential implications of a supreme being having all perfections (e.g., Sobel, pp. 32–39), doing so fundamentally changes the semantics of the argument by placing both the premises and the conclusion of Descartes ontological argument in the world of synthetic propositions. As shown by Sobel, when recast in synthetic terms, Descartes’s first premise (recast on page 32 as ‘A supremely perfect being has every perfection.’) either trivially presupposes the conclusion or yields an invalid argument depending on whether the first premise is taken to mean that there exists a supreme being with all perfections (from which the conclusion trivially follows) or whether it is taken to mean that if there is a supreme being it would have all perfections (in which case the conclusion does not follow). In addition to showing the futility of trying to produce a non-trivially valid ontological argument with synthetic premises, Sobel’s analysis shows how changes in term semantics can impact a logical argument. Descartes’s original argument is, I believe, more sophisticated than accounted for by Sobel’s synthetic rendering. This is because Descartes’s first premise (as per his fifth meditation cited above) is that of an a priori idea in the mind.7 And Descartes’s goal was to demonstrate that from an a priori idea (i.e., a premise) in the mind we can nonetheless deduce the existence of God in reality. Descartes’s ‘trick’ was the inclusion of an existential commitment (necessary existence) as a second premise in a seemingly non-controversial way, that by virtue of the first premise is accorded both an a priori and synthetic status. The semantics of an ontological argument are also impacted by the definitions of the terms that (remain in and) make up the argument. Traditional approaches to the ontological argument implicitly treat the semantics (or meanings) of salient terms as a part of common intuition and so not requiring explicit treatment. Intuition says that ‘God has all perfections.’ must be analytic. But nowhere in Anselm’s original argument or in Descartes’s meditations or in Kant’s reflection on the arguments is there any treatment of the semantic relationship between crucial terms like God and perfect. When asserting the premise that ‘God has all perfections’, what is the presumed relation between God and all perfections? Could God have

7 Independent

of the metaphysical status one may wish to ascribe to Descartes’s postulation of a supreme being having all perfections e.g., whether one wishes to treat it as a clear and distinct idea—namely the idea of a supreme being having all perfections or as simply a definition of same, Descartes needs to link these premises that are not empirical (regardless whether ideas or a priori or analytic definitions) with something that one might call real or ‘in the world’ or ‘empirical’ or ‘synthetic a posteriori’. His method of linking was his second premise that existence (in the world) is a perfection. Since there are no epistemically neutral terms that can be used to describe definitions/ideas or empirical/synthetic statements, the author made every attempt to use extent terms in non-controversial ways so as to support the major thesis in the paper on the value of a Tractarian style logic to resolving ontological arguments without entering into epistemic debates that are beyond the scope of this paper.

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any imperfections? If so, then the assertion that God has all perfections becomes synthetic. And Descartes’s argument becomes invalid.

7.3 Tractarian Logic Before introducing the relevant aspects of Tractarian logic, it is useful to briefly summarize the background consensus now called First Order Logic or FOL, against which Wittgenstein was arguing. That consensus can be exemplified in two parts. First, by Frege’s characterization of an assertion8 in terms of f(a) where ‘a’ denotes an N-adic logical subject presumed to refer to an ordered set of n objects and ‘f’ denotes what Frege called a function (now also called a predicate) presumed to refer to a set of objects possessing whatever is the predicate. Common also nowadays is the notion that for a true proposition, what is referred to by the logical subject is a subset of what is referred to by the predicate (e.g., The assertion ‘The book is blue’ can be interpreted as stating that the book referred to by the logical subject is a subset of the set of blue things). The referential nature of the ‘f’ and the ‘a’ has been a central feature of logic from the time of Aristotle to today. Second, FOL (and even its non-classical offshoots, e.g., Lambert (2003), Oppenheimer and Zalta (1991), Priest (2008)) treat individual declarative expressions (whether called wffs, or sentences) as corresponding to individual assertions/propositions (i.e., with a 1-1 relationship between them).

7.3.1 Tractarian Depiction of the Relationship Between the Logical Subject and Predicate of a Proposition in Terms of Computational Sequence In contrast, the Wittgenstein of the Tractatus ‘TLP’ (1977) voiced concerns about the referential interpretation of the components of a proposition f and a. (3.333). Wittgenstein also distinguished wff/sentences from propositions (5.4733) and suggested a sequential process of evaluation for distinguishing wff that are propositions from those that are not. Legitimate propositions in this scheme maintain their bivalence (4.023). The process of sequenced evaluation suggested by Wittgenstein is inherently non-commutative (because, as will be shown, different sequences have different truth conditions) and so breaks with the tradition originally espoused by Boole

8 Whether

in conjunction with an existential quantifier (e.g., “There exists an X such that f(x)”), or a universal quantifier, (e.g., “For all ‘x’ f(x)”), Hilbert, Peano, Russell, Carnap, Quine and Putnam to name but a few, all used a symbolism based on the notion of a predicate ‘f’ and N-adic logical subject ‘(a)’ to formally denote and reason about propositions.

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Fig. 7.1 The fact of a white sheep

(1854). Given Boole’s profound influence on modern logic, and in order to provide a more tangible reference point for understanding the unconventional aspect to Wittgenstein’s approach, I succinctly review Boole’s major idea below. From ‘The Laws of Thought’, Chap. II on signs and their laws, Boole treats multi-part descriptors as set operations. And in the case of multi-term adjectives or verbs associated with an object, he treats them as intersections. Thus, he writes “If x alone stands for white things and y for sheep, let xy stand for white sheep” [p. 28]. This is normally understood in an extensional sense, (often with the help of a Venn diagram) as the intersection of the set of white things and the set of sheep things. According to Boole, the intersection is indifferent to order of operation: xy = yx. In Chap. IV division of propositions, he treats propositions with the same methods. He distinguishes logical subjects and predicates. But there is no treatment of the sequential aspects to computation. To explain Wittgenstein’s alternative view based on sequenced evaluations, let us now turn to the complex fact, illustrated in Fig. 7.1, of a white sheep. In (5.5423), Wittgenstein states that two individuals might see different propositions or logical arrangements in the same fact/complex.9,10 Since the complex fact is represented by two terms: “sheep” and “white”, we can form two distinct propositions: “White(sheep)” abbreviated as ‘W(s)’ and “Sheep(white)” abbreviated as ‘S(w)’, from the one sentence or wff ‘The sheep is white’.11 Since, as shown in the detailed example below, the truth conditions for these two assertions are

9 The

complexity of the fact, what Wittgenstein calls “logical multiplicity” (4.04) governs the number of distinct propositions that can be generated for a single fact/complex. 10 Wittgenstein’s views on the multiplicity of propositions that can be generated from a single complex fact can be traced back to Aristotle’s (n.d-a, n.d-b) original dialectic (De Interp. (20b 22–31), Categoriae, 2a 4–10, 13b 10–12) which begins by looking at an assertion as the answer to a question; not simply as a declarative statement. Aristotle situated logic within the context of an affirming/denying game (the “dialectic”, in its original sense), and defined assertions (i.e., propositions) as the primitive units of this game. He further diagnosed a certain compositeness of type as their defining character, distinguishing that which an assertion was asserting from that of which the assertion was being made. 11 The presence of a name (“Sheep”) where a predicate generally occurs, and the presence of a predicate (“white”) where a name generally occurs in “Sheep(white)” will be dealt with soon.

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Table 7.1 Complexes and supported propositions

1 2

The specific fact (verbally represented) A white sheep A blue sheep

Purported proposition pp #1 W(s) True proposition False proposition

3

A white cow

4

A blue cow

logical subject not found; predicate is unevaluable logical subject not found; predicate is unevaluable

Purported proposition pp #2 S(w) True proposition logical subject not found; predicate is unevaluable False proposition logical subject not found; predicate is unevaluable

different, the issue cannot be brushed off as mere surface grammar. Consider Table 7.1. Table 7.1 specifies (1) two alternate purported propositions ‘pp’ that can be generated from the one sentence/wff ‘The sheep is white’ (labeled as purported propositions because we cannot assume they will evaluate); (2) a series of facts beginning with the fact pictured in Fig. 7.1 corresponding to row 1 of the table; (3) a collection of related facts in rows 2–4; (4) in the cells that form the intersection of a fact and a pp, the evaluation of the pp when applied to the fact; Fact elements that match pp elements (in either pp) are highlighted in italics. Studying Table 7.1, notice that the facts specified in rows 2–4 diverge from the fact in row 1 that made both pp. true propositions. Note also that the two pp differ in terms of the conditions by which their status changes from True to False or from True to unevaluable. Thus in row 2, the fact varies by one element from the original fact: The sheep is blue instead of white. For pp #1, this makes the pp a false proposition. This is because the logical subject, namely “sheep”, still matches the fact. But the factual attribute is blue not white so the pp which asserts that the attribute is white is a false proposition. In contrast for pp #2, its logical subject no longer matches an element of the fact. Nothing corresponding to the logical subject ‘white’ is found. So the predicate “sheep” never gets to evaluate and the pp. as a whole is considered unevaluable, because the logical subject fails to match any part of the fact. This is because from pp #2, we cannot find an instance of white from where we can evaluate the predicate that identifies what is white. In row 3 the original fact is altered by a different element, namely the identity of the object that is white. So here pp #1 becomes unevaluable because its logical subject, “sheep”, cannot match any element of the fact; whereas pp #2, which had been unevaluable relative to fact 2, is now simply a false proposition. In row 4, the original fact is completely altered. And neither of the two pp are evaluable. Since the same facts that make some pp. false propositions make others unevaluable (and thus not propositions in the classical bi-valent view) and vice versa, we conclude that the sentence as is (e.g., The sheep is white.), is inherently ambiguous; and the logical (dare I say computable) meaning of a sentence can only be captured by a sequenced process of evaluation that (1) differentiates between pp that do and do not evaluate as propositions, and (2) maintains classical bivalency for

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those that do evaluate as propositions (e.g., pp #1 is a true proposition for fact 1 and a false proposition for fact 2; while pp #2 is a true proposition for fact 1 and a false proposition for fact 3). It is important to distinguish false and unevaluable; otherwise one would not be able to distinguish between a fact that negates an assertion (e.g., the fact in row 2 for pp #1) and a fact that has no impact on the truth or falsity of the assertion (the fact in row 4 for both pp #1 and pp #2). Stated alternatively, given the proposition “White(sheep)”, it is clear how the fact of a blue sheep would negate the assertion: “No, the sheep isn’t white; it’s blue.” But it is not clear why an independent fact such as ‘The cow is blue’ should negate the proposition “White(sheep)”: “No the sheep isn’t white. The cow is blue”. If the fact that the cow is blue is allowed to negate “the sheep is white”, why not the fact that French is the language of France, or 7 + 2 = 9? Yet, this is exactly what would happen if false is not distinguished from unevaluable. The door would have been opened to allow any non-matching fact to negate an assertion. One could argue that Wittgenstein is doing no more than suggesting a threevalued logic, e.g., Lukasiewicz (1970) and Kleene (1952): a wff is either true or false or unevaluable. But this would ignore Wittgenstein’s strong commitment to Bivalence (4.023) and his two-phase sequenced evaluation process. For Wittgenstein, true and false depend on the prior establishment of a pp being a genuine proposition (i.e. on the logical subject matching an aspect of the fact thereby allowing for the predicate to evaluate). It would be hard to overestimate the degree to which this sequenced approach to evaluation is a radical departure from conventional approaches to logic and has significant consequences for reasoning over real world information domains (Thomsen 2002) and, as will be shown, on resolving the ontological arguments previously described. If an assertion is true, the process of evaluating the predicate must succeed for just that location identified by the logical subject which had to previously succeed in matching some aspect of the fact(s).

7.3.2 The Tractarian Relationship Between Logic and the World and Resulting Explicit Incorporation of Term Semantics In (3.33) Wittgenstein writes that “in logical syntax the meaning of a sign should never play a role”. Thus, for Wittgenstein, logical syntax (or grammar) is orthogonal to semantics, in the sense that categories of logical syntax (e.g., logical subjects or predicates) can be arbitrarily correlated with meanings (e.g., objects, processes, attributes or relations). As shown above, the single fact of a white sheep could be the veridical source for two distinct propositions: “White(sheep)” and “Sheep(white)”. The two propositions are each comprised of an object ‘sheep’ and an attribute ‘white’. Though it may not be standard to treat an attribute as the logical subject and an object as the predicate, no logical rules are violated by doing so. In fact, each

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proposition represents an answer to a legitimately distinct question. “White(sheep)” answers the question “What is the color of the sheep?”. “Sheep(white)” answers the question “What is it that is white?” The fact that Sheep and white can each figure in the logical subject or predicate role of a proposition, means that neither logical subjects nor predicates refer in isolation to any specific kind of thing in the world (e.g., logical subjects need not refer to some general object).12 Only when the logical subjects and predicates have been associated with semantic variables (e.g., specific objects and attributes) do the ensuing propositions refer. Thus, there are no semantic (or ontological or real world) implications to the fact that “sheep” is treated as a logical subject in a proposition. For in another it may be a predicate. The propositional roles of logical subject and predicate are thus orthogonal to whatever semantic types are (extra-logically) postulated as comprising the world. I believe that Wittgenstein’s view of propositions highlights not only the orthogonal relationship between logic and semantics (and the world), but also the treatment of logical variables as computable elements based on their type of semantics. (For example, if the logical subject were a region of Earth, its type might be comprised of a longitude, latitude and altitude.) To use modern software terminology, Wittgenstein saw logic as a strongly typed system for representing and reasoning with information regardless of origin or meaning (Thomsen 2002). By separating logic from the world, the Tractarian approach is freed from needless existential implications and is capable of fine tuning the expression of ontological commitments based on the specific ontological realization of the computation (e.g., in your head, in your computer or in the remainder of the real world).

7.4 Resolving the Ontological Debate Using Tractarian Logic Now it is time to revisit the ontological arguments described in Sect. 7.2 to see how Tractarian Logic provides a clean resolution to both sides of the existential debate. Let’s begin with what was called an EG argument.

(2) An EG Argument Premise: (1) God is perfect. Conclusion: There exists an X (e.g., ‘God’) such that X is perfect. For a Tractarian approach, there is no issue with the assumption that the premise ‘God is perfect.’ is true. Nor is there any issue with treating that truth as analytic. 12 See

sections 3.314–3.317: “And the only thing essential to the stipulation is that it is merely a description of symbols and states nothing about what is signified” (3.317).

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There are implications for how the terms ‘God’ and ‘perfect’ are defined to support those assumptions; but not with the assumptions themselves. Based on the desired conclusion, (the existence of God in the world), it is reasonable to assume that ‘God is perfect’ has the form Perfect(God). And it is reasonable to assume that what is perfect is defined as a supertype of God. This is because predicating a supertype of a type (e.g., mammal(dog), liquid(water), color(blue)) is true by definition (i.e., is an analytic truth) whereas predicating a subtype of a type (dog(mammal), water(liquid), blue(color)) or predicating a type of an independent type (blue(book), hot(water)) is contingently true. The Tractarian treatment of the premise as an analytic truth means that the logical subject of the proposition (i.e., God) has been found and the predicate, ‘is perfect’, successfully evaluates relative to God. But it doesn’t mean that God exists in the world. The existential implications follow from where the proposition is evaluated: in the external world of facts or our internal worlds of definitions. The only way for Perfect(God) to have the status of an analytic truth about the world would be for God to have been found through empirical means and for what is perfect to have been found through empirical means and for what is perfect to be—by definition—a supertype of God. Otherwise, the only way to keep the analyticity of the asserted premise ‘God is perfect.’ is to restrict its evaluation to the internal world of definitions. In this case, the logical subject ‘God’ is only committed to being found, when evaluated, amongst the definitions. As a result, the conclusion (which, by definition, is about God in the world) would be invalid. In other words, from the premise ‘The term ‘God’ is associated with the term ‘perfect’, I can infer that the term ‘God’ exists amongst my definitions. But no existential claim can be made about the world. In this sense, Tractarian logic does not fall into the ontological trap of existential generalization. Finally, let us turn our attention to Descartes’s more sophisticated predicatebased ontological argument.

Descartes’s Argument Premise (#1): I have an idea of a supreme being that has all perfections; Premise (#2): Necessary existence is a perfection. Conclusion (#3): A supreme being (i.e., God) necessarily exists. Tractarian logic has no issue with Descartes’s first premise. Nor, does it need to disallow premise #2 (or existential predications more generally) as did Kant. Rather, from a Tractarian perspective, the big mistake in Descartes’s argument is implicitly treating premise #2 in two mutually exclusive ways: one way definitional and one way empirical. The definitional way allows him to combine premise #1 and premise #2. The empirical way allows him to combine premise #2 and the conclusion. But no single interpretation of premise #2 can connect to both premise #1 and the conclusion. Let’s look at this more closely.

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Premise #1 has the form of a propositional attitude. The outer assertion “I have an idea that x”, is what allows the inner assertion “a supreme being that has all perfections” to be treated as a definitional truth. (Remove the outer assertion and Descartes’s argument falls prey to Sobel’s criticism as described earlier.) Descartes is not asserting that a supreme being exists that has all perfections; only that he has the idea of such a being. One could argue that Descartes’s having an idea is a contingent (or empirical) statement. And this would be true if made in the third person (e.g., Descartes’s mother said that Descartes had an idea about a supreme being. Maybe he did; maybe he didn’t). But Descartes is making the assertion in the first person. And, being aware that one has an idea is arguably as immune from doubt as being aware that one feels pain. Premise #2 can be evaluated in two different ways. Definitionally, there simply needs to be a term for ‘necessary existence’, a term for ‘perfection’, a term for all, and terms for individual perfections. Necessary existence can then be associated with one of the individual perfection terms that comprise the more encompassing term ‘all perfections’. Premise #2, that necessary existence is a perfection, would then support an analytic evaluation (presumably by Descartes) which would consist in no more than finding the term ‘necessary existence’ amongst his definitions and testing whether it is defined as an individual perfection. Of course, this analytic interpretation of premise #2 would make it impossible to conclude that a supreme being necessarily exists in the world. A modified version of Descartes’s argument that makes clear the definitional interpretation of premise #2 would look as follows Premise (#1): I have an idea of a supreme being that has all perfections; Premise (#2): By my definition, ‘necessary existence’ is a perfection. Conclusion (#3): A supreme being (i.e., God) necessarily exists. The conclusion in this case would be invalid as neither premise #1 or #2 have in any way asserted anything about supreme beings or God in reality. Alternatively, premise #2 can be synthetically interpreted. As a synthetic premise one would have to find something in the world (and more than just non-sentient being as that, per the discussion of Lewis above, is trivial) whose existence was necessary. Only when found could one then go back to premise #1 and change its status from definitional (or analytic) to empirical (or synthetic) at which point one could justify the conclusion. A modified version of Descartes’s argument that makes clear the synthetic interpretation of premise #2 would look as follows. Premise (#1): I have an idea of a supreme being that has all perfections; Premise (#2): Something with necessary existence (which is a perfection), exists. Conclusion (#3): A supreme being (i.e., God) necessarily exists.

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This interpretation of premise #2 would yield a valid interpretation for Descartes argument. But premise #2 is now contingent.13 It cannot combine with premise #1 until it has been empirically determined to be true. Since we have no guarantee of the truth of premise #2, we cannot combine it with premise #1 to support the conclusion. The argument as a whole would be valid but with unknown soundness.

7.5 Conclusion This paper demonstrated that literature on the ontological argument has flip flopped on the pivotal point of classical logic’s existential implications. And that the failure to resolve the issue can be traced to a foundational flaw shared by both classical and non-classical logics; namely their referential approach to the components of a proposition which includes the tying of existential claims to what is referred to (or assumed to be referred to) by the proposition’s terms (e.g., God, perfections, necessary existence) instead of tying it to how the proposition is evaluated (e.g., in the mind’s realm of definitions or in our shared external world). This paper then introduced an approach to logic based on Wittgenstein’s Tractatus and showed that with its notion of sequenced evaluations and its separation of computation and reference, it appears able to resolve both the predicate- and logical subject-based approaches to the ontological argument.

References Anselm, St. 1965. In Proslogion, in St. Anselm’s Proslogion, ed. M. Charlesworth. Oxford: OUP. [Available online, in the Internet medieval sourcebook, ed. Paul Halsall. Fordham University Center for Medieval Studies. Trans. David Burr]. Aristotle. n.d-a. Categories, 350 BCE. Translated by E. M. Edgehill at http://classics.mit.edu/ Aristotle/categories.1.1.html. Aristotle. n.d-b. On interpretation, 350 BCE. Translated by E. M. Edgehill at http:// classics.mit.edu/Aristotle/interpretation.html. Aristotle. On Interpretation, 350 BCE. Translated by E. M. Edgehill at http://classics.mit.edu/ Aristotle/interpretation.html. Bacigalupo, G. 2017. A study on existence. Newcastle upon Tyne: Cambridge Scholars Publishing. Bacigalupo, G. 2017. A study on existence. Newcastle upon Tyne: Cambridge Scholars Publishing. Berto, F. 2013. Existence as a real property: The ontology of meingongianism. Vol. 356. Dordrecht: Springer. Boole. 1854. An investigation of the laws of thought: On which are founded the mathematical theories of logic and probabilities. Mineola, NY: Dover Publications.

13 There

are numerous ways one could modify Descartes’s argument to highlight an empirical interpretation for premise #2. The specific way is not important to the argument.

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Descartes, R. 1968. Discourse on method and the meditations, translated with an introduction by F Sutcliffe. Harmondsworth: Penguin. [Translation of The Meditations by John Veitch, LL.D. available online]. Frege, G. 1980. The foundations of arithmetic. Trans. J. L. Austin; 2nd ed. Evanston, IL: Northwestern University Press. Kant, I. 1933. Critique of pure reason, 1787, 2nd ed. Trans. N. Kemp-Smith, London: Macmillan. Kleene, S.C. 1952. Two papers on the predicate calculus. Providence: American Mathematical Society. Lambert, K. 2003. Free logic: Selected essays. Cambridge: The Press Syndicate of the University of Cambridge. Copyright Karel lambert. Leibniz, G. 1896. New essay concerning human understanding, 1709, Trans. A. Langley, New York: Macmillan. Lewis, D. 1970. Anselm and actuality. Noûs 4: 175–188. Lukasiewicz, J. 1970. Selected works. Studies in logic and the foundations of mathmatics. Amsterdam: North–Holland Publishing Company. Translated by O. Wojtasiewicz. Mion, G. 2018. On Kant’s ambivalence toward existence in his critique of the ontological argument. Journal of Applied Logic 5: 7. Modenato, F. 1995. Meinong’s theory of objects. In Grazer Philosophische Studien, vol. 50. Amsterdam: Rodopi. Nagasawa, Y. 2011. The existence of god. Oxford: Routledge. Oppenheimer, P., and E. Zalta. 1991. On the logic of the ontological argument. In Philosophical perspectives 5: The philosophy of religion, ed. J. Tomberlin, 509–529. Ridgeview: Atascadero. Oppy, G. 1995. Ontological arguments and belief in god. New York: Cambridge University Press. Paoletti, M. 2013. Meinong strikes again, Humana. Mente Journal of Philosophical Studies 25: I–XIV. Plantinga, A. 1974. The nature of necessity. Oxford: Oxford University Press. Priest, G. 2008. The closing of the mind: How the particular quantifier became existentially loaded behind our backs. The Review of Symbolic Logic 1(1): 42–55. ———. 2016. Towards non-being. 2nd ed. Oxford: Oxford University Press. ———. 2017. Logic: A very short introduction. 2nd ed. Oxford: Oxford University Press. Routley, R. 1980. Exploring Meinong’s jungle and beyond. An investigation of noneism and the theory of items. Canberra: Research School of Social Sciences, Australian National University. Russell, B. 1905. On denoting mind. Vol. 14, 479–493. Oxford: Oxford University Press. ———. 1946. History of western philosophy. New York: Routledge. Sobel, J. 2004. Logic and theism. New York: Cambridge University Press. Spinoza, B. 1955. The ethics, 1677, translation of 1883 by R. Elwes. New York: Dover. [Available online, prepared by R. Bombardi, for the Philosophy Web Works project, Middle Tennessee State University]. Thomsen, E. 1990. A tractarian approach to information modeling. In Wittgenstein-towards a reevaluation. Wien: Verlag Holder-Pichler-Tempsky. Thomsen, E. 2002. OLAP solutions: building multidimensional information systems, 2nd ed. New York: Wiley. Wittgenstein, L. 1977. Tractatus Logico-Philosophicus. Trans. D. F. Pears and McGuinness. London: Routledge & Kegan Paul.

Erik Thomsen is a world authority in the field of multidimensional information systems and the author of several reference books including OLAP Solutions: Building Multidimensional Information Systems 2nd edition, which is used around the world in both graduate computer science and MBA programs. He has spent a career pushing the envelope of logically- and ontologically-grounded intelligent software technologies emphasizing applications for sustainable development. Two software companies he co-founded were acquired in unsolicited acquisitions. He is currently CTO at Blender Logic, Inc. in Cambridge MA USA.

Chapter 8

Some Thoughts on the Logical Aspects of the Problem of Evil Ricardo Sousa Silvestre

8.1 Introduction The problem of evil is one the most debated issues in analytic philosophy of religion. Plantinga (1974, p. 164), for instance, has famously described it as “the most formidable objection to theistic belief.” He continues: A multitude of philosophers have held that the existence of evil is at least an embarrassment for those who accept belief in God. And most contemporary philosophers who hold that evil constitutes a difficulty for theistic belief claim to detect logical inconsistency in beliefs a theistic typically accepts. (Plantinga 1974, p. 164)

Besides presenting the problem in relation to the rationality of theistic belief, Plantinga also characterizes it in terms of logical inconsistency. In fact, one of the most traditional descriptions of the problem of evil—John Mackie’s—explicitly uses the notions of inconsistency and contradiction: In its simplest form the problem is this: God is omnipotent; God is wholly good; yet evil exists. There seems to be some contradiction between these three propositions, so that if any two of them were true the third would be false. But at the same time all three are essential parts of most theological positions; the theologian, it seems, at once must adhere and cannot consistently adhere to all three. (Mackie 1955, p. 200)

This is similar to some descriptions of the so-called evidential problem of evil (more on that below): [ . . . ] rather than being formulated as a deductive argument for the very strong claim that it is logically impossible for both God and evil to exist, (or for God and certain types, or instances, or a certain amount of evil to exist), the argument from evil can instead be

R. S. Silvestre () Federal University of Campina Grande, Campina Grande, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_8

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formulated as an evidential (or inductive/probabilistic) argument for the more modest claim that there are evils that actually exist in the world that make it unlikely—or perhaps very unlikely—that God exists. (Tooley 2012, p. 3)

I could fairly rephrase Michael Tooley as follows: besides being characterized as an inconsistency problem, the problem of evil can also can be characterized as an evidential incompatibility problem, in the sense that the evils that exist in the world seem to make it unlikely that God exists. The following characterization would then probably find some sympathy among analytic philosophers of religion: the problem of evil is the supposed incompatibility that exists between the proposition that (G)

The world was created and is ruled by an omnipotent, omniscient and unlimitedly good being whom we call God,

and one that says that (S)

There is evil and suffering in our world.

If we take the word “incompatibility” to mean inconsistency, we get the logical problem of evil; if we take it to mean evidential incompatibility—in the sense of the existence of evil and suffering standing as evidence against the existence of God— we get the evidential or inductive problem of evil. Despite its simplicity and intuitiveness, it is not obvious that this characterization captures all that has been done under the umbrella of problem of evil. There is a considerable variety of concepts involved in the debate, as well as disagreement over their exact meaning and tenability. The very concept of problem of evil, for instance, has been seen with suspicion: Evil, it is often said, poses a problem for theism, the view that there is an omnipotent, omniscient, and perfectly good being, ‘God’ for short. This problem is usually called ‘the problem of evil’. But this is a bad name for what philosophers study under that rubric. They study what is better thought of as an argument, or a host of arguments, rather than a problem. (Howard-Snyder 1996, p. xi)

Indeed, while the notion of argument from evil seems straightforward—we could define it simply as an atheistic argument premised on at least one proposition about the existence of evil and suffering—, the notion of problem of evil might seem a little vague. In response to that, one could say that everything dealing with the issue that there is an incompatibility between (G) and (S) belongs to what we call problem of evil. This would include everything related to arguments from evil—conception, attempts of refutation, response to alleged refutations, etc.—but also theist’s efforts to build satisfactory theodicies, which might not address specific arguments from evil. Howard-Snyder acknowledges this, as he continues: Of course, an argument from evil against theism can be both an argument and a problem. Some people realize this for the first time when they assert an argument from evil in print and someone publishes a reply in which numerous defects and oversights are laid bare for the public eye. An if turns out that there is a God and He doesn’t take kindly to such arguments, then an argument form evil might be a big problem, for one who sincerely propounds it. (Howard-Snyder 1996, p. xi)

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The notion of the evidential problem of evil (and evidential argument from evil as well) is also problematic. First, there is great diversity of approaches falling under this label. For instance, while on one hand there are completely qualitative approaches such as Rowe’s (1979), on the other there are comparative (Draper 1989) and quantitative (Rowe 1996) approaches. The variety of terms used to refer to this family of approaches does not help either; besides “evidential problem of evil”, terms such as “inductive problem of evil”, “probabilistic problem of evil”, “empirical problem of evil” and “a posteriori problem of evil” have been used. Thirdly, unlike the logical problem of evil and the notion of logical inconsistency, there is no consensual theory, be it qualitative, comparative or quantitative, of what it means to say that a set of propositions makes it likely that proposition A is true. Be it as it may, the fact that there is a distinction, however fuzzy it may be, between the concepts of problem of evil and argument from evil on one hand, and a logical approach and an evidential approach on the other, gives rise to at least four conceptual categories: logical problem of evil and logical argument from evil on one side, and evidential problem of evil and evidential argument from evil on the other. And there is more. Depending on the kind of suffering which one decides to emphasize, different problems of evil and arguments from evil, both logical and evidential, will emerge. For example, it is common to distinguish between moral evil and natural evil: Moral evil is evil what we human beings originate: cruel, unjust, vicious, and perverse thoughts and deeds. Natural evil is the evil that originates independently of human actions: in disease bacilli, earth quakes, storms, droughts, tornadoes, etc. (Hick 1985, p. 12)

From that, eight categories will emerge: natural logical problem of evil, moral logical problem of evil, natural logical argument from evil, moral logical argument from evil, and so on and so forth. It seems to me that this variety of categories should be taken into account by any attempt of comprehensively characterizing the problem of evil. Another thing that must definitely be taken into account is the distinction between defense and theodicy. Plantinga has famously used the term “defense” to distinguish his endeavor (which he called the Free Will Defense) from the one associated with a theodicy: Augustine tries to tell us what God’s reason is for permitting evil [ . . . ] Such an attempt to specify God’s reason for permitting evil is what I earlier called a theodicy [ . . . ] A theodicist, then, attempts to tell us why God permits evil. Quite distinct from a Free Will Theodicy is what I shall call a Free Will Defense. Here the aim is not to say what God’s reason is, but at most what God’s reason might possibly be. (Plantinga 1977, pp. 27–28)

Since then, the distinction has been mentioned and used in professional articles and books on God and evil. Nonetheless, contrary to what Plantinga says, the concepts of defense and theodicy are not that distinct from each other. Their task is basically the same: to give the reasons why God allows evil and suffering. As he himself admits (Plantinga 1977, p. 28), finding such reasons is tantamount to solving the logical problem of evil (which is the context inside of which he presents the distinction).

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Of equal importance is the connection that many people see between defense and the logical problem of evil on one side, and theodicy and the evidential problem of evil on the other side: Just as we have classified the two major versions of the problem of evil into the logical and evidential formulations, we may also classify the two main responses to the problem as defense and theodicy [ . . . ] Defense has come to be the theistic strategy most closely associated with discussions of the logical formulation of the problem of evil, whereas theodicy has come to be associated with the evidential formulation. (Peterson 1998, p. 33)

But why is this so? Does not a theodicy also solve the logical problem of evil? And is it really true that a defense cannot solve the evidential problem of evil? Needless to say, in order to answer these questions, one must have a minimally precise characterization of the concepts involved, especially the concept of evidential problem of evil. My purpose in this chapter is to take seriously the idea that the problem of evil is an incompatibility between G and S. This, I believe, is worthy doing. First, as I have said, it (the idea that the problem of evil is an incompatibility between G and S) is in accordance with much of the literature on the problem of evil. Second, it takes the problem at face value, that is to say, it sees it as a logical and incompatibility problem.1 Third and more important, it might allow for a comprehensive and more elegant account of the key concepts involved: despite contrary appearances, these concepts—the concept of problem of evil itself, the concept of argument from evil, the concepts of logical and evidential problems of evil and the concepts of theodicy and defense—can be seen and defined from the standpoint of the same general idea about the problem of evil. Here is what I will do. First, I will refine this characterization and give a more precise definition of the concept of problem of evil. Then I will slightly modify one of its parameters to arrive at the notions of logical problem of evil and evidential problem of evil. This will be done in the next section. In Sect. 8.3 I will deal with the concepts of theodicy and defense. I then move, in Sect. 8.4, to the concept of argument from evil. Finally, in the last section, I lay down some concluding remarks. I will follow here what might be termed a semi-formal approach: despite not using a fully developed logical theory, I shall use the standard notation and a couple of results from the field of formal logic.

8.2 Logical and Evidential Problems of Evil I say that the problem of evil is the claim of incompatibility between (G)

1 The

The world was created and is ruled by an omnipotent, omniscient and unlimitedly good being whom we call God,

two senses of the world “incompatibility” I am taking here—inconsistency and evidential incompatibility—can be seen as conceptually related, in the sense that inconsistency is the extreme case of evidential incompatibility. Propositions α and β are inconsistent when the support given by α against β is so strong that it proves β to be false.

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and some proposition about the existence of evil and suffering in our world, say (S)

There is evil and suffering in our world.

In symbols, it is the claim that (E)

{G, S} ⊥

where ( is such that if is a set of propositions and α is a proposition, then (α means that α can be inferred from or is a consequence of ; ⊥ is the contradiction symbol. As I said, if we take the word “incompatibility” to mean inconsistency, we get the logical problem of evil; if we take it to mean evidential incompatibility—in the sense of the existence of evil and suffering standing as evidence against the existence of God—we get the evidential problem of evil. Let be the inferential relation of classical logic and  an inductive or evidential inferential relation. The distinction then can be put as follows: while the logical problem of evil amounts to the claim that (El )

{G,S} ⊥,

the evidential problem of evil amounts to the claim that (Ee )

{G,S}⊥.

Due to the diversity of existing approaches to inductive reasoning, I take  as generally as possible. It might mean for example the inferential aspect of conditional probability (so that in the case where the probability of β given α—in symbols: P(β/α)—is high, or at least not too low, we write αβ) but also an intuitive and pre-theoretical notion of evidential support dissociated from any formal theory of inference. From a minimal viewpoint,  must allow us to read α as “ stands as evidence for α”, “ confirms α” or “ supports the reasonableness or plausibility of α”. Despite this, I shall suppose in the course of the chapter that  satisfies the following principles (which I take here as intuitive and reasonable principles of inductive reasoning, taken as generally as possible): (1 ) (2 ) (3 )

Supraclassicality: if

α then α. Inductive contradiction: if ∪{α}⊥ then ¬α. Transitivity: if α and {α}β then β.

In addition to them, I shall use the following deductive principle: ( 4 )

Reduction ad absurdum: If ∪{α} β and ∪{α} ¬β then

¬α.

Notice that (1 ), (2 ) and (3 ) all hold if we replace  by , in the case of which I use the markers ( 1 ), ( 2 ) and ( 3 ), respectively. Getting back to the problem of evil, I have mentioned how the variety of kinds of evil and suffering can give rise to several different problems of evil. As one would expect, this variety is not restricted to the notions of moral and natural evil:

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There are, I think, four different things a theodicy might aim at doing, each more difficult than its predecessor. First, a theodicy might seek to explain why O [God] might permit any evil at all. Second, a theodicy might endeavor to explain why there are instances of the various kinds of evil we find in our world – animal pain, human suffering, wickedness, etc. Third, a theodicy might endeavor to explain why there is the amount of evil (of these evils) that we find in our world. And, finally, a theodicy might endeavor to explain certain particular evils that obtain. (Rowe 1988, p. 131)

As far as my characterization of the problem of evil is concerned, while the notions of natural and moral evil can be expressed in terms of proposition as follows: (S2 ) (S3 )

There are instances of natural evil. There are instances of moral evil.

the different kinds of evil which Rowe talks about might be expressed as follows: (S4 ) (S5 ) (S6 ) (S7 )

There are instances of animal pain. There are instances of human suffering. There is the amount of evil and suffering we find in our world. There is this, and this, and this ... instance of evil and suffering in our world.

It is not difficult to see how this diversity of problems of evil would be coped with in the simple symbolism I have introduced. To each S-proposition there will be different logical and evidential problems of evil. Regarding S2 , S3 and S6 , for example, we would have as follows: (El-natural evil ) (Ee-natural evil ) (El-moral evil ) (Ee-moral evil ) (El-amount of evil ) (Ee-amount of evil )

{G,S2 } ⊥ {G,S2 }⊥ {G,S3 } ⊥ {G,S3 }⊥ {G,S6 } ⊥ {G,S6 }⊥

8.3 Theodicy and Defense The use of the word “claim” in my definition is important. Saying that the problem of evil is the claim of incompatibility between G and a S-proposition gives room to at least two distinct, and quite obvious indeed, attitudes towards it: one may try to show that the claim is in fact true, by providing arguments showing that there is really such an incompatibility, or that it is false. Let me take a closer look at what this second movement would look like. Take an arbitrary evidential problem of evil, say the claim that (Ex )

{G,Sx }⊥

From it (through 2 ) we get (E

x)

Sx ¬G

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If (E x ) is false so is (Ex ). One way to show that (E x ) is false is to exhibit a proposition R such that (T)

{G, R}Sx

{G, R} is a consistent set and RSx. 2 Why is this so? Suppose that (E x ) is true. From (T) and (E x ), we can use the transitivity of  (3 ) and conclude {G, R}¬G. But this is absurd, for even if R somehow supports ¬G, this support must be defeated by G. Therefore, our supposition (E x ) is false and so is (Ex ). R however seems to do more than merely showing (Ex ) to be false. By showing that G and R might serve as evidence for Sx , it seems that (T) shows that the existence of evil and suffering is exactly what one would expect given the truth of G. This, incidentally, is one of the main criteria a theodicy should satisfy (van Inwagen 1991, p. 139). Traditionally, the chief task of a theodicy must be to “explain how the universe, assumed to be created and ultimately ruled by an unlimitedly good and unlimitedly powerful Being, is as it is, including all the pain, suffering, wickedness, and folly that we find around us and within us.” (Hick 1981, p. 38). Therefore, if R along with G makes Sx plausible, it seems reasonable to say that R explains the existence of evil and suffering in the world given God’s existence. This connection between explanation and inference is not new. There is a long tradition in the philosophy of explanation which takes inference as a key aspect of explanation.3 In the context of the problem of evil, sometimes this is put in terms of the calculus of probability4 : A theodicy, let us say, is the conjunction of theism with some “auxiliary hypothesis” h that purports to explain how S could be true, given theism. Let us think for a moment in terms of the probability calculus. It is clear that if a theodicy is at all interesting, the probability of S on the conjunction of theism and h (that is, on the theodicy) will have to be high—or at least not too low. (van Inwagen 1991, p. 139)

Here, while our G would correspond to what van Inwagen calls theism, his auxiliary hypothesis h corresponds to R. We can then say that R is at the very least a serious candidate to represent the reasons that would justify and explain why God permits Sx . Taking the idea a little further and generalizing about which kind of problem of evil is at stake, I will say that a theodicy addressing a specific problem of evil (Ex )

{G, Sx }(⊥

is a pair which solves (Ex ); otherwise said, is such that (T)

{G, R}(Sx ,

{G, R} is a consistent set and RSx . 2 RS is there to guarantee that G is indeed a relevant x 3 See Salmon (1989) and Ruben (1990), for instance. 4 Recall

component of (T).

that the inferential aspect of conditional probability is supposed to be encompassed in our inductive relation .

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Although an important one, this is not of course the only condition a pair should satisfy in order to be considered a theodicy. For instance, when one proposes a theodicy for explaining Sx , she wishes to find not only an explanation for why Sx is the case, but also an explanation able to morally justify why God allows or would allow Sx. 5 Therefore, there should be some guarantee that R provides such kind of explanation.6 One familiar with the literature on the problem of evil should have noticed that the reconstruction I have given of how a theodicy solves the problem of evil is very similar to Plantinga’s characterization of what a defense should do: The Free Will Defence is an effort to show that (1) God is omnipotent, omniscient, and wholly good (which I shall take to entail that God exists) is not inconsistent with (2) There is evil in the world. That is, the Free Will Defender aims to show that there is a possible world in which (1) and (2) and both true. No one way to show that a proposition p is consistent with a proposition q is to produce a third proposition r whose conjunction with p is consistent and entails q. r, of course, need not be true or known to be true; it need not be so much as plausible. (Plantinga 1974, p. 165)

In fact, he himself admits that a defense and a theodicy do almost exactly the same thing (Plantinga 1977, p. 28); the difference is that while a theodicy puts R as God’s actual reason, in a defense R needs only be possible (here (1) is the same as G and (22) is a S-sentence): The Free Will Theodicist and Free Will Defender are both trying to show that (l) is consistent with (22), and of course if so, then set A is consistent. The Free Will Theodicist tries to do this by finding some proposition r which in conjunction with (1) entails (22); he claims, furthermore, that this proposition is true, not just consistent with (l). He tries to tell us what God’s reason for permitting evil really is. The Free Will Defender, on the other hand, though he also tries to find a proposition r that is consistent with (I) and in conjunction with it entails (22), does not claim to know or even believe that T is true. And here, of course, he is perfectly within his rights. His aim is to show that (I) is consistent with (22); all he need do then is find an r that is consistent with (1) and such that (1) and (r) entail (22); whether r is true is quite beside the point. (Plantinga 1977, p. 28)

But why is this so? If theodicy and defense are, from a logical point of view, indistinguishable from each other (that is to say, if they both solve in the same manner the problem of evil), why two different terms? And what difference does the claim (or even the fact) that R is true (and not merely possible) make? Put in terms of , the answer is none. Regarding the logical problem of evil, there is no difference at all if R is claimed to be true (or even is true) or is a merely possible proposition. Let us consider an arbitrary logical problem of evil: (El )

5 That

{G, Sx } ⊥

these reasons should give a moral justification is clear. The purpose of a theodicy is to conciliate the existence of evil with the existence of an omnipotent, omniscient and perfectly good being. If the reasons given do not have a moral character, then one feature, namely perfect goodness, would be left out. It is not, as one might think, an asymmetry regarding the treatment given to the three features; instead, it is an exigence of the problem of evil itself. 6 For more on the conditions a theodicy must satisfy. See Silvestre (2017).

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If we have R such that {G, R} Sx ,

(Tl )

{G, R} is a consistent set and RSx , then (El ) is false. The reasoning is the same as above. From (El ) and 2 we get (E l )

Sx ¬G.

Trivially, if (E l ) is false so is (El ). Now Suppose that (E l ) is true. From (Tl ), (E l ) and 3 we get {G, R} ¬G, which is false. Therefore, (E l ) is false and so is (El ). What matters here are the logical relations that R holds with G and Sx . It is completely irrelevant whether R is true or only possibly true. Therefore, from the perspective of the logical problem of evil and its solution, the concepts of theodicy and defense are logically indistinguishable from each other.7 The situation is a bit different when we deal with the evidential problem of evil. We can see this by taking a look at the whole quotation by van Inwagen I have shown above: [ . . . ] whether a theodicy is interesting depends not only on the probability of S on the conjunction of theism and h, but also on the probability of h on theism. Note that the higher P(h/theism), the more closely P(S/theism) will approximate P(S/theism & h). On the other hand, if P(h/theism) is low, P(S/theism) could be low even if P(S/theism & h) were high. (Consider, for example, the case in which h is S itself: even if P(S/theism) is low, P(S/theism & S) will be 1—as high as a probability gets.) The task of the theodicist, therefore, may be represented as follows: find an hypothesis h such that P(S/theism & h) is high, or at least not too low, and P(h/theism) is high. (van Inwagen 1991, p. 139)

The idea here is that the stronger the evidential support given by G to R is, the closer the amount of evidential support given by G to S gets to the amount of evidential support given by G and R to S. Therefore the need of P(h/theism) be high. From a purely qualitative point of view, this means that if G stands as evidence for R, and G, along with R, stands as evidence for Sx , then G itself stands as evidence for Sx . In symbols: if GR and G∪{R}Sx then GSx. 8 Therefore there should be the following additional condition: (P)

GR.

7 Notice

that my claim is circumscribed to the problem of evil as a problem, that is to say, as something that must (or might) be solved. It is only in that sense that I am claiming that the concepts of theodicy and defense are logically indistinguishable from each other. From a more general perspective, it is obvious that they are different. R and possibly R are obviously not equivalent. Therefore, showing that R is God’s possible reasons for allowing evil does not imply that R is God’s actual reasons. However, and that is my point, this does not make a slight difference to the fact that the logical relations that R holds with G and Sx prove that El is false. 8 Incidentally, this is the cut property, which in general holds in deductive logics and is seen as a desirable property of commonsense and non-monotonic reasoning. See Makinson (1994, p. 39).

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As pointed out by van Inwagen, the absence of (P) could give rise to ad hoc uses of  as an explanatory instrument, as in the case mentioned where R (van Inwagen’s h) is Sx. 9 Now we can make some sense of Plantinga’s distinction. As far as the evidential problem of evil is concerned, it really makes a difference that R is true, for if this is so, then G R is also true, and by (1 ) (P) is satisfied. Consequently, we could also make some sense of the connection which has often been made between defense and the logical problem of evil on one hand, and theodicy and the evidential problem of evil on the other. However, and despite this, the distinction as it stands is still untenable. First, because it does not apply equally to both logical and evidential problems of evil, as Plantinga announces. Second, because it is at least controversial to say that all theodicists put R as being God’s actual reason for allowing evil and suffering in the world. Third, as an evaluation criterion for the satisfactoriness of theodicies, how to know that R is indeed true? Most, if not all the reasons so far given by theodicists in principle cannot be known to be true. Finally, if from a logical point of view the two notions are indistinguishable from each other, why have it? Why not simply to drop this theoretically useless criterion that R should be true and have instead only one concept to refer to the theodicist’s attempt to solve a logical problem of evil? That is my proposal: to take the notion of theodicy as defined above and see it as the theist’s response to the problem of evil, be it logical or evidential.10 According to this, Plantinga’s free-will defense (Plantinga 1974, 1977) would be a theodicy aimed at answering the logical problem of evil. As far as the notion of defense is concerned, I propose to take it at face value and define it as an attempt to refute a specific argument from evil, be it logical or evidential. According to this definition, while, for instance, Nelson Pike’s “Hume on Evil” article (Pike 1963) would be a defense against Hume’s logical argument, Stephen Wikstra’s (Wykstra 1984) well-known CORNEA critique would be a defense against Rowe’s 1979 evidential argument (Rowe 1979).

8.4 Argument from Evil Here is what I have done so far. I have defined the problem of evil as a claim of incompatibility between G and some S sentence; taking the word “incompatibility”

van Inwagen’s view, (P) is a necessary condition for the explanatory soundness of . Others who seem to require the same thing are Hasker (1988, p. 5) and Tooley (2002, p. 22). While this is not completely false, the satisfaction of (P) is by no means the only—nor is it the best—way to prevent this type of ad hoc use. By invoking the restriction imposed on (T) that RSx , we obtain the same result in a much simpler way without requiring the satisfaction of a condition as strong as (P). 10 As far as R’s ontological and epistemological status are concerned, one can see them as distinguishing parameters for theodicies. See Silvestre (2017). 9 In

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to mean inconsistency gives us the logical problem of evil; taking it to mean evidential incompatibility gives us the evidential problem of evil. Then I pointed out that two movements might be taken to answer the challenge of incompatibility. While the first one tries to make the incompatibility as explicit as possible by building atheistic arguments premised on at least one proposition about the existence of evil and suffering—these are the so-called arguments from evil—, the second tries to show that the incompatibility is merely apparent and that there are in fact reasons that would morally justify God in allowing the evil and suffering we find in our world. The goal of a theodicy, which is the theist’s movement, is to exhibit these reasons. I defined a defense as an attempt to reply to a specific argument from evil. As far as the notion of argument from evil is concerned, a simple but very fundamental point comes out from what I have said so far: there is a distinction between the notions of problem of evil and argument from evil, for one can successfully answer a problem of evil, that is to say, a specific claim of incompatibility between G and Sx , without taking into account a specific argument which tries to show that there is really such an incompatibility. Another thing which comes out is this: since there is a distinction between the logical and the evidential problems of evil, there must also be a distinction between logical arguments from evil and evidential arguments from evil. In order to cope with these distinctions, I will say that, from a general viewpoint, an argument from evil is an atheistic argument premised on at least one known proposition about the existence of evil and suffering and logically connected to some (Ex ), in the sense that showing (Ex ) to be false amounts to refuting the argument. A logical argument from evil then is an argument from evil which is refuted by refuting (El ) or some of its variations. In its turn, an evidential argument from evil is an argument from evil which is refuted by refuting (Ee ) or some of its variations, but is not refuted by refuting the corresponding (El ) variation. This qualification is needed to prevent that every logical argument be also an evidential argument: by (1 ), (Ee ) follows from (El ). Let me try to show now how some existing arguments from evil fit in my definition. For the logical problem of evil the task is a straightforward one. As shown by Pike (1963) and Plantinga (1974), in order to prove (El )

{G,Sx } ⊥.

one has to show that G → ¬Sx is an analytical truth, or equivalently, to provide an argument for (A)

G → ¬Sx

such that each premise of is either analytically true or a non-controversially known definitional postulate. Such an argument would justify the use of G → ¬Sx in the trivially deductively valid argument below: (A )

{G → ¬Sx , Sx } ¬G.

Although (A) and (A ) together would make up what I am calling here a logical argument from evil, the crucial part is of course (A) (possibly accompanied

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by auxiliary arguments trying to show that the members of are indeed either analytically true or non-controversially known definitional postulates). It is not difficult to see that such argument fits into my definition of logical argument from evil. If one successfully shows (El ) to be false, she has also shown (A) to be false: if (El ) is false, then there cannot be such that each premise of

is either analytically true or a non-controversially known definitional postulate and

G → ¬Sx . Things are not that easy with evidential arguments from evil. As I said in the beginning, there is a considerable variety of arguments which are accepted as evidential (probabilistic or inductive) arguments from evil. Howard-Snyder states this in a quite dramatic way: It is customary to distinguish two families of arguments from evil, calling one ‘logical,’ ‘deductive’, or ‘a priori’ and calling the other ‘evidential’, ‘inductive’, ‘empirical’, ‘probabilistic’ or ‘a posteriori’. But these too are poor names for that to which they refer. [ . . . ] evidential arguments involve quite a bit of logic, both deductive and inductive, [ . . . ] And every undeniably logical argument is superlative evidence against theism, if it is a good argument. Moreover, every undeniably logical argument has a premise that can only be known a posteriori, by empirical means, namely, a premise about evil, e.g., that it exists. And every undeniably evidential argument has a premise that can only be known a priori, e.g., a premise about what counts as good evidence or what we rightly expect from God in the way of preventing evil. And many ‘inductive’ arguments from evil are, on the face of it, deductively structured. (Howard-Snyder 1996, p. xii)

Inspite of this, there have been attempts to give a taxonomy for all these kinds of evidential arguments. Tooley (2002), for example, distinguishes between three kinds of evidential arguments from evil: direct inductive arguments, indirect inductive arguments and probabilistic or Bayesian arguments. While direct inductive arguments are those which try to show that theism is unlikely to be true given evil and suffering, indirect inductive arguments try to establish some alternative hypothesis logically incompatible with theism and more probable than theism given the existence of evil. In its turn, probabilistic or Bayesian formulations start out from probabilistic premises and then attempt to show that it follows deductively, via axioms of probability theory, that it is unlikely that God exists. Examples of these evidential arguments are Rowe (1979), Draper (1989) and Rowe (1996), respectively. As far as my definition is concerned, I will only show here that one of those formulations—Rowe’s 1979 formulation—fits in it. In order to do the same for the other kinds of evidential arguments, I would need to lay down and argumentatively support some presuppositions on the relations between my qualitative inferential relation  and comparative and numerical probability, which would require a space that I do not have here. I shall therefore postpone such an undertake to a future work. Here is Rowe’s argument (1979): (P)

An omnipotent, omniscient and unlimitedly good being (God) would prevent the occurrence of any intense suffering it could, unless it could not do so without thereby losing some greater good or permitting some evil equally bad or worse.

8 Some Thoughts on the Logical Aspects of the Problem of Evil

(S ) (S )

(¬G)

133

There exist instances of intense suffering for which we have found no greater goods which would be lost or evils equally bad or worse which would be permitted if God were to prevent those instances of suffering. [Therefore] There exist instances of suffering which God could have prevented without thereby losing a greater good or permitting an evil equally bad or worse. [Therefore] There does not exist an omnipotent, omniscient and unlimitedly good being that created and rules the world.

The argument is in fact composed by two arguments, a deductive one. 

 P, S ¬G.

and an inductive one, meant to support the second premise of the deductive argument: S | S . Although the main argument is deductive, one of its premises (S ) is inductively supported by S , which is the known proposition about evil. That is why the whole argument is seen as inductive. In order to see how this argument is logically related to the claim that {G,S }⊥, notice that by (1 ) we have {G,S }⊥, for {G,S } ⊥. But since S S , by (3 ) we have that 

 G, S |⊥ .

If we prove that {G,S }⊥ then by (3 ) either {G,S }⊥ or S S . If {G,S }⊥ then the first argument is invalid, which we know is not true. Therefore if {G,S }⊥ then S S . Finally, it has to be said that in the same way I did with the problem of evil, here too for each S-sentence there will be different kinds of logical and evidential arguments from evil.

8.5 Conclusion In this chapter I have taken seriously the idea that the problem of evil is an incompatibility claim between. (G)

The world was created and is ruled by an omnipotent, omniscient and unlimitedly good being whom we call God,

and some proposition about the existence of evil and suffering in our world, say. (S)

There is evil and suffering in our world.

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I have then defined some key concepts of the contemporary debate on God and evil, namely the concept of problem of evil itself, the concept of argument from evil, the concepts of logical and evidential problems of evil and the concepts of theodicy and defense. I have followed what I termed a semi-formal approach; despite not using a fully developed logical theory, I used the standard notation and a couple of results from the field of formal logic. I believe that the modest results I have achieved point to the fruitfulness of my approach. The definition of problem of evil I gave provides the base from which we can arrive at a comprehensive and logically articulated understanding of some key concepts involved in the debate. In special, the definitions of theodicy and defense are given from a logical point of view much more coherent than the definitions found in the literature. As a drawback, there is an obvious limitation of my semi-formal approach concerning its qualitative nature. As I have said in the previous section, in order to consider other evidential arguments from evil, such a qualitative approach has to be enlarged in such a way as to be able represent quantitative and comparative formulations. This shall be done in a future work.

References Draper, P. 1989. Pain and pleasure: An evidential problem for theists. Nouns 23: 331–350. Hasker, W. 1988. Suffering, soul-making, and salvation. International Philosophical Quarterly 28: 3–19. Hick, J. 1981. An irenaean theodicy. In Encountering evil: Live options in theodicy, ed. S. Davis, 39–52. Edinburgh: T & T Clark. Hick, J. 1985. Evil and the god of love. London: Macmillan Press. Howard-Snyder, D. 1996. The evidential argument from evil. Bloomington: Indiana University Press. Mackie, J.L. 1955. Evil and omnipotence. Mind 64: 200–212. Makinson, D. 1994. General patterns in non-monotonic reasoning. In Handbook of logic in artificial intelligence and logic programming, ed. D. Gabbay et al., vol. 3, 35–110. Oxford: Oxford University Press. Peterson, M. 1998. God and evil. Boulder: Westview Press. Pike, N. 1963. Hume on evil. The Philosophical Review 72: 180–197. Plantinga, A. 1974. The nature of necessity. Oxford: Oxford University Press. ———. 1977. God, freedom, and evil. Grand Rapids: Eerdmans. Rowe, W. 1979. The problem of evil and some varieties of atheism. American Philosophical Quarterly 16: 335–341. ———. 1988. Evil and theodicy. Philosophical Topics 16: 119–132. ———. 1996. The evidential argument from evil: A second look. In The evidential argument from evil, ed. D. Howard-Snyder, 262–285. Bloomington: Indiana University Press. Ruben, D. 1990. Explaining explanation. New York: Routledge. Salmon, W. 1989. Four decades of scientific explanation. In Minnesota studies in the philosophy of science, ed. P. Kitcher and W. Salmon, vol. XII. Minneapolis: University of Minnesota Press. Silvestre, R. 2017. On the concept of theodicy. Sophia 56: 207–226. Tooley, M. 2002. The problem of evil. In The Stanford encyclopaedia of philosophy, ed. E. Zalta. Stanford: Stanford University Press.

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van Inwagen, P. 1991. The problem of evil, the problem of air, and the problem of silence. Philosophical Perspectives 5: 135–65. Wykstra, S. 1984. The humean obstacle to evidential arguments from suffering: On avoiding the evils of appearance. International Journal for Philosophy of Religion 16: 73–93.

Ricardo Sousa Silvestre holds a Ph.D. in Philosophy from the University of Montreal. He has been Visiting Scholar at the Universities of Oxford (UK), Notre Dame (USA) and Québec (Canada). He is the author of several papers on Logic and Philosophy of Religion and guest-editor of a couple of special issues on the field of Logic and Religion, namely the special issues on the Concept of God (2019) and Formal Approaches to the Ontological Argument (2018) of the Journal of Applied Logics (College Publications), the special issue on Logic and Philosophy of Religion (2017) of Sophia: International Journal of Philosophy and Traditions (Springer) and the special issue on Logic and Religion (2017) of Logica Universalis (Springer). He is one of the creators and main organizers of the World Congress on Logic and Religion series. He is presently Associate Professor at Federal University of Campina Grande (Brazil).

Chapter 9

The Logic of the Trinity and the Filioque Question in Thomas Aquinas: A Formal Approach Fábio Maia Bertato

9.1 Introduction In this chapter, I aim to show that, once understood and developed, Thomas Aquinas’ insights on the Trinitarian Theology can positively address and solve the Filioque question. In order to accomplish this, besides the very solution given by Aquinas, I provide a formal system for the treatment of the doctrine of the Trinity. I aim to show that within this formal system the Filioque assertion is a theorem, that is to say, that the Filioque logically follows from the main presuppositions of the doctrine of the Trinity. Such formal approach can be considered a case of applied logic, more specifically a case of Logic of Religion, since it is an example of logic applied to religious discourse.1 In Sect. 9.2, I sketch the theological context of the Filioque question and its implications. In Sect. 9.3, I briefly present Aquinas’ discussion of the Filioque issue and his solution. In Sect. 9.4, I introduce a formalized version of the Thomistic oriented Trinitarian Theology called The Thomistic Logic of the Trinity . The formal system  is constructed as a First Order Theory. Finally, in Sect. 9.5, a proof of the formal formula equivalent to the Filioque statement is provided. Additionally, other definitions and results are also offered.

1 On

the application of logic to religious domains and the Logic of Religion cf. Bochenski (1965) and (Weingartner 1999). On the Logic of Trinity, see Thom (2012).

F. M. Bertato () CLE—Unicamp, Campinas, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_9

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9.2 The Filioque Question An important point of distinction between Christians and other monotheists is the belief in the Holy Trinity and in the dual nature of Christ.2 The central mystery of the Christian faith is the mystery of the Holy Trinity.3 This mystery is considered as a fundamental part of the Christian Revelation and source of all other mysteries. The mainstream Western Christian view on this subject can be expressed summarily in these terms: The Church expresses her trinitarian faith by professing a belief in the oneness of God in whom there are three Persons: Father, Son, and Holy Spirit. The three divine Persons are only one God because each of them equally possesses the fullness of the one and indivisible divine nature. They are really distinct from each other by reason of the relations which place them in correspondence to each other. The Father generates the Son; the Son is generated by the Father; the Holy Spirit proceeds from the Father and the Son.4

In Christian philosophy and theology, there exists only one God, who is one according to the essence (truth achieved by both reason and Revelation) and three according to the Persons (revealed truth), i.e., the only essence of divinity, in indivisible form, exists in three distinct persons: Father, Son, and Holy Spirit. In the Latin tradition, the Persons are distinguished not by essence or substance,5 but by relations of origin: generation (generatio, γ ε´ ννησ ις ) and spiration or procession simpliciter (spiratio, π νευσ ˜ ις). The causes of the distinction of three real hypostases (Father, Son, and Holy Spirit) are the Divine Processions, which properly constitute the mystery of the Holy Trinity. By “procession” (processio) is meant the origination of one Divine Person from another, an operation of God “ad intra”. A major point of disagreement between the Latin Church and the Greek Church is that, for the former, the Holy Spirit proceeds from the Father and the Son (“ex Patre Filioque procedit”), while for the latter, the Holy Spirit proceeds from the Father ` εκπ ᾿ τ oυ˜ Π ατ ρ oς ᾿ oρευ oμενoν”), alone (“εκ ´ and not from the Son, emphasizing the fact that the Father is the source of all processions. Such divergence, called the Filioque question,6 is present in the Latin and Greek versions of the Nicene-

2 The

dual nature of Christ will not be treated in this text. Cf. Ratzinger (2000).

3 Mystery means a revealed truth that surpasses the powers of natural reason, i.e., a truth that cannot

be demonstrated only from natural principles. Cf. Catechism of the Catholic Church (1997), 234, 237. 4 Compendium of the Catechism of the Catholic Church, 48. Cf. CCC, 249–256. 5 The terms “essence”, “nature”, and “substance” are different in general predication when one applies them to beings others than God; however, they are synonymous when applied to God. The Trinitarian dogma can be summarized in the following way: In essence, nature, and substance there is but one God. However, the Divine Nature subsists in three really distinct Persons, so much so that one is not the another. Cf. Summa Theologica I, 27, a.1; 41, a.1; Pohle and Preuss (1925), p. 6. 6 Filioque is translated as “and [from] the Son”.

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Constantinopolitan Creed (Nicaea, 325; Constantinople, 381).7 Based on Canon VII of the Council of Ephesus (431), which declared anathema whoever changes the propositions laid down in the Council of Nicaea (including the Profession of Faith of the Creed), the Greek Church accused the Latin Church of heresy. On the other hand, the Latin Church considered a certain unanimous recognition of the Filioque in ancient traditions, even among some authors of the Eastern Church. Much discussion and controversy about the Filioque occurred, along with other cultural and political issues, between the ninth and 11th centuries, culminating in the Great East-West Schism (1054).8 Regarding the question of the Filioque, one can say that the issue is due particularly to a semantic problem: Greek and Latin Traditions do not interpret in the same , ´ and the respective noun “procesway the verb “to proceed” (procedo, εκπ oρευω) sion” (processio, εκπ oρευσ ´ ις ). The Greek Tradition interprets εκπ oρευσ ´ ις in a strict sense: the Father as the origin of all processions. Eκπ oρευσ ´ ις “signifies only the relationship of origin to the Father alone as the principle without principle of the Trinity”. The Latin Tradition interprets processio in a broader sense: the term (terminus) proceeding from the principle. Processio “is a more common term, signifying the communication of the consubstantial divinity from the Father to the Son and from the Father, through and with the Son, to the Holy Spirit.”9 When it was assumed that processio is equivalent to εκπ oρευσ ´ ις the theological problems arose, culminating in the heated historical debate. Processio corresponds to the more common term πρo¨ιε´ ναι (“to send”), which indicates the communication of the divinity to the Holy Spirit from the Father and the Son in their consubstantial communion.10 Concerning to the Creed, one way of reconciliation is to understand “qui ex Patre ` δια´ τ oυ˜ Yι῾oυ˜ εκπ ᾿ oρευ oμενoν” ᾿ τ oυ˜ Π ατ ρ oς ´ (“who Filioque procedit” as “τ o` εκ ` ᾿ τ oυ˜ Π ατ ρ oς proceeds from the Father through or by the Son”) instead of “τ o` εκ ᾿ oρευ oμενoν”, ´ given that such a sentence seems to mean “who κα ι` τ oυ˜ Yι῾oυ˜ εκπ

7 The

specific article of the Creed is “We believe in the Holy Spirit, the Lord, the giver of life, who proceeds from the Father and the Son.” The exact words according to the Greek version ` ᾿ τ oυ˜ Π ατ ρ oς are “Kα ι` ει᾿ς τ o` Π νευμα ˜ τ o` ῞Αγ ιoν [N], τ o` κ υριoν, ´ τ o` ζ ωoπ oιoν, ´ τ o` εκ ᾿ oρευ oμενoν εκπ ´ [C] [...]”. [N] and [C] indicate where the main part of the text ends in the Nicene and Constantinopolitan versions of the Creed, respectively. The Latin version is “Et in Spiritum Sanctum, Dominum, et vivificantem: qui ex Patre Filioque procedit [...]”. One can see that Filioque is in the Latin version, while the corresponding “κα ι` τ oυ˜ Yι῾oυ” ˜ is not present in the Greek version. 8 The literature on this issue is obviously huge. To reconstruct these debates in detail is not a task I should set here. As an introduction to controversies surrounding the Filioque question see Siecienski (2010); Mateo-Seco Mateo-Seco (2005), pp. 338–345. For considerations on philosophical approaches see Mateo-Seco (2005), pp. 367–406; Tuggy (2016). 9 The Greek and Latin Traditions regarding the Procession of the Holy Spirit - Pontifical Council for Promoting Christian Unity. L’Osservatore Romano, Weekly Edition in English, 20 September 1995, page 3. 10 “The Spirit proceeds (π ρ oεισ ´ ι) from the Father and the Son; clearly, he is of the divine substance, proceeding (π ρo¨ιoν) ´ substantially (oυσ ιωδ ως) ´ in it and from it” (St Cyril of Alexandria, Thesaurus, PG 75, 585 A).

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proceeds from the Father and the Son as the origin of all processions”. Many theologians of Latin and Greek tradition have accepted such a semantic solution to the Filioque problem. In the Catholic-Orthodox Joint Declaration of 1965, Pope Paul VI and Orthodox Patriarch Athenagoras expressed regret for many of the actions that led up to the Great Schism and lifted the mutual excommunications. It is noteworthy that nowadays the Roman Catholic Church’s practice includes the Filioque clause when reciting the Creed in Latin, but not when reciting the Creed in Greek, especially when a Pope and a Patriarch recite it jointly. It may be asked whether even in the broader sense of Latin, it is necessary to point out the procession of the Holy Spirit from the Father and the Son. Aquinas’ solution shows that it is since such procession is a condition for distinguishing the Persons of the Trinity.

9.3 Aquinas on the Filioque Thomas Aquinas, based on an ancient tradition, argues that it is possible to comprehend real relations in God through the understanding of the inner processions. According to him, there are only two processions: the procession of the Word (processio verbi), an operation of the intellect, called Generation (Generatio); and the procession of love (amoris processio), an operation of the will, called Spiration, (Spiratio).11 In each procession, there are two opposite relations12 : the relation of a principle to the term (terminus), and the relation of the term (terminus) proceeding from a principle. In other words, the origin of the Persons from one another forms four real relations: active and passive Generation (generare, generari), and active and passive Spiration (spirare, spirari).13 They are called, respectively, Paternity, Filiation, Active Spiration and Passive Spiration.14

11 In

this sense, one can understand that the Father is “God knowing himself”, the Son is “God’s knowledge of himself” and the Holy Spirit is “God’s love for himself”. For an appreciation of Aquinas’s insights into God and the Trinity, see Garrigou-Lagrange (1952). To a contemporary introduction to the Trinity, see Marmion & Van Nieuwenhove (2011); Méndez (2006). 12 In this text, “opposite” and “opposed” will be used as synonymous. 13 “Relation” and “Passive” here do not mean relatio and passio in the accidental Aristotelian sense (σ χ ε´ σ εις and π ασ ´ χειν), because, in the Christian view, God is pure actuality. When the concept of relation is applied to God, it is considered that relation is a pure “esse ad”, a simple and total reference to another. 14 Cf. Summa Theologiae, I, 28, a. 4. The four relations are not merely logical. They are considered real, since that a relation is real if, and only if, the elements of the relation are real. They are “according to reality”.

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In this context “Person” (persona) means the relation while reality subsisting in the divine nature.15 Since there are several real relations in God, it follows that there are several Persons in God. The real distinction between the divine relations comes from the relative opposition. Two opposite relations must refer to two Persons. If two relations belong to the same Person, then they are not opposite. Paternity and Filiation are opposite relations, but if the Filioque clause is valid, then Active Spiration does not oppose them because both Father and Son can satisfy this relation. In this case, the Persons constituted by Paternity and Filiation are subjects of the Active Spiration. Therefore, Active Spiration does not constitute Person. Passive Spiration does constitute, then there are only three Divine Persons.16 Aquinas answers the question whether the Holy Spirit proceeds from the Son, in his two most famous works, Summa Theologica (ST), and Summa contra Gentiles (SG).17 One can say that the solutions presented in both texts are the same. The quotations below serve to record such a solution. ST: I answer that, it must be said that the Holy Spirit is from the Son. For if He were not from Him, He could in no way be personally distinguished from Him; as appears from what has been said above. For it cannot be said that the divine Persons are distinguished from each other in any absolute sense; for it would follow that there would not be one essence of the three persons: since everything that is spoken of God in an absolute sense, belongs to the unity of essence. Therefore, it must be said that the divine persons are distinguished from each other only by the relations. Now the relations cannot distinguish the persons except forasmuch as they are opposite relations; which appears from the fact that the Father has two relations, by one of which He is related to the Son, and by the other to the Holy Spirit; but these are not opposite relations, and therefore they do not make two persons, but belong only to the one person of the Father. If therefore in the Son and the Holy Spirit there were two relations only, whereby each of them were related to the Father, these relations would not be opposite to each other, as neither would be the two relations whereby the Father is related to them. Hence, as the person of the Father is one, it would follow that the person of the Son and of the Holy Spirit would be one, having two relations opposed to the two relations of the Father. But this is heretical since it destroys the Faith in the Trinity. Therefore, the Son and the Holy Spirit must be related to each other by opposite relations. Now there cannot be in God any relations opposed to each other, except relations of origin, as proved above. And opposite relations of origin are to be understood as of a “principle,” and of what is “from the principle.” Therefore, we must conclude that it is necessary to say that either the Son is from the Holy Spirit; which no one says; or that the Holy Spirit is from the Son, as we confess.18 SG: [...] the conclusion remains that one divine Person is not distinguished from another except by the opposition of relation: thus, the Son is distinguished from the Father

15 Cf. Summa Theologiae, I, 29, a. 1; a. 4; 30, a. 1. To subsist means to exist separately, as substance and a subject. “Subsistence” may be considered a simple translation of the Greek hypostasis. 16 Summa Theologiae, I, 30, a. 2. 17 Summa Theologiae, I, 36, a. 2; and Summa contra gentiles, l. 4, c.24. Cf. Aquinas (1990) and Aquinas (2001). 18 Adapted Translation from Aquinas (2007).

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consequently to the relative opposition of father and son. It is because in the divine Persons there can be no relative opposition except, consequently, on origin. [...] Therefore, if the Holy Spirit is distinguished from the Son, He is necessarily from the Son, for we do not say that the Son is from the Holy Spirit, since the Holy Spirit is, rather, said to be of the Son and given by the Son. Again, the Son is from the Father and so is the Holy Spirit. Therefore, the Father must be related both to the Son and the Holy Spirit as a principle to that which is from the principle. He is related to the Son by reason of paternity, but not to the Holy Spirit; for then the Holy Spirit would be the Son, because paternity is not said except of a son. There must, then, be another relation in the Father by which He is related to the Holy Spirit; and spiration is its name. In the same way, since there is in the Son a relation by which He is related to the Father, the name of which is sonship [=filiation], there must also be in the Holy Spirit another relation by which He is related to the Father, and this is called procession [=passive spiration]. And thus, in accord with the origin of the Son from the Father, there are two relations, one in the originator, the other in the originated: to wit, paternity and sonship; and there are two others in reference to the Holy Spirit: namely, spiration and procession. Therefore, paternity and spiration do not constitute two Persons, but pertain to the one Person of the Father, for they have no opposition to one another. Therefore, neither would sonship and procession constitute two persons, but would pertain to one, unless they had an opposition to one another. But there is no opposition to assign save that by way of origin. Hence, there must be an opposition of origin between the Son and the Holy Spirit so that the one is from the other.19

In a kind of reductio ad absurdum, Aquinas shows that if one supposes that the Holy Spirit does not proceed from the Son, then no personal distinction can be posited between them. Since the Son and the Holy Spirit are personally distinguished from each other, they are distinguished by opposite relations. Therefore, either the Son proceeds from the Holy Spirit or the Holy Spirit proceeds from the Son. But the Son does not proceed from the Holy Spirit. Consequence: The Filioque is valid. It is possible to interpret mathematically Aquinas’ concept of the Triune God, via a system or relational structure in which it is logically necessary that the Holy Spirit proceeds from the Son (cf. Bertato, 2014). But, in an equivalent approach, we use the tools of the First Order Logic without identity. In the next section, I provide a formal apparatus to verify how the Persons of the Trinity can be simultaneously equal and distinct in different senses and to show how the Filioque proposition follows from the main presuppositions of the doctrine of the Trinity.

9.4 The Thomistic Logic of the Trinity  In this section, I introduce a formalized version of the Thomistic oriented Trinitarian Theology called The Thomistic Logic of the Trinity . The formal system  is constructed as a first-order theory, therefore classical predicate logic with the usual rules is used. Axioms, definitions, and theorems are denoted by bold letters A, D, and T, respectively. Some proofs of theorems are 19 Translation

from Aquinas (1955).

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given in quite a bit of detail, while others are only sketched. Since the Trinitarian Theology depends upon Revelation and Tradition, there are biblical verses and formulations from the Tradition (Councils, Creeds, liturgy, private prayers of the Christian faithful, and discussions of the Fathers, etc.) supporting each axiom. However, for the sake of brevity, I present only a few of them in this chapter. The language L of the classical first-order predicate logic without identity is extended to form the language L in the following way. The symbols x, y, z, etc. variable symbols. The definition of a formula is the usual one with the expected extensions. The set of proper symbols of the theory  is {π, ϕ, σ, P, F, SA , SP , =sub }. The symbols π, ϕ and σ are constant (or object) symbols of . The symbols P, F, SA , SP , =sub are symbols of binary relations. For ease of reading, I have included parentheses. With “e” some natural language interpretations for each formal formula. Therefore, the following shall be considered as abbreviations: π := ‘Father’ ϕ := ‘Son’ σ := ‘Holy Spirit’ xPy := ‘x is Father of y’ P := ‘Paternity’ xFy := ‘x is Son of y’ F := ‘Filiation’ xSA y := ‘x spirates y’ SA := ‘Active Spiration’ xSP y := ‘x is spirated by y’ SP := ‘Passive Spiration’ (x=sub y) := ‘x is equal to y according to the substance (or nature)’ =sub := ‘substantial identity’

9.4.1 Proper Axioms and Theorems As it was said above, the origin of the Divine Persons from one another forms the basis of a double set of opposite relations. It is clear that Active Spiration and Passive Spiration are opposite relations. Paternity is active Generation (generare) and Filiation is passive Generation (generari). Let us assume the following axioms: A1. ∀x ∀ y (xPy ←→ yFx). (P and F are opposite relations.) A2. ∀x ∀ y (xSA y ←→ ySP x). (SA and SP are opposite relations.) A relation can be a relation of reason (relatio rationis) or a real relation (relatio realis). The relation of reason is a relation according to reason alone, which exists only in the intellect; it can be distinguished only by reason. A reflexive relation (relation of the same to the same) is a relation of reason. Since the four considered relations are real, then we assume the following axiom: A3. (Scheme) For any relation R (except for =sub ),

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∀x(¬(xRx)). (The relation of the same to the same is not real.) A very important symbol of Faith is The Atanasian Creed or Quicumque. For many years accepted as a composition attributed to St Athanasius of Alexandria. The following articles,20 give us a foundation to assume some axioms and to confirm some theorems. (4) neque confundentes personas, neque substantiam separantes: neither confounding the Persons, nor dividing the Substance. Cf. A5. (5) alia est enim persona Patris, alia [persona] Filii, alia [persona] Spiritus Sancti; for there is one Person of the Father, another of the Son, and another of the Holy Spirit. Cf. A7, T7, T9 and T10. [...] (21) Pater a nullo est factus nec creatus nec genitus; The Father is made of none, neither created, nor begotten. [¬(ϕ Pπ) ∧ ¬(σ Pπ)]. Cf. A4. (22) Filius a Patre solo est, non factus nec creatus, sed genitus; The Son is of the Father alone; not made, nor created, but begotten. [¬(σ Pϕ)]. Cf. A4 and T1. (23) Spiritus Sanctus a Patre et Filio, non factus nec creatus nec genitus, sed procedens. The Holy Spirit is of the Father, and of the Son neither made, nor created, nor begotten, but proceeding. [ ¬ (π Pσ)∧ ¬(ϕ Fσ)]. Cf. A4 and A6.

The underlined sentences indicate the main theological facts that are totally or partially contemplated by the related definitions, axioms, and theorems. The next axiom and theorem contemplate the opposition between Paternity and Filiation. A4. (π Pϕ) ∧ ¬(ϕ Pπ) ∧ ¬(π Pσ) ∧ ¬(σ Pπ) ∧ ¬(ϕ Pσ) ∧ ¬(σ Pϕ). (Only π related to ϕ belongs to the relation Paternity.) T1.  (ϕ Fπ) ∧ ¬( π Fϕ) ∧ ¬(π Fσ) ∧ ¬(σ Fπ) ∧ ¬(ϕ Fσ) ∧ ¬(σ Fϕ). (Only ϕ is related to π belongs to the relation Filiation.) Proof Follows from A4 and A1.  A4 asserts that the subsisting Paternity is the Person of the Father alone. Accordingly, T1 asserts that the subsisting Filiation is the Person of the Son alone. The next axiom indicates that the relation =sub captures the unity of essence or substance in God. A5. ∀x ∀ y (x=sub y). (Any x and y are equal according to the substance.) From this axiom it is easy to prove the following theorems: T2.  ∀ x (x=sub x). 20 Extracted

from Denzinger-Hünermann’s Enchiridion Symbolorum, 75. Cf. Denzinger & Hünermann (1999). A selection of the discussions about The Trinity in the Fathers of the Church can be seen in Pons (1999).

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(The relation =sub is reflexive.) T3.  ∀ x ∀ y ((x=sub y) −→ (y=sub x)). (The relation =sub is symmetric.) T4.  ∀ x ∀ y ∀ z ((x=sub y) ∧ (y=sub z) −→ (x=sub z)). (The relation =sub is transitive.) The theorems T2, T3 and T4 guarantee that =sub is an equivalence relation on the set of objects. It is easy to see that. T5.  (π =sub ϕ ) ∧ (π =sub σ) ∧(ϕ =sub σ). The first definition tries to capture the idea that the Persons are distinguished only by relations of opposition: D1. x op y :←→ (xPy ∨ xFy ∨ xSA y ∨ xSP y). (x is opposed to y according to the relations or simply x is opposed to y.) It is easy to prove that. T6.  ∀ x ∀ y ((x op y) −→ (y op x)). (The relation op is symmetric.) T7.  π op ϕ. (π is opposed to ϕ.) Proof Follows from A4 and D1.  As a foundation to A6 and T8, we can consider the following verse of the Gospel of John (Joh, 15, 26):” ῞ αν δ ε` ελθ ῝ εγ ᾿ ω` π ε´ μψω υμ˜ ῾ ιν π αρ α` τ oυ˜ π ατ ρ oς, “oτ ῎ ῃ o῾ π αρ ακλητ ´ oς oν ´ τ o` Π νευμα ˜ ` εκπ ᾿ ᾿ oρε υετ ᾿ ινoς μαρτ υρ ησ τ ης ˜ αληθε´ ιας, o῝ π αρ α` τ oυ˜ π ατ ρ oς ´ αι, εκε˜ ´ ει π ερ ι` ᾿ υ·” εμo ˜ Vulgata: “cum autem venerit paracletus quem ego mittam vobis a Patre Spiritum veritatis qui a Patre procedit ille testimonium perhibebit de me.” “But when the Comforter is come, whom I will send unto you from the Father, even the Spirit of truth, which proceedeth from the Father, he shall testify of me:”

A6. π SA σ. (π spirates σ.) T8.  σ SP π . (σ proceeds from π.) Proof Follows from A6 and A2.  Based on article (5) of the Quicumque, we assume the following axiom: A7. ϕ op σ. (ϕ is opposed to σ.) T9.  (π op ϕ) ∧ (π op σ) ∧(ϕ op σ). (π is opposed to ϕ, π is opposed to σ and ϕ is opposed to σ.) Proof Follows from T7, A6, D1 and A7.  The following theorem (obtained from A2 and T8) is in line with the so-called Fundamental Law of the Trinity (or Anselm’s axiom): “In Deo Omnia sunt unum, ubi non obviat relationis oppositio” (“In God all things are one except where there is opposition of relation”):

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T10.  ∀ x ∀ y ((x=sub y) ∧ (π op ϕ) ∧ (π op σ) ∧ (ϕ op σ)). Since the Holy Spirit is the term (terminus ad quem) of the relation Active Spiration, it is not proper to assume that the Holy Spirit can be the subject (terminus a quo) of the same relation: A8. ∀x(¬(σ SA x)). (There is no x spired by σ. It is not proper of σ to be spirator.)

9.5 The Filioque Theorem In this section, I offer a formal formula equivalent to the Filioque clause and a proof of it is provided. In addition, other definitions and results that contemplate important parts of the Trinitarian Theology are also offered. The next definition is an abbreviation of the assertion that the Holy Spirit proceeds from the Father and the Son, i.e., the Filioque clause: D2. FILIOQUE:←→ ((σ SP π) ∧ (σ SP ϕ)). (σ proceeds from π and ϕ.) Now it is possible to prove the Filioque clause as a theorem within this theory: T11.  FILIOQUE. Proof 0. 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12.

21 (0)

¬FILIOQUE ¬((σ SP π) ∧ (σ SP ϕ)) ¬(σ SP π) ∨ ¬ (σ SP ϕ) (σ SP π) −→ (¬(σ SP ϕ)) σ SP π ¬(σ SP ϕ) σ op ϕ ¬(σ Pϕ) ∧ ¬ (σ Fϕ)∧ ¬(σ SA ϕ) ∧ ¬ (σ SP ϕ) ¬((σ Pϕ) ∨ (σ Fϕ) ∨ (σ SA ϕ) ∨ (σ SP ϕ)) ¬(σ op ϕ) (σ op ϕ) ∧ (¬(σ op ϕ)) ¬FILIOQUE −→ (σ op ϕ) ∧ (¬(σ op ϕ)) FILIOQUE

: Hypothesis. : D2 : De Morgan’s law : Equivalence : T8 : Modus Ponens 4;3 : A7; T6 : A4; T1; A8; 5 : De Morgan’s law : D1 : Conjunction 6; 9 : Lines 0-10 : Contradiction. 21

Suppose for the sake of contradiction that the Filioque clause is not true, i.e., (1) assume that “the Holy Spirit proceeds from the Father and the Son” is false. Then, (2) either the Holy Spirit does not proceed from the Father or the Holy Spirit does not proceed from the Son. Therefore, (3) if the Holy Spirit proceeds from the Father, then the Holy Spirit does not proceed from the Son. But, according to T8, (4) the Holy Spirit proceeds from the Father. Then, by modus ponens, (5) the Holy Spirit does not proceed from the Son. Additionally, (6) The Holy Spirit is opposed to the Son, by A7 and T7. Now, by A4, T1, A8 and line 5, (6 and 7) we have that the Holy Spirit is not subject in any of the relations Paternity, Filiation, Active Spiration, and Passive Spiration, in which the Son is the term. Therefore, (9) the Holy Spirit is not opposed to the Son, by D1. It follows from lines 6 and 9 that (10) the Holy Spirit is and is not opposed to the Son. Contradiction! Therefore, the Filioque must be true. 

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The proof above is a reductio ad absurdum, which is in line with the solution to the Filioque question presented by Aquinas. T11 asserts that the Son is subject of the relation Active Spiration. In this context, it is not adequate for the Son to be the term too. Moreover, it is proper of the Son being begotten and it is proper of the Holy Spirit to be spirated. Applying similar reasoning and by the fact that the Father has no origin, we can assume the axioms A9 and A10. A9. ¬(π SA ϕ). (π does not spirate ϕ.) A10. ¬(ϕ SA π). (ϕ does not spirate π.) It is worth noting that the axioms A9 and A10, the following definitions, and theorems are not necessary for the resolution of the Filioque question. However, they are consistent with other results of Trinitarian Theology, showing that the formal approach cast new light on the reflection on these issues. By conjunction, we obtain the following theorem: T12.  (π SA σ) ∧ (ϕ SA σ) ∧ ¬(ϕ SA π) ∧ ¬(σ SA π) ∧ ¬(π SA ϕ) ∧ ¬(σ SA ϕ). (Only π and ϕ are subject in the relation Active Spiration.) Proof Follows from A6, T11, A2, A10, A8, and A9.  Since the relations SA and SP are opposite, the following theorem is valid: T13.  (σ SP π) ∧ (σ SP ϕ) ∧ ¬(π SP ϕ) ∧ ¬(π SA σ) ∧ ¬(ϕ SA π) ∧ ¬(ϕ SA σ). (Only σ is subject in the relation Passive Spiration.) Proof Follows from T12 and A2.  T13 asserts that the subsisting Passive Spiration is the Person of the Holy Spirit alone. The following definition and theorems capture the fact that the non-opposition is a kind of identity. D3. x ¬ op y :←→ ¬(x op y). (x is not opposed to y according to the relations or simply x is not opposed to y .) It is easy to prove the following theorems: T14.  ∀ x (x ¬ op x). (The relation of non-opposition ¬op is reflexive.) T15.  ∀ x ∀ y ((x ¬ op y) −→ (y ¬ op x)). (The relation ¬op is symmetric.) T16.  ∀ x ∀ y ∀ z ((x ¬ op y) ∧ (y ¬ op z) −→ (x ¬ op z)). (The relation ¬op is transitive.) The theorems T14, T15, and T16 guarantee that ¬op works as an equivalence relation on the set of objects. The next theorem shows that every object is identical to itself according to the “relation” non-opposition ¬op: T17.  (π ¬ op π ) ∧ (ϕ ¬ op ϕ) ∧(σ ¬ op σ). Proof Follows from A3 and D1. 

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The next definitions and theorems try to capture the connection (a kind of identity) between the three Persons and the Three Relations Paternity, Filiation, and Passive Spiration. D4. Let R be a relation, then. (R E x) : ←→ (∃y(xRy)). (The relation R belongs to x if and only if x is related to some y according to the relation R.) D5. (R ¬ E x) : ←→ (¬(R E x)). (The relation R does not belong to x.) D4 and D5 formalize the notion that a relation belongs to a Person if, and only if, the Person is a subject of the relation. Theorems T18, T19, T20, and T21 establish which relations belong to whom. T18.  (P E π) ∧ (P ¬ E ϕ) ∧ (P ¬ E σ). (The relation P belongs only to π.) Proof Follows from A4.  T19.  (F ¬ E π) ∧ (F E ϕ) ∧ (F ¬ E σ). (The relation F belongs only to ϕ.) Proof Follows from T1.  T20.  (SA E π) ∧ (SA E ϕ) ∧ (SA ¬ E σ). (The relation SA belongs to π and ϕ but not to σ.) Proof Follows from T12.  T21.  (SP ¬ E π) ∧ (SP ¬ E ϕ) ∧ (SP E σ). (The relation SP belongs only to σ.) Proof Follows from T12.  The following definition formalizes the notion of opposite relations. D6. Let R1 and R2 be relations, then (R1 OP R2 ) :←→ ∀x ((R1 E x) −→ (R2 ¬E x)) . (The relations R1 and R2 are opposed.) From D6, we have: T22. For any relations R1 and R2 , we have

Θ (R1 OP R2 ) ←→ (R2 OP R1 ) . (OP is symmetric.) T23.  (P OP F) ∧ (F OP SP ) ∧ (P OP SP ). (P, F and SP are mutually opposed relations.) Proof Follows from T9, T18, T19 and T21.  T24.  (¬(P OP SA )) ∧ (¬(F OP SA )).

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(SA is not opposed to both P and F.) Proof Follows from T9 and T20.  T23 and T24 assert that Paternity, Filiation, and Passive Relation are the only relations mutually opposed. The following definition formalizes the notion of relatio personificae, that is to say, when a relation is identified with a Person. D7. Given an object x and a relation R, one defines (R ≡ P ers(x)) :←→ ((R E x) ∧ ∀y ((y op x) −→ (R¬E y))) . (The relation R coincides with x or R is relatio personificae of x.) The following theorems establish that there are only three relationes personificae T25.  (P ≡ Pers(π)) ∧ (F ≡ Pers(ϕ)) ∧ (SP ≡ Pers(σ)). (There are three relationes personificae.) Proof Follows from T18, T19, T21, T16, and T9.  T26.  ¬ ((SA ≡ Pers(π)) ∨ (SA ≡ Pers(ϕ)) ∨ (SA ≡ Pers(σ))). (SA is not a relatio personificae.) From T25 and T26, we conclude that Paternity, Filiation, and Passive Spiration are the only relationes personificae. At this point, we can say that a good part of the treatment of the Trinitarian Theology is formalized. This allows us to perceive the subtleties of reasoning in this subject and to understand some of the crucial points of the Christian discourse about the unity of God, the distinction of the Persons, the identification of the relations and the Persons of the Trinity and the validity of the Filioque, according to the Latin interpretation of the term “procession”.

9.6 Final Remarks The approach above allows us to see how to express the identity of the Divine Persons according to the substance and their distinction according to the relations of origin. This was done preserving the principles of the Classical Logic, since that it is used a First Order Theory. As a consequence, from this system, it is logically concluded the Filioque clause. Moreover, it is possible to express the connection between the Persons and the relationes personificae. Such formal approach can be considered an interesting case of Logic of Religion, since it is logic applied to a particular religious domain, being an example of Formal Theology, a discipline in which the formal methods of logic are combined with principles of theology. Now, we turn to a brief comment on the consistency of this theory, providing a model for it. A theory is said consistent if, and only if, it contains no contradictions. According to a version of Gödel’s Completeness theorem, a first-order theory is consistent if,

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and only if, it has a model. To prove that the Thomistic Logic of the Trinity  is consistent, I offer the following set-theoretical model: Let us consider the set O = {p, f, s}, the set R = {Π , Φ, Σ, Ξ } of relations on O, where Π = {(p, f )}, Φ = {(f, p)}, Σ = {(p, s), (f, s)}, and Ξ = {(s, p), (s, f )}, and a function interpretation I such that, I(π) = p, I(ϕ) = f, I(σ) = s, I(P) = Π, I(F) = Φ, I(SA ) = Σ, I(SP ) = Ξ , and I(=sub ) = O2 . It is easy to see that such structure K = O, I is a model for . Since  has a model, then  is consistent. According to the general Christian view, it is not necessary to prove that the Trinity is positively conceivable and therefore real. Actually, since the Trinity is a mystery, it is considered impossible to demonstrate its validity. The task the Christian thinker should undertake is to use human reason to refute the objections against the possibility of the Trinity, which classify it as absurd or inconsistent. I think that this chapter successfully shows that such thinker may use the formal methods of logic to accomplish this task. Acknowledgments This research was made possible through the support of grant No. 61108 (“Formal approaches to philosophy of religion and analytic theology”) from the John Templeton Foundation. The opinions expressed in this chapter are those of the author and do not necessarily reflect the views of the John Templeton Foundation.

References Aquinas, Thomas. 1955. On the truth of the catholic faith: Summa contra gentiles. Book 4: Salvation. Trans. James F. Anderson. New York: Image Books. ———. 1990. Suma Contra os Gentios. Vol. I and II. Trans. Odilão Moura and Ludgero Jaspers Porto Alegre: Escola Superior de Teologia de São Lourenço de Brindes. ———. 2001. Suma Teológica. Vol. 1, 1–43. São Paulo: Loyola. ———. 2007. Summa Theologica. Vol. I. New York: Cosimo. Bertato, F.M. 2014. Sobre a Definição Matemática de Sistema: Alguns aspectos Históricos, novas propostas e Lógicas Sistêmicas Associadas. In Auto-Organização: estudos interdisciplinares, ed. Bresciani Filho, Ettore, et al., vol. 66, 55–100. Campinas: Coleção CLE. Bochenski, Joseph M. 1965. The logic of religion. New York: New York Press. Catechism of the Catholic Church. 1997. Revised in accordance with the official latin text promulgated by Pope John Paul II. 2nd ed. Vatican City: Libreria Editrice Vaticana. Denzinger, H., and P. Hünermann. 1999. El Magisterio de la Iglesia. Enchiridion Symbolorum definitionum et declarationum de rebus fidei et morum. Barcelona: Herder. Garrigou-Lagrange, R. 1952. The trinity and god the creator. A commentary on St. Thomas’ theological summa. Trans. Frederic C. Eckhoff, 27–119. St. Louis: B. Herder Book. Marmion, D., and R. Van Nieuwenhove. 2011. An introduction to the trinity. Cambridge: Cambridge University Press. Mateo-Seco, L. 2005. Dios Uno y Trino. 2nd ed. Pamplona: Eunsa. Méndez, G.L. 2006. Deus Uno e Trino. Manual de Iniciação. Lisboa: Diel. Pohle, J., and A. Preuss. 1925. The divine trinity: A dogmatic treatise. 5th ed. London: B. Herder Book. Pons, G. 1999. La Trinidad en los Padres de la Iglesia. Madrid: Ed. Ciudad Nueva.

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Ratzinger, Joseph. 2000. Einführung in das Christentum: Vorlesungen über das Apostolische Glaubensbekenntnis. München: Kösel Verlag GmbH & Co. Siecienski, A.E. 2010. The Filioque: history of a doctrinal controversy. Oxford: Oxford University Press. Tuggy, Dale. 2016. Trinity. In The Stanford encyclopedia of philosophy, ed. Edward N. Zalta. Stanford: Stanford University. https://plato.stanford.edu/entries/trinity/. Accessed 18 August 2017. Thom, Paul. 2012. The logic of the Trinity: Augustine to Ockham. New York: Fordham University Press. Weingartner, Paul. 1999. Bochenski: Attempts to apply logic to problems of religion. Philosophia Scientiæ 4: 175–198.

Fábio Maia Bertato has a PhD in philosophy from the University of Campinas (Unicamp), Brazil. He is presently Researcher at Unicamp and Associate Director of the Centre for Logic, Epistemology, and the History of Science (CLE - Unicamp), as well as the Managing Editor of the Brazilian Journal on the History of Mathematics (RBHM) and Associate Editor of the book series Coleção CLE.

Chapter 10

Contradictions and Rationality: An Analysis of Two Biblical Cases Susana Gómez Gutiérrez

10.1 Introduction In the literature about biblical exegesis we find diverse viewpoints about how to read the biblical writings (Cooper 1983; Filson 1948; Hamilton 1965; Quinn 1962; Weber 1901; Whiterup 2013). Some exegetes and biblical scholars recommend to read the scripture as a compilation of historical texts that speak about real facts that happened in the way they are described. Others, propose to go beyond the literal meaning and take the scripture—or at least some of its passages—as having a mythological dimension or as being metaphorical texts, product of poetic imagination. Others suggest to read the biblical writings as written with the intention of exhorting the people of the time to believe in a God who intervenes in the human affairs and, in the case of the New Testament, to believe in the divinity of Jesus Christ. And yet others, consider them as allegories written with the purpose of

A previous version of this chapter was discussed with colleagues and friends at the Corporación Universitaria Minuto de Dios and at the Pontificia Universidad Javeriana in Bogota, Colombia. It was published as part of the book Racionalidad, lenguaje y acción. Aproximaciones analíticas (2016) with the title “Contradicciones y Racionalidad: Susana y los viejos jueces y la resurreción de Jesús. Un estudio de casos.” Regarding the present version, I am thankful to the anonymous referees for their useful comments and to the people who contributed to the enrichment of the ideas developed here through the discussion of some central thoughts and technical proceedings. Among them I include Evandro Luis Gomes, Tomás Barrero, Jose Andrés Forero, Miguel Ángel Pérez, Carlos Miguel Gómez and Luis Fernando Múnera. I am especially grateful to Professor Graham Priest for inviting me to participate as a visiting research scholar in the academic community of CUNY, where several of the ideas presented here were refined and reformulated. Finally, as always, I am more than grateful to God, the Virgin Mary, the angels and the saints for everything. S. Gómez Gutiérrez () Faculty of Philosophy, Pontificia Universidad Javeriana, Bogotá, Colombia © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_10

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exhorting people to morally and spiritually behave in settled ways. Some authors propose to consider the historical circumstances in which the texts were written in order to decide how credible they are or how precise their descriptions may be. Others, propose to base the readings on linguistic studies in order to find the real meaning of the texts. In this chapter, I call for reading two particular biblical passages taking the sentences involved in their literal meaning. I do not intend to stand up for an exegetic reading or biblical study. Rather, I propose to look at those passages through a philosophical lens that uses logical analysis as a form of filter that offers the reader a different outlook on certain literary, historical and, as in this case, biblical passages that seem to contradict each other when taken prima facie. My intention here, then, is neither to produce an exegetic study of the texts selected nor discover their true meaning. Instead, I use them to examine two philosophical problems: (a) how to deal logically with contradictions and (b) what are the consequences that follow from the alternatives examined regarding the notion of rationality.1 To this end, I present the story of Susanna and the Elders in the Old Testament and the passages about the resurrection of Jesus Christ in the New Testament as cases that exemplify different ways of dealing with contradictions. On the one hand, I take the story of Susanna in the Book of Daniel as a case where contradictions between the Elders’ versions of Susanna’s alleged act of infidelity serve as evidence that convinces Daniel that their statements about Susanna’s adultery are false. On the other, I take the passages about the resurrection of Jesus Christ as a case in which an imagined “true believer” refuses to accept that the sentence “Jesus rose from the dead” is false, despite the contradictory accounts presented by the four Evangelists. Considering both cases, and assuming a perspective in which rationality is closely connected to a certain idea of logicity, I ask what is the most appropriate logic to follow in order to evaluate both beliefs as rational. To be clear, rationality has historically been understood as the feature that differentiates human beings from other species. Nevertheless, philosophers, psychologists, economists, social scientists, and others interpret the term rationality in different ways. Contemporary authors such as Karni and Schmeidler (1986) relate rationality to self-preservational actions; for others, like Mackie (1977), Millgram (2005), Korsgaard (1986) and Williams (1979), rationality relies on the nature of the criteria—either external or internal—that lead the individual to hold one belief or

1I

must clarify that my intention is not to reduce the study of the biblical texts to a logical reading. I am simply exploring this option and trying to make explicit some consequences that follow from choosing it. Taking this path implies making some assumptions, such as accepting the biblical sentences analyzed here as sentences that can be true or false (which I sometimes refer to here as “sentences” and sometimes as “propositions” disregarding any possible philosophical differences between these terms), and not simply expressions of feelings or desires, or sentences produced with a solely exhortative or directive evangelical purpose. I obviously recognize that other biblical sentences may have different uses depending on the context in which they are immersed, but I assume that the particular sentences analyzed here have the descriptive character that I attribute to them.

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act in one way instead of another. For authors like Plantinga (1993), Alston (1991) and Plantinga and Wolterstorff (1983), the use of the term implies causal and noncausal views, where what is important is either the kind of process that produces, sustains, and backs up a certain belief, or the capacity (potential) to defend it. Recognizing this variety of uses, the philosopher Searle (2001) presents what he believes to be the set of ideas that constitute the starting point for discussions on rationality. Said author designates the sum of these ideas “the classical model.” Summarizing Searle’s account, the classical model posits rationality as a matter of obeying rules: “The special rules that make the distinction between rational and irrational thought and behavior” (p. 8). As Searle suggests, this process does not necessarily mean that people in everyday life, when acting or holding some belief, are always conscious about those rules or even aware that they are following them. It simply means that the task of the theoreticians is “to make explicit the inexplicit rules of rationality that fortunately most rational people are able to follow unconsciously” (p. 8). Among other things, this perspective implies that rationality is essentially a matter of reasoning logically; and being rational or irrational—applied to beliefs and actions—either as the result of, or as supported by a reasoning that follows certain logical rules. One corollary of this view is a prohibition of any contradictions in the set of beliefs and/or desires that give rise to a particular reasoning. As we shall soon see, from a classical point of view, contradictions must be avoided because a contradiction could imply anything whatsoever, so that if there is a contradiction in the initial set of beliefs and desires, anything whatsoever could follow, and in this sense any action or belief whatsoever could be supported by the reasoning. Scholars and philosophers have widely discussed the relevance of these assumptions with regard to both practical rationality (related to the motivation and/or justification of actions) and theoretical rationality (related to the motivation and/or justification of beliefs), but I do not delve into such discussions here. Rather, I subscribe to Searle’s general notion that rationality is a matter of following rules— mainly logical rules. However, I discuss one of its corollaries; namely, the idea that contradictions must be avoided in relation to rationality. In accordance with this position, my starting point is that in order for a particular belief (or action) to be considered rational, it must be logical. By “logical”, in this specific case, I mean that it is necessarily the result of or is supported by a certain kind of reasoning that either implicitly or explicitly presupposes a specific set of logical rules, something that can be called a “logical system.”2

2 Logic

is normally understood to be the study of valid arguments. From this perspective, we can say that a logical system or a logic is a language provided with a syntax (vocabulary, rules for the formation of sentences and axioms, and rules of inference that allow us to make the transition from one sentence to another), as well as a semantics (models that provide us with the truth values of the sentences and the tools for assessing the validity of the arguments). Logical systems can either be axiomatic (if they include axioms and rules of inference) or natural deduction systems (if they only include rules of inference). One example of an axiomatic system is Russell and

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My first aim in this chapter is to show that when we read the story of Susanna and the Elders and consider the rationality of Daniel’s belief in Susanna’s chastity, we may presuppose a notion of rationality that fits perfectly into the classical model. My second is to show that, when reading the New Testament passages about the resurrection of Jesus and considering that belief in relation to other pivotal Christian beliefs, as well as its own rationality, it seems that we could introduce a different approach to the relationship between rationality and contradictions. Regarding the latter I contend that, to be considered rational, the believer in the resurrection could support that belief with a reasoning based on some form of paraconsistent logic. I focus here on da Costa’s C1 system and the LFI1 system of Carnielli, Marcos, and de Amo, but other paraconsistent systems could also be used. In the second section, I present the cases of Susanna and the Elders on the one hand, and the resurrection of Jesus on the other. I show that while Daniel’s belief in the sentence “Susanna did not commit adultery with a young man in the garden” is supported by an argument that follows the rules of classical logic, the Christian believer’s refusal to accept the falsity of the sentence “Jesus rose from the dead” violates those very same rules. In the third section, I introduce a way of understanding the relationship between logic and rationality within the context of the classical model and show that, given said relationship, the Christian believer’s refusal to accept the falsity of the sentence about the resurrection is irrational. In the fourth section, I present the argument of triviality as one of the arguments against contradictions, along with a paraconsistent solution to this problem. In the fifth section, I briefly introduce da Costa’s particular solution to the problem of triviality and show how, by using C1 as the underlying logic in both Daniel’s argument and the believer’s argument, we can solve the problem of the rationality of the beliefs in each of these cases. In the sixth section, I show some of the advantages of adopting LFI1 instead of C1 as the underlying logic. In closing, I present some problems that emerge when a perspective like this one is adopted.

10.2 Susanna and the Elders and the Resurrection of Jesus Chapter 13 of the Book of Daniel in the Bible tells the story of Susanna, a young woman who is married to a rich young Jewish man named Joachim. Every day at noon, Susanna goes for a walk through the large garden of the house where she lives. On one occasion, she asks her maids to bring oils and perfumes and to close the door of the garden so that she can bathe in privacy. Two Elders, charmed by her

Whitehead’s Principia Mathematica (1910), while one of a natural deduction system is E. J. Lemmon’s Beginning Logic (1965). (Haack 1978).

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beauty, come to spy on her. When she is alone, they approach her and threaten her, saying that unless she agrees to have sex with them they will accuse her of being unfaithful to her husband with a young stranger in the garden. When Susanna refuses to submit to their demand, the Elders make good on their threat. They accuse her of adultery and call for her execution. Susanna prays to God to save her and her prayers are answered. A boy named Daniel defends her by seeking proof of her innocence. He asks each of the Elders, separately, under which tree she had sex with the young man. The first says “under an acacia,” the second says “under a holm oak.” As a result of their prevarications, the Elders are convicted of perjury. From this perspective, one can say that Susanna is saved because Daniel proves the falsity of the Elders’ accusation by showing how they contradict one another regarding what actually happened in the garden. One of them affirms q (Susanna had sex with the young man under an acacia) while the other affirms r (Susanna had sex with the young man under a holm oak), and r implies not-q (Susanna did not have sex with the young man under the acacia). Regarding the contradictions in their accounts, Daniel could either have suspended judgment, or simply declared that one version was true and the other false, and in both cases Susanna would have been condemned to death. Instead, he proves that their allegation of Susanna’s adultery is false. Resorting to tradition, this procedure can be understood as a reductio ad absurdum. We want to prove the falsity of p, or, what is the same in this context, we want to prove that not-p is true. In order to prove not-p, we suppose that p is true and show that this supposition implies a contradiction, which means that p implies q and not-q. By extension, we show the falsity of p or the truth of not-p. In the case of the Elders, what could be considered false would be their sentence p: “Susanna committed adultery with a young man in the garden.” We can express this postulate in axiomatic propositional classical logic as follows:3 1. (a→b)→((a→¬b)→¬a) Or, in the form of natural deduction: a→b, a→¬b

2.

¬a

Proceeding by natural deduction, we can reconstruct the argument maintaining Susanna’s innocence in the context of classical propositional logic as follows:

3I

understand classical logic to consist of the logical systems whose axioms, and deductive rules are completely in accordance with the fundamental principles of the deductive thoughts of Frege and Russell-Whitehead.

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1)

p→q

Premise

2)

p→¬q

Premise6

3)

p

Hypothesis

4)

q

Modus ponens (MP) 1, 3

5)

¬q

MP 2, 3

6)

q∧¬q

Adjunction 4, 5

7)

¬p

Reductio ad absurdum 3–6

Now let us consider the four Evangelists’ narrations of the resurrection. Close examination of the details reveals that they have several elements in common. For example, in all four accounts, the people who visit the sepulcher do so very early in the morning on the first day of the week; Mary Magdalene is always among the visitors; and the body of Jesus is not in the sepulcher—the tomb is empty. However, there are also some significant differences among the narratives. For example, in the Gospel according to Matthew, two women visit the sepulcher; according to Mark, three women visit it; according to Luke, many women visit it; according to John, Mary Magdalene is the only one to go there. In three of the four accounts, the sepulcher stone has already been removed by the time the women arrive, while in the fourth, the angel moves it in front of their eyes. One of the Gospels speaks of a single angel; another speaks of a young man; another of two young men; another of two angels. In three of the accounts, either the angels or the young men send the women to tell the disciples that Jesus has risen from the dead. According to Mark, Jesus encounters Mary Magdalene and James’ mother just after they visit the tomb and sends them to tell the disciples to meet him in Galilee. According to John, after a short conversation in which Jesus talks to Mary Magdalene about being the Son of God, he sends her to tell the disciples that he is returning to his Father. These testimonies evidently contradict each other with respect to certain details. Leaving aside explanations based on the fact that the Gospels were written in different circumstances, and the obvious problems that translations can generate, if we apply the same kind of reasoning we did in the case of Susanna, we must conclude that the sentence “Jesus rose from the dead” is false. This inference is likely to present a serious problem for the Christian believer, since God’s salvific plan in which Jesus is the Savior—one of the pillars of Christian faith—depends on the resurrection of Christ. Bart Ehrman, an American New Testament scholar, proffers an example of just such a dispiriting realization. In Jesus, Interrupted, Ehrman (2009) explains that one of the reasons why he abandoned his belief in Jesus as the Son of God was the many contradictions he found in the New Testament—including those related to the resurrection itself. This dramatic loss of faith is not what happens to the type of believer I want to consider here. Said believer will not admit that the sentence “Jesus rose from the dead” is false despite the contradictory accounts of the resurrection, in which some assert p, while others assert either not-q or something that implies not-q.

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Based on this case and that of Susanna, the question I want to pursue is whether or not, assuming the notion of rationality as being related to logic, we can claim that the beliefs adopted in both cases are rational.4

10.3 Contradictions and Rationality The perspective on contradictions that I describe in Susanna’s story conforms to the rules of classical logic. As seen above, classical logic maintains that if a sentence p implies a contradiction—q and not-q—then p is false. In that case, we could say that the belief that results from accepting Daniel’s argument—“Susanna did not commit adultery with a young man in the garden”—is rational. But does this inference mean that the believer’s refusal to accept the falsity of the sentence “Jesus rose from the dead” is irrational? After all, the believer is dealing here with testimonies that prima facie contradict each other, but nevertheless does not accept the conclusion that would necessarily follow from that situation. This inquiry warrants considering the relationship intuitively established between logic and rationality. We have assumed that logical conformity is a necessary condition for rationality. Thus, if a certain belief is rational in this context, it must be logical, since it must implicitly or explicitly comply with the presuppositions of a certain logical system. In his Ensaio sobre os fundamentos da Lógica (Da Costa 1980), N. C. A. da Costa identifies this as the “principle of systematization.”5 In that book, the above-mentioned Brazilian logician and philosopher presents three pragmatic principles of reason that he considers the formal features of rationality, i.e., the principle of systematization, the principle of unity and the principle of adequacy. According to the principle of systematization, there are different kinds of knowledge, which must be conceptually ordered and obtained through inferences if they are to be considered rational. This means that “reason always expresses itself through a logic,” i.e., “in rational contexts we always find, implicitly or explicitly, a logical system” (Da Costa 1980, p. 45). The principle of adequacy holds that “The logic subjacent to a given context must be the one that best adapts to it” (Ibidem p. 46). This means that, given a certain domain of objects, the chosen underlying logic, i.e., the logic in which the validity

4 Assuming a sort of fideism, some readers of previous versions of this chapter have asserted that—

generally speaking— religious believers do not believe in response to a logical argument, but rather because they are moved by feelings and emotions. I think this is an interesting point that deserves discussion. Even though I would like to clarify that my concern here is not the way believers actually adopt a certain belief, but whether or not a belief once adopted can be shown to be rational, if we assume that rationality is a matter of following logical rules. In this sense, my question, as will be clarified in the next section, is whether a belief that consists of a proposition involving a contradiction can be rational, given that what is considered rational in this context involves a logic that holds contradictions to be false. 5 The translation of all Portuguese and Spanish quotes into English is my own.

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of the arguments is evaluated, must be the one that best adapts to that domain. The choice would be made using pragmatic criteria such as simplicity, facility, economy, and so on (Bobenrieth 1996, pp. 371–372). Considered together, the principles of systematization and adequacy allow us to claim that (a) every rationality has an underlying logic and (b) the underlying logic changes according to the needs of the context. The principle of unity is also important insofar as it works as a restraining principle. It establishes that “in a given context, the underlying logic is unique” (da Costa, p. 46), which means that once we have adopted one logic, that logic must be preserved. Regarding Susanna’s chastity, we accepted the reductio ad absurdum as a postulate and have thus taken classical propositional logic as our underlying logic, with its principle of the Law of Non-Contradiction (LNC). Axiomatically speaking, the reductio ad absurdum and the LNC can both be taken as primitive, but there is a sense in which the reductio is a derivative of the LNC, if we take into account that the reductio can be understood as a Modus Tollens, in which the falsity of the contradiction—which is what the LNC states—in the consequent leads to the falsity of the antecedent. I will speak a little more about this principle later. At this point, suffice it to say that the LNC has been formulated in many different ways. Perhaps the most common is the Aristotelian formulation, which reads: “For the same thing to hold good and not to hold good simultaneously of the same thing and in the same respect is impossible” (Aristotle 1993, 1005b, 18–20). In more technical terms, this formula can be expressed either as saying, “It is not possible that there is an object a and a property F such that Fa ∧ ¬Fa,” or “It is not possible for there to be a proposition a, such that a ∧ ¬a” (Priest 2006, p. 8). Once we assume this option, if we do not admit that the sentence “Jesus rose from the dead” is false, we are in fact being irrational by not accepting the conclusion that necessarily follows from contradictory premises. But the question is why we are obliged to accept the LNC as a universally and necessarily valid principle.

10.4 Contradictions and Nontriviality Ideas about contradictions have been a topic of discussion for philosophers throughout history, but Aristotle is perhaps both the first great thinker to refer to the topic and the most influential one to ever contemplate its implications. Even today the predominant idea that contradictions are unacceptable stems from Aristotle’s works on metaphysics and logic, especially Metaphysics (Book IV), in which the Estagirite defends the LNC by showing, via “proof by refutation” (Aristotle 1993, 1006a, pp. 10–15), that this is “the firmest of all principles” (Aristotle, 1005b 20–25). In “What’s So Bad About Contradictions?” (Priest 2004), Graham Priest presents a list of the most frequent reasons that Aristotle and defenders of the LNC give for rejecting contradictions. The unifying idea is that (a) they entail everything and (b) they cannot be true. The problem of whether or not there can be true contradictions

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is of great importance, especially in the realm of theology, but I do not intend to go into that matter here. Instead, I will focus on the problem mentioned in (a), which can be called “the argument of triviality.” According to Priest, the supposition behind (a) is that rational belief is closed under entailment, which means that someone who believes in a proposition p is committed to all of the propositions that issue from it. Thus, if a contradictory proposition such as a ∧ ¬a entails everything, the believer would be committed to believing everything, which the author says, “is too much” (Priest, p. 24). Taking Priest’s explanation into account, we can summarize the argument of triviality by saying that contradictions cannot be accepted or rationally believed because they entail everything and in so doing, they commit the believer to believing everything. In logical terms, a theory is trivial or explosive when “the distinction between demonstrable and non-demonstrable propositions would no longer hold, given that all syntactically correct sentences will be ‘true’” (Da Costa 1958, p. 6). The assertion that every sentence produced in the language can be deduced from a contradiction corresponds to the principle called either ex contradictione sequitur quodlibet (Bobenrieth 1996, p. 103) or ex falso sequitur quodlibet (Gomes 2013, p. 4),6 both of which can be abbreviated as EQL, and symbolically expressed as follows: 4. ¬a → (a → b) Or it can be expressed as follows by using the law of exportation, which is valid in different systems (Principia Mathematica, Hilbert, Łukasiewicz): 5. (¬a ∧ a) → b (Bobenrieth 1996, pp. 103–104; Gomes 2013, p. 5) This can be read saying that from a contradiction ¬a ∧ a, b follows, where b is any sentence that may be constructed using the formation rules of the system. The argument according to which contradictions must be rejected because they make the theory trivial is found in most discussions about contradiction, but it only began to receive serious consideration in the middle of the twentieth century, with the development of so-called paraconsistent logics (Da Costa 1963; Priest 1979; Routley and Routley 1972). These are logical systems in which the LNC is not valid and even though no triviality is produced. By means of these systems, paraconsistent logicians have shown that contradictoriness and triviality are independent notions, and that the LNC is not a universal principle in this context. From a syntactical viewpoint, paraconsistent logicians separate the LNC and triviality by changing some rules of traditional deductive systems and deriving consequences from the new rules. One way of doing this consists of partially restricting EQL, which means that some formulations of EQL are valid in these systems, and the paraconsistency in this process is called lato sensu paraconsistency. Another way consists of making EQL generally invalid, which means that no

6 Although authors such as Carnielli and Marcos (2002) separate ex contradictione sequitur quodlibet and ex falso sequitur quodlibet, we will maintain the equivalence between these two principles in this work.

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formulation of EQL is valid in these systems, and this kind of paraconsistency is known as stricto sensu paraconsistency (Gomes 2013).

10.5 Da Costa’s C1 System Da Costa’s Cn Systems, in particular the propositional paraconsistent system C1, take the latter approach. That is, they avoid triviality by excluding from the system all implicative, conjunctive and biconditional forms of EQL as well as both forms of reductio ad absurdum, the introduction of double negation, and both forms (the conjunctive and the implicative) of the LNC. Da Costa must also prevent deduction of EQL from the system. For that purpose, the rule of inference known as disjunctive syllogism (DS) is also excluded. If DS were valid and our system included a contradiction, we would have: 6. (1) (2) (3) (4) (5)

¬a ∧ a Supposition a by simplification in 1 a ∨ b by addition in 2 ¬a by simplification in 1 b by disjunctive syllogism between 3 and 4.

This case, known as Lewis’s argument (Bobenrieth 1996), maintains that if we accept DS, we deduce from the system (¬a ∧ a)→b, which is exactly what we are trying to avoid. Thus, da Costa’s strategy is to reject DS in order to have a logical system that accepts contradictions and avoids triviality. Now, if we subscribe to the idea introduced above that for a particular belief (or action) to be considered rational, it must be supported by an inference that follows the rules of a particular logical system (whatever it may be), we can say that the Christian believer’s refusal to accept that the sentence “Jesus rose from the dead” is false when faced with contradictory accounts of the resurrection is, at least in principle, rational. This is so because there is a logical system (in this case da Costa’s C1 system) in which the falsity of that sentence does not follow. I will explicate this idea further, but for now it is worthwhile asking what happens in Susanna’s story. Does this situation invalidate Daniel’s argument against accusing Susanna of adultery? Does it reestablish the credibility of the Elders’ testimonies and save them from death? After all, from the perspective we are considering here, the contradiction between their testimonies does not imply that the sentence “Susanna committed adultery with a young man in the garden” is false. Fortunately for Susanna, a negative answer to these questions remains possible, at least in principle. One of the most important characteristics of da Costa’s C1 system (Da Costa 1963) is that it does not fully reject the consistency requirement of the system. Instead, it subsumes the consistent system into a wider one that includes contradictions, while preserving some sectors in which the LNC is still valid. This is made possible by the introduction of so-called “well-behaved formulas.” These are formulas for which the LNC and its derivatives continue to be valid. A well-

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behaved formula is represented by adding a little white ball at the upper right side of the symbol and is defined as follows: 7. bo =def ¬ (b ∧ ¬b) According to this definition, a well-behaved formula is equivalent to the symbolic expression of the LNC. Thus, as shown above, from a certain perspective reductio ad absurdum is derived from the LNC, and reductio is valid for well-behaved formulas. This can be symbolically expressed as follows: 8. bo → ((a→b) → ((a→¬b) → ¬a)) (Bobenrieth 1996, p. 190) Or, in the form of natural deduction: 9.

bo, a→b, a→¬b ¬a

The presence of well-behaved formulas permits us to say that a theory or a corpus of knowledge with a C1 underlying logic can include sentences that are constrained by the LNC and can be refuted using reductio ad absurdum—as in the case of Susanna—as well as sentences that do not seem to be constrained by the LNC and for which reductio does not hold, as in the case of the resurrection of Jesus. Thus, we can conclude that both of these beliefs are rational because there is a unique logical system (in this case, da Costa’s C1 system) or, as we will see in the next section, the LFI1 system of Carnielli, Marcos and de Amo (Carnielli et al. 2000; Carnielli and Marcos 2002), which allows us to deal with them logically.

10.6 A Discussion of the Resurrection of Jesus Argument: Introducing LFI1 Take p as the proposition “Jesus rose from the dead”; q as the proposition “Only one woman went to the sepulcher at the third day and found it empty”; r as the proposition “More than one woman went to the sepulcher at the third day and found it empty.” We can understand that r implies not-q. These propositional letters permit us to symbolize the two premises of the argument about the resurrection as follows:7 10. p→q, p→¬q Supposing that we are in the context of da Costa’s C1 system, a question emerges about what conclusion we should draw from these two premises. Given that in C1 ,

7 As

in the case of Susanna, the only support here for the sentence “Jesus rose from the dead” are the sentences that constitute the testimonies of the women. Based on this, I take q and ¬q as necessary and sufficient conditions of p, and take p→q and p→¬q as premises of the argument.

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both the LNC and reductio ad absurdum are only valid for well-behaved formulas, we can say that ¬p is not a possible conclusion. The problem now is that we cannot derive any conclusion in C1 from premises that include contradictions because this system has no way to deal with them. Thus it does not have any axioms or theorems that allow us to derive conclusions from formulas that are not well-behaved. In this case, we can shift to another system that does allow us to do so. A good option is the LFI1 system (Carnielli, de Amo and Marcos, Carnielli et al. 2000; Carnielli and Marcos 2002), which is a C-System in that it contains the same axioms and theorems of da Costa’s C-Systems and allows us to prove the same formulas that are proved in C1. Furthermore, it includes axioms and theorems that allow us to deal with inconsistent formulas. The notions of “inconsistency” and “consistency” are formalized using the black ball operator “• ” in opposition to the white ball operator “o ” as follows: 11. 1. • a for “a is inconsistent” 2. o a for “a is consistent” In this sense, • a and o a are abbreviations of the formulas: 12. o adef = ¬(a∧¬a) • a = a∧¬a def In the context of our example, we have two premises: p→q and p→¬q. In the absence of reductio ad absurdum in LFI1, we cannot have ¬p, and considering the set of axioms and inference rules of the system, we cannot have p from them either, but we can prove other things. For example, we can prove (• (p→q)∨• (p→¬q))→p. For that, we suppose that each of these premises is an inconsistent formula, which means that we have • (p→q) and • (p→¬q) as premises. As LFI1 is presented in an axiomatic way, I leave aside natural deduction and proceed to prove axiomatically. The axioms and inference rules of LFI1 used for this proof are the following: Axiom 1: a∧b→a Axiom 2: • (a→b) ≡ (a∧• b) Axiom 3: (a→c)→[(b→c)→((a∨b)→c)] Modus Ponens (MP) Deduction Theorem (DT) 13.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

• (p→q)

Supposition ∗ p∧ • q Axiom 2 in 1 (p∧ • q)→p Axiom 1 p MP 2, 3 • (p→q)→p DT 1-4 • (p→¬q) Supposition ∗ p∧ • ¬q Axiom 2 in 6 (p∧ • ¬q)→p Axiom 1 p MP 7, 8 (p→¬q)→p DT 6-9

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(11) (• (p→q)→p)→[(• (p→¬q)→p)→((• (p→q)∨ • (p→¬q))→p)] Axiom 3 (12) [(• (p→¬q)→p)→((• (p→q)∨ • (p→¬q))→p)] MP 5, 11 (13) (• (p→q)∨ • (p→¬q))→p MP 10, 12 This formal result can be interpreted as saying that, supposing we have contradictory information like “Only one woman went to the sepulcher at the third day and found it empty” and “It is not the case that only one woman went to the sepulcher at the third day and found it empty,” whichever testimony we accept implies the truth of the sentence “Jesus rose from the dead.” We have not proved the truth of the sentence “Jesus rose from the dead” as such, i.e., the conclusion of the argument is not p simpliciter. The conclusion is a conditional sentence in which p is the consequent, and the antecedent is a disjunctive sentence. On the one hand, this seems to be a bad result because it makes the conclusion weaker. But it reflects better what happens in relation to the biblical case since it can be read to mean that the testimonies are one means of allowing us to assert that Jesus rose from the dead, and that whatever testimony we choose permits us to assert that the resurrection occurred. By using LFI1, it is also possible to prove that adding p to the premises does not trivialize the theory and, moreover, the proof itself allows us to observe that q is not well-behaved. Thus, if the supposition “Jesus rose from the dead” is true, the theory is still not trivialized, even though it implies contradictory information. The axiom used here is: Axiom 4: a→(b→(a∧b)) 14.

(1) (2) (3) (4) (5) (6) (7) (8) (9) (10)

p→q Supposition p→¬q Supposition p Supposition ∗ q MP 1, 3 ¬q MP 2, 3 q→(¬q→(q∧¬q)) Axiom 4 ¬q→(q∧¬q) MP 4, 6 q∧¬q MP 5, 7 • q Def • p→ • q DT 3-98

As indicated, we have found a logical system—in this case, LFI1—that authorizes us to draw some conclusions from the contradictory information in the Gospels without trivializing the theory. We avoid trivialization because we cannot deduce everything within it. For example, as we have seen, we could not conclude ¬p, and probably could not conclude p either, by using this system.

8I

owe proofs 13 and 14 as well as two additional proofs not included here to Tomás Barrero, who also suggested the idea of introducing LFI1 into the discussion. I am extremely grateful to him for these important contributions to the present work.

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Adopting LFI1 does not constitute any drastic change in relation to da Costa’s C1 system. Indeed, given that LFI1 is an extension of C1 , we can say that everything we have proved in C1 can be proved in LFI1. This result permits us to say that we have found a unique logical system that preserves the distinction between the domain of the classical formulas and the domain of the paraconsistent or inconsistent formulas while at the same time allowing us to derive some consequences from contradictions. The first domain supports our claim that Daniel’s belief in Susanna’s chastity, due to the Elders’ contradictory testimonies, is, at least in principle, rational. The second domain supports our claim that the Christian believer’s refusal to accept that the sentence “Jesus rose from the dead” is false, despite the fact that the Gospel accounts of the resurrection are contradictory, is also rational at least in principle.

10.7 Some Problems to Consider At this point we may seem to have solved all of our problems, whereas in fact we find ourselves in a new territory of complications, some of which are considered below. First problem: The logical approach to the notion of rationality proposed here appears to be a completely useless detour. The question is why we should accept that “Susanna did not commit adultery with a young man in the garden” is part of an argument involving well-behaved formulas and that is not the case with “Jesus rose from the dead.” Relying on the form of the sentences is apparently not very useful in this situation or in any other of this type because, as we have seen, the formalization of the sentences is exactly the same in both cases. One good alternative might be to base the decision on an interpretation that takes contextual elements into account. For example, in the case of believing in Susanna’s chastity, it is worth considering Daniel’s remark that the Elders were judges moved purely by self-interest, who used to condemn innocent people and save those who were actually guilty. In the case of Jesus, considering the details upon which the different narrations agree can be productive; or, we could base our decision on the fact, suggested by all four Gospels, that the news of the resurrection and Jesus appearing to his disciples thoroughly transforms them. For example, these events eradicate the cowardice that made them abandon Jesus to hide from the authorities, and turns them into convinced witnesses and fervent preachers of his resurrection and promise of forgiveness. Their feelings of painful disappointment and hopelessness over the loss of their leader are changed into the exact opposite: feelings of unremitting happiness and joy (Huffman 1945). This context-based answer would seem to be useful, but it apparently imperils the logical strategy. When we rely on contextual elements, we clearly base our decision on a certain “coherence” between them and the sentence. We decide that the sentence “Jesus rose from the dead” corresponds to the paraconsistent category and is thus rational because it is more coherent with the contextual elements than not. As long as “coherent” is generally understood as synonymous with “non-

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contradictory,” or at least as implying non-contradictoriness, we actually seem to be adopting the latter as the criterion for our decision. But if this is indeed the case, i.e., if non-contradictoriness is the criterion, we must ask why the excursus through paraconsistent logic for evaluating the rationality of the belief would be necessary. Another possibility to be considered is arbitrarily deciding on the domain of the sentences in advance. This solution is apparently problematic since it makes the decision about the rationality of a belief ultimately independent of the argument that supports it and consequently independent of the underlying logic, which can lead us to question the relationship between rationality and logic that we presumed above. Nonetheless, I do not think this difficulty breaks that relationship. It simply shows that, if we assume a notion of rationality like the one we have been considering here, logicality cannot be taken as a necessary and sufficient condition for rationality, but only as a necessary one (see the second section of this work). Being logical does not make the belief in a particular sentence rational. Many sentences can be supported by valid arguments and yet belief in them might be irrational for other reasons. However, when a sentence is not supported by an argument with clear rules, we may have to question its rationality. Second problem: I subscribe here to a notion of rationality that establishes a close relationship between rationality and logic. But, as some of the literature shows, this connection is not really as evident as it may first appear to be. In fact, for authors like Harman (1986), Cherniak (1986), and Brandom (1994), among others, logic imposes a number of restrictions on reasoning that may be effective with idealized agents (whose conditions of reasoning include perfect information or rationality), but not when real, imperfect agents who deal with imperfect information are involved. One way to answer these objections would be to say the criticism these authors advance influences the above-mentioned relationship only in cases where classical logic is considered the unique logical system. This solution assumes that those criticisms are ruled out when other logical systems are introduced. Third problem: There is a possible relativization of the notion of rationality. Given that we have conceived of rationality as being connected to a certain underlying logic, and because many logical systems have been developed in the last 70 years, we might consequently consider any belief or action to be rational. One conceivable answer to this objection could consider the way logical pluralism is understood. The conventionalist viewpoint understands both the construction of different logical systems and the choice of one particular system among many as totally arbitrary. Evidently, this total arbitrariness may tend toward a relativistic conception of rationality, while a more moderated position on logical pluralism might be helpful in trying to avoid this consequence. I would mention, for instance, lectures 23 and 24 of Lectures on the Foundations of Mathematics (Wittgenstein 1989) and paragraph 520 of Philosophical Investigations (Wittgenstein 1968), where Wittgenstein suggests that the change from one logical system (or one technique) to another must be understood not as an arbitrary change of rules—as the conventionalist perspective supposes—but rather as something that is somehow regulated by the use we want to make of those systems, i.e., by their applications in our forms of life. I cannot pursue this idea

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here, but a similar line of thought can also be found in Gómez (2007). Nevertheless, I do want to suggest that an approach to logical pluralism like the one Wittgenstein proposes may point toward a perspective on rationality that is much less problematic than the one that seems to follow from a merely conventionalist approach.

References Alston. 1991. Perceiving God: the epistemology of religious experience. Cornell University Press. Aristotle. 1993. Metaphysics, books Γ , Δ and E. 2nd ed. Trans. Christopher Kirwan. Oxford, UK: Clarendon Press. Bobenrieth, A. 1996. Inconsistencias ¿por qué no? Un estudio filosófico sobre la lógica paraconsistente. Bogotá: Colcultura. Brandom, R.B. 1994. Making it explicit, reasoning, representing, and discursive commitment. Boston, MA: Harvard University Press. Carnielli, W., and J. Marcos. 2002. A taxonomy of C-systems. In Paraconsistency: The logical way to inconsistency, ed. Walter A. Carnielli, M.E. Coniglio, and I.M. Loffredo D’Ottaviano, 1–94. New York, NY: Marcel Dekker. Carnielli, W., J. Marcos, and S. de Amo. 2000. Formal inconsistency and evolutionary databases. Logic and Logical Philosophy 8: 115–152. Cherniak, C. 1986. Minimal rationality. Boston, MA: MIT Press. Cooper, A. 1983. Mythology and exegesis. Journal of Biblical Literature 102 (1): 37–60. Da Costa, N.C.A. 1958. Nota sobre o conceito de contradição. Anuário da Sociedade Paranaense de Matemática. 1: 6–8. ———. 1963. Sistemas Formais Inconsistentes. Curitiba: Universidade Federal do Paraná. ———. 1980. Ensaio sobre os Fundamentos da Lógica. São Paulo: Hucitec e Editora da Universidade de São Paulo. Ehrman, B. 2009. Jesus, Interrupted. In Revealing the hidden contradictions in the Bible (and why we don’t know about them). New York, NY: HarperCollins. Filson, F. 1948. Theological exegesis. Journal of Bible and Religion 16 (4): 212–215. Gomes, E.L. 2013. Sobre a história da paraconsistência e a obra de da Costa: A instauração da lógica Paraconsistente (Unpublished doctoral dissertation). São Paulo: Universidade Estadual de Campinas. Gómez, S. 2007. Language and logical pluralism: some aspects of a Wittgenstenian perspective on the nature of logic. Logic Journal of the IGPL 15 (5–6): 603–619. Haack, S. 1978. Philosophy of logic. Cambridge: Cambridge University Press. Hamilton, N. 1965. Resurrection tradition and the composition of mark. Journal of Biblical Literature 84 (4): 415–421. Harman, G. 1986. Change in view: Principles of reasoning. Boston, MA: MIT Press. Huffman, N. 1945. Emmaus among resurrection narratives. Journal of Biblical Literature 64 (2): 205–226. Karni, E., and D. Schmeidler. 1986. Self-preservation as a foundation of rational behaviour under risk. Journal of Economic Behaviour & Organization 7 (1): 71–81. https://doi.org/10.1016/ 0167-2681(86)90022-3. Korsgaard, C. 1986. The sources of normativity. Cambridge: Cambridge University Press. Lemmon, E.J. 1965. Beginning logic. London: Chapman & Hall. Mackie, J.L. 1977. Ethics: Inventing right and wrong. New York: Penguin Books. Millgram, E. 2005. Ethics done right: Practical reasoning as a foundation for moral theory. Cambridge: Cambridge University Press. Plantinga, A. 1993. Warrant and proper function. New York: Oxford University Press.

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Plantinga, A., and N. Wolterstorff. 1983. Faith and rationality. Notre Dame, IN: University of Notre Dame Press. Priest, G. 1979. The logic of paradox. Journal of Philosophical Logic 8: 219–241. ———. 2004. What’s so bad about contradictions? In The law of non-contradiction, ed. G. Priest, J.C. Beall, and B. Armour-Garb, 23–39. Oxford: Clarendon Press. ———. 2006. Doubt truth to be a liar. Oxford: Clarendon Press. Quinn, D. 1962. John Donne’s principles of biblical exegesis. The Journal of English and Germanic Philology 61 (2): 313–329. Routley, R., and V. Routley. 1972. Semantics of first degree entailment. Noûs: 335–359. Russell, B., and A.N. Whitehead. 1910. Principia mathematica. Cambridge: Cambridge University Press. Searle, J.R. 2001. Rationality in action. Boston, MA: MIT Press. Weber, W. 1901. The resurrection of Christ. The Monist 11 (3): 361–404. Whiterup, R. 2013. Raymond E. Brown, S. S., and catholic exegesis in the twentieth century: A retrospective. U.S. Catholic Historian 31 (4): 1–26. Williams, B. (1979). Internal and external reasons. Reprinted in Moral luck. (pp. 101–113). Cambridge: Cambridge University Press. 1981. Wittgenstein, L. 1968. Philosophical investigations. Trans. G. E. M. Anscombe, Oxford: Basil Blackwell. ———. 1989. Lectures on the foundations of mathematics. Chicago: The University of Chicago Press.

Susana Gómez Gutiérrez has been adjunct professor at the Corporación Universitaria Minuto de Dios (UNIMINUTO) and also Visiting research scholar at the Graduate Center of the City University of New York (CUNY) under the supervision of Professor Graham Priest. She is presently PhD candidate in the Faculty of Philosophy at the Pontificia Universidad Javeriana, Bogotá, Colombia.

Part IV

Computational Philosophy and Religion

Chapter 11

Talmudic Norms Approach to Mixtures with a Solution to the Paradox of the Heap: A Position Paper Esther David, Rabbi S. David, Dov M. Gabbay, and Uri J. Schild

11.1 Background and Orientation This paper uses methods of Talmudic norms as applied to mixtures, and applies it to offer a solution to the paradox of the heap (Sorites paradox). To achieve this aim we first need to form a common point of view between the formation of heaps and the formation of liquid mixtures. To do this, let us imagine two vessels, a small vessel A and a very big vessel B, with B resting on top of A. Imagine that A is either 1. containing one grain of sand (in this case we can refer to the container A with the one grain of sand in it as A1 ), or 2. containing some liquid say Kosher wine (in which case we refer to container A with the wine in it as A2 ); and imagine that that B contains another substance, say respectively either 1. containing a large quantity of fine grains of sand (which respectively we use the notation B1 ), or

E. David · R. S. David Ashkelon Academic College, Ashkelon, Israel D. M. Gabbay () University of Luxembourg, Esch-sur-Alzette, Luxembourg e-mail: [email protected] U. J. Schild Bar Ilan University, Ramat Gan, Israel © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_11

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2. containing a large quantity (very much larger than what A contains) of nonKosher wine (which respectively we use the notation B2 ). Further assume that there is an aperture between A and B, closed by a valve which can be instantly opened at a push of a button. The aperture can be one of two types. A very small aperture allowing the contents of B to drip slowly into A, or a large aperture, allowing the contents of B to gush down (fall) almost instantly into A. Let us use the terminology that the aperture (respectively small/large) allows the contents of B to respectively drip/fall into A. Let us denote by A the container A with the new contents, created after all of the contents of B drips/falls into A. (Respectively we get A1 and A2 , depending on whether the contents is sand or wine. We shall not use the subscripts when the context is clear.) We are familiar with such constructions. In the case 1 of sand this is a sand clock. In the case 2 of wine and small aperture it is the kind of drip you find in hospitals. We can now ask a logical question. If the contents of A satisfies property P and the contents of B satisfies property ¬P, then what property does the new contents of A satisfy? Is it P or is it ¬P? The question can be asked equally in the case when the contents of B fell into A via a large aperture, and it can also be asked in the case that the contents of B dripped slowly into A via the small aperture. In both cases we have that the new content of A (which we called A ) equals the union/sum of the old contents of A and of B, (namely A = (A + B)). Let us write A = (A +small B), for the case of small aperture, and A = (A +large B), for the case of large aperture. Now the Talmud would consider a property, say “Kosher” which can be applied to the contents of A, say it is Kosher to drink the wine in A. Also assume that the wine in B is not Kosher to drink. After the mixture is formed, (i.e. after all the contents of B drips or falls into A), we can ask whether the the mixture in A is Kosher or not? So in the case of the wine we ask Qwine If the wine in A is Kosher and the wine in B is not Kosher, then is the mixture wine in A kosher or not? In the case of sand, let us use the property “Small Heap”. The contents of A (being just one grain of sand) is not a small heap of sand. If B is a very big container of sand (and therefore the sand it it can be considered to be a small heap), we can ask when and whether the new contents of A (after the “dripping” or “falling” of the sand from B into A) can be described as a small heap. So in the case of sand we ask: Qsand If the quantity of sand in A is not a small heap and the quantity of sand in B is a small heap then is the quantity of sand in A a small heap or not? In logic the case of sand and a small aperture allowing one grain of sand to pass one at a time, leads to a paradox. We reason that

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• one grain of sand in A is not a small heap of sand, and • at any step n, if the contents of A (at step n) is not a heap then in step n + 1, when only one more grain of sand is added, it is still not a heap (assuming the dripping from B to A is grain by grain). Therefore • The contents of A after dripping (all of the contents of B) is not a heap, contradicting the fact that we considered the contents of B before dripping to be a heap. In the Talmudic case of wine, we ask a similar question, namely is the mixture of wine in container A after dripping (i.e. A2 , and we assume small drips, very much smaller relative to the quantity of wine in A) of all of the wine from B Kosher or not? We might think that the Talmud would face a similar paradox because it says that a small drip of non-Kosher wine falling into a lot of Kosher wine does not make the mixture non Kosher. However, the Talmud has a different way of looking at mixtures. In this case the Talmud says that we need to ask what is the size of the aperture/the size of the of the drops. The mixture may be Kosher or not Kosher depending on this size. The Talmudic view is that if the non-Kosher wine drips into the Kosher wine in small drops then it basically becomes part of the Kosher wine and this drop of wine is “converted” and becomes Kosher, and so no matter how much non-Kosher wine comes from B into A (in very small drops), they are all “converted” one by one and the resulting mixture will still be Kosher in A. However, if the entire contents of B flows down into A all at once, then the wine in A is non-Kosher. Applying this Talmudic principle to the wine problem we get that the wine in A after dripping becomes Kosher. The Talmud does not see this as contradictory or paradoxical. This is because the key idea being is that we ask how the mixture was formed, before we decide whether it is Kosher or not. To explain exactly what we mean here let us take a Talmudic point of view and apply it to the grains of sand paradox. Assume container A contains 20 grains of sand and assume container B contains 100 grains of sand. Assume that A is not a small heap while B is a small heap. (We are abusing notation, using A and B respectively to denote both the containers and the sand in them). Now move 40 grains of sand one at a time from B to A, to form the new A and  B. Now both A and B  contain 60 grains of sand each. First assume the size of the aperture is very small and allows the passage of one grain of sand at a time. The Talmudic view would be that A is not a small heap (because it was formed from A by adding single grains of sand one by one) while B  is a small heap (because it was formed from B by removing single grains of sand one by one). The idea being that adding or removing one grain from a quantity does not change the truth value of

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the predicate “small heap” assigned to it. Here the Talmud agrees with the reasoning of the paradox of the heap. The Talmud does not get a paradox, however, because its view takes into account how A and B  were formed and so the objects which receive the predicate “Small Heap” are ordered pairs of the form X = (Set S of grains of sand, Algorithm how S was formed). Suppose now that A and B  were formed using a much wider aperture/shutter allowing for the one off moving 40 grains of sand from B to A all at once. In this case the Talmud might say, for example, that A is a heap as well as B  . So the Talmudic contribution to the paradox of the heap is the addition of the component of how the heap was formed. If we just look at the quantity of sand in container A and in container B, they are the same (60 grains) and so we cannot say that one is a heap and one is not, but if we have the additional parameter of how they were formed then we can distinguish between them. We can ask the question what if we do not know how a quantity of sand was formed is it a heap or not? The Talmud will take a default position depending on the nature of the mixture. The case of sand and heap was never discussed in the Talmud, but the case of wine was discussed and the default position is that it is non-Kosher. Let us write slightly more formally the Talmudic rules governing the drip from the small aperture. We have kosher wine in A, so we can write Kosher(A). We have non-Kosher wine in B, so we write ¬Kosher(B). We let the non-Kosher wine drip from B to A, through the small aperture. So we write Asmall = (A +small B). The Talmudic rules are the following: • • • •

Kosher(A) ∧¬Kosher(B) ⇒ Kosher (A +small B). Kosher(A) ∧ Kosher(B) ⇒ Kosher (A +small B). ¬Kosher(A) ∧¬Kosher(B) ⇒ ¬Kosher (A +small B). ¬Kosher(A) ∧ Kosher(B) ⇒ ¬Kosher (A +small B).

So let us give a quick example of a Talmudic model for sand and heap. Let us consider two properties of a quantity A of sand, Heap(A) and Cement(A). Heap (A) means we consider the quantity in A as a heap and Cement(A) means that this sand in A can be used for making cement (and not, say, for making sand-pits for children in kindergarten). Let us be given initial containers {Ai } and initial default predicates H 1 (Ai ) and C 1 (Ai ). We require consistency for these predicates as follows 1. If A is a heap and B contains more sand than A then B is also a heap 2. If A is not a heap and B contains less sand than A then B is also not a heap 3. If A and B are both suitable (respectively both not suitable) for cement and we move sand only from one to the other then the result is also suitable (respectively not suitable) for cement.

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4. If A is suitable (respectively not suitable) for cement and B is not suitable (respectively suitable) for cement, and we move some of the quantity of sand from B to A, then the question of whether the new A is suitable for cement or not depends on how much sand we moved from B to A. Suppose we start moving single grains of sand from one container to another, recording step by step what we are doing. Let us stop after a 100 steps. We get new mixtures in the containers. Call them (A∗i ). Then depending on the agreed rules involved how much quantities moved affect suitability and affect being a heap, we get answers as to whether we get heaps and whether there is suitability for cement. Call the new predicates H i100 and C i100 . The above example shows that we are now at a stage where it is obvious that we need to present a simple logical system which contains atomic objects (different grains of sand, different atomic building blocks for more complex objects, etc.) as well as a variety of constructors/ways of forming new objects from aggregating atomic objects and see how properties of such aggregations can change. We shall give a proposal in later sections. Meanwhile our plan for this section is as follows: 1. Explain the Heap paradox by quoting from the Stanford Encyclopedia of Philosophy 2. Explain the Talmudic view of mixtures in more detail 3. Choose a familiar action language, endow it with construction capabilities and make it suitable for modelling Talmudic mixture ideas. Note that this part goes beyond solving the paradox of the heap, it goes in the direction of modelling the theory of Talmudic mixtures. Also note that the specific language we choose, based on the blocks world is just a matter of convenience, giving us a quick first attempt at modelling the theory of Talmudic mixtures.

11.1.1 Component 1: Paradox of the Heap (Sorites) This paradox is well known since the time of the Greeks and many books and articles have been written about it by many very eminent logicians and philosophers. We rely in our exposition here on the article in the Stanford Encyclopedia of Philosophy (SEP) Hyde and Raffman (2018). We simply quote what we need. (The quote is in a box, the numbering is local, but the references are adjusted to the end of our paper. We prefer not to paraphrase the description in our own words. The encyclopedia is a classic and we do not think we can do better)

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2. Its Paradoxical Forms A common form of the sorites paradox presented for discussion in the literature is the form discussed above. Let F represent the soritical predicate (e.g., is bald, or does not make a heap) and let the expression an (where n is a natural number) represent a subject expression in the series with regard to which F is soritical (e.g., a man with n hair(s) on his head or n grain(s) of wheat). Then the sorites proceeds by way of a series of conditionals and can be schematically represented as follows: Conditional Sorites F a1 If F a1 then F a2 If F a2 then F a3 ... If F ai−1 then F ai F ai (where i can be arbitrarily large) Whether the argument is taken to proceed by addition or subtraction will depend on how one views the series. ... ... ... 3. Responses The various responses to soritical reasoning can be most easily catalogued by focussing on that form most commonly discussed in the literature—the conditional form. As with any paradox, four responses appear to be available. One might: 1. deny that logic applies to soritical expressions. According to this response the problem cannot legitimately be set up in the first place. On the other hand one might accept that the sorites paradox constitutes a legitimate argument to which logic applies and deny its soundness by: 2. denying some premise(s), or 3. denying its validity. Finally, seemingly as a last resort, one might embrace the paradox and 4. accept it as sound. (continued)

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3.6 Our Comment: Solution Approach 3.6—Embracing the Paradox A final option is to simply embrace the paradox. (See Dummett 1975; Wright 1975.) Conditional sorites paradoxes are, contrary to appearances, sound. For example, no amount of grains of wheat makes a heap. This initial claim in favour of a universal type (4) response immediately runs into difficulty, however, with the realisation that, as noted at the outset, such paradoxes come in pairs. There are negative and positive versions depending on whether the soritical predicate is negated or not. To accept all sorites as sound requires assent to the additional claim that, since one grain of wheat makes a heap, any number do. A radical incoherence follows since there is a commitment to all and any number both making a heap and not making a heap. Similarly, everyone is bald and no-one is; everyone is rich and no-one is, and so on. The problem is that the soundness of any positive conditional sorites undercuts the truth of the unconditional premise of the corresponding negative version, and vice versa. Unless one is prepared to countenance the almost total pervasiveness of contradictions in natural language, it seems that not all sorites can be sound. Unger (1979) and Wheeler (1979) propose a more restricted embrace. Following dissatisfaction with responses of type (1) and (3) one accepts the applicability and validity of classical norms of reasoning. Nonetheless, dissatisfaction with responses of type (2) considered so far— rejecting some conditional premise—leaves open the possibility of either rejecting the unconditional premise or accepting it and, with it, the soundness of the paradox. What is advocated is the soundness of those sorites which deny heapness, baldness, hirsuteness, richness, poverty, etc. of everything—a type (4) response—and the corresponding falsity of the unconditional premise of all respective positive variants of the argument—a type (2) response. Terms like heap, bald, hirsute, rich and poor apply to nothing. It is admitted that they apply to everything if they apply to anything, but the all-or-nothing choice is resolved in favour of the latter option. (See Williamson 1994, Ch. 6.)

The SEP article discusses several solutions to the paradox, one of them, quoted above as ‘3.6. Embracing the paradox’, is the one directly related to Talmudic norms. This solution says that indeed if one grain of sand does not make a heap (F a1 ) and adding grains of sand one by one retain this property, then any huge number of grains of sand is not a heap, F an . Say for example that a collection X of 100100 grains is also not a heap. The weakness of this approach is that if we start with a very huge collection Y of grains of sand with, say N > 100100 grains which we do consider to be a heap, then if we take out of the collection one grain we still have a heap, but if we keep taking out more and more grains, we reach a collection X of 100100 grains, but now it will be considered still a heap. We thus have

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(*) The question of whether X is a heap depends on how it was formed. It is at this junction that the Talmud connects with the paradox.

11.1.2 Component 2: Talmudic Norms Regarding Mixtures We first note that the Talmud and its interpretations is a debate about norms to guide our daily lives which has been and still is ongoing for over 2000 years. It is a practical debate and so addressing paradoxical arguments and counterarguments cannot remain in the theoretical realm, but must give an actionable practical conclusion. So the Talmud has to offer conclusions of the paradox of the heap. How does the paradox arise? Imagine two bottles of wine. One bottle we keep for ourselves for religious ceremony. The second bottle is given to a priest from another religion to use in his place of worship. The priest friend decants the wine, and after use, brings it back to keep in the fridge on a shelf above our own wine, which is also in a bowl. So the top bowl of wine is not kosher (no religion x would use the wine of religion y if x = y), but the bottom bowl is kosher. Unfortunately, over several days the top non-kosher wine drips, drop by drop, slowly into the bottom bowl. The practical question is whether the wine in the bottom bowl is kosher or not? There are many opinions of Talmudic scholars about this question. One of these opinions of Rav Dimi’s on behalf of Rabbi Yochanan says as follows: 1. The initial wine in the bowl was kosher. 2. If non-kosher wine drips into a larger quantity of kosher wine then the combined quantity is kosher. ( So the actual drop converts and becomes kosher). 3. Therefore any quantity of wine obtained in this way is kosher. So we can end up with a huge quantity of wine, 99% of which was non-kosher, but since this 99% was dripping slowly into the initial 1% which was kosher, the entire lot is now kosher. OK We can now ask what happens if this 99% of non-kosher wine did not drip slowly drop by drop into the 1% of kosher wine but the entire quantity of nonkosher wine just flooded into the kosher wine in one go? The answer is that the whole lot will now be not kosher. We thus get that given a bowl of mixed quantities of wine, the question of whether the wine in the bowl is kosher or not, depends on how the wine in the bowl was assembled/created as a collection. This is similar to the problem (*) of the philosophical component 1. What is the difference between the philosophical position and the Talmudic position?

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1. The philosophical position sees a candidate for a heap X and has a problem in accepting that it is both a heap and not a heap, all depending on how our philosophical positions looks at it. 2. The Talmudic position of Rav Dimi’s on behalf of Rabbi Yochanan would simply ask how was X created. You answer that question, and you will be told whether it is a heap or not. But this means that we look at an object not as it is but also as to how it was constructed. We make two comments: (c1) The idea that part of the properties of any object is also how it was constructed was put forward and discussed by Gabbay and Siekmann already in 2008, see (Gabbay and Siekmann 2010). (c2) We said that the Talmud is practical, so if we go to Rav Dimi’s on behalf of Rabbi Yochanan with a bowl of wine and ask whether it is kosher or not then he will ask us how it was constructed. If we say we do not know, what will Rav Dimi’s on behalf of Rabbi Yochanan judge it to be, kosher or not? What is the default position? Answer There is a whole body of thought about mixtures, depending on the materials getting mixed. In this case (wine) the default position is not kosher. We now have the following task ahead of us. (T) Export the Talmudic options to the philosophy community and offer some methodological solutions to the paradox of the heap. In fact we can export a Talmudic calculus of Sorites. Find correlations between Talmudic opinions and philosophers positions, and offer solutions to these philosophers. What do we say to this community? We say, you are working with the wrong logic. You need a logic which can express how objects in the logic are constructed. So do not use classical logic, use more appropriate logics suitable for expressing the paradox dynamics.1 Fortunately, such systems exist in the area of Artificial Intelligence. The AI community have been constructing devices and object for years and modelling them. The most famous and simple model is the Blocks World and this model can be enriched and modified and adapted to deal not only with the sorites paradox but also to model in general the Talmudic normative theory of mixtures. (Actually develop a Calculus of Sorites, to model Talmudic reasoning on Sorites as well as a model for interactions of Vague predicates.) The next component will describe the blocks world model.

1 If you read the discussion of the various philosophical positions to the paradox , as summarised in

the Stanford Encyclopedia of Philosophy, you will see that they are all motivated to preserve and stay in classical logic.

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11.1.3 Component 3: Blocks World2 The Blocks World model is taught in most textbook on Artificial Intelligence. We have a table, call it t. We also have blocks, call them a, b, c, . . .. We have a relation On(x, y) saying x is on y. We have the equality symbol =. So using this language we can say 1. Every block is either on the table or on another different block. ∀x(On(x, t) ∨ ∃y[On(x, y)) ∧ y = x] 2. We cannot have two items on the same block unless it is the table, nor a block on two other blocks. On(x, z) ∧ On(y, z) → (x = y ∨ z = t).

On(x, z) ∧ On(x, y) → z = y. 3. The table is not On anything ¬∃yOn(t, y). We also have an action predicate called move(x, y), (this is a meta-predicate, it is a higher level than the first order predicates On and =),3 which means block x is moved and put on top of block y. We can do this only if there is nothing on top of x and nothing on top of y or y = t. Let Clear(x) = def.¬∃yOn(y, x) ∨ x = t. Then the action rule is 4. Clear(x) ∧ [Clear(y) ∨ y = t] ∧ Move(x, y) ⇒ On(x, y). 5. Clear(x) ∧ On(x, u) ∧ move(x, y) ⇒ Clear(u). The important point is that the ‘move’ predicate is an action predicate and the ‘⇒’ in (4) and (5) is not implication but a connecting symbol between the precondition and post-condition of the action ‘move’. We use the pre-condition when

formalise the blocks world in a way where ⇒ is not an implication. This is the correct way to look at the blocks world and is what we need for our paper. We note that one could be using McCarthy’s and Reiter’s situation calculus https://en.wikipedia.org/wiki/Situation_calculus which would be perhaps more appropriate for some people, since this would be based on (classical) logic. In fact in some instances of this formalism a situation is indeed dependent how it was generated, just as we want here. The current paper is a position paper, and so in the full paper we shall examine this option in detail. We shall check whether the situation calculus is the right formalism to use. (It is also strongly related to planning, which we mention at the end of the paper.) 3 The language with On, =, and table is used to describe a frame/picture of how the blocks are arranged as well as constraints on such arrangements. The move predicate tells us how to move from one frame/picture to another. 2 We

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we look at a given frame and want to act on it. Then we can do that if the precondition holds. Once we do the move action then we get a new frame which is obtained by minimal change resulting from the action and in which we are guaranteed that the post-condition holds (among whatever else which holds as a result of this minimal change). This language allows us to record how blocks are piled up. To express the heap/mixture problem we need to extend the language. Our current language has only blocks as elements. So we cannot regard two blocks on on top of the other as a new unit, an entity in the blocks world which can have properties attributed to it. To do this properly we need a constructor in the language, a function symbol, say ‘+’, which will take two blocks x and y and form a new entity, the combined ‘block’ (x + y). If we assume that ‘+’ is associative, (i.e. ((x + y) + z) = (x + (y + z)) then we can write the new ‘blocks’ entities as (x1 , . . . , xn ) meaning (x1 +, . . . , +xn ) . The operation x + y, just combines two elements x and y into a new entity, it does not imply that x is on y or anything else. However, intuitively x + y + z can form part of a tower if indeed we have them one on top of another. Let Tower(x1 , . . . , xn ) mean that indeed the element ((x1 +, . . . , +xn ) satisfies the property that On(x1 , x2 ) ∧ On(x2 , x3 )∧, . . . , On(xn−1 , xn ) ∧ On(xn , t).4 The following holds, for example Tower(x1 , . . . , xn ) ∧ Tower(y1 , . . . , yk ) ∧ (xn = yk ) →



xi = yj .

To connect with sand and heaps, let us assume that each block is a container containing one grain of sand. When we put several blocks on top of each other, this means we put the containers on top of each other and open the apertures and let the sand drip to the base block. Let us allow for a new predicate, the heap predicate on series, Heap(x1 , . . . , xn ). We can write On(x1 , t) → ¬Heap(x1 ) Tower(x1 , . . . , xn ) ∧ ¬Heap(x1 , . . . , xn ) ∧ On(y, x1 ) → ¬Heap(y, x1 , . . . , xn ) We can also write Heap(x1 , . . . , xn ) ∧ Tower(x1 , . . . , xn ) → Heap(x2 , . . . , xn ). Since we are acting on frames and moving blocks around, we need to have a starting arrangement of blocks. We use the predicate ‘Initial’ to describe our starting frame. We now say that every tower is either initial or constructed and so we need to use the predicate Initial. 4 Note

that we record the tower by starting the list from the top element downwards. There is no mathematical reason for it, only a matter of intuitive exposition, since we interact with a tower by putting elements on the top or taking elements from the top.

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Initial(x1 , . . . , xn ) → ¬Initial(xi , . . . , xn )∧ ¬Initial(y1 , . . . , yk , x1 , . . . , xn ), for i > 1. Initial(x1 , . . . , xn ) → Tower(x1 , . . . , xn ) Tower(x1 , . . . , xn ) ∧ ¬Initial(x1 , . . . , xn ) → [∃y(Tower(y, x1 , . . . , xn ) ∧ Move(y, z) ∧ z = x1 )]∨ Tower(x2 , . . . , xn ) ∧ Move(x1 , x2 ). If we start with initial positions of towers, and an initial specification of which tower is a heap or not, then under the correctly specified rules, we can have that every tower will either be a heap or not and every constructed tower will be either a heap or not, depending on how it is constructed. We can have that, for example, in the initial position, one tower of two blocks is a heap while another tower of two blocks is not a heap. So we need more axioms to say something like: Tower(x1 , . . . , xn ) ∧ Tower(y1 , . . . , ym ) ∧Heap(x1 , . . . , xn ) ∧ ¬Heap(y1 , . . . , ym ) ∧Initial(x1 , . . . , xn ) ∧ Initial(y1 , . . . , ym ) → n is much greater than m

11.2 Talmudic Calculus of Sorites—Preliminary Presentation Let us build on the discussion of Section 1 and present (still informally) a Talmudic calculus of sorites. We begin with an example. Example 11.2.1 Assume we have four blocks, {a, b, c, d} organised at the initial position as in Fig. 11.1. Note that the element d + c + b + a which can be written as (d, c, b, a) is not a tower in the figure. To turn it into a tower we need to move the elements around. As we move blocks around , we get new figures/arrangements of the elements. Suppose we have Kosher(b, a) and ¬Kosher(d, c). Let us look at the tower (d, c, b, a). We ask is it kosher or not? The implicit assumption (additional axiom) is that if a series is kosher (resp. non kosher) then each of its blocks is kosher (resp. non kosher) and vice versa. Fig. 11.1 Two towers

b

d

a

c table t

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Well, that depends on the construction. Suppose we follow the sequence A. Sequence A from Initial Position 1. Move(d, t). We have ¬Kosher(d), ¬Kosher(c). 2. Move(c, b). We have Kosher(c, b, a) 3. Move(d, c). We have Kosher(d, c, b, a). Sequence B from Initial Position 1. Move(b, d). We have ¬Kosher(b, d, c) 2. Move(a, b). We have ¬Kosher(a, b, d, c) 3. Move(a, t), ¬Kosher(a) 4. Move(b, t), ¬Kosher(b) 5. Move(d, t), ¬Kosher(c) ∧ ¬Kosher(d) 6. Move(b, a), ¬Kosher(b, a) 7. Move(c, b), ¬Kosher(c, b, a) 8. Move(d, c), ¬Kosher(d, c, b, a) We now have ¬Kosher(d, c, b, a). Imagine now several predicates for sorts, K1 , . . . , Km and axioms (infinite set), saying for each four numbers α, β, i, j what happens if we move α blocks (x1 , . . . , xα ) of sort Ki (i.e. Ki (x1 , . . . , xα ) holds), on top of β blocks (y1 , . . . , yβ ) of sort Kj ? We get a sequence of height β +α of sort r = M(i, j, α, β) (i.e. we have that Kr (x1 , . . . , xα , y1 , . . . , yβ ) holds). where M is a table giving us properties of mixtures (sorites table). Note that this goes much beyond the philosophical sorites considerations, and is more suitable for modelling Talmudic sorites norms. Example 11.2.2 Our modified blocks world language can also easily accommodate another philosophical approach to Sorites, the so-called ideal language approach. We quote some more from the Stanford Encyclopedia of Philosophy.

3.1 Ideal Language Approaches Committed as Frege and Russell were to ideal language doctrines, it is not surprising to find them pursuing response (1). (See especially Russell 1923.) A key attribute of the ideal language is said to be its precision; the vagueness of natural language is a defect to be eliminated. Since soritical terms are vague, the elimination of vagueness will entail the elimination of soritical terms. They cannot then, as some theorists propose, be marshalled as a challenge to classical logic. A modern variation on this response, promoted most notably in Quine (1981), sees vagueness as an eliminable feature of natural language. The class of vague terms, including soritical predicates, can as a matter of fact be dispensed with. There is, perhaps, some cost to ordinary ways of talking, but (continued)

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a cost that is nonetheless worth paying for the simplicity it affords—namely, our thereby being able to defend classical logic with what Quine describes as its sweet simplicity.

We can retain our model language, it is precise and we can in addition simply stipulate a cut off point, and say that 100100 grains of sand is not a heap but 100100 + 1 grains is a heap. This is making ‘heap’ a precise legal term. Like a cut off point for the allowed amount of alcohol in one’s blood when driving. In our model we say ¬Heap(x1 , . . . , xn ), for n ≤ 100100 .

Heap(x1 , . . . , xn ) for n > 100100 . In this case the notion of heap would be independent of how Tower(x1 , . . . , xn ) was constructed. Going back to Talmudic opinion, there is another view about mixing wine which is worth mentioning for the sake of comparison and also for showing what our model can do. If non-kosher wine drips into a container of kosher wine, then this new view says that the result is kosher until a cut-off point where the quantity that dripped in is larger than the original quantity of kosher wine. In which case the entire mixture becomes non-kosher. We conclude this section by highlighting the main point of what we are saying. We are saying that natural language looks vague because it suppresses mentioning the way objects are constructed and so some predicates such as heap which depend on the construction, look vague to us. We also claim that we should recognise the fact that part of the properties of objects is how they are constructed (Gabbay and Siekmann 2010), and encourage philosophers to give up, when necessary, their unquestioning adherence to classical logic.

11.3 Formal Presentation of a Simple Model for the Talmudic Calculus of Sorites (TCS) To present our system, we need to start with several explanatory remarks. Remark 11.3.1 This remark is about the nature of the function symbol ‘+’. The traditional Blocks World has a domain of blocks and the two predicates On(x, y) and x = y, as well as the constant t. So if we have only two blocks {a, b},

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there are three possible models for these elements, given the agreed axioms of the previous section. These are M1 : On(a, t), On(b, t) M2 : On(a, b), On(b, t) M3 : On(b, a), On(a, t) The meta-predicate Move can take us from one model to the other, but there are constraints. Without going in to detail, we can move in one step as follows: 1. 2. 3. 4.

M1 M1 M2 M3

⇒ M2 ⇒ M3 ⇒ M1 ⇒ M1

We cannot move directly in one step from M2 to M3 nor from M3 to M2 . We can leave the ‘move’ function as listed above in abstract form, but we can also go into detail and explain why we cannot move in one step from M3 to M2 . The reason is that we envisage the move executed by moving clear blocks onto other clear blocks or table one at a time. So we write the move as Move(x, y) and we have: 1. 2. 3. 4.

M1 M1 M2 M3

⇒ (by Move(a, b) ⇒ M2 ⇒ (by Move(b, a) ⇒ M3 ⇒ (by Move(a, t) ⇒ M1 . ⇒ (by Move(b, t) ⇒ M1

In order to handle sorites, we need to add the function symbol ‘+’ to the language. We can now theoretically form, out of the two elements {a, b}, an infinite number of new elements, such as a + a, a + b, a + b + a, . . .. However, we want to accept only what we called towers as new elements. So we want to allow only (in the case of two elements {a, b}) the additional elements a + b and b + a. We add the axioms that (") We accept xn +, . . . , +x1 as an element in the domain only if On(xn , xn−1 ), . . . , On(x2 , x1 ), On(x1 , t) all hold. Now the following is a subtle point. If we allow only a + b and b + a, and ("), why do we need ‘+’? We still have the models M1 , M2 , M3 , as the only possible models. The difference is that we want to add unary predicates on the elements. Say we add a predicate K(x). We can, in the model M2 say ±K(a), ±K(b), ±K(a + b), ±K(t). Without ‘a + b’ being in the model, we cannot say ±K(a + b), which should be independent of ±On(x, y), ±K(x), where x, y are blocks. We now have another problem. If (a + b) is a new element and say c is another one, we need to give meaning to On(a + b, c) and On(c, a + b). This is easy. If

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(a + b + c) is available, then On(a + b, c) holds. If (c + a + b) is available then On(c, a + b) is true. Remark 11.3.2 We now discuss the nature of the Move(x, y) meta-predicate in the presence of + and another unary predicate K(x). Consider the situation in Fig. 11.1. This model, when considered in the traditional blocks language without +, has four elements {a, b, c, d}. When we move block b on top of block d, we are just reorganising the model of the figure. We are not creating new elements. When we also have + in the language, the elements in the model are now six elements. We also have b + a and d + c. The candidate element c + d + b is not acceptable in this model. When we move b on top of c, the element b + c becomes acceptable in the new model and the element b + a is not acceptable. So when we execute Move(b, c) in the model of Fig. 11.1, it can be viewed in two ways 1. We go to a new model with new properties. This is regarded just as a mapping from one place to another. We do not say how it is done. 2. We actually move b on top of d and thus we ‘kill’ some old elements, namely b + a, ‘create’ new elements, namely b + d + c and change the values of some predicates. Remark 11.3.3 We mentioned that in the language with + and predicates K(x), we move from one frame to another according to some rules. We create new elements and destroy old elements and change their K properties. The purpose of this remark is to show what language we use to describe the rules governing this move action. We use a prolog program. Thus the language is logic programming. Different Rabbis use different logic programming sets of rules. Let us illustrate by looking at Fig. 11.1. Suppose we have + in the language and the predicate B(x). The meaning of B(x) is x looks Blue. We have that anything on the tower (b, a) looks blue and anything on the tower (d, c) does not look blue. Dov and Uri are allowing actions on the blocks. Uri knows that the tower (b, a) looks blue because of a strong blue spot light shining on it. Dov does not know that. Dov thinks that blocks b and a are actually blue. Dov and Uri allow for moving blocks but they have different rules about B(x). , for the initial state of Fig. 11.1. Dov thinks a sequence (x1 , . . . , xn ) is blue iff one of them is blue. Uri thinks it is blue iff all of them are blue. So according to Dov, if we move a block around, it does not change colour. So if we move block b which Dov believes is genuinely blue and put it on any tower, it will make the entire tower blue. Uri, in comparison, knows that anything put on top of block b will look (i.e. become) blue, because of the blue light shining on that spot. This is reflected in Uri’s rule 2. We are not giving here a complete set of rules for Dov and Uri, only a sample. Had Dov and Uri been Talmudic scholars giving rules for mixtures, they would have had to give rules in such a way that no matter how we move these blocks it can always be determined what is blue and what is not. Actually Dov’s rules are complete but not Uri’s. If Uri will not allow for block a to be moved at all then his rules will be complete.

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Dov Program, Sample Rules 1. B(x) ∧ Clear(x) ∧ Clear(y) ∧ Move(x, y) ⇒ On(x, y) ∧ B(x, y) 2. Clear(x) ∧ ¬B(x) ∧ Move(x, b) ⇒ On(x, b) ∧ ¬B(x) Uri Program, Sample Rules 1. B(x) ∧ Clear(x) ∧ Clear(y) ∧ Move(x, y) ⇒ On(x, y) ∧ ¬B(x, y). 2. Clear(x) ∧ ¬B(x) ∧ Move(x, b) ⇒ On(x, b) ∧ B(x) In general, each person will have rules of the form: 

 ±Ki (x) ∧ Move(x, y) ∧ ±On(uj , vj ) ∧ Clear(x) ∧ Clear(y) ⇒   On(x, y) ∧ ±Ki (y) ∧ ±On(ui , vi )

the pattern is 

 ± Predicates ∧ Move(x, y) ∧ ±On(xi , yi ) ⇒   ∧ ± Predicate ∧ ±On(uj , vj ). The Heap Paradox ¬Heap(x1 ) ¬Heap(xn , . . . , x1 ) → ¬Heap(y, xn , . . . , x1 ) For any y1 , . . . , ym , m ≥ 1, ¬Heap(ym , . . . , y1 ) The way elements are constructed is seen from the history of rule application. Remark 11.3.4 We now propose a basic TCS, the Talmudic calculus of sorites, to be used for modelling various Talmudic norms relating to properties of mixtures as well as address modern theories of vague predicates. 1. Our basic predicate language connects variables and constants for elements (call them blocks). The predicate On(x, y), x, y blocks and equality x = y. 2. We also have a function symbol constructor x + y, which we write also as (x, y) forming associatively sequences of elements. Because of associativity ((x, y), z) = (x, (y, z)), we can simply write (x, y, z). We also have a constant t. These new elements are subject to the discussion in the previous two remarks. 3. We have a predicate Tower(z) meaning z = (x1 , . . . , xn ) and On(xn , t) for some n. Capitals Z, X, . . . will range over sequences of blocks. 4. To make this clear: (a) x, y, z range over blocks (b) t is a constant for table (c) a Tower is a predicate which applies to new and old elements of the form sequence of blocks sitting on the table. We use capital letters to range over towers or simply say Tower(Z1 ). Note that Z = (t) is not a tower. Note that

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we can form sequences Z = (x1 , . . . , xn ), and even if On(xi , xi+1 ) all hold but if On(xn , t) does not hold, then it is not a tower, but it is a ‘floating’ element (sequence) of the domain and can satisfy predicates. 5. There is given a variety of predicates on towers and on elements, satisfying a variety of predicates. For example Kosher (x1 , . . . , xn ), Initial(x1 , . . . , xn ). We must make an important point here. Some predicates are defined axiomatically essentially on atomic unit blocks (not sequences of blocks). The definitions on sequences may be extended axiomatically by rules. For example, we may stipulate that Kosher(x1 ), Kosher(x2 ), ¬Kosher(x3 ). We add the rule  Kosher(x1 , . . . , xn ) iff i Kosher(xi )  ¬Kosher(x1 , . . . , xn ) iff i ¬Kosher(xi ) In this case we cannot prove whether Kosher(x1 , x3 ) or ¬Kosher(x1 , x3 ) because we have an equal number of kosher and non-kosher blocks. We have this because we want Kosher(x1 , . . . , xn ) to depend on how (x1 , . . . , xn ) was constructed. So in the intuitive spirit of Section 2, let us consider two possible constructions: Construction c1 We construct first (x3 ) then (x2 , x3 ) and finally (x1 , x2 , x3 ) we cannot prove Kosher(x1 , x2 , x3 ), nor can we prove ¬Kosher(x1 , x2 , x3 ). The Talmud is practical and would never allow for such a situation, so any Rabbi expressing his own view would have enough rules to resolve this situation. For example one can rule that equal number of kosher and non-kosher blocks make the sequence non-kosher. Construction c2 But if we construct first (x3 ) second (x1 , x2 ) an then (x1 , x2 , x3 ) then Kosher(x1 , x2 , x3 ) would hold. To be strict, we should write Kosher((x1 , x2 , x3 ), c2 ). If we want to remain in classical two-valued logic, we must always give a truth value. 6. We now make an important observation regarding item (5) above. The Talmud and any Talmudic option is absolutely committed to making sure that any system of predicates say K1 (x), K2 (x), . . . , Km (x) which is used and which is given extensions among the blocks, can be extended uniquely to have (truth value) extensions for all elements sequences. In other words, the underlying logic is two valued. We can regard the assignment of extensions to the predicates as given by joint induction. Step 1 Give the extension of Ki (x) or ¬Ki (x) for each KI and each atomic block x. Inductive Step which May Vary According to the Opinion of Different Talmudic Rabbis Give rules to extend by mutual step by step recursion of Ki (x1 , . . . , xn ) for any Z = (x1 , . . . , xn ). Note the following:

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(a) Each Rabbi is absolutely committed that for each Z and each Ki we get a unique answer for Ki (Z) or ¬Ki (Z). (b) the answer may depend on how Z is constructed. So the two valued logic applies to the predicate Ki (Z, c) where c indicates how Z is constructed. Going back to item (5) above, we need to say what value Kosher((x1 , x2 , x3 ), c1 ) would get. So our specification in this case is not complete. Rav Dimi on behalf of Rabbi Yochanan would actually say in this case not-kosher. He has more rules. According to his additional rules, already ¬Kosher(x2 , x3 ) and so also ¬Kosher(x1 , x2 , x3 ). We now deal with the inductive rules which generate the extensions of the predicates. To explain the idea, let us look at classical predicate logic. Suppose we look at the domain of all people living or dead since 1066 in the UK (of today). This is a domain D. Suppose we want to define the predicate Noble(x) for x ∈ D. We can do it inductively with the help of other predicates: King(y), Make-Noble(x, y), Father(x, y). 1. 2. 3. 4. 5.

King (William the Conquerer) King (y)∧ Make -Noble (y, z) → Noble (z) King (x) → Noble (x) Noble (x)∧ Father (x, y) → Noble (y) King (ai ), where ai are all the kings.

The above axioms generate the extension of the predicate of Noble (x). It tells you who are the initial position where we list/know who are the nobles and also tells you how to generate/make more nobles. We can also list rules which allow us to cancel being noble. We do a similar thing here. (a) We use the language with the atomic predicates K1 (x), . . . , Km (x), On (x, y), x = y to define more complex formulas #(Z1 , . . . , Zr ). We start from an initial model for the atomic predicates. We then use the wffs # to serve as preconditions for actions and as postconditions for actions which move us from the initial model to new models. (b) The meta-predicate Move(#1 , #2 ) can denote the actions. The meaning of move is as follows. Suppose we already constructed Z1 , . . . , Zr and we already determined using the rules of Rav Dimi on behalf of Rabbi Yochanan that #1 (Z1 , . . . , Zr ) holds then we activate Move(#1 , #2 ). This action might create new elements U1 , . . . , Us and the formula #2 (U1 , . . . , Us ) holds for these elements (Fig. 11.2). The reader can revisit example 11.2.2.

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x1 x2 x3 Tower (x1 , x2 , x3 )

t= table

11.4 Conclusion We presented our basic initial ideas of how to model principles of mixtures according to Talmudic normative reasoning. We made use of an enhanced Blocks World model, which also contains predicates and function symbols. The mixing rules are logic programs in the extended language. This language allows us to record the way the current model is constructed from the initial model. This is what computer scientists have been doing all the time in planning. As a byproduct of this we get a model for Sorites and the paradox of the heap.

References Dummett, M. 1975. Wang’s paradox. Synthese 30:301–24. Reprinted in his Truth and Other Enigmas, reprinted in Keefe and Smith, 1996. Hyde, Dominic and Raffman, Diana. 2018. Sorites paradox. In The stanford encyclopedia of philosophy. Summer 2018 edition, ed. Edward N. Zalta. https://plato.stanford.edu/archives/ sum2018/entries/sorites-paradox/. Gabbay, D., and J. Siekmann. 2010. Algorithms in cognition, informatics and logic. A position manifesto. Logic Journal of IGPL 18(6):763–768. https://doi.org/10.1093/jigpal/jzq004. Quine, W.V. 1981. What price bivalence? Journal of Philosophy 77:90–95. Russell, B. 1923. Vagueness. The Australian Journal of Philosophy and Psychology 1:84–92. Reprinted in Keefe and Smith, 1996. Unger, P. 1979. There are no ordinary things. Synthese 41:117–154. Wheeler, S.C. 1979. On that which is not. Synthese 41:155–194. Williamson, T. 1994. Vagueness. London: Routledge. Wright, C. 1975. On the coherence of vague predicates. Synthese 30:325–365.

Esther David is an associate professor at the Department of Computer Science, Ashkelon Academic College (Israel). She received her Ph.D. from the Department of Computer Science and Mathematics at Bar-Ilan University. Her research interests include AI techniques, Automated Auctions, Machine Learning applications for Intelligent Tutoring Systems (ITS), Outlier Detection algorithm fusion, Electronic Commerce, Decentralized Agent-based Systems, Recommendation and Broadcasting Systems. She has published over 50 works, including publications in selected manuscripts (i.e., AIJ , MAGS, and DSS) and international conferences. She was a senior research fellow in the Intelligence, Agents and Multimedia (IAM) Laboratory at Southampton University, UK, co-PI in the Artificial Intelligence and Learning algorithms project of the Israeli-Taiwanese

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scientific research cooperation (founded by the Israeli Ministry of Science) and organizer of the international workshop on Agents Mediated Electronic Commerce (AMEC) and of three annual conferences of the Israeli Association of Artificial Intelligence. Rabbi S. David is a Research Rabbi dedicated to the Study of the Talmud. He helps the Talmudic Logic project by collecting the relevant data for modelling a variety of logical themes. Dov M. Gabbay is an international logician. He is Emeritus Professor at Bar Ilan University in the Department of Computer Science, he is Augustus De Morgan Professor Emeritus of Logic at the Department of Informatics, King’s College London and Visiting Professor at the University of Luxembourg. Gabbay has authored over five hundred research papers and over thirty research monographs. He is editor in chief of several international journals, and many reference works and Handbooks of Logic, including the Handbook of Philosophical Logic, the Handbook of Logic in Computer Science, and the Handbook of Artificial Intelligence and Logic Programming. Uri J. Schild has a Ph.D. in Computer Science from Imperial College, London. He is a professor in the Department of Computer Science at Bar Ilan University in Israel. His main research interests are in the application of Artificial Intelligence to the legal domain, Logic and Law, and the development and formulation of requirements for computer systems.

Chapter 12

A Case Study on Computational Hermeneutics: E. J. Lowe’s Modal Ontological Argument David Fuenmayor and Christoph Benzmüller

12.1 Part I: Introductory Matter The traditional conception of logic as an ars iudicandi sees as its central role the classification of arguments into valid and invalid ones by identifying criteria that enable us to judge the correctness of (mostly deductive) inferences. However, logic can also be conceived as an ars explicandi, aiming at rendering the inferential rules implicit in our socio-linguistic argumentative praxis in a more orderly, more transparent, and less ambiguous way, thus setting the stage for an eventual critical assessment of our conceptual apparatus and inferential practices. The novel approach we showcase in this chapter, called computational hermeneutics, is inspired by Donald Davidson’s account of radical interpretation (Davidson 2001c, 1994). It draws on the well-known principle of charity and on a holistic account of meaning, according to which the meaning of a term can only be given through the explicitation of the inferential role it plays in some theory (or argument) of our interest. We adopt the view that the process of logical analysis (aka. formalization) of a natural language argument is itself a kind of interpretation, since it serves the purpose of making explicit the inferential relations between

D. Fuenmayor () Freie Universität Berlin, Berlin, Germany e-mail: [email protected] C. Benzmüller Freie Universität Berlin, Berlin, Germany University of Luxembourg, Esch-sur-Alzette, Luxembourg e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_12

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concepts and statements.1 Moreover, the output of this process, a logical form, does not need to be unique, since it is dependent on a given background logical theory, or, as Davidson has put it: [. . .] much of the interest in logical form comes from an interest in logical geography: to give the logical form of a sentence is to give its logical location in the totality of sentences, to describe it in a way that explicitly determines what sentences it entails and what sentences it is entailed by. The location must be given relative to a specific deductive theory; so logical form itself is relative to a theory. (Davidson 2001a, p. 140)

Following the principle of charity, while engaging in a process of logical analysis, requires us to search for plausible implicit premises which would render the given argument as being logically valid and also foster important theoretical virtues such as consistency, non-circularity (avoiding ‘question-begging’), simplicity, fruitfulness, etc. This task can be seen as a kind of conceptual explication.2 In computational hermeneutics we carry it out by providing definitions (i.e. by directly relating a definiendum with a definiens) or by introducing formulas (e.g. as axioms) which relate a concept we are currently interested in (explicandum) with some other concepts which are themselves explicated in the same way (in the context of the same or some other background theory). The circularity involved in this process is an unavoidable characteristic of any interpretive endeavor and has been historically known as the hermeneutic circle. Thus, computational hermeneutics contemplates an iterative process of ‘trial-and-error’ whereby the adequacy of some newly introduced formula or definition becomes tested by computing, among others, the logical validity of the whole formalized argument. In order to explore the very wide space of possible formalizations (and also of interpretations) for even the simplest argument, we have to test its validity at least several hundreds of times (also to account for logical pluralism). It is here where the recent improvements and ongoing consolidation of modern automated theorem proving technology, in particular for higher-order logic (HOL), become handy. A concrete example of the application of this approach using the Isabelle/HOL (Nipkow et al. 2002) proof assistant to the logical analysis and interpretation of an ontological argument will be provided in the last section. This chapter is divided in three parts. In the first one, we present the philosophical motivation and theoretical underpinnings of our approach; and we also outline the landscape of automated deduction. In the second part, we introduce the method of computational hermeneutics as an iterative process of conceptual explication. 1 In

recent times, this idea has become known as logical expressivism and has been championed, most notably, by the adherents of semantic inferentialism in the philosophy of language. Two paradigmatic book-length expositions of this philosophical position can be found in the works of Brandom (1994) and Peregrin (2014). 2 Explication, in Carnap’s sense, is a method of conceptual clarification, aimed at replacing an unclear ‘fuzzy’ pre-theoretical concept: an explicandum, by a new more exact concept with clearly defined rules of use: an explicatum, for use in a target theory. While Carnapian in spirit, our idea of explication focuses mostly on the activity of conceptual explicitation by the means of formal logic.

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In the last part, we present our case study: the computer-assisted logical analysis and interpretation of E. J. Lowe’s modal ontological argument, where our approach becomes exemplified.

12.1.1 Philosophical and Religious Arguments Is it possible to find meaning in religious argumentation? Or is religion a conversation-stopper? Do religious beliefs provide a conceptual framework through which a believer’s world-view is structured to such an extent, that the interpretation of religious arguments becomes a hopeless case? (given the apparent incommensurability between the conceptual schemes of speakers and interpreters of different creeds). The answer to these questions boils down to finding a way to acknowledge the variety of religious belief, while recognizing that we all share, at heart, a similar assortment of concepts and are thus able to understand each other. We argue for the role of logic as a common ground for understanding in general and, in particular, for theological argumentation. We reject therefore the view that deep religious convictions constitute an insurmountable obstacle for successful interreligious communication (e.g. between believers and lay interpreters). Such views have been much discussed in religious studies. Terry Godlove, for instance, has convincingly argued in Godlove (1989) against what he calls the “framework theory” in religious studies, according to which, for believers, religious beliefs shape the interpretation of most of the objects and situations in their lives. Here Godlove relies on Donald Davidson’s rejection of “the very idea of a conceptual scheme” (Davidson 2001b). Davidson’s criticism of what he calls “conceptual relativism” relies on the view that talk of incommensurable conceptual schemes is possible only on violating a correct understanding of interpretability, as developed in his theory of radical interpretation, especially vis-à-vis the well-known principle of charity. Furthermore, the kind of meaning holism implied by Davidson’s account of interpretation suggests that we must share vastly more beliefs than not with anyone whose words and actions we are able to interpret. Thus, divergence in belief must be limited: if an interpreter is to interpret someone as asserting that Jerusalem is a holy place, she has to presume that the speaker holds true many closely related sentences; for instance, that Jerusalem is a city, that holy places are sites of pilgrimage, and, if the speaker is Christian, that Jesus is the son of God and lived in Jerusalem—and so on. Meaning holism requires us, so goes Godlove’s thesis, to reject the notion that religions are alternative, incommensurable conceptual frameworks. Drawing upon our experience with the computer-assisted reconstruction and assessment of ontological arguments for the existence of God (Benzmüller and Woltzenlogel Paleo 2014, 2016; Fuenmayor and Benzmüller 2017; Benzmüller et al. 2017), we can bear witness to the previous claims. While looking for the most appropriate formalization of an argument, we have been led to consider further unstated assumptions (implicit premises) needed to reconstruct the argument as

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logically valid, and thus to ponder how much we may have departed from the original argument and to what extent we are still doing justice to the intentions of its author. We had to consider issues like the plausibility of our assumptions from the standpoint of the author and its compatibility with the author’s purported beliefs (or what she said elsewhere).3 Reflecting on this experience, we have become motivated to work out a computer-assisted interpretive approach drawing on the notion of semantic holism, which is especially suited for finding meaning in theological and metaphysical discourse. We want to focus our inquiry on the issue of understanding a particular type of arguments and the role computers can play in it. We are thus urged to distinguish the kinds of arguments we want to address from others that, on the one hand, rely on appeals to faith and rhetorical effects, or, on the other hand, make use of already well-defined concepts with univocal usage, like in mathematics. We have already talked of religious arguments in the spirit of St. Anselm’s ontological argument as some of the arguments we are interested in; we want, nonetheless, to generalize the domain of applicability of our approach to what we call ‘philosophical’ arguments (for lack of a better word), since we consider that many of the concepts introduced into these and many other kinds of philosophical discussions remain quite fuzzy and unclear (“explicanda” in Carnap’s terminology). We want to defend the view that the process of explicating those philosophical concepts takes place in the very practice of argumentation through the explicitation of the inferential role they play in some theory or argument of our interest. In the context of a formalized argument (in some chosen logic), this task of conceptual explication can be carried out systematically by giving definitions or axiomatizing conceptual interrelations, and then using automated reasoning tools to explore the space of possible logical inferences. This approach, which we name computational hermeneutics, will be illustrated in the case study presented in the last section.

12.1.2 Top-Down Versus Bottom-Up Approaches to Meaning Above we have discussed the challenge of finding meaning in religious arguments. Determining meanings in philosophical contexts, however, has generally been considered a problematic task, especially when one wants to avoid the kind of ontological commitments resulting from postulating the existence, for every linguistic expression, of some obscure abstract being in need of definite identity criteria (cf. Quine’s slogan “no entity without identity”). We want to talk here of the meaning of a linguistic expression (particularly of an argument) as that which the

3 More

specifically, Eder and Ramharter (2015) propose several criteria aimed at judging the adequacy of formal reconstructions of St. Anselm’s ontological argument. They also show how such reconstructions help us gain a better understanding of this argument.

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interpreter needs to grasp in order to understand it, and we will relate this to such blurry things as the inferential role of expressions. In a similar vein, we also want to acknowledge the compositional character of natural and formalized languages, so we can think of the meaning of an argument as a function of the meanings of each of its constituent sentences (premises and conclusions) and their mode of combination (logical consequence relation).4 Accordingly, we take the meaning of each sentence as resulting from the meaning of its constituent words (concepts) and their mode of combination. We can therefore, by virtue of compositionality, conceive a bottom-up approach for the interpretation of an argument, by starting with our pre-understanding (theoretical or colloquial) of its main concepts and then working our way up to an understanding of its sentences and their inferential interrelations.5 The bottom-up approach is the one usually employed in the formal verification of arguments (logic as ars iudicandi). However, it leaves open the question of how to arrive at the meaning of words beyond our initial pre-understanding of them. This question is central to our project, since we are interested in understanding (logic as ars explicandi) more than mere verification. Thus, we want to complement the atomistic bottom-up approach with a holistic top-down one, by proposing a computer-supported method aimed at determining the meaning of expressions from their inferential role vis-à-vis argument’s validity (which is determined for the argument as a whole), much in the spirit of Donald Davidson’s program of radical interpretation.6

4 Ideally,

an argument would be analyzed as an island isolated from any external linguistic or pre-linguistic goings-on, to the extent that its validity would depend solely on what is explicitly stated (premises, inference rules, etc.); and, for instance, when implicit premises are brought to our attention, they should be made explicit and integrated into the argument accordingly – which must always remain an intersubjectively accessible artifact: a product of our socio-linguistic discursive practices. In the same spirit, it is also reasonable to expect of all sentences to derive their meaning compositionally (in particular, we see no place for idioms in philosophical arguments). Unsurprisingly, these demands are never met in their entirety in real-world arguments. 5 There is a well-known tension between the holistic nature of inferential roles and a compositional account of meaning. In computational hermeneutics, we aim at showing both approaches in action (top-down and bottom-up), thus demonstrating their compatibility in practice. For a theoretical treatment of the relationship between compositionality and meaning holism, we refer the reader to Pelletier (2012) and Pagin (1997, 2008). 6 The connections between Davidson’s truth-centered theory of meaning and theories focusing on the inferential role of expressions (e.g. Brandom 1994; Harman 1987; Block 1998) have been much discussed in the literature. While some authors (Davidson included) see both holistic approaches as essentially different, others (e.g. Williams 1999, Horwich 1998, p. 72) have come to see Davidson’s theory as an instance of inferential-role semantics. We side with the latter.

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12.1.3 Radical Interpretation and the Principle of Charity What is the use of radical interpretation in religious and metaphysical discourse? The answer is trivially stated by Davidson himself, who convincingly argues that “all understanding of the speech of another involves radical interpretation” (Davidson 1994, p. 125). Furthermore, the impoverished evidential position we are faced with when interpreting metaphysical and theological arguments corresponds very closely to the starting situation Davidson contemplates in his thought experiments on radical interpretation, where he shows how an interpreter could come to understand someone’s utterances without relying on any prior understanding of their language.7 Davidson’s program builds on the idea of taking the notion of truth as basic and extracting from it an account of translation or interpretation satisfying two general requirements: (i) it must reveal the compositional structure of language, and (ii) it can be assessed using evidence available to the interpreter (Davidson 1994, 2001c). The first requirement (i) is addressed by noting that a theory of truth in Tarski’s style (modified to apply to natural language) can be used as a theory of interpretation. This implies that, for every sentence s of some object language L, a sentence of the form: «“s” is true in L iff p» (aka. T-schema) can be derived, where p acts as a translation of s into a sufficiently expressive metalanguage used for interpretation (note that in the T-schema the sentence p is being used, while s is only being mentioned). Thus, by virtue of the recursive nature of Tarski’s definition of truth (Tarski 1956), the compositional structure of the object-language sentences becomes revealed. From the point of view of computational hermeneutics, the sentence s is interpreted in the context of a given argument. The object language L thereby corresponds to the idiolect of the speaker (natural language plus some technical terms and background information), and the metalanguage is constituted by formulas of our chosen logic of formalization (some expressive logic XY) plus the turnstyle symbol

XY signifying that an inference (argument) is valid in logic XY. As an illustration, consider the following instance of the T-schema used for some theological argument about monotheism: «“There is only one God” is true [in English, in the context of argument A] iff A1 , A2 , . . . , An H OL “∃ x. God x ∧ ∀ y. God y → y = x”», where A1 , A2 , . . . , An correspond to the formalization of the premises of argument A and the turnstyle H OL corresponds to the standard logical consequence relation in higher-order logic (HOL). By comparing this with the T-schema («“s” is true in L iff p») we can notice that the used metalanguage sentence p can be paraphrased in the form: «“q” follows from the argument’s premises [in HOL]» where the mentioned sentence q corresponds to the formalization (in some chosen logic) of the object sentence s. In this example we have considered a sentence playing the role of a conclusion which is being supported by some premises. It is 7 For

an interesting discussion of the relevance of Davidson’s philosophy of language in religious studies, we refer the reader to Godlove (2002).

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however also possible to consider this same sentence in the role of a premise: «“There is only one God” is true [in the context of argument A] iff A1 , A2 , . . . , “∃ x. God x ∧ ∀ y. God y → y = x”,. . . , An H OL C»; now the truth of the sentence is postulated so that it can be used to validate some conclusion C.8 Most importantly, this example aims at illustrating how the interpretation of a sentence relates to its logical formalization and the inferential role it plays in a background argument. The second general requirement (ii) states that the interpreter has access to objective evidence in order to judge the appropriateness of her interpretations, i.e., access to the events and objects in the ‘external world’ that cause sentences to be true (or, in our case, arguments to be valid). In our approach, formal logic serves as a common ground for understanding. Computing the logical validity of a formalized argument constitutes the kind of objective (or, more appropriately, intersubjective) evidence needed to secure the adequacy of our interpretations, under the charitable assumption that the speaker follows (or at least accepts) similar logical rules as we do. In computational hermeneutics, the computer acts as an (arguably unbiased) arbiter deciding on the truth of a sentence in the context of an argument. In order to account for logical pluralism, computational hermeneutics targets the utilization of different kinds of classical and non-classical logics through the technique of semantical embeddings (see e.g. Benzmüller and Paulson 2013; Benzmüller 2013), which allows us to take advantage of the expressive power of classical higher-order logic (as a metalanguage) in order to embed the syntax and semantics of another logic (as an object language). Using the technique of semantical embeddings we can, for instance, embed a modal logic by defining the modal operators as meta-logical predicates. A framework for automated reasoning in different logics by applying the technique of semantical embeddings has been successfully implemented using automated theorem proving technology (Fuenmayor and Benzmüller 2017; Benzmüller 2017). Underlying his account of radical interpretation, there is a central notion in Davidson’s theory: the principle of charity, which he holds as a condition for the possibility of engaging in any kind of interpretive endeavor. In a nutshell, the principle says that “we make maximum sense of the words and thoughts of others when we interpret in a way that optimizes agreement” (Davidson 2001b). The principle of charity builds on the possibility of intersubjective agreement about external facts among speaker and interpreter and can thus be invoked to make sense of a speaker’s ambiguous utterances and, in our case, to presume (and foster) the validity of the argument we aim at interpreting. Consequently, in computational hermeneutics we assume from the outset that the argument’s conclusions indeed

8 We

may actually want to weaken the double implication in this case, or work with an alternative notion of logical consequence. Moreover, other roles can be conceived for such a sentence in the context of an argument, for instance, it can also play the role of an unwanted conclusion: a sentence which we want to make sure it remains false no matter how we analyze the argument.

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follow from its premises and disregard formalizations that do not do justice to this postulate.

12.1.4 The Automated Reasoning Landscape Automated reasoning is an umbrella term used for a wide range of technologies sharing the overall goal of mechanizing different forms of reasoning (understood as the ability to draw inferences). Born as a subfield of artificial intelligence with the aim of automatically generating mathematical proofs,9 automated reasoning has moved to close proximity of logic and philosophy thanks to substantial theoretical developments in the last decades. Nevertheless, its main field of application has mostly remained bounded to mathematics and hardware and software verification. In this respect, the field of automated theorem proving (ATP) has traditionally been its most developed subarea. ATP involves the design of algorithms that automate the process of construction (proof generation) and verification (proof checking) of mathematical proofs. Some extensive work has also been done in other nondeductive forms of reasoning (inductive, abductive, analogical, etc.). However, those fields remain largely underrepresented in comparison. There have been major advances regarding the automatic generation of formal proofs during the last years, which we think make the utilization of formal methods in philosophy very promising and have even brought about some novel philosophical results (e.g. Benzmüller and Woltzenlogel Paleo 2016). We will, on this occasion, restrain ourselves to the computer-supported interpretation of existing arguments, that is, to a situation where the given nodes/statements in the argument constitute a coarse-grained “island proof structure” that needs to be rigorously assessed. Proof checking can be carried out either non-interactively (for instance as a batch operation) or interactively by utilizing a proof assistant. A non-interactive proofchecking program would normally get as input some formula (string of characters in some predefined syntax) and a context (some collection of such formulas) and will, in positive cases, generate a listing of the formulas (in the given context) from which the input formula logically follows, together with the name of the proof method10 used and, in some cases, a proof string (as in the case of proof generators). Some proof checking programs, called model finders, are specialized in searching for models and, more importantly, countermodels for a given formula.

9 For instance, the first widely recognized AI system: Logic Theorist, was able to prove 38 of the first 52 theorems of Whitehead and Russell’s “Principia Mathematica” back in 1956. 10 For instance, some of the proof methods commonly employed by the Isabelle/HOL proof assistant are: term rewriting, classical reasoning, tableaus, model elimination, ordered resolution and paramodulation.

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This functionality proves very useful in practice by sparing us the thankless task of trying to prove non-theorems. Human guidance is oftentimes required by theorem provers in order to effectively solve interesting problems. A need has been recognized for the synergistic combination of the vast memory resources and information-processing capabilities of modern computers, together with human ingenuity, by allowing people to give hints to these tools by the means of especially crafted user interfaces. The field of interactive theorem proving has grown out of this endeavor and its software programs are known as proof assistants.11 Automated reasoning is currently being applied to solve problems in formal logic, mathematics and computer science, software and hardware verification and many others. For instance, the Mizar Library12 and TPTP (Thousands of Problems for Theorem Provers) (Sutcliffe and Suttner 1998) are two of the biggest libraries of such problems being maintained and updated on a regular basis. There is also a yearly competition among automated theorem provers held at the CADE conference (Pelletier et al. 2002), whose problems are selected from the TPTP library. Automated theorem provers (particularly focusing on higher order logics) have been used to assist in the formalization of many advanced mathematical proofs such as Erdös-Selberg’s proof of the Prime Number Theorem (about 30,000 lines in Isabelle), the proof of the Four Color Theorem (60,000 lines in Coq), and the proof of the Jordan Curve Theorem (75,000 lines in HOL-Light) (Portoraro 2014). The monumental proof of Kepler’s conjecture by Thomas Hales and his research team has been recently formalized and verified using the HOL-Light and Isabelle proof assistants as part of the Flyspeck project (Hales et al. 2017). Isabelle (Nipkow et al. 2002) is the proof assistant we will use to illustrate our computational hermeneutics method. Isabelle offers a structured proof language called Isar specifically tailored for writing proofs that are both computer- and human-readable and which focuses on higher-order classical logic. The different variants of the ontological argument assessed in our case study are formalized directly in Isabelle’s HOL dialect and, for the modal variants, through the technique of shallow semantical embeddings (Benzmüller and Paulson 2013).

12.2 Part II: The Computational Hermeneutics Method It is easy to argue that using computers for the assessment of arguments brings us many quantitative advantages, since it gives us the means to construct and verify proofs easier, faster, and much more reliably. Furthermore, a main task of this work

11 A survey and system comparison of the most famous interactive proof assistants has been carried

out in Wiedijk (2006). The results of this survey remain largely accurate to date. proofs and their corresponding articles are published regularly in the peer-reviewed Journal of Formalized Mathematics.

12 Mizar

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is to illustrate a central qualitative advantage of computer-assisted argumentation: It enables a different, holistic approach to philosophical argumentation.

12.2.1 Holistic Approach: Why Feasible Now? Let us imagine the following scenario: A philosopher working on a formal argument wants to test a variation on one of its premises (or definitions) and find out if the argument still holds. Our philosopher is working with pen and paper and she follows some chosen proof procedure (e.g. natural deduction or sequent calculus). Depending on her calculation skills, this may take some minutes, if not much longer, to be carried out. It seems clear that she cannot allow herself many of such experiments on such conditions. Now compare the above scenario to another one in which our working philosopher can carry out such an experiment in just a few seconds and with no effort, by employing an automated theorem prover. In a best-case scenario, the proof assistant would automatically generate a proof (or the sketch of a countermodel), so she just needs to interpret the results and use them to inform her new conjectures. In any case, she would at least know if her speculations had the intended consequences, or not. After some minutes of work, she will have tried plenty of different variations of the argument while getting real-time feedback regarding their suitability.13 We aim at showing how this radical quantitative increase in productivity does indeed entail a qualitative change in the way we approach formal argumentation, since it allows us to take things to a whole new level (note that we are talking here of many hundreds of such trial-and-error ‘experiments’ that would take weeks or months if using pen and paper). Most importantly, this qualitative leap opens the door for the possibility of automating the process of logical analysis for naturallanguage arguments with regard to their subsequent computer-assisted critical evaluation.

12.2.2 The Approach Computational hermeneutics is a holistic iterative enterprise, where we evaluate the adequacy of some candidate formalization of a sentence by computing the logical validity of the whole argument. We start with formalizations of some simple statements (taking them as tentative) and use them as stepping stones on the way 13 The

situation is obviously idealized, since, as is well known, most of theorem-proving problems are computationally complex and even undecidable, so in many cases a solution will take several minutes or just never be found. Nevertheless, as work in the emerging field of computational metaphysics (Oppenheimer and Zalta 2011; Alama et al. 2015; Rushby 2013; Benzmüller and Woltzenlogel Paleo 2014, 2016) suggests, the lucky situation depicted above is not rare.

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to the formalization of other argument’s sentences, repeating the procedure until arriving at a state of reflective equilibrium: A state where our beliefs and commitments have the highest degree of coherence and acceptability.14 In computational hermeneutics, we work iteratively on an argument by temporarily fixing truthvalues and inferential relations among its sentences, and then, after choosing a logic for formalization, working back and forth on the formalization of its premises and conclusions by making gradual adjustments while getting automatic feedback about the suitability of our speculations. In this fashion, by engaging in a dialectic questions-and-answers (‘trial-and-error’) interaction with the computer, we work our way towards a proper understanding of an argument by circular movements between its parts and the whole (hermeneutic circle). A rough outline of the iterative structure of the computational hermeneutics approach is as follows: 1. Argument reconstruction (initially in natural language): a. Add or remove sentences and choose their truth-values. Premises and desired conclusions would need to become true, while some other ‘unwanted’ conclusions would have to become false. Deciding on these issues expectedly involves a fair amount of human judgment. b. Establish inferential relations, i.e., determine the extension of the logical consequence relation: which sentences should follow (logically) from which others. This task can be done manually or automatically by letting our automated tools find this out for themselves, provided the logic for formalization has been selected and the argument has already been roughly formalized (hence the mechanization of this step becomes feasible only after at least one outermost iteration). Automating this task frequently leads to the simplification of the argument, since current theorem provers are quite good at detecting idle axioms (see e.g. Isabelle’s Sledgehammer tool (Blanchette et al. 2013)). 2. Selection of a logic for formalization, guided by determining the logical structure of the natural-language sentences occurring in the argument. This task can be partially automated (using the semantical embeddings technique) by searching a catalog of different embedded logics (in HOL) and selecting a candidate logic (modal, free, deontic, etc.) satisfying some particular syntactic or semantic criteria.

14 We

have been inspired by John Rawls’ notion of reflective equilibrium as a state of balance or coherence between a set of general principles and particular judgments (where the latter follow from the former). We arrive at such a state through a deliberative give-and-take process of mutual adjustment between principles and judgments. More recent methodical accounts of reflective equilibrium have been proposed as a justification condition for scientific theories (Elgin 1999) and objectual understanding (Baumberger and Brun 2016), and also as an approach to logical analysis (Peregrin and Svoboda 2017).

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3. Argument formalization (in the chosen logic), while getting continuous feedback from our automated reasoning tools regarding the argument’s correctness (validity, consistency, non-circularity, etc.). This stage is itself iterative, since, for every sentence, we charitably (in the spirit of the principle of charity) try several different formalizations until getting a correct argument. Here is where we take most advantage of the real-time feedback offered by our automated tools. Some main tasks to be considered are: a. Translate natural-language sentences into the target logic, by relying either on our pre-understanding or on provided definitions of the argument’s terms. b. Vary the logical form of already formalized sentences. This can be done systematically (and even automatically) by relying upon a catalog of (consistent) logical variations of formulas (see semantical embeddings) and the output of automated tools (ATPs, model finders, etc.). c. Bring related terms together, either by introducing definitions or by axiomatizing new interrelations among them. These newly introduced formulas can be translated back into natural language to be integrated into the argument in step (1.a), thus being disclosed as former implicit premises. The process of searching for additional premises with the aim of rendering an argument formally correct can be seen as a kind of abductive reasoning (‘inference to the best explanation’) and thus needs human support (at least at the current state of the art). 4. Are termination criteria satisfied? That is, have we arrived at a state of reflective equilibrium? If not, we would come back to some early stage. Termination criteria can be derived from the adequacy criteria of formalization found in the literature on logical analysis (see e.g. Baumgartner and Lampert 2008; Brun 2014; Peregrin and Svoboda 2013, 2017). An equilibrium may be found after several iterations without any significant improvements.15 Furthermore, the introduction of automated reasoning and linguistic analysis tools makes it feasible to apply these criteria to compute, in seconds, the degree of ‘fitness’ of some candidate formalization for a sentence (in the context of an argument).

15 In

particular, inferential adequacy criteria lend themselves to the application of automated deduction tools. Consider, for instance, Peregrin and Svoboda’s (2017) proposed criteria: (i) The principle of reliability: “φ counts as an adequate formalization of the sentence S in the logical system L only if the following holds: If an argument form in which φ occurs as a premise or as the conclusion is valid in L, then all its perspicuous natural language instances in which S appears as a natural language instance of φ are intuitively correct arguments.” (ii) The principle of ambitiousness: “φ is the more adequate formalization of the sentence S in the logical system L the more natural language arguments in which S occurs as a premise or as the conclusion, which fall into the intended scope of L and which are intuitively perspicuous and correct, are instances of valid argument forms of S in which φ appears as the formalization of S.” (Peregrin and Svoboda 2017, pp. 70–71).

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12.3 Part III: Lowe’s Modal Ontological Argument In this section, the main contribution of this chapter, we illustrate the computersupported interpretation of a variant of St. Anselm’s ontological argument for the existence of God, using Isabelle/HOL.16 This argument, which was introduced by the philosopher E. J. Lowe in an article named “A Modal Version of the Ontological Argument” (Lowe 2013), serves here as an exemplary case for an interesting and sufficiently complex, systematic argument with strong ties to metaphysics and religion. The interpretation of Lowe’s argument thus makes for an ideal showcase for computational hermeneutics in practice. Lowe offers in his work a new modal variant of the ontological argument, which is specifically aimed at proving the necessary existence of God. In a nutshell, Lowe’s argument works by first postulating the existence of necessary abstract beings, i.e., abstract beings that exist in every possible world (e.g. numbers). He then introduces the concepts of ontological dependence and metaphysical explanation and argues that the existence of every (mind-dependent) abstract being is ultimately explained by some concrete being (e.g. a mind). By interrelating the concepts of dependence and explanation, he argues that the concrete being(s), on which each necessary abstract being depends for its existence, must also be necessary. This way he proves the existence of at least one necessary concrete being (i.e. God, according to his definition). Lowe further argues that his argument qualifies as a modal ontological argument, since it focuses on necessary existence, and not just existence of some kind of supreme being. His argument differs from other familiar variants of the modal ontological argument (like Gödel’s) in that it does not appeal, in the first place, to the possible existence of God in order to use the modal S5 axioms to deduce its necessary existence as a conclusion.17 Lowe wants therefore to circumvent the usual criticisms to the S5 axiom system, like implying the unintuitive assertion that whatever is possibly necessarily the case is thereby actually the case. The structure of Lowe’s argument is very representative of methodical philosophical arguments. It features eight premises from which new inferences are drawn until arriving at a final conclusion: the necessary existence of God (which in this case amounts to the existence of some necessary concrete being). The argument’s premises are reproduced verbatim below: (P1) (P2) (P3) (P4) 16 We

God is, by definition, a necessary concrete being. Some necessary abstract beings exist. All abstract beings are dependent beings. All dependent beings depend for their existence on independent beings.

refer the reader to Fuenmayor and Benzmüller (2017) for further details. That computerverified article has been completely written in the Isabelle proof assistant and thus requires some familiarity with this system. 17 As shown in Benzmüller and Woltzenlogel Paleo (2014), modal logic KB actually suffices to prove Scott’s variant of Gödel’s argument; this was probably not known to Lowe though.

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No contingent being can explain the existence of a necessary being. The existence of any dependent being needs to be explained. Dependent beings of any kind cannot explain their own existence. The existence of dependent beings can only be explained by beings on which they depend for their existence.

We will consider here only a representative subset of the argument’s conclusions, which are reproduced below: (C1)

All abstract beings depend for their existence on concrete beings. (Follows from P3 and P4 together with definitions D3 and D4.) (C5) In every possible world there exist concrete beings. (Follows from C1 and P2.) (C7) The existence of necessary abstract beings needs to be explained. (Follows from P2, P3 and P6.) (C8) The existence of necessary abstract beings can only be explained by concrete beings. (Follows from C1, P3, P7 and P8.) (C9) The existence of necessary abstract beings is explained by one or more necessary concrete beings. (Follows from C7, C8 and P5.) (C10) A necessary concrete being exists. (Follows from C9.) Lowe also introduces some informal definitions which should help the reader to understand some of the concepts involved in his argument (necessity, concreteness, ontological dependence, metaphysical explanation, etc.). In the following discussion, we will see that most of these definitions do not bear the significance Lowe claims. (D1) (D2) (D3) (D4) (D5)

x is a necessary being := x exists in every possible world. x is a contingent being := x exists in some but not every possible world. x is a concrete being := x exists in space and time, or at least in time. x is an abstract being := x does not exist in space or time. x depends for its existence on y := necessarily, x exists only if y exists.

In the following sections we use computational hermeneutics to interpret iteratively the argument shown above (by reconstructing it formally in different variations and in different logics). We compile in each section the results of a series of iterations and present them as a new variant of the original argument. We want to illustrate how the argument (as well as our understanding of it) gradually evolves as we experiment with different combinations of definitions, premises and logics for formalization.

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12.3.1 First Iteration Series: Initial Formalization Let us first turn to the formalization of premise P1: “God is, by definition, a necessary concrete being”.18 In order to shed light on the concept of necessariness (i.e. being a necessary being) employed in this argument, we have a look at the definitions D1 and D2 provided by the author. They relate the concepts of necessariness and contingency (i.e. being a contingent being) with existence:19 (D1) (D2)

x is a necessary being := x exists in every possible world. x is a contingent being := x exists in some but not every possible world.

The two definitions above, aimed at explicating the concepts of necessariness and contingency by reducing them to existence and quantification over possible worlds, have a direct impact on the choice of a logic for formalization. They not only call for some kind of modal logic with possible-world semantics but also lead us to consider the complex issue of existence, since we need to restrict the domain of quantification at every world. The choice of a modal logic for formalization has brought to the foreground an interesting technical constraint: The Isabelle proof assistant (as well as others) does not natively support modal logics. We have used, therefore, a technique known as semantical embedding, which allows us to take advantage of the expressive power of higher-order logic (HOL) in order to embed the syntax and semantics of an object language. Here we draw on previous work on the embedding of multimodal logics in HOL (Benzmüller and Paulson 2013), which has successfully been applied to the analysis and verification of ontological arguments (e.g. Benzmüller and Woltzenlogel Paleo 2016, 2014; Benzmüller et al. 2017; Fuenmayor and Benzmüller 2017). Using this technique, we can embed a modal logic K by defining the  and ♦ operators using restricted quantification over the set of reachable worlds (using a reachability relation R as a guard). Note that, in the following definitions, the type wo is declared as an abbreviation for w⇒bool, which corresponds to the type of a function mapping worlds (of type w) to boolean values. wo thus corresponds to the type of a world-dependent formula (i.e. its truth set). consts R::w⇒w⇒bool (infix R) — Reachability relation abbreviation mbox :: wo⇒wo (-) 18 When the author says of something that it is a “necessary concrete being” we will take him to say

that it is both necessary and concrete. Certainly, when we say of Tom that he is a lousy actor, we just don’t mean that he is lousy and that he also acts. For the time being, we won’t differentiate between predicative and attributive uses of adjectives, so we will formalize both sorts as unary predicates; since the particular linguistic issues concerning attributive adjectives don’t seem to play a role in this argument. In the spirit of the principle of charity, we may justify adding further complexity to the argument’s formalization if we later find out that it is required for its validity. 19 Here, the concepts of necessariness and contingency are meant as properties of beings, in contrast to the concepts of necessity and possibility which are modals. We will see later how both pairs of concepts can be related in order to validate this argument.

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where ϕ ≡ λw.∀ v. (w R v)−→(ϕ v) abbreviation mdia :: wo⇒wo (♦-) where ♦ϕ ≡ λw.∃ v. (w R v)∧(ϕ v) The ‘lifting’ of the standard logical connectives to type wo is straightforward. Validity is consequently defined as truth in all worlds and represented by wrapping the formula in special brackets (−). abbreviation valid::wo⇒bool (-) where ψ ≡ ∀ w.(ψ w) We verify our embedding by using Isabelle’s simplifier to prove the K principle and the necessitation rule. lemma K: ((ϕ → ψ)) → (ϕ → ψ) by simp — Verifying K principle lemma N EC: ϕ ⇒ ϕ by simp — Verifying necessitation rule Regarding existence, we need to commit ourselves to a certain position in metaphysics known as metaphysical contingentism, which roughly states that the existence of any entity is a contingent fact: some entities can exist at some worlds, while not existing at some others. The negation of metaphysical contingentism is known as metaphysical necessitism, which basically says that all entities must exist at all possible worlds. By not assuming contingentism and, therefore, assuming necessitism, the whole argument would become trivial, since all beings would end up being trivially necessary (i.e. existing in all worlds).20 We hence restrict our quantifiers so that they range only over those entities that ‘exist’ (i.e. are actualized) at a given world. This approach is known as actualist quantification and is implemented, using the semantical embedding technique, by defining a world-dependent meta-logical ‘existence’ predicate (called “actualizedAt” below), which is the one used as a guard in the definition of the quantifiers. Note that the type e characterizes the domain of all beings (i.e. existing and non-existing entities), and the type wo characterizes sets of worlds. The term “isActualized” thus relates beings to worlds. consts isActualized::e⇒wo (infix actualizedAt) abbreviation f orallAct::(e⇒wo)⇒wo (∀ A ) where ∀ A % ≡ λw.∀ x. (x actualizedAt w)−→(% x w) abbreviation existsAct::(e⇒wo)⇒wo (∃ A ) where ∃ A % ≡ λw.∃ x. (x actualizedAt w) ∧ (% x w) The corresponding binder syntax is defined below. abbreviation mf orallActB::(e⇒wo)⇒wo (binder∀ A )

20 Metaphysical

contingentism looks prima facie like a very natural assumption to make; nevertheless an interesting philosophical debate between advocates of necessitism and contingentism has arisen during the last years, especially in the wake of Timothy Williamson’s work on the metaphysics of modality (see Williamson 2013).

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where ∀ A x. (ϕ x) ≡ ∀ A ϕ abbreviation mexistsActB::(e⇒wo)⇒wo (binder∃ A ) where ∃ A x. (ϕ x) ≡ ∃ A ϕ We use a model finder (Isabelle’s Nitpick tool (Blanchette and Nipkow 2010)) to verify that actualist quantification validates neither the Barcan formula nor its converse. For the conjectured lemma, Nitpick finds a countermodel, i.e. a model (satisfying all axioms) which falsifies the given formula. The formula is consequently non-valid (as indicated by the Isabelle’s “oops” keyword). lemma (∀ A x. (ϕ x)) → (∀ A x. ϕ x) nitpick oops — Countermodel found : formula not valid lemma (∀ A x. ϕ x) → (∀ A x. (ϕ x)) nitpick oops — Countermodel found : formula not valid Unrestricted (aka. possibilist) quantifiers, in contrast, validate both the Barcan formula and its converse. lemma (∀ x.(ϕ x)) → (∀ x.(ϕ x)) by simp — Proven by Isabelle s simplifier lemma (∀ x.(ϕ x)) → (∀ x.(ϕ x)) by simp — Proven by Isabelle s simplifier With actualist quantification in place we can: (i) the concept of existence becomes formalized (explicated) in the usual form by using a restricted particular quantifier (≈ stands for the unrestricted identity relation on all objects), (ii) necessariness becomes formalized as existing necessarily, and (iii) contingency becomes formalized as existing possibly but not necessarily. definition Existence::e⇒wo (E!) where E! x ≡ ∃ A y. y ≈ x definition Necessary::e⇒wo where Necessary x ≡ E! x definition Contingent::e⇒wo where Contingent x≡ ♦E! x ∧ ¬Necessary x Note that we have just chosen a logic for formalization: a free quantified modal logic K with positive semantics. The logic is free because the domain of quantification (for actualist quantifiers) is a proper subset of our universe of discourse (so we can refer to non-existing objects). The semantics is positive because we have placed no restriction regarding predication on non-existing objects, so they are also allowed to exemplify properties and relations. We are also in a position to embed stronger normal modal logics (KB, KB5, S4, S5, etc.) by restricting the reachability relation R with additional axioms, if needed. Having chosen our logic, we can now turn to the formalization of the concepts of abstractness and concreteness. As seen previously, Lowe has already provided us with an explication of these concepts: (D3) (D4)

x is a concrete being := x exists in space and time, or at least in time. x is an abstract being := x does not exist in space or time.

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Lowe himself acknowledges that the explication of these concepts in terms of existence “in space and time” is superfluous, since we are only interested in them being complementary.21 Thus, we start by formalizing concreteness as a primitive world-dependent predicate and then derive abstractness from it, namely as its negation. consts Concrete::e⇒wo abbreviation Abstract::e⇒wo where Abstract x ≡ ¬(Concrete x) We can now formalize the definition of Godlikeness (P1) as follows: abbreviation Godlike::e⇒wo where Godlike x ≡ Necessary x ∧ Concrete x We also formalize premise P2 (“Some necessary abstract beings exist”) as shown below: axiomatization where P 2: ∃ A x. Necessary x ∧ Abstract x Let us now turn to premises P3 (“All abstract beings are dependent beings”) and P4 (“All dependent beings depend for their existence on independent beings”). We have here three new terms to be explicated: two predicates “dependent” and “independent” and a relation “depends (for its existence) on”, which has been called ontological dependence by Lowe. Following our linguistic intuitions concerning their interrelation, we start by proposing the following formalization: consts dependence::e⇒e⇒wo (infix dependsOn) definition Dependent::e⇒wo where Dependent x ≡ ∃ A y. x dependsOn y abbreviation I ndependent::e⇒wo where I ndependent x ≡ ¬(Dependent x) We have formalized ontological dependence as a primitive world-dependent relation and refrained from any explication (as suggested by Lowe).22 Moreover, an entity is dependent if and only if there actually exists an object y such that x depends for its existence on it; accordingly, we have called an entity independent if and only if it is not dependent.

21 We

quote from Lowe’s original article: “Observe that, according to these definitions, a being cannot be both concrete and abstract: being concrete and being abstract are mutually exclusive properties of beings. Also, all beings are either concrete or abstract [. . .] the abstract/concrete distinction is exhaustive. Consequently, a being is concrete if and only if it is not abstract.” 22 An explication of this concept has been suggested by Lowe in definition D5 (“x depends for its existence on y := necessarily, x exists only if y exists”). Concerning this alleged definition, he has written in a footnote to the same article: “Note, however, that the two definitions (D5) and (D6) presented below are not in fact formally called upon in the version of the ontological argument that I am now developing, so that in the remainder of this chapter the notion of existential dependence may, for all intents and purposes, be taken as primitive. There is an advantage in this, inasmuch as finding a perfectly apt definition of existential dependence is no easy task, as I explain in ‘Ontological Dependence.”’ Lowe refers hereby to his article on ontological dependence in the Stanford Encyclopedia of Philosophy (Lowe 2010) for further discussion.

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As a consequence, premises P3 (“All abstract beings are dependent beings”) and P4 (“All dependent beings depend for their existence on independent beings”) become formalized as follows. axiomatization where P 3: ∀ A x. Abstract x → Dependent x and P 4: ∀ A x. Dependent x → (∃ A y. I ndependent y ∧ x dependsOn y) Concerning premises P5 (“No contingent being can explain the existence of a necessary being”) and P6 (“The existence of any dependent being needs to be explained”), a suitable formalization for expressions of the form: “the entity X explains the existence of Y” and “the existence of X is explained” needs to be found.23 These expressions rely on a single binary relation, which will initially be taken as primitive. This relation has been called metaphysical explanation by Lowe.24 consts explanation::e⇒e⇒wo (infix explains) definition Explained::e⇒wo where Explained x ≡ ∃ A y. y explains x axiomatization where P 5: ¬(∃ A x. ∃ A y. Contingent y ∧ Necessary x ∧ y explains x) Premise P6, together with the last two premises: P7 (“Dependent beings of any kind cannot explain their own existence”) and P8 (“The existence of dependent beings can only be explained by beings on which they depend for their existence”), were introduced by Lowe in order to relate the concept of metaphysical explanation to ontological dependence.25 axiomatization where P 6: ∀ x. Dependent x → Explained x and P 7: ∀ x. Dependent x → ¬(x explains x) and P 8: ∀ x y. y explains x → x dependsOn y Although the last three premises seem to couple very tightly the concepts of (metaphysical) explanation and (ontological) dependence, both concepts are not meant by the author to be equivalent.26 We have used Nitpick to test this claim.

23 Note

that we have omitted the expressions “can” and “needs to” in our formalization, since they seem to play here only a rhetorical role. As in the case of attributive adjectives discussed before, we first aim at the simplest workable formalization; however, we are willing to later improve on this formalization in order to foster argument’s validity, in accordance to the principle of charity. 24 This concept is closely related to what has been called metaphysical grounding in contemporary literature. 25 Note that we use non-restricted quantifiers for the formalization of the last three premises in order to test the argument’s validity under the strongest assumptions. As before, we turn a blind eye to the modal expression “can”. 26 Lowe says: “Existence-explanation is not simply the inverse of existential dependence. If x depends for its existence on y, this only means that x cannot exist without y existing. This is

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Since a countermodel has been found, we have proven that the inverse equivalence of metaphysical explanation and ontological dependence is not implied by the axioms (a screenshot showing Nitpick’s text-based representation of such a model is provided below). lemma ∀ x y. x explains y ↔ y dependsOn x nitpick[user-axioms] oops For any being, however, having its existence “explained” is equivalent to its existence being “dependent” (on some other being). This follows already from premises P6 and P8, as shown above by Isabelle’s prover. lemma ∀ x. Explained x ↔ Dependent x using P 6 P 8 Dependent-def Explained-def by auto The Nitpick model finder is also useful to check axioms’ consistency at any stage during the formalization of an argument. We instruct Nitpick to search for a model satisfying some tautological sentence (here we use a trivial ‘True’ proposition), thus demonstrating the satisfiability of the argument’s axioms. Nitpick’s output is a text-based representation of the found model (or a message indicating that no model, up to a predefined cardinality, could be found). This information is very useful to inform our future decisions. The screenshot below (taken from the Isabelle proof assistant) shows the model found by Nitpick, which satisfies the argument’s formalized premises: lemma T rue nitpick[satisfy, user-axioms] oops

not at all the same as saying that x exists because y exists, or that x exists in virtue of the fact that y exists.”

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In this case, Nitpick was able to find a model satisfying the given tautology; this means that all axioms defined so far are consistent. The model found consists of two individual objects a and b and a single world w1 , which is not connected via the reachability relation R to itself. We furthermore have in world w1 : b is concrete, a is not; a depends on b and itself, while b depends on no other object; b is the only object that explains a and a explains no object. We can also use model finders to perform ‘sanity checks’: We instruct Nitpick to find a countermodel for some specifically tailored formula which we want to make sure is not valid, because of its implausibility from the point of view of the author (as we interpret him). We check below, for instance, that our axioms are not too strong as to imply metaphysical necessitism (i.e. that all beings necessarily exist) or modal collapse (i.e. that all truths are necessary). Since both would trivially validate the argument. lemma ∀ x. E! x nitpick[user-axioms] oops — Countermodel found : necessitism is not valid lemma ϕ → ϕ nitpick[user-axioms] oops — Countermodel found : modal collapse is not valid Model finders like Nitpick are able to verify consistency (by finding a model) or non-validity (by finding a countermodel) for a given formula. When it comes to verifying validity or invalidity we use provers. Isabelle comes with various different provers tailored for specific kinds of problems and thus employing different approaches, strategies and heuristics. We typically make extensive use of Isabelle’s Sledgehammer tool (Blanchette et al. 2013), which integrates several state-ofthe-art external theorem provers and feeds them with different combinations of axioms and the conjecture in question. If successful, Sledgehammer returns valuable dependency information (the exactly required axioms and definitions to prove a given conjecture) back to Isabelle, which then exploits this information to (re)construct a trusted proof with own, internal proof automation means. The entire process often only takes a few seconds. By using Sledgehammer we can here verify the validity of our partial conclusions (C1, C5 and C7) and even find the premises they rely upon.27 (C1)

All abstract beings depend for their existence on concrete beings.

theorem C1: ∀ A x. Abstract x → (∃ y. Concrete y ∧ x dependsOn y) using P 3 P 4 by blast (C5) 27 We

In every possible world there exist concrete beings.

prove theorems in Isabelle here by using the keyword “by” followed by the name of an Isabelle-internal and thus trusted proof method (generally, some computer-implemented algorithm). Some methods commonly used in Isabelle are: simp (term rewriting), blast (tableaus), meson (model elimination), metis (ordered resolution and paramodulation) and auto (classical reasoning and term rewriting). As explained, these methods were automatically suggested and applied by the Sledgehammer tool. The interactive user in fact does not need to know, or learn, much about these methods in the beginning (he will benefit a lot though, if he does).

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theorem C5: ∃ A x. Concrete x using P 2 P 3 P 4 by blast (C7)

The existence of necessary abstract beings needs to be explained.

theorem C7: ∀ A x. (Necessary x ∧ Abstract x) → Explained x using P 3 P 6 by simp The last three conclusions are shown by Nitpick to be non-valid even in the stronger S5 logic. S5 can be easily introduced by postulating that the reachability relation R is an equivalence relation. This exploits the Sahlqvist correspondence which relates modal axioms to constraints on a model’s reachability relation: reflexivity, symmetry, seriality, transitivity and euclideanness imply axioms T , B, D, I V , V respectively (and also the other way round). axiomatization where S5: equivalence R — We assume T : ϕ→ϕ , B: ϕ→♦ϕ and 4: ϕ→ϕ (C8)

The existence of necessary abstract beings can only be explained by concrete beings.

lemma C8: ∀ A x.(Necessary x ∧ Abstract x)→(∀ A y. y explains x→ Concrete y) nitpick[user-axioms] oops (C9)

The existence of necessary abstract beings is explained by one or more necessary concrete (Godlike) beings.

lemma C9: ∀ A x.(Necessary x ∧ Abstract x)→(∃ A y. y explains x ∧ Godlike y) nitpick[user-axioms] oops (C10)

A necessary concrete (Godlike) being exists.

theorem C10: ∃ A x. Godlike x nitpick[user-axioms] oops Note that Nitpick does not only spare us the effort of searching for non-existent proofs but also provides us with very helpful information when it comes to fix an argument by giving us a text-based description of the (counter-)model found. We present below another screenshot showing Nitpick’s counterexample for C10: By employing the Isabelle proof assistant we have proven non-valid a first formalization attempt of Lowe’s modal ontological argument. This is, however, just the first of many series of iterations in our interpretive endeavor. Based on the information recollected so far, we can proceed to make the adjustments necessary to validate the argument. We will see how these adjustments have an impact on the inferential role of all concepts (necessariness, concreteness, dependence, explanation, etc.) and therefore on their meaning.

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12.3.2 Second Iteration Series: Validating the Argument I By carefully examining the above countermodel for C10, it has been noticed that some necessary beings, which are abstract in the actual world, may indeed be concrete in other reachable worlds. Lowe has previously presented numbers as an example of such necessary abstract beings. It can be argued that numbers, while existing necessarily, can never be concrete in any possible world, so we add the restriction of abstractness being an essential property, i.e. a locally rigid predicate. axiomatization where abstractness-essential: ∀ x. Abstract x → Abstract x theorem C10: ∃ A x. Godlike x nitpick[user-axioms] oops — Countermodel found Again, we have used model finder Nitpick to get a counterexample for C10, so the former restriction is not enough to prove this conclusion. We try postulating further restrictions on the reachability relation R, which, taken together, would amount to it being an equivalence relation. This would make for a modal logic S5 (see Sahlqvist correspondence), and thus the abstractness property becomes a (globally) rigid predicate.

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axiomatization where T -axiom: ref lexive R and — ϕ → ϕ B-axiom: symmetric R and — ϕ → ♦ϕ I V -axiom: transitive R — ϕ → ϕ theorem C10: ∃ A x. Godlike x nitpick[user-axioms] oops — Countermodel found By examining the new countermodel found by Nitpick, we noticed that at some worlds there are non-existent concrete beings. We want to disallow this possibility, so we make concreteness an existence-entailing property. axiomatization where concrete-exist: ∀ x. Concrete x → E! x We carry out the usual ‘sanity checks’ to make sure the argument has not become trivialized.28 lemma T rue nitpick[satisfy, user-axioms] oops — Model found : axioms are consistent lemma ∀ x. E! x nitpick[user-axioms] oops — Countermodel found : necessitism is not valid lemma ϕ → ϕ nitpick[user-axioms] oops — Countermodel found : modal collapse is not valid Since Nitpick could not find a countermodel for C10, we have enough confidence in its validity to ask another automated reasoning tool: Isabelle’s Sledgehammer (Blanchette et al. 2013) to search for a proof. theorem C10: ∃ A x. Godlike x using Existence-def Necessary-def abstractness-essential concrete-exist P 2 C1 B-axiom by meson Sledgehammer is able to find a proof relying on all premises but the two modal axioms T and IV. Thus, by the end of this series of iterations, we have seen that Lowe’s modal ontological argument depends for its validity on three unstated (i.e. implicit) premises: the essentiality of abstractness, the existence-entailing nature of concreteness, and the modal axiom B (ϕ → ♦ϕ). Moreover, we shed some light on the meaning of the concepts of abstractness and concreteness, as we disclose further premises which shape their inferential role in the argument.

28 These

checks are constantly carried out after postulating axioms for every iteration, so we won’t mention them anymore.

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12.3.3 Third Iteration Series: Validating the Argument II In this iteration series we want to explore the critical potential of computational hermeneutics. In this slightly simplified variant (without the implicit premises stated in the previous version), premises P1 to P5 remain unchanged, while none of the last three premises (P6 to P8) show up anymore. Those last premises have been introduced by Lowe in order to interrelate the concepts of explanation and dependence in such a way that they play somewhat opposite roles, without one being the inverse of the other. Nonetheless, we will go all the way and assume that explanation and dependence are indeed inverse relations, for we want to understand how the interrelation of these two concepts affects the validity of the argument. axiomatization where dep-expl-inverse: ∀ x y. y explains x ↔ x dependsOn y Let us first prove the relevant partial conclusions. theorem C1: ∀ A x. Abstract x → (∃ y. Concrete y ∧ x dependsOn y) using P 3 P 4 by blast theorem C5: ∃ A x. Concrete x using P 2 P 3 P 4 by blast theorem C7: ∀ A x. (Necessary x ∧ Abstract x) → Explained x using Explained-def P 3 P 4 dep-expl-inverse by meson However, the conclusion C10 is still countersatisfiable, as shown by Nitpick. theorem C10: ∃ A x. Godlike x nitpick[user-axioms] oops — Countermodel found Next, let us try assuming a stronger modal logic. We can do this by postulating further modal axioms using the Sahlqvist correspondence and asking Sledgehammer to find a proof. Sledgehammer is in fact able to find a proof for C10 which only relies on the modal axiom T (ϕ → ϕ). axiomatization where T -axiom: ref lexive R and — ϕ → ϕ B-axiom: symmetric R and — ϕ → ♦ϕ I V -axiom: transitive R — ϕ → ϕ theorem C10: ∃ A x. Godlike x using Contingent-def Existence-def P 2 P 3 P 4 P 5 dep-expl-inverse T -axiom by meson In this series of iterations we have verified a modified version of the original argument by Lowe. Our understanding of the concepts of ontological dependence and metaphysical explanation (in the context of Lowe’s argument) has changed after the introduction of an additional axiom constraining both: they are now inverse

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relations. This new understanding of the inferential role of the above concepts of dependence and explanation has been reached on the condition that the ontological argument, as stated in natural language, must hold (in accordance to the principle of charity). Depending on our stance on this matter, we may either feel satisfied with this result or want to consider further alternatives. In the former case we would have reached a state of reflective equilibrium. In the latter we would rather carry on with our iterative process in order to further illuminate the meaning of the expressions involved in this argument.

12.3.4 Fourth Iteration Series: Simplifying the Argument After some further iterations we arrive at a new variant of Lowe’s argument: Premises P1 to P4 remain unchanged and a new premise D5 (“x depends for its existence on y := necessarily, x exists only if y exists”) is added. D5 corresponds to the ‘definition’ of ontological dependence as put forth by Lowe in his article (though only for illustrative purposes). As mentioned before, this purported definition was never meant by him to become part of the argument. Nevertheless, we show here how, by assuming the left-to-right direction of this definition, we get in a position to prove the main conclusions without any further assumptions. axiomatization where D5: ∀ A x y. x dependsOn y → (E! x → E! y) theorem C1: ∀ A x. Abstract x → (∃ y. Concrete y ∧ x dependsOn y) using P 3 P 4 by meson theorem C5: ∃ A x. Concrete x

using P 2 P 3 P 4 by meson

theorem C10: ∃ A x. Godlike x using Necessary-def P 2 P 3 P 4 D5 by meson In this variant, we have been able to verify the conclusion of the argument without appealing to the concept of metaphysical explanation. We were able to get by with just the concept of ontological dependence by explicating it in terms of existence and necessity (as suggested by Lowe). As a side note, we can also prove that the original premise P5 (“No contingent being can explain the existence of a necessary being”) directly follows from D5 by redefining metaphysical explanation as the inverse relation of ontological dependence. abbreviation explanation::(e⇒e⇒wo) (infix explains) where y explains x ≡ x dependsOn y lemma P 5: ¬(∃ A x. ∃ A y. Contingent y ∧ Necessary x ∧ y explains x) using Necessary-def Contingent-def D5 by meson

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In this series of iterations we have reworked Lowe’s argument so as to get rid of the somewhat obscure concept of metaphysical explanation, thus simplifying the argument. We also got some insight into Lowe’s concept of ontological dependence vis-à-vis its inferential role in the argument (by axiomatizing its relation with the concepts of existence and necessity in D5). There are still some interesting issues to consider. Note that the definitions of existence and being-dependent (axioms “Existence-def” and “Dependent-def” respectively) are not needed in any of the highly optimized proofs found by our automated tools. This raises some suspicions concerning the role played by the existence predicate in the definitions of necessariness and contingency, as well as putting into question the need for a definition of being-dependent linked to the ontological dependence relation. We will see in the following section that our suspicions are justified and that this argument can be dramatically simplified.

12.3.5 Fifth Iteration Series: Arriving at a Non-Modal Argument In the next iterations, we want to explore once again the critical potential of computational hermeneutics by challenging another of the author’s claims: that this argument is a modal one. A new simplified version of Lowe’s argument is obtained after abandoning the concept of existence altogether and redefining necessariness and contingency accordingly. As we will see, this variant is actually non-modal and can be easily formalized in first-order predicate logic. A more literal reading of Lowe’s article has suggested a simplified formalization, in which necessariness and contingency are taken as complementary predicates. According to this, our domain of discourse becomes divided in four main categories, as exemplified in the table below.29

Necessary Contingent

Abstract Numbers Fiction

Concrete God Stuff

consts Necessary::e⇒wo abbreviation Contingent::e⇒wo where Contingent x ≡ ¬(Necessary x) consts Concrete::e⇒wo abbreviation Abstract::e⇒wo where Abstract x ≡ ¬(Concrete x) 29 As

Lowe explains in the article, “there is no logical restriction on combinations of the properties involved in the concrete/abstract and the necessary/contingent distinctions. In principle, then, we can have contingent concrete beings, contingent abstract beings, necessary concrete beings, and necessary abstract beings.”

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abbreviation Godlike::e⇒w⇒bool where Godlike x≡ Necessary x ∧ Concrete x consts dependence::e⇒e⇒wo (infix dependsOn) abbreviation explanation::(e⇒e⇒wo) (infix explains) where y explains x ≡ x dependsOn y As shown below, we can even define being-dependent as a primitive predicate (i.e. bearing no relation to ontological dependence) and still be able to validate the argument. Being-independent is defined as the negation of being-dependent. consts Dependent::e⇒wo abbreviation I ndependent::e⇒wo where I ndependent x ≡ ¬(Dependent x) By taking, once again, metaphysical explanation as the inverse relation of ontological dependence and by assuming premises P2 to P5 we can prove conclusion C10. axiomatization where P 2: ∃ x. Necessary x ∧ Abstract x and P 3: ∀ x. Abstract x → Dependent x and P 4: ∀ x. Dependent x → (∃ y. I ndependent y ∧ x dependsOn y) and P 5: ¬(∃ x. ∃ y. Contingent y ∧ Necessary x ∧ y explains x) theorem C10: ∃ x. Godlike x using P 2 P 3 P 4 P 5 by blast Note that, in the axioms above, all restricted (actualist) quantifiers have been changed into unrestricted (possibilist) quantifiers, following the elimination of the concept of existence from our argument: Our quantifiers now range over all beings, because all beings exist. Also note that modal operators have disappeared; thus, this new variant is directly formalizable in classical first-order logic.

12.3.6 Sixth Iteration Series: Modified Modal Argument I In the following two series of iterations, we want to illustrate the use of the computational hermeneutics approach in those cases where we must start our interpretive endeavor with no explicit understanding of the concepts involved. In such cases, we start by taking all concepts as primitive without stating any definition explicitly. We will see how we gradually improve our understanding of these concepts in the iterative process of adding and removing axioms, thus framing their inferential role in the argument. consts Concrete::e⇒wo consts Abstract::e⇒wo consts Necessary::e⇒wo

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consts Contingent::e⇒wo consts dependence::e⇒e⇒wo (infix dependsOn) consts explanation::e⇒e⇒wo (infix explains) consts Dependent::e⇒wo abbreviation I ndependent::e⇒wo where I ndependent x ≡ ¬(Dependent x) In order to honor the original intention of the author, i.e., providing a modal variant of St. Anselm’s ontological argument, we are required to make a change in Lowe’s original formulation. In this variant we will restate the expressions “necessary abstract” and “necessary concrete” as “necessarily abstract” and “necessarily concrete” respectively. With this new adverbial reading we are no longer talking about the concept of necessariness, but of necessity instead, so we use the modal box operator () for its formalization. It can be argued that in this variant we are not concerned with the interpretation of the original natural-language argument anymore. We are rather interested in showing how the computational hermeneutics method can go beyond simple interpretation and foster a creative approach to assessing and improving philosophical arguments. Premise P1 now reads: “God is, by definition, a necessarily concrete being.” abbreviation Godlike::e⇒wo where Godlike x ≡ Concrete x Premise P2 reads: “Some necessarily abstract beings exist”. The rest of the premises remains unchanged. axiomatization where P 2: ∃ x. Abstract x and P 3: ∀ x. Abstract x → Dependent x and P 4: ∀ x. Dependent x → (∃ y. I ndependent y ∧ x dependsOn y) and P 5: ¬(∃ x. ∃ y. Contingent y ∧ Necessary x ∧ y explains x) Without postulating any additional axioms, C10 (“A necessarily concrete being exists”) can be falsified by Nitpick. theorem C10: ∃ x. Godlike x nitpick oops — Countermodel found An explication of the concepts of necessariness, contingency and explanation is provided below by axiomatizing their interrelation to other concepts. We will now regard necessariness as being necessarily abstract or necessarily concrete, and explanation as the inverse relation of dependence, as before. axiomatization where Necessary-expl: ∀ x. Necessary x ↔ (Abstract x ∨ Concrete x) and Contingent-expl: ∀ x. Contingent x ↔ ¬Necessary x and Explanation-expl: ∀ x y. y explains x ↔ x dependsOn y Without any further constraints, C10 becomes again falsified by Nitpick.

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theorem C10: ∃ x. Godlike x nitpick oops — Countermodel found We postulate further modal axioms (using the Sahlqvist correspondence) and ask Isabelle’s Sledgehammer tool for a proof. Sledgehammer is able to find a proof for C10 which only relies on the modal axiom T (ϕ → ϕ). axiomatization where T -axiom: ref lexive R and — ϕ → ϕ B-axiom: symmetric R and — ϕ → ♦ϕ I V -axiom: transitive R — ϕ → ϕ theorem C10: ∃ x. Godlike x using Contingent-expl Explanation-expl Necessary-expl P 2 P 3 P 4 P 5 T -axiom by metis

12.3.7 Seventh Iteration Series: Modified Modal Argument II As in the previous variant, we will illustrate here how the meaning (as inferential role) of the expressions involved in the argument gradually becomes explicit in the process of axiomatizing further constraints. We follow on with the adverbial reading of the expression “necessary” but provide an improved explication of necessariness (and contingency). We think that this explication, in comparison to the previous one, better fits our intuitive pre-understanding of the concept of being a necessary (or contingent) being. Thus, we will now regard necessariness as being necessarily abstract or concrete. (As before, we regard here metaphysical explanation as the inverse of the ontological dependence relation.) axiomatization where Necessary-expl: ∀ x. Necessary x ↔ (Abstract x ∨ Concrete x) and Contingent-expl: ∀ x. Contingent x ↔ ¬Necessary x and Explanation-expl: ∀ x y. y explains x ↔ x dependsOn y These constraints are, however, not enough to ensure the argument’s validity, as confirmed by Nitpick. theorem C10: ∃ x. Godlike x nitpick oops — Countermodel found After some iterations, we see that, by giving a more satisfactory explication of the concept of necessariness, we are also required to (i) assume the essentiality of abstractness (as we did in a former iteration), and (ii) restrict the reachability relation by enforcing its symmetry (i.e. assuming the modal axiom B). axiomatization where abstractness-essential: ∀ x. Abstract x → Abstract x and B-Axiom: symmetric R — ϕ → ♦ϕ

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theorem C10: ∃ x. Godlike x using Contingent-expl Explanation-expl Necessary-expl P 2 P 3 P 4 P 5 abstractness-essential B-Axiom by metis In each of the previous versions we have seen how our understanding of the concepts of being-necessary (necessariness), being-contingent (contingency), explanation, dependence, abstractness, concreteness, etc. has gradually evolved thanks to the iterative holistic method made possible by the real-time feedback provided by Isabelle’s automated proving tools. We think that, after this last series of iterations, the use of the computational hermeneutics method has been illustrated adequately. We do not claim that this formalization of Lowe’s argument is its best or most adequate one; it is just a consequence of the path we have followed by coming up with new ideas and testing them with the help of automated tools. In our view, while the third variant may be the closest one to Lowe’s original formulation, it is this latter (seventh) variant the one which strikes the best balance between interpretation and critical assessment of this argument. We encourage the reader to continue with this process until arriving to his/her own reflective equilibrium (possibly by building upon our computerverified work (Fuenmayor and Benzmüller 2017) available at the Archive of Formal Proofs).30

12.4 Conclusion We have argued for the role of formal logic as an ars explicandi and the possibility of applying it to foster our understanding of rational arguments (in particular metaphysical and theological ones). We understand the give-and-take process aiming at an adequate formal reconstruction of a natural-language argument in itself as a kind of interpretive endeavor. Moreover, we have argued that, by using automated reasoning technology to systematically explore the many different inferential possibilities latent in a formalized argument, we can make explicit the inferential role played by its constituent expressions and thus better understand their meaning in the given interpretation context. As a computer-assisted method, computational hermeneutics aims at complementing our human ingenuity with the data-processing power of modern computers and at using this synergy to make interpretation more effective. In a similar vein, we currently work on how to apply this approach in the computer science field of natural language understanding. Specifically, we want to tackle the problem of formalization: how to search methodically for the most appropriate logical form(s) of a given natural-language argument, by casting its individual statements into expressions of some sufficiently expressive logical language. Being able to 30 The Archive of Formal Proofs (www.isa-afp.org) is a collection of proof libraries, examples, and larger scientific developments, mechanically checked using the Isabelle proof assistant. It is organized in the way of a scientific journal and submissions are refereed.

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automatically extract a formal representation for some piece of natural-language discourse, by taking into account its holistically-determined logical location in a web of possible inferences, is an important step towards the deep semantic analysis and critical assessment of non-trivial natural-language discourse. Further applications in areas like knowledge/ontology extraction, semantic web and legal informatics are currently being contemplated. Acknowledgement Author “Christoph Benzmüller” was funded by Volkswagen Foundation.

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David Fuenmayor is a Ph.D. candidate at the Department of Mathematics and Computer Science at Freie Universität Berlin (Germany), where he also carried out undergraduate studies in Philosophy and Anthropology (2018). David’s previous studies include an engineer’s degree (2009) and M.Sc. (2012) in Mechatronics from the National University of Colombia and the Karlsruhe University of Applied Sciences (Germany), respectively. Christoph Benzmüller is a professor in artificial intelligence/computer science and mathematics at Freie Universität Berlin (Germany). He is also a visiting scholar of the University of Luxembourg (Luxembourg). Christoph’s prior research institutions include the universities of Stanford (USA), Cambridge, Birmingham, Edinburgh (all UK), the Saarland (Germany), and CMU (USA). He received his PhD (1999) and his Habilitation (2007) from Saarland University, his PhD research was partly conducted at CMU. In 2012, he was awarded with a Heisenberg Research Fellowship of the German National Research Foundation (DFG).

Chapter 13

A Mechanically Assisted Examination of Vacuity and Question Begging in Anselm’s Ontological Argument John Rushby

13.1 Introduction I assume readers have some familiarity with St. Anselm’s Eleventh Century Ontological Argument for the existence of God (Anselm 1077); a simplified translation from the original Latin of Anselm’s Proslogion is given in Fig. 13.1, with some alternative readings in square parentheses. This version of the argument appears in Chapter II of the Proslogion; another version appears in Chapter III and speaks of the necessary existence of God. Many authors have examined the Argument, in both its forms; in recent years, most begin by rendering it in modern logic, employing varying degrees of formality. The Proslogion II argument is traditionally rendered in first-order logic while propositional modal logic is used for that of Proslogion III. More recently, higher-order logic and quantified modal logic have been applied to the argument of Proslogion II. My focus here is the Proslogion II argument, represented completely formally in first- or higher-order logic, and explored with the aid of a mechanized verification system. Elsewhere, I use a verification system to examine renditions of the argument in modal logic (Rushby 2019b), and also the argument of Proslogion III (Rushby 2019a).

This research was partially supported by SRI International. J. Rushby () Computer Science Laboratory, SRI International, Menlo Park, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_13

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We can conceive of [something/that] than which there is no greater If that thing does not exist in reality, then we can conceive of a greater thing—namely, something [just like it] that does exist in reality Thus, either the greatest thing exists in reality or it is not the greatest thing Therefore the greatest thing exists in reality [That’s God] Fig. 13.1 The Ontological Argument

Verification systems are tools from computer science that are generally used for exploration and verification of software or hardware designs and algorithms; they comprise a specification language, which is essentially a rich (usually higherorder) logic, and a collection of powerful deductive engines (e.g., satisfiability solvers for combinations of theories, model checkers, and automated and interactive theorem provers). I have previously explored renditions of the Argument due to Oppenheimer and Zalta (1993) and Eder and Ramharter (2015) using the PVS verification system (Rushby 2013, 2016), and those provide the basis for the work reported here. Benzmüller and Woltzenlogel-Paleo (2014) have likewise explored modal arguments due to Gödel and Scott using the Isabelle and Coq verification systems. Mechanized analysis confirms the conclusions of most earlier commentators: the Argument is valid. Attention therefore focuses on the premises and their interpretation. The premises are a priori (i.e., armchair speculation) and thus not suitable for empirical confirmation or refutation: it is up to the individual reader to accept or deny them. We may note, however, that the premises are consistent (i.e., they have a model), and this is among the topics that I previously subjected to mechanized examination (Rushby 2013) (as a byproduct, this examination demonstrates that the Argument does not compel a theological interpretation: in the exhibited model, that “than which there is no greater” is the number zero). The Argument has been a topic of enduring fascination for nearly a 1000 years; this is surely due to its derivation of a bold conclusion from unexceptionable premises, which naturally engenders a sense of disquiet: Russell (2013, page 472) opined “The Argument does not, to a modern mind, seem very convincing, but it is easier to feel that it must be fallacious than it is to find out precisely where the fallacy lies”. Many commentators have sought to identify a fallacy in the Argument or its interpretation (e.g., Kant famously denied it on the basis that “existence is not a predicate”). One direction of attack is to claim that the Argument “begs the

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question;”1 that is, it essentially assumes what it sets out to prove (Rowe 1976; Walton 1978). This is the primary charge that I examine here. Begging the question has traditionally been discussed in the context of informal or semi-formal argumentation and dialectics (Barker 1976, 1978; Sanford 1977; Walton 1991, 1994, 2006), where it is debated whether arguments that beg the question should be considered fallacious, or valid but unpersuasive, or may even be persuasive. Here, we examine question begging in the context of fully formal, mechanically checked proofs. My purpose is to provide techniques that can identify potential question begging in a systematic and fairly unequivocal manner. I do not condemn the forms of question begging that are identified; rather, my goal is to highlight them so that readers can make up their own minds and can also use these techniques to find other cases. A secondary charge that I will consider is one of vacuity: most of the formalized arguments examined here entail variants that apply no interpretation to “than which there is no greater.” We are therefore free to apply any interpretation and in this way can reproduce the “lost island” parody that Gaunilo (1079) used to claim refutation of the original argument. The chapter is structured as follows. In the next section, I introduce a strict definition of “begging the question” and show that a rendition of the Argument due to Oppenheimer and Zalta (1993) is vulnerable to this charge. Oppenheimer and Zalta use a definite description (i.e., they speak of “that than which there is no greater”) and require an additional assumption to ensure this is well-defined. Eder and Ramharter (2015, Section 2.3) argue that Anselm did not intend this interpretation (i.e., requires only “something than which there is no greater”) and therefore dispense with the additional assumption of Oppenheimer and Zalta. In Sect. 13.3, I show that this version of the argument does not beg the question under the strict definition, but that it does so under a plausible weakening. I then turn to the topic of vacuity and, in Sect. 13.4, show that this version of the argument has a variant that applies no interpretation to “than which there is no greater” and is thus vulnerable to Gaunilo’s refutation; I argue that the original formulation shares this defect. In Sect. 13.5, I consider an alternative premise due to Eder and Ramharter and show that this does not beg the question under either of the previous interpretations, but I argue that it is at least as questionable as the premise that it replaces because it so perfectly discharges the main step of the proof that it seems reverse-engineered. I suggest a third interpretation for “begging the question” that matches this case. In Sect. 13.6, I consider the higher-order treatment of Eder and Ramharter (2015, Section 3.3) and a variant derived from Campbell (2018); these formalized proofs are more complicated than those of the first-order treatments but I show how the third interpretation for “begging the question” applies to them. I also show that all these versions have vacuous variants. In Sect. 13.7, I compare my interpretations to existing, mainly informal, accounts of what it means to “beg the

1 This phrase is widely misunderstood to mean “to invite the question.” Its use in logic derives from

medieval translations of Aristotle, where the Latin form Petitio Principii is also employed.

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question.” Finally, in Sect. 13.8, I summarize and show that all the formalizations examined can be generated as elaborations of a manifestly vacuous and circular starting argument.

13.2 Begging the Question: Strict Case “Begging the question” is a form of circular reasoning in which we assume what we wish to prove. It is generally discussed in the context of informal argumentation where the premises and conclusion are expressed in natural language. In such cases, the question-begging premise may state the same idea as the conclusion, but in different terms, or it may contain superfluous or even false information, and there is much literature on how to diagnose and interpret such cases (Barker 1976, 1978; Sanford 1977; Walton 1994, 1991, 2006). That is not my focus. I am interested in formal, deductive arguments, and in criteria for begging the question that are themselves formal. Now, deductive proofs do not generate new knowledge—the conclusion is always implicit in the premises—but they can generate surprise and they can persuade; I propose that criteria for question begging should focus on the extent to which either the conclusion or its proof are “so directly” represented in the premises as to vitiate the hope of surprise or persuasion. The most elementary instance is surely when one, or a collection, of the premises is equivalent to the conclusion. But if some of the premises are equivalent to the conclusion, what are the other premises for? Certainly we must need all the premises to deduce the conclusion (else we can eliminate some of them); thus we surely need all the premises before we can establish that some of them are equivalent to the conclusion. Hence, criteria for begging the question should apply after we have accepted the other premises. Thus, if C is our conclusion, Q our “questionable” premise (which may be a conjunction of simpler premises) and P our other premises, then Q begs the question in this elementary or strict sense if C is equivalent to Q, assuming P : i.e., P C = Q. Of course, this means we can prove C using Q: P , Q C, and we can also do the reverse: P , C Q. Figure 13.2 presents Oppenheimer and Zalta (1993) treatment of the Ontological Argument formalized in PVS, using the notation of Eder and Ramharter (2015, Section 3.2). I will not describe this formal specification in detail, since it is done at tutorial level elsewhere (Rushby 2013), but I will explain the basic language and ideas. Briefly, the specification language of PVS is a strongly typed higher-order logic with predicate subtypes. This example uses only first order but does make essential use of predicate subtypes and the proof obligations that they can incur (Rushby et al. 1998). The uninterpreted type beings is used for those things that are “in the understanding”. Note that a question mark at the end of an identifier is merely a convention to indicate predicates (which in PVS are simply functions with return type bool). A predicate in parentheses denotes the corresponding predicate subtype, so that > is an uninterpreted relation on beings that satisfies the predicate trichotomous?, which is part of the “Prelude” of standard theories built in

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OandZ: THEORY BEGIN beings: TYPE x, y: VAR beings >: (trichotomous?[beings]) God?(x): bool = NOT EXISTS y: y > x re?(x): bool ExUnd: AXIOM EXISTS x: God?(x) Greater1: AXIOM FORALL x: (NOT re?(x) => EXISTS y: y > x) God_re: THEOREM re?(the(God?)) %---------------- Question Begging Analysis ---------------------Greater1_circ: THEOREM God_re IMPLIES Greater1 END OandZ

Fig. 13.2 Oppenheimer and Zalta’s Treatment, in PVS

to PVS.2 PVS generates a proof obligation (not shown here) called a Typecheck Correctness Condition, or TCC, to ensure such a relation exists, which we discharge by exhibiting the everywhere true relation. The predicate God? recognizes those beings “than which there is no greater”; the axiom ExUnd asserts the existence of at least one such being; the(God?) is a definite description that identifies this being. PVS generates a TCC (not shown here) to ensure this being exists and is unique (this is required by the predicate subtype used in the definition of the, which is part of the PVS Prelude), and ExUnd and the trichotomy of > are used to discharge this obligation. The uninterpreted predicate re? identifies those beings that exist “in reality” and the axiom Greater1 asserts that if a being does not exist in reality, then there is a greater being. Note that the string IMPLIES and the symbol => are entirely equivalent in PVS (also the string AND and symbol &); we use whichever seems most readable in its context. The theorem God_re asserts that the being identified by the definite description the(God?) exists in reality. The PVS proof of this theorem is accomplished by the following commands. (typepred "the(God?)")

2 Trichotomy

(use "Greater1") (grind)

PVS Proof

is the condition FORALL x, y: x > y OR y > x OR x = y.

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These commands invoke the type associated with the(God?) (namely that it satisfies the predicate God?), the premise Greater1, and then apply the standard automated proof strategy of PVS, called grind. Almost all the proofs mentioned subsequently are similarly straightforward and we do not reproduce them in detail. As first noted by Garbacz (2012), the premise Greater1 begs the question under the other assumptions of the formalization. We state the key implication as Greater1_circ (PVS Version 7 allows formula names to be used in expressions as shorthands for the formulas themselves) and prove it as follows. (expand "God_re") (expand "Greater1") (typepred "the(God?)") (grind :polarity? t) (inst 1 "x!1") (typepred ">") (grind)

PVS proof

The first two steps expand the formula names to the formulas they represent, the typepred steps introduce the predicate subtypes associated with their arguments (namely, that the(God?) satisfies God? and that > is trichotomous) and the other steps perform quantifier reasoning and routine deductions. Given that we have proven God_re from Greater1 and vice-versa, we can easily prove they are equivalent. Thus, in the definition of “begging the question” given earlier, C here is God_re, Q is Greater1 and P is the rest of the formalization (i.e., ExUnd, the definition of God? and the definite description the(God?), and the predicate subtype trichotomous? asserted for >). Notice that we can also prove the premise ExUnd from the conclusion God_re, so it looks as if ExUnd begs the question, too. However, ExUnd is used to discharge the TCC that ensures the definite description operator the is used appropriately. Hence, ExUnd is strictly prior to God_re (because the specification is not accepted until its TCCs are discharged), thereby breaking the circularity. Hence, ExUnd cannot be considered to beg the question in this case.

13.3 Begging the Question: Weaker Case Eder and Ramharter (2015, Section 2.3) claim that Anselm’s Proslogion does not employ a definite description and that a correct reading is “something than which there is no greater.” A suitable modification to the previous PVS theory is shown in Fig. 13.3; the differences are that > is now an unconstrained relation on beings, and the conclusion is restated as the theorem God_re_alt. As before, this theorem is easily proved from the premises ExUnd and Greater1 and the definition of God?. And also as before, the premise ExUnd can be proved from the conclusion God_re_alt, so this premise strictly begs the question (unlike the version of Fig. 13.2, there are no TCCs here to break the circularity). However, Greater1

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EandR1: THEORY BEGIN beings: TYPE x, y: VAR beings >(x, y): bool God?(x): bool = NOT EXISTS y: y > x re?(x): bool ExUnd: AXIOM EXISTS x: God?(x) Greater1: AXIOM FORALL x: (NOT re?(x) => EXISTS y: y > x) God_re_alt: THEOREM EXISTS x: God?(x) AND re?(x) %---------------- Question Begging Analysis ---------------------Greater1_circ1: THEOREM trichotomous?(>) IMPLIES God_re_alt => Greater1 Greater1_circ2: THEOREM (FORALL x, y: God?(x) => x>y or x=y) IMPLIES God_re_alt => Greater1 END EandR1

Fig. 13.3 Eder and Ramharter’s First Order Treatment, in PVS

is no longer strictly begging because it cannot be proved from the conclusion God_re_alt. We can observe, however, that this specification of the Argument is very austere and imposes no constraints on the relation >; in particular, it could be an entirely empty relation. We demonstrate this in the theory interpretation EandR1interp shown in Fig. 13.4, where beings are interpreted as natural numbers, all beings exist in reality, and none are > than any other; thus, any natural number satisfies God?. PVS generates proof obligations (not shown here) to ensure the axioms of the theory EandR1 are theorems under this interpretation, and these are trivially true. Such a model seems contrary to the intent of the Argument: surely it is not intended that something than which there is no greater is so because nothing is greater than anything else. So we should require some minimal constraint on > to eliminate such impoverished models. A plausible constraint is that > be trichotomous; if we add this condition, as in Greater1_circ1, then the premise Greater1 can again be proved from the conclusion God_re_alt. A weaker condition is to require only that beings satisfying the God? predicate should stand in the > relation to others; this is stated in Greater1_circ2 and is also sufficient to prove Greater1 from God_re_alt.

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EandR1interp: THEORY BEGIN IMPORTING EandR1{{ beings := nat, re? := LAMBDA (x: nat): TRUE, > := LAMBDA (x, y: nat): FALSE }} AS model END EandR1interp

Fig. 13.4 The Empty Model for Eder and Ramharter’s First Version

In terms of the abstract formulation given at the beginning of Sect. 13.2, what we have here is that the conclusion C can be proved using the questionable premise Q: P , Q C, but not vice versa. However, if we augment the other premises P by adding some P2 , then we can indeed prove Q: P , P2 , C Q, and also the equivalence of C and Q: P , P2 C = Q. Thus, Q does not beg the question C under the original premises P but does do so under the augmented premises P , P2 . In this case, we will say that Q weakly begs the question, where P2 determines the “degree” of weakness. In this example, the question begging premise fails our definition of strict begging because it is used in an impoverished theory, and weak begging compensates for that. Another way a premise Q can escape strict begging is by being stronger than necessary and one way to compensate for that is to strengthen the conclusion by conjoining some S so that P , (C ∧ S) Q and P , Q (C ∧ S). However, it may be difficult to satisfy both of these simultaneously and the first is equivalent to weak begging with P2 = S; hence, we prefer the original, more versatile, notion of weak begging. The rationale for introducing weak begging is that it exposes strict begging that is otherwise masked by an impoverished theory or a strong premise. But with enough deductive power we can always construct a P2 and thereby claim weak begging; the question is whether this additional premise is plausible and innocuous in the intended interpretation, and this is a matter for human judgment.

13.4 Trivializing the Argument, and Gaunilo’s Refutation Notice that the right side of the implication in Greater1 of Fig. 13.3 is equivalent to NOT God?(x), so that Greater1 can be rewritten as Greater1_triv, as shown below. Greater1_triv: LEMMA FORALL x: (NOT re?(x) => NOT God?(x))

PVS fragment

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But now we can prove the theorem without opening the definition of God?. We do this as follows. (lemma "ExUnd") (lemma "Greater1_triv") (grind :exclude "God?")

PVS proof

Here, we install the premises ExUnd and Greater1_triv, and then invoke the general purpose strategy grind, instructing it not to open the definition of God?. Of course, we hardly need mechanized theorem proving to verify this: ExUnd says there is some being satisfying God?, the contrapositive of Greater1_triv says such a being satisfies re?, and we are done. This is not only a trivial argument, but it does not depend on the meaning attached to “God” and so we could replace it by any other term and interpretation. In particular, we could substitute “the most perfect island” and thereby reproduce the “lost island” parody by Gaunilo, a contemporary of Anselm, who used the form of Anselm’s argument to establish the (absurd) existence of that most perfect island (Gaunilo 1079). Anselm and other authors defend the informal Proslogion II argument against Gaunilo’s parody,3 but the formalization of Fig. 13.3 with Greater1_triv is indefensible, since it is true for all interpretations. We will say that such an argument is vacuous. For later reference, we show this vacuous form of the argument in Fig. 13.5; Greater1_vac is the contrapositive of Greater1_triv. We should now ask whether the vacuity of Fig. 13.5 and of Fig. 13.3 with Greater1_triv also applies to the original specification with Greater1. My opinion is that it does, because we can systematically transform Fig. 13.5 into

Vacuous: THEORY BEGIN beings: TYPE x, y: VAR beings God?(x): bool re?(x): bool ExUnd: AXIOM EXISTS x: God?(x) Greater1_vac: AXIOM FORALL x: God?(x) => re?(x) God_re_alt: THEOREM EXISTS x: God?(x) AND re?(x) END Vacuous

Fig. 13.5 A vacuous version of the argument in PVS

3 One approach

asserts that a contingent object, such as an island, can always be improved and thus could never be one than which there is no greater.

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Fig. 13.3: we simply apply an interpretation to God? and open up its appearance in Greater1_vac to reveal that interpretation, then take the contrapositive and thereby obtain Greater1. It is irrelevant what the interpretation is, so the symbol > and its reading as “greater” are entirely specious, and we cannot attach any belief to Greater1 once we see this derivation. I will return to this topic in the conclusion, Sect. 13.8. It is worth noting that Oppenheimer and Zalta’s specification of Fig. 13.2 cannot be reduced to a similarly vacuous form because it becomes impossible to discharge the TCC proof obligation that is required to ensure the definite description is unique.

13.5 Indirectly Begging the Question Eder and Ramharter (2015, Section 3.2) consider Greater1 an unsatisfactory premise because it does not express “conceptions presupposed by the author” (i.e., Anselm) and says nothing about what it means to be greater other than the contrived connection to exists in reality. They propose an alternative premise Greater2, which is shown in Fig. 13.6. This theory is the same as that of Fig. 13.3, except that Greater2 is substituted for Greater1, and a new premise Ex_re is added.

EandR2: THEORY BEGIN beings: TYPE x, y: VAR beings >(x, y): bool God?(x): bool = NOT EXISTS y: y > x re?(x): bool ExUnd: AXIOM EXISTS x: God?(x) Ex_re: AXIOM EXISTS x: re?(x) Greater2: AXIOM FORALL x, y: (re?(x) AND NOT re?(y) => x > y) God_re_alt: THEOREM EXISTS x: God?(x) AND re?(x) END EandR2

Fig. 13.6 Eder and Ramharter’s adjusted first order treatment, in PVS

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Before we proceed to examine question begging in this version, we can note that the right side of the implication in Greater2 is equivalent to NOT God?(y), so the premise can be rewritten as follows. PVS fragment Greater2_triv: LEMMA FORALL x, y: (re?(x) AND NOT re?(y) => NOT God?(y))

This variant premise, plus ExUnd and Ex_re can be used to prove the conclusion God_re_alt without opening the definition of God?. Thus, the theory with Greater2_triv in place of Greater2 is vacuous and, by the same reasoning as in Sect. 13.4, I argue that the original Fig. 13.6 is too. Next, we return to the original premises with ExUnd, Ex_re and Greater2, and note that these also prove the conclusion God_re_alt and that ExUnd and Ex_re strictly beg the question. These three premises also entail Greater1 of Fig. 13.3, so there is circumstantial evidence that Greater2 is question begging. However, it is not possible to prove Greater2 from God_re_alt and the other premises, nor have I found a plausible augmentation to the premises that enables this. Thus, it seems that Greater2 does not beg the question under our current definitions, neither strictly nor weakly. However, when constructing a mechanically checked proof of God_re_alt using Greater2 I was struck how neatly the premise exactly fits the requirement of the interactive proof at its penultimate step. To see this, observe the PVS sequent shown below; we arrive at this point following a few straightforward steps in the proof of God_re_alt. First, we introduce the premises ExUnd and Ex_re, expand the definition of God?, and perform a couple of routine steps of Skolemization, instantiation, and propositional simplification. God_re_alt :

PVS Sequent A

[-1] re?(x!1) |------{1} x!1 > x!2 [2] re?(x!2)

PVS represents its current proof state as the leaves of a tree of sequents (here there is just one leaf); each sequent has a collection of numbered formulas above and below the |--- turnstile line; the interpretation is that the conjunction of formulas above the line should entail the disjunction of those below. Bracketed numbers on the left are used to identify the lines, and braces (as opposed to brackets) indicate this line is new or changed since the previous proof step. Terms such as x!1 are Skolem constants. PVS eliminates top level negations by moving their formulas to the other side of the turnstile. Thus the sequent above is equivalent to the following.

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Variant Sequent

[-1] re?(x!1) [2] NOT re?(x!2) |------{1} x!1 > x!2

We can read this as re?(x!1) AND NOT re?(x!2) IMPLIES x!1 > x!2

and then observe that Greater2 is its universal generalization. PVS has capabilities that help mechanize this calculation. If we ask PVS to generalize the Skolem constants in the original sequent A, it gives us the formula FORALL (x_1, x_2: beings): re?(x_2) IMPLIES x_2 > x_1 OR re? (x_1)

Renaming the variables and rearranging, this is FORALL (x, y: beings): (re?(x) AND NOT re?(y)) IMPLIES x > y

which is identical to Greater2. Thus, Greater2 corresponds precisely to the formula required to discharge the final step of the proof. I will say that a premise indirectly begs the question if it supplies exactly what is required to discharge a key step in the proof. Unless they are redundant or superfluous, all the premises to a proof will be essential to its success, so it may seem that any premise can be considered to indirectly beg the question. Furthermore, if we do enough deduction, we can often arrange things so that the final premise to be installed exactly matches what is required to finish the proof. My intent is that the criterion for indirect begging applies only when the premise in question perfectly matches what is required to discharge a key (usually final) step of the proof when the preceding steps have been entirely routine. It is up to the individual to decide what constitutes “routine” deduction; I include Skolemization, propositional simplification, definition expansion and rewriting, but draw the line at nonobvious quantifier instantiation. The current example does require quantifier instantiation: a few steps prior to Sequent A above, the proof state is represented by the following sequent. God_re_alt :

PVS Sequent

{-1} God?(x!1) {-2} re?(x!2) |------[1] EXISTS x: God?(x) AND re?(x)

The candidates for instantiating x are the Skolem constants x!1 or x!2. The correct choice is x!1 and I would allow this selection, or even some experimentation with different choices, within the “obvious” threshold, though others may disagree.

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I claim that the sequent constructed by the PVS prover following routine deductions is a good representation of our epistemic state after we have digested the other premises. If the questionable premise supplies exactly what is required to complete the proof from that point (by generalizing the sequent), then it cannot be understood independently and therefore satisfies the “epistemic” criterion for question begging (Walton 1994), to be discussed in Sect. 13.7. Furthermore, its construction appears reverse-engineered and this eliminates any hope of surprise or persuasion and thereby satisfies another characteristic of question begging. My description of indirect begging is very operational and might seem tied to the particulars of the PVS prover, so we can seek a more abstract definition. After we have installed the other premises, the PVS sequent is a representation of P ⊃ C (we use ⊃ for material implication). The proof engineering that reveals Q indirectly to beg the question shows that Q is what is needed to make this a theorem, so

Q ⊃ (P ⊃ C). But more than this, it is exactly what is needed, so we could suppose Q = (P ⊃ C) and then take this as a definition of indirect begging. Notice that strict begging implies this definition, but not vice-versa. However, a difficulty with this definition is that the direction (P ⊃ C) ⊃ Q is generally stronger than can be proved. The proof engineering approach to indirect begging can be seen as an operational way to interpret and approximate this definition: we use deduction to simplify P ⊃ C and then ask whether Q is its universal generalization. In simple cases, the proof engineering approach is straightforward and makes good use of proof automation, but it may be difficult to apply in more complex proofs where a premise is employed as part of a longer chain of deductions. In the following section I show how careful proof structuring can, without undue contrivance, isolate the application of a premise and expose its question begging character.

13.6 Indirect Begging in More Complex Proofs In search of a more faithful reconstruction of Anselm’s Argument, Eder and Ramharter (2015, Section 3.3) observe that Anselm attributes properties to beings and that some of these (notably exists in reality) contribute to evaluation of the greater relation. They formalize this by hypothesizing some class P of “greatermaking” properties on beings and then define one being to be greater than another exactly when it has all the properties of the second, and more besides.4 This treatment is higher order because it involves quantification over properties, not merely individuals. This is seen in the definition of > in the PVS formalization 4 Eder

and Ramharter mistakenly state that this is a partial order, but it is not reflexive. In fact, it is a total order (i.e., irreflexive and transitive), but may be sparse or unconnected (i.e., not trichotomous). To see the latter point, suppose that all beings exist in reality, but have no other properties. Then no being is greater than any other, but each is something “than which there is no greater.”

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EandRho: THEORY BEGIN beings: TYPE x, y, z: VAR beings re?: pred[beings] P: set[ pred[beings] ] F: VAR (P) >(x, y): bool = (FORALL F: F(y) => F(x)) & (EXISTS F: F(x) AND NOT F(y)) God?(x): bool = NOT EXISTS y: y > x ExUnd: AXIOM EXISTS x: God?(x) Realization: AXIOM FORALL (FF:setof[(P)]): EXISTS x: FORALL F: F(x) = FF(F) God_re_ho: THEOREM member(re?, P) => EXISTS x: God?(x) AND re?(x) END EandRho

Fig. 13.7 Eder and Ramharter’s Higher Order Treatment, in PVS

of Eder and Ramharter’s higher order treatment shown in Fig. 13.7. Notice that P is a set (which is equivalent to a predicate in higher-order logic) of predicates on beings; in PVS a predicate in parentheses as in F: VAR (P) denotes the corresponding subtype, so that F is a variable ranging over the subsets of P. A tutorial-level description of this PVS formalization is provided elsewhere (Rushby 2016). Before examining question begging in this version, note that Realization can be used to prove the following premise. PVS fragment Greater_triv: LEMMA member(re? P) IMPLIES FORALL x: God?(x) => re?(x)

Given this and ExUnd, it is trivial to prove the conclusion God_re_ho without opening the definition of God?. Thus, the theory with Greater_triv in place of Realization is vacuous and, by similar reasoning to Sect. 13.4, we argue that the original Fig. 13.7 is as well. The strategy for proving God_re_ho in Fig. 13.7 is first to consider the being x introduced by ExUnd; if this being exists in reality, then we are done. If not, then we consider a new being that has exactly the same properties as x, plus existence in reality—this is attractively close to Anselm’s own strategy, which is to suppose that very same being can be (re)considered as existing in reality. In the PVS proof this is accomplished by the proof step

13 A Mechanically Assisted Examination of Vacuity and Question Begging. . .

(name "X" "choose! z: FORALL F: F(z) = (F(x!1) OR F=re?)")

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PVS Proof Step

which names X to be such a being. Here, x!1 is the Skolem constant corresponding to the x introduced by ExUnd and choose! is a “binder” derived from the PVS choice function choose, which is defined in the PVS Prelude. This X is some being that satisfies all the predicates of x!1, plus re?. Given this X, we can complete the proof, except that PVS generates the subsidiary TCC proof obligation shown below to ensure that the application of the choice function is well-defined (i.e., there is such an X). EXISTS (x: beings): (FORALL F: F(x) = (F(x!1) OR F = re?))

PVS TCC

This proof obligation requires us to establish that there is a being that satisfies the expression in the choose!; it is generated from the predicate subtype specified for the argument to choose.5 Eder and Ramharter provide the axiom Realization for this purpose; it states that for any collection of properties, there is a being that exemplifies exactly those properties and, when its variable FF is instantiated with the term { G: (P) | G(x!1) OR G=re? }, it provides exactly the expression above. In other words, Realization is a generalization of the formula required to discharge a crucial step in the proof. Thus, I claim that the premise Realization indirectly begs the question in this proof. This seems appropriate to me, because Realization says we can always “turn on” real existence and, taken together with ExUnd and the definition of >, this amounts to the desired conclusion, whose “hiding place” (see Sect. 13.7) is thereby revealed. An alternative and more common style of proof in PVS would invoke the premise Realization directly at the point where name and choose! are used in the proof described above. The direct invocation obscures the relationship between the formal proof and Anselm’s own strategy, and it also uses Realization as one step in a chain of deductions that masks its question begging character. Thus, use of name and choose! are key to revealing both the strategy of the proof and the question begging character of Realization. Note that the deductions prior to the name command, and those on the subsequent branch to discharge the TCC should be routine if Realization is to be considered indirectly question begging, but those on the other branch may be arbitrarily complex. 5 This

is similar to the proof obligation generated for the definite description used in Oppenheimer and Zalta’s rendition of Fig. 13.2: there, we had to prove that the predicate in the is uniquely satisfiable; here we need merely to prove that the predicate in choose! is satisfiable. The properties of the definite description, the choice function, and Hilbert’s ε are described and compared in our description of Oppenheimer and Zalta’s treatment (Rushby 2013).

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Campbell (2018) adopts some of Eder and Ramharter’s higher order treatment, but rejects Realization on the grounds that it is false. Observe that we could have incompatible properties6 and Realization would then provide the existence (in the understanding) of a being that exemplifies those incompatible properties, and this is certainly questionable. A better approach might be to weaken Realization to allow merely the addition of re? to the properties of some existing being. This is essentially the approach taken below. Campbell’s full treatment (Campbell 2018) differs from others considered here in that he includes more of Anselm’s presentation of the Argument (e.g., where he speaks of “the Fool”). The treatment shown in Fig. 13.8 is my simplified

Campbell: THEORY BEGIN beings: TYPE x, y, z: VAR beings re?: pred[beings] P: set[ pred[beings] ] F: var (P) >(x, y): bool = (FORALL F: F(y) => F(x)) & (EXISTS F: F(x) AND NOT F(y)) God?(x): bool = NOT EXISTS y: y > x ExUnd: AXIOM EXISTS x: God?(x) quasi_id(D: setof[(P)])(x,y: beings): bool = FORALL (F:(P)): NOT D(F) => F(x) = F(y) jre: setof[(P)] = singleton(re?) Weak_real: AXIOM NOT re?(x) => (EXISTS z: quasi_id(jre)(z, x) AND re?(z)) God_re_ho: THEOREM member(re?, P) => EXISTS x: God?(x) AND re?(x) END Campbell

Fig. 13.8 Simplified version of Campbell’s treatment, in PVS

6 Eder

and Ramharter are careful to require that all the greater-making properties are “positive” so directly contradictory properties are excluded, but we can have positive properties that are mutually incompatible (Himma 2005). Examples are being “perfectly just” and “perfectly merciful”: the first entails delivering exactly the “right amount” of punishment, while the latter may deliver less than is deserved.

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interpretation of Campbell’s approach, scaled back to resemble the other treatments considered here, and is based on discussions prior to publication of his book (Campbell 2016). Campbell adopts Eder and Ramharter’s higher order treatment, but replaces Realization by (in my interpretation) the axiom Weak_real which essentially states that if x does not exist in reality, then we can consider a being just like it that does. A being “just like it” is defined in terms of a predicate quasi_id introduced by Eder and Ramharter (2015, Section 3.3) and is true of two beings if they have the same properties, except possibly those in a given set D. Observe that the PVS specification writes this higher order predicate in Curried form. Here, D is always instantiated by the singleton set jre containing just re?, so we always use quasi_id(jre). A couple of routine proof steps bring us to the following sequent. God_re_ho : {-1} P(re?) |------[1] EXISTS y: y > x!1 [2] re?(x!1)

Our technique for discharging this is to instantiate formula 1 with a being just like x!1 that does exist in reality, which we name X. (name "X" "(choose! z: quasi_id(jre)(z, x!1) AND re?(z))")

The main branch of the proof then easily completes and we are left with the obligation to ensure that application of the choice function is well-defined. That is, we need to show EXISTS (z: beings): quasi_id(jre)(z, x!1) AND re?(z)

under the condition NOT re?(x!1). This is precisely what the premise Weak_real supplies, so we may conclude that this premise indirectly begs the question. We should also note that Weak_real can be used to prove Greater_triv, as discussed for Fig. 13.7, and is therefore vulnerable to the charge of vacuity. The higher order formalizations considered in this section have slightly longer and more complex proofs than those considered earlier. This means that the indirect question begging character of a particular premise may not be obvious if it occurs in the middle of a chain of proof steps. Use of the name and choose! constructs accomplishes two things: it highlights the strategy of the proof (namely, it identifies the attributes of the alternative being to consider if the first one does not exist in reality), and it isolates application of the questionable premise to a context where its indirect question begging character is revealed.

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13.7 Comparison with Informal Accounts of Begging the Question There are several works that examine the Ontological Argument against the charge that it begs the question. Some of these, including the present chapter, employ a “logical” interpretation for begging the question, which is to say they associate question begging with the logical form of the argument and not with the meaning attached to its symbols. Others employ a “semantical” interpretation and find circularities in the meanings of the concepts employed by the Argument prior to consideration of its logical form. Roth (1970), for example, observes that Anselm begins by offering a definition of God as that than which nothing greater can be conceived; Roth then claims that greatness already presupposes existence and is therefore question begging. McGrath (1990) criticizes Roth’s analysis and presents his own, which finds circularity in the relationship between possible and real existence. Devine (1975) (who was writing 15 years earlier than McGrath but is not cited by him) asks whether it is possible to use “God” in a true sentence without assuming His existence and concludes that it is indeed possible and thereby acquits the argument of this kind of circularity. All these considerations lie outside the scope considered here. We treat “greater than,” “real existence,” and any other required terms as uninterpreted constants, and we assume there is no conflict between the parts they play in the formalized Argument and the intuitive interpretations attached to them. We then ask whether the formalized argument begs the question in a logical sense. Many authors consider logical question begging in semi-formal arguments. Some consider a “dialectical” interpretation associated with the back and forth style of argumentation that dates to Aristotle’s original identification of the fallacy (as he thought of it), while others consider an “epistemic” interpretation in the context of standard deductive arguments. Walton (1994) outlines a history of analysis of begging the question, focusing on the dialectical interpretation, while Garbacz (2002) provides a formal account within this framework. Walton (2006) contends that the notion of question begging and the intellectual tools to detect it are similar in both the dialectical and epistemic interpretations, so I will focus on the epistemic case. The intuitive idea is that a premise begs the question epistemically when “the arguer’s belief in the premise is dependent on his or her reason to believe the conclusion” (Walton 2006, page 241). Several authors propose concrete definitions or methods for detecting epistemic question begging. Walton (2006), for example, recommends proof diagrams (as supported in the Araucaria system (Reed and Rowe 2004)) as a tool to represent the structure of informal arguments, and hence reveal question begging circularities. He illustrates this with “The Bank Manager Example”: Manager: Smith: Manager: Smith:

Can you give me a credit reference? My friend Jones will vouch for me. How do we know he can be trusted? Oh, I assure you he can.

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Our interest here is with formal arguments and as soon as one starts to formalize The Bank Manager Example, it becomes clear that the argument is invalid, for it has the following form. Premise 1: Premise 2: Premise 3: Conclusion:

∀a, b : trusted(a) ∧ vouch - for(a, b) ⊃ trusted(b) vouch - for(Jones, Smith) vouch - for(Smith, Jones) trusted(Smith)

The invalidity here is stark and independent of any ideas about question begging. Walton describes other methods for detecting question begging in informal arguments but most of the examples are revealed as invalid when formalized. While these methods may be of assistance to those committed to notions of informal argument or argumentation, our focus here is on valid formal arguments, so we do not find these specific techniques useful, although we do subscribe to the general “epistemic” model of question begging, and will return to this later. Barker (1978), building on (Barker 1976; Sanford 1977), calls a deductive argument simplistic if it has a premise that entails the conclusion; he claims that all and only such (valid) arguments are question begging. Our definition for strict begging includes this case, but also others. For example, Barker considers the argument with premises p and ¬q and conclusion p to beg the question, whereas that with premises p ∨ q and ¬q, and the same conclusion does not, which seems peculiar to say the least. Both of these are question begging by our strict definition. Now one might try to “mask” the question begging character of an argument that satisfies Barker’s definition by adding obfuscating material, so he needs some notion of equivalence to expose such “masked” arguments. However, it cannot be logical equivalence of the premises because the conjunction of premises is identical in the two cases above, yet Barker considers one to be question begging and the other not. Barker proposes that “relevant equivalence” (i.e., the bidirectional implication of relevance logic (Dunn 1986)) of the premises is the appropriate notion. The examples above are not equivalent by this criterion (¬q ⊃ p and ¬q illustrate premises that are equivalent to the second example by this criterion) and so the question begging character of the first does not implicate the second, according to Barker. As noted, all these examples strictly beg the question by my definition and I claim this is as it should be. Recall that a premise strictly begs the question when it is equivalent to the conclusion, given the other premises. Now, the essence of the epistemic interpretation for begging the question is that truth of the premise in question is difficult to know or believe independently of the conclusion, and I assert that this judgment must be made after we have digested the other premises (otherwise, what is their purpose?). Thus, if ¬q is given (digested), then p ∨ q and p are logically equivalent and we cannot believe one independently of the other and p ∨ q is rightly considered to beg the question in this context. Barker judges p ∨ q and p in the absence of any other premise and thereby reaches the wrong conclusion, in my opinion.

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My proposal for strict begging differs from those in the literature but is not unrelated to proposals such as Barker’s. However, my proposals for weak and indirect begging depart more radically from previous treatments. I consider a premise to be weakly begging when light augmentation to the other premises render it strictly begging. Human judgment must determine whether the augmentation required is innocuous or contrived and this can be guided by epistemic considerations: if the augmentation is required to establish a context in which the questionable premise(s) is plausible (as in our example of Fig. 13.3, where we certainly intend the > relation to be nonempty), then the questionable premise(s) surely beg(s) the question in the informal epistemic sense as well as in our formal weak sense. Indirect begging arises when the questionable premise supplies (a generalization of) exactly what is required to make a key move in the proof. Provided we have not applied anything beyond routine deduction, I claim that the proof state (conveniently represented as a sequent) represents our epistemic state after digesting the other premises and the desired conclusion. An indirectly begging premise is typically (a generalization of) one that can be reverse engineered from this state, and belief in such a premise cannot be independent of belief in the current proof state; hence such a premise begs the question in the informal epistemic sense as well as in our formal indirect sense. Most authors who examine question begging in the Ontological Argument implicitly apply an epistemic criterion, and do so in the context of modal representations of the argument (which I examine elsewhere, Rushby 2019b). Walton (1978), however, does discuss first-order formulations in a paper that is otherwise about modal formulations. Walton begins with a formulation that is identical (modulo notation) to that of Fig. 13.6. He asserts that the premise Greater2 (his premise 2) is implausibly strong because it “would appear to imply, for example, that a speck of dust is greater than Paul Bunyan.”7 I would suggest that a better indicator of its “implausible strength” is the fact that it indirectly begs the question, as described in Sect. 13.5. Walton then proposes that premise Greater1 of Fig. 13.3 (his premise 2G) may be preferable but worries that our reason for believing Greater1 must be something like Greater2. It is interesting that Walton does not indicate concern that Greater1 might beg the question, whereas our analysis shows that it is weakly begging, and becomes strictly so in the presence of premises that require a modicum of connectivity in the > relation (recall Sects. 13.2 and 13.3). Thus, I suggest that the formulations and methods of analysis proposed here are more precise, informative, and checkable than Walton’s and other informal interpretations for begging the question. My three criteria for begging the question—strict, weak, and indirect—identify question begging premises in a fairly unequivocal manner.8 They provide formal

7 Paul

Bunyan is a giant lumberjack character in American folklore. begging admits some equivocation in the choice of augmenting premises, and indirect begging in the amount of deductive effort expended.

8 Weak

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interpretations for the informal notion of “epistemic” begging, but it is not immediate from either this derivation or their own definitions that these kinds of question begging should be considered defects. Eder and Ramharter (2015, Section 1.2(5)) observe that the conclusion to a deductively valid argument must be implicit or “contained” in the premises (otherwise, the reasoning would not be deductive), but an argument can only be persuasive or interesting “if it is possible to accept the premises without already recognizing that the conclusion follows from them. Thus, the desired conclusion has to be ‘hidden’ in the premises.” In a footnote, they aver “Sometimes, proofs of the existence of God are accused of being questionbegging, but this critique is untenable. It is odd to ask for a deductive argument whose conclusion is not contained in the premises. Logic cannot pull a rabbit out of the hat.” Eder and Ramharter are, of course, correct that the conclusion must be “contained” in the premises, but they are also correct that it should be “hidden”, and so I challenge their claim that accusations of question begging are untenable. I suggest that tests for question begging should expose the “hiding place” of the conclusion among the premises: if this is revealed as inadequate or contrived, then our interest in the argument, and its persuasiveness, are diminished. I claim that my criteria perform this function. Strict begging tells us that the conclusion is barely hidden at all. However, a legitimate criticism of strict begging is that because it applies after we have accepted the other premises, it can indict premises that are merely there “to get the argument off the ground”; typically these are simple existential premises that provide Skolem constants. Examples are ExUnd, which appears in all the formalizations considered, and Ex_re, which appears in Fig. 13.6. Human judgment exonerates these. However, strict begging does correctly indict the premise Greater1 in Fig. 13.2 and weak begging does the same in Fig. 13.3. This premise obfuscates the obviously question-begging premise Greater1_vac of Fig. 13.5 by expanding the definition of God? and using the contrapositive, and strict and weak begging expose this subterfuge. Indirect begging identifies premises that are equivalent to those that would be constructed by reverse-engineering from the conclusion and other premises: to my mind, it reveals contrivance. One reason for the enduring interest in the Ontological Argument is surely that its premises seem innocuous, yet its conclusion is bold. But when a premise is revealed as indirectly begging, we see how the “trick” is performed and this must eliminate our surprise and diminish our delight. In summary, my criteria for begging the question are consistent with, and give formal expression to, informal “epistemic” interpretations. That is, a question begging premise is one that is difficult to understand or believe independently of the conclusion. In addition, my criteria expose premises that “hide” the conclusion or a key step in its proof in ways that can suggest contrivance or obfuscation.

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13.8 Conclusion Once we go beyond the “simplistic” case (Barker 1978), where the conclusion is directly entailed by one of its premises, the idea of begging the question is open to discussion and personal judgment. A variety of positions are contested in the literature on argumentation and were surveyed in Sect. 13.7, but I have not seen any discussion of question begging in fully formal deductive settings. My proposal is that a premise may be considered to beg the question when it is equivalent to the conclusion, given the other premises (strict begging), or a light augmentation of these (weak begging), or when it directly discharges a key step of the proof (indirect begging). The intuition is that such premises are so close to the conclusion or its proof that they cannot be understood or believed independently of it, and their construction seems “reverse-engineered” or otherwise contrived so that surprise and interest in the argument is diminished. I have shown that several first- and higher-order formalizations of the Ontological Argument beg the question, illustrating each of the three kinds of question begging. I suspect that all similar formulations of the Argument are vulnerable to the same charge. Separately (in work performed after that described here), I have examined several formulations of the argument in quantified modal logic (including that of Rowe (1976), who explicitly accuses the Argument of begging the question, and those of Adams (1971) and Lewis (1970), who also discuss circularity) and found them vulnerable to the same criticism (Rushby 2019b). Begging the question is not a fatal defect and does not affect validity of its argument; identification of a question begging premise can be an interesting observation in its own right, as may be identification of the augmented premises that reveal a weakly begging one. However, I think most would agree that the persuasiveness of an argument is diminished when its premises are shown to beg the question. Furthermore, revelation of question begging undermines any delight or surprise in the conclusion, for the question begging premise is now seen to express the same idea. Indirect begging is perhaps the most delicate case: it reveals how exquisitely crafted—one is always tempted to say reverse-engineered—is the questionable premise to its role in the proof. To my mind, it casts doubt on the extent to which the premise may be considered analytic in the sense that Eder and Ramharter (2015, Section 1.2(7)) use the term: that is, something that the author “could have held to be true for conceptual (non-empirical) reasons”. I have also shown that all the formalizations of the argument considered here entail variants that are vacuous: that is, apply no interpretation to “than which there is no greater” (formally, they leave the predicate God? uninterpreted). I think this a more serious and overarching defect than question begging. To see this, observe that all the formalizations considered here can be reconstructed by the following procedure. The first four steps construct a vacuous formalization similar to Fig. 13.5. 1. Introduce uninterpreted predicates God? and re? over beings 2. Introduce a premise similar to ExUnd and, optionally, one similar to Ex_re (Fig. 13.6).

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3. Specify a conclusion, similar to God_re_alt, that some being satisfies both God? and re?. 4. Reverse-engineer a premise Greater that entails the conclusion, given the other premises: something like Greater1_vac or Greater2_triv (Sect. 13.5). At this point, we have a reconstruction similar to Fig. 13.5 that is valid but vacuous, because there is no interpretation for God?. Hence, it remains valid when any interpretation is supplied, including Gaunilo’s most perfect island. The reverse-engineered premise Greater indirectly begs the question because (due to its method of construction) it cannot be understood or believed independently of the conclusion and the other premises (so it is question begging “in the epistemic sense”). 5. Supply an interpretation (i.e., a definition) for God? and replace some or all appearances of this predicate by its definition. The interpretation may require additional terms and premises, such as an interpreted or uninterpreted > relation, and higher-order constructions. 6. Optionally, adjust the resulting reconstruction (e.g., by Boolean rearrangement, by adding terms, or by adding variables and adjusting quantification), taking care that it remains valid (typically an adjusted premise will entail the original). The reconstruction following step 4 is vacuous and begs the question and I maintain that the adjustments made in steps 5 and 6 cannot remove these characteristics (although they can obfuscate them) and we cannot attach any belief to the premises once we see how they are constructed. Sections 13.4 through 13.6 provided evidence that all the formalized arguments examined there can be reconstructed by this procedure. I do not claim the original authors of those formalized arguments took this route—they were surely sincere and constructed premises that they considered both analytic and faithful to Anselm’s intent—but its existence exposes the hollow nature of all these formalizations. It is, of course, for individual readers to form their own opinions and to decide whether the forms of question begging and vacuity identified here affect their confidence, or their interest, in the various renditions of Anselm’s Argument, or in the Argument itself. What I hope all readers find attractive is that these methods provide explicit evidence to support accusations of question begging that can be exhibited, examined, and discussed, and that may be found interesting or enlightening even if the accusations are ultimately rejected. Observe that detection of the various kinds of question begging requires exploring variations on a specification or proof. This is tedious and error-prone to do by hand, but simple, fast, and reliable using mechanized assistance. I hope the methods and tools illustrated here will encourage others to investigate similar questions concerning this and other formalized arguments: as Leibniz said, “let us calculate”. Acknowledgments I am grateful to Richard Campbell of the Australian National University for stimulating and dissenting discussion on these topics, to my colleagues Sam Owre and N. Shankar for many useful conversations on PVS and logic, and to anonymous reviewers of this and a previous version for very helpful comments

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References Adams, Robert Merrihew. 1971. The logical structure of Anselm’s arguments. The Philosophical Review 80(1):28–54. Barker, John A. 1976. The fallacy of begging the question. Dialogue 15(2):241–255. Barker, John A. 1978. The nature of question-begging arguments. Dialogue 17(3):490–498. Benzmüller, Christoph, and Bruno Woltzenlogel-Paleo. 2014. Automating Gödel’s ontological proof of God’s existence with higher-order automated theorem provers. In 21st European Conference on Artificial Intelligence (ECAI 2014), 93–98, Prague, Czech Republic. Campbell, Richard. 2018. Rethinking Anselm’s arguments: A vindication of his proof of the existence of God. Leiden: Brill. Campbell, Richard J. 2016. Personal communication. Devine, Philip E. 1975. Does St. Anselm beg the question? Philosophy 50(193):271–281. Dunn, J. Michael. 1986. Relevance logic and entailment. In Handbook of philosophical logic, 117– 224. Berlin: Springer. Eder, Günther, and Esther Ramharter. 2015. Formal reconstructions of St. Anselm’s Ontological Argument. Synthese 192(9):2795–2825. Garbacz, Paweł. 2002. Begging the question as a formal fallacy. Logique & Analyse 177–178:81– 100. Garbacz, Paweł. 20122. PROVER9’s simplifications explained away. Australasian Journal of Philosophy 90(3):585–592. Gaunilo, a Monk of Marmoutier. In behalf of the fool. Internet Medieval Sourcebook. Fordham University (in English, the original Latin is from around 1079). Himma, Kenneth. 2005. Ontological argument. In Internet encyclopedia of philosophy, eds. James Fieser and Bradley Dowden. Lewis, David. 1970. Anselm and actuality. Noûs 4(2):175–188. McGrath, P.J. 1990. The refutation of the Ontological Argument. The Philosophical Quarterly 40(159):195–212. Oppenheimer, Paul E. and Edward N Zalta. 1991. On the logic of the Ontological Argument. Philosophical Perspectives 5:509–529. Reprinted in The Philosopher’s Annual: 1991, Volume XIV (1993): 255–275. Reed, Chris and Glenn Rowe. 2004. Araucaria: Software for argument analysis, diagramming and representation. International Journal on Artificial Intelligence Tools 13(4):961–979. Roth, Michael. 1970. A note on Anselm’s Ontological Argument. Mind 79:271. Rowe, William L. 1976. The Ontological Argument and question-begging. International Journal for Philosophy of Religion 7(4):425–432. Rushby, John. 2013. The Ontological Argument in PVS. In Fun with formal methods, ed. Nikolay Shilov. St Petersburg. Workshop in association with CAV’13. Rushby, John. 2016. Mechanized analysis of a formalization of Anselm’s ontological argument by Eder and Ramharter. CSL technical note, Computer Science Laboratory, SRI International, Menlo Park. Rushby, John. 2019a. Mechanized analysis of Anselm’s modal Ontological Argument. Technical report, Computer Science Laboratory, SRI International, Menlo Park. Rushby, John. 2019b. Mechanized analysis of modal reconstructions of Anselm’s traditional Ontological Argument. Technical report, Computer Science Laboratory, SRI International, Menlo Park. Rushby, John, Sam Owre, and N. Shankar. 1998. Subtypes for specifications: Predicate subtyping in PVS. IEEE Transactions on Software Engineering 24(9):709–72. Russell, Bertrand. 2013. History of Western philosophy: Collectors edition. London: Routledge. Sanford, David H. 1977. The fallacy of begging the question: A reply to Barker. Dialogue 16(3):485–498. Anselm, S. 1077. Proslogion. Internet Medieval Sourcebook. Fordham University (in English, the original Latin is dated 1077).

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Walton, Douglas. 1978. The circle in the Ontological Argument. International Journal for Philosophy of Religion 9(4):193–218. Walton, Douglas. 2006. Epistemic and dialectical models of begging the question. Synthese 152(2):237–284. Walton, Douglas N. 1991. Begging the question: Circular reasoning as a tactic of argumentation. New York: Greenwood Press. Walton, Douglas N. 1994. Begging the question as a pragmatic fallacy. Synthese 100(1):95–131.

John Rushby has one of the oldest BSc degrees in Computer Science (1971) and also a PhD from the University of Newcastle upon Tyne in England. Since 1983, he has been with the Computer Science Laboratory of SRI International (formerly Stanford Research Institute) in Menlo Park, California, where he is currently a Distinguished Senior Scientist and SRI Fellow. He works mainly on the development of formal verification methods and tools and their application to the assurance of critical systems, for which he received the IEEE Harlan D. Mills Award in 2011 and, together with his colleagues Sam Owre and N. Shankar, the Computer-Aided Verification (CAV) Award in 2012.

Part V

Logic, Language and Religion

Chapter 14

Logic and Religion: The Essential Connection Benjamin Murphy

14.1 Introduction The continued existence of analytical philosophy of religion demonstrates that there are plenty of philosophers who think that evaluating religious claims by using tools of logical analysis is a worthwhile activity, but there is also no shortage of writers who believe that religion and logic do not make good partners. Donald Wiebe argues that rationality is logically incompatible with the mythopoeic thinking of religion (Wiebe 1991, pp. 199–209). Ronald Davidson argues that Buddhist monks made an error in the seventh century CE when they attempted to demonstrate the truth of Buddhism by appealing to universal standards of reason. Henceforward, instead of asking what was authentic Buddhist teaching, they asked what was rational according to standards accepted by Hindus and Buddhists, thus ceasing to negotiate with other ideologies from a position of strength (Davidson 2002, p. 104). John Caputo argues that logical thinking is inimical to religion because, by its nature, logic is a heartless discourse (Caputo 2006, p. 104). The Jewish philosopher Lev Shestov insisted that religious thinkers should question the inviolability of the law of non-contradiction, which since at least the time of Aristotle has been taken as the fundamental basis of logic (Shestov 1966, p. 417). Anyone familiar with contemporary logic should be aware of the existence of paraconsistent logics, in which violations of the principle of non-contradiction do not result in triviality. We can no longer assume that a violation of a fundamental principle of classical logic is a violation of logic as such. Any discussion of the role that logic should or should not play in religious thinking needs to take into account the variety of systems that are described as logics by current practitioners of the discipline.

B. Murphy () Florida State University – Panama, Panama City, Republic of Panama e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_14

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But there is another assumption that I wish to draw attention to, more subtle than Shestov’s assumption that classical logic is the only form of logic, and one that I think is shared by all four writers mentioned here. This is the assumption that logic can simply be identified with the activities performed by logicians. As JeanYves Béziau has pointed out, the word ‘logic’ is ambiguous, in the same way that ‘history’ is ambiguous. Both may refer to an academic discipline, or to the object that the academic discipline studies (Béziau 2010, p. 176). If one ignores the second sense of ‘logic’, then a religion that originates in an environment where the academic discipline of logic does not exist will have no logic. If the environment changes, and the academic discipline of logic comes to be important (or if the religion spreads to a new environment) then a choice must be made about whether the religion will benefit from forming a partnership with the academic discipline. So, Wiebe thinks that in the twelfth century, when universities started to play an important part in European life, monastic theology, in which knowledge of logic was not required, gave way to scholastic theology, in which theological thinking was subordinated to general rules of thought that prevailed across all academic disciplines (Wiebe 1991, p. 201). Accepting the imposition of logical standards was the price that was paid for the incorporation of theology into the university curriculum. If we accepted this way of framing the situation, we could then debate whether the price that was paid was too high, as Wiebe seems to think, or whether perhaps the advantages outweighed the cost, just as we could argue about whether adding synchronized sound to movies was a good idea. But if we consider logic in the first sense, that is the object that logicians study, then we must consider the possibility that logic might have been part of a religion even when no member of that religion was interested in the study of logic. I will argue that logic in this first sense is an essential part of any religion, and those who evaluate religious thinking based on logical standards are not smugglers bringing in a new animal that will upset a delicate eco-system, but zoologists studying a creature in its natural environment. The creature that is being studied, the object of logic, is, so I will argue, commitment. My thesis is that logicians study validity, which is a relationship that one set of commitments can have to another, and a religion cannot exist without commitments. So logicians are studying something that is an essential part of any religion. In order to establish this conclusion, it is not sufficient to show that logicians can be said to study commitments and that religions are based on commitments. I must demonstrate that the word ‘commitment’ is being used univocally in these two contexts. In Sect. 14.2, I examine the connection between the concept of a commitment and intuitive validity. In Sect. 14.3, I argue that a prayer can be an example of a commitment in the logical sense. In Sect. 14.4, I argue that a religion cannot exist without commitments. This does not imply that a religion has a fixed and unchanging essence: in Sect. 14.5, I argue that it is precisely because the essence of any religion is flexible, by which I mean that members of a religion engage in debates that establish but also alter the boundaries of religious identity, that logic has an important role to play in religious discourse. In Sect. 14.6, I argue that in

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such debates logic is not merely a tool of conservatives who want to enforce existing standards. In Sect. 14.7, I consider the implications for philosophy of religion if my version of logical pluralism is correct.

14.2 The Essence of Logic My thesis is that logic is an essential part of any religion. This thesis is interesting because it justifies the application of techniques developed by logicians to religious issues, but such a justification is only possible if the claims I make about the nature of logic are accurate claims about the object that is studied by contemporary logicians. Typically, when students start to study logic today, they are first taught classical propositional calculus. However, students who advance beyond a basic introductory class will soon discover that there are alternatives to classical logic. The fact that logicians study so many different systems could be seen as a sign of a crisis in logic—as though logicians are no longer sure what logic is—but it could also be seen as a sign of strength—perhaps the study of logic should lead to the study of many systems. Logical pluralism is the view that there is more than one correct system of logic (therefore the proliferation of logical systems can be regarded as a good sign). Any attempt to spell out precisely what is meant by logical pluralism must explain what it means to say of some set of rules that it is intended as a logic, and what it means to say of a system intended as a logic that it is correct, that it succeeds in doing what a logic should do. Unless we explain these points, logical pluralism will be a vacuous thesis. The suggestion that I support is that a system is intended as a logical system if its purpose is to enable us to keep track of validity. This implies that validity is something that exists independently of the logical systems that are designed to measure it, and a logical system is more or less successful insofar as it succeeds or fails in detecting validity. The term ‘intuitive validity’ is used to emphasize the fact that validity precedes the logical systems. This proposal is not original, and that can be regarded as a minor point in its favour— an entirely novel proposal about the nature of logic that had no connection with anything said by contemporary logicians would be much harder to defend. I cannot attempt a complete survey of different versions of logical pluralism here, (for such a survey see Cook (2010)), but three prominent books that defend a version of logical pluralism that invokes the concept of intuitive validity are Haack (1978), Beall and Restall (2006) and Shapiro (2014). As has already been indicated, my thesis about the connection between logic and religion is based on the idea that intuitive validity, the object that logical systems are intended to measure, is a relationship between commitments. My argument will be that the concept of a commitment, suitably defined, clarifies the concept of intuitive validity, giving us a more precise understanding of logical pluralism than can be found in any of the authors I cite. Haack states that formal systems must be faithful to some prior extra-systemic standard of validity, but the formal systems refine and so modify ‘an idea that is by no means fully precise.’ (Haack 1978, p. 229) Beall and Restall state the

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logician’s task is to present precise standards and rigorous standards that sharpen a pre-theoretic notion of logical consequence, which ‘does not have sharp edges’, and since different sharpenings of a vague idea can be acceptable, so too, different and incompatible logical systems can be equally legitimate (Beall and Restall 2006, p. 28). Shapiro sets out the case for pluralism in a slightly different matter, saying of logical consequence and validity that A number of different, closely related notions go by those names . . . there are different, mutually incompatible, but equally legitimate ways to sharpen or further articulate the intuitive notion(s). (Shapiro 2014, p. 2)

Whereas Haack, Beall and Restall write of there being a single intuitive idea of validity which is vague and so can be sharpened in different ways, Shapiro, by adding an ‘s’ in parentheses, introduces the possibility that there is more than one intuitive notion to begin with. This is a significant difference, and I will argue that we do need a variety of intuitive concepts of validity, and that we can state very clearly what they all have in common. The existence of intuitions about validity is obvious to anyone who has taught an introductory class in symbolic logic. The study of symbolic logic usually begins with classical propositional calculus. Students will be told that the arguments that will be considered consist of sentences that are determinately true or false, and never both true and false. Entities that are either true or false can also be called truth-bearers or propositions. An argument consists of a set of propositions that are called premises, and another proposition that is called the conclusion, and it is considered to be valid if and only if it is contradictory for the premises to be true and the conclusion false. Since contradictions cannot be true (something most students consider to be self-evident), if an argument is valid in the sense defined here, and the other rules about truth and falsity are also accepted, a valid argument will be an argument in which it is impossible for the premises to be true and the conclusion false. At this stage, students are ready to see how a symbolic system, in which wellformed formulae can be used to stand for propositions, can play a useful role in demonstrating that certain arguments are valid. Some students find this definition to be objectionable or surprising, and since they find it counter-intuitive, they must have some intuitions in place. One common objection is fairly trivial. Many students believe that it is wrong to call an argument valid if the premises are not true. It is not difficult to get students to see that there are advantages in separating an inquiry into whether the truth of the premises would imply the truth of the conclusion from an inquiry into whether the premises are in fact true, and therefore the value of having a word that states specifically that the premises have the right kind of connection to the conclusion. They are then willing to accept that logicians have chosen the word ‘valid’ to cover that relationship. However, even when they see that an argument can be valid even if the premises are false, many students still have a problem in saying that an argument with contradictory premises is valid whatever the conclusion. When they see that this rule of ex contradictione quodlibet follows from the definition of validity they have

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been given, some of them will see this as a reason for doubting the definition. This is far from being a trivial objection: some alternatives to classical logic have been motivated by an objection to this rule. I regard it as a point in favour of my definition of intuitive validity that it can explain why our intuitions become hazy over this particular issue, for reasons that will be given in due course. Haack’s logical pluralism is, of course, a reaction to exactly this type of situation. A symbolic system is to be evaluated according to whether it is true to the pretheoretical notion of validity. But when, as in this case, intuitions differ, two different systems can both claim to represent that vague intuitive notion: classical logic preserves the validity of ex contradictione quodlibet, whereas relevant logics, such as First Degree Entailment, do not (Priest 2001, pp. 138–167). Both of these systems are exact, but the exact concepts of validity that each of them contain cannot be exactly the same as the intuitive concept of validity, since a precise concept cannot be exactly the same as a vague concept. Haack never tries to tell us precisely what the intuitive notion of validity is, since her whole point is that logicians deal with precise, sharpened notions of this inherently vague concept. Consequently, in her opinion, the boundaries of logic are fuzzy: she tells us that she doubts that there is such a thing as the essence of logic (Haack 1978, p. 6). Haack thinks that the intuitive notion of validity has to do with transmission of truth from premises to conclusion. She admits that this does not cover imperative sentences, and creates difficulties for systems that do not assume bivalence—propositions that are not determinately true or false may not qualify as truth-bearers at all, and so cannot transmit truth from premises to conclusion. However, she is aware that validity can be defined in other ways. An imperative, ‘Close the door!’ is satisfied if the imperative is obeyed, and for imperatives, validity can be defined as transmission of satisfaction from premises to conclusion. Haack concludes that [ . . . ] developments such as non-bivalent or imperative logic may call for changes or extensions of the intuitive conception of validity to which the standard logical apparatus gives formal expression. (Haack 1978, p. 85)

So Haack allows that there are legitimate forms of logic that are concerned not with intuitive validity, but with concepts that result from extending or changing the intuitive concept of validity. This certainly supports Haack’s claim that there is no such thing as the essence of logic, because she has placed no kind of limit on the extent to which the intuitive concept may be altered. It also gives the impression that whereas classical logicians are keeping track of something, intuitive validity, that they did not invent, adherents of imperative logic, intuitionistic logic and other logical systems that are not concerned with transmission of truth from premises to conclusion are inventing new objects to study. But the existence of an intuitive concept of validity matters because it is what gives significance to the claim that some proposed system of logic succeeds, and is thus an acceptable system. A system is said to be acceptable because it enables us to keep track of intuitive validity, not because it enables us to keep track of an artificial notion that was invented along with the system. This gives us a reason to wonder whether Haack’s idea of intuitive validity is too narrow.

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If Haack is correct that logic lacks an essence, then it cannot be true to claim, as I have claimed, that there is an essential connection between logic and religion. However, Beall and Restall have a proposal for a clarified version of logical pluralism, one in which it does make sense to talk about the essence of logic. Their starting point is the same as Haack’s: intuitive validity is a vague notion that may be sharpened in various ways by different logical systems. They then propose that a system that is proposed as a sharpening of the intuitive concept of validity must satisfy a certain definition, the Generalized Tarski Thesis: An argument is validx if and only if in every casex in which the premises are true, so is the conclusion.

Logical pluralism is then defined as the doctrine that there are at least two nonequivalent but equally admissible instances of systems that satisfy the Generalized Tarski Thesis (Beall and Restall 2006, p. 29). One problem with the Generalized Tarski Thesis is that it includes the concept of a ‘case’, and Beall and Restall admit that they cannot give clear individuation conditions for a ‘case’ beyond saying that a case is clearly a thing in which a claim may be true (Beall and Restall 2006, p. 89). It is clear from the examples that they give that the Generalized Tarski Thesis is satisfied by supplying a formal semantics for some logical system, and that cases are part of those formal semantics. For example, in simple propositional calculus, truth-tables can be used to supply a semantic test for validity, and a line of a truth-table would constitute a case. In modal semantics, a possible world would constitute a case. But even the one condition that Beall and Restall place on cases— that a case is a thing in which something may be true or false—is problematic. This restriction means that, like Haack, Beall and Restall think that validity is a matter of transmission of truth from premises to conclusion. However, whereas Haack recognized that this is problematic for systems in which bivalence does not hold, and proposed that we need to extend the notion of validity beyond its intuitive sense, Beall and Restall do not. Beall and Restall consider intuitionistic logic to be an example of a system whose formal semantics fits the Generalized Tarski Thesis. Here is how they define intuitionistic validity: Given a system of stages, we say that an argument is valid (with respect to the system of stages) if and only if, at any stage in which the premises are true, so is the conclusion. An argument is intuitionistically or constructively valid if and only if it is valid in all systems of stages. (Beall and Restall 2006, p. 64) [ . . . ] we need to show how intuitionistic logic stands as an instance of the schema provided by the Generalised Tarski Thesis. We do this by indicating how the stages of the semantics for intuitionistic logic can serve as the precisifications of ‘case’. A stage in an intuitionistic model serves as the right unit of evaluation. An argument is intuitionistically valid if and only if, for any stage (in any model) at which the premises are satisfied, so is the conclusion. We have a straightforward instance of GTT. (Beall and Restall 2006, p. 68)

So, they write of a premise being true at a stage or satisfied at a stage, as if the two are interchangeable. But this is not the way the stages are described within intuitionistic logic. Here is a typical quotation from Dummett’s Elements of Intuitionism:

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A statement proved at any stage is true; or rather, more exactly, from a proof that A has been proved at some stage we can obtain a proof of A. (Dummett 2000, p. 236)

Dummett’s self-correction is, of course, a pedagogical device. We are being reminded of the need to consciously break with our habit of treating A as a statement that A is true, and instead treating it as a statement that we have a proof of A. In fact, Dummett’s position on this point evolved. In early work, he was clear that in intuitionistic logic it is proof, not truth, that is transmitted from premises to conclusion (Dummett 1978a, p. 18). In later work, he argued for retaining the idea that truth is transmitted from premises to conclusion, by saying that the existence of a proof is what makes a statement true (Dummett 2004, p. 32). However, in that case, the truth that the intuitionist transmits from premises to conclusion is not the same thing as the truth that the classicist transmits from premises to conclusion (Dummett 1978b, p. 239). So we can say that classical and intuitionistic logicians both want to transmit truth only if we equivocate over the meaning of truth. The basic point is not difficult to grasp, even for readers who are unfamiliar with intuitionistic logic. In classical logic, the following is a valid argument: Premise (1) Premise (2) Conclusion:

If P then Q. If Not-P, then Q. Therefore Q.

This argument is valid because, in classical logic, P is determinately true or false (the principle of bivalence), and if P is false, Not-P is true. But in intuitionistic logic, to assert that P is to assert that one has a proof for P, and to assert ‘If P then Q’ is to assert that a proof of P will yield a proof of Q. ‘Not-P’ asserts not merely that there is no proof of P, but that there is a proof that there will never be a proof of P. There is no guarantee that, for any arbitrary proposition, we will have either a proof or a proof that there can be no proof, and therefore the principle of bivalence does not apply in intuitionistic logic. For the intuitionist, a proof of the first two premises of this argument would not count as a proof of Q, and therefore the argument is not valid. So, Beall and Restall say that intuitionistic logic meets their Generalized Tarski Formula, but in order to demonstrate this, they have to describe intuitionistic logic in a way that obscures the central difference between intuitionistic and classical logic. Haack admits that intuitionistic logic is not based on a form of validity that involves preserving truth, and so introduces the possibility of extending and altering the intuitive concept of validity. Haack, Beall and Restall all agree that pluralists should recognize intuitionistic logic as a form of logic, but, so it seems to me, they only succeed in granting it official recognition by fudging on key issues. There is an alternative approach. It will be recalled that Shapiro introduced into consideration the possibility that there might be more than one intuitive concept of validity, and I think this is a fruitful suggestion. Just as Restall and Beall offered a generalized definition of semantic validity, I propose a Generalized Definition of Intuitive Validity:

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An argument is validx if and only if for some type of commitmentx , violating the commitmentx to the conclusion violates the commitmentx to at least one of the premises.

According to this definition, there are different types of intuitive validity because there are different types of commitment. This is based on Robert Brandom’s suggestion that a good inference is one that preserves commitment (Brandom 1994, p. 112). One can be committed to the truth of a proposition, or one can be committed to there being a proof of a proposition. Classical logic concerns the transmission of a commitment of truth from premises to conclusion, intuitionistic logic concerns the transmission of proof from premises to conclusion. Performative utterances can also involve commitments. If I promise to be in New York all day tomorrow, and promise to dance at some time tomorrow, I have promised to dance in New York tomorrow. This is because a failure to dance in New York tomorrow would mean that I had reneged on at least one of the two initial promises. I think that this is an obvious fact that can be recognized without any special training. It is an example of an inference that is intuitively valid where what is transmitted from premises to conclusion is not truth. My definition makes use of the concept of a commitment. A commitment is a speech-act for which there is a recognised form of failure. This failure need not be intentional. If I let you use my car, and assure you that it is in good condition and the car breaks down, my assurance has failed and I have reneged on the commitment, even if I had no way of knowing that the car was faulty. In order to set up valid arguments, we must be capable of recognising when a failure to meet one commitment is a failure to meet another—the argument is valid if a failure to meet the commitment of the conclusion means there has been a failure to meet the commitment of at least one of the premises. We can keep the word ‘proposition’ for anything that could serve as the premise or conclusion of an argument, but propositions need not be, as in classical logic, truth-bearers; they are any units of discourse to which one can make a commitment of an appropriate type. I noted that it is counter-intuitive to say that an argument with contradictory premises is valid whatever the conclusion. In such a case, we cannot really say that reneging on the conclusion would undermine the premises, rather, the premises undermine themselves. So, according to my definition of intuitive validity, ex contradictione quodlibet is not a valid principle of inference, and it is hardly surprising that people reject it as counter-intuitive. It is important to point out that not every pre-theoretic judgment about validity counts as an example of intuitive validity as I have defined the term. Many students believe that from ‘If P then Q’ and ‘Not-P’ they can deduce ‘Not-Q.’ This is a common error, a tempting error, but of course, reneging on ‘Not-Q’ by asserting ‘Q’ does not mean reneging on commitments to ‘Not-P’ and ‘If P then Q’. Logical systems are supposed to be faithful to intuitive validity, but this does not mean that every uneducated hunch about which arguments are valid counts as an example of something that is intuitively valid. Having a definition of intuitive validity helps us to distinguish situations where logical systems fail to be true to intuitive validity from common errors of reasoning.

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My main focus has been to show that the Generalized Definition of Intuitive Validity provides a better explanation of why intuitionistic logic is a form of logic than does the Generalized Tarski Thesis. This is one reason for my claim to be offering an improved version of logical pluralism. A satisfactory version of logical pluralism should enable us to understand why a system is a correct form of logic. Haack, Beall and Restall recognize that intuitionistic logic is a correct form of logic, but cannot provide a satisfactory explanation as to why that is. A referee for this book pointed out to me there are obvious similarities between the approach to logic suggested here and the work of Searle and Vanderveken on illocutionary logic (1985). Searle and Vanderveken do not feel the need to present a lengthy justification of the claim that their system of illocutionary logic really is a form of logic, and so they offer no equivalent of the Generalized Definition of Intuitive Validity, that is they do not offer an explicit criterion that can be used to demonstrate that the system they present really is a system of logic. However, it is not difficult to discern an implicit idea about the nature of logic in the following passage: A theory of illocutionary logic of the sort we are describing is essentially a theory of illocutionary commitment as determined by illocutionary force. The single most important question it must answer is simply this: Given that a speaker in a certain context of utterance performs a successful illocutionary act of a certain form, what other illocutions does the performance of that act commit him to? (Searle and Vanderveken 1985, p. 6)

Searle and Vanderveken are offering a theory of logic that encompasses illocutionary acts that are not truth-bearers, so it is no surprise that they focus on the transmission of commitment from one speech-act to another rather than transmission of truth. They introduce a symbol, which will be represented here as precisely to indicate transmission of commitment from one illocutionary act to another: An illocutionary act A1 commits the speaker to an illocution A2 (for short A1  A2 ) iff it is not possible for the speaker to perform A1 without being committed to A2. (Searle and Vanderveken 1985, p. 81)

Clearly, if ‘A1  A2 ’ then there is an intuitively valid argument from A1 to A2 , although we cannot say that illocutionary commitment is equivalent to intuitive validity, since the former is a binary relation and the latter permits multiple premises. So, insofar as Searle and Vanderveken have anything to say about what makes their system a form of logic, they are implicitly appealing to something like the Generalized Definition of Intuitive Validity. To say that the Generalized Definition of Intuitive Validity gives the essence of logic is to say that someone is engaged in the study of logic if and only if they are engaged in the study of some form of intuitive validity. This may be done by using an existing system of logic to evaluate arguments for validity, or by developing new systems that enable us to study forms of intuitive validity that have been neglected. The most likely form of objection I can anticipate to my claim about the essence of logic is for someone to find a form of logic that does not fit the Generalized Definition of Intuitive Validity, just as I have claimed intuitionistic logic does not fit

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the Generalized Tarski Thesis. However, although such an objection would require me to drop the claim that people are studying logic only if they are studying intuitive validity, and thus to drop the claim about the essence of logic, it would not require me to drop the claim that if someone is studying intuitive validity, they are studying logic.

14.3 Intuitive Validity and Prayer But does intuitive validity, as I have defined it, have anything to do with religion? John Caputo, as we have seen, thinks that logic and religion should not be mixed. I cannot do justice to his complex writings here, but there is one simple idea suggested by his work that is worth mentioning, because responding to it will help me explain just how it is that intuitive validity is already embedded in religious discourse. Caputo frequently draws attention to the fact that typically, religious discourse consists of prayers and thanksgivings, which are not constatives (intended to describe the way things are), but performatives (i.e. the words themselves are actions, as when someone names a ship) (Caputo 1997, pp. 297 and 316; 2013a, p. 54; 2013b, p. 37). He believes that, unlike science, religion is not concerned with propositional truth, that is with the kind of truth that consists in getting an accurate description of the world, and, therefore, logic should be kept out of religion (Caputo 2001, pp. 109–116; 2006, pp. 103–122). This argument rests on the assumption that logic is only concerned with truth-bearers, not with any other kind of speech act. That is a reasonable assumption if logic is limited to classical logic, and Caputo has tradition on his side. Aristotle claimed in De Interpretatione (17a:1-5) that prayers should be studied as part of poetry or rhetoric, because they cannot be true or false. However, Searle and Vanderveken include prayers within the scope of illocutionary logic (Searle and Vanderveken 1985, p. 205) and, I will argue, they are right to do so. In Hamlet, Act III Scene III, Claudius, who has murdered his brother to attain the crown, experiences anguish at his own inability to pray: [ . . . ] But, O, what form of prayer Can serve my turn? ‘Forgive me my foul murder?’ That cannot be; since I am still possess’d Of those effects for which I did the murder.

Claudius perceives that he cannot ask God for forgiveness and also ask to keep his kingdom. He cannot pray ‘Forgive my foul murder’ sincerely without true penitence, and penitence requires expressions of commitment, such as ‘I renounce my kingdom.’ In other words, we have exactly the ingredients that I identified as necessary features of intuitive validity. ‘Forgive my foul murder’ and ‘I renounce my kingdom’ are both speech acts that involve a commitment. To renege on the commitment to ‘I renounce my kingdom’ is also to renege on the commitment to

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‘Forgive my foul murder’, so, according to the Generalized Definition of Intuitive Validity, there is a valid argument from ‘Forgive my foul murder’ to ‘I renounce my kingdom.’ This is why Claudius recognizes that his unwillingness to give up his kingdom precludes him from praying for forgiveness. Neither ‘Forgive my foul murder’ nor ‘I renounce my kingdom’ is a proposition in the classical sense of being either a truth-bearer. They are propositions in my revised sense: we can establish that one of them validly implies the other. Searle and Vanderveken have at least laid the groundwork for examining such implications. They argue that all illocutionary acts have sincerity conditions. A prayer is a type of directive (an illocutionary act aimed at influencing the future), and a directive can only be sincere if the speaker wants those results that the directive is aimed at producing (Searle and Vanderveken 1985, p. 56). Unfortunately, although they discuss prayers, Searle and Vanderveken have nothing to say about forgiveness. However, an obvious thought is that to forgive is to remove the lingering effects of a sin, restoring the sinner to their previous state. Claudius, of course, does not want to give up the effects of his sin. It should be noted that the test of Claudius’ sincerity is whether he is willing to relinquish the effects of his sin, not whether he has a genuine conviction that there is a divine entity listening to his prayer. Caputo is correct that an atheist may pray to God, and that the contents of such prayer can be deeply revealing (Caputo 1997, pp. 291–299). My point is that prayers are revealing precisely because there is such a thing as the logic of prayer.

14.4 The Essence of Religion Although I noted Haack’s doubts about whether there is an essence of logic, I tried to demonstrate that logic does have an essence by demonstrating what that essence is. When it comes to the essence of religion, I tend to side with the sceptics. However, I hope to demonstrate that one can be wary of the trend to essentialize religions, and yet still think that there is an essential connection between logic and religion. The boundaries of religion, and the boundaries of any particular religion, are subject to negotiation. In that sense, there is no such thing as the essence of religion, nor any such thing as the essence of Hinduism, or Buddhism and so on, at least not if we consider an essence to be a fixed and changeless entity. But the very negotiations that are grounds for denying the existence of fixed unchanging essences are also demonstrations of the indispensable role of logic. The kind of evidence I need to demonstrate that negotiations over the boundaries of religion have a logical element comes from examining certain very contentious debates. Of course, I am not trying to solve these debates, only to provide a sense of what is at issue. Because I am not trying to solve these debates, I have had no compunction about offering a highly selective and simplified view of history. Readers who are familiar with this history are asked to excuse the simplification, and readers who are unfamiliar are warned that this is the Reader’s Digest version of events. This tendency towards over-simplification comes with the territory. Tomoko

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Masuzawa points out that those who teach Introduction to World Religions often complain That such a comprehensive treatment of the subject in one course [ . . . ] is impossibly ambitious or inexcusably simplistic [ . . . ] unless, perhaps, one begins with the scholastically untenable assumption that all religions are everywhere the same in essence, divergent and particular only in their ethnic, national or racial expressions. (Masuzawa 2005, p. 9)

As Masuzawa then points out, although in the United States classes and textbooks still refer to ‘World Religions’, the term ‘World Religion’ has lost any vestiges of scientific meaning—originally it was used as a way of stating the claim that Christianity had a unique status as the only universal religion (Masuzawa 2005, p. 22). Decisions that were made in the nineteenth century about what is and what is not a religion remain in place, in part, through deference to tradition. For example, although there is room for doubt about whether or not Confucianism counts as a religion, if I were to publish a textbook on World Religions that did not mention Confucianism, people would request a refund. Debates about what should be classified as a religion are debates about the essence of religion, and I wish to focus, very briefly, on debates about the status of Confucianism and Buddhism. Anna Sun has devoted a whole book to the question Is Confucianism A World Religion? (2013) She provides a detailed history of debates about the status of Confucianism, but comes to no conclusion. According to Sun, it was Max Müller who first established the status of Confucianism as a world religion when he decided to include translations of the Confucian Classics in the Sacred Texts of the East series (Sun 2013, p. xii). For centuries, candidates taking the Civil Service Entrance Exam in China were required to study the Confucian Classics, arguably making those books the closest Chinese equivalent to the Bible, thus suitable for inclusion in Müller’s series. But just because a set of texts can be considered sacred it does not follow that they have a religion associated with them. One piece of evidence that Sun points out that counts against the classification of Confucianism as a religion is the fact that although there are many traditional rituals that outsiders call Confucian (Sun 2013, p. 117), there is no ceremony by which a person can officially become a Confucian, (Sun 2013, p. 122) although in an attempt to reestablish Confucianism as a religious identity, new rituals have been adopted, such as Confucian marriage ceremonies (Sun 2013, p. 170). However, if Sun is right, I could not wake up tomorrow, decide to renounce my current religious identity and become a Confucian, and then undergo some appropriate initiation ceremony after which my status as a Confucian would no longer be in doubt. Contrast this with Buddhism. The usual way of expressing a commitment to Buddhism is by declaring that one takes refuge in the Buddha, the Dharma and the Sangha. Some version of this formula is used in Theravada, Mah¯ay¯ana and Mantrayana Buddhism, that is to say in all of the main forms of Buddhism (Harvey 1990, pp. 176–179). The Sangha refers to the community of celibate men and women who choose a life of poverty in order to devote themselves to meditation, study, and passing on the Dharma, that is the Buddha’s teaching. Taking refuge in the Sangha means

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offering it support and deferring to its teaching. The Sangha is organized in exactly the same way as those organizations called religious orders in Europe, and it is from these religious orders that the word ‘religion’ is derived. This is perhaps why Müller used Buddhism as an example of an atheistic religion. The idea that there could be a religion without God was deeply counter-intuitive in the nineteenth century, but everything about the social organization of Buddhism suggests that it should be classified as a religion, and so whatever the content of the Dharma, it has to be described as religious teaching (Müller 1907, p. 28). So, we have here a reason for saying that Buddhism is clearly a religion whereas Confucianism may not be: to say that someone is a Buddhist is to say that they have participated in a speech-act that was an expression of a set of commitments, and we can say of other speech-acts that they express commitments that are incompatible with Buddhism. The study of religions would be simpler if we could say that making a commitment to the three refuges of Buddhism precludes making a commitment to any other religion. But this is the kind of over-simplification against which Masuzawa quite rightly warns. It is true that taking the three refuges can be interpreted as an act in which allegiance to any other religion is repudiated. Bhimrao Ambedkar, regarded as the founder of Navayana Buddhism, formally converted to Buddhism by making a public declaration that he was taking the three refuges in 1956, and, in his first act as a Buddhist, he immediately presided over the conversion of 400,000 Untouchables (Vajpeyi 2012, p. 220). Ambedkar believed, and taught his followers to believe, that a commitment to following the Buddha’s Dharma was incompatible with worship of Hindu gods (Lokamitra 1991, p. 1303). However, not all Buddhists share Ambedkar’s strict views about Hindu gods. Richard Gombrich reports that a Theravada Buddhist monk once told him that although gods exist, they have nothing to do with religion (Gombrich 2006, p. 24). Gombrich explains this attitude by introducing a distinction between two types of religion: communal religion, which marks our passage through life, and salvific religion, which tells us how to be saved. In the west, we are used to thinking of these as a single package, but in Sri Lanka, they can be considered to be separate, and it is only the quest for salvation that is regarded as truly religious (Gombrich 2006, p. 25). From a Theravada perspective, Buddhism is atheistic only in the sense that Buddhism teaches the path to Nirvana, and gods are irrelevant to this path, leaving Buddhists free to worship gods for non-religious reasons. From a Navayana perspective, Buddhism is atheistic in the stronger sense that worship of a god is detrimental to pursuing the path of Buddhism. It is clear that Theravada and Navayana Buddhists do not agree about what the essence of Buddhism is. This also demonstrates a potential disagreement about what the essence of religion is. Ambedkar announced his intention to break away from Hinduism in 1936 because of his opposition to the caste system, and only decided to convert to Buddhism much later (Vajpeyi 2012, pp. 213–215). A Theravada Buddhist might argue that Ambedkar’s relatively strong form of atheism was a consequence of his desire to turn his followers away from anything associated

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with Hinduism and, therefore, that it was the result of a political rather than a religious impulse. But what reason do we have for saying that a political motivation cannot also be religious? It is clear that religions involve commitments and that not every commitment is a religious commitment, but what is it that makes a religious commitment distinctive? Arnal and McCutcheon suggest that the question we need to ask is why the word ‘religion’, originally used by Europeans to classify communities they encountered, has become a word that so many communities want to claim as part of their self-designation (Arnal and McCutcheon 2013, p. 50). In the case of Theravada Buddhists in Sri Lanka, Gombrich provides an answer. In 1815, when the British were invited by members of the aristocracy in Sri Lanka to take control of the government, they signed a treaty promising to respect the Buddhist religion, although starting in 1824, the British gradually severed the official recognition of Buddhism (Gombrich 2006, p. 174). By presenting certain rules and demands as religious, the local population improved their chances that those rules would be respected by the imperialists. To say ‘This is part of my religion’ is to say ‘Doing this is required to be a member of a community with which I identify, a community bound by certain commitments, including this one. Prevent me from doing this, and you are attacking the existence of that community, and can expect resistance from the whole community.’ I do not mean that the only motive for saying ‘This is part of my religion’ is to demand respect in this manner, but a religion is something about which such a demand can be made.

14.5 Drawing Religious Boundaries If this is more or less correct, that does not mean that where there is talk of a religion, there will be unanimous agreement about the commitments required of members. It is more likely that where there is talk of a religion, there will be negotiations about exactly what commitments are required. Islam provides a good example, because if there is any religion that, at first sight, appears to have an essence, it is Islam. A standard text-book will tell students that there are five pillars of Islam, the first of which is the Shahadah, the declaration of faith, above all faith in the unity of God. They will then learn that the worst sin in Islam is shirk, idolatry, which includes any denial of divine unity. Students who learn this much may believe that they now understand something about the essence of Islam—to worship a god other than Allah is to renounce Islam. Rejection of shirk would be a necessary condition of being a Muslim, and would serve to explain, in many cases, why Muslims behave in the way that they do. But things are not so simple. Gayatri Reddy, in her study of hijras (emasculated men) in Hyderabad reveals that these hijras identify themselves as Muslims but worship a Hindu goddess, Bahuchara Mata (Reddy 2005, p. 58). It is true that in

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determining religious identity in India, orthopraxy counts for more than orthodoxy, but worshipping Bahuchara Mata seems like as clear an example of shirk as one could wish for, and as such an offense against correct Muslim practice as well as correct belief. According to Zia Jaffrey, Muslim rulers chose non-Muslims to become hijras because castration is forbidden for Muslims (Jaffrey 1996, p. 20). Reddy has a lot to say about why hijras do not find it difficult to reconcile their worship of Bahuchara Mata with a Muslim identity. She explains, for example, that in Hyderabad one of the most important differences between Muslims and Hindus is that Muslim men are circumcised and Hindu men are not. Some hijras then argue that the ritual of castration that is central to their collective identity can be seen as the most extreme form of circumcision (Reddy 2005, p. 103). Rather than seeing themselves as having participated in the forbidden ritual of castration, they see themselves as having taken the required ceremony of circumcision to its ultimate conclusion. So, there is a logic to the hijras’ position, but it is not one that could be anticipated based on a text-book definition of Islam in terms of the five pillars. A student who thinks that hijras cannot really be Muslims because they do not meet the criteria laid down in an introductory text-book attributes the wrong kind of authority to the text-book. At the same time, a student who fails to realize why the hijras’ claim to be Muslim worshippers of Bahuchara Mata is problematic and worthy of discussion displays an ignorance of basic facts about Islam. Another example of Muslim syncretism is provided by Michael Cook. In Indo-China, the Cham people professed to be Muslims, but worshipped a Hindu goddess alongside Allah. However, as he also notes, in the nineteenth century a Muslim pilgrim who passed through Cham on the way to Mecca persuaded Cham Muslims in three villages to abandon their worship of goddesses: The pilgrim from Mecca had clearly put the Chams on the spot. But in a world in which there really was no such thing as Islam, just many local Islams, there would have been no spot for him to put them on. (Cook 2014, p. xvii)

People who claim to be Muslims must negotiate to decide upon criteria for Muslim identity. There is no fixed, unchanging essence of Islam, in the sense that there was nothing that prevented the Cham Muslims from worshipping a local goddess. But there are negotiations about Muslim identity—their claim to be Muslims meant that they were subject to criticisms from the visiting pilgrim. Such debates about what is compatible with Muslim identity have produced a flexible and changing essence of Islam, a simplified and static version of which may be presented to students when they first begin to study Islam. Logic in the sense of intuitive validity, the object of logical investigation, is present in the expressions of commitment that are characteristic of a religion. Logic in the sense of the investigation of the links between different commitments comes to be important when the boundaries of such commitment are being negotiated.

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14.6 The Role of Logic in Religious Debates So, I have argued that a religion exists when a group of people are united by a set of shared commitments. If they are truly united, they should know what is incompatible with those commitments. In practice, there will be disagreements about the content of those commitments, disagreements that may be resolved by one side admitting its errors, or that may lead the community to split. According to the argument set out in Sect. 14.2, these commitments are propositions, and any attempt to explore their consequences is an exercise in logic. In his more ironic mood, Caputo acknowledges that there is value in analytical philosophy of religion (Caputo 2000, p. 557), however, when writing in a more polemical mode he has expressed a willingness to make enemies of logicians (Caputo 2006, p. 103) and theologians who insist on enforcing orthodoxy (Caputo 2013a, p. 25). He expresses disdain for philosophers like Kant who see their role as being that of an intellectual police force (Caputo 2006b, p. 29). It is possible that he fears that logicians will inevitably be allies of the enforcers of orthodoxy—in his mind, both groups try to control the way people think. Whether Caputo actually fears this or not, it is worth dispelling the idea that such an alliance is inevitable. In a debate about religious identity, we should not expect that we will necessarily find a logical position on one side and an illogical position on the other. Rather, the role that logic may play is that of clarifying both positions, enabling clearer decisions. Robert Brandom presents the following example. A bank employee may put on a tie every day because he works at a bank, and that’s what bank employees do. This need not be based on any explicitly formed belief that ‘All bank employees wear ties. I am a bank employee, therefore I wear a tie.’ In fact, one way that the bank employee might come to question the necessity of wearing a tie is precisely by formulating such an explicit statement—it is much harder to challenge the unwritten and unspoken rules than to rebel against something explicitly stated (Brandom 1994, pp. 247–248). Following Brandom, I think that in disputes about group identity, whether the issue is bankers wearing ties or women wearing burqas, both sides can, and should, make use of logic. Suppose I establish that ‘If P, then Q’. I establish this based on my knowledge of exactly what is involved in a commitment to P and a commitment to Q. If I can show that anything that undermines a commitment to Q also undermines a commitment to P, then I know that ‘If P, then Q’. This gives me a choice: I can deny Q, or I can affirm P, but I cannot do both. Suppose P is ‘Church Councils teach only the truth’, and Q is ‘It is right to sell indulgences.’ A Catholic can reason that because Church Councils teach only the truth, selling indulgences is acceptable, a Protestant that because selling indulgences is unacceptable, Church Councils can err. If I say modus ponens when you say modus tollens, we are in agreement about the logic, but I am more committed to P and you to the denial of Q. If nothing else, this might give both of us a reason to part company and call the whole thing off.

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Ambedkar’s judgment was that it was impossible to practice Hinduism without thereby committing one’s self to the caste system. He may have been wrong about this—that was the subject of his debate with Gandhi—but the claim was not intrinsically absurd. Neither Hinduism nor the caste system is a declarative sentence, and so neither one is the kind of thing that ‘P’ and ‘Q’ normally stand for in propositional calculus. But they are both things to which someone can be committed, and there is clearly some connection between these two commitments such that it at least made sense to argue that whatever undermines the caste system also undermines Hinduism. Ambedkar presented Hindus with a statement of the form ‘If P then Q’ where ‘P’ is Hinduism and ‘Q’ is the caste system, hoping that other would respond, as he did, with a modus tollens argument. Because we can take Hinduism and the caste system to be the referents of ‘P’ and ‘Q’ in a conditional sentence such as this they are both, in my sense, propositions. Luther and Ambedkar are significant figures in the history of religion because they faced up to dilemmas in their own lives, made difficult decisions, and articulated their reasons in such a way as to force other people to decide as well. I am not suggesting that Luther and Ambedkar had logic on their side—the sale of indulgences and the caste system may both have been reprehensible, but not everything reprehensible is illogical. My suggestion is that logic revealed what the two sides were, because all participants were confronted with a proposition of the form ‘If P then Q.’

14.7 Conclusion As I observed at the start of this chapter, analytical philosophers do not need to be persuaded that logic has an important role to play in the evaluation of religious claims. This chapter can be seen to a certain extent as a vindication of analytical philosophers’ reliance on logic. Still, there are lessons to be drawn about how analytical philosophy of religion should be practiced. In this chapter, I have argued for a form of logical pluralism, so this vindication of logic should not be taken as a vindication of classical logic—analytical philosophers of religion should be aware of the existence of alternatives to classical logic, and be willing to use them where appropriate. If the refined form of logical pluralism that I have been defending is correct, although there is no such thing as the One True Logic, a particular logic may be appropriate or inappropriate for a particular application. There are different forms of logic because there are different kinds of commitment, so a philosopher of religion must try to understand what kind of commitment is involved in a particular religious claim, and find the logic that is appropriate to this commitment. The work of Searle and Vanderveken is an important starting point. This task requires an understanding of logic, but also sympathetic attention to the history of the community whose commitments are being studied. If I am right that a commitment is religious because of the kind of community that is committed to it rather than because the commitment itself is inherently religious, then there is no

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shortcut that will enable us to find the system of logic that is appropriate for all religious claims. There is plenty of scope for future work. If I am wrong, the best I can hope for is that someone else will point out my errors. But I have at least made an effort to avoid some errors that have been ascribed to analytical philosophers of religion (Lewis 2015, pp. 16–23). I have not focussed exclusively on one religion, I have tried to avoid an ahistorical approach and I have tried not to intellectualize religion, meaning that I have focussed on religions as attempts to live in the world rather than attempts to explain what reality is. I have argued that logic has a role to play in religious discourse because logicians study the relationship between commitments and there is no such thing as a religion without commitments. Readers who still reject the idea that logic has a role to play in religious discussions may have a different idea about what religion is, but it may also be that they have a narrower idea than I do about the scope of logic. Anyone who rejects the idea that there is a connection between logic and religion must rise to the challenge of explaining what logic is, and that requires some attention to recent developments in the study of logic. Acknowledgments I am very grateful to have been given an opportunity to present a version of this paper at the 1st World Congress on Logic and Religion in 2015, and particularly grateful to have been given a chance to publish it in this volume even though I was unable to attend the congress for personal reasons. In October 2016 FSU-Panama provided me with funding to give a presentation on intuitive validity at a conference on ‘The Bounds of Logic Reloaded’, organized by the Higher School of Economics in Moscow, and I am grateful for the feedback I received there. Daniel von Wachter read an earlier draft of this chapter, and his comments were very helpful, as were the comments by Ricardo Silvestre and the referee for this volume. Without the support of my wife Rocío this chapter would not have been written.

References Arnal, William, and Russell T. McCutcheon. 2013. Words, words, wordbooks or everything old is new again. In The sacred is the profane: The political nature of religion, 31–56. Oxford: Oxford University Press. Beall, J.C., and Greg Restall. 2006. Logical pluralism. Oxford: Oxford University Press. Béziau, Jean-Yves. 2010. Logic is not logic. Abstracta 6 (1): 73–102. Brandom, Robert. 1994. Making it explicit. Cambridge, MA: Harvard University Press. Caputo, John. 1997. The prayers and tears of Jacques Derrida: Religion without religion. Bloomington: Indiana University Press. ———. 2000. Philosophy and prophetic postmodernism: Toward a Catholic postmodernity. American Catholic Philosophical Quarterly 74: 549–568. ———. 2001. On religion. London: Routledge. ———. 2006. The weakness of God: A theology of the event. Bloomington: Indiana University Press. ———. 2006b. Philosophy and theology. Nashville: Abingdon Press. ———. 2013a. The insistence of God: A theology of perhaps. Bloomington: Indiana University Press. ———. 2013b. Truth: Philosophy in transit. London: Penguin.

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Cook, Michael. 2014. Ancient religions, modern politics: The Islamic case in comparative perspective. Princeton: Princeton University Press. Cook, Roy T. 2010. Let a thousand flowers bloom: A tour of logical pluralism. Philosophy Compass 5: 492–504. Davidson, Ronald M. 2002. Indian Esoteric Buddhism: A social history of the tantric movement. New York: Columbia University Press. Dummett, Michael. 1978a. Truth. In Truth and other enigmas, 1–25. London: Duckworth. ———. 1978b. The philosophical basis of intuitionistic logic. In Truth and other enigmas, 215– 248. London: Duckworth. ———. 2000. Elements of intuitionism. 2nd ed. Oxford: Clarendon Press. ———. 2004. Truth and the past. New York: Columbia University Press. Gombrich, Richard. 2006. Theravada Buddhism: A social history from ancient Benares to modern Colombo. 2nd ed. London: Routledge. Haack, Susan. 1978. Philosophy of logics. Cambridge, UK: Cambridge University Press. Harvey, Peter. 1990. An introduction to Buddhism: Teachings, history and practices. Cambridge, UK: Cambridge University Press. Jaffrey, Zia. 1996. The invisibles. New York: Pantheon Books. Lewis, Thomas A. 2015. Why philosophy matters for the study of religion – and vice versa. Oxford: Oxford University Press. Lokamitra, Dhammachari. 1991. Ambedkar and Buddhism. Economic and Political Weekly 26: 1303–1304. Masuzawa, Tomoko. 2005. The invention of world religions: Or, how European Universalism was preserved in the language of pluralism. Chicago: University of Chicago Press. Müller, Max. 1907. Collected works of the right honourable. In Natural religion, ed. F. Max Müller, Vol. I. London: Longmans, Green. Priest, Graham. 2001. An introduction to non-classical logic. Cambridge, UK: Cambridge University Press. Reddy, Gayatri. 2005. With respect to sex: Negotiating Hijra identity in South India. Chicago: University of Chicago Press. Searle, John, and Daniel Vanderveken. 1985. Foundations of illocutionary logic. Cambridge, UK: Cambridge University Press. Shapiro, Stewart. 2014. Varieties of logic. Oxford: Oxford University Press. Shestov, Lev. 1966. Athens and Jerusalem. Trans. Bernard Martin. Athens, OH: Ohio University Press. Sun, Anna. 2013. Is Confucianism a world religion? Princeton: Princeton University Press. Vajpeyi, Ananya. 2012. Righteous republic: The political foundations of modern India. Cambridge, MA: Harvard University Press. Wiebe, Donald. 1991. The irony of theology and the nature of religious thought. Toronto: McGillQueen’s University Press.

Benjamin Murphy studied Philosophy and Theology at Oxford University, graduating in 1993. He returned in 1995 for postgraduate studies in Philosophical Theology, completing his doctorate in 2000. He is now Professor in Philosophy and Religious Studies in Florida State University’s branch campus in the Republic of Panama.

Chapter 15

Logic in Islam and Islamic Logic Musa Akrami

15.1 Introduction Logic has a central place both in Islam as a religion and in the Islamic system of thought. It has found such a place according to its place in reasoning and inference of various relevant judgments in diverse fields, particularly in religious precepts, theology, and philosophy. Qur’¯an, as the canonical sacred book (with its so-called revealed origin), has shown some important manifestations of reasoning and logical inference of some judgments of religious practice and theoretical theology that concerns with God’s existence, origin of the world, origin of the man, and life after death. Commonsensical reasoning as well as some traditions of genuine logical reasoning in the Arabia region and its neighbors helped Muslim religious leaders to use logic for their various religious aims. Islam was flourished within a cultural context shaped by some Greek, Christian, Jewish, and Iranian effective elements. Before Islam, there had been struggles between two deploys of widespread trust and widespread distrust towards Greek philosophy and logic within both Jewish and Christian societies. Researches on the history of the entry of Greek philosophy and logic into those societies have reported a “genre of literature – the defense raisonnée of the religious acceptability of philosophical studies”, that have been “developed on account of this antagonism” (Suy¯ut¯ı 2008, p. 55). Due to its appealing potential and possibilities, the Greek heritage in logic was broadly translated into other languages (for Arabic translation of Aristotle’s writings see (Walzer 1962)). The leading role of logic gave rise to studying, learning, and

M. Akrami () Department of Philosophy of Science, Islamic Azad University, Science and Research Branch of Tehran, Tehran, Iran e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_15

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teaching it in various religious and philosophical circles. In spite of some antiphilosophical trends, it is not surprising that the translation of Greek works in logic was welcomed by most scholars with their own strong Islamic religious and theological attitudes. Such an attitude led to the emergence of important logicians and extension of Greek logic in the Islamic world. This chapter is arranged in three main sections to show the place of 1. reason and reasoning in Islam as a religion, 2. logic in Islam as a religion, and 3. the well-known “Islamic logic”, a tradition in the history of logic that was developed in the Islamic world largely by Muslim logicians, in relation to logic in Islam as a religion that has its own sacred texts, on the one hand, and the theoretical and practical issues that possibly makes using logic necessary, on the other hand.

15.2 Reason and Reasoning in Islam Islam is a religion that has its own rich system of theology and jurisprudence. The use of reason and reasoning in theoretical and practical issues within various mainstream schools of theology and jurisprudence is arguably unique to Islam, compared to any other religion. We try to give a short, sufficient description of the place of reason and reasoning in Islamic sacred texts and in theoretical subjects of theology and practical issues of jurisprudence.

15.2.1 Types of ‘Aql in Religious Texts, Theology, and Metaphysics Depending on the field of study, one may find different terms for the faculty of reasoning in the Islamic tradition of logico-philosophical thought, the most important word of which is “‘aql”. We may choose the English terms “Intellect”, “Intelligence”, or “Reason”. ‘Aql appears as a canonical concept in the large part of philosophy, theology, logic, jurisprudence, and, even, religious sacred texts, so that one may find various manifestations of these terms in various texts on Qur’¯anic commentary, theology, philosophy, and jurisprudence. The main usages of the terms can be listed as follows: Speculative Reason, Practical Reason, Particular Reason, Universal Reason, Common Reason, Sacred Reason, Human Reason, Angelic Reason, Divine Intelligence, First Intellect, Potential Reason, Actualized/Actual Reason, Passive Intellect, Active Intellect, Material Intellect, Pure Reason, and Acquired Reason (cf. Fakhry 1997, pp. 44, 53; Fakhry 2002, pp. 72–73).

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15.2.2 ’Aql in Qur’¯an One may find reason and reasoning in Qur’¯an, the most authoritative sacred text for all Muslims, in various cases; they may be classified in three categories: 1. Words derived from the Arabic “’aql” (= Intellect/Intelligence /Reason), which have been used 49 times in Qur’¯an (e.g. 2: 75, 10: 16, 13: 4, 67: 10); 2. Words derived from different infinitives with meanings more or less close to the meaning of “to reason” (e.g. “to reflect” in 2: 19 (“tatafakkar¯un” = “perhaps you may reflect”), “to understand” in 2: 269 (“¯ulul-alb¯ab” = “men of understanding”), and in 20: 28 (“yafqah¯u qowl¯ı” = “that they may understand my saying”), and some other infinitives such as “to think”, “to grasp”, and “to comprehend”); 3. Cases of reasoning, arguing, and inferring a conclusion on the basis of some premise(s), in part concerned with theological issues. Another important word is “‘ilm” (= knowledge, awareness) the frequency of which is 80 (e.g. “knowledge” in 2: 32 (“l¯a ‘ilm-a lan¯a” = “we have no knowledge”)). The frequency of other words derived from “’ilm” as a root is 667 (e.g. 2: 30 (“inn¯ıa’lam-o m¯al¯ata’lamun” = “surely I know what you do not”) and 2: 77 (“av-a l¯a ya’lam¯un-a ann-all¯ah-a ya’lam-o . . . ” = are they then unaware that God knows . . . ”)) ‘Aql plays important roles in Qur’¯an, some of which are as follows: 1. As a means to know God, to prove His existence and His oneness; 2. As a means to know the universe, its origin, and its laws; 3. As a means to know the commands of God (i.e. religious rules and decrees originated from God); 4. As a means to believe in life after death; and 5. As a means to judge human actions according to the Divine laws. All of these cases are based on a commonsensical approach to the roles of reason and reasoning in convincing the readers and listeners to accept the doctrines manifested in the Qur’¯an’s verses. Such an approach makes possible 1. to use reasoning in Islamic jurisprudence, and 2. to find examples for conciliating the sacred texts and the Greek logic by theologians and religious scholars such as al-Ghaz¯al¯ı (c. 1058–1111 AD, the anti-philosopher Ash’ar¯ı Jurisprudent, theologian, and S¯uf¯ı).

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15.2.3 ‘Aql in Immaculate Holy Religious Leaders’ Sayings Islam, as a religion, has been comprised of two types of doctrines: 1. Theoretical theological doctrines concerning important issues such as God, Being, the Universe, the Origin of Man, and the Life after Death (and Resurrection); 2. Practical doctrines concerning various issues of individual and collective lives of the believers and of those non-believers who live in an Islamic society; the collection of these doctrines makes the Islamic jurisprudence. Qur’¯an, as the so-called revealed holy book respected by all Muslims, is the first and most fundamental source for both types of doctrines. However, according to some epitomes and seemingly vague verses in Qur’¯an, there are a lot of differences concerning the interpretation of its relevant verses. It is of no surprise that the sayings (= had¯ıths in Arabic) and practices of the first religious leaders have had the potential of being an additional source (along with Qur’¯an) for both theoretical and practical doctrines. Accordingly, there are two sacred texts as the sources of various doctrines: 1. Qur’¯an (with its absolute authenticity and authority for all Muslims); and 2. Collections of the sayings and practices of the immaculate holy religious leaders. The differences in views concerning various issues (including the leader who is entitled to be considered as immaculate) have led to various religious branches, the most important of which are Sunn¯ıism and Sh¯ı’¯ıism with their own internal theological and legal divisions. According to Sunn¯ıism, there is only one immaculate holy religious leader, who is the Prophet Mohammad, while there are several immaculate holy religious leaders for Sh¯ı’¯ıism: the Prophet and Im¯ams (with the difference in number according to the beliefs of various sub-branches). Sunn¯ıs (as the largest branch of Islam) have four principal legal sub-branches: Hanbal¯ısm, Sh¯afi’¯ısm, Hanaf¯ısm, and M¯alik¯ısm. The most important sub-branch of Sh¯ı’¯ısm is Twelver Im¯am¯ı Sh¯ı’¯ıism. Now we introduce the “authentic” collections (i.e. the Had¯ıth books) of sayings and practices of the immaculate holy religious leaders, for the main Islamic branches. For the Sunn¯ıs there are six authoritative or canonical collections of reports concerning both sayings and practices of the immaculate holy religious leader who is the Prophet. These books are: 1. 2. 3. 4. 5.

Sah¯ıh al-Bukh¯ar¯ı, Sah¯ıh al-Muslim, Sunan-e Ab¯u D¯aw¯ud, J¯ami’-e at-Tirmiz¯ı, Sunan as-Sughr¯a, and

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6. Sunan-e Ibn M¯ajah (it must be said that, for M¯alik¯ıs, Sunan-e Ibn M¯ajah is replaced by Muw¯att¯a of M¯alik). For Sh¯ı’¯ıs, the authoritative canonical books of had¯ıth are four: 1. 2. 3. 4.

Kit¯ab al-K¯af¯ı, Tahz¯ıb al-Ahk¯am, Istibs¯ar, and Man l¯a Yahzuruhu al-Faq¯ıh.

In both sets of the Had¯ıth books of above main Islamic religious legal branches, there are hundreds, and even thousands, of sayings concerning “‘aql”, attributed to the Prophet Mohammad and/or some of the Im¯ams in Sh¯ı’¯ıism, the most significant of which is: “The first thing God created was reason”. This saying has been mentioned in the main authoritative texts of Sunn¯ıs. It has been accepted in Sh¯ı’¯ıi texts too, though in some different versions. As mentioned, one of the four authoritative Sh¯ı’¯ı collections of twelve Im¯ams’ sayings and traditions is al-K¯af¯ı (of Kit¯ab al-K¯af¯ı = the Sufficient Book), having been compiled by Mohammad ibn Ya’qub ibn Ish¯aq al-Kulayn¯ı, with 34 parts. The collection of the first eight parts of al-K¯af¯ı, called Us¯ul al-K¯af¯ı (= the Principles of al-K¯af¯ı or the Sufficient Principles) contains the sayings and traditions that deal largely with theoretical issues concerning epistemology, Qur’¯an, theology, and ethics. The collection of the other 26 parts, called Fur¯u’ al-K¯af¯ı (= the Offshoots of al-K¯af¯ı or the Sufficient Offshoots), contains the sayings and traditions that deal largely with practical issues concerning various individual and collective religious practices. The first part of Us¯ul al-K¯af¯ı is called Kit¯ab al-‘Aql wa al-Jahl (= the Book of Intellect and Ignorance) that has 36 important sayings. The second part is called Kit¯ab fadl al-‘ilm (= the Book of the Merit of Knowledge), containing 176 important sayings. It is interesting that Mull¯a Sadr¯a, as one of the leading Twelver Sh¯ı’¯ı philosophers, has a very important philosophico-mystico-theological commentary on Us¯ul al-K¯af¯ı, with significant notes of the first book (particularly ‘Aql (= “intellect”)). He has very important, more or less neo-Platonic, philosophical comments on the sixth Im¯am’s saying that “God created al-‘aql and it is the First Creature God created among the ‘spiritual beings’” (see Mull¯a Sadr¯a [= Sadr ad-D¯ın Muhammad IbnIbr¯ah¯ım al-Sh¯ır¯az¯ı] 1981, vol. 1, 400ff). Most of the Muslim philosophers before Mull¯a Sadr¯a, for example al-F¯ar¯ab¯ı and Ibn S¯ın¯a, particularly influenced by both Plotinus philosophy and the Prophet’s sayings, have asserted that the first creature is “Intellect” (see Badav¯ı & Or-Rahm¯an 1977, pp. 135–136; Mull¯a Sadr¯a [= Sadr ad-D¯ın Muhammad IbnIbr¯ah¯ım al-Sh¯ır¯az¯ı] 1366, vol. 1, p. 217). The above saying, attributed to the immaculate holy religious leader(s), has had vast implications and far-reaching consequences for philosophy, logic, theology, Qur’¯anic commentary, mysticism, and jurisprudence in both Sunn¯ıism and Sh¯ı’¯ıism.

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15.2.4 ‘Aql in Jurisprudence (Fiqh) Fiqh (literally meaning, in Qur’¯an, “deep understanding”) is the “Science of Inferring the Ordinances of Religious Law” from relevant authoritative (Islamic) religious sources. According to the mainstream Islamic jurisprudence, all human actions are of one of the five types: 1. Obligatory/necessary (W¯ajib = must be done), 2. Prohibited/forbidden (Har¯am = must not be done (one will be punished for doing it)), 3. Recommendable/recommended (mostahab = it is better to be done (one will be compensated for doing it but will not be punished for not doing it)), 4. Undesirable (Makr¯uh = it is better not to be done (it is better not to be done, one will not be compensated for not doing it but will be punished for doing it)), and 5. Permissible (Mub¯ah = there is no difference between it being done and it not being done (there is no compensation for doing it and no punishment for not doing it)). In recognizing the appropriate ordinances concerning human actions, one may be confronted with three cases: 1. The ordinances for these actions may exist directly and explicitly in the sacred texts, 2. They may exist indirectly and implicitly in the sacred texts, and 3. They may not exist at all in the sacred texts. The jurisprudents have the duty of codifying the ordinances of the first type and inferring the ordinances of the second and third types. They have, according to their beliefs in their religious branch, their own appropriate rules and tools (i.e. sources and fundamentals) to do their duties.

15.2.4.1

Sources and Fundamentals of Jurisprudence (Fiqh) According to the Sunn¯ıism

According to four leading Sunn¯ı schools (i.e. Hanaf¯ısm, Hanbal¯ısm, Sh¯afe’¯ısm, and M¯alik¯ısm), the sources of deriving the ordinances of Islam are 1. Qur’¯an, 2. Prophet’s sayings and practices, and 3. Consensus. Ab¯u Han¯ıfeh (the founder of the Hanaf¯ı School) has accepted a fourth source, i.e. Judicial Reasoning by Q¯ıy¯as (= Analogy, similar to analogy in logic). M¯alik¯ıs and Hanbal¯ıs pay no attention to judicial reasoning by analogy, whereas the Sh¯afi’¯ıs have an intermediate position.

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There are some differences among the schools concerning the interpretations and details of consensus, the conditions of judicial reasoning by analogy, as well as the possibility to consider other sources such as Induction, Authorizing the Previous State, Juristic Preference, Custom or Common Sense, the Predecessors’ Laws, and the (Prophet’s) Companions’ Adjudges. Moreover, scrutinizing on Q¯ıy¯as, the scholars undertook a logical debate concerning the definition and characteristics of Q¯ıy¯as, comparing it with analogical reasoning, inductive reasoning or categorical syllogism.

15.2.4.2

Sources and Fundamentals of (Islamic) Jurisprudence (Fiqh) According to the Sh¯ı’¯ıism

First of all, we must say that according to Sh¯ı’¯ı jurisprudents the analogy accepted by Sunn¯ı jurisprudence is wrong and false because 1. It adheres to guess, opinion and supposition, and 2. The principal doctrines and teachings of both Qur’¯an and immaculate holy religious leaders are sufficient for answering any questions concerning religiously right actions. In Im¯am¯ı-Twelvers Sh¯ı’¯ı School, the sources for inferring the ordinances of Fiqh are 1. 2. 3. 4.

Qur’¯an, The Prophet’s and immaculate holy Im¯ams’ sayings and practices, Consensus (on the basis of the Immaculate’s sayings and practices), and Reason.

Of course, there are some tendencies (e.g. among the Traditionalist) to insist on the Book (i.e. Qur’¯an) and Tradition (i.e. the Immaculate’s sayings and practices), without any belief in consensus and reason. Sh¯ı’¯ı jurisprudents (with the exception of the Traditionalist) hold that the authority of reason means that a judgment is authoritative if the intellect’s judgment concerning it is certain.

15.2.4.3

‘Aql in Science of the Principles of (Islamic) Jurisprudence (‘Ilm-e Usul-e ¯ Fiqh)

The Science of the Principles of Jurisprudence (‘Ilm-e Us¯ul-e Fiqh) is the science of the rules and tools that are used to derive the commands of jurisprudence. It teaches the safe and right method of inferring the commands from the sources of jurisprudence. It is, therefore, a normative or prescriptive science, being closer to technique than to science (speaking of “ought” rather than of “is”). There is a section in Mu’tazil¯ı and Sh¯ı’¯ı “Science of the Principles of Jurisprudence” that speaks of “rational good and evil”, according to which the necessity

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of obeying the commands and prohibitions of the Divine Lawgiver is a rational necessity. In this regard, the specialists in the first principles recourse to “the rule of correlation/mutual implication”: “all judgment of reason would be the judgment of the divine religious law too”, and vice versa. (see, e.g., Muzaffar 1393, p. 250) ‘Abd al-Jabb¯ar, the Mu’taz¯ı judge, says that the proof for authority of the Book (i.e. Qur’¯an), the Immaculate’s Tradition, and Consensus is a proof by reason. He holds that, in inferring the religious ordinances, reason is prior to other three sources (see ‘Abd al-Jabb¯ar 1384, p. 88)

15.2.4.4

The Sciences Necessary for Muslim Jurisprudents

Muslim jurisprudents usually study a certain curriculum as necessary preliminaries for inferring the ordinances in jurisprudence. This curriculum consists of courses as follows: 1. 2. 3. 4. 5. 6. 7. 8.

Arab Literature, Morphology, Philology, Syntax, Semantics, Figurative expression, Exegesis, Science of traditions (i.e. sayings and practices of the immaculate religious leaders), 9. Biography and authority of narrators of the traditions, 10. Science of fundamentals and methodology of jurisprudence, and 11. Logic. Of course, different schools or sub-schools have their own authoritative texts, some of which have been adopted as classic texts with their own claimed authenticity.

15.2.5 Capability of Reason in Understanding the Depth of the Doctrines and Justifying Them As mentioned, according to some schools in the science of the principles of jurisprudence (particularly Mu’tazilism and Sh¯ı’¯ısm) reason has a central status in both understanding the doctrines and inferring the ordinances, so that all rational statements are among the valid religious statements. In such an approach, most, if not all, of religious doctrines and decrees are intelligible (ma’qul al-ma’n¯a). It has been said by religious authorities that the first principles are understandable on the

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basis of reason and logic, so that being a Muslim or embracing Islam must be on such a basis. Thus, those who want to enter into Islam must be able to make use of their cognitive faculty to recognize and/or infer the ordinances. Accordingly, though in theory not in practice, the prerequisites for entering into Islam are 1. Having reached the age of puberty and 2. Being sane. Moreover, the dogmatic orders and prohibitions are valid for those people who 1. Have reached the age of puberty, and 2. Are sane. However, there are some doctrines, and some decrees manifested in the sacred texts, that cannot be comprehended by reason, though (in spite of such an inability) reason has no disagreement with them. These doctrines and decrees are dogmatic (ta’abbud¯ı) ones that have been revealed by God and must be obeyed without need of any rational justification. Here, some opponents have found an Achilles’ heel to criticize Islam for its probably unjustifiable/unreasonable orders and prohibitions, though the defenders and apologists give their own explanations and justifications. These explanations and justifications are based on the thesis that Qur’¯an is the word of God, and, accordingly, everything in it is correct, even though any mind other than the Divine mind (including the human mind) cannot comprehend something that has appeared in Qur’¯an.

15.3 Logic in Islam By “logic in Islam” we mean 1. Logic in sacred books (particularly Qur’¯an), 2. Logic in Islamic theology as a tool for reasoning and argumentation for theological issues (e.g. existence of God), and 3. Logic in jurisprudence for deriving religious orders and prohibitions. We try to give short accounts of each topic.

15.3.1 The Word for “Logic” in Pre-Islamic Persia and the Islamic world: Its Root and History It is believed that the term “Logic” is from the Latin (ars) logica, from the ancient Greek logike (techne), meaning “reasoning (art)”, from fem. of logikos

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“pertaining to speaking or reasoning” from logos, meaning “reason, idea, word” (see, e.g., {www.etymonline.com/index.php?l=l&p=22&allowed_in_frame=0}). It can be noted that the meanings of the ancient Greek logos are “word, thought, idea, argument, account, reason, or principle”. Similarly, logike has meanings such as “possessed of reason, intellectual, dialectical, argumentative” (see Marmura 1963, the entry “logikos”). It is clear that Iranians of the S¯ass¯an¯ıd era (224–651 AD) had been acquainted with Plato’s and Aristotle’s works, so that they had taught and learned both philosophy and logic. According to different researchers, referring to different documents (from Khurdeh Avest¯a to Persian Paul’s treatise on Logic), there was several words for logic in pre-Islamic Persia (particularly S¯ass¯an¯ıd Persia): 1. Sukhun, from “sukhan”, meaning speech or word, having the same relation to Sukhun as the relation of “Logic” to “Logos”; 2. Çim-guwag¯ıh (çim means “meaning” and “gowagih” means “reasoning” or “rationality”), 3. Tarkeh (a Sanskrit word), with its Arabicized forms “tarq” and “tarqa”; 4. Mant¯ag (from “man” = to know, and “t¯ag”, a suffix which means “concomitant” and “compatible”; and 5. V¯ır as it is seen in Khordeh Avest¯a. The Arabic word accepted for “Logic” is “Mantiq”, probably from the word “nutq”, meaning “speech”, “utterance”, and “oration”. The literal meaning of “mantiq” is “speech” and “language”. Referring to texts such as Khurdeh Avest¯a (Avest¯a 1993, pp. iii and 104), some scholars argue that the root of the apparently Arabic word “mantiq” is not “nutq” but its root is the Avestaic word “m¯anthra” which means “spell”, or “secret mystery”. In Islam (as a religion) and Islamic culture, or, better, in Islamic-Arabic religious and nonreligious literature, mantiq has various meanings and usages, some of which are as follows: 1. Reason in religious tradition, or reasonable/rational action compatible with intuition or common sense; 2. Foundation or doctrinal principle for moving towards truth or rightness; 3. Religious justification or explanation of (a) some unusual statements in theology or exegesis of Qur’¯an, and (b) some practical affairs or rituals; it is argued that most of these justifications or explanations show that such statements and affairs are reasonable; it is in such a framework that the experts in Islamic jurisprudence believe that one is entitled to speak of “the logic behind Islamic rules/laws”; 4. “Islamic logic” as the whole of Muslim logicians’ activities and achievements in logic from about 900 AD to about 1400 AD on the basis of Greek logic (both Aristotelian and Stoic), as well as some innovations.

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15.3.2 “Mantiq” and Logic in the Sacred Texts The words derived from “nutq” (= speech) have been used 12 times in Qur’¯an, one of which is “mantiq” itself in the term “Mantiq ot-Tair” (= the language of birds that Solomon had been taught {Qur’¯an, al-Naml: 16}) Depending on the degree by which one believes in Qur’¯an’s stance on logic, one may recognize three points of view concerning the meanings and usages of “mantiq” and logic. We give below a short report on these three views.

15.3.2.1

Logic in Qur’¯an According to Minimalist Views

The main Islamic theological and/or legal groups that weaken the role of reason in interpretation of Qur’¯an are as follows: 1. Literalists (or Z¯ahir¯ıs, i.e. those who, generally belonging to Sunn¯ıism, rely on the apparent and literal meaning of the words of Qur’¯an and the Prophet’s sayings and usually reject the derivation of verdicts through reasoning); 2. Traditionists (Akhb¯ar¯ıs, those who, belonging to Twelver Sh¯ı’¯ısm, deny the capability of reason and believe in Qur’¯an and the immaculate Twelve Im¯ams’ sayings as the only sources of deriving the religious verdicts); 3. Segregationists (Tafk¯ık¯ıs, those Twelver Sh¯ı’¯ıs who reject any use of philosophy and other non-revealed knowledges in interpreting Qur’¯anic verses), and 4. Ash’ar¯ıs (those who, opposing Mu’tazilism, rely on old interpretations of Qur’¯an). According to these groups, generally speaking, the Divine word and the Immaculates’ speech are incomparable with human speech, human reason and human logic, even if one may find some cases of similarities between some statements of the sacred texts and human reasoning.

15.3.2.2

Logic in Qur’¯an According to Moderate Views

According to the scholars having a moderate position, there are examples of logical techniques in both Qur’¯an and the Immaculate’s sayings. However, such a phenomenon is not a rule but rather an exception so that one is not entitled to look at such texts as manifestations of logic and rational argumentation.

15.3.2.3

Logic in Qur’¯an According to Maximalist Views

The only two leading schools of thought in theology and /or jurisprudence, as rationalist schools, are:

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1. Usulites (= foundationalists, those who, opposing to Akhbariism, believe in the conclusive role of reason and reasoning in deriving the verdicts; they make the mainstream of Twelver Sh¯ı’¯ı jurisprudence); 2. Mu’tazil¯ıs (those Sunn¯ıs who, opposing to Ash’ariism, believe in the central conclusive place of reason in theology). Generally speaking, according to these rationalist views: 1. Qur’¯an regards itself in plain Arabic (e.g. Y¯usuf : 2), so that it is in accordance with Aristotelian commonsensical rules of logic as the rules of natural thinking in natural language; 2. Qur’¯an speaks of contemplation and reasoning, and addresses those who are possessed of reason; 3. There are a lot of examples of reasoning on the basis of logical inference, so that a fair reader may find in Qur’¯an some cases of reasoning, methods of reasoning, deductive reasoning, inductive reasoning, analogical reasoning, exceptive syllogism (both conjunctive and disjunctive), implicit/abridged syllogism, reduction ad absurdum, inference on the basis of resemblance, poetics, rhetoric, dialectics (a variety of cases), demonstration, argumentation, and invitation to rational disputation; 4. One may find various arguments for existence, oneness, and attributes of God in Qur’¯an, the most important of which are as follows: (a) Teleological argument (i.e. argument based on design, many examples), (b) Cosmological argument (i.e. argument on the basis of necessity and contingency), (c) Ontological argument (with proponents like Mull¯a Sadr¯a, as the most leading Sh¯ı’¯ı philosopher, and Muhammad Hussayn Tab¯atab¯ay¯ı, as the most famous contemporary Iranian Islamic philosopher; they have posed their ontological argument as addendum of the verse 53rd of Fussilat), (d) Argument from human primordial (innate) disposition, (e) Argument from absolute, pure being, (f) Mutual hindering or mutual antagonism argument (Tam¯ano’ argument), (g) Arguments on the basis of causality or motion and change, comprehensive harmony of the universe . . . , and (h) Denying the idolaters’ and polytheists’ arguments as being fallacies; 5. There are two terms in Qur’¯an that have been interpreted by some logicians as logic: M¯ız¯an (Scales), and Qist¯as (Balance). The word M¯ız¯an (Scales) has been used 9 times in Qur’¯an, in two times of which it is accompanied by “al-Kit¯ab” (“the Book”), being sent by God (42: 17 and 57: 25). One of the meanings of M¯ız¯an is “criterion”. Ibn S¯ın¯a(c. 980—June 1037) has regarded logic as “M¯ız¯an”. Al-Ghaz¯al¯ı has made use of the term “M¯ız¯an” for naming the various syllogisms appearing in Qur’¯an. Mull¯a Sadr¯a, agreeing with al-Ghaz¯al¯ı, has interpreted “M¯ız¯an” as (a) “rational criterion” in general and

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(b) logic in particular. Muhammad Husayn Tab¯atab¯ay¯ı (16 March 1903 – 7 November 1981), has chosen the title “Al- M¯ız¯an” for his “Qur’¯anic Commentary” that has been considered by a majority of scholars as the most comprehensive and important Sh¯ı’¯ı Qur’¯anic commentary over the past 1400 years (see Walzer 1962). Qist¯as (Balance) has been used 2 times in Qur’¯an in the form of the term “alQist¯as al-Mostaq¯ım” (= the Just Balance). Al-Ghaz¯al¯ı has chosen this term for his book on logic in which he has tried to show that one can see the various forms of syllogism in Qur’¯an as the divine source of logic.

15.3.2.4

Attitudes Towards Some Apparent Contradictions in Qur’¯an

Some scholars (belonging to both Islamic and non-Islamic societies, from old ages to the present) have shown some contraries and contradictions among Qur’¯anic verses concerning both theoretical theological issues and practical legal orders. Muslim apologists have tried to show that 1. The contradictions and contraries are apparent, so that they have no reality; and 2. There are some justified ways for believing in consistency among the verses. The opponents have spoken of some internal inconsistencies (among some parts of Qur’¯an) as well as contradictions between some verses and some scientific theories. Both opponents and defenders make use of the logical reasoning and principles of contradiction and identity to, respectively, criticize or justify such phenomena (via, e.g., recourse to metaphor or allegorical/exoteric interpretation). There is a concept in Qur’¯an that is used by Muslim apologists to solve the evident contradictions reflected in Qur’¯an. This concept is “abrogation” (= naskh), according to which God has decided to reveal some verses that have superseded corresponding earlier ones, so that one finds in the existing Qur’¯an not only the abrogator (= n¯asikh) verses but also the abrogated (= mans¯ukh) ones, bringing about some cases of evident contradiction between the coupled (abrogatorabrogated) verses. The 106th verse of the S¯urah al-Baqarah confirms this phenomenon in a phrase that the critics usually do not regard as a justified explanation: “Nothing of our revelation (even a single verse) do we abrogate or cause to be forgotten, but we bring (in place) one better or the like thereof. Knowest thou not that Allah is Able to do all things?” (Qur’¯an, 2: 106) The defenders, e.g. Muhammad Hussayn Tab¯atab¯ay¯ı, had regarded the phenomenon of “abrogation” not an example of contradiction but an indication of “addition and supplementation” (Walzer 1962, vol. 1). Such a justification has been challenged by the opponents, arguing that it is irrational to ascribe such a thing to the so-called omniscient and omnipotent “God”. They usually regard such an explanatory concept as a way to justify the contradiction manifested in various mutually contradictory verses appearing in the book attributed to God.

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Finally, there are also some long-established debates concerning “revelation” and its relation to reason, which constitutes 1. 2. 3. 4.

The nature of revelation, Its being real or fictional, Coherency or incoherency between reason and revelation, and Priority of reason and revelation with respect to each other.

15.3.3 Transmission of Aristotelian and Stoic Logic into the Islamic World and Development of Islamic Logic The new religious Islamic society had its own appropriate needs in argumentation and reasoning concerning various issues, questions, and problems in various fields such as theology, ideology, teaching the doctrines, and both derivation and justification of practices within the framework of Islamic legal principles and norms. In spite of some antagonistic attitudes towards logic in the Islamic society, surely there is no other religious society in the history that can be compared to this society in the extent of the need and making use of logic. Here we provide an overview of how logic was introduced to this society, particularly in connection to religious prescriptions for using logic.

15.3.3.1

Development of Islamic Logic

The historical development of Islamic logic can be seen in the following order: 1. 2. 3. 4.

The introduction of Greek logic into the neighbors of the Arabic region, The introduction of Greek logic into the Islamic world, Translation of Greek logic into Arabic language, Some elements of the Indian and Iranian traditions of logic entered into the Islamic world, 5. Assimilation of imported logic, 6. Genuine achievements in logic and logicography, and 7. Introduction of logic into the so-called Islamic sciences. During the great movement of translation in the Islamic world, Aristotle’s books, in particular his books on logic, were translated from Greek and/or Syriac into Arabic. His Organon, consisting of six treatises, were accepted as the books related directly to logic: 1. Categories, 2. On Interpretation, 3. Prior Analytics,

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4. Posterior Analytics, 5. Topics, and 6. Sophistical Refutations. These books, as well as Poetics, Rhetoric and Porphyry’s Isagoge were accepted as the classical parts of an Aristotelian nine-partite logic in the Islamic world. Ibn S¯ın¯a, in particular, has been an influential force in opening doors to Aristotelian logic, as one can see the manifestation of such an effort in his as-Shif¯a (see S¯ın¯a 1373). Moreover, Islamic logicians became familiar with Stoic logic as well as some theories in linguistics (for a thorough introduction to logic in the Islamic world see (Inati 1996; Rescher 1964; Suy¯ut¯ı 2008)).

15.3.3.2

Muslims’ Innovations in Logic

There are some innovations in the works of some Muslim logicians, some reports of which can be found in various papers and books written by Muslim or non-Muslim scholars across the globe. The main innovations encompass issues such as 1. 2. 3. 4. 5. 6. 7. 8.

The nature, aims, and tasks of logic; The relation between logic and language; The overlap of logic and metaphysics in some subject matters; The elements and structure of logic; Changes in logicography; Some exact analyses of topics such as propositions and syllogisms; Study of hypothetical syllogism; and Introducing temporal modal logic.

A more comprehensive report on the innovations and achievements of Muslim logicians is beyond the scope of this chapter (for a short account of Ibn S¯ın¯a’s innovations in the structure of the system of logic and logicography see (Akram¯ı 2015)).

15.3.4 Logic in Jurisprudence and the Science of the Principles of Jurisprudence Prima Facie, logic, as the science of thought and correct reasoning, was extensively welcomed in the Islamic world by logicians, theologians, and philosophers, as well as Qur’¯anic commentators, jurisprudents, and the scholars working on the fundamentals and first principles of jurisprudence. All of the schools of jurisprudence that believe, in one way or another, in deriving the religious verdicts from Qur’¯an and the Immaculate’s sayings (either directly or indirectly), or from other relevant principles or rules or verdicts, have been in need of using logic in its extensive form.

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There is a rich literature of applying logic in both Sunn¯ı and Sh¯ı’¯ı jurisprudence. Similarly, the theoretical theologians of both Sunn¯ı and Sh¯ı’¯ı schools have had a deep and extensive interest in the logical bases of their speculative debates and argumentations.

15.4 Qur’¯anic Reading of Logic in the Islamic World as a Specific Trend in the Works of Some Leading Muslim Theologians and Metaphysicians Logic has had a central place in the Islamic world of thought, having a strong connection with metaphysics, theology, mathematics, natural philosophy, Qur’¯anic commentary, ethics, and jurisprudence. We give an account of the impact logic has had on both leading classic philosophers and prominent classic anti-philosophers, in specific al-Ghaz¯al¯ı. All important philosophical systems have made use of logic. The main philosophical systems are: 1. Peripatetic Philosophy (with representatives such as ai-F¯ar¯ab¯ı and Ibn S¯ın¯a, largely Aristotelian in logic), 2. Illuminationist Philosophy (with Suhraward¯ı as the representative, largely critical of Aristotelian logic), and 3. Transcendental Theosophist Philosophy (with representatives such as Mull¯a Sadr¯a).

15.4.1 Logic in Leading Classic Philosophical Systems Peripatetic philosophy has made use of Aristotelian (and, in some cases, Stoic) heritage in logic and has extended it to new borders. Al-F¯ar¯ab¯ı has been one of the pioneers in logic and linguistic theory. He, adopting Aristotelian logic, has written significant commentaries on Aristotle’s books and has had some innovative writings particularly on the relation of logic and language (for a study of al-F¯ar¯ab¯ı’s logical writings and their Greek sources see (Abed 1991; Liddell and Scott 1935); for an example of al-F¯ar¯ab¯ı’s reading of Aristotle’s philosophy see (Al-F¯ar¯ab¯ı 1969)). Al-F¯ar¯ab¯ı wrote a treatise in defense of logic on the basis of the Prophet’s sayings, similar to John of Damascus who wrote within the tradition of “the apologia for logic [ . . . ] in defense of logical studies [ . . . that] worked powerfully for acceptance of this discipline among the Syriac-speaking Christians” (Rescher 1964, p. 55). In his important book on classification of sciences, he has tried to locate logic among the techniques and sciences (see Al-F¯ar¯ab¯ı 1968). He calls logic “the Header/Head of the Sciences” (for al-F¯ar¯ab¯ı’s and Ibn S¯ın¯a’s positions on the place of logic among sciences (see Al-R¯az¯ı and Ad-D¯ın 1373, p. 48). Ibn S¯ın¯a, as the

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most leading logician in the Islamic world over the centuries, has called logic “the Science of Scales”, or “the Science of Balance” (S¯ın¯a 1353, p. 10; for Ibn S¯ın¯a’s logic see S¯ın¯a 1952; S¯ın¯a 1984). Suhraward¯ı has criticized Aristotelian logic in various respects. Al-Ghaz¯al¯ı, in spite of having a counter-philosophical view, has tried to show that logic has a divine origin (see below). His central place in the Islamic world on the one hand, and his approach to logic and enthusiastic religious support of it played an important role in the acceptance of logic even within some radical religiously biased communities (for al-Ghaz¯al¯ı’s attitude towards logic see (Momen 1985)). Mull¯a Sadr¯a, as the most leading Sh¯ı’¯ı philosopher, has tried to combine philosophy, theology, mysticism and Qur’¯anic doctrines. He has been influenced by al-Ghaz¯al¯ı in his reading of Qur’¯an in the light of logic. It may be said that he has passed alGhaz¯al¯ı in the field of tracing most of the elements of classical logic in Qur’¯an as the “word of Allah”!

15.4.2 Counter-Logical Attitudes in Leading Muslim Scholars One may find the most famous of anti-logicians among traditionalists who were offensive against Greek heritage and rational attitudes towards Divine Revelation. There are some documents from the 3rd Islamic century (i.e. 9th Christian century) showing moderate or, even, radical disagreements with logic and its teaching or learning (e.g. Ibn Sharsh¯ır or N¯ash¯ı-ye Akbar and Hassan Ibn M¯us¯a Nawbakht¯ı). There are some reports on accusing al-F¯ar¯ab¯ı of heresy because of his works on philosophy and logic. Excommunication of Ibn S¯ın¯a by al-Ghaz¯al¯ıis was reflected in his Incoherence of the Philosophers (Tah¯afut al-Fal¯asifa). There was a widely used proverb to defame the logicians (indeed, the rationalists) in the era of flourishment of teaching and learning philosophy and logic: “whoever uses logic would become a libertine/impious” (“Man Tamantaq-a Tazandaq-a”). According to Jal¯al al-D¯ın al-Suy¯ut¯ı, philosophy, theology, and, above all, logic would weaken people’s faith. Thus he wrote a book to prohibit teaching and learning these disciplines: al-Qawl al-Mashriq f¯ı Tahr¯ım al-Ishtigh¯al bi al-Mantiq (= “The Uncontestable Verdict on the Proscription of Having a Preoccupation in Logic”) (Tab¯abtab¯ay¯ı 2017). Muslim scholars’ proscriptive attitudes towards reason and reasoning in general and Greek logic and rationalism in particular have an interesting and instructive long history from the early Islamic era to the contemporary Islam. One may give a sketchy list of some leading scholars who have tried to criticize and reject logic as manifested largely in Aristotelian logic: the Mu’tazil¯ıs such as Ab¯u Sa’¯ıd S¯ır¯af¯ı, ‘Abd al-Jabb¯ar, Jabb¯ay¯ı, and Abu al-Q¯asim Ans¯ar¯ı, the Ash’ar¯ıs such as Baqil¯an¯ı, the Sh¯ı’¯ıs such as Hassan Ibn M¯us¯a Nawbakht¯ı (one of the most leading scholars who plays an important role in consolidation of Im¯am¯ı school of theology, with a book under the title ar-Radd ‘al¯a Ahl al-Mantiq (= Refutation of the Logicians),

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¯ an the Salaf¯ıs such as Ibn Taym¯ıyyah with a book under the title Nas¯ıhat Ahl al-Im¯ f¯ı ar-Radd ‘al¯a Mantiq al-Y¯un¯an (= Advice to the Believers in Refutation of Greek Logic), in which he argues against usefulness of syllogistic logic (not, of course, against its validity). It must be said that some critics of Aristotelian syllogism (e.g. Fakhr ad-D¯ın R¯az¯ı and, even Ibn Taym¯ıyyah) have defended some kind of inductive reasoning. Nevertheless, logic has been widely accepted and welcomed by the majority of leading theological and legal schools of the Islamic world, so that many of the scholars from both metaphysical and Islamic legal traditions have written books on logic, either as encyclopedic and textbooks or as treatises with some innovations.

15.4.3 Our Case Study: al-Ghaz¯al¯ı as a Classic Anti-Philosopher Logician, and the Implications of His Attitude There have been some leading scholars in the Islamic community who have had antagonistic attitudes towards Greek sciences, particularly Greek philosophy. Some of them have disagreed with logic too. Some of the anti-philosophers have accepted logic as an important and, even, necessary tool for reasoning in various fields, in particular theology and jurisprudence. Al-Ghaz¯al¯ı may be considered as the most leading anti-philosopher who made an enthusiastic defense in the framework of his own reading of the sacred texts. We try to give a short account of his views on logic in the light of Qur’¯an.

15.4.3.1

Al-Ghaz¯al¯ı on the Relation of Logic and Qur’¯an

Al-Ghaz¯al¯ı, adopting the Greek heritage in logic as presented and extended by Muslim logicians (particularly by Ibn S¯ın¯a), is a good case for 1. The most leading theologian and religious authority in Ash’ari school, the only person who had the title Hojjat al-Islam (= “Proof of Islam”), 2. The most leading anti-philosopher all over the the Islamic world, 3. One of the most proponents of logic who has written some books on logic; in one of which he has borrowed a Qur’¯anic phrase as the title: al-Qist¯as al-Mostaq¯ım (= “the Just Balance”), appearing in Qur’¯an 17: 35 and 26: 182. This book is the most leading example of “the Islamization of logic”, in which al-Ghaz¯al¯ı tries to show that logic, as introduced in its mainstream manifestation in the Islamic world, is rooted in Divine Revelation (i.e. Qur’¯an). We give below an overview of al-Ghaz¯al¯ı’s main theses on the relation of logic and Qur’¯an (for details of al-Ghaz¯al¯ı’s views on logic and syllogism in the light of

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reflections on Qur’¯an, see (Al-Ghaz¯al¯ı and H¯amid 1959)/(‘Azm¯ı 1981; Al-Ghaz¯al¯ı and H¯amid 1978)). 1. He searches for “certain knowledge” and tries to find the Qur’¯anic genuine criterion for certainty. 2. He pays great attention to the epistemological grounds of certainty. 3. He speaks of the relation of certain knowledge and logic, as well as the method of achieving it. 4. He believes in the role of syllogism in achieving certainty. 5. He studies the relation between theology (= Kal¯am) and certainty. 6. He tries to show Qur’¯an’s classification of the types of arguments according to the different classes of people. 7. He argues for the necessity of derivation of general rules of Islamic jurisprudence from revelation on the basis of the Qur’¯an and the Prophet’s use of logic and other rational tools in his sayings and practices. 8. He derives five types of syllogism from Qur’¯an, calling them the criteria of Qur’¯an (its Qur’¯anic term is M¯ız¯an which is interpreted by al-Ghaz¯al¯ı as the logical rule of inferences). According to al-Ghaz¯al¯ı (and, following him, Mull¯a Sadr¯a) there are three primary kinds of criterion in Qur’¯an: (a) The criterion of “equivalence” (Ta’¯adul), (b) The criterion of “concomitance” (Tal¯azum), and (c) The criterion of “opposition” (Ta’¯anud); the criterion of equivalence is itself in three sub-kinds: Major, Middle, and Minor. 9. According to al-Ghaz¯al¯ı (and, following him, Mull¯aSadr¯a), in the process of teaching logic to humankind, God, Gabriel and Prophet are, respectively, the first teacher, the second teacher, and the third teacher (Muzaffar 1393, vol. 9, p. 300).

15.4.3.2

Five Sound Syllogisms Adopted by al-Ghaz¯al¯ı

Let us take a look at the five criteria. First of all, we speak of two types of syllogism: “conjunctive” (= iqtir¯an¯ı) and “exceptive” (= istisn¯a’¯ı). In Islamic logic, syllogism is a form of deductive inference in which two premises, taken jointly, give rise to the conclusion (a statement that cannot be more general than the premises). According to the explicit/implicit presence of the conclusion or its contradictory in the premises, the syllogism is divided into two kinds: conjunctive syllogism and exceptive syllogism. Conjunctive syllogism is composed of two premises and a conclusion, so that the conclusion or its contradictory does not exist explicitly/actually in one of the premises but is distributed implicitly/potentially in them. Predicative conjunctive syllogism is a syllogism both premises and conclusion of which are predicative statements in which a predicate is predicated to a subject.

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Thus, each part of the three parts of a predicative syllogism contains two predicative terms such that each of the premises has one term in common with the conclusion. Accordingly, one finds three terms in a predicative syllogism: (a) Minor Term, being the Subject of the Conclusion, (b) Major Term, being the Predicate of the Conclusion (it is more general than the minor term as the subject), and (c) Middle Term (or Common Term, which connects the subject and the predicate of the conclusion), being a Subject or Predicate repeated in both Premises, which is eliminated in the Conclusion. Each of the premises is in the form of one of the four forms “All S are P” (with the code A), “Some S are P” (with the code I), “No S are P” (with the code E) or “Some S are not P” (with the code O), where “S” is one term and “M” is another. “All S are P”, and “No S are P” are called universal propositions; “Some S are P” and “Some S are not P” are called particular propositions. A set of statements that have no Middle Term (or Common Term), cannot give rise to a conclusion. Such statements are called “Strange Statements”. According to the various positions of the Middle Term, predicative conjunctive syllogism may appear in four figures (figure is the conjunctive syllogism according to the position of the middle term with respect to two premises): 1. The 1st figure, in which the middle term is the predicate of minor premise and the subject of the major premise; 2. The 2nd figure, in which the middle term is the predicate of both minor and major premises; 3. The 3rd figure, in which the middle term is the subject of both minor and major premises; and 4. The 4th figure, in which the middle term is the subject of the minor premise and the predicate of the major premise (this figure does not exist in Aristotle’s Organon); Exceptive syllogism (having been elaborated by Ibn S¯ın¯a in ash-Shif¯a’) is a kind of syllogism in which the conclusion or its contradictory exists explicitly/actually in the premises, and one of its premises is “hypothetical”, either “connective” or “separative”. This kind was discussed for the first time by Theophrastus (after Aristotle’s death). The premise containing the conclusion must be a conditional statement. The other premise is an exceptive statement containing a word showing an exception. One of the conditional/hypothetical sides or its contradictory is excluded so that its contradictory is proven. According to conjunctivity or disjunctivity of the conditional premise, the exceptive syllogism is divided into two sub-kinds: conjunctive and disjunctive.

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Qur’¯an’s Five Criteria According to al-Ghaz¯al¯ı

In al-Ghaz¯al¯ı’s view, the five Qur’¯anic criteria that are in accordance with the three first figures of predicative/categorical syllogism and two exceptive syllogisms are as follows: 1. The major criterion of equivalence: the first figure of categorical/conjugate syllogism. One of the examples of this criterion in Qur’¯an is the Ibrahim’s argument against Namr¯ud (Baqarah: 258). 2. The middle criterion of equivalence: the second figure of categorical/conjugate syllogism; One of the examples of this criterion in Qur’¯an is the Ibrahim’s argumentation for recognizing God as the real and true deity (al-An’¯am: 76-77). 3. The minor criterion of equivalence: the third figure of categorical/conjugate syllogism. One of the examples of this criterion in Qur’¯an is the refutation of the Jews’ claim against the prophecy of Muhammad, using the fact of prophecy of Musa as a man (al-An’¯am: 91) 4. The criterion of concomitance: connective exceptive syllogism, the conclusion or its contradictory of which exists explicitly and one of its premises is hypothetical, either conjunctive or disjunctive; One of the examples of this criterion in Qur’¯an is the 22nd verse of the S¯ura Anb¯ıy¯a’ in which it is argued for oneness of Allah. 5. The criterion of opposition: separative exceptive syllogism. One of the examples of this criterion in Qur’¯an is an argument for the claim that Qur’¯an is the word of God, otherwise one would find many discrepancies in it (an-Nis¯a’: 82).

15.4.3.4

Mull¯a Sadr¯a and the Five Criteria

Mull¯a Sadr¯a has adopted al-Ghaz¯al¯ı’s attitude towards logic in Qur’¯an and manifestation of the fundamentals of syllogism in it. His confidence in both the authoritative texts of Islam and logic as elaborated by Ibn S¯ın¯a (on the basis of Greek logic, particularly Aristotelian logic) is seen in the following quotation, as a significant testimony of an attempt to find a strong justification for classic logic in Religious texts, on the one hand, and a strong evidence for the rationality of such texts on the other hand: “If one gains the knowledge of the five criteria that God has revealed to His Prophet, (s)he will certainly be guided; and if one does not make use of them and acts arbitrarily, (s)he will be aberrant.” (Mull¯a Sadr¯a [= Sadr ad-D¯ın Muhammad IbnIbr¯ah¯ım al-Sh¯ır¯az¯ı] 1981, vol. 1, p. 553–554). Such a position is an indication of entertaining logic that was Islamicized and established by al-Ghaz¯al¯ı and has been continued in the Islamic culture, including in the Sh¯ı’¯ı culture of Iran. Such a view on logic has had some positive and negative implications for logic itself as well as for other fields such as theology, metaphysics, Qur’¯anic commentary, and jurisprudence.

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The Positive and Negative Roles of al-Ghaz¯al¯ı’s and Mull¯a Sadr¯a’s Attitudes

Al-Ghaz¯al¯ı’s attitude and, following him, Mull¯a Sadr¯a’s attitude could play the role of a double-edged blade. Such an attitude has played a positive role in giving a warm welcome to logic in the Islamic Sunn¯ı and Sh¯ı’¯ı seminaries. However, the dogmatic fundamentals of logic which have relatively superficially derived from a so-called sacred revealed text might play a negative role in withholding any attempt to change basically the adopted logical system. Now, the defenders of the attitude of an Ash’ar¯ı anti-philosopher such as alGhaz¯al¯ı, as well as those of the attitude of a Sh¯ı’¯ı theosophist like Mull¯a Sadr¯a, have a great responsibility concerning the negative role of such an attitude. They should take a clear position on the relation of Qur’¯an and Modern Logic with its growing achievements in new fields. One may speak of various approaches towards such a problem: 1. There may be some anti-logician Muslims (or the Muslims disagreeing with logical reading of Qur’¯an) who will emphasize an “irreconcilability of logic and Qur’¯an”, insisting on the superior status of Qur’¯an compared with logic as a man-made discipline; 2. There may be some irreligious people who will find evidence for “antagonism between logic and Qur’¯an”, insisting on illogicality and irrationality of Qur’¯an; 3. There may be some dogmatic apologists insisting on classical syllogism as the genuine manifestation of logic inspired by God and reflected in Qur’¯an; 4. There may be some religious modernists who insist on “reconcilability of logic and Qur’¯an”, trying to find examples of modern logic in Qur’¯an; and 5. There may be some religious modernists who will be indifferent towards the relation between logic and Qur’¯an with a slogan “anything goes”, having no particular distinct position concerning the existence or non-existence of a plain relation between them.

15.4.4 Logic as a Part of Curriculums in Schools and Professions Logic, at least as a trustworthy instrumental science, has been a widespread introduction to all rational, argumentative, and inferential sciences all over the Islamic world (except within some small circles, adversary of the Greek tradition and opposing to using reason in reading religious texts). Accordingly, one may easily understand the importance of logic in curriculums of all schools, particularly all over Persia, so that it has been widely taught and learned in most of the religious schools on the basis of Ibn S¯ın¯a’s, Suhraward¯ı’s, T¯us¯ı’s, Abhar¯ı’s, Qutb ad-D¯ın R¯az¯ı’s, Taft¯az¯an¯ı’s, and ‘All¯amah Hell¯ı’s books. This

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tradition has powerfully continued until now, as one may find a large number of books on traditional logic published in Qum after the Islamic Revolution of 1979.

References ‘Abd al-Jabb¯ar. 1384 Lunar Hejirah. Sharh-e Us¯ul al-Khamsa (= Commentary on the five principles), edited by ‘Abd al-Kar¯ım ‘Usm¯an, Cairo. ‘Azm¯ı, T. 1981. as-Sayyed Ahmad, al-Ghazali’s Views on Logic, Thesis presented for the Degree of Doctor of Philosophy in the Faculty of Arts, University of Edinburgh (its file in pdf is accessible via internet). Abed, S.B. 1991. Aristotelian logic and the arabic language in al-F¯ar¯ab¯ı. Albany, NY: State University of New York Press. Akram¯ı, M¯usa. 2015. Ibn S¯ın¯a’s Two-partite versus nine-partite logicography. In The road to universal logic, ed. Arnold Koslow and Arthur Buchsbaum, 1–112. Switzerland: Springer (Birkhauser). Al-F¯ar¯ab¯ı. 1968. In Ihs¯a al-Ul¯um (= Enumeration of the sciences), ed. U. Amin, 3rd ed. Cairo: Librairie Anglo-Égyptienne. ———. 1969. Falsafah Aristutalis (Philosophy of Aristotle). Translated and edited by M. Mahd¯ı In Al-F¯ar¯ab¯ı’s Philosophy of Plato and Aristotle, Ithaca, NY: Cornell University Press. Al-Ghaz¯al¯ı and Ab¯u H¯amid. 1959. al-Qis¸ta¯ s al-Mustaq¯ım (= The correct balance), translated by Richard McCarthy, S. J.. http://www.Ghazali.org/works/qastas.htm ———. 1978. al-Qis¸ta¯ s al-Mustaq¯ım (The just balance), translated by D. P. Brewster. Pakistan, Lahore: Sh. Muhammad Ashraf. Al-R¯az¯ı and Fakhr Ad-D¯ın. 1373 Solar Hejirah. Sharh-e ‘Uyun al-Hikma, edited by Ahmad Hejazi As-Saqqa. Tehran: Mu’assesa as-Sadiq. Khurdeh Avest¯a. 1993. Translated into English by Ervad Maneck Furdoonji Kangama, Avesta Org., Kasson MN, USA. http://www.avesta.org/kanga/ka_english_kanga_epub.pdf Badav¯ı and ‘Abd Or-Rahm¯an. 1977. Eflutin ‘end al-Arab (= Plotinus among Arabs). Kuwait: Vekalatol-Matbu’at. Fakhry, M¯ajid. 1997. Islamic theology, philosophy, and mysticism: A short introduction. Oxford: Oneworld Publications. ———. 2002. Al-F¯ar¯ab¯ı, Founder of Islamic neoplatonism: His life, works and influence. Oxford: Oneworld Publications. Inati, S. 1996. Logic. Chap. 48 in History of Islamic philosophy, S.H. Nasr and O. Leaman, 802– 823. London: Routledge. Liddell, Henry George, and Robert Scott. 1935. Greek-english lexicon. Oxford: Oxford University Press. Marmura, M.E. 1963. Studies in the history of Arabic logic. Pittsburgh, PA: University of Pittsburgh Press. Momen, Moojan. 1985. An introduction to Sh¯ı’¯ı Islam: The history and doctrines of TwelverSh¯ı’¯ısm. Oxford: G. Ronald. Mull¯a Sadr¯a [= Sadr ad-D¯ın Muhammad IbnIbr¯ah¯ım al-Sh¯ır¯az¯ı]. 1981. al-Hikmat al-Muta’¯al¯ıyyah f¯ı Asf¯ar al-‘Aql¯ıyyah al-Arba’ah (= Trabscendent theosophy in the four rational journeys), Beirut: D¯arIhy¯a’ al-Tur¯ath al-‘Arab¯ı. Muzaffar, Muhammad Riz¯a. 1393. Us¯ul-e Fiqh (= The principles of Jurisprudence), translated into ¯ Persian by ‘Abb¯as Zer¯a’at and Hamid Masjedsar¯ay¯ı, Payam-e Now-Avar, Vol. 3. ———. 1964. The development of Arabic logic. Pittsburgh, PA: University of Pittsburgh Press. Ibn S¯ın¯a. 1353 Solar Hejirah. D¯anishn¯ame-ye ‘Al¯ay¯ı (= ‘Al¯ay¯ı Encyclopedia), Mantiq, Edition with Introduction by Muhammad Mu’in and Muhammad Meshkat, Tehran: Dehkhoda Book Shopping.

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———. 1373 Solar Hejirah. Burh¯an (= Demonstration), translated into Persian by M. Qav¯am Safar¯ı, Tehran: Fekr-e R¯uz Publishing. ———. 1952. ash-Shifa’ (Healing), al-Madkhal (Isagog), ed. G. Anawati, M. El-Khodeiri and F. al-Ahwani, rev. I. Madkour, Cairo: al-Matba’ah al-Amiriyyah. ———. 1984. al-Isharatwa al-Tanb¯ıh¯at (= Remarks and admonitions), Part one: Logic, translated by S.C. Inati, Toronto: Pontifical Institute of Mediaeval Studies. Suy¯ut¯ı, Jal¯al Ad-D¯ın. 2008. al-Qawl al-Mashriqf¯ı Tahr¯ım al-Mantiq (= “The uncontestable verdict on the proscripition of logic”), edited by Sayyid Muhammad ‘Abd al-Wahh¯ab, Cairo: D¯ar alHad¯ıth. Tab¯abtab¯ay¯ı, Muhammad Husayn. 2017. al-M¯ız¯an (in Arabic Language, 12 Vols), Vol. 2, Various Printings. Walzer, Richard. 1962. Greek into Arabic: Essays on Islamic philosophy. Cambridge: Harvard University Press.

Musa Akrami has studied both physics and philosophy. He is presently (full) professor at the Science and Research Branch of Islamic Azad University, Tehran, Iran. He has written/translated many papers, treatises, book-chapters, and books, in various fields such as cosmology, metaphysics, philosophy of science, philosophy of logic, political philosophy and mysticism. The PhD program in “Philosophy of Science” has been established by professor Akrami for the first time in Iran (2006). Personal University Webpage: http://faculty.srbiau.ac.ir/m-akrami/fa#employmentdata; Personal Weblog: www.musaakrami.blogfa.com.

Chapter 16

Thinking Negation in Early Hinduism & Classical Indian Philosophy Purushottama Bilimoria

16.1 Preamble In this chapter I examine a number of different kinds of negation and negation of negation developed in Indian thought, from ancient religious texts to classical philosophy. I shall be drawing from R.g Veda, the M¯ım¯am . s¯a, Ny¯aya, Jaina schools to Buddhist theorizing on the various forms and permutations of negation, denial, nullity, nothing and the possibility of nothingness. I will explore the logical elements of consistency, non-contradiction, and parsimony in their respective accounts; or consider whether they might be suggesting alternative perspectives on these very principles of negational reasoning. The present analysis follows the function of negation/the negative copula, nãn, and dialetheia in Indian grammar and logic. There is less of a concern with reductively parring affirmation and negation (or denial) with truth and falsity (the pairs are not coterminous as we are dealing with the structure of speech-acts, semantic fields and expressions). The main thesis argued for is that in the broad Indic tradition, negation cannot be viewed as a mere classical operator turning the true into the false (and conversely). Moreover, in the first two schools (M¯ım¯am . s¯a and Ny¯aya) the logical form of sentences are transformed into more complex syntax in order to disentangle their sense; whereas the Jaina and Buddhist school present either relative or non-discursive forms of assertions which cannot be reduced to the mainstream Boolean dichotomy: 1 vs 0. Since few of the negative expressions contravene or defy conventional meaning, certain rules of classical logic, such as obversion, conversion, or contraposition, would not be universally applicable. Still, the critical point to be made is that there is a certain ambiguity in the tradition’s handling of negation and that without adequate

P. Bilimoria () University of California at Berkeley, Berkeley, CA, USA e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_16

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formalization advanced by the respective schools the emphasis veers more towards linguistics than logic of the sophisticated kinds that have been developed in Western philosophy. While the chapter is reticent to enter into the latter territory in this preliminary exploration, some humble efforts nevertheless are made at comparative alignments. I begin with R.g Veda X. 129 (N¯asad¯ıyas¯ukta) which in Sanskrit has this structure: n¯asad¯as¯ın no sad¯as¯ıt tad¯an¯ım . .... : Then there was not non-existent nor existent: there was no realm of air, no sky beyond it. What covered in, and where? and what gave shelter? was water there, unfathomed depth of water? He, the first origin of this creation, whether he formed it all or did not form it, Whose eye controls this world in highest heaven, he verily knows it, or perhaps he knows it not.1 Note the occurrence of the negation twice in the conjunct: asat, ‘non-existence’ and na ‘neither’. It is singularly lacking in affirmation of anything positive; the values are compounded negatives. Various hermeneutical devices, including abduction (not unlike Peirce’s) have been deployed over two millennia to unpack this arresting statement, for its meaning. Pacitti (1991) considers ten different possibilities in respect of the function of the negation (and its negation) in the above s¯ukta; it collapses with some modification into four possibilities: 1. 2. 3. 4. 5.

There was not what-is-not, and there was not what-is, then Both what-is-not was not and what-is was not. ¬(what-is-not was there) and ¬ (what-is was there) It can neither be asserted nor denied that what-is-not was nor that what-is was. What was neither what-is-not nor what-is (= neither non-existence nor existence) (emphasis added)

What is going on here? Is denial tantamount to an assertion of negation? Is it a rejection of double negation via assertion of a single negation? Or, declaring the impossibility of linguistic assertion (4)? Perhaps inconsistent readings arise only when one takes properties of the constituent parts as though they are properties of the whole; reasoning in either direction yields different kinds of negation. Many scholars have been rather apologetic, but I wish to argue from a retrospective genealogy that tracks back through the M¯ım¯am . s¯a’s treatment of exclusionary negation.2 On the cosmogenic side, one could connect the beginning in prior asat

1 The

Rig Veda: An Anthology, transl. Wendy Doniger O’Flaherty, Penguin Classics, New York, NY (1981), p. 25. 2 It may be noted that this s¯ ukta along with various Upanis.adic binomial statements such as ‘It [Brahman] moves and it does not move’ resonate with the sevenfold predication of Jaina logic

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or non-existent (noting the reverberation in a literal reading of creatio ex nihilo); indeed, in response to the question in Ch¯andogya Upanis.ad on ‘how could anything come out of nothing’ it is suggested that some people have held such a view. It is thus established that the R.g Veda X.72.3 and X.129 along with a variety of other sources up to and beyond Ch¯andogya Upanis.ad (ChUp III, 19) taught that nonbeing, asat, “was” in the beginning, and that sat (being) arose from nonbeing in, as it were, a quantum flash.3

16.2 Mim¯am . s¯a and Ny¯aya on Negations The focus of Part I is on the Mim¯am . s¯akas’ theory of negation, otherwise known as abh¯ava or ‘absence’, and may be read as n¯asti or non-existent. Reference will also be made to the adaptation and variation in the Ny¯aya system.4 I contend that in its Bh¯at.t.a (i.e. Kum¯arila Bhat.t.a’s5 ) formulation this doctrine ought not to be reduced to anupalabdhi as it only achieves this latter re-naming and ´ alikanath Mi´sra. Both these commentaarticulation with P¯arthas¯arathi Mi´sra and S¯ tors are influenced by the criticisms of Buddhist logicians, especially Dharmak¯ırti (a contemporary of Kum¯arila, circa seventh century C E) and Prabh¯akara (the other doyen of the Mim¯am . s¯a), and indeed responses to the Naiy¯ayikas. The latter themselves did not begin with the kind of radical doctrine of negation that they end ´ up with, particularly, in Vallabha and Raghun¯ath Siroman . i. I have three aims in Part I. First I should like to outline the logical theory of negation in the Mim¯am . s¯a. Second, I make a connection with the Mim¯am . s¯a hermeneutic of moral judgments (which for them are inscribed in s´ruti, the Vedas, in the form of Vedic injunctions, ‘the ought to do’ type of positive propositions— vidhis). The most important part of the discussion here is the Mim¯am . s¯a treatment of negative propositions. Before that, however, a word on Mim¯am . s¯a (for those who may not be familiar with this particular school of classical Indian thought, perhaps in some ways even pre-classical with its roots in the Br¯ahman.as of the Vedas), to which also arguably belongs the genesis of the Ny¯aya or logic school of Indian thought. The term ‘mim¯am . s¯a’ literally signifies ‘commeasurement’, from the roots ‘m¯ım’, ‘to align, to sound . . . ’, and ‘man’ to think’, and this implies, in Zilberman’s

(Bhargava 1973); and that had also later preoccupied many an Indologist in Europe and India and recently the US. The Jaina theory is discussed in Part II. 3 For further discussion of Wilhelm Halbfass on this observation, see (Bilimoria 2012). Ch¯ andogya Upanis.ad (ChUp III, 19) in Olivelle, Patrick Upanis.ads, Oxford University Press, New York (1998). 4 Some further discussions on ‘Absence (abh¯ ava)’, see (Bilimoria et al. 2016) Some of the analyses on Negation in this Part have been taken with adaptation from Bilimoria (2016). 5 Slokav¯ ´ arttika (=SV), with Ny¯ayaratn¯akara of P¯arthas¯arathi Mi´sra. Tara Publications, Varanasi (1979).

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words, “achievement of consistency in reasoning [naya], as a necessary precondition of making the meanings of words or sentences (for Sanskrit it is all the same) ´ comprehensible” [28, 288]. According to the definition given by Sabara in his 6 bh¯as.ya on Jaimini-s¯utra I.1.2, “[ . . . ] interconnected words or sentences which instruct in methods of cooperation or congruency in actions are known as p¯urvaMim¯am . s¯a, or karma-Mim¯am . s¯a.” (Ibid.) To elaborate on this, I will use Zilberman’s succinct descriptors here. Actions are subdivided into two classes: prescribed and prohibited ones. Normative injunctions (vidhi), derived from the Vedic sentences, induces or entices one to accomplish certain actions, which can be 1. Regular, imperative, deontological (duty-based) or non-exemptive (nitya); 2. Imposed by circumstances (e.g. filial exigencies or social contingencies) (naimittika); 3. Optative, i.e. performed with the purpose of achieving some desirable award or effect (k¯amya). These together constitute dharma (this is the Bhat.t.a view; Prabh¯akara extend dharma only to the first two). The inducement is in the form of a generated verbal energy in the so-compelled, excited or inspired initiate by the vidhi-proposition, generally of the form: ‘Let him be induced to do this and this’, and it abides in the subjective intention (bh¯avana). It is rendered by the imperative or optative form of the verb (li˙n) and conceals the deontic modality of the statement which proposed something as hitherto not existent (non-existent), but as due to be done (Zilberman 1988) (p. 289). The reference here to ‘deontic’ need not extend to ‘non-formal’ logic, as in ‘aesthetical logic’. Now, coming to the semantics of negation—and this is a crux of the argument— the prior non-existent is the logical negation of what has not yet been actualized. But there are other possible negations as well. The problem then that faces the philosopher is to interpret all parts of the vidhis and the larger passages in which they are embedded, and then to sift through the kinds and degrees of permitted, proscribed or prohibited, and excluded incentives toward a particular action. The way this is done is by focusing primarily on the verbal displayment that abides in the impersonal Vedic words (being authorless, there is no question of inquiring into the intentions of the supposed author); and that focus narrows down to the verbal intentionality (´sa¯ bd¯ıbh¯avana) that presents a specific transcendental function as the specific property of the intension of the verbal meaning.7 It follows that among the injunctions there may well be resemblances, dissimilarities, or oppositions; generating an order among these becomes of paramount importance.

6 See

footnote 9 below.

7 Here Zilberman cites Mandana Mi´sra as his authority for this rendering of ‘´sabda bh¯ avana’, as he

.. calls it, which otherwise would be s´a¯ bd¯ıbh¯avana, as ‘verbal energy’ (Zilberman 1988). Arindam Chakraborty disputes this is a correct rendering of s´a¯ bd¯ıbh¯avana, for the bh¯avana presumably is not in the verbal formation but rather is a disposition within the hearer which propels him forthwith into action (seminar response).

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In the analysis that ensues, two categories become important: affirmative and negative meanings attached to the verbal expressions. Balancing the effects, the consequent, of these would mean first examining their antecedent casual potency (´sakti)—how much good or damage they can do; and this calls for dialectical mediation, and it may mean a recourse to deontic, subjective, aesthetic, causal, etc. considerations or markers. It may be noted that, in fact, the Mim¯am . s¯a were the first to introduce negative kinds of propositions in Indian philosophy. Jaimini specifically discusses pratis.eda (VI.5.15, VIII.7,10),8 and it may be noted that this is well before the Ny¯ayaVai´ses.ika and the Jaina and Buddhists took up these concepts and developed them in rather different and indeed logically rigorous ways. Although the grammarian P¯anin.i had already laid out some of the terms and rules for interpreting opposition and tensions between contrary expressions; his rules however did not embed the same degree of logical articulation, particularly in respect of non-indicative sentences, which the Mim¯am . s¯a would herald in. Again, the Mim¯am . s¯a needed this in order to sort out the different kinds of negative injunctions, which Jaimini undertakes so that all kinds of b¯adhas or handicaps and contradictions in and between the injunctions, mantras, and their supplementations (arthav¯adas)—and the moot distinctions between these two as well—could be put in place. So here is an application of their thinking on negative propositions and how one is to ‘read’ these in respect of their ramifications for the intended purport of the Vedic codan¯as or vidhis. For those with an interest in finding other ways to talk about contradictions— a divergence from Aristotelian logic—this earliest of Indian classical treatment may well be instructive. From (3), in the fourfold M¯ım¯am . s¯a negation, following the Grammarian commentator Patañjali, one is able to derive pratis.edha, ‘mutual prohibition’, and vipratis.edha9 where “two rules with different meaning apply to one (word),” could be read as “opposition (between two propositions) of equal force” (Staal 1962) (pp. 112–113). The question is how to resolve the tension. The Mim¯am . s¯a for their part begin by grouping the negations into three categories, which I will call limiting negations. These are now described: vipratis.edha, nis.edha, pratis.edha: permissible, prohibited, and excluded negations, respectively. The first kind is a contingent opposition, applying to individual instances but is not considered to be universal (for there is no j¯ati—genus—that pervades across the two expressions). When a jar is destroyed, not all jars are destroyed. You may be seen with the umbrella this morning even though it is not raining, for it may rain in the afternoon. Nevertheless, it has more of a force in prescriptive sentences than 8 Jaiminis¯ utra In: M¯ım¯am . s¯adar´sana edited by Gajanana Sastri Musalgaonkar, Bharatiya Vidya Prakashan, Varanasi (1979). 9 Technically signifies a conflict of two rules of which the latter prohibits the former.

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in indicative ones. Strict prohibitions belong to the other two. I will comment on pratis.edha before coming to nis.edha. Pratis.edha admits of two further types, paryud¯asa and prasajya: by negative implication ‘p implies ¬ q’—‘If x is a ks.atriya she is not a brahmin’; ‘if it is grey it is not white’, ‘if y is an asura, he is not a god’. The others are prasajya-pratis.eda: ‘snow is not black’ ¬ [F(x)]; and paryud¯asapratis.edha, ‘the jar is not (here)’, F (¬ x). This is as far as the grammarians went, but now we’ll see how these are tightened up when the negative functor is added in the hands of the Mim¯am . s¯a. According to the Mim¯am . s¯a philosophers each of the elements of a sentence, such as verb, verbal ending, or noun, can be negated, seamlessly and defeasibly. (This is in contrast to its singular prohibition in Western linguistic philosophy.) An obligation operator Ø is introduced along with a prohibition operator (ج). Thus: ج [F(x)]; and Ø[F (¬x)] respectively. The difference between the two is that in the former—paryud¯asa—there is a residue of the essentially positive still lingering on, while the negative is secondary or of second order, a semantic negation that strikes at the last member, not the verb itself, which it would if the statement had said: ‘He is not being a brahmin, or a good Brahmin at least’. Whereas in the latter—prasajya-pratis.edha—the essential assertion is of a negation and there need not be any positive residue; it does not apply to the last member of the negative compound but strikes at the verbal ending (kriyay¯a saha yatra nañ; nañ [= n] signifies the negation). So the ‘paryud¯asa’ type of negation may be called ‘exclusion type of negation’, whereas prasajya-pratis.edha is a ‘prohibition type’ of negation. And this is also called ‘nis.edha’ which can spell out more radical forms of negation depending on which part of the verbal intension the n-factor is seen to strike at. Generally speaking, in indicative sentences, in the case of paryud¯asa type of negation the negative is connected with the verbal root or the noun, but not with the verbal ending. Whereas for the Mim¯am . s¯aka, who are primarily concerned with Vedic injunctions which contain a verb in the optative mood, the principal part is the optative verb form, and the principal part of the optative verb form is the (verbal) ending, not the verbal root. That is the structure of positive injunctions, but it came to be transferred to negative injunctions or types of prohibitions and limiting negative sentences as well. Hence we have to understand the nis.edha type of negation where the negative is connected with the verbal ending: because it can provide prohibitory force without limits. To give one prominent example of nis.edha: the negative injunction na bhaks.ayet is a negation of the positive injunction bhaks.ayet, ‘he shall eat’ that is denoted by Ø[F(x)], where F(x) denotes ‘he eats’ (and modally, ‘it is necessary that he eats’). Now a prohibition (nis.edha) or negation of this injunctive sentence then is to be symbolized not by Ø[¬F(x)] (even though this is its logical negation), but by (¬Ø) [F(x)] just as—and Staal points this out rather perceptibly—the negation of ‘he shall

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eat’ is not ‘he shall not-eat’ but ‘he shall-not eat’ (colloquially, eating is a ‘no-no’). Here also ¬ Ø[F(x)] denotes (¬Ø)[F(x)] (‘there is no mandate for eating’). The expression Ø[¬F(x)], even though a negation of the positive injunction, is in itself not an injunction, and even though not well formed, still has meaning not unlike F(¬x), it could mean ‘it is permissible that she does not eat’. It belongs to the paryud¯asa, ‘exclusionary’ rather than the strictly prohibitive. In other words, one form of negation may suggest that doing x is not all right at this moment, but it may be permissible at another moment, or by someone else. ‘She shall not-eat’, inscribes into its propositional structure the permissibility of eating at other times (e.g. crossing communities, it would apply to the Muslim observance of Ramzaan, where eating in the day hours is prohibited, but permitted after sunset). The Mim¯am . s¯akas were interested in delineating the kinds of negation that are prohibitory without a residue of permissibility in any part of the semantic field, for one can easily say, ‘Do not indulge in sex’, but if one got married tomorrow it may become obligatory not to avoid sex, in the interest of progeny. ´ udra should not even as much as be permitted to hear the Vedas Likewise, ‘a S¯ recited’ stands as a prohibition; but it does not necessarily exclude a son born of the intermarriage of a Brahmin and a s´u¯ dra, or the other castes, from being present at a sacrifice, and so on. The Mim¯am . s¯akas are in search of the equivalence of the strictly obligatory in the negative kind (adverting to the injunction ‘Ought not to be locked, at least not this door’: Ø(¬ [F(x)] → [F(¬x)]); we elaborate this formalization below). Clearly this kind of negation, if it be admitted as a valid kind of negation which I presume it would not be in Aristotelian logic (unless one brings in some other abstract operator10 ) the Mim¯am . s¯a classify under paryud¯asa, ‘exclusion’. Yatrauttara-padena nañ, ‘where the negative is connected with the next word’—denoting here, ‘other than the verbal ending’, which as we saw was the mark ascribed to pratis.edha (above), kriyay¯a saha yatra nañ. And the ‘next word,’ uttarapada as suggested earlier, denotes the second member of a negative compound (tatpurus.a or bahuvr¯ıhi, different kinds of nominal compounds in Sanskrit, but not ruling out verbs) can be either a verbal root or a noun, but such a negation does not strike at the core of the injunction (because it excludes the verbal ending), hence it is not properly tantamount to a negative injunction. In other words, it has a qualificative limiting function rather than an absolute prohibitory function, and therefore it does not fit the fourth kind of negation simpliciter, atyant¯abh¯ava. 10 My

suggestion of substituting here Church’s λ, lambda operator, has not gone down well with reviewers.

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In any event, for whatever it is worth, let us explore the application of the exclusionary or paryud¯asa kind. Now Staal takes an example from the Mim¯am . s¯aNy¯aya-Pr¯ak¯as´a (ably translated by Edgerton), of the sentence, ‘neks.eta’, ‘he shall not look’. Curiously, there is positive injunction here, because there is a preceding phrase ‘his vows are . . . ’. (Staal 1962) Hence, technically speaking, nothing is prohibited; there is no ¯ıks.ana-virodh¯ı (opposition to looking) for he never thought of looking (in the direction of Dharmak¯ırti’s distractingly dancing mistress)! He has not been given a desirous option: the expression is bereft of an optative ending in the negative, which is different from saying you can be enjoined to do W [work/duty/kartavya] even where W cashes out into ¬ F(x) = ‘not-look’. You are still doing your work. Hence this is represented by the formula, Ø[(¬ F(x)]. We could go on and find sentences where one of the nouns is negated, drawing on (3), the anyony¯abh¯ava negation (here the negation is of difference of an entity under the mode of identity relation). If the positive injunction of the expression F(x,y) is Ø[F(x,y)], [y is contra x, so other than y], its negation or paryud¯asa is Ø[F(¬x.y)], but it still remains positive in its injunctive force (´sa¯ bd¯ıbh¯avana). Bringing this part of the discussion to a conclusion, the innovative element in the Mim¯am . s¯a approach to prohibition and exclusion vis-à-vis the grammarians of the three types of negation, is that negation can be applied to either Ø or F or all parts of a sentence, which is impossible in grammar. This may not seem very novel from our modern, post-classical point of view in logic, but at a time when logic was totally in the control of a tool used principally by grammarians to structure the determinants of proper speech, this indeed is quite a break-through, for it attempts to mirror the world outside—of real kriy¯as, action and things, in speech—and it also tries to understand without compromising the meaning of certain a priori negations in the Vedic corpus, which the M¯ım¯am . s¯a took to be unquestionably valid (because of its apaurus.eyatva, freedom from personal errors, including that of a possible deity). So here is finally the difference, in formal expression (again, Ø is the operator for the deontic or obligatory force) and the pair obeyed-disobeyed is the counterpart of true-false:11 Grammarian s¯utra (event) prasajya-pratis.edha

F(x) Ø [¬ F(x)]

paryud¯asa

Ø F(¬x)

The door is locked (i) You may not unlock this door (ii) (negation of the predicate, as in Aristotelian logic; governed by principle of noncontradiction) You may unlock not this, the other door (iii) (not well formed negation; not governed by principle of noncontradiction, as in Quine also)

11 I am grateful to Staal—when I was very confused by the M¯ım¯ am . s¯a formulations, I went to Staal personally back in 1981, and we discussed some M¯ım¯am . s¯a texts together in Berkeley many moons ago.

16 Thinking Negation in Early Hinduism & Classical Indian Philosophy Mim¯am . s¯a (event) vidhi (positive injunction) nis.edha (strict prohibition) paryud¯asa I (exclusion) (embeds prasajya-nis.edha) paryud¯asa II (inclusive exclusion) (embeds prasajya-pratis.edha) The anomaly or ‘wild-card’

F(x) Ø[F(x)] Ø[¬ F(x)] Ø[¬F(x)]

The door is locked (iv) The door ought to be locked (v) The door ought-not to be locked (vi) The door need not be locked (vii)

Ø[F(¬x)]

Not this, another door needs to be locked (viii) You told me not to lock the door (ix) (ambiguous/disobeying)

¬ Ø[F(x)]

309

The Naiy¯ayikas, from what I understand to be their thinking, reduce negation to two—at most three—main kinds of negation. From the list given earlier by Kum¯arila Bhat.t.a, they accept the ‘not-yet’ (pr¯agabh¯ava), ‘no-more’ (dhvam . sa), and—if J L Shaw is right12 —‘never’ (atyant¯abh¯ava). I leave it to the more able and vociferous scholars of Ny¯aya to elaborate on the Ny¯aya theory of negation and its difference from the Mim¯am . s¯aka’s theory that I have sketched above, and also to show what the Ny¯aya owes to the Mim¯am . s¯a in developing its particular view in all its symbolic and logical sophistication, particularly when it comes to double negation and contradictions.

16.3 Jaina Anek¯antav¯ada and Sy¯adv¯ada ´ Somewhere during the antiquity the Jaina tradition—a ‘dual’ system in the Sraman .a paradigm (forest-based ascetical yoga-orthopraxy) with Buddhism, though preceding it—enunciated a radical critique in recognition of the epistemological instability in our ability to capture and grasp all of reality, or definitively even any point within it, for reality is more complex and multidimensional than we might realize or in our positivist hubris acknowledges. Some have averred that Jainism is more “accepting of all views” than Buddhism is—characteristically “rejecting of all views”; but as we shall see Jainism betrays a more exclusion-inclusive or conditional denial position

12 Shaw

(1988) (pp. 144–145) classes them under ‘relational absences’, which certain caveats built into the “temporal relation as the limiting relation of the property of being the counterpositive.” A counterpositive is the negatum of a negation. Although in his other papers on negation, Shaw limits Ny¯aya negation to two main kinds: relational absence and mutual absence, represented respectively by (1) x is not in y, or x does not occur in y, or the absence of x occurring in y; and (2) x is not y, or x is different from y; where ‘x’ and ‘y’ are non-empty terms, and their counterpositive are: (1’) x is in y, or x occurs in y, and (2’) x is y. I am indeed grateful to Shaw for sharing his papers on negation with me, and I have drawn liberally from with his permission. See See also Shaw (1981) (pp. 57– 78); Shaw (1978); Shaw (1988) (p. 144). See also J L Shaw on double negation that pertains to this discussion (Shaw 2016c) (pp. 224–237).

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along the lines of prasajya-paryud¯asa (II) of the M¯ımam . s¯a, though somewhat more complex than even that. A well-known parable is presented—viz., of an elephant and five blind-folded men (never having encountered the creature before) who are asked to describe the species; each men touches different parts of the animal and each reports that the elephant is whatever part of the animal each happens to be feeling: trunk, legs, earflaps, tail, torso (i.e. the particular characteristic). In other words, each came up with a partial truth as none could feel the animal in its magnificent fullness or conjure up a 3-dimensional image from all angles as those blessed with vision are able to, most often. We cannot know anything with absolute certainty (tattv¯anartha); we are forever compromised in our knowledge and fall short of the truth as it is. What is possible though is to make known reality—or the truth in a given situation—through a series of partially true statements, without committing ourselves to any one among them exclusively. This may sound like a distant echo of the fanciful post-modern obsession with relativism; perhaps so. But what this points to counterfactually is the perniciousness of the absolutist and authoritarian tendencies in our lived reality. We tend to tenaciously hold on to the view, even with little foreknowledge or scrutinizing investigation, we come up with or receive from our own kind, or from adulterated sources—such as the media and our political leaders -, and often out of pride or vanity are not able to let go of the view. In this way we render partial truths, even untruths, into full truths, all because we want to believe and not readily confess to our faults or any shortcomings in our epistemological forays. The ontological component of the anek¯antav¯ada thesis proceeds via its logical corollary of a seven-step formulation, sapta-bhan.g¯ı, within a form of dialectical reasoning or, better, conditioned predication, called syadv¯ada, (semantics of possibilities)13 Technical details aside, this is what it looks like (adapted from Matilal 1981), (adopt operator S, Sy¯at: somehow or from a point of view): 1. Sy¯ad asti (S1, somehow, i.e., from some particular point of view, a thing may be said to exist as itself) 2. Sy¯ad n¯asti (S2, somehow, from another point of view, the thing does not exist as itself, but possibly as something else) 3. Sy¯ad asti n¯asti (S3, taking both angles of (1) and (2) together, there is affirmation of existence as itself from one point of view and of non-existence of itself from another) 4. Sy¯ad avaktavya (S4, despite both views being present—i.e. existent and nonexistent simultaneously—somehow the thing is indescribable) 5. Sy¯ad asti avaktavya (S5, a combination of the first and the fourth forms of predication; somehow the thing is itself and still indescribable or indeterminate)

13 There

is tendency sometimes to collapse anek¯antav¯ada with sy¯adv¯ada, but I believe it makes for more precision to keep them apart and see them as interactive reasoning.

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6. Sy¯ad n¯asti avaktavya (S6, a combination of the second and the fourth forms; somehow there is non-existence of itself and is still indescribable or indeterminate) 7. Sy¯ad asti n¯asti avaktavya (S7, a combination of the first, second and fourth forms of judgment; somehow the thing is existent as itself, is non-existent as itself, and yet is describable or indeterminate). Here one has to postulate ‘is’, ‘is not’ and ‘indescribable’ as primitive operators. There is an objection from some quarters that the argument is well and truly over with step 4, as it violates the law of contradiction and therefore renders 5-onwards as redundant speculations or obfuscations. However, that would be to misunderstand the function of the operator ‘indescribable’: it is because a contradiction is discerned that the move is made towards overcoming the contradiction, or turning it onto a quasi-paradoxical trope (where the negation of the excluded middle instead is anticipated, and then the negation of that too). And this is done not by affirming one side or the other—and jettisoning off the flawed side as though we were locked inside the limits of binary or simple opposites—but rather by locating another standpoint that might transcend the ground (Grund) on which the contradiction has apparently occurred; or be ‘gifted’ another viewpoint, however indeterminate. Thus the table might be brown from John’s perspective and blue from Jill’s perspective; but if there is no observer we cannot say with any definiteness whether the table is brown or blue. And hence it makes sense to go over the steps again combining the fourth with the first, second and third, to see which one if any of the perspectives is valid or even privileged. In the end, perhaps none is: that is the predominant reading. The long and short of it is that the Jaina logician is here attempting to capture a lacuna in our efforts at knowing anything in its nakedness as it were; reality is multidimensional or varied and even with all the strides we may have made in phenomenology and the sciences we are still not anywhere near uncovering the full truth about anything or any situation let alone in respect of the whole of metaphysics and cosmology. We may hear whispers of Heisenberg’s uncertainty principle read as indeterminacy in Jaina nondualism (implying that, the element of subjective intervention in any attempted objective observation cannot be ruled out: in measuring time, the instrument of measure—human consciousness and the “eye” of the machine—may have an impact on the outcome as much as time “out there” will have). The same goes for our beliefs in ethics, religion and aesthetics; and all the more confounding when we are trying to “observe” certain supposedly existent but absent phenomena: such as “dark matter” and “dark energy”, or neutrinos. It is sheer hubris to maintain that we have the full-throttle on knowledge in any of these areas: we are strangers in our own homes; how much more in the homes of others? This phenomenology of ‘strangification’ (German, Verfremdung), whereby we translate language-game A into terms understandable in language-game B, despite seemingly incommensurable differences, should be welcomed by us as a heuristic possibility and an invitation to explore further and deeper beliefs and practices of others; or

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as Nietzsche too reminded us, we should open ourselves to perspectivism.14 Each perspective captures different aspects of the ultimate reality and none can claim to be in possession of the whole truth. So syadv¯ada—sometimes mistakenly rendered as ‘maybe-ism’, a ‘perhaps-so’ perspective—supplements and reinforces its corollary, leading to anek¯antavada, which literally means ‘No one-view’, and it embeds the moral lesson: not holding on to any one view, inconsistencies notwithstanding, for the contrary might be true in another possible world in another time, etc. “This is my cup”, as a statement of what I might be holding in my hand, may just turn out be a false statement: that it is a fools-cap that cannot hold water but you are hypnotized into believing it to be a cup; or alternatively, my statement was intended to point to the absence of tea in the vestibule more than a description of what the object in my hand is. Toleration, it follows, is an implicative virtue derived from the combined operations of anek¯antav¯ada and syadv¯ada. So summing up this part, the Jain standpoint is called nayav¯ada. With sy¯adv¯ada there is shift to the semantic structure of thinking, conditionality of assertions. So here we are given several partial (incomplete) views; the tensions and oppositions between them is tolerated and no attempt is made toward a grand synthesis to iron out the differential views. No commitment is made to any one particular view: multiple (modal) possibilities are admitted. E.g. if we say ‘Here is a jar of x; if we remove its x-ness, it must cease to exist’, the conclusion might be true in some cases, though not in others, for x may represent only such non-essential qualities or things as butter, or some living being’s name. One way of interpreting is to see the background, viz: 1. No distinction is made between general and special properties of an existential being asserted. 2. Deals exclusively with the general qualities of things. 3. The particular: the standpoint of particularity. 4. Things as they exist in the present, and without regard to their past and future aspects. 5. Exclusive attention to number, gender, tense, etc., of the words employed. 6. Distinguishes between synonymous words on etymological grounds. 7. Such like, hence the point of view which describes things by words expressing their special functions, e.g., to call a man a devotee because of his being engaged in devotion. Fallacies will be involved if we fail to adhere to the proper function of predication (e.g. mixing up general with particular properties; incorporating naturalistic fallacies, causistry and so on). In any event, in a simplified, more prosaic formulation, this would look like thus: 1. A thing is existent—from a certain point of view. 2. It is non-existent—from another point of view. 3. It is both existent and non-existent in turn—from a third point of view. 14 On

the difference between ‘perspectivism’ and ‘relativism’ as understood in contemporary thought, see Bilimoria (2008).

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4. It is indescribable—that is, both an existent and non-existent simultaneously— from a fourth point of view. 5. It is existent and indescribable—from a fifth point of view 6. It is non-existent and indescribable—from a sixth point of view. 7. It is both existent and non-existent and indescribable—from a seventh point of view.

16.4 Derivations of Buddhist Dialetheia; The Work of catus.kot.i I now wish to relate the foregoing disquisition and formulae explored to the logic of cat.uskot.i in Madhyamaka Buddhism (with some references to Therav¯ada, P¯ali nik¯ayas as well). Each kot.i is regarded as a proposition, and for this reason since the 1930s the catus.kot.i has come to be known as the tetralemma (or quadruple proposition or only just if at all four-valued semantics as distinct from semantic of conditionals as in relevant logic).15 The transition proceeds in this manner. The nature of postulated entity and relation to its predicate is investigated. All (‘imaginable’ conceptual) possibilities are exhausted in the four-cornered alternatives, otherwise known as catus.kot.i (fourfold negation): 1. 2. 3. 4.

There is a positive affirmation (p) A negative retraction or negation (¬p) Conjunction of the positive and the negation (p v ¬p) (Bi)negation of both the positive and the negation ¬ (p v ¬p). Comment.

• The concluding lemma is of the pratis.eda form (that we saw in the M¯ım¯am . s¯a discussion above). • If it is a simple negation (relational) then it is of the paryud¯asapratis.eda (anony¯abh¯ava) form. • If it is absolute negation then it is of the prasajyapratis.eda (atyant¯abh¯ava) form. The tetralemma is applied to unfurl the truth (coherence or incoherence) of concepts such as causation, self, and all such factors at conventional level (samvr.ti) of reality (tattva), of the ultimate (param¯artha), the existence of tath¯agata (Buddhanature) and nirv¯an.a or emptiness (in Mah¯ay¯ana). Applied to the first, we get : something “x is (i) Caused by itself (p) (ii) From another (¬p) 15 For

further discussion, see Williams [27, 246] here Robinson is cited; and especially the chapter by D. Seyfort Ruegg [18, 213-277]; see also Garfield [7, 46].

314

(iii) From both (p v ¬p) (iv) From neither p nor not p (no cause or inexplicable)

P. Bilimoria

¬ (p v ¬p)”

This is what is said from conventional reality, but none is assertable from the standpoint of ultimate reality (meaning, there is no absolute negation). Hence, from the point of view of ultimate reality, which is devoid of conceptualization and description, none of the above holds. Another way of describing the reasoning here is that N¯ag¯arjuna rejects both identity and difference. Since judgement is a unit of thought, it takes the form ‘S is P’. If identity is asserted, then either the nature of both is lost, and hence false, or we have not said anything. If S and P are different, then identity is asserted; hence it is false. If both identity and difference are asserted, then it contradicts. Since there is no other category than these two as a relation between them, assertion of it does not arise.16 Applying this analysis to the case of cause and effect, the assertion of identity between cause and effect leads to a paradox. Similarly, assertion of difference leads to a paradox. The assertion of both is an explicit contradiction. Since there is no other category different from identity and difference, there is no fourth alternative: they all stand rejected. Hence reality is free from each of the four alternatives. A question that has plagued most commentators, classical and modern, is whether there is an affirmation of the contradictory (virodha) of what has been negated (qua iii) (which might be acceptable in paryud¯asa negation in the syntactical form qua the grammarians)? Commentator Candrak¯ırti (eighth century C E) said ‘no’; the Buddha preached prat¯ıtyasamut.p¯ada, entities arise in dependence on causes and conditions; so he has basically denied all eight negations. We are urged to look upon the schemata as a prasajya-pratis.edha type of negation; not absolute negation but relational negation (Fenner 1984) (p. 188). Modern commentators have variously argued that contrary to appearances, N¯ag¯arjuna (second century CE Indian Buddhist philosopher par excellence) does not reject the principles of non-contradiction and excluded middle; what he rejects is the view that entities or essences possess own being, self-nature (svabh¯ava) (e.g. as maintained by the Abhidharma and Ved¯anta schools), or that there is absolute nonexistence. So then we get N¯ag¯arjuna stating for the Buddha: All is just so or not just so, both just so and not just so, neither just so nor not just so: this is the graded teaching of the Buddhas. Here N¯ag¯arjuna is not denying the truth merely of universal or categorical statements of the form “Every/No . . . is/is not”, but doing more.

16 I

am grateful to J L Shaw for this summary from Inada (1993).

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Does this lead to a paradox, or a rejection of non-contradiction? Paraconsistent philosophers, such as Graham Priest and Jay Garfield (Garfield and Priest 2003) are not worried about either: paradoxes can be resolved, and contradictions are no longer a concern (all contradictions are true or bare out some truth), and besides, should not be seen in isolation of the larger project at stake. Same goes for the teachings of self, no-self, ‘there is no self and no non-self.’ (Candrak¯ırti) But the teaching of the self adverts to other schools in the larger Indian philosophical system, not the teachings of the Buddha as such. It is a view toward weakening attachment to a real substantial self so that one can see emptiness of the view. But about ‘no non-self’ = no an¯atman?: that too can be an attachment (or taking an-an¯atman as an essence of some universal natural kind), when the truth is indeterminate (dependent-origination might be excluded thereby, the bundle of aggregates, skandha, view, which is the Buddha’s teachings). In this way, the Buddha is said to have avoided both the vikalpa or conceptualization that posits the extreme of eternalism (onto-theo-logos) and nihilism (the other extreme); which the Middle Way eschews; hence the middle position (madhyamaka) is taken (Gunaratne 1986; Ruegg 1977). It is very interesting that the tropes of sat and asat, that we saw operating in R.g Veda X.129 [n¯asad¯ıya], re-appear, e.g. in Candrak¯ırti’s formulation of binomial negation: • sad asat sadasac ceti sadasan neti ca kramah. • es.a prayojyo vidvadbhir ekatv¯adis.u nitya´sah. As Ruegg explains aptly: ‘This type of analysis of a problem thus constitutes one of the basic methods used by the M¯adhyamikas to establish the inapplicability of any imaginable conceptual position—positive, negative, or some combination of these—that might be taken as the subject of an existential proposition and becomes one of the set of binary (binomial) doctrinal extremes (antadvaya).’(Ruegg 1977) (So its negation resonates with M¯ım¯am . s¯a’s antyant¯abh¯ava.) I believe reading backwards, Madhyamakas have solved the paradox or conundrum that the bards of R.g Veda began with, to underscore the indeterminacy of the Big Questions haunting any deep inquiry. From this reasoning utilizing ‘neither . . . nor’ N¯ag¯arjuna is able to present at least one interpretation of emptiness: Whatever exists in dependence [on a cause] is not that [cause] nor is it different [from that cause]. Therefore [the cause on which there is dependence] is neither destroyed nor [is it] eternal. (Garfield and Madhyamaka 2014; Inada 1993; Ruegg 1977)

Reality might be otherwise than what we tend to dichotomize in discursive conceptualization, multiplying entities and factors which may not be so warranted, from the view of the enlightened. So the strategy of the ‘neither . . . nor’ articulation is to reveal the irrationality of something, that is, the antinomic and logically inconsistent character of causality operating with the concept of bh¯avas [entities] endowed [as if] with the real

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nature (svabh¯ava). (Ruegg 1977) [this is from Paul Williams ed. Abhidharma and Madhaymaka, pp. 218–221] But N¯ag¯arjuna cautions that even ‘emptiness’ runs the danger of becoming another absolute designation (like eternalism or nihilism), or that emptiness has self-nature, is inherently an existent, etc. Using modus tollens in conventional dichotomized disposition, to be empty entails ‘not empty’, both and neither; yes, for the sake of designation [such] is stated. ´ unyat¯a applies to all dharma, all entities. S¯ So suppose ‘empty’ is the null consequence; would you say that of the Buddha too? The answer is: Yes, and no. To this there is a firm response also, we draw from Tibetan commentaries for their clarity; so we have Garfield elaborating (via Tsong-khapa’s restatement) to surmise on the intent of this tetralemma, thus: We do not assert “empty”; we do not assert “nonempty”; we neither assert both nor neither. They are asserted only for the purposes of designation (Garfield and Madhyamaka 2014) (p. 45). Empty does not entail non-existent; people might mistake it to be so; nor do we say non-empty, that is, he exists inherently (so that designation has to be avoided) And not both either, nor neither (that he exists nor does not exist ultimately); ‘none of the four alternatives can be maintained’ [ibid]. Or, ‘N¯ag¯arjuna reminds us that emptiness is the relinquishing of all view, and that anyone for whom emptiness becomes a view is hopeless’. [Ibid] But, just for conventional purposes we say empty, non-empty, both and neither: for purposes of designation for those people to whom we are speaking, as long as we are clear that emptiness means the emptiness of essential nature as an understanding of ultimate truth, for there is no ultimate reality other than the conventional reality (ultimate truth is the truth about the inapplicability of certain designation to conventional reality—such as the extremes of inherent existence, eternalism or reification, and nihilism. Dependent existence does not imply nonexistence, only that no entity exists independently of conditions and causes, which too are dependent on other conditions and causes, and so on and so forth (See Sebastian 2016). In Vigraha-vyav¯artan¯ı, N¯ag¯arjuna is clear that he holds no views; even his statements are all empty, and this does not nullify his argument that all conventional designations are empty, all views are empty; but does not make just anything true, nor that nothing whatever is true. In the remainder of the chapter, I wish to backtrack a little in (contemporary) time and consider some alternative readings of the cat.uskot.i. K N Jayatilleke, working from outside the Mah¯ay¯ana tradition (the Therav¯adan tradition to be precise), tried to demonstrate that the cat.uskot.i is propositional and disjunctive, that is to say, in respect of propositions iii. and iv. the assertions are not a conjunction of i. and ii, and nor their bi-negation either, respectively (Ruegg 1977) (p. 48); the claim he makes is that N¯ag¯arjuna’s “method is intended to reveal the limitations of logic and language, and the principle of negative application of the cat.uskot.i derives from the conceptual scheme which consists of the four dyadic propositions such that their totality leads

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to contradiction and hence to total annihilation. The breakdown of symbolism.” (Jayatilleke 1967) (pp. 47–48) The negative use of cat.uskot.i (in Ruegg’s reading of Jayatilleke) was then developed for a religious reason, to transcend the limits of logical and linguistic conventions [11,50], “demonstrating the self-contradictory nature of the referential object of the proposition”. Jayatilleke is unabashed in not wishing to rescue N¯ag¯arjuna from charges of irrationality, only that it serves a ‘higher order’ purpose. R. H. Robinson, for his part, proffered the unusual suggestion that the 4th lemma should be read as: All x is A and all x is not-A He varied it to read ‘No x is A and no x is not-A’ (Aristotelean form; true ‘when x is null’) (Robinson 1957) Jayatilleke had tried to save this puzzling formulation by suggesting it be read as: Some x is A and some (other) x is not-A Or S is partly A and partly not-A So the third lemma is a conjunction of first and second lemmas; whereas the fourth is a conjunction of the contrary-wise or conflicting conjuncts of the third lemma, to yield, e.g., ‘the universe is finite in one dimension and infinite in another S’. Likewise, Frits Staal did not believe there is much point in trying to save cat.uskot.i from the charges of inconsistency, which only appears so if cat.uskot.i is taken as a statement, while in fact it is a “pedagogical and therapeutic device”; still, he is eager to maintain that M¯adhyamikas are not throwing the baby out with the bathwater, i.e., rejecting the principle of non-contradiction. What they in fact are doing is rejecting the principle of excluded middle. So he explains (paraphrased): If we reject the fourth clause [cat.uskot.i], as the M¯adhyamika philosophers did, we are free to accept the principle of excluded middle. But we don’t have to, since denying the denial of the excluded middle only implied the excluded middle if we accept the principle of double negation, which is itself equivalent to the excluded middle (Staal cited (Ruegg 1977) (p. 48). Staal concluded: When the M¯adhyamika philosophers negates a proposition, it does not follow that he himself accepts the negation of the proposition. Accordingly, there are other alternatives than A and not-A, the principle of the excluded middle does not hold (Ruegg 1977) (p. 44). Staal was palpably wrong; excluded middle is essential to cat.uskot.i, and also to the aligned prasa˙nga (reductio); two contradictories will result in null. Even if the excluded middle is rejected, this does not block the further rejection of contradiction. Staal’s attitude of the sacrosanct status of contradictions in the Indic tradition stands at the critical edge and is discarded by the later developments, such as in Priest and Garfield’s reading of the work of cat.uskot.i (Garfield and Priest 2003). Bimal K. Matilal If we interpret cat.uskot.i as prasajya kind (exclusion negation), then according to Matilal “the apparent contradiction of the joint negation of the four-fold alternatives

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will disappear” (Matilal 1971) (p. 164), for “denial of a position does not necessarily involve commitment to any other position.” But Matilal overlooks that contradictoriness is essential to the dialectical synthesis (especially with respect to ii and iv). Must contradictions disappear? Our humble response is that dialetheism prevails: P is true, so is non-P; contradictions are true about some state of affairs (e.g. greed in respect of capitalism and, more so, corporatism). D. Seyfort Ruegg In his early thinking on the matter Seyfort Ruegg had emphasized the point that, contrary to the opinion of some scholars, Madhyamaka reasoning (yukti) is based on the twin pillars of the principles of non-contradiction and excluded middle; and the negation of the four kot.is then serves to bring to a stop all discursive thinking consisting of conceptual development (prapañca) and dichotomizing conceptualization (vikalpa) involving the solidarity of complementary opposites expressed as affirmation (vidhi) and negation (pratis.edha). When the four kot.is— taken as being exhaustive of all imaginable positive and negative positions within discursive thought—have been used up, there remains no ‘third’ (indeterminate or putatively dialectical) position between the positive and negative which discursive thought could then seize on and cling to; and the mind therefore becomes still. This exhaustion—and ‘zeroing’ is of all the discursively conceivable extreme positions by means of the negation of all four kot.is corresponds to the Middle Way, and to reality understood by the M¯adhyamika as emptiness of own being and nonsubstantiality of all factors of existence. Finally then, the questions we are left with, are the following: • Has Madhyamaka school moved to three, even four-valued logic, or is it still functioning within two-valued logic? Here one might observe that each of the kot.i does not correspond to single truth-value, especially since four-valuedness matches with, e.g., Belnap & Dunn’s system of First Degree Entailment where T = “true (only)”, F = “false (only)”, B = “both (true and false)”, and N = “none” (neither true nor false); indeed, each of the four kot.is is a denial of the preceding four-valued assertions. Such as observation has led logicians such as Priest to suggest that catus.kot.i corresponds to a rejection of the preceding four values and an ensuing affirmation of a fifth value of “silence” (whereforth one cannot speak).17 • Has it rejected or, more strongly, denied the principle of non-contradiction? • And also excluded middle? • What are the logical and epistemological consequences if either or both moves are made?

17 I

am grateful to one of the anonymous reviewers for this input by way of clarification (reference source undisclosed).

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These questions are troubling only if we take cat.uskot.i as propositionally referential, i.e. logical and not as arresting ontological speculations, the ‘views’ of essential nature (svabh¯ava), reification, annihilation, etc. And it makes better sense when we read cat.uskot.i as a gem within a large encrustment of an over 2 millennia old pondering on the nature of reality and Nothingness—as with Heidegger against the grain of Biblical creatio ex nihilo.

16.5 Concluding Comment For my purposes, what I take away from the disquisition we have engaged in this chapter is the preparedness in ancient Hindu, Jaina and Buddhist thought as they evolved to take seriously the possibility of negation in its various forms and permutations as a ground, or the lurking empty space, in which conventional truths arise that is descriptive of an equally conventional reality, albeit without definite determinacy. Acknowledgement I wish to express gratitude for the immense guidance I received from Jayshankar Lal Shaw and Anand Vaidya toward revising the chapter. I also thank (the late) Frits Staal, (the late) Bimal K. Matilal, Stephen Phillips, my students in Berkeley. The editors of this volume and an anonymous referee have also provided invaluable feedback from the time of the conference to this point in the process.

References Bhargava, Dayananda. 1973. The Jaina Tarka Bhasha of Acharya Yashovijaya. Delhi: Motilal Banarsidass Publishers. Bilimoria, P. 2008. Nietzsche as ‘Europe’s Buddha’ and Asia’s Superman. Sophia (Guest Issue on Continental Philosophy of Religion) 47(3); 359-376. ———. 2012. Why is there nothing rather than something? An essay in the comparative metaphysic of nonbeing. Special issue on Max Charlesworth: Crossing the philosophy and religion divide. Sophia 51 (4): 509–530. ———. 2016. Negation (Abh¯ava), Non-existents, and a distinctive pram¯an.a in the Ny¯ayaM¯ım¯am . s¯a. In Comparative Philosophy and J L Shaw, Sophia Series, ed. P. Bilimoria and M. Hemmingsen, 183–202. Dordrecht/London: Springer. Bilimoria, P., A. Vaidya, and J.L. Shaw. 2016. Absence – An Indo analytic study. Sophia (Special Issue on 25th Anniversary of the Demise of Professor Bimal K Matilal) 55 (4): 491–513. Fenner, P. 1984. A study of the relationship between analysis (vic¯ara) and insight (prajña) based on the Madhyamak¯avat¯ara. Journal of Indian Philosophy 12 (2): 139–198. Garfield, J., and L. Madhyamaka. 2014. Nihilism, and the emptiness of emptiness. In Nothingness in Asian philosophy, ed. JeeLoo Lieu and Douglas L. Berger, 44–54. NY/London: Routledge. Garfield, J.L., and G. Priest. 2003. ‘N¯ag¯arjuna and the limits of thought. Philosophy East & West 53: 1–21. Gunaratne, R.D. 1986. Understanding N¯ag¯arjuna’s Cat.uskot.i. Philosophy East & West 36 (3): 213–234.

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Inada, K.K. 1993. N¯ag¯arjuna: A translation of his M¯ulamadhyamakak¯arik¯a with an introductory essay. Delhi: Sri Satguru Publications. Jayatilleke, K.N. 1967. The logic of four alternatives. Philosophy East & West 17: 69–83. Matilal, Bimal K. 1971. Epistemology, logic and grammar in Indian Philosophical analysis. The Hague: Mouton. ———. 1981. Central philosophy of Jainism: Anek¯antav¯ada. L D Institute of Indology, Ahmedabad ———. 1991. Anek¯anta: Both yes and no. Journal of the Indian Council of Philosophical Research 8: 1–12. Pacitti, Domenico. 1991. Negation in Sanskrit: Rig Veda X,129. The nature of the negative, edited by Pacitti, D, Pisa: Giardini. http://www.pacitti.org/books_00199102.htm (accessed 25/02/2015) Robinson, R.H. 1957. Some logical aspects of N¯ag¯arjuna’s system. Philosophy East & West 6: 291–308. ———. 1969. Review of K. N. Jayatilleke, early Buddhist theory of knowledge. Philosophy East & West 19: 69–81. Ruegg, D. 1977. Seyfort. The uses of the four positions of Catus.kot.i and the problem of the description of reality in Mah¯ay¯ana Buddhism. Journal of Indian Philosophy 5: 1–71. Sebastian, C.D. 2016. The cloud of nothingness: The negative way in N¯ag¯arjuna and John of the cross. In Sophia studies in cross-cultural philosophy of traditions and cultures. New Delhi/Dordrecht: Springer. Shaw, J.L. 1978. Negation and the Buddhist theory of meaning. Journal of Indian Philosophy6 (1): 59–77. ———. 1981. Negation: some Indian theories. In Studies in Indian philosophy, ed. D. Malvania and N.J. Shah, 57–78. Ahmedabad: L D Institute of Indology. ———. 1988. The Ny¯aya on double negation. Notre Dame Journal of Formal Logic 29: 139–154. ———. 2016a. Austin on falsity and negation. In The collected writings of Jaysankar Lal Shaw: Indian analytic and Anglophone philosophy, 197–200. London: Bloomsbury Academic. ———. 2016b. Descriptions: Contemporary philosophy and the Nyaya. In The collected writings of Jaysankar Lal Shaw: Indian analytic and Anglophone philosophy, 328–361. London: Bloomsbury Academic. ———. 2016c. The Ny¯aya on double negation. In The collected writings of Jaysankar Lal Shaw: Indian analytic and anglophone philosophy, 224–237. London: Bloomsbury Academic. Staal, Frits J.(1962) 1988. Negation and the law of contradiction in Indian thought: A comparative study. Bulletin of the School of Oriental and African Studies, 52-71. Reprinted, in Staal, F. J. Universal Studies in Indian logic and linguistics. Chicago: The University of Chicago Press, pp. 112–113 Williams, P., ed. 2005. Buddhism critical concepts in religious studies. Vol. Vol. IV. London: Abhidharma and Madhyamaka. Routledge. Zilberman, D.B. 1988. Birth of meaning in Hindu thought (Boston series in philosophy of science). Dordrecht: D Reidel & Co.

Purushottama Bilimoria PhD is presently a Fulbright-Nehru Distinguished Fellow in India affiliated with Ashoka University. He is also lecturer with Legal Studies in the University of California, Berkeley, and serves as a senior fellow in Indian Philosophy with the Center for Dharma Studies at Graduate Theological Union, in Berkeley. He is otherwise an Honorary Research Professor of Philosophy and Comparative Studies at Deakin University and Senior Fellow at University of Melbourne, in Australia. He is a Permanent Fellow with the Oxford Centre for Hindu Studies in Oxford University, and past visiting scholar at All Souls College, University of Oxford, Harvard University, Emory University and UC Santa Barbara, and Visiting Professor in two universities in Brazil. He is a Co-Editor-in-Chief of Sophia, international journal of philosophy & traditions and of the Journal of Dharma Studies, both published by Springer.

Chapter 17

Is God Paraconsistent? Newton C. A. da Costa and Jean-Yves Beziau

Deus é a congruência dos opostos Dele se podendo afirmar e negar tudo Lars Eriksen

N. C. A. da Costa Federal University of Santa Catarina and Federal University of Rio de Janeiro, Brazilian Research Council, Rio de Janeiro, Brazil J.-Y. Beziau () Federal University of Rio de Janeiro, Rio de Janeiro, Brazil e-mail: [email protected] © Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6_17

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17.1 Paraconsistent Negations and Paraconsistent Things How and in which sense something can be paraconsistent? Paraconsistent is first of all a quality that applies to negation. A negation is paraconsistent only if there is a formula F such that F and ¬F are true together. It is not an “iff”, just a negative criterion, positive criteria are requested to ensure we are dealing with a negation, not with any kind of unary operator (see Beziau 2000). And it is not enough to use the sign “¬” to guarantee we have a negation. The name, the symbol, is not at all a sufficient condition. Moreover, there is not only one paraconsistent negation but different ones. Given a paraconsistent negation ¬ (no quotation marks: the sign is not the thing), P is a paraconsistent proposition iff P and ¬P can be true together. In the three-valued paraconsistent logic of Asenjo/Priest (cf. Asenjo 1966; Priest 1979) all propositions are paraconsistent, quite strange! (see Beziau 2016b). In the paraconsistent logic C1 (Da Costa 1963), all atomic propositions are paraconsistent, but not all the molecular ones. In C1 a proposition typically not paraconsistent is P&¬P. In the paraconsistent logic Z (Beziau 2006, 2016a) all propositions are paraconsistent excepted tautologies and antilogies. And there are some paraconsistent logics where only atomic propositions are paraconsistent, like Sette’s logic P1 (Sette 1971).

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In all these cases therefore, an atomic proposition like “A triangle has three sides” is paraconsistent, because according to these logics “A triangle has three sides” and “A triangle has not three sides” can both be true. This may look strange and in fact it is strange. But classical logic also is odd: “A triangle has three sides” can be true and “A triangle has three sides” can be false. Anyway, a mistake cannot be corrected by another mistake. We will not correct this mistake here but the present discussion may shed some light on it and urge to develop better paraconsistent logics, or better logics tout court. A thing t can be defined as paraconsistent if there is a property Π , such t has Π and does not have Π . In other words: the two propositions “t is Π ” and “t is not Π ” are true, for a given paraconsistent negation corresponding to the “not” of the second proposition. It can be objective or subjective in the sense that the property Π can be a way to conceive the thing or an intrinsic quality of the thing. Consider the following picture:

The cylinder appears as a square and as a circle. “It is a square” and “It is not square” are two propositions true about it considering that a circle is not a square. The cylinder is paraconsistent from this point of view, or better from these two points of view. This does not necessarily mean that the cylinder itself is intrinsically paraconsistent. With the picture below we have a more ambiguous situation: an old woman intertwined with a young woman. But anyway, we don’t have here a woman which is at the same time both young and old. At best, we have a “thing” that can be seen quasi-simultaneously as an old lady or as a young lady.

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17.2 The Paraconsistency of God God is first of all a word. Like many things . . . Most of the time our mind is governed by words. Words can be useful tools but we have to be cautious not to just paddle into the linguist pond. Maybe we could spend a few hours, not to say a couple of academic years, discussing the question if “God” is a proper name (hence the capital letter) or just a common name (not so big). Reality is also important. But how to reach or deal with reality beyond words? You may look at the Sun, and this is the Sun! For God that’s not obvious, unless you identify God with the Sun.

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God is not the only thing that cannot directly be perceived. What about Time? Not so easy, unless identifying your Rolex with Time. Time has indeed been deified: as Chronos, or as Money.

It is not because time cannot be directly perceived that there is nothing beyond the word, that time does not exist. On the other hand, the idea associated with the word “time” may be confused if not inconsistent, the same with the ideas associated with words like “life” or “existence”. Something may be real and difficult to conceive. It is true in fact of many things of the world we are merged in. But if the perception of a given object is rather clear and objective, we can associate a word to it without knowing exactly what it is, and that’s the meaning of the word. That’s the case of the Sun. For most of the people the idea of the sun is nothing more than its appearance, despite the fact that there is a scientific theory explaining what it is in a more sophisticated way: a star among black holes in a mad universe.

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In the case the thing does not explicitly show up the situation is not that simple, it is not necessarily easy to know what we are talking about. We have to be careful not to be only playing with words. Some words are really ambiguous in the sense that we may have the impression they have a meaning corresponding to a certain reality. At the end they are maybe just an illusion, a way of speaking, leading nowhere, or a highway to hell . . . Investigation of the meaning of a word may lead to the discovery that there is no real meaning. A problematic case is for example the one of “causality”, related with God’s case: God has been considered as the first cause or/and the cause of everything (see Beziau 2015).

Many things are complex, they cannot be understood right away, we have to develop a sophisticated theory to catch them. This is typically the case of the intimate nature of physical reality or of living beings. In modern physics there is the duality wave/particle. Roughly speaking microscopical reality can be viewed from two different opposite viewpoints. One may argue that to save objective reality, we need to use a paraconsistent logic. Something can be considered as a wave and as a particle. It does not mean that it is a wave and a particle. It can be seen as a wave and it can be seen as a particle, not simultaneously, but from two different points of views. Reality is not necessarily in itself paraconsistent. But paraconsistent logic may preserve the co-existence of two opposite theories to describe reality, without properly catching it.

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Microscopical reality can be considered as a manifestation, no to say a product, of God. But let us turn to a more dramatical aspect of the divinity and check if it can be saved by paraconsistent logic and at what price, to speak the language of the cheap philosophy nowadays dominating the market. On November 1st 1755 there was a terrible Earthquake in Lisbon destroying one of the most flamboyant cities of the time. How such a phenomenon is compatible with the goodness of God? This earthquake was used by the candid Voltaire to ridicule Leibniz’s theory of the best of all possible worlds (originally due to Malebranche, and later on artificially revived by Kripke’s possible worlds semantics—see (Beziau 2010a)).

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The contradiction is not straightforward. To consider we have a contradiction here, we must suppose that: 1. God is good 2. Lisbon’s earthquake is bad (2a) and is the product of God (2b). We will not discuss (2b) here, a question related to the omnipotence of God. Just let us point out that Lisbon’s earthquake is completely different from global warming or the Shoah, two other hot phenomena. It would be difficult to argue that it has been produced by human beings. If (2b) is not assumed, then there is no explicit contradiction, but space for Hazard, not to say Devil. “God is good” is a nice stipulation, but it has not necessarily the absolute obviousness of an axiom, like “the number one is the first” (also indeed not so obvious, see (Beziau 2017)). “God is good” is not tautological like “If the sky is green then the sky is green”. Is it analytical like “a bachelor is unmarried”? We quite understand the meaning of “unmarried”, even without being a lawyer or a priest. But the meaning of “good” is not so clear, even being a lawyer and a priest. We can say that an apple, a book or a music is good. But it is not exactly the same meaning as when saying a woman is good. Goodness has many aspects. It is easier to argue an earthquake is not good than God is good. If we don’t specify too much what God is, it is difficult to find any contradiction. On the other hand, if we want to know all the details of its configuration, size, color, gender, etc., it is easy to find some contradictions. As it is known the devil wallows in details . . . At this evil stage maybe we need paraconsistency, if we don’t want to only have a ghostly or phantasmagoric conception/vision of god. But it is not because something cannot be defined precisely that it does not exist. It makes sense to say that reality does not reduce to thought, even without assuming any noumenal dark side of reality. Many real things are difficult to define and reality itself is difficult to define, nevertheless we have a word for it. We can talk about it even if don’t know what it is. To talk about God is thus not necessarily a problem. If we want to reason about it/with it, we have to see what kind of logic is the best. Gödel used quantified first-order modal logic to prove the existence of God. Is it sufficient or necessary?

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17.3 The Divinity of Paraconsistency Paraconsistent logic is a scientific theory, part of mathematical logic. Logic can be understood both as reasoning and as theory of reasoning. In (Beziau 2010b) this distinction has been literally expressed respectively using Logic and logic. As pointed out, this is the same problem as with history. That’s the reason why a Capital difference similar to History and history was made, There is an interaction between the two sides. History is based on history. The Historical action of the Emperor of France is based on the vision he had of the history of France. Similarly, our theory of reasoning may shape our way of reasoning. Paraconsistent logic is first of all, like classical logic, relevant logic, modal logic, erotetic logic, polar logic, etc., a theory of reasoning. This theory develops through the construction and the study of a class of logical systems. Someone may believe that classical logic is a description of the way we are reasoning, like Newton physics is a description of physical reality. But an important difference between logic and physics is that, since the beginning of the theory of reasoning with Aristotle, there is an important normative aspect. Classical logic can be considered as the way we should reason. This clearly shows the interaction between Logic and logic. There is a system, product of a theory and this system can direct reasoning, like a road traffic sign.

A logical system can be different from another one in different ways. Going on with traffic analogy: there is a difference like the one between the highway code in England and the highway code in France. Cars don’t run in the same side of the road. It is impossible and dangerous to combine the two. The difference between the classical logical system and some paraconsistent logical systems, or other nonclassical systems, may appear like that. But the difference can also be different. In a traffic road system with only two lights red and green, red means stop, green go. It is a bit dangerous: if you arrive

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with your car at a high speed and the light turns from green to red, you may suddenly brake and your car will skid and/or bump into the next Rolls Royce. In case of a traffic light with three lights, the most common one, the third orange light prevents this problem making the traffic smoother. The difference between two-valued logic and three-valued logic can be seen as similar. Many, but not all, paraconsistent logics are based on three-value semantics (see Asenjo 1966; D’Ottaviano and Da Costa 1970; Arieli and Avron 2015; Priest 1979; Beziau et al. 2007, 2015; Beziau 2016a).

Another example is between traffic circle and traffic cross with light. This is compatible in two different ways. You can have a traffic system with both separately and also you can mix the two. In a paraconsistent system you may have side by side a paraconsistent negation and a classical negation and/or the classical negation can be defined from/with the paraconsistent negation.

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With a good system of logic, we can better direct our mind and have a better understanding of reality. This can be related with the divine in two connected ways: the mind works better, is more intelligent, and consequently this permits a better understanding of reality. In the Bible there is identification between God and the Logos:

We can consider that logic, as reasoning and as the theory of reasoning, which itself is also reasoning, is the manifestation of the divinity. But there is no reason to reduce logic or metalogic to classical logic, even if we believe in only one logic. On the other hand, even if we don’t believe that True Logic is described by a paraconsistent system, in many ways paraconsistent logic is richer and more subtle than classical logic and that’s a good point. However, we have to be careful to develop beautiful and meaningful systems, not just fashionable nonsense.

Acknowledgements This paper is based on a talk presented by the second author at the First World Congress on Logic and Religion (1st WoCoLoR) which took place in João Pessoa, Brazil, April 1-5, 2015 - about the history of this event, see (Beziau and Silvestre 2017). We would like to thank all the participants of this event.

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Da Costa, N.C.A. 1963. Calculs propositionnels pour les systèmes formels inconsistants. Comptes rendus de l’Académie des Sciences Paris 257: 3790–3793. Priest, G. 1979. The logic of paradox. Journal of Philosophical Logic 8: 219–241. Sette, A.M. 1971. On the propositional calculus P1. In Notas e comunicacões de matemática, vol. vol. 17. Recife: Universidade Federal de Pernambuco, Departamento de Matemática.

Netwon C. A. da Costa is the creator of paraconsistent logic, the main promotor of logic in Brazil since the 1950s and the most famous logician and philosopher from South America. He is the first researcher from South America to have been elected to the International Institute of Philosophy in Paris. He was invited researcher and gave talks in many countries around the world, in particular: Australia, France, Poland, Russia, United States. He published more than 200 papers in the most famous journals of logic and philosophy. Jean-Yves Beziau has a Ph.D. in Mathematics (University of Paris 7) and a Ph.D. in Philosophy (University of São Paulo, Brazil). He has done research in France, Brazil, Poland, California (UCLA, Stanford, UCSD) and Switzerland. He is presently professor of logic in Rio de Janeiro at the University of Brazil, and the President of the Brazilian Academy of Philosophy. He is the promoter of Universal Logic as a general theory of logical structures, the founder and Editor-inChief of the journal Logica Universalis and book series Studies in Universal Logic, both published by Birkhäuser/Springer, Basel. He has organized a series of events on universal logic around the world (Montreux 2005, Xi’an 2007, Lisbon 2010, Rio de Janeiro, 2013, Istanbul 2015, Vichy 2018). He has renewed the study of the square of opposition, organizing interdisciplinary world events on the topic (Montreux 2007, Corsica, 2010, Beirut 2012, Vatican 2014, Easter Island 2016, Crete 2018) and the publication of special issues of journals and books. In 2019 he launched on January 14 the 1st World Logic Day which was celebrated in 60 locations around the world and was subsequently approved as an international day of UNESCO.

Index

A Abd al-Jabb¯ar, 293 Abduction, 302 Abductive reasoning, 206 Abhar¯ı’, 298 Abu al-Q¯asim Ans¯ar¯ı, 293 Ab¯u Han¯ıfeh, 282 Ab¯u Sa’¯ıd S¯ır¯af¯ı, 293 Acquired Reason, 278 Active Intellect, 278 Active Spiration, 140, 141, 143, 147 Actualist quantification, 210, 211 Actualized/actual reason, 128–130, 210, 278, 304 ¯ 287 Akhb¯aris, Al-Fr¯ab¯ı, 292, 293 Al-Ghaz¯al¯ı, 279, 288, 289, 292–295, 297, 298 All¯amah Hell¯ı, 298 Ambedkar, B, 269, 273 Analytic philosophy of religion, 4, 9, 10, 78, 79, 212, 257, 273 Angelic Reason, 278 Animal suffering, 29, 32–36, 44–46 A priori contention that the belief in miracles is irrational, 79 Aql, 278–283 Aquinas, 7, 9, 12, 19, 50–55, 59–61, 67, 69, 71, 72, 137–150 Argument formalization, 206 Argument from consciousness, 20 Argument from evil, 11, 12, 29, 31, 32, 36, 37, 39–41, 72, 73, 121–124, 130–132, 134

Aristotelian logic, 292, 293, 297, 305, 307, 308 Aristotle, 7, 111, 112, 160, 231, 246, 257, 266, 277, 286, 290, 292, 296, 329 Arnal, W., 270 ars explicandi, 195, 199, 225 ars iudicandi, 195, 199 Ash’aris, 287, 293 Authorizing the previous state, 283 Automated reasoning, 198, 201–203, 206, 218, 225 Automated theorem proving (ATP), 196, 202–204, 215

B Barcan formula, 211 Basic TCS, 189 Beall, J.C., 259, 260, 262, 263, 265 Begging the Question: Strict Case, 232–234 Begging the Question: Weaker Case, 234–236 Begs the question, 232, 234, 236, 240, 243, 246–248, 251 Béziau, J.-Y., 3–14, 258, 321–331 Bible, 35, 50, 53, 75, 76, 78, 156, 268, 331 Biblical, 12, 50, 52, 79, 143, 153–168, 319 Blocks World, 13, 177, 181–183, 185, 186, 192 Book of Daniel, 12, 154, 156 Book of Job, 51, 52, 71 Boolean dichotomy, 14, 301 Brandom, R.B., 167, 196, 199, 264, 272 Buddhism, 4, 257, 267–270, 309, 313 Buddhist logic, 14, 303

© Springer Nature Switzerland AG 2020 R. S. Silvestre et al. (eds.), Beyond Faith and Rationality, Sophia Studies in Cross-cultural Philosophy of Traditions and Cultures 34, https://doi.org/10.1007/978-3-030-43535-6

335

336 C C1 , 156, 162–164, 166, 176, 181, 190, 191, 208, 215, 218–220, 322 Campbell’s treatment, 244 Caputo, J., 257, 266, 267, 272 Characterization principle, 109 Christianity (Christian), 4, 7, 11, 12, 19, 20, 24, 29, 34–36, 39, 50, 51, 69, 71, 72, 76, 82, 84, 87, 88, 99, 138–140, 143, 149, 150, 156, 158, 162, 166, 197, 268, 277, 292, 293 Classical logic, 13, 107, 118, 125, 149, 156, 157, 159, 167, 181, 182, 185, 186, 201, 203, 257–259, 261, 263, 264, 266, 273, 293, 302, 323, 329, 331 Common Reason, 278 Common sense, 283, 286 Commonsensical reasoning, 277 Companions Adjudges, 283 Compositionality, 199 Computational hermeneutics, 13, 195–225 Concept of God, 9, 10, 12, 32, 41, 44, 45, 47, 81, 93, 95, 103, 105 Conceptual analysis, 5, 6, 8, 12 Conceptual explication, 96, 198 Conceptual relativism, 197 Conceptual scheme, 197, 316 Conjunctive syllogism, 295, 296 Consensus, 63, 111, 282–284 Consistency, 41, 45, 150, 162, 164, 176, 196, 206, 214, 215, 289, 301, 304 Contradictions, 12–14, 19, 24, 37, 38, 42, 45, 47, 80, 125, 146, 150, 153–168, 179, 212, 260, 261, 264, 289, 305, 308, 309, 311, 314, 315, 317, 318, 328 Cook, M., 259, 271 Criterion of concomitance, 295, 297 Criterion of opposition, 295, 297 C-Systems, 164 Custom, 283

D Da Costa, N.C.A., 14, 156, 159, 161–164, 166, 321–331, 333 Davidson, D., 195–197, 199–201 Davidson, R., 257 Defense, 12, 32, 71, 123, 124, 126, 128–131, 134, 277, 292, 294 Definitional Problems with Miracles, 76–78 Descartes, R., 106–111, 116–117 Determinism, 44, 45 Devine Intelligence, 278

Index Divine laws, 279 Dummett, M., 179, 262, 263

E Eder, G., 198, 230–232, 234–236, 238, 241–245, 249, 250 Epistemic, 5, 21, 40, 41, 47, 72, 73, 77, 110, 241, 246–249, 251 Epistemological problems with miracles, 79–84 Essence, 35, 36, 78, 93, 96, 98, 106, 107, 138, 141, 144, 247, 258, 259, 261, 262, 265–271, 314, 315 Essence of Hinduism, 267 Essence of Islam, 271 Essence of logic, 259–267 Essence of religion, 267–270 Eternity, 25, 32, 86 Evidential (or inductive) problem of evil, 12, 121–126, 131 Evil, 9, 19, 29, 49, 85, 283, 328 Exceptive syllogism, 295–297 Existential entailments, 12, 107–108 Explicating, 53, 198, 209, 220

F Faith, 3–14, 138, 141, 144, 158, 198, 270 Fakhr ad-D¯ın R¯az¯ı, 294 Filiation, 140–144, 147–149 Filioque, 12, 137–150 Fiqh, 282, 283 First Intellect, 278 Fixed unchanging essences, 257, 267, 271 Formalization, 13, 82, 94, 95, 103, 166, 195–197, 200–206, 208, 209, 211–214, 216, 221, 223, 225, 232, 234, 237, 241, 245, 247, 249–251, 302, 307 Formal methods, 4, 11, 150 Formal system, 95, 137, 142, 259 Foundational, 84, 107, 118, 288 Free will, 26, 32, 34–36, 45, 49, 62, 86, 123, 128, 130 Free will (freedom), 26, 62 Free-will defense, 36, 86, 123, 128, 130

G Gaunilo’s refutation, 231, 236 Gayatri, R., 270 Generalized Definition of Intuitive Validity, 265, 267 Generalized Tarski Thesis, 262, 263, 265, 266

Index Global Suffering, 34 Gödel’s argument for God’s existence, 93 God’s existence, 11, 49–51, 54, 72, 80, 93, 127 Gombrich, R., 269, 270 Goo , 96–99, 101, 103 Goodness, 26, 50, 52, 54–56, 59, 85, 86, 128, 327, 328 GO¬P, 101, 102 Greek philosophy, 277, 294

H Haack, S., 156, 259–263, 265, 267 Hamlet, 266 Hanaf¯ısm, 280, 282 Hanbal¯ısm, 280, 282 Hermeneutic circle, 196, 205 Higher-order logic (HOL), 196, 200–203, 205, 207, 209, 229, 242 Higher order treatment, 231, 242, 244, 245 Hijras, 270, 271 Hinduism, 14, 267, 269, 273, 301–319 Human reason, 150, 278, 287 Hume, D., 7, 9, 72, 76–80, 82, 85, 87–89, 130

I Ibn S¯ın¯a, 281, 288, 291–294, 296–298 Ibn Taym¯ıyyah, 294 Ideal language approaches, 185 Illocutionary logic, 265 Illuminationists, 292 Ilm, 279 Ilm-e Us¯ul-e Fiqh, 283 Imâms, 280, 281, 283 Impossibility, 35, 37–39, 302 Inconsistency, 9, 37, 40, 41, 121–125, 131, 164, 312, 317 Indeterminism, 44 Indian philosophy, 14, 301–319 Indian religions, 14, 271, 301–319 Indirectly Begging the Question, 238 Induction, 39, 80, 190, 283 Induction (inductive arguments), 20, 39, 80, 132, 190, 283 Inferential adequacy criteria, 206 Inferential role, 195, 198, 199, 201, 216, 218, 220–222, 225 Informal Accounts of Begging the Question, 246–249 Intellect, 58, 59, 62, 140, 143, 278, 279, 281, 283 Intelligence, 30, 43, 83, 181, 182, 202, 278, 279

337 Intuitionistic logic, 261–265 Intuitive concept of validity, 260–263 Isabelle, 203, 205, 207, 209–211, 214–216, 218, 224, 225, 230 Isagoge, 291 Islam, 4, 14, 270, 271, 277–299 Islamic jurisprudence, 279, 280, 282, 283, 286, 295 Islamic philosophy, 278 Islamic theology, 285 Islamic world, 14, 278, 285, 290–294, 298 Istibs¯ar, 281

J Jabb¯ay¯ı, 293 Jaffrey, Z., 271 Jain, 312 Jaina anek¯antav¯ada, 309 J¯ami’-e at-Tirmiz¯ı, 280 Jean-Yves, B., 14 Jesus, 12, 13, 20, 28, 75, 86, 87, 153, 154, 156–160, 162, 163, 165, 166, 197 Jurisprudence, 278–288, 291, 292, 294, 295, 297 Juristic Preference, 283

K Kant, I., 9, 106–108, 110, 116 Khurdeh Avest¯a, 286 Kit¯ab al-K¯af¯ı, 281

L Lambert, K., 109, 111 Law of Non-Contradiction (LNC), 160–164 Laws of nature, 27, 33, 43, 44, 76 Leaman, O., 59 Lewis, D., 106, 117, 250 LFI1, 156, 163–166 Literalists, 287 Local suffering, 32, 35 Logic, 3, 30, 94, 105, 125, 137, 154, 174, 195, 229, 257, 277, 301, 322 Logical philosophy of religion, 4, 8, 10, 11 Logical pluralism, 167, 168, 196, 201, 259, 261, 262, 265, 273 Logical problem of evil, 12, 123–125, 128–131 Logical reasoning, 14, 288, 289 Logicography, 290, 291 Lokamitra, D., 269 Love of God, 46–47, 52 Luther, 273

338 M Maimonides, 57–61, 65, 69, 72 Major criterion of equivalence, 297 Major Term, 296 Mâlikîsm, 280, 282 Man lâ Yahzuruhu al-Faqîh, 281 Masuzawa, T., 268, 269 Material Intellect, 278 McCutcheon, R.T., 270 Meaning, 7, 11, 13, 28, 38, 76–78, 81, 87, 94, 99, 110, 113–115, 122, 153, 154, 183, 187–189, 191, 195, 197–199, 216, 218, 220, 224, 225, 237, 246, 263, 268, 274, 279, 282, 285–288, 302, 304, 305, 307, 308, 314, 325, 326, 328, 331 Meaning holism, 197, 199 Metalanguage, 200, 201 Metaphysical contingentism, 210 Metaphysical necessitism, 210, 215 Middle criterion of equivalence, 297 Middle term, 296 Mîmâmsa, 303–309 Minor criterion of equivalence, 297 Mion, G., 106 Miracles, 9, 11, 69, 75–89 Modal logic, 94, 100, 201, 207, 209, 211, 217, 219, 250, 291, 328, 329 Model finders, 202, 206, 211, 214, 215, 217 Mohammad, P., 55, 78, 280–283, 287, 292, 295, 297 Mohammad ibn Ya’qub ibn Ishâq al-Kulaynî, 281 Mullâ Sadrâ, 281, 288, 292, 293, 295, 297, 298 Müller, M., 268, 269 Mu’tazilis, 284, 287

N Nâgârjuna, 314–317 Natural theology, 19, 20, 103 Negation, 14, 80, 94, 95, 98, 99, 103, 162, 210, 212, 222, 239, 301–309, 311, 313–319, 322, 323, 330 Nitpick, 211–219, 223, 224 Nyâya, 4, 14, 301, 303–309

O Object language, 200, 201, 209 Omnipotence, 24, 25, 37, 44, 45, 328 Omniscience, 25, 32, 37, 44, 45 Omputational metaphysics, 204

Index Ontological arguments, 9–13, 15, 93–103, 105–118, 195–226, 229–251, 288 Ontological dependence, 207, 208, 212–214, 219–222, 224 Opposition, 16, 141, 142, 144–146, 164, 269, 295, 297, 305, 308, 312, 333

P Paraconsistency, 14, 161, 162, 324–331 Paraconsistent, 13, 14, 156, 161, 162, 166, 167, 257, 321–331, 333 Paraconsistent logic, 13, 156, 161, 167, 257, 322, 323, 326, 327, 329–331, 333 Paraconsistent systems, 156, 162, 330, 331 Paradox of the heap, 13, 173, 176–181, 192 Particular Reason, 155, 278 Passive Intellect, 278 Passive Spiration, 140, 141, 143, 147–149 Paternity, 140–144, 148 Paul, P., 286 Performatives, 264, 266 Performative utterances, 264 Peripatetics, 292 Persons, 5, 20–22, 24–27, 30, 32, 36, 42, 46, 53–56, 60–66, 68, 71–73, 77–79, 81, 83, 86, 89, 117, 138, 140–145, 148–150, 189, 250, 268, 274, 294, 304, 308 Philosophical Success, 30–32, 36, 37, 39 Philosophy, 4, 28, 30, 50, 76, 107, 121, 138, 177, 196, 257, 277, 301, 327 Platinus, 281 Plato, 7, 286 Porphyry, 291 Possible worlds, 41, 43–45, 101, 102, 207–210, 215, 217, 262, 312, 327 Possible-world semantics, 209 Potential Reason, 278 Practical Reason, 278 Prayers, 14, 53, 75, 78, 81, 85, 143, 157, 258, 266, 267 Predecessors Laws, 283 Predicative syllogism, 296 Priest, G., 106, 107, 109–111, 153, 160, 161, 180, 261, 315, 317, 318, 322, 328, 330 Priestian dialetheism, 301, 313–319 Principle of charity, 69, 195–197, 200, 201, 206, 209, 213 Probability, 20, 22, 37, 46, 63, 72, 80–82, 125, 127, 129, 132

Index Problem of evil, 9, 11, 12, 28–73, 85, 86, 121–134 Procession, 138–140, 142, 149 Proof assistants, 13, 202–204, 207, 209, 214, 216, 225 Proof checking, 202 Pure Reason, 108, 278 PVS, 230, 232–245, 251

Q Qistâs, 288, 289, 294 Qur’ân, 14, 277–289, 291–295, 297, 298 Qur’ânic commentaries, 281, 289, 291, 297 Qutb ad-Dîn Râzî, 298

R Radical interpretation, 195, 200–202 Ramharter, E., 198, 230–232, 234–236, 238, 241–245, 249, 250 Rationality, 3–14, 34, 66, 79, 80, 82, 83, 89, 121, 153–168, 257, 286, 297, 298, 315, 317 Reason, 4, 13, 14, 20, 24–28, 32, 34, 35, 38, 40, 45, 47, 53, 58, 59, 61, 62, 64, 67, 69–73, 77, 80, 81, 84, 85, 88, 111, 123, 128, 130, 138, 142–144, 150, 159, 174, 183, 187, 246, 248, 249, 257, 261, 265, 269, 270, 272, 278, 279, 281, 283–288, 290, 293, 298, 313, 317, 328, 329, 331 Reasoning, 10, 12–14, 20, 79, 105, 109, 114, 115, 125, 129, 147, 149, 155, 156, 158, 167, 176, 178, 180, 181, 192, 198, 201–203, 206, 215, 218, 225, 228, 232, 234, 239, 242, 249, 264, 277–283, 285–291, 293, 294, 301, 302, 304, 310, 314, 318, 329, 331 Reddy, G., 270, 271 Reflective equilibrium, 13, 205, 206, 220, 225 Relationes personificae, 149 Religion, 4, 29, 50, 75, 121, 122, 137, 180, 197, 257, 277, 311, 331 Religious argumentation, 197, 198 Restall, G., 259, 260, 262, 263, 265 Resurrection, 12–14, 20, 28, 52, 54, 61, 70, 75, 154, 156–159, 162, 163, 165, 166 Resurrection of Jesus, 75, 154, 156–159, 163–166 Reverse engineered, 231, 241, 248–251 Rg Veda, 301–304, 307, 315

339 S Saadia, G., 11, 49–73, 305 Sacred Reason, 278 Sahîh al-Bukhârî, 280 Sahîh al-Muslim, 280 Sahlqvist correspondence, 217, 219, 224 Scott version of Gödel’s argument, 95 Searle, J.R., 155, 265–267, 273 Segregationists, 287 Semantical embedding, 201, 203, 205, 206, 209, 210 Sequent, 204, 239–241, 245, 248 Shapiro, S., 259, 260, 263 Shestov, L., 257, 258 Shî’îism, 287 Shî’îs, 287 Skolem constants, 240, 243, 249 Sledgehammer, 205, 215, 218, 219, 224 Speculative Reason, 278 St. Anselm, 13, 198, 207, 223, 229 St. Anselm’s ontological argument, 13, 198, 207, 223 Stoic logic, 290, 291 St. Paul, 19 Substance, 5, 21, 22, 24, 25, 28, 65, 138, 139, 141, 143–145, 149, 173 Suhrawardî, 292, 293, 298 Sun, 21, 23, 52, 60, 268, 280, 324, 325 Sun, A., 268 Sunan as-Sughrâ, 280 Sunan-e Abû Dâwûd, 280 Sunnîism, 280–283, 287, 288, 292, 298 Syllogism, 162, 283, 288, 289, 291, 294–298

T Tafkîkîs, 287 Taftâzânî, 298 Tahzîb al-Ahkâm, 281 Talmudic calculus of sorites (TCS), 181, 184–192 Talmudic norms as applied to mixtures, 173 Talmudic norms regarding mixtures, 180–184 Tarski, 200, 262, 263, 265, 266 Teleological argument, 20, 288 Theism, 8, 21, 24–26, 28, 33, 77, 81–83, 88, 89, 122, 127, 129, 132, 200, 269, 318 Theodicy, 12, 32, 50, 51, 53–55, 57, 59–69, 71, 85, 86, 88, 123, 124, 126–131, 134 Theology, 4, 9, 11, 12, 15, 19, 20, 50, 64, 93, 103, 137, 138, 142, 143, 146, 147, 149, 150, 161, 258, 277, 278, 281, 285–288, 290, 292–295, 297

340 The ontological argument, 9–13, 15, 93–103, 105–118, 195–198, 203, 207, 209, 212, 216, 218, 223, 229–251, 288 Theophrastus, 296 Theory of truth, 200 The posteriori argument, 79 Thomistic, 12, 137, 142–146, 150 Tractatus, 12, 107, 111, 118 Traditionists, 283, 287 Treatment, 12, 54, 61, 76, 106, 110, 112, 115, 116, 128, 137, 149, 199, 231–233, 235, 238, 241–245, 248, 268, 303, 305 Trinitarian theology, 12, 137, 146, 147 Trinity, 12, 24, 137–150 Typecheck Correctness Condition (TCC), 233, 234, 238, 243

Index U Universal Reason, 278 Upanis.ads, 303 Usulites, 288 V Vacuity, 229–252 Vajpeyi, A., 269 Vanderveken, D., 265–267, 273 Verification systems, 229, 230 W Wiebe, D., 257, 258 Wittgenstein, 4, 12, 107, 111, 112, 114, 115, 118, 167, 168