God, Truth, and other Enigmas 9783110418934, 9783110419955

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God, Truth, and other Enigmas

Philosophische Analyse/ Philosophical Analysis

Herausgegeben von/Edited by Herbert Hochberg, Rafael Hüntelmann, Christian Kanzian, Richard Schantz, Erwin Tegtmeier

Band/Volume 65

God, Truth, and other Enigmas Edited by Miroslaw Szatkowski

ISBN 978-3-11-041995-5 e-ISBN (PDF) 978-3-11-041893-4 e-ISBN (EPUB) 978-3-11-041899-6 ISSN 2198-2066 Library of Congress Cataloging-in-Publication Data A CIP catalog record for this book has been applied for at the Library of Congress. Bibliographic information published by the Deutsche Nationalbibliothek The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at http://dnb.dnb.de. © 2015 Walter de Gruyter GmbH, Berlin/Munich/Boston Printing: CPI books GmbH, Leck ♾ Printed on acid-free paper Printed in Germany www.degruyter.com

Contents Acknowledgements | XI Mirosław Szatkowski A Guide to the Book: God, Truth, and other Enigmas | XIII

Part I (God’s) Existence/Non-Existence C. Anthony Anderson Logical Necessity, Conceptual Necessity, and the Ontological Argument | 3 1 What is Logical Necessity (Logical Modality)? | 3 2 Conceptual Modality | 5 3 Terse Formalization of a Hartshornian Modal Ontological Argument | 6 4 A “Fitchy” Argument for Premise (2) | 10 5 The Crucial Premise | 11 Seweryn Blandzi Problem of the Origins of Ontotheology | 15 1 Towards the Theologization of Being | 16 2 “Craftsman” or “Creator”? | 19 3 The Unnamed, and yet Named “the Being One”: the Premises of Ontotheology | 21 4 God as Esse Absolutum | 24 Robert E. Maydole On the Anti-Ontological Doom Argument | 29 Peter van Inwagen Nothing Is Impossible | 33 1 Preliminary Matters | 35 1.1 A Miscellany of Terms, Concepts, Definitions, and Assumptions | 35 1.2 Kinds | 36 1.3 Explanation | 37 1.4 Three Principles | 39

VI | Contents 1.5 2 2.1 2.2 3

A Premise about Kinds | 42 The Arguments | 43 The Arguments | 43 A Possible Objection to the Arguments | 48 “The Mystery of Existence” | 53

Jan Woleński God and Good: Does God’s Existence Imply that Anything is Good | 59

Part II Omniscience Stamatios Gerogiorgakis Gaps, Gluts and God | 71 1 Gaps and Gluts | 71 2 Milne’s Paradox | 72 3 Strengthening the Milne Sentence | 75 4 Conclusion | 76 Jason Megill Fitch’s Paradox and the Existence of an Omniscient Being | 77 1 The Knowability Principle and Fitch’s Paradox | 77 2 An Argument for an Omniscient Being | 79 3 How Plausible is the Knowability Principle? | 83 4 Concluding Remarks | 87 Elisa Paganini Vagueness and Omniscience | 89 1 Introduction | 89 2 Cooperation by an Omniscient Being: First Proposal | 90 3 Cooperation by an Omniscient Being: Objection to the First Proposal | 91 4 Cooperation by an Omniscient Being: Second Proposal | 92 5 Cooperation by an Omniscient Being: Objection to the Second Proposal | 93 6 Some Conclusive Remarks | 95

Contents

| VII

Mirosław Szatkowski God’s Omniscience and Logical Virtue | 97 1 Introduction | 97 2 Cantor’s Argument and the Complete Knowledge of God | 99 3 The Liar and the Sound Knowledge of God | 102 4 McTaggart’s Paradox, and Various Semantical Approaches to Time | 104 5 Anderson-type Semantics Taking into Account the Time and the Knowledge of Truths | 108 6 Conclusion | 111

Part III Truth Srećko Kovač Logic and Truth in Religious Belief | 119 1 Introduction | 119 2 Logic and Religious Belief | 119 3 Faith Pragmatics and Truth | 122 4 Appearance and Truth in Religious Belief (Semantics) | 124 4.1 Logic QB Modified | 124 4.2 Formal Analysis of John 3 | 128 Anna Lemańska Absolute Truth and Mathematics | 133 Alexander Pruss The Divine Belief Theory of Truth: Might It Work? | 141 1 The Theory | 141 2 Statement and Some Merits | 141 2.1 Beliefs by a Perfect Beings | 141 2.2 Sentential Truth | 142 2.3 Some Merits | 142 3 Objections | 143 3.1 The Euthyphro Objection | 143 3.2 Knowledge | 145 3.3 Epistemological Circularity | 147 3.4 Atheists and Truth | 148

VIII | Contents 3.5 3.6 4

The Liar Paradox | 148 Similarity to Occasionalism | 149 Conclusions | 150

Peter Simons Makers and Models: Two Approaches to Truth, and their Merger | 153 1 Introduction: Tarski and Nominalism | 153 2 Truth, Consequence, and Models | 154 3 Makers | 156 4 Models | 159 5 Multitudes | 160 6 Models from Multitudes | 162 7 Consequences for Nominalists | 163 8 Open Questions and Conclusion | 165

Part IV Metaphysical Enigmas Christopher Daly and David Liggins Agnosticism about Material Composition | 169 1 Introduction | 169 2 Agnosticism about Material Composition | 170 3 Conceptual Analysis | 171 4 Common Sense | 173 5 Science | 175 6 Simplicity | 177 7 Philosophical Argument | 178 8 Reflections and Recriminations | 179 Christian Kanzian Existential Dependence and other Formal Relations | 183 1 Formal Relations | 184 2 Ontological Dependence | 187 3 Existential Dependence | 189 4 Existential Dependence and the Three-Categorial Ontology | 191 5 Formal Relations and Their Possible Function in Philosophical Theology | 194

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Duncan H. Pritchard Wittgenstein on Faith and Reason: The Influence of Newman | 197 1 Introductory Remarks | 197 2 Wittgenstein on the Structure of Reasons | 199 3 Newman on Faith and Reason | 204 4 On Certainty | 208 5 Concluding Remarks | 211 Scott A. Shalkowski Necessity, Worlds, and God | 217 1 Introduction | 217 2 A Most Common Strategy | 218 3 Necessity and Worlds | 219 4 Necessity and God | 221 5 Be Careful What You Wish For | 227 6 How Not to Argue for the Plurality | 229 7 Rods for Their Own Backs | 232 8 God, Necessity, and Limits | 237 9 Conclusion | 239 Bartłomiej Skowron The Explanatory Power of Topology in the Philosophy of God | 241 1 Goal | 241 2 Mathematical Theology | 241 3 Mathematica Theologiae Ancilla | 242 4 Topology – Basic Ideas and Concepts | 244 5 Topological Explanations in the Philosophy of God | 245 5.1 God’s Suffering vs. God’s Infinite Power. Is God a Topological Space? | 246 5.2 God as a Closure of the World | 246 5.3 God-Topology: Connected and not Metrizable? | 248 5.4 Topological Analysis of Unity of God | 249 6 The Explanatory Power and Limits of Topological Explanations | 250 Authors of Contributed Papers | 255 Author Index | 261 Subject Index | 265

Acknowledgements The publication of this book was preceded by the 2nd International God Oriented Conference, entitled: God, Truth, and other Enigmas, which took place in Warsaw on 17–19 September 2013. The conference was organized under the auspices of the Institute of Philosophy and Sociology at the Polish Academy of Sciences (IFiS PAN), within the research grant Existence of God. Research on the ontological proofs – obtained in the competition “OPUS 4” of the National Science Center (decision DEC-2012/07/B/HS1/01978). So I would like to thank the National Science Center for funding this conference. Moreover, I would like to acknowledge IFiS, for acting as a unit in implementing the project. Special thanks also go to Ms. Grażyna Drążyk from IFiS PAN, who helped me in different ways throughout the preparation of the proposal to the National Science Center, as well as in the organization of the conference. Not all speakers who took part in the conference submitted their papers to the present volume and vice versa – not all authors of the volume participated in the conference. I would like to express my sincere gratitude to all eighteen authors who accepted the invitation to contribute to the volume and showed great understanding and patience during the publication process. I would also like to thank all of the speakers at the conference. I am grateful to Dr. Bartłomiej Skowron for faithfully supporting my initiatives by editing the websites of both International God Oriented Conferences and International Ontological Workshops. He also provided much-appreciated help in bringing the earlier version of this manuscript is its present form. Last, but certainly not least, I would like to thank my brother, Sławomir Szatkowski, for his voluntary help in preparing this volume for print. I have tried to make the best use of all his suggestions and corrections.

Mirosław Szatkowski

Mirosław Szatkowski

A Guide to the Book: God, Truth, and other Enigmas The book God, Truth, and other Enigmas is a collection of eighteen essays that are subsumed under four headings: (God’s) Existence/Non-Existence, Omniscience, Truth, and Metaphysical Enigmas. The final phrase in the title of the book, ‘and other Enigmas’ suggests that the terms that precede it ‘God’ and ‘Truth’ are enigmatic. Indeed, both terms have been and are among the most discussed terms in the philosophical literature. And certainly, they will be discussed for a long time to come, if not indefinitely. The essays vary widely in topic and tone, perhaps as a result of the fact that the contributors submitted them independently of one another rather than as a result of group work. The essays are engaging and certainly provide the reader with a review of contemporary philosophical approaches to the subjects in the title of the book. What comes to mind when we think about God? Philosophical reflection on what is meant by the term ‘God’ and whether such a God exists tries to use rational methods to clarify this problematic situation. The only way to know what God is, is to know His attributes. The attributes of God such as, for example, omniscience, omnipotence, and perfect goodness are distinguishing characteristics of the divine nature and are crucial to understanding the truth about God. And it is obvious that only after recognizing the nature of God can one argue for His existence or non-existence, although there is also a view in the philosophical literature that since God is neither an entity in the universe nor a mathematical object, it is not obvious what kinds of arguments/proofs are relevant to God’s existence or non-existence. Even if the concept of scientific proof were not problematic, the fact that there is no conclusive scientific proof of the existence, or non-existence, of God mainly demonstrates that the existence or non-existence of God is not a scientific question. Indeed, it would be difficult to find an argument for the existence or non-existence of God in the philosophical literature whose premises and line of reasoning would be undisputed. Yet, this does not mean that the endeavours to construct new rational arguments for the existence or non-existence of God should be given up, or that the study of the rationality of the existing arguments should cease. It should be added, though that defining rationality criteria is almost as difficult as defining God. Another reason why one should not be overcome with pessimism is that it would be difficult to find a philosophical issue that does not raise controversies among philosophers.

XIV | Mirosław Szatkowski The second of the terms present in the title of the book – namely, ‘truth’ — also gives rise to many difficult questions, the most fundamental of which are the following: What is truth? and What sorts of things can be true? (or, more generally, What sorts of things can have truth values?). From among many theories of truth, the most significant ones are: (i) The Correspondence Theory. According to this theory, a proposition is true provided there exists a fact corresponding to it; or alternatively, for any proposition p, p is true if and only if p corresponds to a fact. Thus truth is a certain relationship that holds between a proposition and its corresponding fact. Two questions arise here: What is this relationship? and What are facts? There is a wide variety of responses, along with their critics, to both these questions. (ii) The Semantic Theory. This theory was formulated by A. Tarski to resolve the Liar Paradox (*): The statetment (*) is false, and similar sentences. The Liar paradox and all other sentences similar to it have a self-referential nature, which means that they express inside a language their own truth-concepts. The selfreference of sentences is, however, in striking disagreement with the Tarski’s indefinability theorem. To solve the Liar paradox, Tarski proposed the hierarchy of not universal object-languages and not universal meta-languages, which fragments the concept of truth. Instead of one concept, we have infinitely many of them, arranged in a hierarchy. Given an object-language, Tarski requires that the concept of truth for this language be contained in its meta-language and not in the objectlanguage itself. A meta-language is much stronger than the object-language, since the latter is assumed either to be a fragment of the meta-language, or to have a preordained translation into the meta-language. Tarski’s theory is very technical. It applies only to any sufficiently strong artificial languages – i.e., to any first-order formal languages with negation, and with sufficient capability for selfreference that Gödel’s diagonal lemma holds. Tarski gave a number of reasons for not extending his theory to natural languages. In contrast, D. Davidson – based on certain ideas of Frege – has expanded Tarski’s theory to any natural language, whereas S. Kripke showed that a natural language can contain its own truth predicate without running into contradiction. He created a language in which truth is not defined for sentences like the Liar Padadox, because these sentences are neither true nor false. The semantic theory of truth is related in many points to the correspondence and deflationary theories, but it is a controversial matter whether Tarski’s semantic theory should be counted as one of them. (iii) The Coherence Theory. According to the coherence theory, a proposition is true if and only if it coheres with some specified set of propositions. Depending on the understanding of the coherence relation and of the set (or sets) of propositions with which true propositions cohere, different versions of the coherence theory can be identified. Some coherentists claim that the coherence relation is simply con-

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sistency – to say that a proposition coheres with a specified set of propositions is to say that the proposition is consistent with the set. Others argue that a coherence relation is some form of entailment – a proposition coheres with a set of propositions if and only if it is entailed by members of the set. According to yet another view, coherence is a mutual explanatory support between propositions. Generally, coherentists state that this specified set of propositions with which true propositions cohere is a set of true propositions. However, they differ on the question how the truthfulness of these propositions is determined. And so, some coherentists claim that the specified set of propositions is the largest consistent set of propositions currently believed by actual people. For others, the specified set consists of those propositions which will be believed when people like us have reached some limit of inquiry. Yet another opinion holds that the specified set contains the propositions which would be believed by an omniscient being. (iv) The Deflationary Theory. According to the deflationary theory of truth, to assert that a statement is true is to assert the statement itself. What is expressed by the equivalence schema: ‘p’ is true if and only if p. Thus, for example, saying that ‘snow is white’ is true (or that it is true that snow is white) is equivalent to saying that snow is white. Depending on different interpretations of the equivalence schema, different variations of deflationism are identified, the most infuential of which are: redundancy theory, performative theory, and prosentential theory. Advocates of redundancy theory claim that truth is a redundant concept, i.e., that ‘truth’ is a mere word that is conventional to use in discourse but not a word that points to anything in reality. According to them, nothing is added to the assertion of the sentence ‘Snow is white’ by quoting it, appending the predicate ‘... is true’, and then asserting the result. That is, they treat the predicate is true as empty. According to the performative theory of truth, ascribing truth to a proposition is not really characterizing the proposition itself, but it is telling something about the speaker’s intentions. So, to say that proposition P is true is to say in a disguised fashion ‘I commend P to you’, or ‘I endorse P’, or something of the sort. Finally, according to the prosentential theory of truth, all uses of the grammatical predicate ‘is true’ are prosentential uses. All uses of ‘true’ can be reduced to uses of either ‘that is true’ or ‘it is true’ or variants of these with other tenses, which are substitutes for the propositions considered. (v) The Pragmatic Theory. Generally, according to the pragmatic theory of truth, utility is the essential mark of truth, i.e., a proposition is true if it is useful to believe in. However, some differences in the understanding of particular problems of truth are pointed out even by the three most famous advocates of this doctrine: Ch. S. Peirce, W. James and J. Dewey. Peirce distinguishes between two methods: scientific and logical. The scientific method is aimed at attaining truth. Its fundamental hypothesis is: “There are real things whose characters are entirely inde-

XVI | Mirosław Szatkowski pendent of our opinions about them.” The new conception involved here is that of reality. Thus, Peirce’s understanding of truth is a means of clarifying the concept of reality. In contrast, the logical method is aimed at attaining clearness. Clearness and truth, according to Peirce, have no necessary connection. James’ views on truth were subject to change. In his earlier writings, James said that an idea is true when it resembles something which I, as a critic, think to be reality. In the book Pragmatism, James used the term ‘truth’ in two senses, although he did not recognize these two meanings of ‘truth’ himself. In the book, one can find passages which can be interpreted as meaning either value or fulfillment, but there are also passages in which the use of value as a criterion seems unmistakable. The difference between the Peirce’s and James’ positions may be stated – inter alia – by the fact that James introduces the factor of value as a criterion for truth – the value of a belief is an apparent evidence for true/the true, while for Peirce the value has no relation to truth. James defined the true also in terms of satisfaction. But he understood satisfaction as satisfaction by rather than satisfaction of. So, for instance, one may be satisfied of the correctness of one’s idea without being at all satisfied by it. Dewey, in turn, states in his Logic: The Theory of Inquiry that the best definition of truth from the logical standpoint which is known to him is that by Peirce: “The opinion which is fated to be ultimately agreed to by all who investigate is what we mean by the truth, and the object represented in this opinion is the real.” But Dewey further defines truth in terms of his preferred concept of warranted assertibility as the ending- view and conclusion of inquiry. Truth is a function not of whim nor purely of social construction, but a quality situated in events. In his opinion, experimentation of various sort is an arbiter of truth. There are a lot of different issues concerning the relationship between God and truth. Generally, it is all about the question: Are the meanings hidden under both these terms compatible with each other? Or: what are the conditions for these meanings to be compatible? Most of these 18 papers directly deal with the issue of the compatibility of these two terms, focusing on different aspects of this problem. But even the papers in which this relationship is not directly addressed, pertain to some ontological questions that are in some way close to the relationship between God and truth. All the 18 papers will be characterized in some detail below.

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1 The issue of the non-existence or existence of either anything or God is the glue that connects all five papers of Part I. C. Anthony Anderson (“Logical Necessity, Conceptual Necessity, and the Ontological Argument”) challenges the idea that there exists such a thing as “logical necessity”: he introduces “conceptual necessity”, or, more generally, “conceptual modality” instead. By using this notion, he introduces a strict formalization of the Hartshornian modal ontological argument (abbreviated to HMOA), adding supporting arguments for the particular premises. Without going here beyond propositional and first-order logic, for instance, a proposition is tautologically (logically) necessary if it is true in virtue of its constituent concepts determined by the connectives, and a proposition is first-order (logically) necessary if its validity is ruled out by the fixed concepts of first-order logic, i.e. by the meanings of the connectives and quantifiers and, perhaps, exemplification. Following Peter van Inwagen, Anderson claims that “logical possibility” is not a kind of possibility, and, in a similar vein, adds that “first-order possibility” is not a kind of absolute possibility. He argues that “a proposition is conceptually necessary if it is necessary simply in virtue of the essential properties of, and relations between, the concepts it contains”. But “conceptual possibility” is not a kind of absolute possibility either. As far as the axiomatization of HMOA is concerned, the primary linguistic terms include conceptual necessity, which is attached only to sentences, whereas defined connectives embrace: conceptual impossibility, conceptual consistency and conceptual entailment. Lambda abstraction symbol, exemplification (for properties), and predicates of: being perfect, being omnipotent, being omniscient, being perfectly good and existence also function as primary terms. All these concepts are characterized by relevant axioms or axiom schemas. A proof for the actual existence of a perfect being is provided. Anderson argues for one of the premises of this proof, i.e. (in words): “That perfection is instantiated (conceptually) necessitates that perfection is (conceptually) necessarily instantiated”. He also justifies the axiom (in words): “It is conceptually coherent to suppose that perfection is instantiated”. Seweryn Blandzi (“Problem of the Origins of Onto-theology”) provided a short outline of his own paper. It is presented here without any changes. And so: Seeking the further and more distant sources of the radical formulation of the issue of metaphysics as onto-theology by M. Heidegger (1957), we find the definitive analyses of Aristotle’s Metaphysics put forward by P. Natorp (1888) and P.

XVIII | Mirosław Szatkowski Aubenque (1983), and in more distant past, Philo’s of Alexandria (around 20 BC40 AD), who was probably inspired by the Book Kappa of this work in his theological interpretation of Being. In modern scholarship, Natorp was the first to demonstrate the contradiction in Aristotle’s “first philosophy” in terms of dual components, as this philosophy was understood, i.e., at the same time as the science of all Being and of the Supreme Being. This contradiction was a result of the failure to recognize that Aristotle did not write certain parts of the Metaphysics (the Book K in particular), on the one hand, and the erroneous interpretation of its genuine content, on the other. Ignoring this important reservation, Heidegger attributed to Aristotle’s metaphysics the domination of the ontological component by the theological one, although this reservation is not totally incorrect with respect to the medieval metaphysics, which identified God with Being itself. This identification is rooted in the imposition of the maximalist concept of Being (originating in Parmenides and Plato) onto a personal God. The first trace of the merging of the two ideas can already be found in the compiler of the Book K in his theological understanding of Aristotle’s formula to on he on (Being as Being), but explicitly the reference of this formula to God (hitherto unnoted by commentators) can be found in Philo, who can be treated as the actual originator of onto-theology. On the other hand, Aristotle’s genuine “first philosophy” could be regarded as having at most two non-equivalent components: an general-ontological one, that focuses on “Being as Being”, and the other one and subordinate to it – the theological one, that focuses on the “highest essence” (timiotaton genos). Robert E. Maydole (“On the Anti-Ontological Doom Argument”) argues that Turri’s Anti-Ontological Argument is false. He calls it “Doom” and expresses it in the following logical form: (1) If you can non-empirically know that a certain person exists now, then you are that person; (2) If any ontological argument can succeed for you, then you can non-empirically know that God exists now; (3) You are not God; (4) If God exists, then God is a person; (5) If God exists, then God exists eternally; (6) If God exists eternally, then God exists now; (7) If no ontological argument can succeed for you, then no ontological argument can succeed for any of us; (8) No ontological argument can succeed for any of us. Maydole challenges premises (1) and (2). He considers two cases: (i) The meanings of ‘non-empirical knowledge’ in (1) and (2) are not identical; (ii) The meanings of ‘non-empirical knowledge’ in (1) and (2) are identical. In case (i), he only allows the possibility that the meaning of ‘nonempirical knowledge’ in (1) should be based on introspection, whereas in (2) – on rational intuition or understanding of concepts. But then Doom is invalid. In case (ii), he takes into consideration three sub-cases: (ii1). The ‘non-empirical knowledge’ means knowledge based on introspection; (ii2) The ‘non-empirical knowledge’ means knowledge based on rational intuition

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or understanding of concepts; and (ii3) The ‘non-empirical knowledge’ means the same thing but it is not knowledge based on inspection or rational intuition or understanding of concepts. If (ii1) occurs, then (2) is a false strict implication because God cannot be known to exist introspectively. If (ii1) occurs, then (1) is a false strict implication because there can exist a possible world in which the antecedent of (1) is true and its consequent is false. And if (ii3) occurs, then Doom is unsound because an ontological argument might succeed for everyone without knowledge, whether empirical or non-empirical, that God exists; thus (2) is false. Maydole also analyses Turri’s arguments for premises (1) and (2), which he considers incomplete or incorrect, and justifies this assessment. Peter van Inwagen (“Nothing Is Impossible”) presents two a priori arguments for the conclusion that it is impossible for there to be nothing. Each argument refers to the rules of modal logic S5, and the conclusion of each argument is the same conditional statetment: if it is possible for there to be something, then it is impossible for there to be nothing. Van Inwagen explains that: “The materials for these arguments are drawn in large part from the modal ontological argument ... and from a certain kind of cosmological argument, the kind based on the principle of sufficient reason.” He first offers a detailed discussion of terms, concepts and principles that play a role in his argument. Here, we will quote only three principles which are premises of both of his arguments: (i) The Principle of Sufficient Reason (PSR): It is a necessary truth that: If beings of a certain kind exist, then there is an explanation of the existence of beings of that kind; (ii) The Principle of the Externality of Explanation (PEE): It is a necessary truth that: If it is contingently true that beings of the kind F exist, then any explanation of the existence of beings of that kind must appeal to or involve beings that are not of the kind F; and (iii) The Principle of Existential Implication (PEI): It is a necessary truth that: For any property, if an explanation (sc. of anything) appeals to or involves beings that have that property, then beings with that property exist. An important premise of both arguments is that “contingent being” is a kind, where a kind of being (or simply a kind) is a property that satisfies the four following conditions: (1) It entails being (or concrescence or concreteness); (2) It is an essential property – a property that can be possessed only essentially (impossible properties are thus trivially essential properties); (3) In every possible world in which anything has it, the “boundary” between the beings that have it and the things (beings or not) that do not have it marks a real division among things; and (4) It is not a negative or disjunctive property. Both arguments proceed by disjunctive dilemma. The arguments are different in that the first argument it has the following form: “In w, contingent beings exist. ∨ In w, no contingent beings exist”, whereas the second argument has this form: – “In w, no necessary beings exist. ∨ In w, necessary

XX | Mirosław Szatkowski beings exist”, where w is any world. Van Inwagen anticipates some objections to these arguments, but then claims: “The ‘possible objection’ I wish to consider is best presented by considering a third argument for our conditional conclusion – an argument that is considerably simpler than the First and the Second arguments. The relative simplicity of the Third Argument (for so I shall call it) is due to the fact that it has as a premise not the thesis that ‘contingent being’ is a kind but that ‘being’ itself, ‘being’ tout court, the property of being a concrete object, is a kind.” In the last section of the article, van Inwagen meditates – as he puts it – on the question: “Why is there anything at all?” Jan Woleński (“God and Good: Does God’s Existence Imply That Anything Is Good?”) poses the question (*): Does A logically imply B?, where A stands for the sentence ‘there is an x such that x = God’ and B refers to the sentence ‘there is an x such that x is good’. After negatively evaluating possible affirmative answers to (*), he gives a negative answer to this question. And so, the strategy in which a premise ’God is good’ is required to derive B from A, is refuted, because – in his opinion – reasons for accepting this premise are not compelling and seem arbitrary. A strategy based on the conclusion – ‘There is an x such that x is good and x ̸ = God’, here denoted by (**), requires that the x in question stands in some relation to God; adequately to standard theological proposals, x is created by God. We presume that Woleński weaves the statement: ‘The creator of a good creation must be good himself’ into the proof for B. Woleński argues against the view that the theory of transcendentals could solve the problem of the entailment of (**) by the assertion that God exists. Additionally, he argues that Hume’s Thesis that Q does not logically entail •Q, where • is a normative operator and Q is a sentence without normative or evaluative ingredients, supports that (**) is not entailed by the sentence ‘God exists’.

2 Part II contains four papers in formal semantics and ontology that focus on problems related to either omniscience in general, or God’s omniscience in particular. Stamatios Gerogiorgakis (“Gaps, Gluts and God”) introduces definitions of “truth-value gluts” and “truth-value gaps”. The first are sentences which are trueand-false, and the second are sentences which are neither-true-nor-false. Scholars who accept truth-value gluts but no truth-value gaps are called dialetheists by him, whereas scholars who accept true-value gluts along with truth-value gaps are called paranormalists. The latter argue that the Liar Paradox, i.e., P: P is false,

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can be solved by accepting both truth-value gaps and truth-value gluts, whereas The Strengthened Liar Paradox, i.e., Q: Q is not true, can be solved only by accepting truth-value gluts. Then Gerogiorgakis focuses on the Milne Sentence, i.e., R: No omniscient being knows that which the sentence R expresses, which is paradoxical. Milne himself shows that the notion of omniscience can be made intelligible only in a dialetheist semantics. Gerogiorgakis disagrees with Milne’s argument, and claims that it can be refuted on the grounds of a paranormal semantics which accepts truth-value gaps together with truth-value gluts. However, he admits that Milne’s argument can be defendend vis-à-vis the Strengthened Milne Sentence, i.e., T: No omniscient being knows truly or indeterminately or half-indeterminately that which the sentence T expresses. Jason Megill (“The Knowability Principle and Fitch’s Paradox”) argues that the following three claims entail the existence of an omniscient being: (a) A fairly basic modal epistemic logic is sound; (b) The Knowability Principle (KP) is true, i.e. all truths are knowable; and (c) If all truths are known, then there is an omniscient being. The argument only tries to show that there is an omniscient being, and does not attempt to establish the existence of any particular God. Megill admits that claims (b) and (c) are controversial, but they are more plausible than some of the principles or claims used in other extant arguments for theism. Consequently, he estimates his argument is better than some other extant arguments for theism that are typically put forth. He relates this argument to Fitch’s Paradox, which demonstrates a contradiction resulting, on the basis of a certain logic, from KP and the Non-Omniscience Principle, i.e., the claim that some truths are unknown. He refutes the Non-Omniscience Principle. Elisa Paganini (“Vagueness and Omniscience”) grapples with the question: What is a good philosophical question to pose an omniscient being whose answer is relevant for a current philosophical debate? Assuming that a semantic theory of vagueness – like supervaluationism – is correct, she considers two proposals to characterize cooperation by an omniscient being. Then she explains why both the proposals are inadequate, in one case adding that the inadequacy depends on the fact that there is no exhaustive definition of cooperation by an omniscient being. From this Paganini infers that an omniscient being may be noncooperative if a semantic theory of vagueness is correct, which has some philosophically problematic consequences. Mirosław Szatkowski (“God’s Omniscience and Logical Virtue”) starts with the following definition: God is omniscient iff he has: (1) complete knowledge (God knows all truths (about God himself, creation, logic and mathematics, etc.)), (2)

XXII | Mirosław Szatkowski sound knowledge (everything that God knows is true), (3) necessary knowledge (God necessarily knows whatever he knows), (4) tenseless knowledge (God knows the past, present and future simultaneously), and (5) the knowledge of unrealized possibilities (God knows all things that could have come to pass but will not). Each of these determinants of God’s omniscience comes with its own set of problems. Szatkowski looks at the individual elements of the definition (1)–(4), postponing the discussion of the element (5) to another occasion, and indicates a way in which one can consistently speak of God’s omniscience. He confronts God’s omniscience with some meta-mathematical theorems and problems from philosophy of mathematics, introduces the issue of the unreality/reality of time, and makes the reader familiar with various types of temporal semantics. Finally, he proposes Andersonlike semantics with ‘temporal’ and ‘knowledge of truths’ dimensions. Going into detail, he claims that it is obvious that the saying that God knows all truths demands that there is a set of all truths, which is inconsistent with Cantor’s power set theorem: ‘for any set X, the cardinality of power set 2X of a set X is always strictly larger that the cardinality of X ′ . On the basis of this theorem,it can be proved that there is no set of all truths, and consequently, there exists no omniscient being. Since the core of this argument is the power set axiom: ‘for every set X there is a power set 2X whose elements are precisely the subsets of X ′ , a natural way to rescue the completeness of God’s knowledge is to adopt an alternative set theory in which such an axiom doesn’t appear. Szatkowski lists such set theories. The argument against the sound knowledge of God refers to the so-called divine Liar (*): God knows that (*) is false. This sentence and all other sentences similar to it have a self-referential nature, and the self-reference of sentences is in striking disagreement with the Tarski indefinability theorem, which says that truth in the standard model of the system cannot be defined within the system. There exist many ways of solving the Liar paradox. Tarski himself offered the hierarchy of not universal object-languages and not universal metalanguages. Szatkowski uses this solution in his paper. Going further, the statement that God knows the past, present and future simultaneously is the presupposition that there really is such thing as time. McTaggart – more than a century ago – argued that time is unreal, finding – up to the present day – both opponents and supporters of his argument. Without insisting that any of these opinions is true, Szatkowski characterises different semantical approaches to time. He marries the branching-time Ockhamist semantics with the partly free semantics for Anderson-like ontological proofs, which he defined in one of his earlier papers. After building a hierarchy of languages Ln , 1 ≤ n < ω, each of which contains the predicate ‘A knowledge of all truths from a given set of truths’, he defines a hierarchy of corresponding model structures Wn , 1 ≤ n < ω. In this way, he gives a fragmentary account of the concept of God’s omniscience.

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Instead of one concept of God’s omniscience, he offers infinitely many of them, arranged in a hierarchy.

3 All four papers of Part III focus on issues related to truth. Srećko Kovač (“Logic and Truth in Religious Belief”) asks the question: What is specific for religious belief with respect to the concepts of belief and knowledge as they are usually understood in epistemic (and doxastic) logic? Assuming that “i believes that ϕ” means that i holds that ϕ is true, he distinguishes the objective side of religious belief: the reference to truth – and this reference is a sort of knowledge, and the subjective side: the subjective trust, confidence, i.e., religious faith. This subjective side – specific for religious belief – is a reason why we do not identify religious belief with knowledge. He argues, on the basis of biblical texts, that logic and religious belief are interconnented in two ways: (i) religious belief includes reasoning, and (ii) religious belief is a pragmatic function applied to logical (syntactical and semantic) forms, and therefore is a part of logic in a wider sense. Analysing the episode of Nicodemus from the Gospel of John 3, Kovač shows how different stages of truth and its knowledge correspond to different stages of religious faith. He gives a formal description of the language of the Gospel passage about Nicodemus and describes a semantics for it. Anna Lemańska (“Absolute Truth and Mathematics”) analyses five mathematical theorems, and shows – contrary to post-modernity – that there exist absolute truths in mathematics. Here are the theorems: (1) The Continuum Hypothesis: The arbitrary infinite subset of the set of real numbers is equipollent with the set of natural numbers or with the set of real numbers; (2) The sum of angles in an arbitrary triangle is equal to the sum of two right angles; (3) 2 + 2 = 4; (4) For n > 2 the equation: x n + y n = z n has no solution within the set of integers greater than 0; and (5). Each even natural number bigger than two is a sum of two prime numbers. Lemańska finishes her article as follows: “These considerations about truth in mathematics can be transferred to a much wider area. So there are truths eternal and absolute. The guarantor of the existence of such truths is Absolute Being – God.” Alexander Pruss (“The Divine Belief Theory of Truth: Might It Work?”) evaluates the Divine Belief Theory of Truth (DBToT) in the context of many objections against it. Assuming that “God” can be a proper name or a definite description,

XXIV | Mirosław Szatkowski Pruss chooses the second approach and takes “God” to be shorthand for “the perfect being”. DBToT then reads: p is true iff the being that has all perfections (:= God) believes p. The two concepts: belief and perfection, present in this principle, are non-logical. In particular, if p stands for a sentence or a proposition, then DBToT says: A sentence s is true iff it expresses a belief of a perfect being and A proposition p is true if and only if it is expressed by a true sentence, respectively. Pruss examines six types of objections against DBToT; here (i)–(vi). (i) The Euthyphro objection. Since DBToT resembles the Divine Command Theory of Obligation, according to which something is obligatory provided that God commands it, thus the Euthyphro objection (EO) to the Divine Command Theory of Obligation is an objection to DBToT. According to EO, relativised to DBToT, two cases are possible: (1) God believes a proposition p because p is true, and (2) The proposition p is true because God believes it. But if (1) holds, then we have an explanatory circularity as what makes a proposition p true is God’s believing them. And if (2) holds, then it is difficult to see what reason God could have to believe p as he does; and if God believes p without reason, then he is irrational. Pruss accepts both (1) and (2), and he adds that we had better not say that God believes it because it is true. And as he says: “the proposition ‘God believes p’ is explained by the proposition p, rather than by the higher-order proposition ‘p is true’.” (ii) Knowledge. An almost complete theory of knowledge, according to Pruss, is expressed by the clause (*): For any agent x, that x knows p is grounded in four conditions: (i) that x believes p, (ii) that p is true, (iii) that x is justified with respect to p, and (iv) that the anti-Gettier condition holds. Problems arise when the clause (*) is to be reformulated with reference to God. From among several solutions considered, Pruss chooses the (**): That Y– knows p is grounded in the satisfaction of four conditions: (i) that Y– believes p, (ii) that the perfect being believes p, (iii) that Y– is justified with respect to p, and (iv) that the anti-Gettier condition holds; where “Y–” is a name of God. The reason to choose the (**) is that it is free from a collapse and applicable to ineffable truths, i.e., truths that are not expressible by any sentence. (iii) Epistemological circularity. Pruss writes: “It seems that I have reason to think that the perfect being believes snow is white because I have reason to believe that it’s true that snow is white, and I have reason to believe on the basis of the arguments for classical theism that there is a unique perfect being and that being believes all truths. But of course if truth just is belief by a perfect being, this reasoning becomes circular.” This circularity can be solved by differentiating between second- and first-order claims. (iv) Atheists and truth. Clearly, if DBToT would be true, then it would be metaphysically impossible for anything to be true if God didn’t exist. Thus, the existence of God would be a criterion of true. Pruss states that “... either theists are believing in a metaphysical impossibility when they believe that God exists, or atheists are believing in a metaphysical impossibility when they believe that

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God does not exist. ” (v) The Liar Paradox. The same reasoning as the standard Liar Paradox can be used to prove the sentence (†): Sentence (†) does not express a belief of the perfect being is paradoxal. But then, the DBToT fails. Pruss asks if there exists a response to the Liar that is consistent with the DBToT. Claiming that there is such a response, he says that meaning is not a function of sentence types but of token speech acts, and (†) does not manage to express a proposition. (vi) Similarity to occasionalism. Pruss writes: “Occasionalism brings God in where intuitively there are explanations on the level of creation, and thereby robs the created order of some of its reality. The heat of the burner doesn’t cause water to boil: God just happens to habitually will water to boil after it is heated.” Suspecting that DBToT could be similarly criticized, he refutes this objection. Peter Simons (“Makers and Models: Two Approaches to Truth, and their Merger”) attached the following abstract to his article: A semantic account of truth along the lines initiated by Alfred Tarski has a number of advantages that render it theoretically attractive. One is that it becomes possible to use the same methods and materials to define both truth and logical consequence. On the other hand the theory comes at a price. One element is that the ontological cost of deploying the tools required to define truth render it unacceptable to those of a non-platonist persuasion, something of which Tarski was only too painfully aware. Another element is that it is not clear how the Tarskian method of delimiting the true from the false connects with the intuitive notion of truth as answering in some way to the way things are in the world. It was for this reason among others that the modern theory of truth-making was initiated. Truth-making, whatever form the correct theory may take, for its part eschews the idea that truth-makers should be in general epistemically and semantically transparent to the user of the idiom whose truth-bearers are thereby rendered true. There thus appears to be a radical disconnect between the modeltheoretic approach to truth and the truth-making account, so that they might as well be about different things. This paper will attempt to bring the two approaches together, show how they interact and complement one another, and can be deployed together to provide a nominalistically acceptable account of truth and consequence.

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4 Part IV contains five studies in which the authors address a variety of metaphysical issues. These issues are substantially different from those discussed in the three preceding parts, but that does not mean that they are neutral towards them. Christopher Daly and David Liggins (“Agnosticism about Material Composition”) examine agnosticism about material composition, in a far wider context, with a view of evaluating the issue of agnosticism about every ontological issue. It is worth emphasizing that ontological issues can be evaluated in three respects: (Q1). Is the issue intelligible?, (Q2). Is the issue substantial?, and (Q3). Is the issue resolvable? The focus is on the principle of composition, which can be described as follows: Any answer to Van Inwagen’s ‘Special Composition Questions’: Under what conditions do some things compose a further thing? is referred to by Christopher Daly and David Liggins as a ‘Principle of Composition’. These principles include the ‘universalist’s principle’ that for any two non-identical things, there is something which they compose, and the ‘nihilist’s principle’ that two or more things never compose something. According to Rosen and Dorr, the most promising or employed sources of evidence for principles about composition should be: conceptual analysis, common sense and science. Questioning these sources, they, in consequence, infer agnosticism about material composition. Bennett adopts a similar strategy, even though she has changed the last two sources into: simplicity and philosophical argument. Daly and Liggins devote a separate section of their paper to each of these sources. In each of these sections, they first characterize the objections of Rosen and Dorr, and Bennett, to the claim that a particular source gives good reasons for principles about composition, and then they criticize these objections. In the last section (“Reflections and recriminations”), they attempt to provide answers to four questions: (1). Have agnostics about material composition presented examples of philosophical debates which they think have been resolved?; (2). Are agnostics about material composition just making the point that there are no knockdown arguments in philosophy?; (3). Is agnosticism supported by the fact that humans have cognitive limitations?; and (4). Why be agnostic? Christian Kanzian (“Existential Dependence and other Formal Relations”) structures his paper as follows: The first section is devoted to general considerations concerning formal relations. The next two sections are focused, respectively, on ontological dependence and existential dependence, as specific types of formal relations. The final two sections pertain to certain possible applications of the presented theory of formal relations to other ontological issues, and to philosophical

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theology, respectively. Going into detail, Kanzian quotes the Mulligan’s conception of internal (also called thin) relations: “... a relation is internal with respect to objects a, b, c etc., just if, given a, b, c etc., the relation must hold between and of these objects”. For Mulligan’s internal relations Lowe interchangeably uses the names ‘founded relations’ or ‘grounded relations’, regards them as “entirely determined by the relata”, and explicitly states that they offer “no additions to reality”. Kanzian says that such relations are neither entities nor elements of beings. According to him, the relations, for example: bigger/smaller, characterization and identity should fall, even though they don’t, under this initial definition. Both Lowe and Kanzian differentiate between formal relations and thin ones. In Kazian’s words: “I prefer calling all grounded relations ‘internal’, in agreement with Mulligan’s terminology. ‘Thin’ should be the label for all the derivative relations without any substantial importance for the understanding of the nature of their relata in their respective ontological functions.” Kanzian claims that ontological dependence is an internal relation determined by its relata, but of a special kind. Its defining mark is that it co-occurs with other formal relations. Existential dependence, in turn – as a special kind of ontological dependence – is a type of substantial and individual dependence where “Substantial dependence occurs if an x depends on a y under an aspect which is essential to x, respectively if it belongs to the nature of x to depend on y”, and – in relation to individual dependence – “things, like me, undeniably depend on some determinate mode of the mode-genus or -determinable weight. Here I, as an entity in space and time, am a generically dependent entity”. Using the concepts introduced above in the threecategorial ontology makes it – according to Kanzian – more plausible. Kanzian also finds it possible to apply these concepts in philosophical theology, and provides relevant examples. Duncan Pritchard (“Wittgenstein on Faith and Reason: The Influence of Newman”) presents two different ways of reading Wittgenstein’s On Certainty (OC). The first reading is inspired by Moore’s remarks on idealism and radical scepticism and therefore devoted to the solution of these problems. The second way of thinking about OC is inspired by John Henry Newman’s writings on faith and reason. According to Pritchard, it is beyond dispute that Wittgenstein was interested in Moorean certainties and the problems of idealism and radical scepticism. Yet, he opts for the reading of OC inspired by Newman’s views. As he very carefully points out: “I hope the foregoing suffices to motivate the thought that viewing OC via Newman’s views about the rationality of religious belief is at least a useful intellectual exercise.” In his opinion , this approach to OC leads to a peculiar quasi-fideistic conception of the epistemology of religious belief. Going into details, Moore chooses perfectly everyday certainties (which Pritchard calls Moore

XXVIII | Mirosław Szatkowski certainties) – for example, that one has two hands – which possess a kind of epistemic groundedness, i.e., more controversial and less certain claims can be extracted from them. Wittgenstein, in contrast, maintains that these Moorean certainties cannot be coherently thought of as rationally grounded, but that this optimal certainty belongs to their nature. According to him, not only are Moorean certainties necessarily groundless, but also rationally doubting them is simply impossible. He argues that both projects, sceptical and anti-sceptical, of offering a wholesale rational evaluation of our beliefs are incoherent. Wittgenstein’s critique of Moore is, according to Pritchard, a Newman-inspired critique. This is indicated by: (1) The way of overcoming the difficulty associated with “realizing the groundlessness of believing”, and (2) “...that Wittgenstein had a conception of the epistemology of religious belief which is very different to that usually attributed to him”, as pointed out by Pritchard. Scott A. Shalkowski (“Necessity, Worlds, and God”) discusses two versions of possible worlds theory: David Lewis’s Genuine Modal Realism and Brian Leftow’s theory, and compares them with each other. Going further into detail, Lewis believes in the truthfulness of the theory of a plurality of worlds, because it is serviceable. In his opinion, it is a theory of greater simplicity, greater theoretical unification, and both greater explanatory and expressive power than the alternatives. Leftow, on the other hand, introduces God, divine ideas, and divine action in place of Lewis’s worlds and sets of individuals. He does not allow for any possible worlds (they are a fiction) or for any self-existing abstract entities of any kind. Shalkowski conveys Leftow’s view as follows: “Everything else depends upon God, whether concrete or abstract, whether actual or merely possible, whether contingent or not. There are acts of creation and of sustaining physical things in existence. There is God participating in the making of all that there is – even in the products of human hands – so that God is rightfully thanked and praised for the good even that we do.” Shalkowski claims that Leftow’s project is ill-conceived, because it does not take into account modal facts which are undoubtedly metaphysically basic. He argues that if Lewis’s theory is treated at its meta-theoretical sense, standard reasons for rejecting theistic accounts are nullified. He also proposes ways of avoiding many philosophical problems, and his suggestions are different than Lewis’s or Leftow’s. Bartłomiej Skowron (“The Explanatory Power of Topology in the Philosophy of God”) writes: “The aim of this article is to examine the explanatory power and the limitations of topological concepts and arguments in philosophical reflection on God.” The author is convinced of the possibility to pursue the so-called mathematical theology – by which he understands the mathematical modeling of some

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of theological issues – and supports his conviction by the authorities of Józef M. Bocheński and Nicholas of Cusa. In topology, which is part of modern mathematics, properties of objects such as being open, closed, dense, nowhere dense, connected (many kinds), compact (many kinds), continuous, hollow, meagre, etc., are analysed. Skowron gives several examples of how certain theological issues were formulated and explained in the language of topology. And so, Hitoshi Ochiai used topological concepts to explain the following theological paradox: ‘how is it possible that God suffers if He has the infinite power?’ Coyne and Heller perceive God as a closure of the world, in other words, He is a complement of the world. Skowron supplements this approach: “God is the closure of the world but God is omnipresent and that is why the world is a part of God. ... God provides some kind of unity and bonding of the world.” Calude, Marcus and Stefanescu use the concepts of filter and cofinite topology for the description of God. According to their approach, God is an extrapolation of the set of all created beings, and connectedness of the space is interpreted as the unity of God. The understanding of God’s unity, as Skowron says, can be twofold: unity as a part of God and unity as the unicity, i.e., there are no other gods. Skowron analyses the first understanding of God’s unity and uses terms of topology to model this unity. Finally, he expresses a view that mathematics, and especially topology, may be helpful in resolving philosophical and theological problems, although there some limitations in this area.

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Part I: (God’s) Existence/Non-Existence

C. Anthony Anderson

Logical Necessity, Conceptual Necessity, and the Ontological Argument There is a fairly widespread belief that there is something appropriately called “logical necessity” and perhaps even that the notion is captured by current formalizations of modal logic. I will argue very briefly that this is a mistake, or at least a confusion. My main goal in that polemic is to clear the ground for a general notion, a generalization of increasingly inclusive notions of logical necessity, which I call “conceptual necessity”. More generally I will speak of “conceptual modality”. Based on this I give a partial formalization of what I take to be an improved version of the modal ontological argument, following the lead of Charles Hartshorne.¹ I give a distinctly metaphysical argument for one of its premises and very briefly sketch a strategy for defending another.

1 What is Logical Necessity (Logical Modality)? The notion of a tautology in the technical logical sense is as clear as one could hope. A formula is a tautology if, fixing the meanings² of the truth-functional connectives, it comes out true no matter the truth-values of its constituent atomic sentences or formulas. We could say, if we like, that every meaningful sentence which is an instantiation of a tautology formula is tautologically necessary or, better, that the proposition which it expresses is tautologically necessary. What this amounts to is that the proposition is true, and necessarily true, in virtue of the concepts expressed by the connectives. It worth noticing, following an observation due to Peter van Inwagen³, that the corresponding “logical possibility” is not a kind of possibility. The negation of a proposition may fail to be tautologically necessary without the proposition being really possible – in the absolute or metaphysical sense. Nevertheless, it is established philosophical usage to speak of such notions as epistemic possibility, even though something that is possible-for-all-we-know need not be absolutely possible. What we have in that case is that the possibility of

1 Hartshorne (1961). 2 Strictly speaking, the meanings of the connectives are not fixed, only their corresponding truthfunctions are fixed. 3 Van Inwagen (1988).

4 | C. Anthony Anderson the truth of the proposition is not ruled out by what we know. “Tautological possibility”, if we care to speak of such a thing, is the property a proposition has when its truth is not ruled out by the concepts of the truth-functions, i.e. its negation is not a tautology. First-order validity is also an extremely clear notion. Essentially, a formula is first-order valid if it would come out true no matter what non-empty domain is the range of the quantifiers and no matter what meanings are assigned to its predicates (with the exception, perhaps, of the identity predicate). Consider a meaningful sentence which is an instance of a first-order validity. The proposition it expresses is true, and indeed necessary, simply in virtue of the meanings of the connectives and quantifiers and, perhaps, exemplification,⁴ the latter being expressed by juxtaposition of predicate and term (or terms). Call such a proposition first-order necessary. Again, “first-order possibility” is not a kind of absolute possibility. A proposition is first-order possible if its possibility is not ruled out by the fixed concepts of first-order logic. It may still be quite impossible. Higher-order logic has been challenged by Quine as mathematics in disguise. I disagree, but will avoid the dispute here. Clearly there are necessary truths that are determined by other concepts than those codified by first-order logic – and they do not differ significantly in kind from these. A meaningful instance of “If most A’s are B’s and all B’s are C’s, then most A’s are C’s” will certainly express a necessary proposition. And there is no plausible reason not to count this as “logically” necessary. Of course there are necessities resulting from other generalized quantifiers.⁵ There seems to be no reason whatever to deny that (some of) these are “logically” and necessarily true. And respectable candidates for logically necessary propositions are codified by the logic of action, deontic logics, the logic of adverbs, and indeed logics not yet studied. The term “logic” has no fixed extension. Therefore, neither does “Necessary in virtue of logic”. John Burgess⁶ takes it that logical necessity is “truth in virtue of form”. On this basis he tries to justify the modal logic S5 as the appropriate logic for logical necessity in this sense. But it all depends on what you include in the form. Identity is a binary predicate and, hence, not part of the form of the sentence or proposition. Is it a mistake to treat this as part of logic? Or consider Alonzo Church’s formaliza-

4 If we think of the predicates as only fixed as to their extensions, perhaps as “denoting” sets, set theoretic membership is expressed by juxtaposition. This is the standard way of defining validity, there being a more or less agreed upon standard theory of sets. There is no such standard theory for properties. 5 It is a bit vague what is to count as a “logical” quantifier. Bell and Slomson (Bell and Slomson (1969)) adopt the view that a quantifier is logical if it is definable in set theory. 6 Burgess (1999).

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tion⁷ of universal statements. The usual ∀xA[x] is formalized instead as Π(λxA[x]), very roughly: “The function from individuals to truthvalues λxA[x] is universal, i.e. yields truth for every argument”. Here universality is treated as a (higher order) predicate. It is thus not part of the “form”. Surely this is a completely acceptable formalization of universality. Is this wrong or misleading because quantification thus treated isn’t part of the form? Other logics treat “seeing to it that”, “It ought to be that”, and so on, as part of the form, and typically they are taken to be operators or connectives. Still, even in those cases it is perfectly sensible to treat these notions as expressed by predicates of propositions (or other intensional entities). The notion of truth or necessity in virtue of form is worse than worthless. Logical constants are, among other things, those expressions whose meanings we decide to keep fixed in order to study their necessary connections to one another. Quine’s obiter dictum that first-order logic is all of logic is just conceptual legislation.⁸ It is clear that there is no fixed notion of logical necessity. But there is a guiding idea behind the various extensions of first-order logic that does deserve to be singled out. We might say that a proposition is conceptually necessary if it is necessary simply in virtue of the essential properties of, and relations between, the concepts it contains.

2 Conceptual Modality I have discussed this notion a bit elsewhere.⁹ I presuppose a kind of Platonism, or at least realism, about concepts. The early G. E. Moore, in “The Nature of Judgment”, expresses the view succinctly: A proposition is composed not of words, nor yet of thoughts, but of concepts. Concepts are possible objects of thought; but that is no definition of them. It merely states that they may come into relation with a thinker; and in order that they may do anything, they must already be something. It is indifferent to their nature whether anybody thinks them or not. They are incapable of change; and the relation into which they enter with the knowing subject implies no action or reaction. It is a unique relation which can begin or cease with a change in the

7 Church (1940). 8 Some authors include “truth in virtue of semantical rules” as part of the characterization of logical necessity, e.g. van Inwagen (Van Inwagen (1988)). I think this a mistake, but will not argue the matter here. 9 Anderson (2011).

6 | C. Anthony Anderson subject; but the concept is neither cause nor effect of such a change. The occurrence of the relation has, no doubt, its causes and effects, but these are to be found only in the subject.¹⁰

Following Alonzo Church’s terminology¹¹ I take concepts to include properties and propositions. There is nothing intrinsically mental about this notion.¹² Conceptual necessity is not always trivial and it is not stipulated. And there is no reason to suppose that grasp of the relevant concepts guarantees knowledge of the necessary connections between them and other concepts. Indeed, I think that a good case can be made that all the necessary truths of mathematics are conceptual necessities.¹³ It is far from obvious that they are all knowable a priori. The negation of a proposition might not be conceptually necessary and yet the proposition might be absolutely (metaphysically) impossible. So “conceptual possibility” is not a kind of absolute possibility: being conceptually possible does not entail being metaphysically or absolutely possible (but the converse does hold). Perhaps the term “conceptual coherence” is somewhat better. “Conceptual consistency” is a reasonably good alternative, except that it may connote ideas about explicit contradiction and provability, notions we do not want to evoke. With this proviso I will use these terms more or less interchangeably.

3 Terse Formalization of a Hartshornian Modal Ontological Argument Charles Hartshorne gave the first modern version of the modal ontological argument. Others¹⁴ have followed, but the present version draws its inspiration from

10 Our discussion will concern only “purely conceptual” propositions. If one includes “Russellian propositions”, some of them will be conceptually true as well. 11 In Church (1940) and Church (1951). 12 At least there is no reference to human mentality. Timothy Williamson observed that a concept that cannot be possibly be grasped seems like an impossibility. Agreed, but if we do not put human limits on the possible grasper, all is well. 13 Cardinal and ordinal numbers are properties (or relations-in-intension). Set theory needs interpreting in a theory of concepts. I discuss this briefly below. 14 Notably Kurt Gödel (Gödel (1970)) and Alvin Plantinga (Plantinga (1972)). Gödel’s version depends on a notion of “positive property” that does not seem (to me) to be robust enough to justify his premises. Plantinga’s proof is simpler, but it uses some technical notions which seem to leave it open to the charge of question begging.

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Hartshorne. We give just enough of a formalization to produce the desired proof.¹⁵ Take as primitive a singulary connective , expressing conceptual necessity. Because there are special problems with the idea of de re conceptual modality, we require that  be attached only to closed well-formed formulas, that is, to sentences. We do not give detailed formation rules. Define: df

⊗A = ¬A Conceptual Impossibility.

To simplify the statement of the proof a bit, we define conceptual coherence (consistency) and conceptual entailment (necessitation or implication). df

A = ⊗¬A Conceptual Consistency. df

A ⇒ B = (A → B) Conceptual Entailment.

We will use lambda abstraction expressions to denote properties.¹⁶ λxA[x] : the property asserted of x when it is said that A[x]. We adopt a predicate of properties, exemplification: Σ(Φ): The property Φ is exemplified. In order to avoid direct talk about possible entities¹⁷, we follow Plantinga¹⁸ in preferring this to its first-order counterpart. The underlying modal propositional logic is as follows. CM0. All tautologies. C1. A → A. C2. (A → B) → (A → B). C3. A →   A. These are the analogues of the standard formulation of the axioms of the (propositional) modal logic S5.¹⁹ We do not formulate a semantics, but it is reasonably

15 It may be worthwhile to give a complete formalization combining conceptual and metaphysical modalities. 16 This is now perhaps the most common interpretation of lambda expressions. Church’s original interpretation was entirely in terms of (extensional) functions. See Church (1940) and Church (1951). 17 In the end we would take possible things to be concepts of certain sorts. 18 Plantinga (1972). 19 The use of conceptual modality avoids the known objections to S5. See Anderson (2011).

8 | C. Anthony Anderson clear that we could produce one using the intuitive idea of a “conceptually coherent world”. The absolutely possible worlds would be a subset of these. On this basis, C1 is easily seen to be valid if one observes that every metaphysically possible world is a “conceptually coherent” world. Then something true in every conceptually coherent world is true in every metaphysically possible world and, hence, is true in the actual world. Informally, and roughly, C3 says that if a proposition is conceptually consistent, then the fact that this is so is due to essential relations (e.g., compatibility) among concepts. Conceptual necessitation of theorems and, of course, Modus Ponens are intuitively valid.²⁰ To save space and patience, we simply postulate rules of inference corresponding to what are certainly valid modal quantificational rules of inference. The formal details of the appropriate quantification theory are omitted, but can easily be supplied by the interested and technically inclined reader. (E1)

From ⊗Σ(λx(A[x] ∧ ¬B[x])) and ⊗Σ(λx(B[x] ∧ ¬C[x])) to infer ⊗Σ(λx(A[x] ∧ ¬C[x])) “If it is inconsistent to suppose the property of property of being an A and not a B is instantiated and it is inconsistent to suppose that the property of being a B and not a C is instantiated, then it is inconsistent to suppose that the property of being an A but not a C is instantiated.”

(E2) From ⊗Σ(λx(A[x] ∧ ¬B[x])) to infer Σ(λx(A[x]) ⇒ Σ(λx(A[x] ∧ B[x]))) “If it is inconsistent to suppose that the property of being an A but not a B is instantiated, then the conceptual necessity of the instantiation of the property of being an A entails the conceptual necessity of the instantiation of the property of being an A and a B.” A little reflection will convince the reader that these are intuitively valid rules of inference. We adopt the following interpreted predicates: Prfct(x): x is perfect. Omnp(x): x is omnipotent. Omnis(x): x is omniscient. Allgd(x): x is all-good (perfectly good). E!(x): x exists.

20 Of course we have not given a formal semantics so that validity is not precisely defined.

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I prefer to use a non-logical predicate to express (actual) existence. E!(x) means “x has spatial-temporal location or is capable of acting or being acted upon”. I do not pause to justify my preference for this terminology. The dissenter who insists that “there is” and “there exists” just have to mean the same thing may just regard my reading of E! as “existence” is eccentric. The proof goes through, and is indeed simpler, if one prefers to use Σ to express instantiation by an actual existent rather than instantiation by something or other. The four premises of this version of the modal ontological argument are: (1) Σ(λxPrfct(x)). “It is conceptually coherent to suppose that perfection is instantiated.” (2)

Σ(λxPrfct(x)) ⇒ Σ(λxPrfct(x)). “That perfection is instantiated (conceptually) necessitates that perfection is (conceptually) necessarily instantiated.”

(3) ⊗Σ(λxPrfct(x) ∧ ¬(Omnp(x) ∧ Omnis(x) ∧ Allgd(x))). “It is conceptually inconsistent to suppose that the property of being perfect, but not omnipotent, omniscient, and all good is instantiated.” (4) ⊗Σ(λ(Omnp(x) ∧ Omnis(x) ∧ Allgd(x)) ∧ ¬E!(x)). “It is inconsistent to suppose that the property of being omnipotent, omniscient and all good, but non-existent is instantiated.” The argument for the actual existence of a perfect being is then as follows: (5) ⊗Σ(λxPrfct(x) ∧ ¬E(x)) (6) Σ(λxPrfct(x)) → Σ(λx(Prfct(x) ∧ E!(x))) (7) Σ(λxPrfct(x)) →   Σ(λxPrfct(x)) (8)   Σ(λxPrfct(x)) → Σ(λxPrfct(x))

(9) Σ(λxPrfct(x) ∧ E!(x)) (10)

Σ(λxPrfct(x) ∧ E!(x))

(3), (4), E1 (5), E2 (2), modal logic T S5 (7), (8), Prop. calc. (9), modal logic T

10 | C. Anthony Anderson This conclusion, (10), says that the property of being perfect and actually existing is instantiated, i.e. there actually exists a perfect being.²¹ Instead of inferring existence directly from perfection, we have essentially assumed that perfection entails omnipotence, omniscience, and all goodness, and postulated that the latter three together entail existence. There are two reasons for this complication. Perfect-being theology will need the assumption that being perfect entails having these three perfections. So we will eventually have to postulate it anyway. And it seems slightly more obvious that being omnipotent, omniscient, and all good, taken together, entail actually existing than that perfection entails actual existence (in our perhaps eccentric sense). Being omnipotent by itself certainly entails being capable of acting. And being omniscient and all good entail having a mind and perhaps being a person. Since the claim that all three perfections together entail actual existence is epistemically weaker than that perfection or, say, omnipotence alone entails existence, we adopt the weaker premise. Admittedly premise (2) is stronger than the corresponding thing with metaphysical necessity (as in Hartshorne’s argument). Still, it seems reasonable that a perfect being should be necessary in the strongest sense.²² The properties involved in perfection, if jointly instantiated, should guarantee that instantiation. However, there is an interesting and very metaphysical argument for premise (2).

4 A “Fitchy” Argument for Premise (2) There is a somewhat exotic argument for premise (2) that may have some merit. It utilizes an idea due to Frederic Fitch and was used by him to formulate an even more exotic modal ontological argument.²³ Suppose that perfection is instantiated. Then a truly perfect being will have to have perfect perfection. Of course the standards of perfection for properties are not the same as those for individuals. Perfection, qua property, if it is perfect, must be not only instantiated, but it must be instantiated in virtue of its essential nature. And this will be conceptually necessary. Of course, if perfection could fail to

21 Thus we rebut Yablo’s (Yablo (1999)) claim that the ontological arguments requires the absolute possibility of the Perfect Being as premise. In this version, from the conceptual possibility we infer the actual existence of such a being. 22 If we accept David Lewis’s claim (Lewis (1970)) that the Ontological Arguer is entitled to whatever standards of perfection that he wishes to adopt, then the matter is settled here also. 23 Fitch (1963).

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be instantiated, then it is not perfect qua property. So we adopt the boldly metaphysical premises: (P1)

Σ(λxPrfct(x)) ⇒ PRFCT(λxPrfct(x)). “If perfection is instantiated, then it is perfect qua property.”

(P2)

PRFCT(λxPrfct(x)) ⇒ PRFCT(λxPrfct(x)). “That the property of being perfect is itself perfect entails that this fact is a conceptually necessity.”

(P3) PRFCT(λxPrfct(x)) ⇒ Σ(λxPrfct(x)). “That the property of being perfect is itself perfect entails that it is instantiated.” (P1) and (P3) claim the necessary equivalence of the instantiation of perfection and its perfection. Upon reflection this seems reasonable. A simple modal argument yields our original premise (2): (2)

Σ(λxPrfct(x)) ⇒ Σ(λxPrfct(x)).

5 The Crucial Premise The real crux of the matter, as the philosophically perceptive will have discerned, is the justification of the conceptual consistency premise (1). As explained, conceptual consistency or coherence is not always a trivial matter. Just imagining a situation, “conceiving” of it, is little evidence that a proposition supposedly true in it is coherent. It is very unlikely that there are any successful “stand alone” arguments in philosophy. And the offered modal ontological argument is certainly not a proof. In particular, as noted, premise (1) is not self-evident and there is little reason to suppose that it can be proved from self-evident propositions.²⁴ Still, it may be possible to offer some evidence which confirms (1).²⁵

24 Given our notion of conceptual possibility, we would not agree with Yablo that the ontological arguer needs metaphysical possibility in premise (1). But then “conceiving” does not justify premise (1) in this sense. 25 Here we mean that it seems to be possible to confirm the coherence premise without first confirming the existence of a perfect being pace van Inwagen (Van Inwagen (2011)). The offered

12 | C. Anthony Anderson The foundational theory of mathematics, Set Theory, is not really intelligible on its face. The obscurities are, in brief, the ideas of a null set and of singleton sets. It can be made sense of by means of a theory of concepts. These concepts are infinitary and not graspable by a finite mind. The assumption of the possibility, in the sense of coherence, of an infinite mind is all that is needed. These are all bold claims and, given their radical nature, must be substantiated in excruciating detail. I hope to do so elsewhere. The general argument will be this: the assumption of the possibility of an infinite mind yields a scheme for making sense of mathematics. This may well be the best available hypothesis for doing so. The consistency of the assumption that there is such a mind follows from the consistency of the assumption that there is a perfect being. To a certain extent the latter hypothesis is confirmed by its application for this philosophical purpose. Now it is true that, given the validity of the Ontological Argument, the possibility of a perfect being and its existence, indeed its necessary existence, are necessarily equivalent. And so standard probability theory would give them the same probability on any evidence. Nevertheless, it seems that one could confirm the hypothesis of a perfect being in the way described without already coming to the conclusion that such a being must exist.²⁶ The stated consequence of probability or confirmation theory is clearly an idealization that ignores our differential knowledge of necessarily equivalent propositions. One might further confirm the hypothesis (of possibility) by consideration of its consequences for value theory and/or ethics. It might even be that rational choice theory is based on the possibility of an ideally rational being. Such confirmation would of course need to be argued in detail.

Bibliography Anderson, C. A. (2011), “Conceptual Modality and the Ontological Argument”, in Szatkowski (2011), 295–307. Bell, J. L., and Slomson, A. B. (1969), Models and Ultraproducts, North Holland Publishing Co. Burgess, J. (1999), “Which Modal Logic is the Right One?”, in Notre Dame Journal of Formal Logic: 40, 81–93. Church, A. (1940), “A Formulation of the Simple Theory of Types”, in Journal of Symbolic Logic: 5, 56–68.

argument shows that the existence of a perfect being and the assumption of the consistency of the existence of a perfect being are equivalent. 26 Thus we challenge van Inwagen’s claim (Van Inwagen (2011)) that one cannot give evidence for the possibility that is not evidence for the proposition.

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Church, A. (1951), “A Formulation of the Logic of Sense and Denotation”, in Henle (et. al.), 3–24. Feferman, S., et. al. (eds.) (1995), Collected Papers of Kurt Gödel, vol. III, New York and Oxford: Oxford University Press. Fitch, F. (1963), “The Perfection of Perfection”, in Monist: 47, 466–471. Gödel, K. (1970), “Ontological Proof”, in [Feferman, et. al., 1995], 403–404. Hartshorne, Ch. (1961), “The Logic of the Ontological Argument”, in Journal of Philosophy: 58, 471–473. Henle, P., et. al. (eds.) (1951), Structure, Method and Meaning. Essays in Honor of Henry M. Sheffer, New York: The Liberal Arts Press. Lewis, D. K. (1970), “Anselm and Actuality”, in Nous: 4, 175–188. Moore, G. E. (1899), “The Nature of Judgment”, in Mind: 8, 173–193. Plantinga, A. (1972), The Nature of Necessity, Oxford: University Press. Szatkowski, M. (ed.) (2011), Ontological Proofs Today, Frankfurt/Paris/Lancaster/New Brunswick: Ontos Verlag. Van Inwagen, P. (1988), “Modal Epistemology”, in Philosophical Studies: 92, 67–84. Van Inwagen, P. (2011), “Three Versions of the Ontological Argument”, in Szatkowski (2011), 143–162. Yablo, S. (1999), “Concepts and Consciousness”, in Philosophy and Phenomenological Research: 59, 455–463.

Seweryn Blandzi

Problem of the Origins of Ontotheology When in 1957 Martin Heidegger published his essay entitled Die onto-theo-logische Verfassung der Metaphysik (see Heidegger (2002b)), the conviction has been established that metaphysics, beginning with Aristotle, is basically an “onto-theology”, i.e., a knowledge that culminates in the distinctive and specified form of Being (summum ens), i.e., the Divine Being. More specifically, metaphysics, according to Heidegger, is ambiguous in its very structure: “When metaphysics thinks of beings with respect to the ground that is common to all beings as such, then it is logic as onto-logic. When metaphysics thinks of beings as such as a whole, that is, with respect to the highest being which accounts for every thing, then it is logic as theo-logic” (Heidegger (2002a), 70–71). In an Introduction (Heidegger (1956)) that in 1949 was added to his lecture What Is Metaphysics (held in 1929 Heidegger (1949a)), Heidegger cites the books Gamma, Epsilon and Kappa of Aristotle’s Metaphysics (Aristotle (1933)). He notes that when metaphysics presents Being as Being, it does so in a twofold manner: “in the sense of the highest and thereby divine be-ing ( υ, [the universal, what is the furthermost, divinity]). The emergence of be-ing was developed in its twofold sense especially in the metaphysics of Aristotle (cf. Aristotle (1933): Γ, E, K). Because it makes being as be-ing an idea, metaphysics in itself is in fact two-in-one: the truth of be-ing in the most general sense and in the highest sense. In its essence it is ontology, in the narrower [scholastic] sense, and theology. This onto-theological essence of authentic philosophy ( ) must indeed be accounted for by the way it brings , that is, as , out into the open” (Heidegger (1956)). What is of paramount importance for Heidegger, however, is not this ambiguity of metaphysics as such, but rather the fact that it is dominated by the theological component that utterly clouds the possibility of thinking the Being (Sein) itself. Being (Seiendes) is linked with God as the Highest Being that “grounds it as the first cause (ratio)”. Being (Seiendes) participates in the Highest Being as the first (summum ens), and this relation is explained in metaphysics by reference to the conception of analogy or participation. Paul Natorp was the first to have discovered this ambivalence in Aristotle. Yet at the same time, the scholar sought to free the Stagirite from it. He did that by pointing to the inauthenticity of the Kappa book and by introducing an important correction to chapter Epsilon 1 that is shown to be of paramount importance for our understanding of “the first philosophy”. The scholar does not regard the chapter as inauthentic, but rather damaged by interpolations and, therefore, misinterpreted. Indeed, one should not subsume under the notion of “the first philos-

16 | Seweryn Blandzi ophy” the two components: the ontological and the theological as well as the science of the Being as such (die allgemeine, für alle grundlegende [...] Wissenschaft) and the particular and supreme Being (vom stofflosen, unwandelbarer Sein, als der vornehmsten Gattung des Seienden) (Natorp (1888a), 49). As Natorp notices, this would have clearly been a contradiction that could not be maintained (unleidlicher Widerspruch). In order for the contradiction to be resolved, it would suffice – according to him – to assume the concept of primacy in various senses: 1) in the sense of universal scope and 2) the highest dignity (value). When the two are combined, a contradiction arises. Heidegger’s charge of the structure of metaphysics as being “two-headed”, i.e., onto-theo-logic can, thereby, be recognized as valid with respect to the medieval metaphysics and not with respect to Aristotle’s first philosophy. In the present paper, we shall bring forth the crucial role of the emergence of the issue of ontotheology in the specific application of Aristotle’s formula of to with reference to God of the Old Testament by Philo of Alexandria. This seems to have been unnoticed so far.

1 Towards the Theologization of Being The teachings of the Old Testament, regarded as revelation, have become the first and most important of the non-Greek sources that have affected the reborn Platonism of the 1st and 2nd century (Plutarch, Numenius). The teachings have been delivered to the Greek thinkers via the interpretation put forward by Philo of Alexandria. Born at the end of the old era and influenced by the Judaic as well as the Hellenic culture, the thinker left a rich production that shows the dominance of the religious aspect and at the same time remains imbued with Greek philosophy. Philo found the Greek language even in the Holy Books, as he used the 3rd century B. C. translation of the Bible, known as the Septuaginta. When reading the Bible, Philo interpreted it allegorically chiefly in the light of Plato’s philosophy; it is highly probable that Philo became acquainted the works of the Platonists that lived in the second half of the 1st century B. C., such as Derkyllides and Eudoros of Alexandria. It was this philosophy that greatly shaped his exegetical views on the nature of God and the spiritual world in particular. Philo was also familiar with the exoteric and the akroamatic production of Aristotle, from who he draws what – at least in his opinion – is consistent with Platonism. Apart from Plato’s philosophy combined with Pythagoreanism, certain influence has been exerted on Philo by the Stoic philosophy. This, however, he sought to differentiate from Platonism in a similar manner that he tried to refine Plato’s

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philosophy from all skeptic contaminations so as to interpret some of its elements in accord with the monotheistic theology. Plato’s absolutization of Ideas as Demiurge-independent algorithms was not particularly appealing to Philo. He accepted though the understanding of ideas as paradigms attributed to God as his thoughts, placing, thereby, God above the ideas. The ideas function here as incorporeal archetypes, i.e., exemplary causes of corporeal things. The most essential element ( ) of their being, namely the archetypal patterns of all qualities in what exists, and on which the form and dimensions of each separate thing was modeled (See De specialibus legibus I, 327, 5–328, I: [...] , , ).

Without them, things would be merely “an amorphous matter” (Ibid., 328, 4: “ ”). For when out of that confused matter God produced all things, He did not do so with His own handiwork, since His nature, happy and blessed as it was, forbade that He should touch the limitless chaotic matter. Instead He made full use of the incorporeal potencies well denoted by their name of Forms to enable each kind to take its appropriate shape. (See Ibid., I, 329; , 1–5: sc. – – ˜ ).

Thus, incorporeal God proves to be transcendent to the world. For not even the whole world would be a place fit for God to make His abode, since God is His own place, and He is filled by Himself, and sufficient for Himself, filling and containing all other things in their destitution and barrenness and emptiness, but Himself contained by nothing else, seeing that He is Himself One and the Whole. (See Legum allegoriae, I, 44, 1–45, 1: ˜ ˜ ˜ ,

).

The divine transcendence contains also the intelligible world that was created by him. Thus, in Philo’s view the ideas are not immortal and unbegotten, as they are in Plato, but rather created by God’s thought. An important hint concerning the nature of the Highest Being is to be found in the second book of the Legum allegoriae (86.9). Several issues are touched upon here. One of them concerns the universal genus that is expressed by the indefinite pronoun (aliquid), which Philo elevates to the rank of a transcendentale: ‘ ’,

18 | Seweryn Blandzi . God, on the other hand, occupies to position above the genus that is expressed by the superlative , which in translations is all too superficially rendered as the “most general”; the second one is the divine Logos: . All other things exist owing to the Logos, which means that they in fact occupy a position close to nothingness: ’ . In the third book of his treaty, Philo does not hesitate to use the term with reference to the transcendental (aliquid): ‘ ’, . In the following sentence the term is also given to the divine Logos: . It is worth noting that the idea of a double above-genus, a logical and an ontological one, is clearly present in Porphyry’s Isagoge. The term , is used here quite often and becomes a technical term. The word, as a superlativus of the adjective , meaning not only “generic”, but also “ancestral”, “original”, signifies in one aspect the highest category that is named (nominalized form of the pronoun ) and in the other it is the very first Being that is most generic. Generally speaking, it is a border concept that express the impossibility of exceeding, i.e., that, on the one hand, there cannot be any other genus above it ( : 1.4.16), and, on the other hand, it points to the ultimate instance that is individual in nature, and that in human genealogy functions as an ancestor, whereas in the universal aspect it refers to the Deity as the ultimate principle ( : 1.5.17). Philo strongly rejects the idea of an anthropomorphic God. He writes in the On the immutability of God (55): now the companions of the soul [...] do not compare the living God ( ) to any species of created beings ( ); but, dissociating it with any idea of distinctive qualities ( ), [...] they, I say, are content with the bare conception of his existence ( ), and do not attempt to invest him with any form. (56) But those who enter into agreements and alliances with the body, being unable to throw off the robes of the flesh, and to behold that nature, which alone of all natures has no need of anything, but is sufficient for itself, and simple, and unalloyed, and incapable of , being compared with anything else ( ), from the same notions of the cause of all things ( ) that they do of themselves. (61) Those, therefore, who have received a fortunate disposition, and an education in all respects blameless, finding the path of life which proceeds in this direction plain and straight, take truth with them as the companion of their journey; by which they are initiated in the true mysteries relating to the living God, and therefore they never attribute any of the properties of created beings to him. [...] (62) But he is not even comprehensible by the intellect ( ), except merely as to his essence ( );

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for his existence ( ), indeed, is a fact which we do comprehend concerning him, but beyond the fact of his existence, we can understand nothing.

According to Philo, it is easier to prove the existence of God “that He is” ( ) than to grasp His nature, “what He is” ( ). Even the name that is ordered by God to Moses to characterize Himself: expresses rather the impossibility of any characteristic. (“I Am who I Am” (Ex 3: 14). [Notice that Philo uses to nominate God the participles and (= ) synonymously, thus the formula should be understood: “I am the truly being” or “I am this true Being”.]) Man should not strive to know His essence, but merely to reasonably confirm in the absoluteness of His existence. The only thing about the divine ( ) that can be comprehended is , “that it is, which is called existence”: (De praemiis et poenis, 40), showing, thereby, not – “what it is”, but merely “that it is”. For his essence is better than the Good, older than the Monad, purer than the One, and cannot be grasped by anything but Himself, because no one else is worth Him: ,

,

praemiis et poenis, 39ff.)

,

. (De

Cf. also De vita contemplativa, 2.8–3.1, where Philo says about those who truly worship God they have been instructed by nature and the sacred laws to serve the living God, who is superior to the good, and more simple than the one, and more ancient than the unit ( .).

2 “Craftsman” or “Creator”? The world that we know results from a divine act. What He created had not existed before: for as he produced that most perfect work, the world, bringing it out of non-being into existence: (De vita Mosis II 267, 2-3).

20 | Seweryn Blandzi [...] who created the whole universe out of things that had no previous existence: [...] (Legum allegoriae III 10, 7).

On the basis of these utterances one cannot, however, ascribe to Philo the idea of creatio ex nihilo. The Greek expresses a relative non-Being: “from something that previously was not what He created”. When Philo uses the term in the De somniis to attribute to God the role superior to that of the Demiurge, this does not entail a creation out of nothing. The Demiurge is likened to the Sun that does not create, but merely reveals with its light things that already exist, but cannot be seen in the darkness. Contrasting this with the function of a “creator” ( ) shows that God is much more than that: And besides all this, as the sun, when he arises, discovers hidden things, so also does God, who created all things, not only bring them all to light, but he has even created what before had no existence, not being their only maker, but also their founder. (De somniis I, 76, 3–77,1:

.)

The use of the negation and not in the expression (“what previously was not”) could be interpreted in the direction of a creatio ex nihilo, but the preponderance of Philo’s utterances suggests the eternity of the matter. Thus, one must agree with the opinion that although “it is sometimes maintained that already Philo spoke about the creation out of nothing, the impression has rather been created by the Christian Alexandrian school which used a philosophical apparatus that was very similar to or even identical with Philo’s terminology” (Domański (1989/1990), 34n). J. Domański reminds us, then, that the idea of a creatio ex nihilo finds its origin in the Vulgate translation of the Old Testament: ex nihilo fecit illa Deus from the Greek fragment of 2 Maccabees of (7, 28): . We should note, however, that on a closer look both formulas (the Greek and the Latin one), which are commonly and uncritically accepted as identical, in fact, do not overlap semantically. The Latin version is more radical, as it speaks of a creation of the world out of nothing (resp. nothingness). The Greek version, on the other hand, is surely closer to the original, but it has a somewhat different sense: “not out of Beings”, i.e., out of ready (actualized things) or elements (for this would entail some sort of reformulation of the already existing world), but out of devoid of qualities and amorphous (not actualized) matter, that is not any Being (for Being is something that is a formed one, sc. something definite), albeit it is not nothingness, either.

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It is clear, then, such a reading is determined by placing at the beginning the negation “not” that is immediately separated by the preposition “out”, which does not allow to connect it directly to the “Being” that would automatically yield its absolute opposition “Non-Being”. Thus, in the Greek version we have the following sequence: “not out of Beings did God make those things”, where the negation is evidently related to the predicate, and not to the Beings! In the Latin version it is the other way round: it is suggested here (and this is a misuse) that God created the world out of nothing (absolute Non-Being), i.e., nothingness. This is also the general understanding of the creatio ex nihilo formula. This change can be explained by the fact the later Christian philosophy tried to make God entirely independent of the matter, rendering, thereby, the latter quite superfluous. The nothingness, on the other hand, is not some metaphysical fore-substrate that would exist as something even more abstract than the unformed matter. The crucial novum is here that the emphasis has been put on the absolute freedom or indeterminacy of God in the act of creation. This “made out of nothing” means: God created the world not out of matter, not out of nothingness as some preexisting substrate, but caused it to emerge by the strength of the divine fiat!, thus, by the infinite strength of his absolute free will that absolutely does not require anything else for this purpose.

3 The Unnamed, and yet Named “the Being One”: the Premises of Ontotheology In Philo’s works, one could show many places where the participle (gen. ) denotes simply God in a closer or further context, and the philosopher uses for that purpose the plain form . God himself uses the term, as is testified by the Greek translation of the Septuagint: (“I Am who I Am”). When Moses points to the difficulty of how to respond to those who ask about the name of the one that sent him ( – De vita Mosis, I, 74, 3), God replies: At first say unto them, I am that I am, that when they have learnt that there is a difference between him that is and him that is not, they may be further taught that there is no name whatever that can properly be assigned to me, who I am the only being to whom existence belongs. (De vita Mosis, I, 75, 1: ,

.)

22 | Seweryn Blandzi Philo explains: Since God alone exists in essence, on account of which fact, he speaks of necessity about himself, saying, I am that I Am, as if those who were with him did not exist according to essence, but only appeared to exist in opinion. (Quod deterius potiori insidiari soleat, 160, . 7–9: « , », .)

When asking the question is there a name for that Being, Moses was perfectly aware that even the name Lord is not at all worthy of him (De somniis, I 230, 3–4: , ).

No specificity applies to Him. Philo elucidates: for the living God is not of a nature to be described, but only to be (De somniis, I 230, 5–231, 1: , ).

God says: I am that I am, that the questioner might know the existence of those things which it was not possible for man to conceive notbeing connected with God. (De somniis, I 231, 1–3: , [...] « », .)

Here are other places in Philo where the naming appears. He speaks of the auton, omy of “He who exists himself by himself alone”: (Quod deus sit immutabilis 110, 2) and full self-sufficiency: (ibid. 181), the goodness of the Being One: (Legum allegoriae III 105, 7) or his grace: (ibid. 214, 2), and precepts: (De migratione Abrahami 195, 10 – 196, 1), and trustfulness, in connection with the man’s attitude to Him: (Legum allegoriae III 173, 1), reverence (“the fear of God”): « » (ibid., I 99, 7), His contemplation: (De migra, tione Abrahami 170, 3), preparing His “tent”: (Quod deterius potiori insidiari soleat 160, 5). We read about the “eye [of the Providence] of the Being”: (De cherubim 97,1), about his “reign”: (ibid., 108, 1), “powers”: (De migratione Abrahami 40, 5–41, 1; Quod Deus sit immutabilis 109, 2; Quod deterius potiori insidiari soleat 159, 3). The Revelation about “the Being” is true: (Quod

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deus sit immutabilis 61, 5–6), what we know about the Being wonderful in comparison to other gods: (De ebrietate 43, 2–3), for the knowledge of the living God having beamed upon it, out-dazzles everything else: (ibid., 44, 5–6), albeit His concept is mysterious and obscure: (De posteritate Caini 14), and ascribing to Him such states as anger, fear, sorrow and pleasure must be seen as metaphorical: (ibidem, 71, 2). How could one gain a , sharp vision of “the One that is” (De mutatione nominum 82, 2–3), and is there someone who could comprehend the final stage of the soul’s journey to Him? (De sacrificiis Abelis et Caini 10, 2–3). The above examples use the participium masculini , showing, thereby, the personal character of God. In other places, Philo employs the abstract neutrum form which expresses the general sphere of the divine transcendence. Such an understanding is expounded further in the treaty On the immutability of God, whose Greek title already points to the auto-referentiality of the divine. Instead of the personal description “God”, Philo prefers to call it “Being”, speaking, for example, about Abraham’s experiencing the unshakable stability of the Being: (4.10). This understanding of immutability by Philo is clearly reminiscent of Parmenides’ Being in Plato’s Sophist (Plato (1971): 248e – 249a): But for heaven’s sake, shall we let ourselves easily be persuaded that motion and life and soul and mind ( ) are really not present to absolute being ( ), that it neither lives nor thinks ( ), but awful and holy ( ), devoid of mind, is fixed ( ) and immovable ( ).

This passage from the Sophist seems to be echoed in the neoplatonic identification of the Being-Idea sphere with the Nous which Philo also anticipates, although he situates God at the eidetic level, lowering, thereby, his position in relation to the trans-transcendent idea of the Good-One that transcends the Being ( ). The immutability ( ) which in Philo’s account is an immanent feature of God-Being as a thinking-nature. In contrast to man, the creator of all-things, possesses constantly the unshakable and a priori powers: reflection (consideration) and decision, controlling, thus, his works: , (34. 1–5).

24 | Seweryn Blandzi

4 God as Esse Absolutum Of special importance for us is the case of application in the De mutatione nominum 27, 1-5, the Aristotelian formula directly to the Absolute. By using that Philo wants to emphasize the self-referentiality of this Being, excluding its any relation to anything else ( ) – the original understanding of Aristotle’s formula is discussed extensively by J. Bigaj in Bigaj (2005). The idea occurs in the context, when Philo argues that in the famous phrase from the Old Testament “I am thy God” the final pronomen possessivum (“thy”) can only be understood metaphorically, since God as an autorelative has no relations to anything, but rather is a Being in itself: does not consist in relation to anything; for he himself is full of himself, and he is sufficient for himself (De mutatione nominum 27, 4–5: ’ ‘ ’ (Gen. 17,1) , ).

and further: for he himself is full of himself, and he is sufficient for himself, and he existed before the creation of the world, and equally after the creation of the universe (De mutatione nominum 27, 4–5: ).

Philo’s argument gains full conclusiveness only when the enthymematic premise reducing the concept of Being to God is revealed. In the next fragment, he introduces yet another synonym, this time of Platonic origin: (“that what is really Being”) and enriched by the Pythagorean-Platonic concept of One-UnityUniqueness [...] [...] (Quod Deus sit immutabilis, 11.4 – 12.1). The syntactically modified phrase that has been so far unnoticed by Philo’s commentators, it is his hapax legomenon. The Alexandrian gives a different, more radical meaning to the expression that it had in the books Gamma and Epsilon of Aristotle’s Metaphysics. It is closer to the inauthentic book Kappa. In this book, the formula refers to an unspecific, although clearly monotheistic deity. It is conceivable that Philo was inspired by the book Kappa, although he introduced the personal God. The Metaphysics might have been known in the Alexandrian circles as Aristotle’s work, but its book Kappa has been show by modern research (Natorp, Aubenque) to definitely be inauthentic – the inauthenticity of the Kappa book has been convincingly demonstrated by P. Natorp in his paper (1888b), 178–193. Some one hundred years later, P. Aubenque summarizes the re-

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sults of the research on this book, stating definitively: “K is later than B E and its purpose is summary. [...] Aristotle is not the author of the summary” (Aubenque (1983), 343). Thus, the theologizing understanding of the formula cannot be attributed to Aristotle. The authentic understanding is to be found in the books Gamma and Epsilon, if – as has been suggested by Natorp – the interpolated sentences are removed from the latter and in the others the correct philological understanding is maintained. From the book Epsilon it does not follow that the (= ) should “dethrone” as the only legitimate subject of the first philosophy – one should note the striking similarity of the following sentences: E 1, 1026a, 20– 22: and K7, 1064a 36 – b 1: , . The major problem of the Epsilon book is the problem of the primacy of philosophy understood as the very first from the abilities (explaining the beings/things/facts distributively), and not some distinctive object that would determine its primacy. Its primacy is not due to its being some “supraphilosophy”, but rather due to the universality of its method, i.e., due to the fact that it can (albeit does not have to) deal with the most dignified Being. The concept and object of philosophy understood as a general ability to explain all things without being limited to one subject discipline is constituted in the opening sentences of the Metaphysics’ book with the use of the expression : There is a science which studies Being ( ) qua Being ( ), and the properties inherent in it in virtue of its own nature. This science is not the same as any of the so-called particular sciences. (Aristotle (1933), 1, 1003a, 21–23: , , . .)

The formula from the book receives the theological sense of naming “The Being that fully is” of transcendent character in a paraphrase of the books E which belongs to the book K, an apocryphal work that originated in II/I century B. C, when the Metaphysics was complied (P. Natorp ascribes the authorship of the book K to “an older Peripatetic” (durch einen älteren Peripatetiker – ibidem, 193). A detailed research makes it nevertheless necessary to postpone the date of the origin of this paraphrase of the books B E to II/I century B. C. For an extensive discussion of this issue see Bigaj (2013), 42–66). It is there that the author flatters himself for having found a philosophically proper description for the transcendent Essence: (“The Being as «Being»”). In this formula, he found an elaboration of the simple by , which in the book expressed only the

26 | Seweryn Blandzi manner of investigating what has been termed as ( ) (One should note that a similar mistake has been notoriously made in rendering the related expression from 1003a, 23–24: (cf. the English translation by H. Tredennick: contemplates generally being qua being)). The author K treated the explanation as a characteristic – and thereby made out of the formula unum, giving it the sense of: “the Being whose essence can be reduced to the «Being»”. This is reminiscent of the Biblical “I am who I am”, which suggests an influence of the Old Testament’s monotheism. In this way, the theological interpretation of the “Being” ( ) “as Being” ( ), would be prepared approximately one hundred years before Philo by the editor of the Metaphysics, which enabled the Alexandrian to view “the Being God”as auto-referential and relationless Being. The editor of the Metaphysics and, at the same time, the author of the K paraphrase, finds in this the distinctive Beingness that is different from everything that is characterized as , introducing, thereby, a certain dualism into the specification . By distinguishing the “inbred” existence that is homogenous and holistic ( ) from the plurality of only analogically called partial “existences” ( ), in a contrast to the later metaphysics that pushes for the view of homogenous “Beingness” in the sense of existence, the ontic boundary between various types of beings, including God (summum ens) and creations, is blurred, and remains only a matter of degree. It is symptomatic that the author of the Kappa, fascinated by the formula that reduces “Being” to itself, omits in his paraphrase entirely the reduction, introduced in the Gamma book and later so crucial for scholasticism of all forms to , in which the deity culminates. According to Franz Brentano (Brentano (1862)) in all Beingness is gathered (E. Berti levels a charge against Brentano because of that: “Ma l’interpretazione di Brentano, secondo cui i molteplici significati dell’essere distinti da Aristotele si riducono tutti all’ , è insoddisfacente per varie ragioni” (La ‘Metafisica’ di Aristotele: «onto-teologia» o «filosofia prima»? In: Berti (2005), 400)), whereas Carl Braig sees the full Beingness in God. In this, view, as has been observed by Heidegger, God becomes the quintessence of Being as the most supreme (suprema concrezione dell’ ), and “ontology transforms into theology” (l’ontologia si converte in teologia) (Berti (2005), 396). In Brentano’s position, one can find a transition from analogia to univocitas entis. Homogeneity, to the point of unequivocality, can also be discerned in Heidegger’s conceptof “Being”, which he refuses to find in Aristotle’s metaphysics. For the K author, on the other hand, the two distinctive spheres remain heterogeneous in their “existence”, the former is transcendent ( ) to the latter one which is included ( ) by the former, showing with it a certain community ( ), under the relation of subordination, i.e., – as one may conjecture

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– inferiority the creation to the Creator. Here, Heidegger erroneously ascribes to this “union” ( ) of Being from the book K a mediation by (Berti (2005), 397–398: “La «riduzione» ( ) in questione, per la quale Heidegger si rifà sopratutto al libro K della Metafisica (dove effettivamente l’ è concepita come un ), viene presentata come fondata sulla scolastica analogia attributionis, intesa come «partecipazione» dei vari significati al primo”), since the term does not occur in the paraphrase of the book not even once (!), as it is consciously and consistently disregarded by the author. That is why God is not presented there – as Heidegger would have it – as summum ens, i.e. “«Being», in whose «das Sein» manifests itself in the highest sense” (ed infine Dio viene presentato come il summum ens, cioè come l’ente nel quale l’essere si manifesta nel senso più alto) (Berti (2005), 398), but merely as “Being itself” ( ) in an exclusive, full and unique sense. The formula brings out of the its , to use a neoplatonic term that accentuates the fullness of the transcendent One-Being. In the sphere of the “Being itself”, one is struck by the similarity to the Parmenidean that designates the transcendent sphere of Being, the domain of the full and immutable truth. Philo’s use of the term with reference to God has been continued by other thinkers, dependent on the Alexandrian theologian, first Plutarch, then Numenius, who both used derivative vocabulary, e.g. with reference to God as the very first cause, which was quite different from the neoplatonic thinkers, for whom it was Pre-Being One that was logically prior to the Being and, thereby, trans-transcendent ( , resp. ).

Bibliography Aristotle (1933), “Metaphysics”, in Aristotle in 23 Volumes, translated by H. Tredennick, Cambridge, MA: Harvard University Press. Aubenque, P. (1983), “Sur l’inauthenticité du livre K de la Métaphysique”, in Zweifelhaftes in Corpus Aristotelicum. Studien zu einigen Dubia, Berlin-New York. Berti, E. (2005), Nuovi studi aristotelici, vil. II, Fisica, antropologia e metafisica, Brescia: Editrice Morcelliana. Bigaj, J. (2005), Zrozumieć metafizykę [To Understand Metaphysics], Vol. I, Part 2, Toruń. Bigaj, J. (2013), Dlaczego «Pseudo-Arystoteles»?, Vol. I, Part 2, Ustrzyki Dolne. Brentano, F. (1862), Von der mannigfachen Bedeutung des Seienden nach Aristoteles, Freiburg. Domański, J. (1989/1990) “‘Stworzenie z niczego’. Kilka uwag o genezie i rozwoju pojęcia”, in Roczniki Filozoficzne: 37-38.

28 | Seweryn Blandzi Heidegger, M. (1949a), “What is Metaphysics?”,¹ It was first translated by R. F. C. Hull and Alan Crick in 1949 and published in Heidegger (1949b), 325–349. Heidegger, M. (1949b), Existence and Being – A Collection of Heidegger’s Essays, edited by W. Brock, Chicago: Henry Regnery. Heidegger, M. (1956), “Introduction to ‘What Is Metaphysics?’. Getting to the Bottom of Metaphysics”, It first appeared in a translation by W. Kaufmann in Existentialism from Dostoevsky to Sartre, New York: Meridian Books. See also the transiation by M. Groth in http://wagner.wpengine.netdna-cdn.com/psychology/files/2013/01/Heidegger-What-IsMetaphysic. Heidegger, M. (2004 ⌈1929/1930⌉), Die Grundbegriffe der Metaphysik. Welt – Endlichkeit – Einsamkeit, Frankfurt am Main: Klostermann. Heidegger, M. (2002a), Identity and Difference, translated and with an introduction by, J. Stambaugh, Chicago: University of Chicago Press. Heidegger, M. (2002b), The Onto-theo-logical Construction of Metaphysics, in Heidegger (2002a), 45–74. Natorp, P. (1888a), “Thema und Disposition der aristotelischen Metaphysik”, in Philosophische Monatshefte: 24. Natorp, P. (1888b), “Ueber Aristoteles, Metaphysik K 1–8, 1065a, 26”, in Archiv für Geschichte der Philosophie: 1, 178–193. Philo, 10 vols. and 2 supplementary vols. (1929–62), translated by F. H. Colson, G. H. Whitaker and R. Markus, Loeb Classical Library, Cambridge: Harvard University Press. Plato (1971), “Sophist”, in Plato in Twelve Volumes, Vol. VII, translated by H. N. Fowler, Cambridge, MA: Harvard University Press.

1 The lecture was presented to the faculties of the University of Freiburg on July 24, 1929 as Heidegger’s inaugural address.

Robert E. Maydole

On the Anti-Ontological Doom Argument Some ontological arguments for the existence of God are dismissed because they allegedly or arguably beg the question, and some because they allegedly or arguably assume that existence is a property, and some because they are allegedly or arguably susceptible to parody, and others because they allegedly or arguably attempt to prove existence a priori. In “Doomed to Fail” (Turri (2012)) John Turri argues that all ontological arguments are doomed to fail because they purport to nonempirically prove that God exists now, and the only thing anyone can nonempirically prove to exist now is that they themselves exist now. Call this argument ‘Doom’. I shall argue that Doom itself fails. Turri defines nonempirical knowledge as knowledge that is not based on external perceptual experience. He says that there are two kinds of nonempirical knowledge: knowledge based on introspection, and a priori knowledge based on rational intuition or understanding. Using virtually Turri’s own words, and a slightly rearranged order, we can express Doom in standard logical form thus: 1. If you can nonempirically know that a certain person exists now, then you are that person. (Premise) 2. If any ontological argument can succeed for you, then you can nonempirically know that God exists now. (Premise) 3. You are not God. (Premise) 4. If God exists, then God is a person. (Premise) 5. If God exists, then God exists eternally. (Premise) 6. If God exists eternally, then God exists now. (Premise) 7. If no ontological argument can succeed for you, then no ontological argument can succeed for any of us. (Premise) 8. No ontological argument can succeed for any of us. (Conclusion) I submit that Doom is either invalid, or valid but not sound. Doom is invalid unless the meaning of ‘nonempirical knowledge’ is identical in premises (1) and (2). But the intended meaning of ‘nonempirical knowledge’ in (1) would seem to be that of knowledge based on introspection; and the intended meaning of ‘nonempirical knowledge’ in (2) would seem to be that of a priori knowledge based on rational intuition or understanding of concepts. So Doom is prima facie invalid.

30 | Robert E. Maydole Suppose, however, that we stipulate that ‘nonempirical knowledge’ in both (1) and (2) means knowledge based on introspection. Then (2) is a false strict implication. For God cannot be known to exist introspectively, and it is possible that an ontological argument might succeed for you in some possible world where you have “mistaken” reasons to unwittingly believe that the argument is sound and not question begging. Likewise, suppose we stipulate that ‘nonempirical knowledge’ in both (1) and (2) means knowledge based on rational intuition or understanding of concepts. Then (1) is a false strict implication. For surely there is a possible world where two people exist now and you can rationally intuit or conceive that someone does. So there is a possible world where the antecedent of (1) is true and its consequent is false, making (1) false. Suppose, we stipulate that ‘nonempirical knowledge’ means the same thing in both (1) and (2), but it is not knowledge based on introspection or rational intuition or understanding of concepts per se. Let us just say with Turri that it is simply knowledge that is not based on external perceptual experience. Doom still appears unsound. For consider the antecedent of (2). An argument succeeds if and only if it is sound and not question begging. In particular, an argument succeeds for you if and only if you believe that the argument is sound and not question begging. Yet, you might mistakenly believe that an argument is sound and not question begging when either it is not or you cannot adequately show or justify that it is. So an argument might then succeed for you, and you might only believe but not know that that it is sound and not question begging, yet think that you do know it. Thus, an ontological argument might succeed for you without it being true that you know that God exists now, nonempirically or not. Premise (2) of Doom is false. Turri, however, argues that Doom is sound. What are his reasons for claiming that its premises are true? He rightfully notes that premises (4), (5) and (6) are simple conceptual truths, and that premise (7) is true because there is nothing special that distinguishes you from others with regard to the success of an ontological argument. And he correctly proclaims that “upon cool reflection” we will agree that premise (3) “needs no defense” (p. 416). Premise (1) is very controversial philosophically. Numerous philosophers from Kant, to Wittgenstein, and Austin, and more recently Putnam and Davidson, have argued that thought and language nonempirically “presuppose” the existence more than one person at some point in time. Turri does attempt to defend (1) by arguing by example that these philosophers and other externalists fail to show that there could not be a moment of time when only one person exists who introspectively knows that she or he exists then. But his defense is a straw man argument against externalism, and derivatively for the truth of (1). Externalists do not espouse the view that Turri refutes. His only

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other defense of (1) is to say that it “is apt to seem uncontroversial, verging on the obvious” (p. 415). I rather doubt that it is obvious to most people. Let us turn finally to Turri’s reasons for premise (2). He says, My second premise is that if any ontological argument can succeed for you, then you can nonempirically know that God exists now. For if it were to succeed, then you would nonempirically know that God exists. And it is trivially obvious that if God exists, then God exists eternally, including now; this is a simple conceptual truth, which you can and do know nonempirically. (pp. 417–418)

I suspect that Turri inadvertently omitted the phrase ‘for you’ from the second sentence of this quotation. I think that Turri must have intended the second sentence to be ‘For if it were to succeed for you, then you would nonempirically know that God exists’. Otherwise the sub-argument for premise (2) of would be blatantly invalid. We can put this amended sub-argument into standard logical form thus: 1s. If any ontological argument can succeed for you, then you can nonempirically know that God exists. (Premise) 2s. If God exists, then God exists eternally. (Premise) 3s. If God exists eternally, then God exists now. (Premise) 4s. If any ontological argument can succeed for you, then you can nonempirically know that God exists now. (Conclusion) The problem is that this amended sub-argument (sub-Doom) begs the question and has a false first premise. Premise (1s) is false for the same reason that premise (2) of Doom is false. Yet sub-Doom is valid, and (2s) and (3s) are surely true. So, suppose that (1s) were true, if only for the sake of argument. Sub-Doom would still beg the question. Consider the rather obvious example of an argument that is sound and begs the question: 1 + 1 = 2 or 1 + 2 = 4 1+2 ̸ = 4 1+1 = 2

(Premise) (Premise) (Conclusion)

Both premises of this valid argument are true. It begs the question because the only possible reason we could have for the first premise must depend on the truth of the conclusion. Consider the similar argument:

32 | Robert E. Maydole If the ontological argument succeeds for you, then God exists. (Premise) If God exists, then God exists now. (Premise) If the ontological argument succeeds for you, then God exists now. (Conclusion) It too is sound, and it begs the question because ‘God exists’ and ‘God exists now’ are equivalent, as are the first premise and the conclusion. Likewise, the only possible reason we could have for the first premise must depend on the truth of the conclusion. Turri’s sub-Doom might seem different from the simple arguments of the last paragraph because sub-Doom contains the opaque expression ‘knows that’ and might therefore not sustain salva veritate all substitutions of equivalences of x in knows that x. On the other hand, it seems plausible to assume that the evidence or reasons for nonempirically knowing that God exists and knowing that God exists now would be the same if the knower knows that ‘God exists’ and ‘God exists now’ are equivalent. Moreover, if you can rationally determine whether or not an ontological argument can succeed for you, and/or minimally understand the ingredients, terms, or logic of an ontological argument, then you should be able to know that ‘God exists’ is equivalent to ‘God exists now’ and then also know that the evidence or reasons for ‘You nonempirically know that God exists’ is the same as the evidence for ‘You nonempirically know that God exists now’. And it surely is the case that sub-Doom presupposes that you can rationally determine whether or not an ontological argument can succeed for you, and/or minimally understand the ingredients, terms, or logic of an ontological argument. We must conclude therefore that sub-Doom begs the question. Since Doom is either invalid, or valid and unsound, or not adequately supported, we must conclude that Doom does not spell doom for ontological arguments.

Bibliography Turri J. (2012), “Doomed to Fail: The sad epistemological fate of ontological arguments”, in Ontological Proofs Today, edited by M. Szatkowski, Frankfurt/Paris/Lancaster/New Brunswick: Ontos Verlag, 413–421.

Peter van Inwagen

Nothing Is Impossible I will present two arguments for the conclusion that it is impossible for there to be nothing. (The arguments involve various modal inferences. Each argument contains a modal inference that is valid only in S5.) More exactly, the conclusion of each argument is a conditional statement: If it is possible for there to be something, then it is impossible for there to be nothing. Alternatively, if there can be something, there has to be something. Alternatively, it is impossible for there to be nothing – unless it is impossible for there to be something. Now I concede that we know from observation that there is something, and concede further that it may be validly deduced from that proposition that it is possible for there to be something. But I am a metaphysician, and the metaphysician eschews any empirical aid and disdains argument a posteriori. The arguments I present will be entirely a priori. If you wish to extend my results by constructing a further argument, an argument whose premises are the common conclusion of my two arguments and the empirical proposition that there is something, and whose conclusion is the proposition that it is impossible for there to be nothing, that is entirely your own affair. For my part, I would dismiss any such project as mere “applied philosophy”. Students of my work may protest at this point that I am on record as having contended that it is impossible to prove that it is impossible for there to be nothing; that it is impossible to prove this conclusion even by an argument that involves a crude appeal to experience. Indeed, they will remind me, I have contended not only that there cannot be a proof that it is impossible for there to be nothing, but that there is no reasonable prospect of even a good or plausible or interesting argument for this conclusion. I expressed this contention in a paper called “Why Is There Anything at All?”¹ The core of that paper was an argument that purported to show that the probability of there being nothing was 0.² (That conclusion is, of course consistent with the proposition that that it is impossible for there to be nothing, but, I pointed out, does not entail it, owing to the fact that there are possible propositions whose probability is 0.) In the introductory parts of that paper, I maintained that the prospects of an interesting argument for the sheer impos-

1 Van Inwagen (1996). 2 Dustin Crummett has recently convinced me that this argument requires revision. The conclusion of the revised argument would be that the probability of there being nothing is either 0 or infinitesimal.

34 | Peter van Inwagen sibility of there being nothing were dim. The reasoning by which I defended this pessimistic conclusion was essentially this: An argument for the impossibility of there being nothing would be an argument for the conclusion that there is something in every possible world. There are two ways, and two ways only, in which there might be something in every possible world. There might be – on the one hand – one or more necessary beings, beings that existed in every possible world. But if – on the other – there were no necessary beings, it might nevertheless be that there were contingent beings in every possible world. It is unlikely that there could be a convincing argument for the existence of a necessary being. Such an argument would, it seems, be either some variant on the so-called modal ontological argument or some variant on the cosmological argument. But, for various reasons, it is doubtful whether there could be a convincing argument of either sort. And as for there being no necessary being but, nevertheless, contingent beings in every possible world – well, no one has ever presented even a candidate for an argument for that conclusion.

I expect you have seen the fallacy in my reasoning. Whether you have seen it or not, here it is: If a proposition p is equivalent to the disjunction of the propositions q and r, and if there is no prospect of a convincing argument for q and no prospect of a convincing argument for r, it hardly follows that there is no prospect of a convincing argument for p. There might, for example, be a convincing argument for the conclusion that a certain disputed painting was either by Giorgione or by Titian and no convincing argument for the conclusion that it was by Giorgione (and not Titian) and no convincing argument for the conclusion that it was by Titian (and not Giorgione). It is just this sort of possibility that is exploited in the arguments I shall present: they are in effect arguments for the conclusion that if it is possible for there to be something, then either there is a necessary being or there are contingent beings in every possible world. You will see that they make extensive use of the inference-form sometimes called disjunctive dilemma. (Each in fact contains a disjunctive dilemma within a disjunctive dilemma.) The materials for these arguments are drawn in large part from the modal ontological argument (in a modest version – a version whose conclusion is only that there is a necessary being, and not the much stronger conclusion that there is a perfect being) and from a certain kind of cosmological argument, the kind based on the principle of sufficient reason. But although my arguments draw on those arguments, their conclusion is much weaker: their conclusion is not that there is a necessary being but rather the conclusion I have announced: that if it is possible for there to be something, it is impossible for there to be nothing. I now turn to various matters that must be attended to before the arguments can be presented – discussions of some terms and concepts and principles that will figure in them.

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1 Preliminary Matters 1.1 A Miscellany of Terms, Concepts, Definitions, and Assumptions We use ‘thing’ as the most general count-noun: everything is a “thing”; a “thing” is anything that can be the referent of a pronoun. ‘Any thing’ is equivalent to ‘anything’, and ‘some thing’ is equivalent to ‘something’. A concrete thing is a thing that can be an agent or a patient: a concrete thing is a thing that can act on or be acted on by other things. An abstract thing is anything that is not concrete. We do not assume that there are abstract things or even that it is possible for there to be abstract things. We leave it an open question whether abstract things exist (or are even possible). That is to say, the existence of abstract things will be neither explicitly affirmed nor explicitly denied in our arguments. In the course of presenting and evaluating the arguments we shall consider, however, we shall freely “quantify over” abstract things – in most cases, properties or attributes, and among them certain special properties I shall call “kinds”. (Consider this analogous case. Phoebe, lecturing to her introductory physics class, proves that the orbital velocity of a planet equals its escape velocity divided by the square root of 2. The proposition that numbers and other mathematical objects exist is not going to figure – explicitly, at any rate – in her demonstration, but she is certainly going to have to quantify over mathematical objects in the course of presenting that demonstration.) By ‘There is nothing’ we understand ‘Nothing is concrete’ or ‘Everything is abstract’. By ‘There is something’ we understand ‘Something is concrete’. Modal terms will be used in their “metaphysical” or ‘unrestricted” sense – that is, the sense to which restrictions are applied to yield the various restricted modalities. For example, ‘It is physically impossible for there to be a 100,000 kg ball of U-235 that exists for more than a fraction of a second’ means, ‘It is metaphysically impossible – or impossible tout court, impossible simpliciter, impossible full stop, impossible period – for the laws of physics to be as they actually are and for there to be a 100,000 kg ball of U-235 that exists for more than a fraction of a second’. It will be assumed that S5 captures the logic of metaphysical or unrestricted modality.

36 | Peter van Inwagen We will use the count-noun ‘being’ as an abbreviation for ‘concrete thing’. And we will use the mass term ‘being’ to denote the property of “being a being” or being a concrete thing.

1.2 Kinds Our arguments will involve the concept of a “kind of being” or simply a “kind”. In this essay, I use ‘kind’ as a term of art and I do not pretend that it represents the usual notion of a kind (or natural kind) – although it is certainly in some way related to that notion. We treat “kinds” as properties (attributes, qualities, characteristics, features . . . ) . By ‘properties’ we understand universals, and platonic not Aristotelian universals, transcendent rather than immanent universals, universals ante res. Properties, being platonic universals, are in no sense constituents of the objects that have them, but reside, as they say, in the Platonic Heaven. Consider, for example, a shamrock with four leaves. It is, of course, green. On the present conception of “property”, the relation between the shamrock and the property greenness or viridity is as abstract and bloodless as the relation between its leaves and the number 4. Properties, on this conception, are much like propositions – but where propositions are true or false simpliciter, properties are true of false of things. Greenness for example, is true of the shamrock but false of the White House. (To say that greenness is true of the shamrock and to say that the shamrock has or instantiates or exemplifies greenness is to say the same thing in different words.) If propositions are what are expressed by closed declarative sentences, properties are what are expressed by declarative sentences in which one variable is free. Properties in this sense are “abundant” rather than “sparse”: a property corresponds to every “precise condition”, to every open sentence in which one variable is free and which is such that it is determinate for every thing whether that thing satisfies that sentence. (With, to be sure, the exception of a few Russellian self-referential monsters.) To say that a being is of the kind F is simply to say that it has F (or that it exemplifies or instantiates F or that F is true of it) – for, kinds are, as I have said, a species of property. We introduce a convenient abbreviation by example. Instead of writing, e.g., ‘If the property of being a horse is a kind ...’ we write ‘If “horse” is a kind ...’. Now assume, simply for the sake of having an example to consider, that “horse” (equinity or horsiness or being a horse or whatever you want to call it) is indeed a kind. Since “horse” is a platonic property, it existed before there were horses and will exist when horses are no more. It exists in all possible worlds, in-

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cluding those worlds (the vast majority, presumably) in which there are no horses – ever. Thus, a kind, since it is an universale ante res, may exist when there is (and if there is never) anything of that kind. In fact, it is a plausible thesis that almost all kinds are of that sort – “empty” kinds so to call them. But if kinds are properties, what properties are they? Perhaps something like a definition is called for. I will propose a definition. (I shall later mention some considerations that suggest that it might need to be revised, and consider some possible revisions.) A kind of being (or simply a kind) is any property that satisfies the following conditions: — It entails being (or concrescence or concreteness) — It is an essential property – a property that can be had only essentially (impossible properties are thus trivially essential properties) — In every possible world in which anything has it, the “boundary” between the beings that have it and the things (beings or not) that do not have it marks a real division among things — It is not a negative or disjunctive property.³

(It may well be a consequence of this definition that all impossible properties are kinds. If the definition does indeed have that consequence, our arguments will not depend on it and it may be ignored, treated as a “don’t care”.) In the following subsection, we shall consider the concept of explanation. In the next subsection but one (subsection 1.4) we shall consider three principles concerning explanation. The concept “kind” will figure in two of those principles.

1.3 Explanation We will make use of the idea of the truth of a proposition’s explaining the truth of a proposition – an idea that comprehends both the idea of a proposition’s explaining its own truth and the idea of a proposition’s explaining the truth of another proposition. We express the “explanation relation” on propositions by sentences of the form The truth of p explains (or is an explanation of) the truth of q. (We shall sometimes abbreviate this as ‘p explains q’.)

3 I am not sure whether the final clause in this definition is necessary. I have, as it were, thrown it in for good measure. It will play no essential role in the arguments I shall present. I would define ‘negative property’ and ‘disjunctive property’ in more or less the way Chisholm does. (See, “Properties and States of Affairs Intentionally Considered” in Chisholm (2007), 141–149.)

38 | Peter van Inwagen An explicit definition of the “explanation” relation will not be given (an unwieldy amount of technical apparatus would be needed to set out a definition that satisfied the use-mention scruples that all post-Quinean philosophers should have). A definition would be a careful and precise development of the “general idea” displayed in the following two examples (the first is an example of what may be called a “why it is the case that” explanation, and the second of what may be called a “how it came to pass that” explanation). The truth of the proposition that the moon pulls harder on one side of the earth than the other explains the truth of the proposition that there are tides. if and only if The statement “(Because) the moon pulls harder on one side of the earth than the other” is a correct and informative answer to the question, “Why are there tides?” The truth of the proposition that the zookeeper left the cage door open explains the truth of the proposition that the lion has escaped from its cage if and only if The statement “The zookeeper left the cage door open” is a correct and informative answer to the question, “How did it come to pass that the lion has escaped from its cage” (or, more idiomatically, “The lion has escaped from its cage – how did that happen?”)

As these examples show, we do not suppose that if p explains q, then p entails q. A correct answer the question, ‘Why are there tides?’ need not consist in the assertion of a proposition that entails the proposition that there are tides. The same point applies, mutatis mutandis to the question in the second example, the “how did it come to pass that” question. (But, to use a familiar pair of terms, an “explanans” may entail its “explanandum”: “Dr Crippen murdered his wife Cora in 1910” explains “Cora died in 1910” – assuming, of course, that Crippen did indeed murder Cora in 1910.) And, of course, we not only do not assume, but we positively deny, the general principle that if a true proposition p entails a proposition q, then p explains q. This principle is inconsistent with obviously true thesis that no contingent proposition explains itself, but that is not the only reason to reject it: it can hardly be supposed, for example, that, in general, a conjunction explains both (or either of) its conjuncts. Note that we use ‘explain’ and ‘explanation’ in their “achievement” senses (we do not use ‘explanation’ in the sense “explanation candidate” or “proposed explanation” – the sense illustrated by ‘The coroner’s explanation of her death later turned out to be wrong’). Note also that if p is an explanation of q, it does not follow that there cannot be a fuller or more satisfactory explanation of q than p. If, for example, ‘The

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zookeeper left the cage door open’ is a correct and informative answer to the question ‘How did the lion escape from its cage?’, that does not prevent ‘The zookeeper left the cage door open in order to test her theory that lions are friendly animals who would never harm a human being’ from also being a correct and informative answer to that question. The present conception of explanation does not presuppose that there is such a thing as an ultimate or final explanation of any truth, and most particularly of any contingent truth. It may be, for example, that the contingent truth “An asteroid with a mass of 3 × 1015 kg with a speed of 20 km/sec struck the Earth about 66 million years ago” has no explanation (although that seems implausible, given our very liberal conception of explanation) and explains the truth of “Dinosaurs were extinct 65 million years ago”. It seems correct in that case to say both that the fact that dinosaurs were extinct 65 million years ago has an explanation and has no explanation that could in any reasonable sense be called an ultimate explanation of that fact. (Indeed it is hard to see how it could be possible for any contingent truth to have an “ultimate” explanation. But we need not address the question whether a contingent truth can have an ultimate explanation – or, indeed, the question of what an ultimate explanation of a contingent truth would be if there were such a thing.) We may define a complete explanation of p as an explanation of p that entails every explanation of p. It would seem that every proposition that has any explanation must have a complete explanation, but that does not imply that a complete explanation of the truth of any contingent proposition is humanly discoverable or could be expressed in a sentence of finite length. In any case, the concept of a complete explanation will play no part in the arguments we shall consider. This conception of explanation does not imply that if p and q are both explanations of r, then it must be that either p entails q or q entails p (that if a truth has two explanations, one must be an elaboration of or subsume the other): it may be that “Her husband murdered her on her twenty-fifth birthday” and “She ingested arsenic at some point during the twenty-sixth year of her life” are both explanations of “She died at age twenty-five”. It is this conception of explanation that figures in the principles that will be presented in the following subsection.

1.4 Three Principles The following three principles – principles about explanation – are premises of both arguments:

40 | Peter van Inwagen The Principle of Sufficient Reason (PSR): It is a necessary truth that: If beings of a certain kind exist, then there is an explanation of the existence of beings of that kind. If, for example, “elephant” is a kind, PSR implies that there is an explanation of the existence of elephants. It does not imply that there is an explanation of the existence of any particular elephant. It does not imply that there is an explanation of the fact that elephants have trunks (unless, perhaps it is a necessary truth that – mature, genetically normal, unmaimed – elephants have trunks). PSR is therefore a considerably weaker proposition than any other proposition that has ever – to my knowledge – been called ‘the principle of sufficient reason’. And this for two independent reasons: (i) the relation between propositions that I call explanation is a much weaker relation than the relation Leibniz had in mind when he used the phrases ratio sufficiens and raison suffisante, and (ii) PSR does not imply that every true proposition is such that some proposition bears even the relatively weak relation “explanation” to it. (It is in fact doubtful whether such a weak principle as this deserves to be called ‘the Principle of Sufficient Reason’. I retain the term because the above principle plays a role in my arguments that is in a certain sense analogous to the role played by “the principle of sufficient reason” in many versions of the cosmological argument.) The Principle of the Externality of Explanation (PEE): It is a necessary truth that: If it is contingently true that beings of the kind F exist, then any explanation of the existence of beings of that kind must appeal to or involve beings that are not of the kind F. If, for example, “elephant” is a kind, and it is contingently true that there are elephants, then any explanation of the existence of elephants must involve beings that are not elephants: God or (inclusive) the evolutionary precursors of elephants or genes or carbon atoms or supernovae. (Note that PEE does not imply that if it is contingently true that beings of kind F exist, then there is an explanation of the existence of beings of the kind F. For all PEE tells us, it may be that “elephant” is a kind, that it is contingently true that elephants exist, and that there is no explanation whatever of their existence. PEE tells us only that if there is an explanation of the – contingent – existence of beings of the kind “elephant”, it’s going to have to appeal to beings that are not elephants.) Note that the sentence (in PEE) ‘it is a contingent truth that beings of the kind F exist’ means ‘it is a contingent truth that beings of the kind F exist at some time or other – past, present, or future’ (and not, e.g., ‘it is a contingent truth that beings that of the kind F now exist’). Colloquially speaking, PEE insists that (given

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that “elephant” is a kind) if there is an explanation of the fact that elephants exist at all, this explanation must involve beings that are not elephants. PEE is consistent with the statement that “Elephants existed in the past, and they have not become extinct” explains “Elephants exist at the present time”. It is easy to see why PEE is at least extremely plausible. An explanation that does not appeal to or involve beings that are not of the kind F must either appeal to or involve no beings at all, or else must appeal to or involve beings that are of the kind F. It is hard to see how a proposition that appeals to or involves no beings at all could explain the existence of beings of some given kind. (In Leslie (1979) and Leslie (1989) and in other publications, John Leslie defends a position that has just that consequence: that the ethical requiredness of the existence of beings with certain properties – consciousness and rationality, for example – explains the existence of such beings – and therefore of a universe to contain them –, but I have never been able to see how that could possibly be true. Analogous remarks apply to the thesis that the laws of quantum mechanics explain the existence of particles – and therefore of a physical universe –, but I will not develop this analogy here.) And, obviously, for no kind F, can an explanation of why there are beings of the kind F (“at all”, “in the first place”) appeal to or involve beings that are of the kind F. Even if Aristotle had been right about the eternality of species, one could not explain why the world contained elephants (“at all”) by saying that every elephant was produced by other elephants. And not because that statement is false – for all we can say a priori it could be true – but for the plain reason that it presupposes the existence of things of the kind whose existence is to be explained. It is, incidentally, necessary that PEE refer to kinds (or at least to essential properties). It is certainly possible to explain why there are beings with a certain property (a certain accidental property) by appealing only to things that have that property. Consider, for example, a simple world in which two bodies, A and B, are, owing to their mutual gravitational attraction, eternally revolving in stable orbits about their common center of mass. Both bodies have (at all times) the property “undergoing acceleration”, and an explanation of why there exist beings that have that property need appeal only to beings – A and B – that have that property. I owe this nice point to Kris McDaniel and Kevin Klement. Note finally that PSR implies that if there is a kind such that there are necessarily beings of that kind, the existence of beings of that kind has an explanation. For example, many people believe that the there are necessarily beings of the kind “Divine Being” (most of them believe that there is only one being of that kind, of course), and their belief and PSR jointly imply that there is an explanation of the existence of Divine Beings. But that belief, PSR, and PEE do not together imply that that there is an explanation of the existence of Divine Beings that appeals to or involve beings that are not Divine Beings, owing to the fact that ‘There are

42 | Peter van Inwagen Divine Beings’ is, if true, not a contingent truth. (If there are necessarily Divine Beings, the explanation of their existence demanded by PSR would presumably be simply that it is not possible for there not to be any.) Finally, we have, The Principle of Existential Implication (PEI): It is a necessary truth that: For any property, if an explanation (sc. of anything) appeals to or involves beings that have that property, then beings with that property exist.⁴ I will make two remarks about the meaning of the phrase ‘appeals to or involves’ as it occurs in PEE and PEI. (i) In the sense that phrase has in those principles, an explanation appeals to or involves beings that that have the property F only if their having F actually figures in the explanation. The ordinary sense of ‘appeals to or involves’ certainly permits statements like ‘The coroner’s explanation of the victim’s death appeals to [involves] an exotic poison that it would have been very difficult for the accused to obtain’. But the present sense of ‘appeals to or involves’ does not permit statements of this sort – or would (in the case used as an example) only in the event that the fact that the poison in question was an exotic one that it would have been very difficult for the accused to obtain actually did somehow figure in the coroner’s explanation of the death. (ii) Consider the following explanation of the fact that cavalry charges did not play a significant role in World War II. “Because a militarily effective cavalry charge in that era would have required horses that could not be killed by machine-gun fire.” This explanation does not “appeal to or involve” horses that cannot be killed by machine-gun fire.

1.5 A Premise about Kinds Our two arguments will depend on the premise that “contingent being” is a kind. This premise seems reasonable. The property “being a contingent being” entails being; it is a property that can be had only essentially – provided, at any rate, that the accessibility relation is symmetrical and transitive; the line dividing contingent beings from other things (from abstract things if there are any such, and

4 C. Anthony Anderson has argued (in conversation) very convincingly for the conclusion that this principle is analytic. I think he is almost certainly right, given that ‘explanation’ is, as it is, being used in its achievement sense and given the two remarks about the intended meaning of ‘appeals to or involves’ that follow in the text.

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from necessary beings if there are any such) certainly cuts the world at its joints; it is not a negative or disjunctive property.⁵

2 The Arguments 2.1 The Arguments The First Argument I will present this argument in the form of a commentary on a diagram (Figure 1; see the next page) that displays its logical structure. The argument is a conditional proof: we assume the antecedent of the conditional ‘If it is possible for there to be something, it is impossible for there to be nothing’ and derive the consequent. Here’s a hint about the structure of the argument. Take a look at that complicated bit of the diagram inside the box where ‘PSR, PEE, PEI’ appear in bold-face to the right of a vertical arrow. That’s the core of the argument, the tricky and difficult and likely-to-be-controversial part. The stuff represented in the other parts of the diagram consists mainly in various items of logical bookkeeping. So: Assume that it is possible for there to be something. If this is so, there are possible worlds (accessible from the actual world) in which beings exist. Let w be any one of these worlds. We have: In w, beings exist And of course it must be the case that In w, contingent beings exist. ∨ In w, no contingent beings exist.

(This statement does not depend on our assumption that in w beings exist: every possible world is such that in that world contingent beings exist or in that world no contingent beings exist.) 5 It might be argued that “being a contingent being” was a negative property, owing to the fact that something is a contingent being if and only if it is not a necessary being. First, the premise of this argument is not true if there are abstract things: the number 4 is not a being at all, and hence is neither a contingent nor a necessary being. Secondly, the argument is not valid; if it were, there would be, for almost every property, a parallel sound argument for the conclusion that that property was a negative property: for almost every property F (one of the exceptions would be “being a set that belongs to itself”) there is a property G such that a thing has F if and only if it does not have G.

Fig. 1. The First Argument

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The argument proceeds by disjunctive dilemma, by showing that, inside the scope of our assumption that in w beings exist, ‘In w, it is impossible for there to be nothing’ follows from either of the two disjuncts of this statement. Assume the second disjunct. Then, since in w beings exist and in w no contingent beings exist, and since every being is either necessary or contingent, in w necessary beings exist (that is, at least one necessary being exists). In the diagram, that proposition is over on the right. The “arrows” indicate the premises from which it is derived. Now assume the first disjunct: In w, contingent beings exist. If that is so, then the “boxed argument” is sound in w: Since contingent beings exist, it is either a necessary truth that contingent being exist or it is a contingent truth that contingent beings exist. We proceed once more by disjunctive dilemma. Suppose it is a necessary truth that contingent beings exist. Then it is impossible for there to be nothing. Suppose, then, that it is a contingent truth that contingent beings exist. Since contingent beings exist, and since “contingent being” is a kind, PSR implies that there is an explanation of the existence of contingent beings. (Remember that PSR states that it is a necessary truth that if beings of a certain kind exist, then there is an explanation of the existence of beings of that kind; PSR therefore implies that the conditional ‘If beings of a certain kind exist, there is an explanation of the existence of beings of that kind’ is true in w. The same point applies, mutatis mutandis, to PEE and PEI.) We are now assuming that it is contingently true that there are beings of the kind “contingent being”. And, by PEE, any explanation of the fact that there are beings of that kind must appeal to or involve beings that are not of that kind. Hence, there is an explanation of something (sc. of the existence of contingent beings) that appeals to or involves beings that are not contingent beings. And it then follows by PEI that beings that are not of the kind “contingent being” exist. That is to say, necessary beings exist, from which it follows that it is impossible for there to be nothing. So, whether it is a necessary truth that contingent beings exist or a contingent truth that contingent beings exist, it is impossible for there to be nothing.

This argument, represented in the diagram as the boxed argument, is therefore sound in w if contingent beings exist in w – which of course implies that if contingent beings exist in w, it is, in w, impossible for there to be nothing. We have already seen that if no contingent beings exist in w, then necessary beings exist in w. And if necessary beings exist in w, then it is, in w, impossible for there to be nothing. So we have our disjunctive dilemma: whether contingent beings exist in w or no contingent beings exist in w, it is, in w, impossible for there to be nothing. (See

46 | Peter van Inwagen how the “inferential lines” on the right-hand side of the diagram lead to ‘In w, it is impossible for there to be nothing’ on the lower right-hand side.) And if, in w, it is impossible for there to be nothing, then, by S5, it is impossible for there to be nothing (impossible without qualification, impossible simpliciter, impossible tout court, impossible full stop, impossible period). For if it is possible simpliciter for there to be nothing, then there is a world ν accessible from the actual world α in which there is nothing. Since w is accessible from α, α is, by symmetry, accessible from w, and, by transitivity, ν is accessible from w.

 w

∴ 

α



ν



∴

And if ν is accessible from w, it is possible in w for there to be nothing, contrary to what has been shown. We have therefore deduced ‘It is impossible for there to be nothing’ from ‘It is possible for there to be something’, and hence have established ‘If it is possible for there to be something, it is impossible for there to be nothing’ by conditional proof.

The Second Argument The Second Argument is presented in the form of a commentary on Figure 2. It begins as the first did. We assume that it is possible for there to be something. It follows that, for some possible world w (accessible from the actual world): In w, beings exist. Again, our argument proceeds by disjunctive dilemma. But we apply disjunctive dilemma to a different disjunction from the one we applied it to in the first argument; we apply disjunctive dilemma to In w, no necessary beings exist. ∨ In w, necessary beings exist.

As with the First Argument, we show that, inside the scope of our assumption that in w beings exist, ‘In w, it is impossible for there to be nothing’ follows from each of the two disjuncts of a true disjunction – the disjuncts in the present case being ‘in w, no necessary beings exist’ and ‘in w, necessary beings exist’.

Fig. 2. The Second Argument

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48 | Peter van Inwagen Assume the first disjunct: assume that in w no necessary beings exist. Then (given our assumption that in w, beings exist), the following argument, the argument that is boxed in the diagram, is sound in w. No necessary beings exist. But beings exist. So contingent beings exist. Now if contingent beings exist, it is either necessarily true that contingent beings exist or it is contingently true that contingent beings exist. Assume that it is contingently true that contingent beings exist. It follows, by the same reasoning that we employed in the First Argument, that beings exist that are not contingent beings – that is to say, it follows that necessary beings exist, which contradicts the premise ‘No necessary beings exist’. It is therefore not contingently true that contingent beings exist. If it is not contingently true that contingent beings exist, then, since contingent beings do exist, it is necessarily true that contingent beings exist. And, therefore, it is impossible for there to be nothing. Therefore, given that beings exist in w and that no necessary beings exist in w, this argument, the boxed argument, is sound in w – and, therefore, (given that beings exist in w) if no necessary beings exist in w, it is, in w, impossible for there to be nothing. But if necessary beings exist in w, it is also impossible in w for there to be nothing. Therefore, by disjunctive dilemma, it is, in w, impossible for there to be nothing.

And, as we have seen, if it is impossible in w for there to be nothing, then, given S5, it is impossible simpliciter for there to be nothing. And, finally, by conditional proof, if it is possible for there to be something, it is impossible for there to be nothing.

2.2 A Possible Objection to the Arguments The “possible objection” I wish to consider is best presented by considering a third argument for our conditional conclusion – an argument that is considerably simpler than the First and the Second arguments. The relative simplicity of the Third Argument (for so I shall call it) is due to the fact that it has as a premise not the thesis that “contingent being” is a kind but that “being” itself, “being” tout court, the property of being a concrete object, is a kind. And it does seem that “being” satisfies the conditions laid down in my definition of ‘kind’: Being certainly entails being – certainly entails itself; it is an essential property, for no concrete thing

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is possibly an abstract thing; the line that divides concrete things from abstract things (if there are any such) certainly cuts reality at the joints; it is not a negative or disjunctive property. That is what the relative simplicity of the Third Argument is due to. What this simplicity mostly consists in, formally speaking, is the fact that, while it makes use of disjunctive dilemma, it does not contain a disjunctive dilemma inside a disjunctive dilemma. As I did when I presented the First and Second Arguments, I will present the Third Argument in the form of a commentary on a diagrammatic representation of the argument.

The Third Argument Assume that it is possible for there to be something – and, therefore, that, for some possible world w (accessible from the actual world): In w, beings exist. This assumption entails that the following argument, the boxed argument, is sound in w. It is either necessarily true that beings exist or contingently true that beings exist. Assume that it is contingently true that beings exist. “Being” is a kind and beings of that kind – i.e., beings – exist. There is, therefore, by PSR, an explanation of the existence of beings. Since it is, as we are assuming, contingently true that there are beings of that kind – the kind “being”, – then, by PEE, any explanation of the fact beings of the kind “being” exist must appeal to or involve beings that are not of that kind. Hence, there is an explanation of something (sc. of the existence of beings) that appeals to or involves beings that are not beings. And, therefore, by PEI, beings that are not beings exist. But that is a contradiction, and it was the assumption that it was contingently true that beings exist that led to this contradiction – and, therefore, it is not contingently true that beings exist. Since beings exist and it is not contingently true that beings exist, it is necessarily true that beings exist – and hence it is impossible for there to be nothing.

Since that argument is sound in w, it is, in w, impossible for there to be nothing. Therefore, by S5, it is impossible – simpliciter, without qualification – for there to be nothing. And, by conditional proof: ing.

If it is possible for there to be something, it is impossible for there to be noth-

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Fig. 3. The Third Argument

This argument may make one suspicious my employment of PSR in any argument for the conclusion that it is impossible for there to be nothing if it is possible for there to be something. PSR is, as I have remarked, a much weaker principle than any of the principles that Leibniz and later philosophers who have used the phrase have referred to as ‘the principle of sufficient reason’. And that is all to the good, for those principles have been much too strong to be at all plausible. (Leibniz’s version, for example, implies that all truths are necessary truths – not perhaps in Leibniz’s idiosyncratic sense of ‘necessary truth’, but in the sense ‘necessary truth’ has in present-day philosophy. Or so, at any rate, I have tried to show elsewhere.)

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But perhaps even my version of PSR – with its liberal notion of explanation, and its restriction of the class of truths that have explanations in this liberal sense to those truths whose content is the assertion of the existence of beings of some specified kind – is too strong. If not so strong that it demonstrably has consequences that most present-day philosophers would consider absurd, at any rate so strong that its role in the First and Second arguments might be thought to render them question-begging. Why might even my weak principle be too strong to be legitimately employed as a premise in an argument that concerns the possibility of there being nothing at all? Well, my statement of PSR contains the word ‘kind’, and that word is supposed, therein, to have the sense provided by my earlier definition of ‘kind’. It may be that PSR is an objectionably strong principle owing to the fact that its constituent term ‘kind’ has the sense my definition has provided. For, given that sense of ‘kind’, PSR implies that if anything at all exists, then that fact – that anything at all exists, that something exists – has an explanation. And it is certainly hard to see what that explanation could be if it were not simply that there not being anything at all, there being nothing, was an impossible state of affairs. And that suggests that employing PSR in a proof of the conditional ‘If it is possible for there to be something, it is impossible for there to be nothing’ begs the question (whatever exactly that widely deplored logical fallacy may be) or comes perilously close to begging the question. What might I say in response to this suspicion raised by the Third Argument, this suspicion that PSR is such a strong principle that to employ it any argument for the common conclusion of the three arguments is to beg the question? I’ll try this. Either PSR has this feature or it doesn’t. If it doesn’t, fine: then – or so I would assume – there’s nothing wrong with the Third Argument, and I am in a position to offer not two but three arguments for my conditional conclusion. But if PSR does have this feature, then, if I am to save the First and Second arguments, I must abandon PSR in favor of some still weaker principle that can do the work it does in those two arguments. And that work is: To ensure that if, in a given world w, contingent beings exist, then there is, in w, an explanation of the existence of contingent beings. Suppose I were to weaken PSR by weakening my definition of ‘kind’ – by weakening it in such a way that “being” is not a kind, and “contingent being” is, so to speak, still a kind. There are various ways in which this might be done. I give one example, just to show that it can be done. Perhaps there are more interesting “weaker” principles that could perform the same function. Let us say that the conditions I have specified define not ‘kind’ but ‘weaksense kind’. And let us say that a set of two or more properties is a partition of

52 | Peter van Inwagen a property F if none of its members is equivalent (i.e. necessarily extensionally equivalent) to F, its members are all logical contraries of one another, and F is equivalent to the disjunction of its members. Say that a partition of a property is kindly if all its members are weak-sense kinds. Then: A property F is a kind (without qualification) if F is a member of a kindly partition of a weak-sense kind. It follows that all kinds are weak-sense kinds, but not all weak-sense kinds are kinds – for (at least if being is a possible property) “being” is a weak-sense kind but not a kind. (For there is no kindly partition of any weak-sense kind such that being is one member of that partition.) “Contingent being”, however, is a kind, since the weak-sense kind “being” can be kindly partitioned into “necessary being” and “contingent being”. That is one way to do it. However we do it, we shall have to have to replace PSR with some principle that entails that in any world in which contingent beings exist, there is, in that world, an explanation of the existence of contingent beings. No principle we employ can be more plausible than that thesis. How plausible is that thesis? And can any modified versions of the First and Second arguments be said to beg the question if they employ a premise that entails that thesis? In brief, what is there to be said for the thesis that in any world in which contingent beings exist, there is, in that world, an explanation of the existence of contingent beings? Well, we certainly take it for granted that the existence of things of any of the kinds we consider in everyday life has an explanation. Consider our old friend the elephant. We all take it for granted – do we not? – that the existence of beings of the kind “elephant” has an explanation (that is, that the truth of ‘The world contains beings of the kind “elephant”’ has an explanation) – even if the explanation is nothing more interesting than ‘Elephants came into existence owing to the chance interplay of the effects of certain biological and environment factors among the members of various species ancestral to elephants – factors such as mutation, genetic recombination, and environmental selection pressure’. (Note that this counts as an explanation in our very liberal sense of ‘explanation’.) Now, leaving the elephants, move up the ladder of kinds – “up” meaning ‘in the direction of increasing abstraction’: proboscidea, placental mammals, mammals, amniota, vertebrates, animals, living things, material things, physical things, concrete things ... . Is there a place where we can draw a line and say, “For all the kinds in our ladder of increasing abstraction that are lower than this line, the existence of beings of those kinds has an explanation, but the existence of beings of any of the kinds that occurs above the line has no explanation – or at

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any rate, there’s no reason to suppose that it does”? Or might one say that at a certain point in the ladder of abstraction, the general terms that figure in our verbal representation of the line simply cease to represent kinds – perhaps because at a certain level of generality, there cease to be any real lines of division in the world, that at that level of abstraction there is no longer any such thing as cutting reality at the joints? Well, perhaps so. Who can say with any hope of certainty? This is metaphysics, after all. Perhaps, like Kant’s dove, we metaphysicians, feeling the resistance of the air to our intellectual wings, have got it into our heads that these wings can take us into airless space – indeed that they will do their job better when they no longer have to work against the resistance of the air. Perhaps. But I do think that the idea that “contingent being” is one of the kinds whose existence – given that they do indeed exist – necessarily has an explanation is a not wholly implausible idea.

3 “The Mystery of Existence” In the remainder of this essay, I’m going to pretend that the conclusion of my arguments was that it is impossible for there to be nothing and not, as it in fact was, a conditional proposition of which that proposition is the consequent. This pretense has no better excuse than that it makes it possible for me to replace some very complicated sentences in the sequel with considerably simpler sentences. However plausible anyone may find the First and Second arguments, no one should regard them as proofs of the impossibility of there being nothing. Philosophy has no proofs to offer, only arguments. But suppose I had proved that it was impossible for there to be nothing. What would be the significance of such a proof? In particular, what would its significance be in relation to the so-called mystery of existence? Such a proof would have some relevance to the “mystery of existence”, wouldn’t it? For – surely? – the proof would have some relevance to the question, “Why is there anything at all?” And not one but two books devoted to that question have borne the title The Mystery of Existence.⁶ (One was a systematic book by Milton K. Munitz, published in the sixties.⁷ The other is a recently published

6 I rather suspect that Tyron Goldschmidt, ed., The Puzzle of Existence: Why Is There Something Rather Than Nothing? (Goldschmidt (2013)) would have been called The Mystery of Existence if it had not been for the book cited in note 8. 7 Munitz (1965).

54 | Peter van Inwagen anthology edited by John Leslie and Robert Lawrence Kuhn.⁸) The remainder of this essay is a sort of meditation on this question. I begin my meditation by asking whether it is possible for a contingent truth to explain a necessary truth. It seems to me that it is at least not obviously false that a contingent truth might explain a necessary truth. For might it not be that there is a proposition that is true in every possible world and also has the following property: there is some world w in which its truth is explained by a proposition that, while it is of course true in w, is false in various other worlds? Here is a very simple argument for the conclusion that that this is in fact the case – that it is in fact the case that there is a necessary proposition whose truth is explained by a contingent proposition. This simple argument is, as simple arguments so often are, wholly unpersuasive, but I think it can be elaborated so as to yield an argument for the conclusion that for all we know a contingent proposition can explain a necessary proposition – an argument that is at least not wholly unpersuasive.

The simple but wholly unpersuasive argument

Consider the proposition that either there are elephants or there are not. This is a necessarily true proposition, and its truth is explained by the contingently true proposition that there are elephants – and so it will be in any world in which there are elephants. (And, of course, in worlds in which there are no elephants, that contingent truth will explain the truth of the proposition that either there are elephants or there are not.)

I call this argument wholly unpersuasive because the statement that the truth of the proposition that there are elephants explains the truth of the disjunction of that proposition with its negation seems implausible – in fact, wholly implausible. Explaining the truth of the disjunctive proposition – surely? – if it’s anyone’s business is the business of logicians or philosophers of logic or metaphysicians; it is certainly not the business of zoologists. But more elaborate cases of disjunction, cases involving some metaphysical components, can provide arguments for at least the epistemic possibility of a contingent proposition’s explaining a necessary proposition. Consider, for example, the following metaphysical fable:

8 Leslie and Kuhn (2013).

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The goddess Ungit is a necessary being and the only one. Ungit has (essentially) the power to create beings ex nihilo and, like the Christian God, she has (and has essentially) free will in the matter of whether she exercises that power: in some possible worlds she freely chooses to create beings ex nihilo (and hence does) and in some she freely chooses not to (and hence does not). It is, moreover, metaphysically necessary that every being she creates be of – let us call it – Kind A, and the only way for a being of Kind A to exist is for it to be created by Ungit. It is also metaphysically necessary that if (and only if) she chooses not to create any beings, beings of another (inferior) sort, beings of Kind B, will emanate from her – as beings do from the neoPlatonic One. (She has no choice about the truth of this conditional, any more than you have a choice about whether your body gives off heat if you’re alive; it’s a consequence of her essence.) The only way for a being of Kind B to exist is for it to emanate from Ungit. Every being in every possible world is either Ungit or of Kind A or of Kind B. Beings of Kind A and Kind B are, as I have said, very different kinds of being, but the beings of both kinds are, unlike Ungit herself, contingent beings. (Their contingency is not an additional supposition; it’s a consequence of what has already been said.)

It follows from this story that it is a necessary truth that there are contingent beings. But it does not seem obviously wrong to say that in some worlds the contingent truth “Ungit creates beings of Kind A” explains the truth of “Contingent beings exist”. It does not seem wholly implausible to say that in a world in which Ungit has created beings of Kind A, the statement “Because Ungit created beings of Kind A” is a correct answer to the question, “How did it come to pass that contingent beings exist?” And, of course, the same holds, mutatis mutandis, for the worlds in which she chooses not to create: it is plausible to suppose that in such worlds, “Because Ungit chose not to create any beings, and, in consequence, beings of Kind B emanated from her” is a correct answer to the question, “How did it come to pass that contingent beings exist?” If it is indeed the case that a contingent truth can explain a necessary truth (I do not claim to have shown that this is the case, or even that it is metaphysically possible), this implies that a proof of the impossibility of there being nothing, should such a proof exist, would not resolve the question whether existence is a mystery. Or, at any rate, it has that implication in respect of any proof of the impossibility of there being nothing that is as, well, as abstract, as my two arguments. I’ll try to explain why I say this by telling another story – a story that is every bit as fanciful as the fable of Ungit, though its fancies are of another sort. The story presupposes that a contingent truth can explain a necessary truth.

56 | Peter van Inwagen The fanciful story In every possible world there is something, and in every possible world there is an explanation of why there is something. That is, In every world w there is something, and there is a proposition that is, in w, a good and informative answer to the question, “Why is there something – and not, rather, nothing?” Let us suppose, however, that the answer to that question differs from world to world; that is, although there is, for every world, a proposition whose truth in that world explains why there is something rather than nothing, it is not the same proposition in every world. (It seems to me that this would entail that any proposition that in any world explains why there is something rather than nothing is a contingent proposition. It seems to me that if a necessary truth were an explanation of why there was something rather than nothing in any world, it would be an explanation of why there was something rather than nothing in every world.) And let us suppose, just to have a number before us, that (given the correct principle of individuation for propositions, whatever it may be) there are exactly 510 propositions that have that property – that is, 510 propositions that satisfy the following condition: For some world w, x is, in w, a good and informative answer to the question, “Why is there something – and not, rather, nothing?” Let S be the set of these propositions. We further suppose, 1. There is at least one answer – at least one good and informative answer – to the question, “Why are those propositions – the 510 members of S – all the possible answers to the question, ‘Why is there something – and not, rather, nothing?’ ” (For every member of S, there is a good and informative answer to the question why that proposition is an answer to the question, “Why is there something – and not, rather, nothing?” in some possible world. For every proposition that is not a member of S, there is a good and informative answer to the question why that proposition is not an answer to the question, “Why is there something – and not, rather, nothing?” in any possible world.) 2. For every proposition p and every world w, if p is, in w, a good and informative answer to the question, “Why is there something – and not, rather, nothing?”, there is in w, no explanation of the truth of p. In every possible world, in other words, the truth of any proposition that is in that world an answer to the question, “Why is there something – and not, rather, nothing?” is, as they say, a brute fact – and a brute contingent fact.

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3. No human being – or any other finite creature – could possibly grasp or understand any member of S – including, of course, those members of S that are in the actual world correct answers to the question, “Why is there something – and not, rather, nothing?” And, a fortiori, no human being (or other finite creature) could possibly understand any answer to the question “Why are those propositions – the 510 members of S – all the possible answers to the question, ‘Why is there something – and not, rather, nothing?’” I ask you to consider this story. I ask you to consider it because, if it were true, it would be – at any rate, this seems evident to me – correct to speak of “the mystery of existence”. That is, the truth of this story would be a sufficient condition for the truth of the thesis that “existence is a mystery”. No doubt there are weaker sets of conditions that would also be sufficient for the truth of this thesis. Notice, however, that the story is consistent with the existence of a proof that it is impossible for there to be nothing – even the existence of a proof that human beings can understand. But that bald statement requires qualification, for, of course, the story may be impossible – no doubt it is impossible – and therefore inconsistent with everything, even itself. Now if the story is possible, then, since it logically implies the impossibility of there being nothing, it will be consistent with any sound argument for the impossibility of there being nothing. One should also note that, for all we know, if sound arguments for the impossibility of there being nothing exist, none of infinitely many such arguments to be found in the Platonic heaven has the right features to count as a proof ; or, if the Platonic heaven does contain such proofs, it may be that none of them is humanly accessible. It may therefore be that nothing is consistent with the existence of a proof of the impossibility of there being nothing or that nothing is consistent with the existence of a humanly accessible proof of that proposition. Let us say, therefore, that the story is logically consistent with the existence of a proof of the impossibility of there being nothing: one cannot logically deduce from the story that there is no proof of the impossibility of there being nothing. And notice that every step in both the First Argument and the Second Argument is logically consistent with everything said in the story. If someone were unwise enough to suppose that those two arguments were proofs of the impossibility of there being nothing, this would provide that person with no reason to suppose that the fanciful and elaborate story I have just told was false. The lesson is that a proof of the impossibility of there being nothing (at least a proof that is as abstract as my two arguments) can do nothing to allay anyone’s conviction that existence – the fact that there is something, and not, rather, nothing – is a mystery. I’m not

58 | Peter van Inwagen saying that this fact should – or should not – be regarded as a mystery. I’m saying only that my arguments, however interesting or valuable or metaphysically penetrating someone may take them to be – are simply irrelevant to the question whether existence is a mystery.

Bibliography Chisholm, R. M. (2007), On Metaphysics, Minneapolis: the University of Minnesota Press. Goldschmidt, T. (ed.) (2013), The Puzzle of Existence: Why Is There Something Rather Than Nothing?, London and New York: Routledge. Leslie, J. (1979), Value and Existence, Oxford: Blackwell. Leslie, J. (1989), Universes, London and New York: Routledge. Leslie, J. and Kuhn, R. L. (eds.) (2013), The Mystery of Existence: Why Is There Anything at All?, Oxford: Wiley-Blackwell. Munitz, M. K. (1965), The Mystery of Existence: An Essay in Philosophical Cosmology, New York: Appleton-Century Crofts. Van Inwagen, P. (1996), “Why Is There Anything at All?”, in Proceedings of the Aristotelian Society: 70, 95–110.

Jan Woleński

God and Good: Does God’s Existence Imply that Anything is Good God’s existence is taken in this paper for granted. Although I am an agnostic (in the sense, that, on my view, proofs of God’s existence or non-existence are impossible) are or even atheist (in the sense, that I do not believe in God), I admit that a neutral ontological language is possible in which questions concerning God’s existence can be rationally discussed. Assume that X is a set of statements about God, for instance, accepted by a (rational) theologian (although the problem of concrete denomination is not relevant, I will work in the frameworks of Christian doctrine). There is nothing wrong in presupposing that X is consistent. If so, X has a model, let say, M. Since ‘God’ is a proper name, it has a reference, at least in the standard semantics within the universe of M. Other possibility consists in considering ‘God’ as a part of the predicate ‘is a God’, which refers to a set (possibly empty) as its reference. Note that such semantic settings do not depend whether God really exists or not. On the other hand, they make some discussions on theology and its existential involvement more easier than debates starting from proofs of God’s existence or their criticism. Assume that A is the sentence ‘there is x such that x = God’ (this form is preferable according to first-order syntax) which is true in M and B refers to the sentence ‘there is x such that x is good’ (for simplicity I will frequently use more ordinary locutions). Now, our problem may be framed in ta fairly simple way: 1. Does A logically imply B? My answer to (1) is negative and this paper tries justifies that is should be such. In order to do that, I will analyze possible affirmative answers to (1). The first way to derive B from A is to say that since God is good, there is something good. Clearly, this strategy requires the premise ‘God is good’ as indispensable. Denote it by the letter C. However, C is not obvious per se. Of course, many theologians take C as granted, but it is hard to agree a priori with this opinion.¹ First of all, C has various interpretations depending on an invoked ethical theory, for instance, the virtue theory (Aristotle, Thomas Aquinas), the duty theory (Kant) and the utilitarian theory (Bentham, Mill). Should we take into account intentions

1 See the discussion in Garcia (2009).

60 | Jan Woleński or consequences? In fact, particular interpretations offer different and partially inconsistent accounts of God’s goodness, even if we agree with theologians that goodness belongs to His basic perfections or even constitutes the most fundamental one. The crucial difficulty in this respect consists in relation of God to evil. Is evil a consequence of admitting human freedom by God or perhaps the state of the world stems from God’s intention to maximize goodness? The pessimist with respect to God’s intention can observe that any inspection of the world and what is going in it motivates saying that our reality was created by the Malissimus (the worst power) but not by the best (or even a good) God. To sum up, reasons for accepting C are not compelling and seem arbitrary. There is still one problem with deriving B from A. It seems that common intuitions suggest the conclusion that God’s existence entails that something is good and is different from God. Formally speaking, we need to obtain a conclusion of the form 2. There is x such that x is good and x ̸ = God. Clearly, (2) requires that x in question stands in a relation to God in order to possess the property of being good (I do not enter into the problem of the nature of good as a property; the only important qualification is that goodness is understood as moral or ethical). Although it is fairly possible that x is eternal and it is good because co-exists with God, the most plausible account of the relation involved in (2) consists, at least according to standard theological proposals, in understanding x as created by God. This view will be constantly assumed in my further analysis. Let me note that no particular idea of creation by God matters in this context. One can think about creatio ex nihilo (creation from nothing) or continuous creation in Whitehead’s sense. In order to justify (2), we can eventually employ the theory of transcendental concepts (transcendentalia).² This monumental ontological theory was elaborated in the Middle Ages, particularly by Thomas Aquinas and it is accepted by his contemporary Neo-Thomist followers. The Schoolmen considered the transcendentals as the most general and overcategorial (transcendentalia omnia genera transcendunt) notions. It was (and still is) controversial how many transcendentals there are. Leaving aside truth (verum), thing (res), one (unum) and beauty (pulchrum), we can concentrate on two notions, namely ens (being) and bonum (goodness) as relevant for our problem. The concept of being is decisively distin-

2 This theory is extensively presented in Aertsen (1996). See also Woleński (1997): 358–370 and Woleński (2004) (or Woleński (2011): 43–50); the second paper is employed in the present essay.

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guished in every version of the theory of transcendentals (TT, for brevity) and other transcendental are compared with it. TT is based on the following general principle: 3. if T and T’ and are transcendentals, they are mutually convertible. Its particular instance is captured by the statement (in Latin): 4. Ens et bonum convertuntur. This last thesis means that being and good are co-extensional. On the other hand, they are not co-intensional. Roughly speaking, bonum in this context expresses a property of being, namely that it is good. Although transcendentals are concepts, they are not universals (denominations of genera). Assume that P is an universal. It entails, that non-P is universal as well. In general, if P and non-P are universals, there exists an universal U such that U is a genus, but P and non-P are its species. This property allows to define universals by genus proximum et differentiam specificam (for instance, man (or Man) is a rational animal). This strategy is not available in the case of transcendentals, because they are not subordinate to more general concepts. Consequently, we cannot define ens and bonum by invoking a property functioning as a differentia specifica. The application of (4) to the discussed problem seems to be straightforward, but it is not such. The reason is that the set X of assertions {God exists, God created being, created being is different from God, being is good} does not entail (2), because God’s existence is a necessary condition of creation, but not sufficient. In order to complete the entire reasoning, one should assume that God had to create being. However, it is a very controversial theological issue. For instance, Catholicism accepts that God’s creation was (I ignore the problem whether a temporal, or rather atemporal aspect of God’s activities can be rendered by the past tense) an absolutely free act. However, there are other alternatives, for instance, that God was necessitated to create contingent being by His very essence or at least motivated to execute the best possibility (the best possible world).³ If God’s existence constitutes a necessary condition of being, then we obtain only the statement that good being could be impossible without God. Yet we have a model with God, but without being different from Him, for instance, instantiated by Spinoza’s pantheism. The adding the premise that creation was necessary, solves the problem, because (2) logically follows from X + {God necessary created being different from

3 This problem is extensively discussed by B. Leftow in Leftow (2012).

62 | Jan Woleński Him}. Other alternatives introduce complications. In particular, if creation by God was absolutely free, it is unclear why God decided to create being. If one say that we should not ask such a question, our theology becomes not quite rational, but if we decide that God was necessitated or motivated, the question remains open and requires an additional explanation, for instance, by adding the premise that God’s goodness as His inherent perfection contributed to the act of creation. Anyway, TT in its classical setting is not sufficient for justifying (2) by God’s existence. TT leads to other problems. Consider the status of evil (malum). Clearly, having the property of being wrong is something opposite to goodness (having the property of being good). Consequently, if ens et bonum convertuntur, then malum is negative (it is convertible with non-being). This means that malum does not exist as being not a being at all. Observe, however, that ‘the opposite’ can be conceived understood either as a negativum (to not white with respect to be white) or as a privativum (to be blind with respect to be have no vision; for instance, vegetable are neither blind nor non-blind). Due to the relative character of negativa and privative, one can say privativa consist in lacking of respective positive properties. Logically speaking, if P is a positivum and P’ acts as its functions as its negativum, they are mutually contradictories, but, on the other hand, if both are related as positiva and privative, they are contraries. The standard way of relating bonum and malum in TT is captured by 5. Malum is the privativum with respect to bonum. This thesis presents the basic content of the theory of evil as a privativum (or the negative theory of malum): evil is the lack of goodness, although it is not only non-goodness. The negative (as privative) theory of evil assumes that we have an universe of items, for instance, the class of human actions, which can be qualified as good or wrong. The statement (5) plays an important role in theology, because it serves to justify the view that God is no responsible for evil, because malum does not exist and, thereby, could not be created by Him. However, this defence of God’s innocence with respect to malum seems to require that goodness and evil are understood metaphysically, not ethically.⁴ Clearly, if (4) holds, malum must be identified (this word is used with some exaggeration here) with nothingness. In fact, the School-

4 Yet we obtain a strange consequence if the creatio ex nihilo is assumed. Since nothing (notbeing) is a malum or even the malum, God has created bonum (= ens) from malum. Although God is omnipotent, this result seems ridiculous, particularly if bonum and malum are understood ethically.

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men distinguished (the same concerns malum) methaphysical bonum (ens as contrasted with non-ens), physical bonum (for instance, having vision as contrasted with being blind with respect to creatures which can use eyes at all) and ethical bonum (for instance, to be just as contrasted to be unjust). There is no problem if the principle (4) is applied to metaphysical or physical bonum and malum (or saying more carefully, one can defend this thesis as ontologically sound). On the other hand, (4) is much less satisfactory when regulates the use bonum and malum as ethical properties. In fact, Thomas Aquinas, perhaps guided by some difficulties (see below), has offered also another approach to bonum and malum, which was based on the idea of proper (right) desire (appetitus). The main point is expressed by 6. x is good if and only if it is an object of a proper desire. This definition taken together with TT implies that malum ethicum is a lack of human desires consisting in aiming at what they should not do (I do not enter into the problem of what is the proper desire). However, the lack of a proper desire does not imply that no other desire is lacking as well. Now we have the following question: are malum in the sense of (5) and malum in the sense (6) coextensive? The answer “no” seems correct, because if something is a bonum metaphysicum, it not automatically an object of honest desire. A more general explanation points out that bonum ethicum and malum ethicum behave as modal predicates or operators (in the broad sense, including alethic, deontic, axiological, expressing various propositional attitudes, etc. modalities).⁵ Consequently, the forms ‘Q is good’ and ‘Q is wrong’ should be understand respectively as ‘it is good that Q’ and ‘it is wrong that Q’ (I consider the letter Q as standing for a proposition, for instance, ‘it is good that people tell truth’; in the case of the predicative version, that is, ‘Q is good (wrong)’ the letter Q stands for a reified sentence, for instance ‘telling truth’). If we consider modal instances for bonum and malum, we have the following mutually exclusive possibilities (I use predicative forms and disregard instantiations of the excluded middle, like ‘everything is P or it not the case that everything is P’): 7. Everything is good and reversely (strong moral optimism, see (5)); 8. Everything is wrong and reversely (strong moral pessimism); 9. Everything is good or wrong (moral dualism);

5 This treatment is similar to Duns Scotus’ theory of disjunctive transcendentalia. For Scotus, not all transcendental are co-extensional with ens.

64 | Jan Woleński 10. 11. 12. 13.

Everything is neither good not wrong (morally neutral) (non-cognitivism); Everything is good or neutral (weak moral optimism); Everything is wrong or neutral (weak moral pessimism); Everything is good, wrong or neutral.

The positions (7), (8), (9), (11) and (12) seem counterintuitive, although (9) has actually occurred in the history as Manicheism. The assertion (10) is coherent with all views, relatively frequent in moral philosophy, that axiological predicated are grounded in feeling, emotions, etc., that is, non-cognitive mental acts. (13) is presumably the most coherent with ordinary intuitions. Note that (13) is the only formula being a logical truth of extended modal logic (see Logical Appendix below for other logical principles of this kind). A more detailed discussion of (7) – (13) would require a closer analysis of ‘good’ and ‘wrong’ as axiological predicates, particularly deciding whether they refer to genuine properties of beings or function as projections of subjective (for instance, emotional) mental states. Since I already noted (see above) that discussing of this issue exceeds this paper, I abstain from further remarks about the semantic status of ‘good’ and ‘wrong’. Fortunately, we can easily see that (7) – (13) are consistent with the statement ‘God exists’. God could create (a) only goodness; (b) only evil; (c) only goodness or evil neutral item; (d) only neutral items; (e) only good or neutral items; (f) only wrong or neutral items; (g) good items, wrong items or neutral items). If someone is a theist, he or she may add that God’s existence is a necessary condition of which possibility from (7) to (13) obtains. On the other hand, God’s existence does not constitute a sufficient condition. Although (7), (9), (11) and (13) state that something is good and different from God, that is, generate models in which (2) is true, His existence does not entail (2). In order to construct a valid inference ending with (2) as the conclusion, we must add some additional premises, for instance, asserting God’s moral perfection, the necessity of creation, etc. Moreover, atheism (understood as denying God’s existence) and agnosticism (understood as suspending the view whether God exists or not) are also perfectly consistent with the listed positions numbered as (7) – (13). This shows that TT does not solve the problem of logically following (2) from the assertion that God exists. In general, modal analysis supports only that being a bonum, a malum or an axiological neutrality implies being an ens. And this conclusion is too weak in order to solve anything in moral philosophy related to God’s existence. The last question to be discussed concerns the Hume thesis (HT) as applied to normative and evaluative statements.⁶ Let the formula •Q represents a modal

6 See Woleński (2006) or Woleński (2011): 155–162.

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normative or evaluative sentence, where • is a normative (like ‘it is obligatory, permitted, indifferent’, etc.) or evaluative (like ‘it is good, wrong, neutral’, etc.) operator, and Q is a sentence without normative or evaluative ingredients (a purely descriptive sentence). HT says that Q does not logically entail •Q. HT additionally supports that (2) is not entailed by the sentence ‘God exists’, because the former is evaluative in its part ‘Something is good’, but the latter is purely descriptive. Thus, in order to obtain (2) from God’s existence, one needs add some evaluative premises, for instance, ‘God is good’ or ‘God creates good beings’, etc. Yet such supplements can be insufficient (see above) for obtaining the required conclusion, that is, the statement (2). The reason is constantly the same: the additions can constitute a necessary condition, but not sufficient one. HT also shows why the famous Dostoyevsky’s reasoning (from The Brothers Karamazov) captured by the conditional ‘if there is no God, everything is permitted’ is incorrect. It is so, because its antecedent has a purely descriptive content, but its consequent is normative. In order to ground Dostoyevsky’s claim, one should assume that God is the highest normative authority and His orders must be obeyed. My general conclusion is there is no way to infer any evaluations or norms from God’s existence as the only premise – the opposite view is committed to the naturalistic fallacy.

A Formal Appendix The modal character of bonum and malum suggests the logical diagram (D) (see the next page). This diagram (it is a generalized logical square for modalities) displays several formal relations between sentences: α – it is good that Q, β – it is wrong that Q, γ – it is not wrong that Q, δ – it is not good that non-Q), ν – is it is good that Q or it is wrong that Q, µ – it is not good that Q and it is not wrong that Q (it is neither good nor wrong that Q; it is indifferent (neutral) that Q), κ – Q, λ – non-Q). Good and wrong are considered here as axiological modalities, which have analogical logic to deontic logic. In particular, we have the following dependencies (I neglect here reductions via interdefinability, for example ‘it is wrong that Q’ is equivalent to ‘it is not good that non-Q’; ⊢ – the symbol of provability): 14. 15. 16. 17. 18.

⊢ ¬(α ∧ β); ⊢ (α ⇒ γ); ⊢ (β ⇒ δ); ¬ ⊢ (κ ⇒ α) (the Hume thesis for goodness); ¬ ⊢ (λ ⇒ β) (the Hume thesis for evil);

66 | Jan Woleński ν

β

α

κ

λ

γ

δ

µ Fig. 1. Diagram (D) 19. 20. 21. 22. 23. 24. 25. 26.

¬ ⊢ (α ⇒ κ) (the converse of the Hume thesis for goodness); ¬ ⊢ (β ⇒ λ) (the converse of the Hume thesis for evil); ⊢ (α ⇔ ¬δ); ⊢ (β ⇔ ¬γ); ⊢ (ν ⇔ ¬µ); ⊢ (µ ⇒ γ); ⊢ (µ ⇒ δ); ⊢ (α ∨ β ∨ γ).

The relations noted in (14)–(26) have their paraphrases in the statements: 27. 28. 29. 30. 31. 32.

No objects is simultaneously good and wrong; If an object is good, it is not wrong; If an object is wrong, it is not good; It is not the case (on logical grounds), that if an object is, it is good; It is not the case (on logical grounds), that if an object is not, it is wrong; It is not the case (on logical grounds), that if an object is good, it is;

God and Good: Does God’s Existence Imply that Anything is Good |

33. 34. 35. 36. 37. 38. 39.

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It is not the case (on logical grounds), that if an object is wrong, it is not; An object is good if and only if it is not wrong; An object is wrong if and only if it is not good; An object is indifferent if and only is neither good nor wrong; If an object is indifferent, it is not good; If an object is indifferent, it is not wrong; Every object is good or wrong or indifferent.

Accepting (D), each of (14)–(24) presents a logical truth on ethical transcendentals. Note that HT is assumed as a logical principle.

Bibliography Aertsen J. (1996), Medieval Philosophy and the Transcendentals. The Case of Thomas Aquinas, Leiden: E. I. Brill. Bilková M. and Tomala O. (eds) (2006), Logica Yearbook 2005, Praha: Filosofia. Garcia L. (2009), “Moral Perfection”, in The Oxford Handbook of Philosophical Theology, edited by T. P. Flint and M. C. Rea, Oxford: Oxford University Press, 217–238. Leftow B. (2012), God and Necessity, Oxford: Oxford University Press. Woleński J. (1997), “Two Theories of Transcendentals”, in Axiomathes: VIII, 367–380. Woleński J. (2004), “Malum, Transcendentalia and Logic”, in Phenomenology and Analysis. Essays on Central European Philosophy, edited by A. Chrudzimski and W. Huemer, Frankfurt am Main: Ontos Verlag, 358–370; reprinted in Woleński (2011): 43–50. Woleński J. (2006), “A Generalization of Hume’s Thesis”, in Philosophia Scientiae: 10, 111–115; reprinted (with additions) in Bilková and Tomala (2006): 215–223. Woleński J. (2011), Essays on Logic and Its Applications in Philosophy, Frankfurt am Main: Peter Lang.

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Part II: Omniscience

Stamatios Gerogiorgakis

Gaps, Gluts and God 1 Gaps and Gluts Since Aristotle, there is probably no harsher criticism of some position, philosophical or other, than to say that it invites a contradiction. The idea behind this criticism is, of course, that, since in classical logic contradictions are by definition false (and nothing but false), then, given the truth-functional definition of implication, a position which implies a contradiction has to be false (and nothing but false) as well. Throughout the history of philosophy, the attempts to revise the Aristotelian idea that all contradictions present false, useless information have been and remain strongly debated. Quite famously, true contradictions have been suggested by G. W. F. Hegel (Hegel (1812/1986): part 1, book 2, section 1, chapter 2, C, 3rd remark) and, quite recently, Graham Priest (Priest (1979)) presented (or, rather, relaunched, since he had some predecessors in this) a Logic of Paradox (LP), a logic system designed to cope with true contradictions. Whereas the truth-functional definitions of the connectives of LP are identical to Kleene’s strong three-valued logic (Ks3 ), there is a huge difference between Ks3 and LP: unlike the interpretation of the intermediate value in Ks3 as neither-true-nor-false, in LP the intermediate value is interpreted as true-and-false, NB a truth-value glut. To Priest’s mind, the value true-and-false is adequately assigned to true contradictions. While the idea that there are true contradictions becomes more and more widespread in logic, differences between the supporters of this idea also emerge. Some of them accept truth-value gluts, i.e., sentences which are true-and-false, but no truth-value gaps, i.e., sentences which are neither-true-nor-false. E.g. Priest’s LP does not account for truth-value gaps. Others scholars, predominantly South-American paraconsistent logicians, accept truth-value gluts along with truth-value gaps.

72 | Stamatios Gerogiorgakis I will call the first group dialetheists in accordance with the usual usage of the term.¹ Following Béziau (Béziau (1999): section 3.5, 11-12), I would like to call the second group paranormalists. Paranormalists argue that the Liar Paradox: P: P is false invites a regress/contradiction (if P is true then it is false because it says so; and if P is false then it is true because it makes the true statement that P is false) which can be solved both by accepting truth-value gaps and truth-value gluts. If we assume that the value of P is a gap, we avoid regress as much as we avoid it if we assume that the value of P is a glut because, in both cases, the assignment of the truth value gap or glut to P does not render the truth value of P into the trueonly/false-only-dichotomy which triggers the regress. However, the Strengthened Liar Paradox: Q: Q is not true can be solved only by accepting truth-value gluts. Assuming that the truth value of Q is a gap triggers an infinite regress: If Q is neither-true-nor-false then it is not true, therefore Q is true. But if Q is true then it is not true because it says so, and so on ad infinitum. But if Q is true-and-false then it is – at least – true and since it says that it is not true, it is false – therefore it is true-and-false. But being false is being not true and this is what Q says of itself. Therefore Q is true – therefore it is trueand-false again. As one sees if Q is true-and-false then it remains true-and-false and no regress can be triggered.

2 Milne’s Paradox Peter Milne (Milne (2007)) formulated a sentence, the Milne Sentence (R) which has paradoxical consequences for the philosophy of religion. R: No omniscient being knows that which the sentence R expresses.

1 The term “dialetheism” gives a hint towards true contradictions. It does not necessarily entail the view that there are no truth-value gaps. According to its technical usage, however, dialetheism is very often taken to mean logical intuitions and philosophical consequences connected with LP, and eo ipso to negate the existence of truth-value gaps.

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Some distinctions have to be made due to the fact that R is ambiguous depending on the understanding of the negation. Like the difference between the sentences ‘It is not the case that there are vampires and that they have compassion’ and ‘Vampires are compassionless’ – the second clearly presupposing the existence of the referent of the subject term, the first clearly not presupposing any existence, there are two different readings of R: 1st reading: the negation in R negates the whole sentence. Clearly, then, if there are no omniscient beings, R is true. However, accepting the First Reading of R while denying the existence of omniscient beings makes no justice to correct English. Normally, sentences of the form: ‘No x has the property Π’ presuppose the existence of at least one x. If there are omniscient beings, this reading of R involves a series of problems (see below). 2nd reading: the negation in R does not negate the whole sentence and presupposes the existence of at least one omniscient being.² If there are no omniscient beings, R under this reading expresses the thought that at least one omniscient being exists (i.e. it presupposes the existence of an omniscient being) who does not know that which R expresses. But since by the hypothesis there are no omniscient beings, R is false. If there are omniscient beings, also this reading of R involves some problems (see below). If there is at least one omniscient being, the following is the case under both readings: if R is true, then no omniscient being knows that which R says because this is what R says. But omniscient beings by definition know everything that any true sentence says. Therefore R is false. But if R is false, then there is at least one omniscient being who knows that which R expresses. But then R is true due to the definition of knowing (i.e., Kp implies p) etc. ad infinitum. Given the existence of omniscient beings, that is, omniscient beings have to accept that R is true-andfalse at the same time: this makes omniscient beings necessarily supporters of true contradictions – Milne says that it makes them dialetheists. Additionally, it implies that existing omniscient beings are omniscient and fail to be omniscient at the same time. Omniscient beings are a contradiction in adjecto – which implies that theists should support true contradictions also. Milne formulated this argument vis-à-vis dialetheism. However, if the regress which is triggered when we reflect on R under the assumption that there is at least one omniscient being, can be avoided in other ways than by accepting truth-value

2 Characteristically, Ulrich Blau (Blau (1977), 49) calls the kind of negation exemplified in the 2nd reading “presupposing”.

74 | Stamatios Gerogiorgakis gluts, like three-valued semantics or truth-value gaps, Milne’s result is unjustified. Milne did not bother to explore this important possibility. Like bombing a shopping mall is too drastic a way to make a pocket thief leave it, accepting that R is a true contradiction is too drastic a way to avoid paradox. It is reasonable to avoid adopting drastic solutions when less drastic ones work equally well. On more detailed investigation, R invites a contradiction if taken together with the following assumptions: A) there is at least one omniscient being; B) R expresses something. R is, under these assumptions, true and false at the same time: if R is true, then R says that there are omniscient beings who fail to know that which the sentence R expresses, therefore it has to be true that omniscient beings fail to know that which the sentence R expresses. But then R is false because everything which an omniscient being does not know is false. But then, R is true again due to the fact that R says itself that R, a false sentence, is not known by an omniscient being and so on... Since the assumption A invites a contradiction when taken in combination with the assumption B, let us negate the assumption B and assume instead that R has nothing to express. R is true then. As one sees, R is a sentence which on different readings of the negation (and different assumptions) can be true or false or true-and-false or have nothing to express. This can make us think to assign R the truth value indeterminate in accordance with to Bas van Fraassen’s supervaluation method. In this case, it is indeterminate whether an omniscient being knows that which R expresses. Being indeterminate, R is not true. But then, R says the truth about itself. But since R says the truth about itself and what R says is that R is not true, i.e., indeterminate, it must be the case that R is not true, i.e., indeterminate etc. We are back in a loop, however in one which does not run from false to true to false to true etc. but from indeterminate to true to indeterminate to true etc. We may feel compelled at this point to super-supervaluate R, to introduce, that is, a fourth value “between” the values: true and indeterminate and to assign it to the iteration instance of R which comes after the first indeterminate iteration instance. Let us call it: half-indeterminate. For analogous reasons a fifth value (say: quarter-indeterminate) has to be introduced between half-indeterminate and true and so on. R becomes less and less

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indeterminate the longer we iterate.³ NB, this is not an infinite regress: as long as you iterate, no truth value assigned to an iteration instance is assigned anew to another. Of course, as Brian Skyrms (Skyrms (1970), 161) pointed out, van Fraassen’s supervaluations entail that whatever we know about the value of indeterminate first-order-liar-sentences has to be indeterminate itself. Is this compatible with the notion of an omniscient being? I think that it is. Even an omniscient being should not be expected to know more on a subject matter than what the subject matter contains. To sum up: One can avoid the regress which Milne saw triggered in R by assigning R a truth-value gap. Once this move towards paranormal logic and away from dialetheism is granted even a theistic supporter of true contradictions who reflects on R is not forced to accept that God is a dialetheist. I gave in Gerogiorgakis (2011) a more elaborated argument for this claim.

3 Strengthening the Milne Sentence However, I shall refrain from celebrating a triumph over Milne here. Like P, R can be strengthened. And this is something which I had not investigated in Gerogiorgakis (2011). Considerations on the Strengthened Milne Sentence: S: No omniscient being knows truly or indeterminately that which the sentence S expresses reveal paradoxical consequences. If S is true and omniscient beings are taken to exist, then they know neither truly nor indeterminately that which S expresses. Therefore, they ignore that which S expresses. But an omniscient being’s ignoring something is an omniscient being’s knowing that this is false. Consequently, that which S expresses is false and, consequently, S is false. But if S is false, then there is at least one omniscient being who knows truly or indeterminately that which S expresses. If truly, then we are back in the loop; if indeterminately, then S is indeterminate. However, the thought that S is indeterminate contradicts to what S says and makes S false – always under the proviso that there is at least one omniscient being. And with this we are back in the loop also by the option that S is indeterminate. A possible way out of the loop would be to take S to be

3 An intuitive grasp of the idea behind this procedure is given in our giving more credibility to people who repeat an information over and over than to people who do not repeat their information. The credibility of the information increases with the number of repetitions.

76 | Stamatios Gerogiorgakis half-indeterminate. This would help to combine the existence of at least one omniscient being with assigning some truth value to S. Our problems with S would be settled then, but (alas!) we would not be able to deal with the Strengthened Milne Sentence: T: No omniscient being knows truly or indeterminately or half-indeterminately that which the sentence T expresses and so on ad infinitum (for a similar point cf. Van Fraassen (1968), 147). If we take S and T and other strengthened sentences of higher order to be are true-and-false instead, embracing thus Milne’s conclusion that omniscient beings are dialetheists, we do avoid paradox.

4 Conclusion Milne produced an argument in order to show that the notion of omniscience can be made intelligible only in a dialetheistic semantic frame. Milne’s reflexions on the Milne Sentence (R) and Milne’s argument on behalf of the claim that omniscient beings are dialetheists, can be refuted with reference to a paranormal semantic frame which accepts truth-value gaps along with truth-value gluts. However, Milne’s conclusion can be defended vis-à-vis strengthened versions of the Milne Sentence.

Bibliography Béziau, J. -Y. (1999), “The Future of Paraconsistent Logic“, in Logical Studies: 7, 1–20. Blau, U. (1977), Die dreiwertige Logik der Sprache, Berlin: Walter de Gruyter. Gerogiorgakis, S. (2011), “Omniscience in Lukasiewicz’s, Kleene’s and Blau’s Three-Valued Logics”, in Polish Journal of Philosophy: 5(1), 59–78. Hegel, G. W. F. (1812/1986), Wissenschaft der Logik, Frankfurt/M.: Suhrkamp. Milne, P. (2007), “Omniscient beings are dialetheists”, in Analysis: 67, 250–251. Priest, G. (1979), “The Logic of Paradox”, in Journal of Philosophical Logic: 8(2), 219–241. Skyrms, B. (1970), “Return of the Liar: Three-Valued Logic and the Concept of Truth”, in American Philosophical Quarterly: 7(2), 153–161. Van Fraassen, B. (1968), “Presupposition, Implication, and Self-Reference”, in The Journal of Philosophy: 65(5), 136–152.

Jason Megill

Fitch’s Paradox and the Existence of an Omniscient Being 1 The Knowability Principle and Fitch’s Paradox The knowability principle, henceforth “KP”, claims that all truths are knowable, i.e., ∀p(p → Kp).¹ Fitch (Fitch (1963)) demonstrated that the conjunction of the KP and the claim that we are not omniscient, i.e., ∃p(p ∧ ¬Kp), entails a contradiction.² Here is a version of his proof.³ First, suppose both the KP and NonOmniscience, ∀p(p → Kp) ∧ ∃ p(p ∧ ¬Kp).

(1)

∃p(p ∧ ¬Kp).

(2)

p ∧ ¬Kp.

(3)

(p ∧ ¬Kp) → K(p ∧ ¬Kp).

(4)

K(p ∧ ¬Kp).

(5)

Given (1), obtain (2) with conjunction elimination,

And now consider a particular instance of (2), a particular unknown truth,

If the KP is true, as we have assumed, we can substitute (3) for p in the KP and obtain a true claim. That is, if we do not know p, it should be possible for us to know that we do not know p; after all, given our assumption (in line (1)), it is possible for us to know anything,

With (3) and (4) infer (5) with conditional elimination,

So, it is possible to know both that p is true and that we do not know p. 1 Throughout, as is customary, “K” is the knowledge operator, which means “it is known by someone at some time that”. 2 The proof was first suggested to Fitch by an anonymous referee. Salerno (Salerno (2009b)) has discovered that the referee was Alonzo Church (Church (2009)). 3 This version of the proof closely follows that given in Brogaard and Salerno (Brogaard and Salerno (2004)).

78 | Jason Megill However, (5) contradicts a claim that one can derive as a theorem, at least given certain plausible assumptions. Suppose, for reductio, that the following is true: K(p ∧ ¬Kp).

(6)

Kp ∧ K¬Kp.

(7)

K¬Kp.

(8)

Given (6), then since knowledge distributes over a conjunction, we can infer,

Use ∧ elimination on (7) to obtain,

Furthermore, we cannot know a falsehood, i.e., knowledge entails truth, so given (8), the following is true, ¬Kp.

(9)

But we can obtain (10) with ∧ elimination on (7): Kp.

(10)

We have a contradiction ((9) and (10)), so our assumption on line (6) is false, ¬K(p ∧ ¬Kp).

(11)

¬K(p ∧ ¬Kp).

(12)

¬K(p ∧ ¬Kp).

(13)

Moreover, (11) is theorem, so it is necessarily true,

And given that ¬¬ is equivalent to , and given double negation elimination, with (12) we can easily obtain,

In short, if we assume both that the KP is true and Non-Omniscience, we obtain a contradiction; to be precise, the assumption that KP is true and that some truth is unknown entails a claim (claim (5)) that contradicts a theorem (claim 13)). So, on the assumption that contradictions are impossible, one faces a choice between three options, Option One: Reject the KP. Option Two: Reject Non-omniscience, i.e., claim that all truths are known.

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Option Three: Reject at least one of the inference rules or principles used in the proof above; e.g., deny that (i) knowledge distributes over a conjunction, or that (ii) knowledge entails truth, (iii) or else revise logic in some manner (e.g., reject conjunction elimination or some other logical inference rule used in the proof). Given that we are not omniscient, and given the plausibility of the principles and logical inference rules used in the proof, option one might appear, at least prima facie, to be the most natural response. However, Fitch’s (Fitch (1963)) proof has received much attention in part because the knowability principle is a fundamental tenet of anti-realism as formulated by, e.g., Dummett in Dummett (1976). Roughly, the anti-realist denies that there are unknowable truths, i.e., she endorses the knowability principle, because such truths would have to be unverifiable and mind-independent in principle. So clearly, a putative proof of the falsity of the KP is of great importance in the realism/anti-realism debate. And there are numerous extant responses to Fitch; these responses are generally motivated by a desire to save the KP, i.e., to avoid option one. For example, Nozick (Nozick (1981)) opts for option three; to be specific, he attempts to refute Fitch’s proof by claiming that knowledge does not distribute over a conjunction. Beall (Beall (2000)) claims that the logic that governs knowledge is paraconsistent, so we cannot infer anything from a contradiction in this context and Fitch’s proof falters. Tennant (Tennant (1997)) argues that the KP should be restricted so that we cannot substitute things that we can prove to be contradictory into it (so line (4) above is blocked). There are further responses to Fitch (see, e.g., Williamson (1982)); for more on various attempts to save the KP from Fitch’s result, see Brogaard and Salerno (2004). But note that these responses are all instances of option three; option two, a rejection of Non-omniscience, is rarely explored.

2 An Argument for an Omniscient Being As noted, there are three possible responses to Fitch’s proof, and most choose either option one, i.e., they reject the KP, or option three, i.e., they reject one of the inference rules or principles used in the proof. Note, however, that classical theists are already committed to option two; given classical theism, there is an om-

80 | Jason Megill niscient being, and so Non-Omniscience is false.⁴ Moreover, one might wonder if Fitch’s paradox can be adapted into an argument for the existence of an omniscient being; after all, there are only three possible responses to the proof, and one of them entails that all truths are known? I now formulate one possible argument; the argument clearly faces some difficulties, but it is arguably better than at least some of the extant arguments for theism that are typically discussed.⁵ There might be numerous ways that Fitch’s proof might be adapted into an argument for theism, but here is one suggestion. Again, there are three possible responses to Fitch’s proof: (1) deny the KP; (2) deny Non-Omniscience; or (3) reject a logical inference rule or principle used in the proof. On pain of contradiction, one must choose one of these responses. And if it could be shown that (2) is the only viable – or at least the best – option, then we must deny Non-Omniscience; i.e., claim that all truths are known. So, any putative argument for theism based on Fitch’s proof cannot endorse (3) (for otherwise the atheist could then simply response (2), in which case not all truths will be known); i.e., any putative argument for theism based on Fitch’s proof will need to be carried out in a logical system of sufficient strength, and specifically, a system that contains all of the logical inference rules and principles used in Fitch’s proof. A fairly weak modal epistemic logic can suffice; we need to assume a system that contains (i) first-order logic, along with (ii) modal system K, the weakest modal system, and an (iii) epistemic logic system, system T, in which knowledge distributes over a conjunction. Here is a list of the relevant inference rules and principles used in Fitch’s proof: (a) conjunction elimination (to obtain lines (2) and (10)); (b) existential instantiation (to obtain line (3)), (c) universal elimination (to obtain (4)); (d) conditional elimination (to infer (5)); (e) knowledge distributes over a conjunction (to obtain (7)); (f) knowledge entails truth (to infer (9)); (g) negation introduction (to infer (11)); (h) theorems are necessarily true (to infer (12)); and (i) the modal operator equivalence rule and (j) negation elimination (to infer (13)).⁶ Recall that we have assumed a particular logical system: a relatively weak modal epistemic logic. And note that all of these various inference rules and principles will hold in that system: (a), (b), (c), (d), (g) and (j) will hold in first-order logic; (e) and (f) will hold in

4 To clarify, the theist could of course also reject the KP and revise logic if she wished; it is simply that she must reject Non-Omniscience because she believes that all truths are known, by God. 5 Note that the argument only tries to show that there is an omniscient being. It does not, for example, try to establish that any particular God exists (e.g., the Christian God). Of course, even so, the claim that there is an omniscient being would still hold some interest for philosophers of religion. 6 Of course, some of the steps in Fitch’s proof ((1) and (6)) are assumptions and so do not depend on the use of any inference rule; these are the missing steps above.

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an epistemic logic system that contains T and allows knowledge to distribute over a conjunction; and (h) and (i) will hold in modal system K. So, given the logical system we have assumed (and need to assume, if we are to construct an argument for theism from considerations involving Fitch’s paradox), (3) will not be an option. And, of course, (1) will not be an option if the KP is true. Therefore, given a particular logical system comprised of first-order logic, K, and a relatively weak epistemic logic, and the KP, (2) is false, i.e., Omniscience. All truths would be known. However, there is an immediate problem with the argument: it is not yet an argument for theism. Saying that all truths are known simply means that for any particular truth, someone (or something) knows it. This is consistent with numerous possibilities; e.g., there is a single omniscient being; there are two beings that are each “half-omniscient” (one of the beings knows half of all truths and the other being knows the other half; so between the two of them, all truths are known); there is a being that knows all and only those truths that are not currently known by humans (so the combination of us with this being entails that all truths are known); and so on. Thus, claiming that all truths are known is consistent with theism, but also with numerous other non-theistic possibilities. Therefore, it appears that the argument must be augmented by an additional claim if it is to provide evidence for theism: one must show that given that all truths are known, the most plausible possibility is that there is a single being that knows all truths. So, the following three claims entail that there is an omniscient being: (a) a fairly basic modal epistemic logic is sound; (b) the KP is true; and (c) if all truths are known, then there is an omniscient being. Frankly, I do not know if (a) – (c) are true, and obviously an atheist might deny any one of these claims. However, in my opinion, these claims are not obviously false either; indeed, I suggest that they are more plausible than they might initially appear. Consider (a), for instance. The atheist can argue that the argument only succeeds in a particular logical system. But isn’t that true of any argument? All arguments make inferences (or at least one inference), and so must be carried out in some logical system or other, even if only implicitly. And the relevant system is indeed fairly weak. Its foundation is classical logic. And K is uncontroversial; given that K is the foundation of the other major modal systems, if one rejects K, one essentially rejects modal logic itself. Moreover, epistemic logic system T is also fairly weak; the only two axioms of T are (1) if we know a conditional, and we know the antecedent, then we know the consequent, and (2) knowledge entails truth. Or the atheist might deny (c). However, I suggest that, upon closer inspection, (c) is more plausible than it might initially appear. Again, (c) claims that if all truths are known, then there is an omniscient being. So, posit an atheist that rejects the argument by only rejecting (c); this atheist grants (a) and (b), but de-

82 | Jason Megill nies (c).⁷ Given (a) and (b), each individual truth is known by at least one knower; indeed, given (a) and (b), all truths are known right now, have always been known (and so were known, e.g., instantly after the Big Bang), and will always be known; if the entailment from (a) and (b) to (c) holds now, it has always held; there is nothing about the entailment that restricts it to only holding in the present moment.⁸ Granted, again, this does not entail that a single being knows all truths; it might be that a knows truth t but not t1 , but b knows t1 but not t, and so on, although for all truths t, t will be known by at least one agent. But it is difficult to see how all truths are (or have been or will be) known at all times without positing an entity that is not “naturalistically acceptable”. For example, how, given naturalism, could all truths have been known an instant after the Big Bang? Naturalism tells us life could not yet have been formed at that moment. Likewise, right now, all truths would be known; but how could any naturalistically acceptable being (or collection of beings) know – right now – the number of particles in the universe at some time t after the big bang, or the exact number of dinosaurs who lived on Earth, or every knowable mathematical truth? Humans don’t know these things; nothing else on Earth does; and even if we desperately posit super intelligent aliens, it is difficult to imagine how they could know these things either.⁹ In short, it appears that if all truths are (and always have been and always will be) known by at least one entity, at least sometimes (e.g., right after the Big Bang), some of these truths (e.g., the number of particles in the universe) must be known by a being that is not naturalistically acceptable. The atheist might object that perhaps there is a team of omniscient beings echoing Hume’s (Hume (1779/1980)) claim that perhaps there was a team of intelligent designers), or perhaps there is a team of beings who are not individually Omniscient but jointly are? But if we are forced to posit at least one naturalistically unacceptable entity, but not forced to posit more than one,

7 Of course, an atheist can deny more than one of (a), (b) and (c), but simply rejecting (c) as opposed to (c) plus either (a) or (b) is a weaker response, and hence has a higher probability of being true. Basically, the probability of ¬(c) is higher than the probability of (¬(c) ∧ ¬(a ∨ b)). 8 In case this is unclear, if we are granting (a), then Fitch’s proof will be valid. So the conjunction of the KP and Non-Omniscience will entail a contradiction. So, one of them must be false. And if we are granting the KP, Non-Omniscience must be false. Therefore, given (a) and (b), Omniscience will be true, i.e., all truths are known. And Fitch’s proof is “timeless”, in the sense that nothing about the argument is “time-indexed” etc.; if the proof holds now, it has always and will always hold. 9 I know this is getting bizarre, but that’s my point. Given (a) and (b), a denial of (c) is very bizarre. Prima facie, (c) looks a lot like some other principles used in arguments for theism, i.e., it looks like it can easily be denied by an atheist (who can either argue that the claim is outright false, or is at least not conclusively established as true); however, upon further examination, a denial of (c) is completely untenable.

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then why posit more than one?¹⁰ In short, if all truths are known, then it appears plausible that there is a single, non-naturalistically acceptable entity that knows all truths; i.e., (c) is true. Perhaps rejecting (c) is not the best strategy for an atheist; perhaps the best strategy is rejecting (b), i.e., deny the KP? I now discuss the KP.

3 How Plausible is the Knowability Principle? The atheist who does not wish to revise logic and agrees that it is difficult to see how all truths could be known (at any moment in the history of the universe) in a naturalistic framework might simply deny that the KP is true. Indeed, the KP is controversial. Frankly, I do not know if the KP is true or false; but in this section, I attempt to gauge its plausibility. Generally, when confronted with a putative principle such as the KP, one might falsify it by producing a counterexample. The KP claims that all truths are knowable; so the opponent of the KP might produce a claim that is true yet unknowable. But this is impossible. Suppose that we know that we have a successful counterexample to the KP; there is a claim that we know is true but unknowable; call this claim, whatever it might be, “c”. That is, K(c ∧ ¬Kc).

(1)

Kc ∧ K¬Kc.

(2)

Kc.

(3)

But since knowledge distributes over a conjunction, we can infer,

And given (2) and conjunction elimination, we can infer,

10 One can of course appeal to Occam’s Razor here. Indeed, one can also appeal to basic probability theory: the probability that one strange, naturalistically unacceptable entity exists is greater than the probability that two or more strange entities exist.

84 | Jason Megill And, K¬Kc.

(4)

¬Kc.

(5)

¬Kc.

(6)

But if we know that it is not possible to know c, as (4) claims, then since knowledge entails truth,

And of course, if it is not possible to know c, as (5) claims, then we do not know it:

But line (3) claims that we do know c; we have a contradiction; therefore our assumption on line (1) is false, i.e., ¬K(c ∧ ¬Kc).

(7)

Therefore, we cannot know that we have a successful counterexample to the KP, for if we assume that we know that some given claim is a counterexample to it, a contradiction results. However, perhaps we can know that the KP is false without producing a particular counterexample to it? Perhaps, but consider the following argument. Suppose that we know that the KP is false although we cannot produce a counterexample; even though we cannot produce a particular example of a true but unknowable proposition, we (somehow) know that they exist. But then we would know, for example, that there is a set S of propositions that are true yet unknowable, whatever the cardinality of this set might be. Moreover, there appears to be no reason why we could not single out a particular member of this set and assign it a name. Consider the following “procedure”: (i) think of a particular set; (ii) pick out or isolate one and only one member from that set by, e.g., using a definite description (or definite descriptions); (iii) give a name to that member of the set; and finally, (iv) make true statements about that set member, even if our knowledge of that set member is severely limited (e.g., even if we could not recognize that set member if it were right in front us). Here is an example: I walk into a large science fair that has a couple of hundred entries. There will be only one winner and this person will receive a $1,000 scholarship to help pay for college. I consider the set of all science fair entrants. I then use the definite description “the winner of this science fair” to pick out a single member of this set. I give this set member a generic name, e.g., “Winner”. I can then say some true things about this person; e.g., “Winner will receive a $1,000”. This statement is true even assuming that

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I know very little (or even nothing) else about Winner. For instance, it is true even if I could not point at Winner because the winner has not even been chosen yet. To offer a different example, think of Jack the Ripper. A killer was singled out of a set (say, the population of London) with definite descriptions, e.g., “the person who committed these murders”. The person was given a name, “Jack the Ripper”. Finally, true statements were made about Jack the Ripper, for instance, “Jack the Ripper killed again last night”. These statements were true despite the fact that no one (at least to our knowledge) aside from Jack the Ripper knew who Jack the Ripper was. In short, at least in some cases, we can pick out a unique member of a set by using definite descriptions, give this set member a name and then use this name to make true statements about the set member. So consider various ways we might pick out one and only one of these unknowable truths in set S with definite descriptions. For example, assuming that (at least some of) these truths could be stated in some natural language, consider “the truth that would be first alphabetically” in that language. There are various ways in which one could use a definite description to pick out a specific truth in S. And we can also give this truth a name. Let’s call this truth, whatever it might be, ”a”. Even though we know little about a, for example, we cannot state a, and we do not know the content of a and so on, we can make at least a couple of true statements involving a. Indeed, we do know at least two things about a. First, we know that a is true; by hypothesis, a is a member of the set S of unknowable truths. Second, we know that a is unknowable; by hypothesis, a is a member of the set S of unknowable truths. That is, we know that a is an unknowable truth, K(a ∧ ¬Ka).

(8)

Ka ∧ K¬Ka.

(9)

K¬Ka.

(10)

¬Ka.

(11)

Ka.

(12)

Since knowledge distributes over a conjunction, given (8) we can infer,

With ∧ elimination and (9) we can infer,

And since knowledge entails truth, (10) entails,

But with ∧ elimination, (9) also gives us,

86 | Jason Megill Given (12) and modal theorem T1 (if p, then it is possible that p), we can infer, Ka.¹¹

(13)

But (11) and (13) contradict one another, so our assumption must be false. We cannot know that the KP principle is false; the KP might be false, but even assuming that it is, we cannot know that.¹² Of course, the atheist can still object that even assuming that we cannot produce a particular counterexample to the KP, or even assuming that we cannot know that the KP is false, the KP still might be false nevertheless. We need some positive reason to believe that the KP is true. Again, theKP claims that ∀p(p → Kp). That is, for any truth p, it is possible to know p; for all p, there is a logically possible world in which p is known. So, if the KP is false, then there is a truth that is logically impossible to know. But suppose, as some do, that consistency entails logical possibility; see, e.g., Szabo-Gendler and Hawthorne (2002), “On a standard sort of characterization, P is logically possible just in case no contradiction can be proved from P using the standard rules of deductive inference ...”. Note that, if we are given p, and we can consistently know p, then if consistency entails possibility, then we can possibly know p. So, if the KP is false, and if consistency entails possibility, then there must be a truth p such that, given p, Kp generates 11 See Hughes and Cresswell (1996): 42, for a derivation of T1, though the theorem is obviously true. A denial of the theorem amounts to the claim that the impossible can be actual, which is incoherent. 12 Here is a more systematic version of the argument. (1) Assume that we know that the KP is false. Assumption for reductio. (2) If we know that the KP is false, then there is a set (call it “S”) of unknowable truths. This is obvious; if we know that the KP is false, then it is false, but then there is at least one unknowable truth, and so there is a set of unknowable truths. (3) There is a set (call it “S”) of unknowable truths. This follows with Modus Ponens and (1) and (2). (4) We can use a procedure to pick out one and only one member of this set and give it a name; call it “a”. We do similar things all of the time, e.g., when referring to Jack the Ripper. (5) K(a ∧ ¬Ka). We know this is true because we know that a is in S; so we know, by hypothesis, that a is true and unknowable. (6) Ka ∧ K¬Ka. From (5), given that knowledge distributes over a conjunction. (7) K¬Ka. From (6) with conjunction elimination. (8) ¬Ka. From (7), given that knowledge entails truth. (9) Ka. From (6) with conjunction elimination. (10) Ka. From (9), with modal axiom T1. (11) We have a contradiction (lines (8) and (10)). So, our assumption is false, i.e., (12) We do not know that the KP is false.

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a contradiction (for if it does not, then we possibly know p, and p cannot be an unknowable truth and so cannot falsify the KP). But consider any random truth p. Given p, there is no way to deduce a contradiction if we are also (only) given Kp. There are only so many ways p and Kp could produce a contradiction: (i) p could entail ¬p; (ii) p could entail ¬Kp; (iii) Kp could entail ¬p; (iv) Kp could entail ¬Kp; or either (v) (p ∧ Kp), (vi) p or (vii) Kp could entail ¬(p ∧ Kp).¹³ But none of these putative logical entailments hold. If (i) held, then there are no truths, for any truth would entail its own falsity. And if (ii) held, then a truth would entail that we do not know it, so there would be no knowledge. Likewise, if (iii) is true, then if we know a truth, then it is false, which is absurd; if (iv) is true, then if we know something, then we would not know it, so again, there would be no knowledge. If (v) holds, then if we know a truth then it is not the case that we know it, so again, there would be no knowledge. And if (vi) holds, then if something is true, then either it is not true or else we do not know it; there would either be no truth or no knowledge. Likewise, if (vii) holds, there would either be no truth or no knowledge. So, for any truth p, if we are given p, we will not be able to logically infer a contradiction if we are also (only) given Kp; so p and Kp will be logically consistent; and if consistency entails logical possibility, then given (any) p, it will be logically possible for us to know that p. That is, the KP is true.

4 Concluding Remarks I argued that the following three claims entail that there is an omniscient being: (a) a fairly basic modal epistemic logic is sound; (b) the KP is true; and (c) if all truths are known, then there is an omniscient being. So we basically have an argument for the existence of an omniscient being. I do not know if the argument is sound, though I did defend (b) and (c), and the logical system in question is not particularly strong. However, although the argument is not conclusive as it stands, I will make the following modest claim in its favor: the argument is stronger, at least in certain respects, than some extant arguments for theism that receive attention, so perhaps it deserves attention as well? While the argument depends upon the truth of the KP and the claim that if all truths are known, there is an omniscient being, these claims appear more plausible than some of the principles or claims used in other extant arguments for theism, e.g., the principle of sufficient

13 These are the only possible options. Given only p and Kp, there are only so many possible ways one can generate a contradiction. I believe the above options exhaust all possibilities.

88 | Jason Megill reason, or the claim that the existence of morality depends upon the existence of God etc.¹⁴ And again, the atheist might reject the logical system used in the proof; the atheist might reject classical logic, the weak epistemic logical system T, or the weakest modal system K. Note, however, that the system used in the proof is clearly weaker than the systems used in some other extant arguments for theism; e.g., modal system K is much weaker than S5, which is used in some contemporary modal ontological arguments. So, while the argument for theism given above might not be conclusive, it is arguably more plausible than some other extant arguments for theism, at least in certain respects. Theists should further explore the possibility that Fitch’s proof can be adapted into an argument for theism.

Bibliography Beall, J. C. (2000), “Fitch’s Proof, Verificationism, and the Knower Paradox”, in Australasian Journal of Philosopohy: 78, 241–247. Brogaard, B. and Salerno, J. (2004), “Fitch’s Paradox of Knowability”, in The Stanford Encyclopedia of Philosophy (Summer 2004 Edition), edited by E. N. Zalta, URL = . Church, A. (2009), “Referee Reports on Fitch’s “A Definition of Value””, in Salerno (2009a), 13–20. Dummett, M. (1976), “What is a Theory of Meaning? (II)”, in Truth and Meaning, edited by G. Evans and J. McDowell, Clarendon Press. Fitch, F. (1963), “A Logical Analysis of Some Value Concepts”, in The Journal of Symbolic Logic: 28, 135–142. Hughes, G. and Cresswell, M. (1996), A New Introduction to Modal Logic, London: Routledge. Hume, D. (1779/1980), Dialogues Concerning Natural Religion, Indianapolis: Hackett. Nozick, R. (1981), Philosophical Explanations, Cambridge, MASS: Harvard University Press. Salerno, J. (ed.) (2009a), New Essays on the Knowability Paradox, Oxford: Oxford University Press. Salerno, J. (2009b), “Knowability Noir: 1945–1963”, in Salerno (2009a), 29–48. Szabo-Gendler, T. and Hawthorne, J. (eds.)(2002), Conceivability and Possibility, Oxford: Oxford University Press. Tennant, N. (1997), The Taming of the True, Oxford: Oxford University Press. Williamson, T. (1982), “Intuitionism Disproved?”, in Analysis: 42, 203–207.

14 Of course, the principle of sufficient reason is often used in Cosmological Arguments.

Elisa Paganini

Vagueness and Omniscience 1 Introduction More or less ten years ago, at a summer school in Crans-Montana, I was asked to formulate a good philosophical question to pose an omniscient being. The idea was to pose an omniscient being a question whose answer is relevant for a current philosophical debate. The request is a difficult one. Ten years later, during my presentation at the God Oriented Conference in Warsaw, C. Anthony Anderson advanced a witty and clever suggestion. He suggested asking: “What is the best question I can ask you? And what is the correct answer to it?” He also suggested the answer an omniscient being would give, i. e., “Your question is the best question you can ask an omniscient being like me and the correct answer is the one I am actually giving you”. I admit that Anthony Anderson’s suggestion is good and funny but I do not see how such an answer would be relevant to any current philosophical debate. I understand instead that Anthony Anderson is a more experienced philosopher than I am, that he immediately realized that the request to come up with a question for an omniscient being is an insidious one to which he found an elegant way out. When I was challenged to provide a good philosophical question for an omniscient being, I was instead rather naïve and made a suggestion I now see as inadequate. I am going to present my suggestion and the reason I now find it inadequate. In a sense it is an account of failure but I believe that the failure is instructive (at least it has been instructive to me) and therefore worth considering. My suggestion was to investigate whether an omniscient being could find a boundary in a sorites series. At the time I made my suggestion, I had already started to work on vagueness and I was aware that not all philosophers working on vagueness agreed on the boundaryless extension of vague predicates. Some philosophers deny that the extension of vague predicates is boundaryless,¹ others instead allow boundaryless extensions.² I thought that if it were possible to investigate whether an omniscient being could find a boundary in a sorites series, this would show whether there actually exists a boundary in the sorites series and

1 See, for example, Williamson (1994) and Soames (2003). 2 See, for example, Fine (1975) and Keefe (2000).

90 | Elisa Paganini would be relevant for the current philosophical debate on vagueness, showing at least that some philosophers should revise their position. As I anticipated, I now think my suggestion was inadequate. The reason is that, at least if a semantic theory of vagueness like supervaluationism were the correct one, an omniscient being could not be cooperative. And this result shows that my suggestion would be ineffective: as long as there is no way to establish in advance whether the semantic theory of vagueness is correct or not, there is no way to know whether an omniscient being could be cooperative; and therefore it is not a good idea to use the omniscient being as a test, there being at least the doubt that she would not be able to be cooperative. The following sections are organized as follows. First, supposing that a semantic theory of vagueness like supervaluationism is correct, I present a way to characterize cooperation by an omniscient being (§2) and I explain why it cannot be adequate (§3); I then present a second way to characterize cooperation by an omniscient being under the same assumption as before (§4) and I explain not only that it is inadequate but that the inadequacy depends on the fact that there is not an exhaustive definition of cooperation by an omniscient being (§5). I therefore conclude that an omniscient being may be non-cooperative if a semantic theory of vagueness is correct and I highlight some philosophically problematic consequences of this result (§6).

2 Cooperation by an Omniscient Being: First Proposal I am now going to present a way to characterize the cooperation by an omniscient being supposing that a semantic theory of vagueness like supervaluationism is the correct one. This characterization is very similar to one presented by John Hawthorne (Hawthorne (2005)) with some variations suggested to me by Peter van Inwagen while I was at the God Oriented Conference in Warsaw.³ Let us suppose that we have in front of us an entire sorites series of men, the first one is two meters tall, the last is one meter tall and there is a very small difference in height between each man and the following one. And imagine that we want to verify whether an omniscient being is able to find the last tall man in the series.

3 For a discussion strictly concerned with Hawthorne’s (Hawthorne (2005)) argument see Paganini (2012).

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Starting from the two-meter tall man we point to each man in the series in succession and for each of them we ask the omniscient being to adopt the following rule allegedly describing cooperation: R. The omniscient being raises her arm iff the man under consideration is tall. Suppose, as supervaluationists do, that there are indeterminate cases of tallness, i. e., men such that it is nether super-true nor super-false that they are tall; suppose moreover that there are doubly indeterminate cases for tallness, i. e., men such that it is neither super-true that it is super-true that they are tall nor supertrue that it is not super-true that they are tall; and suppose that indeterminate cases may be increased ad infinitum. We can expect from this definition that, so long as we go through the sorites series starting from the two-meter tall man, the arm of the omniscient being would lower imperceptibly until she would not raise her arm at all. This definition of cooperation is clearly intended to support the boundaryless extension of vague predicates and to show that as long as there is no precise boundary in a sorites series, there is no precise boundary in the cooperative performances of the omniscient being either. I believe, instead, that the definition of cooperation (R) is not adequate and I am going to explain my reasons in the next section.

3 Cooperation by an Omniscient Being: Objection to the First Proposal Let us suppose that we are confronted with a man who is an indeterminate case for tallness: it is super-true that it is neither super-true nor super-false that the man is tall. Let us call him Tom. And let us imagine pointing to Tom and expecting the reaction of the cooperative omniscient being. If the omniscient being follows (R), her arm gesture would be such that it would be neither super-true nor super-false that she raises her arm. Let us now consider whether this behaviour would be cooperative. By definition an omniscient being knows everything that is super-true. Therefore she knows that Tom is a borderline case for tallness. Even assuming that we do not expect the omniscient being to talk to us, we may expect that her cooperative behaviour will be such that when she is confronted with a definite borderline case, i. e., a case definitely distinct from a clear case of tallness, her behaviour is clearly different from that adopted before; for example we may expect her not to raise her

92 | Elisa Paganini arm in such a case, so that we can understand that this is a case that is definitely different from a definite case of tallness. Bearing this example in mind, let us reconsider (R). In such a case we expect a cooperative omniscient being not to raise her arm when it is neither supertrue nor super-false that Tom is tall. In such a case, the bi-conditional (R) would not be super-true even for supervaluationist standards: the left-hand side of the bi-conditional would be super-false, while the right-hand side would be neither super-true nor super-false. It follows that (R) cannot be the correct definition of cooperation.

4 Cooperation by an Omniscient Being: Second Proposal It may be believed that we could improve the previous definition of cooperation in the following way: let us ask the omniscient being to raise her arm only when she is confronted with a case such that it is super-true that the man under consideration is tall. For any of the infinite higher-orders of vagueness for tallness the man under consideration may instantiate, we may expect the omniscient being not to raise her arm. Some may complain that the omniscient being would not really be cooperative in such a case because she does not inform us of everything she knows about the man under consideration: her behaviour should differ for each of the infinite higher orders of vagueness a man under consideration may instantiate. The objection would be relevant if we were interested in distinguishing among the different higher orders of vagueness. But we are interested, instead, in establishing whether an omniscient being can at least find a boundary between the men such that it is super-true that they are tall and the borderline cases for any of the infinite higher orders of vagueness. It may therefore be sufficient for the research we are doing for the omniscient being to adopt two clearly different performances when confronted with the super-true cases of tallness on the one hand and the borderline cases on the other. Let us now consider how we may characterize the cooperative behaviour by an omniscient being we have just envisaged. It is evident that a bi-conditional like (R) cannot be adopted as we have seen that it is inadequate. The best way I can think of to characterize such behaviour is to give a meta-linguistic definition of

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cooperation.⁴ Let us suppose that we adopt a meta-language, i.e., a language in which we talk about sentences in another language and their truth-values, and we characterize cooperation through inference rules as follows: P. The omniscient being raises her arm ⊣⊢ “the man under consideration is tall” is super-true. The omniscient being does not raise her arm ⊣⊢ it is super-true that “the man under consideration is tall” is neither super-true nor super-false. The omniscient being does not raise her arm ⊣⊢ it is super-true that “the man under consideration is tall” is neither super-true that it is super-true nor it is super-true that it is not super-true. ...... The set of instructions for cooperative behaviour by an omniscient being is infinite, but we may try to sum them up as follows: P*. The omniscient being raises her arm ⊣⊢ “the man under consideration is tall” is super-true. The omniscient being does not raise her arm ⊣⊢ it is super-true that “the man under consideration is tall” instantiates any of the infinite higher orders of vagueness. Anyone who is trying to apply this definition of cooperation by an omniscient being probably expects to see the omniscient being lower her arm at a certain point in the sorites series, indicating that she recognizes a precise boundary in the sorites series.

5 Cooperation by an Omniscient Being: Objection to the Second Proposal In fact, this conjecture concerning the actual behaviour of the omniscient being does not allow it to be cooperative. The reason is that if a sorites series were really boundaryless, an omniscient being would not be able to cooperate according to the above rules (P*).

4 I try to be as colloquial as possible in this work. For a more rigorous presentation of the second proposal see Paganini (2012).

94 | Elisa Paganini In order to understand the previous observation, it is worth considering what a boundaryless sorites series would be like and then consider why the previous definition (P*) cannot be an exhaustive definition of cooperation in such a case. Let us consider the sorites series of men once again: the first is two meters tall, the last is one meter tall and there is a very small difference in height between each man and the following one. If such a sorites series were boundaryless in terms of the extension of “tall”, then there would not be a man such that it is super-true that he is tall, followed by a man who is a clear borderline case of tallness for any of the infinite higher orders of vagueness. But if this were not the case, what would the sorites series be like? In my opinion, we should expect there to be at least one man in the sorites series such that there is not anything super-true which can be said concerning his tallness: it cannot be said that it is super-true that he is tall, it cannot be said to be a borderline case of tallness for any of the infinite higher orders of vagueness, in general nothing super-true can be said concerning his relation with tallness. We may be a bit incredulous as to the existence of such a man, but it is inevitable that a man like him should be present in a sorites series if such a series is boundaryless. If instead something super-true could be said concerning the relation of any man in the sorites series and tallness, then there would be a difference between at least two man in the sorites series (a difference concerning their relation to tallness), and there would be a boundary in the extension of the predicate “tall’ in the sorites series. Now let us suppose that the man such that nothing super-true can be said concerning his relation to tallness is called Jim and that the omniscient being is confronted with him. Let us now consider what the cooperative behaviour of an omniscient being would be when confronted with Jim. Suppose that we want to ask her to clap her hands when confronted with a case like Jim. If this instruction were possible, we would be able not only to recognize the boundary between the definite cases of tallness and the cases like Jim, but also the boundary between the cases like Jim and the definite borderline cases. As far as I know, there is no way to convey such an instruction in order for an omniscient being to be cooperative when confronted with a case like Jim. Let us remember that we would like to ask the omniscient being to clap her hands when confronted with a case like Jim. Let us now try to implement the definition of cooperation in (P*) by filling in the gap in the following meta-linguistic instruction: (J) The omniscient being claps her hands ⊣⊢ it is super-true that “the man under consideration is tall”......

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Now, if we were able to fill in the gap accounting for a case like Jim, we could convey an exhaustive definition of cooperation for an omniscient being when confronted with the sorites series. The problem is that in the case of Jim nothing supertrue can be said concerning his relation to tallness, therefore there is no adequate way to fill in the gap in (J). Let us now consider why the fact that there is no way to fill in the gap in (J) does not allow an omniscient being to be cooperative. The situation of an omniscient being confronted with a sorites series would be such that there are instructions which regulate her cooperation when confronted with some men in the series, but there is no instruction which regulates her cooperation when confronted with some other men in the series (i.e., men like Jim): this is the same as saying that it is impossible for an omniscient being to be cooperative. And if there is at least the possibility that an omniscient being cannot be cooperative, she will not be a good test for establishing whether there is a boundary in the sorites series.

6 Some Conclusive Remarks I have argued so far that, if a semantic theory of vagueness like super-valuationism is correct, an omniscient being cannot be cooperative in establishing whether there is a boundary in the sorites series. And as long as we cannot exclude the supervaluationist theory being the correct one, we cannot exclude that the omniscient being may not be able to be cooperative. This result has consequences for any questions that could be asked an omniscient being, as long as they may be invalidated because of vagueness: as it is well known vagueness characterizes most of our language and it is quite difficult to isolate an interesting philosophical question which does not use vague words. This shows that the suggestion by C. Anthony Anderson presented at the beginning of this work is probably the best one if we are encouraged to interrogate an omniscient being, knowing in advance that the answer would not be relevant for any current philosophical debate. As I anticipated, the result is not only disappointing – as it shows that there may not be any interesting philosophical question to ask an omniscient being – but it is also instructive. The information to be drawn from this result is in a sense puzzling: the possibility of an omniscient being not being cooperative depends on the presence of a person like Jim in a sorites series and the possibility of a person like Jim in a sorites series is the interesting conundrum of the result presented above. Jim is such that nothing super-true can be said concerning his relation to tallness: it cannot be said that such a relation is indeterminately instantiated or

96 | Elisa Paganini indeterminately indeterminately instantiated or ... whatever else may be thought of. The presence of Jim in a sorites series is a puzzling consequence for a semantic theory, because it is a case where no semantic notion can be offered in order to characterize the semantic status of a sentence concerning Jim and tallness: this not only shows that semantic notions are themselves vague, but also that their vagueness leaves room for inexpressibility. And the possibility of inexpressibility is itself an interesting and puzzling consequence to be reflected on.

Bibliography Fine K. (1975), “Vagueness, Truth and Logic”, in Synthèse: 30, 301–324. Hawthorne J. (2005), “Vagueness, and the Mind of God”, in Philosophical Studies: 122, 1–25. Keefe R. (2000), Theories of Vagueness, Cambridge: Cambridge University Press. Paganini E. (2012), “God’s Silence”, in Philosophical Studies: 197, 287–298. Soames S. (2003), “Higher-order vagueness for partially defined predicates”, in Liars and Heaps, edited by J. C. Beall, Oxford: Clarendon Press, 128–150. Williamson T. (1994), Vagueness, London: Routledge.

Mirosław Szatkowski

God’s Omniscience and Logical Virtue 1 Introduction Generally, it can be said that omniscience is understood as being all-knowing. Thus, to say that God is omniscient is to say that he knows absolutely everything. But what does it mean to say that God knows absolutely everything? There is the question of what exactly our human knowledge is. Whether or not an understanding of human knowledge can be applied to God? There is also the question of what exactly the word “everything” in the phrase “God knows absolutely everything” is supposed to mean. For example, does God know everything which is actual but not all that is possible? Does God know the future, and if so, how exactly? Philosophers do not start with an empirical observation of God’s attributes and then proceed to come to terms with how divine omniscience should be understood. But they begin with some high concept of God, for example, with St. Anselm’s idea that God is a being than whom no greater can be conceived, or with Rene Descartes’s idea that God is a being who has all the perfections, or the claim that God is a being who is worthy of whole-hearted worship, and then they go on to speculate about what properties a being must have in order to satisfy this high conception.¹ Supposing that knowledge is a good property (perfection) and that it is the better to have the more knowledge, it seems attractive to conclude that the greatest conceivable being, a perfect being, etc., must possess an all-encompassing knowledge, with nothing left out. In the most radical form, the doctrine of divine omniscience is thought of as comprising the following five claims: (1) God’s knowledge is complete, i.e., God knows all truths (about God himself, creation, logic and mathematics, etc.); (2) God’s knowledge is sound, i.e., everything that God knows is true; (3) God’s knowledge is tenseless, i.e., God knows the past, present and future simultaneously; (4) God’s knowledge is necessary, i.e., God necessarily knows whatever he knows; (5) God’s knowledge also ranges over unrealized possibilities, i.e., God knows all things that could have come to pass but will not.

1 Cf., for example, Hill (2005) and Nash (1983).

98 | Mirosław Szatkowski This theory simply states that God is omniscient iff he has complete knowledge, sound knowledge, necessary knowledge, tenseless knowledge and the knowledge of unrealized possibilities. Each of these determinants of God’s omniscience comes with its own set of problems. We will look at the individual elements of the definition (1)–(4) in turn, postponing the discussion of the element (5) of the definition on another occasion.² We think in fact that firstly unrealized possibilities should be thoroughly studied with the ontological and semantic point of view. The main aim in this paper is an indication of the way – one of many – in which one can consistently speak of God’s omniscience. This final result of the article is preceded by considerations confronting God’s omniscience with some meta-mathematical theorems and problems with philosophy of mathematics. The issue of unreality/reality of time is also briefly outlined and the multiplicity of temporal semantics is listed. The paper will be structured as follows. In section 2, we demonstrate difficulties associated with the completeness of God’s knowledge and point out different ways to solve them. In section 3, we focus on difficulties associated with the soundness of God’s knowledge. In section 4, we briefly describe McTaggart’s argument for the unreality of time and – after informing about the existence of a plurality of

2 But a brief comment about unrealized possibilities is in order here. In spite of the claim (5), God knows not only the entire realm of actuality (past, present and future), but he also knows the entire realm of possibility, for example, he knows a certain member of a species of animal that will never exist. Since God is the creator of universe, he ruled its evolution by laws and initial conditions, knowing what will occur and what will not occur according to them. Therefore, God also knows what is not, in the sense of what is either impossible or incompatible with laws of nature. Thomas Aquinas argues that things which are not actual are in the power either of God or of a creature, and because God knows his power and the power of his creatures, therefore he has knowledge also of things that are not, see Thomas Aquinas (1981) (I,14,9): Now it is possible that things that are not absolutely, should be in a certain sense. For things absolutely are which are actual; whereas things which are not actual are in the power either of God Himself or of a creature, whether in active power, or passive; whether in power of thought or of imagination, or of any other manner of meaning whatsoever. Whatever therefore can be made, or thought, or said by the creature, as also whatever He Himself can do, all are known to God, although they are not actual. And in so far it can be said that He has knowledge even of things that are not. Of course, there exist critics of this strong request that God knows unrealized possibilities. Among them, Ch. Hartshorne – cf., Creel (1982) – holds that God knows all actualities as actualities and all possibilities as possibilities, but he adds that possibilities are by nature vague and indeterminate so that, for example, God could – in the best case – only have known of the possibility of new members of a species of animal, without knowing precisely what they would be like.

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counter-arguments to it and the arguments supporting it – give an overview of the different semantical approaches of time. In the section 5, after describing the formal syntax, we propose Anderson-like semantics with ‘temporal’ and ‘knowledge of truths’ dimensions. And finally, in the conclusion, we summarize the results of previous sections, and indicate in what sense we can consistently speak of God’s omniscience.

2 Cantor’s Argument and the Complete Knowledge of God The saying that God knows all truths demands that there is a set of all truths. This statement is, however, inconsistent with Cantor’s power set theorem which states that for any set X, the cardinality of power set 2X of a set X is always strictly larger that the cardinality of X. More precisely, the argument against the existence of a set of all truths is as follows, cf., P. Grim (Grim (1983)), (Grim (1984)) and (Grim (1988)), and P. Grim and A. Plantinga (Grim and Plantinga (1993)): Suppose there is a set T of all truths, and consider the power set 2T of T. To each element X of this power set 2T will correspond a truth, for example, a truth of the form: ‘t belongs to X or t does not belong to X’ where t is an element of T. There will then be at least as many truths as there are elements of the power set 2T . But according to Cantor’s Theorem, the cardinality of the power set 2T is strictly larger than the cardinality of the original T. There will then be more truths than there are members of T, and for any set of truths T there will be some truth left out. It follows that there can be no set of all truths. And consequently, if there were an omniscient being, then all which that being would know would constitute a set of all truths. But there can be no set of all truths, and so there can be no omniscient being.

The core of this correct argument is the power set axiom: For every set X there is a (power) set 2X (P(X)) whose elements are precisely the subsets of X. Thus, a natural way to rescue the completeness of God’s knowledge is to adopt an alternative set theory in which such an axiom doesn’t appear. And so, according to Ch. Menzel (Menzel (1986)), the notion of a set of all truths can be saved in ZF (ZermeloFraenkel) set theory minus the power set axiom. He says: By adopting ZF-Power (or some similarly conservative set theory), then, and adjusting it appropriately to allow for the existence of large sets (and urelements), the world-story theorist is free to postulate the existence of his world-stories without fear of paradox, at least by way of [the Cantorian] argument; for that argument depends essentially on there being a power

100 | Mirosław Szatkowski set of the world-story S; but there is simply no way of generating the full classical power set of an infinite set in ZF-Power. (Menzel (1986), p. 71).³

Similarly, S. Bringsjord (Bringsjord (1989)) and R. E. Maydole (Maydole (2012)) take the power set axiom to be responsible for difficulties in the matter, and adopt alternative set theories in which such an axiom doesn’t appear. The first of them appeals to ZF-Power (or, for that matter, VNCB (von Neumann-Bernays-Gödel set theory), KPU (Kripke-Platek set theory with urelements))⁴, and the second – to ML (Quine’s set theory, where ML is short for Mathematical Logic).⁵

3 Let us quote the Grim’s opinion concerning the Menzel’s argument: ...the most telling objection against ZF-power is simply that it doesn’t ultimately avoid the central Cantorian difficulties for which Menzel invokes it. It turns out that because a set of all truths (or the like) is at issue, sacrifice of even the power set axiom is not enough. (Grim (1991), p. 112). 4 By P. Grim (Grim (1990), p. 273), Bringsjord’s appeal to ZF-Power proves insufficient, because by the axiom schema of separation, taken from ZF, it can be proved that there are more truths than elements of the set of all truths T, and thus T cannot be a set of all truths. He says: Something above and beyond the power set axiom would have to be sacrificed. Perhaps additional restrictions on the axiom schema of separation are called for. Perhaps the language of the system must exclude or restrict the notion of truths about sets of truths – though since there clearly are truths about any sets of truths this would seem far from satisfactory. Or perhaps we should simply conclude that there is no set of all truths. 5 In Maydole (2012), p. 368, R. E. Maydole writes: There are many possible ways to object to Grim’s Cantorian argument, the most notable being: (1) Reject Cantor’s theorem outright by rejecting the very idea of different levels of infinity. (2) Deny the existence of the power set of each set. (3) Show it as a disguised Liar Paradox. (4) Link omniscience to some vaguely defined multiplicity or totality of truths other than the set all truths. (5) Redefine omniscience to avoid quantification over any kind of multiplicity, totality or collection. (6) Adopt a non-classical logic without the principle of excluded middle. (7) Use a set theory other than the set theory that undergirds Grim’s reasoning.

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Of course, the choices between different set theories are and will remain highly debatable in the philosophy of mathematics.⁶ Most of ordinary mathematics requires to have a theory of natural numbers. In fact, most of ordinary mathematics can be indirectly developed in the theory of second-order arithmetic. And since the characterization of the set of natural numbers requires the use of the power set axiom, most of ordinary mathematics makes use of the power set axiom. But there are also mathematical theories that reject the power set axiom. For example, ZFC (Zermelo-Fraenkel set theory with the axiom of choice) minus the power set axiom was studied by U. Abraham (Abraham (2010)), T. Jech (Jech (2003)), A. Kanamori (Kanamori (2003)), V. Gitman (Gitman (2011)), J. D. Hamkins and T. A. Johnstone (Hamkins and Johnstone (2010)), I. Neeman (Neeman (2010)) and A. Zarach (Zarach (1982)). Rejection of the power set axiom is one of the fundamental features of the predicative mathematics, where such a mathematics is a way of doing mathematics without allowing impredicative definitions and, informally, a definition is impredicative if it refers to a totality which includes the thing being defined. Among many others, both Myhill’s (Myhill (1973)) and Aczel’s (Aczel (1978)) constructive set theories are predicative; in the first of them, the classical power set axiom is very weakened, in the other – it is replaced by the so-called collection scheme. For many set theories without the power set axiom it was proven that they are weaker than may be supposed and that they are inadequate to prove a number of basic facts that are often desired and applied in them context. Such theories may also have unintuitive consequences; A. Zarach in Zarach (1982) initiated the program of establishing unintuitive consequences of set theory without power set axiom. Another solution in order to avoid the difficulties associated with the completeness of God’s knowledge can be a redefinition of God’s omniscience. Instead of referring to the set of all truths in the definition of an omniscient being it’s just used something similar to the following quantificational form: ‘for all p, if p is true then ... p...’, and then, a coherent notion of omniscience is defined by: ‘A being x is omniscient iff for every proposition p, if p is true then x knows that p’ (S. Bringsjord in Bringsjord (1989) proposes also such a solution); or, ‘A being x is omniscient iff for every proposition p, if p is true iff x knows that p’ (cf., Grim (1991), pp. 113-122). And again, P. Grim in (Grim (1991), pp. 120-121), argues against this propositional quantification.

6 P. Grim, for example, argues that any adoption of: an Quine-like set theory (those of Quine’s New Fundations or Mathematical Logic), von Neumann-Bernays, Kelley-Morse and Ackerman to save a “set of all truths” proves to be a very desperate move, because the costs of this set theories turn out to be enormous; cf., Grim (1991), pp. 99-110.

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3 The Liar and the Sound Knowledge of God In what follows, we want to outline the argument against the sound knowledge of God. For this purpose, let us consider the sentence of the so-called divine Liar: (*) God knows that (*) is false.

Supposing now that (*) is true, we obtain that it is true that God knows that (*) is false. But supposing that (*) is true, we are forced to conclude that God holds a false knowledge. So, on such a supposition God can not be qualified as omniscient. Supposing now that (*) is false, we have that it is not the case that God knows that (*) is false. But our supposition is that (*) is false, therefore it must be a truth that (*) is false that God does not know. So again, God can not be qualified as omniscient. The name ‘divine Liar’ emphasizes the relationship with the Liar paradox: (A) The sentence (A) is false.

What is here important to emphasize is that the Liar paradox and all other sentences similar to it have a self-referential nature, that means, they express inside a language their own truth-concepts. That is not to say that the self-reference is either necessary or sufficient for the paradoxicality; see, N. Tennant (Tennant (1982) and Tennant (1995)), and S. Yablo (Yablo (1985) and Yablo (1993)). The self-reference of sentences is, however, in striking disagreement with the Tarski’s indefinability theorem. Intuitively, this theorem says that arithmetical truth cannot be defined in arithmetic; see, Tarski (1935/6). But this theorem applies to any sufficiently strong formal system – i.e., to any first-order formal languages with negation, and with sufficient capability for self-reference that Gödel’s diagonal lemma holds – showing that truth in the standard model of the system cannot be defined within the system. J. L. Bell and M. Machover in (Bell and Machover (1974), pp. 330-331), comment the philosophical significance of this theorem as follows: This suggests that there cannot exist a formal language which – under some "standard" interpretation – could adequately serve as its own metalanguage; for the syntax and semantics of a formal language cannot be adequately expressed within the language itself. In particular, the dream of certain philosophers, that some day a precise formal language will be constructed in which all scientific notions and theories would be expressible, is most probably unrealizable.

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We want, however, to express Tarski’s indefinability theorem more formally. For this aim, suppose that L be a first-order formal language with negation, and with the capability for self-reference that Gödel’s diagonal lemma holds. Thus, for every formula ϕ of L, ϕ is a term of L, called Gödel’s number, code number or quote. Let M be a model structure for L. Let TR denote the set of L-sentences true in M, and TR the set of code numbers of the sentences in TR. We can now formulate Theorem 3.1 (Tarski’s indefinability theorem). which defines TR.

There is no L-formula True(x)

Proof. By reductio ad absurdum. Suppose that there exists such a formula True(x). Hence, for every L-sentence ϕ, ϕ ∈ TR if and only if True(ϕ) ∈ TR, and consequently, True(ϕ) ↔ ϕ ∈ TR. But, using Gödel’s diagonal lemma, we can construct a Liar L-sentence ψ such that ¬True(ψ) ↔ ψ ∈ TR, which stands in contradiction to True(ψ) ↔ ψ ∈ TR. Finally, no such True(x) exist in the language L. Tarski’s theorem shows very clearly that the Liar problem is closely related to the problem of the universality of languages. Where, in Tarski’s sense, a language is universal if it can say everything there is to be say. For solving the Liar paradox, Tarski offered the hierarchy of not universal object-languages and not universal metalanguages – see, Tarski (1935/6) – which fragments the concept of truth. Instead of one concept, we have infinitely many, arranged in a hierarchy. Tarski insists that any object-language should be distinguished from its meta-language. Given an object-language, he requires that the concept of truth for this language be contained in its meta-language and not in the object-language itself. A metalanguage is much stronger than the object-language, since the latter is assumed either to be a fragment of the meta-language or to have a preordained translation into the meta-language. It seems fair to conclude that a huge number of solutions have been proposed for understanding the nature of the Liar paradox and for the exclusion of inconsistency that is generated by these sentences; see, for example, Gupta (1982), Gupta and Belnap (1993), Ketland (2003), Kripke (1975), Leitgeb (1999), Martin and Woodruff (1975), Perlis (1988), Perlis (1988), Perlis and Subrahmanian (1994), Visser (1984), Yablo (1985) and Yablo (1993). We do not have here the space to compare and to evaluate this many solutions.

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4 McTaggart’s Paradox, and Various Semantical Approaches to Time What is required in the statement that God knows the past, present and future simultaneously is the presupposition that there really are such properties as being past, being present and being future. Unfortunately, this presupposition is not universally recognized. In the paper McTaggart (1908) (see, also, McTaggart (1988)), published in 1908, J. M. E. McTaggart argues that there is no such thing as time.⁷ The McTaggart’s work provoked a lively discussion in the philosophical literature, finding both opponents and supporters of the conclusion of the McTaggart’s argument. According to McTaggart, the things in time can be ordered in two ways: (i). There is the B-series which orders events and times in terms of the tenseless relation earlier∖later than, and (ii). There is the A-series which orders events and times of the tensed properties of being past, present, and future. McTaggart claims, first, that the B-series presupposes the A-series, and, second, that the assumption that there is an A-series leads to a contradiction. Therefore, he concludes that time is unreal. Many philosophers do not accept McTaggart’s final conclusion that time is unreal, they differ as to the part of the argument they reject. Some of them denies the claim that the B-series presupposes the A-series.⁸ M. J. Loux in (Loux (2006), p. 205), recognizes concisely the arguments of this group: They insisted that the B-series is a properly temporal framework all by itself. They took time to be just a dimension along with the three spatial dimensions; they held that all times and their contents are equally real; and they insisted that tensed language can be translated into tenseless language.

Other of them reject the claim that the A-series is contradictory.⁹ And again, M. J. Loux says concisely (ibid):

7 Let us add that many other philosophers, before and since 1908, have also argued for the unreality of time. 8 For this group, for example, belong: W. V. Quine (Quine (1960), J. J. C. Smart (Smart (1963)), D. C. Williams (Williams (1951)). For the characterization of this group see Gale (1967), Section II. 9 For this group, for example, belong: C. D. Broad (Broad (1923) and Broad (1938)), A. Prior (Prior (2001)) and R. Taylor (Taylor (1963)).

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They held that time is inherently tensed, and they attacked the B-theorists’ [= the first group¹⁰] attempts to reduce tensed language to tenseless language. Their attacks on the attempt to eliminate tensed language were compelling and led many to reject the B-theory.

And he adds (ibid): Then in the 1980s, a new breed of B-theorists [= the first group] appeared on the philosophical scene. They endorsed the metaphysical claims of the old B-theory, but rejected its claim that tensed language is eliminable. They argued that while tensed language is ineliminable, the states of affairs that constitute the truth conditions for tensed sentences are just the tenseless states affairs making up the B-series.¹¹

It is impossible here to present, compare and evaluate the arguments for and against McTaggart’s one. But we should add, the rejection of McTaggart’s argument is of peculiar interest to us since we are launching an inverse argument from the reality of time to the tenseless knowledge of God. The fact that there are different semantical approaches of time – see, Figure 1 –, so that no one set of criteria suffices to capture all of them, suggests that we should make here a review of these approaches and decide which of them are appropriate for our purposes. 1. Linear-time semantics 1.1 Kamp semantics 1.2 T×W-semantics 2. Branching-time semantics 2.1 Standard (⟨Moment, History⟩) semantics 2.1.1 Priorean semantics 2.1.2 Peircean semantics 2.1.3 Ockhamist semantics 2.2 Bundled-Trees semantics 2.2.1 Priorean semantics 2.2.2 Peircean semantics 2.2.3 Ockhamist semantics Fig. 1. Types of temporal semantics

10 It is Szatkowski’s supplement. 11 The literature – defending and attacking – this new theory can be found in Oaklander and Smith (1994) and in LePoidevin (1998). See, also, Mellor (1981) and Mellor (1998).

106 | Mirosław Szatkowski In linear-time semantics, according to determinism, time is viewed as a sequence T = ⟨T, t, |= Lϕ iff W, a, t, w |= ϕ for all v ∈ W such that wR t v, |= ϕ iff W, a, t, w |= ϕ for all v ∈ W such that wRv.

Note that in the case of Kamp frames, ‘W, a, t, w |= ϕ’ is only defined if t ∈ T w , and the clause for  is dropped. In the context of branching-time semantics, time is represented as a tree T = ⟨T, t, (iii) T , a, t, h |= Lϕ iff T , a, t, h1 |= ϕ for all h1 ∈ H t .

Since Peircean language contains no modal operators, therefore Peircean semantics don’t evaluate modal operators in the usual sense. They are adjusted to interpret the future operators:  and , and the past operators:  and . For any ψ ∈ {ϕ, ϕ, ϕ, ϕ}, the truth relation T , a, t |= ψ (read: the formula ψ is valid in the tree T , under the evaluation a, at the moment t) in standard-tree Peircean semantics is defined as follows: (i) (ii) (iii) (iv)

T T T T

, a, t , a, t , a, t , a, t

|= ϕ |= ϕ |= ϕ |= ϕ

iff iff iff iff

∀h ∈ H t ∀t1 ∈ h : t < t1 and T , a, t1 |= ϕ, ∃h ∈ H t ∀t1 ∈ h : t < t1 and T , a, t1 |= ϕ, T , a, t1 |= ϕ for all t1 < t, T , a, t1 |= ϕ for some t1 < t.

Obviously, the dual operators:  = ¬¬,  = ¬¬ and  = ¬¬ are also interpretable in Peircean semantics. Peircean language can be viewed as a fragment of the Ockhamist one because the operators: , ,  and  can be expressed as: L, M,  and M, respectively. The Priorian language contains only the operators:

108 | Mirosław Szatkowski  and , and their truth conditions are the same as in Peircean semantics. This means that Ockhamist semantics correspond to the most expressive language of the three. For this reason, we will concentrate in the next section only with Ockhamist semantics. Looking, however, from a philosophical point of view, the Priorean, Peircean and Ockhamist semantics represent different, the most typical but not the only, treatments of the non-deterministic future possibility and necessity operators; see, for example, Prior (1967). The quantifications over histories H t in standard-tree Ockhamist and Peircean semantics is replaced in the bundled-tree Ockhamist and Peircean ones by a quantifications over a given set of bundles Bt , respectively, which can be thought of as the set of admissible histories; see, for example, Burgess (1979), Zanardo (1991), Zanardo (1996) and Zanardo (1998). Bundled-trees semantics are more general than standard semantics – there are formulas that are valid in all trees but not in all bundle trees, as it has been proved by R. Thomason in (Thomason (1984)). Of course, the question is which of these two types of semantics is more natural. The answer to this question depends greatly on ontological commitments about the nature of time and whether moments and histories on the one hand, or bundles on the other hand, should be accepted as primitive temporal entities. We do not take this discussion here, but we inform the reader that a discussion about the philosophical background of bundled trees semantics can be found, for example, in Belnap, Perloff and Xu (2001), Section 7.

5 Anderson-type Semantics Taking into Account the Time and the Knowledge of Truths In this section, we constitute together the partly free semantics for Andeson-like ontological proofs (see, Szatkowski (2011)) and the branching-time Ockhamist ones.

– – – – – –

The language L of temporal Anderson-like ontological proofs is equipped with: a denumerable infinite set of 1st order variables: x, y, z, . . .; a denumerable infinite set of 2nd order variables: α, β, γ, . . .; the 2nd order unary predicate P, where P(...) is to be read: ... is positive; the Boolean operator: − (complementation); the logical symbols: ∧, ¬ (conjunction, negation); the existence determinator E;

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– the operators: historical necessity L (it is historically necessary that. . . ), strong future  (it will always be that. . . ), and strong past  (it has always been that. . . ); – a universal quantifier ∀ for both sorts of variables. The only terms of the 1st sort are variables of the 1st sort and terms of the 2nd sort are formed from variables of the 2nd sort by applying complementation any finite (possibly zero) number of times. Thus, the set of terms of the 2nd sort and the set of formulas are given by the grammars: ⃒ df A = α⃒ −A ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ ⃒ df ⃒ ϕ = ⃒ E(x) ⃒ A(x) ⃒ P(A) ⃒ ϕ ∧ ψ ⃒ ¬ϕ ⃒ Lϕ ⃒ ϕ ⃒ ϕ ⃒ ∀xϕ ⃒ ∀αϕ

The definitions of G(x), A  B, α Ess x and NE(x) adopted in Anderson-like theories are the same as proposed by Anderson (Anderson (1990)):

[︀ ]︀ df G(x) = ∀α P(α) ↔ Lα(x)

(1)

Informally: God is any being that necessarily has a property α if and only if α is a positive property, [︀ ]︀ df A  B = L∀x A(x) → B(x)

(2)

Informally : A entails B if and only if necessary for every object x,

if x has the property A then it has also the property B,

[︀ ]︀ df A Ess x = ∀β Lβ(x) ↔ (A  β)

(3)

Informally : A property A is an essence of an object x if and only if the following condition holds: x necessarily has all and only these properties which are entailed by A, [︀ ]︀ df NE(x) = ∀α α Ess x → L∃yα(y)

(4)

Informally: An object x has the property of necessarily existing if and only if its essence is necessarily exemplified.

We are not here cataloging axioms of temporal Anderson-like ontological proofs. By a model structure taking into account the time we mean a 4-tuple of the form W = ⟨T , D1 , D2 , G⟩, where: • T = ⟨T, 1 and for every non-empty subset X ⊆ Th(W ), k < n G(x) → LKX k Informally: For any level n > 1 and for any set of truths from any level lower than n, God necessarily knows the truths.

are valid in the model structures Wn and provable in the Anderson-like theories corresponding to Wn , n > 1, is of great significance for our purposes. Finally, it may be instructive to mention that on the basis of the other types of model structures for Anderson-like ontological proofs – examined, respectively, in Szatkowski (2005), Szatkowski (2007), Szatkowski (2011), and Szatkowski (2012) – also other types of Anderson-type semantics taking into account the time and the knowledge of truths can be defined.

6 Conclusion Cantor’s Theorem and the divine Liar paradox undermine the claim that there is an omniscient being. Consequently, one should reject the belief in God who

112 | Mirosław Szatkowski is qualified just as omniscient. Patterning ourself after the Tarski’s solution to the Liar paradox, we have offered the hierarchy of languages and metalanguages which gives a fragmentary account of the concept of God’s omniscience. Instead of one concept of God’s omniscience, we have infinitely many of them, arranged in a hierarchy. Apart from the earliest language in the hierarchy, all other languages are expansions of their own predecessors by adding the families of predicate constants of the 2nd sort: a knowledge of all truths from a given set of truths. Andersonlike systems, defined in all languages of the hierarchy, seem to be too poor in the sense that they do not express relationships between these knowledge predicates. Another important task is to prove (strong) completeness theorems for Andersonlike theories with respect to corresponding model structures. Investigating these matters is, however, a task to be faced in future research.

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Part III: Truth

Srećko Kovač

Logic and Truth in Religious Belief 1 Introduction Logic and religious belief are narrowly interconnected. This is so in the sense of the consistency required in religious belief, and in the sense of perseverance in belief and in its consequences. We aim to show, on the ground of biblical texts and using logical tools, that there is a gradation in implementing religious faith in logic and that it is reflected in a gradation of knowledge and the corresponding notion of truth. The role of logic in religious belief consists not only in the correctness of reasoning, but also in the semantic interplay of appearance and truth in the process of the building-up of a religious belief. Preliminarily, let us briefly outline what is specific for religious belief with respect to the concepts of belief and knowledge as they are usually understood in epistemic (and doxastic) logic. In general, “i believes that ϕ” means that i holds that ϕ is true. Here we can, first, distinguish the objective side, the reference to truth. In the case of religious belief (in some fully realized sense), this reference to truth is in fact a sort of knowledge. This may be seen in the example of the Roman centurion from the Gospels, who, coming to Jesus, obviously knows (according to Mt 8:8) that his belief will be realized: “say the word and my servant will be healed” (cf. a slightly different formulation in Lk 7:7). However, it seems that the reason why we usually do not identify religious belief with knowledge is precisely the strong subjective side of religious belief, as can be seen, for instance, from Jesus’ words to his disciples in a storm: “Why are you terrified? Do you not yet have faith?” (Mt 8:26, Mk 4:40). Religious belief essentially depends not only on truth, but also on subjective trust, confidence, i.e. on religious faith.

2 Logic and Religious Belief¹ We aim to show the interconnectedness of logic and religious belief in two ways: first, religious belief includes reasoning; secondly, religious belief is a pragmatic function applied to logical forms, and hence is a part of logic in a wider sense.

1 This section is a further elaboration of section 1 and of a part of section 4 of Kovač (2011).

120 | Srećko Kovač As for the first aspect, we indicate with two well-known examples in which sense logical reasoning is included in religious belief. In the first example (Lk 10:30-37), Jesus leaves his disciples to judge for themselves which of three people, encountering a robbed half-dead man on the way, really fulfils the law, i.e. the commandment of neighborly love (“Love your neighbor as yourself”). As we know from the story, a priest, “when he saw [the man], ... passed by on the opposite side”. A Levite did the same. A Samaritan traveler, however, “was moved with compassion at the sight” and took care of the man. Logically formulated, Jesus’ question is about who instantiated the general law, or who was consistent, comparing their knowledge of the law and their behavior in the situation? The second example is the “Golden Rule”, to which Jesus refers and, in a general premise, wants to be applied: “Do to others whatever you would have them do to you” (Mt 7:12; Lk 6:31). This entails the commandment “Love your enemies” (Lk 6:27, 35), assuming that you would have others love you. Also: “Forgive and you will be forgiven. Give and gifts will be given to you” (Lk 6:37-38). Obviously, the converse of the Golden Rule is assumed, too (in a contrapositive formulation): “Do not do to others what you would not have them do to you”. Instantiations are, for example, “Stop judging and you will not be judged. Stop condemning and you will not be condemned” (Lk 6:37, Mt 7:1). In the next example, in the parable of the sower, we outline in which way religious belief (precisely, religious faith) can be conceived as a function applied to logic and language (as a part of logic). In the parable, a word (rh˜ ema, logos) is put in connection with faith, precisely: the word of God with religious faith (Mt 13:3-23; Mk 4:3-20; Lk 8:4-15). Here, the sown seed is compared to the spoken word. The tertium comparationis, common to the seed and word, and enabling the comparison, is giving and receiving. The giving is sowing and saying the seed and the word, respectively. The receiving is the receiving of the seed in the ground and the receiving (accepting) of the word in the mind (“heart”). In the parable, four grades of religious faith are distinguished: (1) according to the inner quality of the reception of the word (I, in analogy with the inner quality of the ground), and (2) with respect to the least outer circumstances it cannot endure (O, in analogy with the outer circumstances for the growth of a plant; see result R). These outer circumstances, which, in a sense, measure the endurance of faith according to its inner quality, are ordered in the following way: (1) the devil (without any special efforts from him) < (2) tribulation, persecution < (3) worldly life and its riches. The corresponding order of the inner quality of faith is the following:

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(1) faith without understanding < (2) rootless faith < (3) faith mixed with anxieties and pleasures < (4) persevering faith. Inner quality (4) can endure all outer circumstances (1)–(3). Table 1 displays the analogy between the sown seed and the spoken word of God with respect to the grades of their reception and the respective results (outcome) in the limiting outer conditions. Table 1 seed

the word of God (of the kingdom)

I O R I

on the path birds trampled, eaten by birds on rocky ground, little soil

O R I

sun, lack of moisture withered among thorns

O R I O R

full-grown thorns choked on the good soil / fruitful

heard, not understood (n) devil, the evil one (d) taken away received with joy, only for a time, no root (o) tribulation, persecution (t) fallen away heard, mixed with anxieties and pleasures (m) worldly life, riches (w) choked, unfruitful embraced in the heart, perseverance (p) / fruitful

According to this table we can conceive religious faith as a function of the sentences of a given language, and of inner and outer circumstances. In the case where a sentence is “the word of God” we get the following values: faith(ϕ, n, d) = no, faith(ϕ, o, t) = no, faith(ϕ, m, w) = no, faith(ϕ, p, x) = yes, where ‘yes’ and ‘no’ are understood as ‘yes’ and ‘no’ to the word of God, respectively (yes is like ‘am¯en’ in the Bible). In the last case, of persevering faith, no outer circumstances x can turn the result to no. In general, inner quality x can match outer circumstances y < x, but cannot match outer circumstances y ≥ x: faith(ϕ, x, y) = yes iff x > y, where ϕ is the word of God. Religious faith can thus be conceived as a function that applies to syntactically and semantically already determined forms (sentences). It is an additional, pragmatic function, which pertains to the actual use of syntactical and semantic forms and to their execution in a context: faith(ϕ, x, y) ∈ {yes, no}.

122 | Srećko Kovač Let us mention that what is understood as consistent faith is the faith that perseveres in all circumstances, so that the third argument can be ignored: faith(ϕ, x) ∈ yes, no. Consistent faith behaves in a standard way (as in classical logic): faith(¬ϕ, x) = yes iff faith(ϕ, x) = no, faith(ϕ ∧ ψ, x) = yes iff faith(ϕ, x) = yes and faith(ψ, x) = yes, etc., and should correspond to some chosen (semantic) model(s). By including religious use (pragmatics) into logic as its part, we get a more comprehensive concept of logic, which may be sketched as in Table 2.² Table 2 LOGIC

Syntax

Semantics

Pragmatics

Language

Formation: linguistic forms

Interpretation: truth conditions

Reasoning

Derivation: correctness

Consequence: truth preservation

Faith: actualization, execution Fruitfulness: perseverance

3 Faith Pragmatics and Truth Let us, at first informally, analyze the episode of Nicodemus from the Gospel of John 3 in order to see how different stages of truth and its knowledge correspond to different stages of religious faith. Corresponding to the gradation (see Table 1) of (1) faith without understanding (merely “hearing”), (2) temporary, rootless faith dependent on natural conditions and threats, and (3) mixed faith within the richness of worldly life, we encounter in John 3 the following stages of knowledge: (1) materialistic knowledge in the sense of being based on outer signs, (2) the bio-naturalistic conception of man,³ and (3) rich (Pharisaic) historicist knowledge. The insufficiency of each of these three kinds of knowledge is uncovered in a dialogue with Jesus. Nicodemus’ initially claimed knowledge that Jesus has come from God turns out to be only apparent knowledge, and is three times in succes-

2 Compare the three-fold structure of logic in Table 2 with the self-defining words of Jesus (Logos, Word): “I am the way and the truth and the life” (John 14:6). 3 I have reclassified some aspects of knowledge as naturalistic the following discussion on the occasion of the conference where the talk from which this paper originates was presented (God, Truth and other Enigmas, Warsaw, September 2013).

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sion reduced to a contradiction, due to incompatibility with knowledge in a true, spiritual sense. Here is a brief summary of what happens in the dialogue of John 3. (1) Nicodemus claims to have knowledge of the kingdom of God, stating that on the ground of the signs Jesus has made, he knows that Jesus is a “teacher who has come from God”. However, according to Jesus’ reply, it is contradictory to claim knowledge of the kingdom of God on the ground of signs: “No one can see (know) the kingdom of God without being born from above” [re-born]. Hence, the pre-condition for knowledge of the kingdom of God is to be “re-born”, or “born from above” (according to the ambiguity of the Greek an¯othen). (2) Now, the requested pre-condition of (1) (“re-birth”) is conceived by Nicodemus naturalistically, leading again to contradiction. Nicodemus wonders how an old person could re-enter his mother’s womb and be born again. Jesus gives further specification of the pre-condition for “knowledge” of God: no one can enter the kingdom of God without being born of water and Spirit. He makes explicit the distinction between the required spiritual and Nicodemus’ naturalistic conceptions: what is born of flesh is flesh and what is born of spirit is spirit. (3) Nicodemus manifests the lack of knowledge about “spiritual re-birth” by asking: how can this happen? His knowledge as “a teacher of Israel” (he is a Pharisee), despite its possible entirety and richness, and although it should be knowledge about the kingdom of God, remains only a historicist knowledge of facts, lacking true understanding. There is therefore a further contradiction, consisting in an attempt to come to the knowledge of God on the ground of historicist knowledge. (4) What remains for Jesus is to instruct Nicodemus in order to lead him to an adequate understanding. Being himself a learned man and a teacher, Nicodemus should perhaps be receptive to such instruction. The dialogue has thus a tree structure, where the claimed knowledge branches (a) on the left with apparent truth, which leads to contradictions, and (b) on the right with the alternative conception, within which the truth and true knowledge should be sought (see Figure 1).

124 | Srećko Kovač N: We know that you are a teacher who has come from God.           J: One must be born from above/anew. J: No one can see   the kingdom of God       without being born   J: One must be born N: One cannot reenter from above/anew. (Materialistic

one’s mother’s womb

knowledge

and be born again.

by outer signs)

(Naturalistic knowledge)

×

×

of water and Spirit.      J. InstrucN. How can this happen?

tion,

(Historicist

interpre-

knowledge)

tation

× Fig. 1. The structure of the dialogue between Jesus and Nicodemus

4 Appearance and Truth in Religious Belief (Semantics) Since different stages of religious faith have their own truth (or apparent truth), it follows that faith pragmatics (associated to a logic) should itself be conceived and presented as some comprehensive semantics. We define such semantics (and appropriate logical language) in order to prepare the ground for a formal semantic analysis of some characteristic aspects of truth and appearance in religious belief according to the Gospel passage about Nicodemus.

4.1 Logic QB Modified We give a formal description of the language and semantics of a variant of firstorder modal logic QB according to Kovač (2013). QB is a general logic of belief by means of which we can formalize contradictions and non-completeness of be-

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liefs, and which we want to show to be suitable to give a formal account of different stages of religious belief (from apparent to true belief) mentioned above. In Kovač (2013), QB was primarily applied to model de re contradictions resulting from using different names of the same object as if they refer to different objects. Here we focus on the ambiguity of names: one name can be used to refer to different, apparent or real, objects. In accordance with the episode about Nicodemus, the objects concerned will be belief and knowledge agents themselves. Hence, we specify QB so as to identify domain objects with belief agents and call this logic QBA. Therein, belief agents with the empty accessibility relation can be understood as non-rational beings (where logic and belief in fact collapse: there are no epistemic possibilities, and hence everything is a vacuously epistemic necessity). The vocabulary of language LQBA contains individual constants c1 , c2 , c3 , . . . (set C, we will also informally use c, d, j), individual variables x, y, z, x1 , . . . (set V), descriptive predicate letters P ni (informally, G1 as well), logical predicates = and E1 , connectives ¬ and ∧, the quantification symbol ∀, the predicate abstractor λ, belief operators B t and parentheses. Symbols ∨, → and ∃ are defined in a usual way. The formulas of LQBA are Φt1 . . . t n , t1 = t2 , Et, ¬ϕ, ϕ ∧ ψ, B t ϕ, ∀x ϕ, and (λx.ϕ)(k), where ϕ and ψ are formulas, Φ n is an n-place description predicate letter, t i a term (a constant or variable), k an individual constant, and (λx.ϕ) a predicate abstract. A subscript occurrence of t in B t of B t ϕ is an occurrence of t in the formula B t ϕ (and if t is a variable, it can be bound by a quantifier). The λ-abstraction renders unambiguous the sense in which an individual constant should be understood. For example, in B c (λx.Px)(d), constant d is dependent on λ and should be understood in the sense in which agent c understands d (de dicto); in B c Pd, constant d is taken objectively and independently of agent c as well as of any other agent (de re); in B x (λy.BzPy)(c), constant c is taken in the sense in which it is understood by agent x (not necessarily also by agent z) (see Kovač (2013)). Definition 4.1 (Frame). Frame F = ⟨W , S, U, Q, {w }w∈W , {Ru }u∈U ⟩, where 1. W is a non-empty set of possible worlds (w ∈ W), 2. S ⊆ W × W (reflexive), 3. U = D ∪ A (a set of objects), where (a) D is a non-empty set (of actual and possible things), (b) A ⊆ D × C (a set of appearances) (d ∈ D, a = ⟨d, k⟩ ∈ A, u ∈ U, in addition, other bold letters instead of d will be used), 4. Q : W → ℘U \ {∅},

126 | Srećko Kovač 5. for each w, w is an equivalence relation (reflexive, symmetric, transitive) on set U, 6. Ru ⊆ W × W (serial, transitive, euclidean). We will use abbreviations: Dw = Q(w) ∩ D, Aw = Q(w) ∩ A, [u]w = {u′ |u′ w u}. Definition 4.2 (Model). Model M = ⟨F, I, I ′ ⟩, where 1. I(k) ∈ D, I(Φ n , w) ∈ ℘Un closed under w , I(E1 , w) = Q(w), informally, we will denote I(k) by ‘k’), 2. I ′ (k, w) is the smallest subset of U such that (a) there is ⟨d, k⟩ ∈ I ′ (k, w), and (b) for each ⟨d, k⟩ ∈ I ′ (k, w), d ∈ I ′ (k, w). Definition 4.3 (Variable assignment). Variable assignment is mapping v : V → U. An x-variant of the variable assignment v is variable assignment v[u/x], differing from v at most in assigning u to x. Definition 4.4 (Denotation of a term). tM,w = I(k) if t = k, v

tM,w = v(x) if t = x, v

where tM,w is the denotation of term t in model M (at world w) for variable asv signment v, and k is an individual constant. Definition 4.5 (Satisfaction). 1. M, w |=Tv t1 = t2 {︃ (∀w′ wSw′ )t1 M,w w′ t2 M,w v v iff (∃w′ wSw′ )t1 M,w w′ t2 M,w v v

if t1 M,w ∈ A and t2 M,w ∈ A, v v otherwise,

M, w |=Fv t1 = t2 ⎧ ⎪ (∀w′ wSw′ )t1 M,w ̸w′ t2 M,w v v ⎪ ⎪ ⎨ iff ⎪ (∃w′ wSw′ )(∃w′′ wSw′′ )(∃u′1 ∈ [u1 ]w′ ) ⎪ ⎪ ⎩ (∃u′2 ∈ [u2 ]w′′ )u′1 ̸w u′2

if t1 M,w ∈A v and t2 M,w ∈ A, v otherwise, where t1 M,w v = u1 and t2 M,w = u2 , v

2. M, w |=Tv Φt1 . . . t n ⎧ M,w M,w ′ ′ ′ ⎪ ⎨ (∀w wSw )⟨t1 v , . . . , t n v ⟩ ∈ I(Φ, w ) iff ⎪ ⎩ (∃w′ wSw′ )⟨t M,w , . . . , t M,w ⟩ ∈ I(Φ, w′ ) n v 1 v

if t i M,w ∈A v (1 ≤ i ≤ n), otherwise,

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M, w |=Fv Φt1 . . . t n ⎧ ⎪ (∀w′ wSw′ )⟨t1 M,w , . . . , t n M,w ⟩∈ / I(Φ, w′ ) v v ⎪ ⎪ ⎨ iff ⎪ (∃w1 wSw1 ) . . . (∃w n wSw n )(∃u′1 ∈ [u1 ]w1 ) . . . ⎪ ⎪ ⎩ (∃u′n ∈ [un ]w n )⟨u′1 , . . . , u′n ⟩ ∈/ I(Φ, w)

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if t i M,w ∈A v (1 ≤ i ≤ n), otherwise, where t i M,w = ui , v

3. M, w |=Tv Et iff tM,w ∈ I(E1 , w), M, w |=Fv Et iff tM,w ∈ / I(E1 , w), v v T F F 4. M, w |=v ¬ϕ iff M, w |=v ϕ, M, w |=v ¬ϕ iff M, w |=Tv ϕ, T T 5. M, w |=v ϕ ∧ ψ iff M, w |=v ϕ and M, w |=Tv ψ, M, w |=Fv ϕ ∧ ψ iff M, w |=Fv ϕ or M, w |=Fv ψ, 6. M, w |=Tv B t ϕ iff (∀w′ wRu w′ ) M, w′ |=Tv ϕ, M, w |=Fv B t ϕ iff (∃w′ wRu w′ ) M, w′ |=Fv ϕ, where u = tM,w , v T 7. M, w |=v ∀xϕ iff (∀u ∈ Uw ) M, w |=Tv[u/x] ϕ, M, w |=Fv ∀xϕ iff (∃u ∈ Uw ) M, w |=Fv[u/x] ϕ, 8. M, w |=Tv (λx.ϕ)(k) iff (∀u ∈ I ′ (k, w)) M, w |=Tv[u/x] ϕ, M, w |=Fv (λx.ϕ)(k) iff (∃u ∈ I ′ (k, w)) M, w |=Fv[u/x] ϕ.

Note that w-equivalent objects (objects in relation w ) need not behave in the same way with respect to the belief operator B t , since w-equivalent objects need not have the same accessibility relation Ru . As remarked in Kovač (2013), the idea of modally relativized formulas (here in relation to S-accessible worlds) is well-known in inconsistency logics (“paraconsistent” logics), as, for example, in Jaśkowski’s discussive logic (Jaśkowski (1969, 1999)), see also Béziau (2005), Béziau (2011) as well as a comprehensive overview and discussion in Nasieniewski (2011)). The idea of universal quantification under the “mode of presentation” (here, formulas (λx.ϕ)(k)) in Definition 4.5, case 8, originates from Ruili Ye (see Ye (1999) and Ye and Fitting (2002)). Definition 4.6 (Satisfiability). A set Γ of formulas is satisfiable iff there is a model M, world w and variable assignment v such that for each formula ϕ ∈ Γ, M, w |=Tv ϕ. Definition 4.7 (Consequence). Γ |= ϕ iff, if M, w |=Tv ψ for each ψ ∈ Γ, then M, w |=Tv ϕ.

128 | Srećko Kovač

4.2 Formal Analysis of John 3 We now describe a concrete model M for section John 3. Operator B t covers belief (pistis) as well as knowledge (seeing; idein, gin¯oskein). Religious belief is the belief at w1 , which includes the belief that Jesus (j) has come from God (Gj). But this Gospel section describes Nicodemus (c) in a state of unrealized religious belief, comparable to the stages 1–3 of the religious faith in the synoptic Gospels (see above). That is, we encounter Nicodemus’ belief as endangered by his materialistic and naturalistic knowledge as well as almost “choked” by his historicist knowledge (Jesus says to him: “You are a teacher of Israel and you do not understand this?”), preventing him from coming to the belief and knowledge proposed by Jesus. In M there are two worlds: 1. world w1 : all of j, j′ , ⟨j, j⟩, ⟨j′ , j⟩ are mutually equivalent (w1 ), they are all and the only members of I ′ (j, w1 ), and they are members of I(G, w1 ). 2. world w2 : – j ̸w2 j′ , j ̸w2 ⟨j′ , j⟩, ⟨j, j⟩ ̸w2 j′ , ⟨j, j⟩ ̸w2 ⟨j′ , j⟩, – I ′ (j, w2 ) = {j′ , ⟨j′ , j⟩}, – j′ ∈ I(G, w2 ), ⟨j′ , j⟩ ∈ I(G, w2 , j ∈/ I(G, w2 ), ⟨j, j⟩ ∈/ I(G, w2 ). Relations Rj , Rs , Rf , Rc for j, s, f and c (Jesus, Spirit, “flesh” and Nicodemus, respectively) as members of U, and S are presented in Figure 2 (S is dashed). Let us analyze a few statements that characterize Nicodemus’ (partly ambivalent) religious belief. M, w1 |=T B c (λx.Gx)(j)

(1)

(‘Nicodemus believes that Jesus has come from God.’) Proof. In each c-accessible world from w1 , i.e. in w2 , (λx.Gx)(j) should be true. That means that for each u ∈ I ′ (j, w2 ), i.e. for j′ and ⟨j′ , j⟩, M, w2 |=Tv[u/x] Gx. Now, according to the definition of satisfaction, for j′ it suffices that in at least one world w that is S-accessible from w2 , j′ ∈ I(G, w), and this holds for w = w1 , as well as for w = w2 . For ⟨j′ , j⟩, ⟨j′ , j⟩ ∈ I(G, w) should hold for each w that is S-accessible from w2 , which is in fact the case in our model M. Therefore, B c (λx.Gx)(j) is true at w1 . M, w2 |=T B c (λx.Gx)(j)

(2)

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Fig. 2. Relations between worlds in M

(‘Possibly, Nicodemus believes that Jesus has come from God.’ ‘Possibly’ refers to possible world w2 .) Proof. Since the only world that is c-accessible from w2 is w2 itself, the rest of the argumentation is the same as in (1). M, w1 |=T B c B c (λx.Gx)(j)

(3)

(‘Nicodemus is aware of his belief that Jesus has come from God.’) Proof. This easily follows from (2) since w2 is the only world c-accessible to w1 .

M, w1 |=T B c ¬(λx.x = j)(j)

(4)

(‘It is not the real Jesus about whom Nicodemus believes that he is Jesus.’) Proof. This means that at each world w that is c-accessible from w 1 , the formula (λx.x = j)(j) is falsified. World w2 , which is the only world c-accessible form w1 , falsifies (λx.x = j)(j) since there is at least one u ∈ I ′ (j, w2 ) for x, namely j′ , which does not satisfy x = j at w2 . This can be shown in the following way: w2 is Saccessible to itself, and on the one side, j′ ∈ [j′ ]w2 since, of course, j′ w2 j′ , while on the other side, analogously, j ∈ [j]w2 ; but j′ ̸w2 j; hence, according to the definition of the falsification of identity formulas, M, w2 |=Fv[j′ /x] x = j. In fact, with j′ for x, formula x = j is falsified in w2 by any S-accessible w (since for each

130 | Srećko Kovač w, j′ ∈ [j′ ]w and j ∈ [j]w ). Thus, (λx.x = j)(j) is falsified and ¬(λx.x = j)(j) verified at w2 , and therefore B c ¬(λx.x = j)(j) is verified in w1 . Moreover, M, w1 |=T B c (λx.¬x = j)(j)

(5)

(‘It is the real Jesus of whom Nicodemus believes that he is not Jesus.’) Proof. In each c-accessible world from w1 , i.e. in w2 , it should be true that (λx.¬x = j)(j). This means that ¬x = j should be satisfied, and hence x = j falsified at w2 by each u ∈ I ′ (j, w2 ), i.e. by j′ and by ⟨j′ , j⟩ as well. For j′ , we have already shown this in the proof of (4). Similarly, x = j is falsified at w2 since ⟨j′ , j⟩  ̸ w2 j (this non-equivalence follows already from ⟨j′ , j⟩ ∈ I(G, w2 ) and j ∈/ I(G, w2 ), because, according to the definition of model, I(Φ n , w) should always be closed under w ). This proves that at w1 , B c (λx.¬x = j)(j) is true. M, w1 |=T ¬B c Gj

(6)

(‘It is not so that about the real Jesus Nicodemus believes that he has come from God.’) Proof. As we have mentioned in the proof of (5), j ∈/ I(G, w2 ), and w2 is caccessible from w1 . It even holds that M, w1 |=T B c ¬Gj, since no world other than w2 is c-accessible to w1 (see the proof of (6)). M, w1 |=T B c ¬j = j

M, w1 |=T B c j = j

(7)

(‘Nicodemus believes about Jesus that he is and that he is not self-identical.’) Proof. The left proposition follows from the facts that ⟨j′ , j⟩ ∈ [j]w1 and j ∈ [j]w1 (j ∈ [j]w2 as well), w1 (and w2 ) being S-accessible to w2 , whereas j′ ̸w2 j. The right proposition follows from the definition of the frame (u w u for any w). M, w1 |=T B j ¬(λx.B c Gx)(j)

(8)

(‘Jesus believes (knows) that it is not about him that Nicodemus believes that he has come from God.’)

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Proof. In w1 (the only j-accessible world to w1 ), sentence (λx.B c Gx)(j) is false since for some u ∈ I ′ (j, w1 ), i.e. for j (as well as for ⟨j, j⟩), Gx is falsified at w2 (the only c-accessible world form w1 ), which is S-accessible to itself. M, w1 |=T B j (λx.B c (λy.¬x = y)(j))(j)

(9)

(‘Jesus believes (knows) that about him Nicodemus believes that he is not Jesus.’) Proof. In each j-accessible world to w1 , i.e in w1 , formula (λx.B c (λy.¬x = y)(j))(j) holds, since for each u ∈ I ′ (j, w1 ), i.e. for j, ⟨j, j⟩, j′ , and ⟨j′ , j⟩ as values of x, B c (λy.¬x = y)(j) is satisfied. The reason is that in the only c-accessible world from w1 , in w2 , objects j, ⟨j, j⟩, j′ , and ⟨j′ , j⟩ for x satisfy (λy.¬x = y)(j). Namely, all four objects for x satisfy ¬x = y, i.e. falsify x = y for each u ∈ I ′ (j, w2 ) as a value for y. These values are j′ and ⟨j′ , j⟩. Thus, we have the following cases: (a) M, w2 |=Fv[j/x,j′ /y] x = y, (b) M, w2 |=Fv[j/x,⟨j′ ,j⟩/y] x = y, (c) M, w2 |=Fv[⟨j,j⟩/x,j′ /y] x = y, (d) M, w2 |=Fv[⟨j,j⟩/x,⟨j′ ,j⟩/y] x = y, (e) M, w2 |=Fv[j′ /x,j′ /y] x = y, (f) M, w2 |=Fv[j′ /x,⟨j′ ,j⟩/y] x = y, (g) M, w2 |=Fv[⟨j′ ,j⟩/x,⟨j′ ,j⟩/y] x = y. The respective reasons are: (a) j ̸w2 j′ ,

̸ w2 j′ , (d) ⟨j, j⟩  ̸ w2 ⟨j′ , j⟩, each of them together with (b) j ̸w2 ⟨j′ , j⟩, (c) ⟨j, j⟩  u ∈ [u]w (for each u and w) and w2 Sw2 ; further, together with w2 Sw1 and j ̸w2 j′ : (e) j ∈ [j′ ]w1 and j′ ∈ [j′ ]w1 , (f) j ∈ [j′ ]w1 and j′ ∈ [⟨j′ , j⟩]w1 , (g) j ∈ [⟨j′ , j⟩]w1 and j ′ ∈ [⟨ j′ , j ⟩ ] w 1 .

In John 3, Jesus’ way of seeing (knowledge, belief) is conceived as spiritual, while Nicodemus’ way is conceived as ambivalent between Spirit and flesh. Formally, we have subsumed j-accessibility under the spiritual, s-accessibility, and c-accessibility partly under f-accessibility. Hence, Nicodemus could achieve true belief, which turns out to be knowledge (with reflexive accessibility), by abandoning f-accessibility and changing his c-accessibility for s-accessibility = {⟨w1 , w1 ⟩}, i.e. in the Gospel words, he should be “re-born” in Spirit.

Bibliography Béziau, J. -Y. (2005), “Paraconsistent logic from a modal viewpoint”, in Journal of Applied Logic: 3, 7–14. Béziau, J. -Y. (2011), “A new four-valued approach to modal logic”, in Logique et Analyse: 54, 109–121. Jaśkowski, S. (1969), “Propositional calculus for contradictory deductive systems”, in Studia Logica: 24, 143–157. Jaśkowski, S. (1999), “A propositional calculus for inconsistent deductive systems”, in Logic and Logical Philosophy: 7, 57–59.

132 | Srećko Kovač Kovač, S. (2007), “Contradictions, objects, and belief”, in Perspectives on Universal Logic, edited by J. -Y. Béziau and A. Costa-Leite, Monza – Milano: Polimetrica, 417–434. Kovač, S. (2011), “Logika i vjera”, in Suvremena znanost i vjera [Contemporary Science and Faith], editor-in-chief Z. Primorac, Mostar: Sveučilište u Mostaru; Ljubljana: Univerza v Ljubljani, 69–84. Kovač, S. (2013), “Kako logički objasniti mnijenje?”, in Nasljeđe antike : ogledi u spomen Maji Hudoletnjak Grgić, edited by D. Pećnjak, P. Šegedin, and K. Zakarija, Zagreb: Kruzak, Institut za filozofiju, 131–152. Nasieniewski, M. (2011), Logiki zdaniowe wyrażalne przez modalność, Toruń: Wydawnictwo naukowe Uniwersytetu Mikołaja Kopernika. Nestle-Aland Novum Testamentum Graece, (2012), B. Aland, K. Aland et al., 28. rev. edition, Stuttgart: Deutsche Bibelgesellschaft. New American Bible, Revised Edition, (2011), Charlotte, NC: Saint Benedict Press – Also: New American Bible, 2002, http://www.vatican.va/archive/ENG0839/_INDEX.HTM Ye, R. (1999), Belief, Names and Modes of Presentation: A First-Order Logic Formalization, PhD Thesis, City University of New York. Ye, R. and Fitting, M. (2002), “Belief, Names, and Modes of Presentation”, in Advances in Modal Logic, vol. 3, edited by F. Wolter et al., New Jersey: World Scientific, 389–408.

Anna Lemańska

Absolute Truth and Mathematics In the contemporary Western culture there has been adopted a relativistic view on values. One of them is truth, which lost its absolute character, and changed from the objective, universal point of reference into one of goods, subject to market rules. The quest for truth no longer matters, because any view can be regarded to be true by someone. The problem of finding the irrefutable criteria for checking what is true and what is false was eliminated in the easiest way: everyone can accept as true the things which suit him best at a given moment. Thus we witness the postmodern crisis of absolute truth. I will try to show that such a pessimistic view of truth is not a full one. Absolute true statements do exist. We find them, for example, in mathematics. Although there are mathematical theorems, which are not absolute true, nonetheless some mathematical theorems depend neither on the time nor on the place or culture. Therefore, there are the absolute, invariable, objective truths in mathematics. To demonstrate this I will analyse the following five mathematical theorems:¹ (1) The Continuum Hypothesis: The arbitrary infinite subset of the set of real numbers is equipollent with the set of natural numbers or with the set of real numbers; (2) The sum of angles in an arbitrary triangle is equal to the sum of two right angles; (3) 2 + 2 = 4; (4) For n > 2 the equation: x n + y n = z n has no solution within the set of integers greater than 0; (5) Each even natural number bigger than two is a sum of two prime numbers. The first sentence was formulated by Georg Cantor in 1878. In 1937, Kurt Gödel (Gödel (1939), 220–224) proved, that the continuum hypothesis is consistent with Zermelo-Fraenkel axiomatic set theory, and in 1963, Paul Cohen (Cohen (1963), 1143–1148; Cohen (1964), 105–110) built up the model for the set theory within which the continuum hypothesis is not true. Thus, the truth of the continuum hypothesis depends on the axiomatic system. What we have here is almost classical case of the coherent understanding of truth: there is no point to talk about the

1 The broader analysis of these sentences is in Lemańska (2011), 95–116.

134 | Anna Lemańska truth of the continuum hypothesis; instead, we can talk about the consistency of the system within which this hypothesis is accepted. The second sentence is the theorem of Euclidean geometry and, as we know, it is equivalent to the so-called fifth Euclid’s postulate.² That’s why talking about the truth of this sentence starts causing troubles. If we accept Euclidean geometry, this sentence is obviously true. Yet, if we accept non-Euclidean geometry, this sentence is no longer true. Again, the coherent nature of truth in mathematics reveals itself here. However, in the case of geometric theorems, we can try to go beyond the axiomatic-deductive system and try to employ the use of geometry in physics with the aim of judging their truthfulness. In the case of the sum of angles in a triangle, there seems that a straightforward test exists: we can simply measure the sum of angles in a triangle and check whether this sum equals the sum of two right angles or not. This measurements were performed many times and they proved that the fragment of space, directly surrounding us, including errors in measurements, is Euclidean. Therefore, the measurement seems to point to the truth of the sentence about the sum of angles in a triangle. What is more, a man, as a representative of Homo sapiens, experiences the surrounding space as the Euclidean one. What is more, recognizing in a correct way the spatial relationships is a necessary condition of survival. Thus, a human being was, in the course of evolution, equipped with the tool of recognizing “geometry structures” in the environment around him. Since, the space of this environment is almost Euclidean, these inborn structures have to reflect the properties of this very space; because otherwise, a human being would have a false picture of his environment and could not survive within it.³ Moreover, some properties of geometric figures within Euclidean geometry are obvious for us, while the properties of the objects from other system contradict our intuitions. It is quite easy to imagine the non-Euclidean surfaces, because we can put them within the Euclidean, three-dimensional space, but to imagine the three-dimensional, non-Euclidean space seems impossible at least to me. Probably, if we lived near some black hole, theorems of Euclidean geometry would seem for us strange and inconsistent with our geometric intuitions, but we live in space

2 The fifth postulate, formulated by Euclid in the Elements of Geometry is as follows: if a straight line falling on two straight lines makes the interior angles on the same side less than two right angles, the two straight lines, if produced indefinitely, meet on that side on which are the angles less than the two right angles. 3 Konrad Lorenz, among others, pays attention to it Lorenz (1973).

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the curvature of which we are unable to notice, thus, the flat Euclidean space is something natural for us. Therefore, the truth of the theorem of the sum of triangle angles is, in a way, confirmed empirically. Thus, there is some additional motivation, from beyond the axiomatic system, which forces us to accept that this theorem is true. The third sentence is often quoted as a sentence illustrating the certainty of mathematical knowledge. Obviously, we can look at this sentence as at some theorem of a given axiomatic-deductive system, e.g. Peano arithmetic. Yet, contrary to the first and second sentence, there exists no other arithmetic system in which 2+2 would not equal 4. If we understand the meaning of each symbol in this equation, we have to acknowledge the fact that it describes the actual state of affairs. What we have here, therefore, is the irrefutable, objective truth, independent of any condition, in particular, of a given axiomatic-deductive system. If within a given algebraic system we use the same symbols, but we get the result that e.g. 2 + 2 = 1, it means nothing more than that we understand the same symbols in a different way, namely, instead of the “ordinary” addition we have here the addition modulo 3, hence the sign “+” means something different from its meaning within the natural numbers arithmetic. The sentence 2 + 2 = 4 is obvious for everyone who understands these symbols. Hence its truth has the source other than the mere consistency of a given system. Moreover, it is possible to build up different geometries; building up new “arithmetic”, within which natural numbers, our “true” natural numbers, would have different properties, is impossible, because the construction of the series of natural numbers by adding one to the previous number, determines the so-called intended, standard model⁴ for the arithmetic of natural numbers. The operation of adding number one is intuitively and empirically tangible for us. Natural numbers are given directly. They occupy special place in mathematics. This seems to be the basis for the conviction of the truth of such sentences as: 2 + 2 = 4. Arithmetic of natural numbers may, obviously, be presented in the form of formal, axiomatic-deductive theory. Moreover, it follows from the Gödel theorem that there are undecidable sentences and non-standard models for Peano arithmetic. Yet, all those non-standard models contain the part including standard natural

4 Intuitionally, the intended model of a given mathematical theory is the domain of such mathematical objects, which are regarded as the “true” objects described by this theory. The intended model for the arithmetic of natural numbers is the natural numbers constructed by adding 1 to each subsequent number starting from 0. It is usually the case that if a given theory has the intended model, it also has many non-intended, non-standard models. Frequently, the structure and elements of non-standard models are substantially different from the intuitions associated with the elements belonging to the standard ones.

136 | Anna Lemańska numbers, i.e., the “true natural numbers”. This part of a non-standard model is identical with the standard one. From this viewpoint, the non-standard numbers are artificial and non-natural. The fourth and fifth sentences strengthen additionally this claim. The fourth sentence is the so-called Fermat’s last theorem. As the problem is interesting and easy to understand, mathematicians (and not only them) made numerous attempts to find the proof of this theorem. They failed, although some partially successful solutions were found. No one found any counterexample, either. In 1995, after various attempts, Andrew Wiles (Wiles (1995), 443–551) published the proof of Fermat’s theorem. Therefore, the truth of Fermat’s theorem was proven. It also means that the negation of this sentence is false. It is worth noting that Wiles’ proof, supplemented by Robert Taylor, uses the theory of elliptic functions,⁵ hence refers to the facts from beyond arithmetic.⁶ So, the problem concerns the properties of natural numbers, yet various branches of mathematics are used while solving it. Although this proof is the deductive one, hence it uses the already proven theorems, (facts known to mathematicians), it is clear that it goes beyond the framework of a certain established axiomatic system. Such a situation is a typical one: mathematicians go beyond the limitations imposed by the axioms of some deductive systems and, proving theorems, they use the results gained sometimes within rather ‘remote’ branches of mathematics. If they have some problem to solve, they look for all the possible ways of finding the solution, not bothering themselves too much to stay within a certain, strictly defined axiomatic system (Murawski (1990), 173–174). Thus, in the case of Fermat’s theorem, it is difficult to talk merely about truth in the sense of coherence, because its proof refers to the facts from various branches of mathematics. What is more, the truth of Fermat’s theorem is objective in character; it is not conditioned by the mere consistency of some axiomatic-deductive system. The search for the proof of Fermat’s theorem shows that mathematicians were convinced of the existence of a clear answer to the question about the truth of this sentence. They were convinced that either the natural numbers have this property, or there is a counterexample. Similar comments can be related to the last sentence. The fifth sentence is the so-called Goldbach’s hypothesis, formulated in 1742 and stating that every even number bigger or equal to 4 is the sum of two primes (Ribenboim (1996)). It is one of the older mathematical problems, not solved so

5 The general form of the equation of elliptic function: y2 = ax3 + bx2 + c, where a, b, c are rational numbers. 6 For the history of searching for the proof of Fermat’s last theorem see, e.g.: Narkiewicz (1993), Aczel (1996), Ribenboim (1999).

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far. For it was neither proved, nor any counterexample was found, its truthfulness or falseness cannot be determined. Nevertheless, this sentence concerns the properties of natural numbers, so it has to have the well-determined logical value, which we still do not know. The continuum hypothesis is the sentence, to which attributing the truth makes no sense. There are, therefore, such fields in mathematics, which are not fully determined either by the theory or model. So to say, different “mathematical worlds” with distinct properties exist side by side. In this context, we are unable to talk about the absolute truth. From this viewpoint, neither the continuum hypothesis nor the theorem of the sum of triangle angles is absolutely true. We can only talk about their consistency with a certain system of formulas or about their truth but only within a given model. This is often referred to as an argument against the absolute truth in mathematics. Moreover, the formalistic approach towards mathematics leads inevitably to understanding truth in a relative way, because there is not any external, beyond the theory, point of reference. However, the approach towards mathematics, which ignores the content of theories, understanding and truth of sentences, concentrating, instead, merely on the consistency of the system, is inadequate. Obviously, the consistency of a theory is the necessary condition of its further development, but it is not a sufficient condition. What matters for mathematicians is, first of all, the meaning of a theory, which makes it interesting for some reason, e.g. the theory has significant applications, extends the scope of research, answers theoretical questions and so on. In all those cases, the most important thing is the truth of the theorems related to some reality other than the language and theory itself. The meaning may appear itself in the standard model, which enables the comparisons of the content of the formula with the reality of mathematical objects; it can also appear itself beyond this model. As far as the continuum hypothesis is concerned, Cantor, who formulated the set theory, thought that there had to be some solution for it. He had an intuitive view of what the set is. He supposed that this view provides the basis for verifying the truth of the continuum hypothesis and other sentences of this type. Nevertheless, Cantor’s expectations cannot be fulfilled and, in the case of continuum hypothesis, we are still on the level of the relative, coherent truth. Yet, in the case of the theorem concerning the sum of angles in the triangle, it would be too much of a simplification to focus just on the coherence of the geometrical system and ignore its truth. This theorem is the consequence of the postulates of Euclidean geometry, which are intuitively obvious for us. Within a non-Euclidean geometry, the sum of the angles in a triangle will be either less or more than the sum of the two right angles. Nevertheless, for an average human being such statements will

138 | Anna Lemańska contradict his geometric intuitions. Thus, acknowledging the second sentence as the true one is based on our human experience. Fermat’s theorem, in turn, is not obvious, as far as our experience is concerned. Its proof is not straightforward, either. Yet, in this case, we do have the point of reference, which we relate its truth to: there exists the standard model of natural numbers. So the question whether there exist (or not), among standard natural numbers, the following natural numbers m, k, l and n > 2, satisfying the equation: m n + k n = l n makes sense. This question has to have one, unambiguous answer. The situation here is different from the one with continuum hypothesis. We cannot answer the question about the existence of the set with the cardinality intermediate between the cardinality of the set of natural numbers and of that of the real numbers, because there does not exist the intended standard model for the set theory. In this case, “constructing” the universe of sets is of no help.⁷ Lack of the intended standard model for set theory is also, as it seems, the source of the differences between continuum hypothesis and the sentence 2 + 2 = 4. If the standard intended model for a given theory exists, the truthfulness or falseness of the independent sentences, the existence of which follows from Gödel’s theorem, can be determined in the course of examining this standard model. In this sense, we can say that from the first Gödel’s theorem there follows the existence of the true sentences with no proof. The first order Peano arithmetic is incomplete, but it has its standard model; the Zermelo-Fraenkel set theory is also incomplete but it does not have its standard intended model, which would be a reference point for the truth of the independent sentences. Within this framework, Goldbach’s hypothesis has its logical value of being true or false, too, yet, for the time being, we are unable to determine it. In the case of natural numbers, the operation of adding 1 is possible to comprehend for us, both intuitively and empirically. The operation of creating the power set for the infinite sets, however, cannot be defined in the way which would enable the precise definition of all the elements of this set. The standard model for natural numbers is possible to comprehend intuitively, while, in the case of set theory, it is difficult to talk about any special intended model. Therefore, the truth can be treated as the provability within a certain system, on the one hand, but on the other, we cannot ignore the questions concerning the relationships and properties of particular mathematical objects, which are given to us, in some way, independently of any formal or axiomatic-deductive systems.

7 This construction is performed e.g. through the operation of taking the power set and through the operation of sum. We obtain the following sets hierarchy (indexed with ordinals): V0 = ∅, ⋃︀ V α+1 = P(V α ) and V λ = α m → |S(n)∖E|) ≤ ℵ0 )}

F is a proper filter. The topology defined over the S is obtained by adding ∅ to F. This topological space is connected, but not Hausdorff (thus it is not metrizable). The family F represents God. The intuition behind this approach is that God is the upper limit of human understanding. The filter F is a set of all the “big enough” sets of meanings: i.e., F will include those sets of meanings in S which are so large, that they define the quasitotality of all human poetics competences. Namely, a subset E of S will belong to F if E includes nearly all meanings perceived by the quasi-totality of human beings. Calude et al. (1995), p. 8.

The quasi-totality of human beings is a set of all human beings without some finite subset, a set E includes nearly all meanings from S(n) if the power of the S(n)∖E is at most countable. The connectedness of the space is interpreted as the unity of God. It is worth noting that this space is compact, locally connected, hyperconnected (has no open disjoint sets) and each point of it is a limit point of any infinite subset E ⊆ S (see Steen and Seebach (1970), pp. 50–51).

6 Authors focus on poetic meanings, but this approach can be easily generalized to meanings in a broad sense, not only poetics.

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5.4 Topological Analysis of Unity of God One can understand God’s unity in two ways: unity as a part (sensu largo) of God and unity as the unicity that is to say there are no other gods. We want to analyse the first sense of God’s unity. E. Husserl in the third of his Logical Investigations distinguished between two fundamental types of being a part: (1) being a dependent part and (2) being an independent part. If we consider a white closed ball then roughly speaking a half of the ball is a piece of it, an independent part of it. But the whiteness of the ball is a dependent part of the ball, as Husserl would say, the moment of the ball. Husserl – what may be surprising at first glance – used the concept of being a part in a wide sense, namely: We interpret the word ‘part’ in the widest sense: we may call anything a ‘part’ that can be distinguished ‘in’ an object, or, objectively phrased, that is ‘present’ in it. Everything is a part that is an object’s real possession, not only in the sense of being a real thing, but also in the sense of being something really in something, that truly helps to make it up: an object in itself, considered in abstraction from all contexts to which it is tied, is likewise a part. Husserl, 2001 (1900), p. 5.

We claim, as Husserl argued in his Logical Investigations, that the unity of the object is the part of it which is founded in all parts of the object. The moment of unity connects all parts of the object, they are all in some sense together. If we accept this analysis of unity, we can try to model it in terms of topology. The first possibility is that the unity of the space under consideration (we assume that God is a topological space) is connectedness of that space. But some of the connected spaces are strange and have undesirable properties. An example is the Topologist’s Sine Curve {(x, sin 1x ) : x ∈ (0, 1]} ∪ {(0, 0)}. In this space there is no path (a path in space X is a continuous map f from the unit interval to X) from the origin to the remainder of the space. We need stronger varieties of connectedness, for example, arc connectedness, path connectedness or simply connectedness (e.g. coffee cup with a handle is arc and path connected but not simply connected because it has a hole). But even strong connectedness is not sufficient for unity. Compactness also appears in the context of research for the unity of God. The compactness of a topological space is associated with finiteness; every open cover of space should have a finite subcover in compact space⁷. As in the case of connectedness there are many

7 This combination of finiteness with the infiniteness is often interpreted in the context of the finiteness of human and infiniteness of God.

250 | Bartłomiej Skowron types of compactness, for example pseudocompactness or countable compactness. It is worth noticing that in n-dimensional Euclidean space Rn compactness is equivalent to both closedness and boundness. If we assume that the unity of the space requires connectedness and compactness, our attention narrows down to space known as continuum⁸.

6 The Explanatory Power and Limits of Topological Explanations Ochiai argues in Ochiai (2010), p. 26, that likening God to a topological space is to consider topological space as a metaphor for God. He argues that this is the way to find hidden attributes that belong to the object in question. By means of metaphors we can discover what we did not know before. In this connection it is important to emphasise that we use metaphors when the object is unknown to us and when we want to apprehend and approach it. When we use topology to model some aspects of God, we can distinguish more differences in the ontological structure of God. For example we can distinguish between different sorts of compactness and connectedness, we can analyse the issue of finiteness/infiniteness of God from the perspective of compactness, we can also, just like Ochiai did, try to resolve the paradoxes. Any attempt to grasp a cognitive field requires the right tools. If we want to know God or come close to knowing Him, we should try all of the possible cognitive tools, even if they have to be just Cusanus’ opposition or Wittgenstein’s ladder. Mathematical (or even topological) theology is in our opinion the mathematical modelling of theological issues, in particular the issue of God, the issue of the creation, etc. Each kind of God’s model is some of His aspects; it is the way we can think of Him. Modelling is some kind of simulation of God and through newer and newer models it is becoming more and more accurate. Mathematics is not just mathematical language, symbols or expressions. Mathematics provides material for thought, i.e. complex models and rich structures through which we obtain real knowledge. The purpose of the use of topology in the philosophy (of God) is not to make philosophy more “scientific”. It means that we do not want to treat philosophy as “hard” science, like physics. But science such as mathematics may be helpful

8 A continuum is a nonempty compact connected space. Sometimes a continuum is required to also be metric or Hausdorff space.

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in resolving philosophical and theological problems, and in particular theological paradoxes. Of course only if we agree to the fact that some of the theological beliefs are intended to mean propositions. There are some limitations and shortcomings of the topological and generally mathematical explanations used in philosophy and theology. First of all, we claim that the motivation for the selection of structures should be clearly presented. Sometimes, however, the topological structures selected to explain a theological paradox have been chosen randomly. One can ask then why this kind and no other structures were selected? There is an enormous number of structures in mathematics. Why is the space of natural numbers chosen by H. Ochiai? In topology there are many infinite and at the same time bounded spaces⁹, a simple example is the line segment (0, 1) ⊂ R. This example is simple but really interesting in this context, because the space (0, 1) is homeomorphic to the real line R. From the topological point of view, these objects are the same, despite the fact that the first is bounded and the second is not. Being bounded is not the topological property; strictly speaking it is not invariant under homeomorphisms. That is the reason why one should take into account not only the considered space, but also all of its homeomorphic equivalents. That is why we claim that the motivation for the selection of structures should be clearly presented. One should be particularly careful in the selection of structures and be aware of all the mathematical consequences of this choice. Another general issue is why we choose the topological and not other mathematical structures? Contemporary mathematics – as it can be argued – provides many other structures such as category from category theory, Banach spaces from functional analysis, probability space from probability calculus, etc. It seems that the choice of the mathematical field is dictated by the ontological characteristics of the object in question and the practical (explanational) purposes. There is some evidence to suggest that the predecessor of applying topology to philosophical meditations (in particular in ontology) was the application of geometry. De docta ignorantia of Nicholas of Cusa and Timaeus of Plato can serve as illustrations of that kind of application. It could be reasonably argued that if Cusanus had lived in our century, he would have used topology instead of geometry. His discussion about spheres, circles, lines, triangles and his treatment of the universe as a “contracted” Maximum is essentially very close to topology. Topology (earlier geometry) is successfully used in ontology because both of them explore the spatial phenomena, most generally speaking. This is not to say that topology is the only

9 In this context, the set A is said to be bounded if it is contained in a ball of finite radius.

252 | Bartłomiej Skowron possible instrument for ontology. Successfully and without any problems one can make use of other mathematical concepts. The one receives the other sides of God. One could make another objection. Suppose that God is topological space ω. In that case what is the ω×ω or disjoint union of ω and ω ∪{ω}? Are they also God? What are then the other spaces nonhomeomorphic with this one? These types of questions are not difficult to reject. If we are claiming that God is a topological space then we do not want to say that God is only the topological space. Ochiai would say that it is only a metaphor. We would say that being a topological space is one of the sides (dependent parts) of God. From that point of view the abovementioned combinations of sides do not have to be a side. And other spaces may be, but do not have to be, other sides of God. Based on an increased number of the known sides of God one can better apprehend Him.

Bibliography Bocheński, J. M. (1965), The Logic of Religion, New York. Broda, B. (2004), “Fizyka i topologia” [“Physics and Topology”], in Postępy Fizyki: 55(3), 120– 122. Bourbaki (1989/1971), Elements of Mathematics. General Topology, Chapters 1–4, Springer. Calude, C., Marcus, S. and Stefanescu, D. (1995), The Creator versus its Creator. A Mathematical Exercise, Preprint. Coyne, G. and Heller, M. (2008), A Comprehensible Universe. The Interplay of Science and Theology, New York: Springer Verlag. Nicholas of Cusa (1990/1440), De docta ignorantie (On Learned Ignorance), trans. by J. Hopkins, Internet source: http://jasper-hopkins.info. Duda, R. (2001), “Początki topologii w Polsce” [“The Origins of Topology in Poland”], in Monografie Komisji Historii Nauki PAU, t. 4, Kraków, 106–112. Fine, K. (2006), “Part–Whole”, in The Cambridge Companion to Husserl, edited by B. Smith and D. W. Smith, Cambridge. Heller, M. (1996), Mechanika kwantowa dla filozofów [Quantum Mechanics for Philosophers], Kraków: Biblios. Husserl, E. (2001/1900), Logical Investigation, vol. II, translated by J. N. Findlay, edited by D. Moran, New York. Kulpa, W. (2009), Topologia a ekonomia [Topology and Economics], Warszawa: Wyd. Univ. Stefana Wyszyńskiego. Kuratowski, K. (1961), Introduction to Set Theory and Topology, translated by L. Boron, Warszawa. Lewin, K. (1936), Principles of Topological Psychology, translated by F. Heider and G. Heider, New York and London. Mormann, T. (2013), “Topology as an Issue for History of Philosophy of Science”, in New Challenges to Philosophy of Science, The Philosophy of Science in a European Perspective 4, edited by H. Andersen et al., Dordrecht, 423–434.

The Explanatory Power of Topology in the Philosophy of God | 253 Ochiai, H. (2010), “Theology of Suffering and Topology as Metaphor”, in Religion a Human Phenomenon, XXth World Congress International Association History of Religion, Book of Abstracts, Toronto, 133–134. Ochiai, H. (2012), “The Theology of Simone Weil and the Topology of André Weil”, in Religious Studies in Japan: 1, 25–38. Runde, V. (2005), A Taste of Topology, Springer. Skowron, B. (forthcoming), “Mereotopology”, in Handbook of Mereology, edited by H. Burkhardt, J. Seibt and G. Imaguire, Philosophia Verlag. Smith, B. and Żełaniec, W. (1996), “Topology for Philosophers”, in The Monist: 79(1). Steen, L. and Seebach, A. (1970), Counterexamples in Topology, New York: Holt, Rinehart and Winston.

Authors of Contributed Papers C. Anthony Anderson is Professor of Philosophy at the University of California at Santa Barbara. He earned his PhD in philosophy from University of California at Los Angeles under the direction of Alonzo Church. He also holds an M.S. in mathematics from the University of Houston, where he earned his undergraduate degree in physics and mathematics. Prof. Anderson’s work over the years has focused primarily in the philosophy of logic and the philosophy of language, although he also works in such areas as the philosophy of religion and has an interest in most areas of traditional philosophy. An example of his work that combines his skill at logic with his interest in the philosophy of religion appears in his papers on Gödel’s ontological proof. Among his publications, about 40 in number, the following five articles are more significant ones: Some New Axioms for the Logic of Sense and Denotation (1980), The Paradox of the Knower (1983), Russellian Intensional Logic (1989) – reprinted in Philosopher’s Annual as one of the ten best philosophical papers of 1989, Some Emendations of Gödel’s Ontological Proof (1990), Alternative: A Criterion of Identity for Intensional Entities (2001). During 2010-11 he was at Hertford College, Oxford University, on a Templeton Fellowship. Seweryn Blandzi is Professor of Philosophy at IFiS PAN (Institute of Philosophy and Sociology Polish Academy of Sciences). He is Head of a Research Group for Ancient Philosophy and History of Ontology and Editor-in-Chief of Archive of the History of Philosophy and Social Thought. Research interests: history of philosophy (mainly ancient philosophy), history of metaphysics and its transformation into modern ontology, German philosophy and hermeneutics. He published (inter alia): Henologia, meontologia, dialektyka: Platońskie poszukiwanie ontologii idei w ‘Parmenidesie’ [Henology, Meontology, Dialectic: Plato’s Searching for Ontology of Ideas in ‘Parmenides’](Warszawa 1992), Platoński projekt filozofii pierwszej [Plato’s Project of the First Philosophy ](Warsaw 2002), Między aletejologią Parmenidesa a ontoteologią Filona. Rekonstrukcyjne studia historyczno-genetyczne [Between Parmenides’ Aletheiology and Philo’s of Alexandria Ontotheology. Reconstructive Historical and Genetic Studies] (Warsaw 2013). Christopher Daly is Professor of Philosophy at the University of Manchester. His principal research interests are in metaphysics, philosophical methodology, and philosophy of language. He is the author of Introduction to Philosophical Methods (Broadview, 2010) and Philosophy of Language: An Introduction (Bloomsbury,

256 | Authors of Contributed Papers 2012). He has published papers on natural kinds, properties, truthmaker theory, grounding, philosophy of mathematics, and fictionalism. He is an Associate Editor of the Australasian Journal of Philosophy. He and David Liggins are currently directing a research project entitled The Foundations of Ontology, funded by the UK Arts and Humanities Research Council. Their contribution to this volume is part of this project. Stamatios D. Gerogiorgakis teaches philosophy and religious studies at the University of Erfurt, Germany. He earned his PhD in philosophy from the University of Munich, and was habilitated at the University of Erfurt. Gerogiorgakis’ main interests are the philosophy of religion and the history of logic. Among others he has published three philosophical monographs: Die Rolle des Schematismuskapitels in Kants Kritik der reinen Vernunft (1998), Necessary Being and Causal Determination (2002 – in Modern Greek), Zeitphilosophie im Mittelalter (2006); and some ten articles which combine topics in logic with religion. Christian Kanzian is a Professor of Philosophy at the University of Innsbruck, Austria. He is, since 2006, the president of Austrian Ludwig Wittgenstein Society (Österreichische Ludwig Wittgenstein Gesellschaft). Kanzians’ main interests are the analytic philosophy, the history of philosophy, and ontology. He is the author of four books: Originalität und Krise – Zur systematischen Rekonstruktion der Frühschriften Kants (1994), Grundprobleme der Analytischen Ontologie (with E. Runggaldier, 1998), Ereignisse und andere Partikularien (2001), Ding – Substanz – Person. Eine Alltagsontologie (2009), and an editor or co-editor of eleven books. He has also published over seventy articles. Srećko Kovač (PhD in Philosophy, University of Zagreb) is affiliated to the Institute of Philosophy (Zagreb, Croatia), doing research in philosophical logic (Logical Structures and Intentionality; Logic, Language and Modalities). Teaches basic and advanced courses in logic at the University of Zagreb. He is the author of Logical Questions and Procedures (with B. Žarnić, 2008), Logical-Philosophical Papers (2005), Logic (1994, 2nd ed. 2004), and Logic as a “Demonstrated Doctrine” (1992), in Croatian. He has published about 40 papers in international and Croatian journals and collections of papers. His current interests include logic of belief and knowledge, logic and ontology, logic of group decision-making, and history of logic. Anna Lemańska is Associate Professor of Philosophy at the Institute of Philosophy, Cardinal Stefan Wyszyński University in Warsaw. She is a head of Department of Philosophy of Nature and a deputy editor of a quarterly Studia Philosophiae

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Christianae. She is the author of the book: Philosophy of Nature vs. Natural Science (ATK Press, Warszawa 1998) and numerous articles on philosophy of mathematics and philosophy of nature. David Liggins is a Senior Lecturer in Philosophy at the University of Manchester. His principal research interests are in metaphysics and the philosophy of language. His published papers have addressed nominalism in the philosophy of mathematics, modal fictionalism, material composition, truthmaker theory, and philosophical method; they have appeared in Mind, Analysis, Philosophical Quarterly, Philosophical Studies, and other journals. David Liggins received his Ph.D. from the University of Sheffield in 2005; he then spent a year as a post-doctoral researcher in Cambridge, supported by the ANALYSIS Trust, before coming to Manchester. In 2009, he became an Associate Editor of the Australasian Journal of Philosophy. David Liggins and Chris Daly are currently directing a research project entitled The Foundations of Ontology, funded by the UK Arts and Humanities Research Council. Their contribution to this volume forms part of the project. Robert E. Maydole is an Emeritus Professor of Philosophy at Davidson College, Davidson, North Carolina. He specializes in logic, philosophy of logic, and philosophy of religion; and he has also taught courses and seminars on a wide range of other topics, including game and decision theory, philosophy of mathematics, philosophy of science, the nuclear arms race, existentialism, modal metaphysics, and Kant’s Critique of Pure Reason. He has published articles on many-valued logic, logical paradoxes, and on ontological, cosmological and modal arguments for the existence of God, including a featured debate with Graham Oppy in Ontological Proofs Today (2012). His current research interests include set theory, the limits of reason, and theories of perfection; and he is slowly writing a book about proving the existence of God. Jason Megill has taught at the University of Colorado at Boulder and Old Dominion University. He primarily works in philosophy of mind and philosophy of religion. Elisa Paganini is Lecturer in Philosophy at the University of Milan (Università degli Studi di Milano). Her principal research interests are in philosophy of language and metaphysics. She is the author of two books: La realtà del tempo [The Reality of Time] (2000) and La vaghezza [Vagueness] (2008). Her published papers have addressed philosophy of time and vagueness; they have appeared in Philosophical Studies, Erkenntnis, Dialectica, and other journals.

258 | Authors of Contributed Papers Duncan Pritchard is Professor of Philosophy at the University of Edinburgh, Scotland. He is also the Director of the University of Edinburgh’s Eidyn Centre for philosophical research. His main field of research is epistemology, and he has published widely in this area, including the books Epistemic Luck (Oxford UP, 2005), The Nature and Value of Knowledge (co-authored, Oxford UP, 2010), and Epistemological Disjunctivism (Oxford UP, 2012). In 2007 he was awarded the Philip Leverhulme Prize for his research in philosophy. In 2011 he was elected to a Fellowship of the Royal Society of Edinburgh. In 2013 he delivered the annual Soochow Lectures in Philosophy in Taiwan. These lectures are currently being developed for publication as a monograph for Princeton University Press, entitled Epistemic Angst: Radical Scepticism and the Groundlessness of Our Believing. Alexander R. Pruss is Professor of Philosophy at Baylor University, USA. He earned his PhD in mathematics from University of British Columbia (1996) under the direction of John J. F. Fournier, and PhD in philosophy from University of University of Pittsburgh (2001) under the direction of Nicholas Rescher. His areas of specialization are: Metaphysics, Philosophy of Religion, Applied Ethics; and areas of competence are: Philosophy of Science, Ancient Philosophy, Aquinas, Leibniz, Truth, Philosophy of Probability. Alexander R. Pruss is the author of three books in Philosophy: The Principle of Sufficient Reason (2006), Actuality, Possibility and Worlds (2011), One Body: An Essay in Christian Sexual ethics (forthcoming); and the author or co-author of over 50 articles in philosophy and mathematics. Scott A. Shalkowski is Senior Lecturer in Philosophy at the University of Leeds. His principal research interests are metaphysics, the philosophy of religion, and the philosophies of logic and mathematics. His (and co-authored) articles have appeared in Philosophy and Phenomenological Research, American Philosophical Quarterly, Faith and Philosophy Philosophical Review, Journal of Philosophy, Mind, Noûs, and Philosophical Studies. He is currently engaged (with Otávio Bueno) in a project on modal epistemology for Routledge. Peter Simons is Professor of Philosophy at Trinity College Dublin. He specializes in metaphysics and logic, with interests also in the history of philosophy and logic in Central Europe. His books include Parts (1987), Philosophy and Logic in Central Europe from Bolzano to Tarski (1992) and some 250 papers. Bartlomiej Skowron is philosopher and mathematician. He defended his Ph.D. thesis, on Mereotopological aspects of philosophical part-whole theory, at University of Wrocław (2012). He received also BA in Mathematics (2011) from the University of Wrocław, Faculty of Mathematics and Computer Science. He was elected as

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Fellow of the Academy of Young Scholars and Artists in 2012. Interests: topological philosophy, formal ontology (part-whole theory, mereotopology), phenomenology, philosophy of morality, axiology, philosophical anthropology, the basis and philosophy of mathematics, applied logic, topology. Mirosław Szatkowski is Professor of Philosophy at the Warsaw University of Technology, Poland. He is also associated with the Department of Philosophy, Theory of Science and Religious Studies at the Ludwig-Maximilians University in Munich, Germany. He earned his PhD in philosophy from the Jagiellonian University in Cracow under the direction of Andrzej Wroński, and was habilitated at the Ludwig-Maximilians University in Munich. Szatkowski’s main fields of research are: logic, the foundations of mathematics, and formal ontology. In logic, he has specialized in non-classical logics and published papers in Studia Logica, Zeitschrift für mathematische Logik und Grundlagen der Mathematik (Mathematical Logic Quarterly), Archiv für Mathematische Logik und Grundlagenforschung (Archive for Mathematical Logic), and Notre Dame Journal of Formal Logic. In formal ontology, his works concerning ontological proofs for the existence of God have been published in Studia Logica, Journal of Applied Non-Classical Logics, Journal of Logic, Language and Information, and Metaphysica. He is currently working on a book about Gödel-type proofs for the existence of God. Peter van Inwagen is an American analytic philosopher and the John Cardinal O’Hara Professor of Philosophy at the University of Notre Dame. He previously taught at Syracuse University and earned his PhD from the University of Rochester under the direction of Richard Taylor and Keith Lehrer. Van Inwagen is one of the leading figures in contemporary metaphysics, philosophy of religion, and philosophy of action. Peter van Inwagen has published seven books: An Essay on Free Will (1983), Material Beings (1990), Metaphysics (1993, and rev. ed. 2002), God, Knowledge and Mystery: Essays in Philosophical Theology (1995), The Possibility of Resurrection and Other Essays in Christian Apologetics (1998), Ontology, Identity, and Modality: Essays in Metaphysics (2002), The Problem of Evil (2006). He also has published over 150 articles on general metaphysics, free will, material objects and human persons, philosophy of religion, logic and language. In 2005 Peter van Inwagen was elected to the American Academy of Arts and Sciences, and in 2011 he received the degree Doctor Divinitatis (honoris causa) from the University of St Andrews. Jan Woleński, Professor emeritus of Jagiellonian University, Kraków, Poland, studied law and philosophy. Member of Polish Academy of Sciences, Polish Academy of Sciences and Arts, and Institut International de Philosophie. In 2013

260 | Authors of Contributed Papers he was awarded the Scientific Prize given by the Fund for Polish Science, called the Polish Noble. He is working in the history of logic, applications of logic to philosophy, epistemology and legal philosophy. Major publications: Logic and Philosophy in the Lvov-Warsaw School (1989), Essays in the History of Logic and Logical Philosophy (1999), Essays on Logic and Its Applications in Philosophy (2011), Historico-Philosophical Essays (2013).

Author Index Abraham U. 101, 112 Ackermann W. F. 161 Aczel A. D. 136, 140 Aczel P. 101, 112 Adams R. M. 220, 240 Aertsen J. 60, 67 Alston W. P. 206, 212 Anderson C. A. 5, 7, 12, 42, 89, 95, 109, 112 Anscombe G. E. M. 197, 215 Anselm 97 Aristotle 15, 16, 24–27, 41, 59, 71, 156, 160, 247 Armstrong D. M. 158, 165 Aubenque P. 24, 27 Audi R. 212 Austin J. L. 30, 202, 212 Ayer A. J. 169, 182 Baldwin T. 212 Barrett C. 207, 212 Beall J. C. 79, 88 Bell J. L. 4, 12, 102, 112 Belnap N. 103, 108, 112, 113 Beltrami E. 156 Bennett K. 170, 173, 177–179, 182 Bentham J. 59 Bernays P. 161 Berti E. 26, 27 Béziau J. -Y. 72, 76, 127, 131 Bigaj J. 24, 27 Bigelow J. 158, 165 Bilková M. 67 Bird D. 212 Blau U. 73, 76 Bocheński J. M. 241, 252 Bolzano B. 154, 155 Boolos G. 163, 165 Bourbaki 252 Brady M. 212 Braig C. 26 Brentano F. 26, 27 Bringsjord S. 100, 101, 112 Broad C. D. 112

Broda B. 245, 252 Brogaard B. 77, 79, 88 Bueno O. 230, 240 Burgess J. 4, 12, 108, 112 Calude C. 248, 252 Cameron R. 220, 240 Cantor G. 133, 137 Carnap R. 169, 182 Carter J. A. 212 Chang C. C. 161 Chisholm R. M. 37, 58 Chomsky N. 180, 182 Chrudzimski A. 67 Church A. 4, 6, 7, 12, 77, 88, 154, 163, 165 Cohen P. 133, 140 Coliva A. 204, 209, 212 Cowling S. 224, 240 Coyne G. 246, 252 Creel R. F. 98, 112 Cresswell M. 86, 88 Crummett D. 33 Davidson D. 30, 201, 213 Dawson J. W. 140 Derkyllides of Alexandria 16 des Rivieres J. 112 Divers J. 220, 224, 240 Domański J. 20, 27 Dorr C. 169, 171–177, 179, 182 Dostoyevsky F. M. 65 Duda R. 244 Dummett M. 79, 88, 112 Euclid 134 Eudoros of Alexandria 16 Evans S. C. 143, 151 Feferman A. B. 153, 165 Feferman S. 13, 140, 153, 165 Fine K. 89, 96, 245, 252 Fitch F. 10, 13, 77, 79, 81, 88 Fitting M. 127, 132

262 | Author Index Flew A. 228, 240 Flint T. P. 67 Forbes G. 220, 240 Fosl P. 212 Frege G. 154, 160, 164, 180 Gale R. 104, 112 Galude C. 248 Garcia L. 59, 67 Gerogiorgakis S. 75, 76 Gitman V. 101, 112 Goldschmidt T. 53, 58 Goodman N. 163, 165 Gödel K. 6, 13, 100, 102, 103, 112, 133, 135, 138–140, 164 Grim P. 99–101, 112 Gupta A. 103, 113 Hamkins J. D. 101, 113 Hartshorne Ch. 3, 6, 13, 98 Hawthorne J. 86, 88, 90, 96 Healey R. 240 Hegel G. W. F. 71, 76 Heidegger M. 15, 26, 27 Heil J. 192 Heller M. 245, 246, 252 Henle P. 13 Hilbert D. 156, 161, 242 Hill D. J. 97, 113 Hirsch E. 169, 174, 182 Hodges W. 156, 165 Horwich P. 149, 151 Howard-Snyder D. 240 Huemer W. 67 Hughes G. 86, 88 Hume D. 82, 88, 179, 210 Husserl E. 249, 252 Jaśkowski S. 127, 131 Jech T. 101, 113 Johnstone T. A. 101, 113 Jordan J. 240 Kanamori A. 101, 113 Kant I. 30, 53, 59 Kanzian Ch. 191, 195 Keefe R. 89, 96

Keisler H. J. 161 Kenny A. 207, 213 Kerr F. 208, 213 Ketland J. 103, 113 Kienzler W. 207, 212, 213 Kleene S. 140 Klein F. 156 Klement K. 41 Kovač S. 125, 127, 132 Kripke S. 103, 113, 149, 151 Kuhn R. L. 54, 58 Kulpa W. 245, 252 Kuratowski K. 244, 252 Kutschera v. F. 106, 113 Leftow B. 61, 67, 217, 221, 222, 224, 225, 227–229, 233, 237, 240 Leibniz G. W. 50 Leitgeb H. 103, 113 Lemańska A. 140 LePoidevin R. 105, 113 Leslie J. 41, 54, 58 Leśniewski S. 153, 155, 160–163 Levesque H. J. 112 Lewin K. 245, 252 Lewis D. 10, 13, 217, 219–221, 224–226, 229–231, 233, 237, 240 Liggins D. 176, 182 Locke J. 187, 195, 204–206, 213 Lorenz K. 134, 140 Loux M. J. 104, 113, 240 Lowe E. J. 184, 186, 188, 195 Lycan W. 223, 240 Łukasiewicz J. 76, 154 Machover M. 102, 112 Malcolm N. 197 Malia J. 220 Marcus S. 248, 252 Martin C. B. 158 Martin J. B. 192 Martin R. M. 103, 113 Maydole R. E. 100, 113 McDaniel K. 41 McGinn C. 223, 240 McGinn M. 204, 213 McTaggart J. M. E. 98, 104, 105, 113

Author Index Melia J. 178, 182, 240 Mellor D. H. 105, 113 Menzel Ch. 99, 113 Mill J. S. 59 Milne P. 72, 74–76 Moore G. 140 Moore G. E. 5, 13, 197, 199, 200, 204, 205, 208, 213 Mormann T. 245, 252 Moyal-Sharrock D. 204, 213 Mulligan K. 153, 165, 184, 186–188, 195 Munitz M. K. 53, 58 Murawski R. 136, 139, 140 Myhill J. 101, 113 Narkiewicz W. 136, 140 Nash R. H. 97, 113 Nasieniewski M. 127, 132 Natorp P. 15, 24, 28 Neeman I. 101, 113 Newman J. H. 198, 204–207, 210, 211 Newman K. 213 Nicholas of Cusa 241, 243, 251 Nielsen K. 211, 213 Nozick R. 79, 88 Oaklander L. N. 105, 113 Ochiai H. 241, 246, 250, 251, 253 Padoa A. 156 Paganini E. 90, 93, 96 Parmenides 23 Paul D. 197 Peano G. 164 Perlis D. 103, 113 Perloff M. 108, 112 Phillips D. Z. 211, 213 Philo of Alexandria 16, 18, 20–24, 26 Plant B. 212 Plantinga A. 6, 7, 13, 99, 113, 143, 151, 206, 213, 220, 226, 240 Plato 16, 23, 28, 146, 242, 243, 247, 251 Priest G. 71, 76 Prior A. 104, 108, 114 Pritchard D. H. 201, 204, 206, 209–211, 213 Proad C. D. 104

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Pruss A. 143, 151 Putnam H. 30, 169, 182 Pythagoras 243 Quine W. V. 4, 5, 104, 114, 161, 163, 165, 176, 182 Rea M. C. 67 Ribenboim P. 136, 140 Richmond A. 210 Rieman B. 156 Robinson A. 155 Rodriguez-Pereyra G. 178, 182 Rosen G. 169, 171–177, 179, 182 Runde V. 244, 253 Russell B. 161, 164, 175, 182 Ryle G. 158 Sainsbury R. M. 149, 151 Salerno J. 77, 79, 88 Seebach A. 244, 248, 253 Shalkowski S. A. 223, 230, 240 Simons P. 153, 160, 161, 165, 189, 194, 196 Skowron B. 241, 245, 253 Skyrms B. 75, 76 Slomson A. B. 4, 12 Smart J. J. C. 104, 114 Smith B. 153, 165, 245, 253 Smith Q. 105, 113 Soames S. 89, 96 Sober E. 181, 182 Socrates 144–146 Solovay R. 140 Spinoza B. 61 Stalnaker R. 220, 240 Steen L. 244, 248, 253 Stefanescu D. 248, 252 Strawson P. F. 201, 214 Stroll A. 204, 214 Stroud B. 202, 214 Subrahmanian V. S. 103, 114 Szabo-Gendler T. 86, 88 Szatkowski M. 13, 108, 111, 114 Tarski A. 102, 103, 112, 114, 149, 151, 153–156, 159, 164, 166 Taylor R. 104, 114, 136 Tennant N. 79, 88, 102, 114

264 | Author Index Thomas Aquinas 59, 60, 63, 98, 114, 185, 194, 196 Thomason R. H. 108, 114 Thomasson A. L. 172, 173, 182 Tomala O. 67 Tredennick H. 26 Turri J. 29, 30 Uzquaino G. 176, 182 Van Fraassen B. 74–76 Van Heijenoort J. 140 Van Inwagen P. 3, 5, 11–13, 33, 58, 90, 146, 151, 170, 176, 180, 182, 228, 240 Visser A. 103, 114 Von Wright G. 197 Weir A. 164, 166 Whitehead A. N. 161, 164 Wiles A. 136, 140

Williams D. C. 104, 114 Williams M. 197, 204, 209, 214 Williams R. 212 Williamson T. 6, 79, 88, 89, 96, 181, 182 Wittgenstein L. 30, 156, 197, 198, 200–205, 207–211, 214 Woleński J. 60, 67 Wolterstorff N. 206, 215 Woodruff P. 103, 113 Wölfl S. 106, 114 Wright C. 204, 215 Xu M. 108, 112 Yablo S. 10, 11, 13, 102, 103, 114 Ye R. 127, 132 Zanardo A. 106, 108, 114 Zarach A. 101, 114 Żełaniec W. 245, 253

Subject Index A-series 104 abstracta 221, 222, 236 agnosticism 169, 179, 180 all goodness 10 analysis – conceptual 172 anti-realism 79 argument – cosmological 34 – ontological 6, 10, 29, 30, 32, 34 – paraphrastic 219 – parity – religious 206 arithmetic – Peano 135, 138 assignment 126 attribute 35 – divine 232, 233 axiom – choice 101 – power set 101 B-series 104 B-theorist 105 beauty 60 being 15, 17, 20–26, 37, 60, 62 – contingent XX, 34, 42, 45, 48, 51–53, 55 – divine 41 – necessary 34 – omnipotent 10 – omniscient 10, 73, 75, 77, 89–95, 111 – perfect 10, 141–143, 145, 147, 150 belief 119 – non-religious 206, 211 – religious 119, 124, 128, 205–207, 210, 211 bonum 60, 63–65 characterization 183, 188, 192 closure 246, 247 – ideal 247 – of the world 247

compactification – Alexandroff 247 compactness 245, 247, 250 – as unity of God 249 – as unity of the space 250 – pseudocompactness 250 composition 181, 183, 194 – material 170, 179 concrescence 37 concreta 236 concreteness 37 condition – anti-Gettier 145, 147 connectedness 249 – of God 248–250 – path 249 – simply 249 – as unity of the space 250 consequence 127 – logical 155, 159 consistency 137, 156 – conceptual 6, 11 constituent 36 constitution 183 contradiction 71 cooperation 93, 94 craftsman 19 creation 184, 195 – out of nothing 20 creator 19, 27 definition – impredicative 101 denotation 126, 159 dependence 183 – existential 183, 189, 191 – ontological 183, 187–189, 191, 192, 195 – generic 189 – individuel 189, 190 – proprial 190 – substantial 190 – strong 189 – weak 189

266 | Subject Index description – definite 84, 141 determinism 106 dilemma – disjunctive 34, 45 discourse – religious 241 disposition 158 Doom 29, 30, 32 ens 60, 64 entailment – conceptual 7 entity 185 – abstract 153, 157 – possible 7 essence 109 evaluation 201, 202, 209 evil 60, 62, 64, 66 existence 59, 62, 64 – actual 10 experience – external perceptual 30 explanation 37–40 – “how it came to pass that” 38 – “why it is the case that” 38 – complete 39 – mathematical 251 – topological 246 extension 159 form – ontological 186 frame 125

– Zermelo-type 161 hypothesis – continuum 133, 137 – Goldbach’s 136, 138 idealism 197, 199 identity 183, 184, 194 impossibilia 224 incommensurability – epistemic 204, 209 indeterminism 106 interpretation 155 kind 35–37, 40, 42, 51, 52 – natural 36 – weak-sense 52 knowledge 78, 85, 87, 119, 122, 145 – a priori 29 – complete 98, 99 – mathematical 135 – necessary 98 – nonempirical 29 – sound 98 – tenseless 98 lemma – Gödel’s diagonal 102 Liar – divine 102 logic – epistemic 80, 81, 87, 119

gap 71 geometry – Euclidean 134 – non-Euclidean 134, 137 glut 71 God 17–19, 21–24, 26, 29–32, 59, 61, 62, 88, 97, 109, 111, 139, 141, 143, 145–150, 195, 221, 222, 224, 227, 228, 232–234, 236–239 goodness 60, 62, 64

malum 62, 64, 65 mathematics – predicative 101 maximalism 157 metaphysics 15, 113, 114 modality 221, 224, 237 – conceptual 3, 5, 7 mode 191, 192 model 59, 61, 64, 126, 155, 156, 159, 160, 162–164 multitude 160–163 mystery of existence 53

hierarchy – languages 103, 110, 112

naturalism 82, 154 nature 186

Subject Index necessity 217 – conceptual 3 – logical 3, 4 nihilism 178 nominalism 153, 160, 163, 165 non-ens 63 non-entity 185 non-omniscience 77, 78, 80 nothing 33–35, 46, 53, 57 number 169, 234 – natural 135 object – abstract 221, 234 – composite 176 – material 169, 191 – composite 169 – mathematical 35 objection – Euthyphro 143 omnipotence 10, 237 omniscience 10, 77, 81, 82, 87, 97, 98, 112, 237 one 60 onto-logic 15 onto-theology 15, 16, 21 opacity – semantic 158 paradox – Fitch’s 80 – Liar 72, 100, 103, 148 – McTaggart’s 104 – Milne’s 72 partition 51 perfection 10, 60, 62 – moral 237 platonism 153 possibilia 224 possibility 34 – epistemic 3 – logical 3 predicate – axiological 64 – knowledge 111, 112 presumption 171

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principle – composition 170, 175, 179 – existential implication 42 – externality of explanation 40 – individuation 56 – knowability 77, 79 – nihilist’s 170 – recombination 223 – suflcient reason 34, 40, 87, 154 – universalist’s 170 property 35, 36, 60, 220, 234 – disjunctive 37 – disjunctuve 49 – essential 37, 41 – impossible 37 – negative 37, 49 – platonic 36 proposition 169, 220, 234 pulchrum 60 question – special composition 170 realism 79 – genuine modal 217 relation – causal 195 – formal 183, 188, 191 – internal 184, 194 relativism – epistemic 198, 203, 208 requiredness – ethical 41 res 60 satisfaction 126, 155 satisfiability 127 scepticism 197, 198, 203, 206, 208 semantics – branching-time 105 – bundled-trees 105 – standard 105 – bundled-trees – Ockhamist 105, 107 – Peircean 105, 107 – Priorean 105, 107

268 | Subject Index – linear-time 105 – Kamp 105, 106 – T×W 105, 106 – partly free – Anderson-type 108 – standard – Ockhamist 105, 107 – Peircean 105, 107 – Priorean 105, 107 – temporal 105 sense – common 173–175 sentence – self-reference 102 set – closed 244 – open 244 set theory – Kripke-Platek 100 – Quine 100 – von Neumann-Bernays-Gödel 100 – Zermelo-Fraenkel 99, 101, 138 simplicity 178, 181, 194 – divine 232, 233 something 33–35 space 134 – Euclidean 134 – three-dimensional 134 – non-Euclidean – three-dimensional 134 – topological 244, 245 – compact 245, 246, 249 – connected 244, 245, 248, 249 state 194 state of affairs 220 structure – model 109, 110 sub-Doom 31, 32 supervaluation 75 supervaluationism 90 theo-logic 15 theology 195, 237, 251 – mathematical 241, 250 – philosophical 184, 189, 194, 195, 232, 237

theorem – Cantor’s 100 – Cantor’s power set 99 – Fermat’s 136 – Gödel’s 138 – Tarski’s indefinability 102, 103 theory – Anderson-like 109 – communicativist 242 – correspondence 156 – divine belief 141, 143, 145, 148–150 – divine command 143 – divine thought 143 – emotionalist 242 – ethical 59 – incomplete meanings 242 – model 155 – non-communicativist 242 – non-propositional 243 – nonsense 241 – propositional 242 – utilitarian 59 – virtue 59 thing 35, 60, 191, 193 – abstract 35, 42, 43, 48, 49 – concrete 35, 36, 49 – material 52 topology 244, 247–251 – cofinite 248 – Euclidean 245 transcendentalia 60 trope 157, 217 truth 60, 119, 133, 135, 153, 154, 156, 159 – analytic 171, 172 – contingent 39, 40, 42, 45, 54, 55 – modal 223 – necessary 40, 42, 45, 50, 54, 55 truth-maker 153, 154, 157, 159, 165 universal 61 – Aristotelian 36 – immanent 36 – platonic 36 – transcendent 36 universalism 173, 178 unum 26, 60, 185, 194 urelement 161, 164

Subject Index vagueness 89, 92, 95 validity 4 verum 60 world 17, 19, 20, 60, 156, 159, 183, 185, 186, 246, 247

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– created out of nothing 21 – intelligible 17 – possible 30, 43, 54–56, 61, 129, 159, 217, 219, 222, 226 – real 165 – spiritual 16