The Liar Speaks the Truth: A Defense of the Revision Theory of Truth

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The Liar Speaks the Truth

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The Liar Speaks the Truth A Defense of the Revision Theory of Truth

ALADDIN M. YAQUB

New York Oxford OXFORD UNIVERSITY PRESS 1993

Oxford University Press Oxford New York Toronto Delhi Bombay Calcutta Madras Karachi Kuala Lumpur Singapore Hong Kong Tokyo Nairobi Dar es Salaam Cape Town Melbourne Auckland Madrid and associated companies in Berlin Ibadan

Copyright © 1993 by Aladdin M. Yaqub Published by Oxford University Press, Inc. 200 Madison Avenue, New York, New York 10016 Oxford is a registered trademark of Oxford University Press All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording, or otherwise, without the prior permission of Oxford University Press. Library of Congress Cataloging-in-Publication Data Yaqub, Aladdin Mahmud. The liar speaks the truth : a defense of the revision theory of truth / Aladdin M. Yaqub. p. cm. Includes bibliographical references and index. ISBN 0-19-508343-1 1. Truth. 2. Liar paradox. I. Title. BD171.Y37 1993 121—dc20 92-37507

246897531 Printed in the United States of America on acid-free paper

To my mother, wife, and daughters And in loving memory of my father

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Acknowledgments

I am greatly indebted to Michael Byrd, who sparked my interest in philosophical logic in general and the problem of truth and semantical paradox in particular. His scholarly advice and sincere encouragement guided me throughout my graduate study in philosophy at the University of Wisconsin-Madison. Professor Byrd also supervised my doctoral dissertation (1991), on which this book is based. I am also grateful to the other members of my dissertation committee, Ellery Eells, Leora Weitzman, and Kenneth Kunen. Professor Weitzman was especially helpful: she spent many hours discussing with me parts of the first draft of this work. That draft was written during the academic year 1990/91. For that year the Graduate School of the University of Wisconsin-Madison awarded me a fellowship that allowed me to devote most of my time to the writing of that draft. I would like to extend my thanks to Anil Gupta who read a large part of the manuscript and wrote several important comments, and to the colleague who advised Oxford University Press on the manuscript—his comments were very useful. I am also thankful to Yvonne Nagel. Mrs. Nagel was of great help in teaching me the nuances of the TEX typesetting system. Angela Blackburn, Robert Dilworth, and Nancy Hoagland of Oxford University Press gave the manuscript their most expert and prompt attention. I owe them a great deal. Numerous friends and colleagues have offered support and advise in various forms. Of these friends, the following stand out as

most generous in their help: Martin Barrett, Russ and Margaret Dancy (who forcefully encouraged me to submit the manuscript for publication), Janet Kelley, Greg Mougin (to whom I owe the title of this book), and Robert and Marilyn Sullivan. By far my greatest debt is to my wife, Connie. Her endless patience and remarkable resourcefulness made it possible for me to endure the graduate program and to complete the project of this book. Our daughters, Mariam and Ranah, made their own contribution by being a source of invigorating joy. Finally, to my family in Baghdad, Iraq, I bow in respect and gratitude. Despite the miserable conditions that were inflicted upon them by two catastrophic wars, they have never failed to express their support and happiness for me.

Contents

Introduction. The Tarskian Schema 1.1. The Fundamental Intuition: Formulation 1.2. The Fundamental Intuition: Motivations 1.3. More Motivations: Tarski's Theory of Truth

3 6 9 19

Chapter One. The TS Conception of Truth 1.1. The Tarskian Schema Defines Truth Completely 1.2. The Tarskian Schema Defines Truth Correctly 1.3. The Circularity Argument

27 27 30 36

Chapter Two. Stability Semantics 2.1. Syntax 2.2. Semantics 2.3. An Example 2.4. The Seven Categories Stability Tables for Chapter Two

43 43 47 53 59 64

Chapter Three. The Original Three Systems 3.1. Describing the Systems 3.2. Their Artifacts 3.2.1. Biconditionals of Liars 3.2.2. Conjunctions and Disjunctions of Liars 3.2.3. Truthteller Cycles and Cyclic Truthtellers 3.2.4. Circular Invariant Sentences 3.3. Philosophical Diagnosis

69 70 79 79 81 82 87 89

Stability Tables for Chapter Three

99

Chapter Four. The Revision System 4.1. Remarks on Constant Limit Rules 4.2. The Limit Rule Assignment V 4.3. Stability Logic

113 114 126 131

Bibliography

141

Symbols

147

Index

149

The Liar Speaks the Truth

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Introduction The Tarskian Schema

The project of this book is to give a thorough defense of the semantical theory that arises from a certain deflationary conception of truth together with some initial conditions.1 The conception is given by the following thesis: TS. The concept of truth is completely and correctly defined by the Tarskian schema, whose instances are all the biconditionals obtained from the phrase 'x is true if and only if p' by substituting any sentence for 'p' and any expression which stands for that sentence for 'x'. In other words, the Tarskian biconditionals (i.e., the instances of the Tarskian schema) are collectively exhaustive and individually constitutive of the concept of truth. The initial conditions describe a bivalent language (i.e., a language in which every sentence is either true or false but not both) that contains its own truth predicate. This project is carried out in three stages: first, I unpack the TS conception to see what theory of truth it entails, second, I show that this theory is consistent (despite the existence of Liar sentences, it does not generate contradictions), and third, I argue that it is materially adequate (it yields all the intuitively expected results). 1

The conception is deflationary but the semantical theory is not. I explain and argue for these claims in Section 1.3. 3

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This introduction is devoted to exploring issues about the Tarskian schema and what it signifies. Here I set the stage, philosophically and historically, for the main argument of the book. I discuss in Section I.I preliminary issues concerning the Tarskian schema as a formulation of our most fundamental intuition about truth. This intuition may be described informally as follows: the conditions under which a sentence is true are what the sentence itself asserts. For instance, a sufficient and necessary condition for the sentence 'Socrates drank hemlock' to be true is that Socrates drank hemlock. In the next two sections (1.2 and 1.3) I demonstrate how fundamental this intuition is by explaining its central role in ordinary discourse and in philosophical reflection. I give an historical and philosophical exposition of several influential theories of truth, such as Frege's, Ayer's, and Tarski's. The primary purpose of the Introduction is to motivate the thesis TS and to familiarize the reader with the issues at hand. In Chapter 1 I carry out the first stage of the project. In Sections 1.1 and 1.2 I give a preliminary defense of the thesis TS and the initial conditions mentioned above. Then in Section 1.3 I argue that the TS conception entails that the concept of truth is circular, for the Tarskian biconditionals of certain sentences specify circular truth conditions for those sentences.2 For example, the Tarskian biconditional of the truth-generalization 'No sentence is both true and false' shows that the truth conditions of this generalization essentially involve reference to truth itself. I also argue that the circularity of the concept of truth in bivalent languages gives rise to a revision process in which the semantical status of each sentence is determined by its Tarskian biconditional once a totality of nonsemantical facts and an initial extension of the truth predicate are posited. This semantical theory is called the revision theory of truth.3 This theory paints a metaphysical picture of truth as 2

A Tarskian biconditional of a sentence is an instance of the Tarskian schema obtained by substituting that sentence for 'p' and any of its names or descriptions for 'x'. 3 Gupta was the first to describe the revision process and to give it that name. In his very insightful paper "Truth and Paradox" (1982), he advanced a semantical system that was intended to formalize the revision process. A fuller and clearer philosophical account of the revision theory of truth was presented in his later paper "Remarks on Definitions and the Concept of Truth" (1989).

The Tarskian Schema

5

a property that cannot be reduced to other properties and whose applicability is given by a revision process rather than by a fixed extension. That is to say that the objects of the world are not neatly divided into those that have the property of truth and those that lack it, but rather they stand in a hypothetical relation to this property: if an object were to have (or lack) the property of truth, would it maintain that status? Chapters 2 through 4 are devoted to showing that the revision theory of truth is consistent and materially adequate. In Section 2.11 define the syntax of the type of language with which I work in the book; in Section 2.2 I introduce its semantics. The formal semantics is called stability semantics. Every system of stability semantics is a formalization of the revision theory of truth. Thus, the consistency of any such system demonstrates that the revision theory is consistent too. In Section 2.3 I give a detailed example to explain the ideas of the previous two sections, and finally in Section 2.4 I discuss the different semantical categories that are generated by any system of stability semantics. Chapter 3 is an expository and critical study of the systems of Herzberger, Gupta, and Belnap (the original three systems of stability semantics). After describing, defining, and comparing these systems in Section 3.1, I show in Section 3.2 that they generate many intuitively erroneous results. I discuss a large sample of examples, many of which are quite surprising. The last section of Chapter 3 contains a philosophical diagnosis of the source of such problems. The conclusion there is that these erroneous results are artifacts of certain formal features of these systems, and hence they pose no danger to the material adequacy of the revision theory of truth. In Chapter 4 I advance my own proposal: the revision system of stability semantics. I argue that this system is free from artifacts and that it represents the revision theory faithfully. Hence, this system establishes that the revision theory of truth is indeed materially adequate. Section 4.1 contains proofs of two pivotal theorems that are needed for the construction of the revision system. The actual construction of this system and the demonstration of See Chapter 3 for a critical discussion of Gupta's formal system, as well as those of Herzberger and Belnap. A more complete picture of the historical context of these works emerges from that discussion.

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its material adequacy are carried out in Section 4.2. In the last section of Chapter 4 (Section 4.3) I define a notion of logical consequence that is based on the type of formal semantics described in Section 4.2. The definition of logical consequence, in effect, introduces a logic of truth, which I call stability logic. This logic has many important properties that serve as further testing grounds for the material adequacy of the revision theory. Having sketched the general structure of this work, we now begin our tour exploring issues about the Tarskian schema and what it signifies. I.1. The Fundamental Intuition: Formulation The Tarskian schema represents the most fundamental intuition about the concept of truth. But this modern quasi-formal schema is only one among many possible formulations of this intuition. We could have chosen a less formal, more familiar formulation. For example, we could have used Aristotle's famous words, "To say of what is that it is not, or of what is not that it is, is false, while to say of what is that it is, and of what is not that it is not, is true" (Metaphysics: 1011b23-27),4 or his biconditional about falsehood, "If it is not the case it is false, if it is false it is not the case" (De Interpretatione: 18bl-5).5 Another choice could have been Plato's statement in the Cratylus, "And a true proposition says that which is, and a false proposition says that which is not" (385b-c).6 But by adopting any such formulation we run the risk of importing with it extra philosophical baggage that is part of the philosophical system to which the adopted formulation belongs. Of course, we need not import familiar formulations of the fundamental intuition from any particular philosophical system; we could have simply chosen a less formal phrasing than the one given on page 3. Two formulations are frequently found in the literature. The first states that a sentence is true if and only if what it asserts is the case. The second is that the condition for the truth of a sentence is given by what is asserted by the sentence itself. It might be true that these two formulations and the one given on 4

The translation is by W. D. Ross. See Barnes (1984, vol. 2). The translation is by J. L. Ackrill. See Barnes (1984, vol. 1). The translation is by B. Jowett. See Hamilton and Cairns (1963).

5 6

The Tarskian Schema

7

page 3 all ultimately express the same thing, and it might be true that these two formulations have more intuitive appeal than the Tarskian schema. Nevertheless, I think that this schema has two advantages. First, it is more readily formalizable and hence more easily adoptable in a system of formal semantics. Second, it does not involve the mention of assertions or the case, which require some clarification to avoid a prior commitment to propositions or to a correspondence notion of truth. The Tarskian schema appears to be a fair and adequate formulation of the fundamental intuition about the concept of truth. It fully represents the "disquotation" feature of this concept yet without carrying implications about any correspondence notion of truth. 7 There is, however, an important thing to note about the way the Tarskian schema is worded on page 3. This wording includes the phrase 'substituting any sentence for' which reveals a commitment to sentences as the bearers of truth. There are many philosophers and logicians who have argued that propositions or statements, rather than sentences, are the appropriate vehicle for truth (see, for example, Austin, 1950; Cartwright, 1962; Barwise and Etchemendy, 1987; Horwich, 1990). We are often told that truth must be attributed to what sentences express and not to the sentences themselves. The argument sketched below contains some of the reasons most frequently offered in support of this position. In ordinary usage truth is attributed both to sentences and to the assertions (or statements, or propositions) that they express. But when we assign truth or falsity to sentences, we do so by examining the assertions that they make. Of course, it does not follow from this alone that assertions are the bearers of truth and sentences are not; as it does not follow from the fact that attributing pain to someone is usually done by examining his behavior, that pain is an attribute of the behavior itself and not of the per7 The disquotation feature of truth is represented by the schema: 'p' is true if and only if p. Hence, it is already included in what I have been calling the fundamental intuition about truth. The term 'disquotation' is used by several philosophers to distinguish this feature from the correspondence conception of truth which roughly says that a sentence is true if and only if what it asserts corresponds to the facts (or to reality). See Field (1986) for an excellent discussion of disquotational and correspondence truth. Also see Quine (1974), Putnam (1978), and Gupta (1982) for helpful remarks on disquotation.

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8

son. But given that truth and falsity are ordinarily attributed to assertions as well as to sentences, it becomes quite reasonable to conclude that assertions are at least the primary bearers of truth, since the truth or falsity of a sentence is ascertained by examining the assertion it makes. The argument continues by attempting to show that hardly any sense can be made of attributing truth to sentences in the first place. For to speak of sentences is to speak of sentence-types or of token sentences (utterances) .8 If it is sentence-types, then surely it does not make much sense to attribute truth or falsity to them. One reason is that sentence-types are coarse: the same sentence-type may be instantiated by several utterances making different assertions which have different truth values.9 Thus, it is utterances of which we must be speaking. But utterances are certain marks or noises with spatiotemporal dimensions. While it seems odd to attribute truth or falsity to noises and marks, it is perfectly sensible to attribute such properties to what these marks and noises express. Hence, the ordinary practice of calling sentences and the assertions they make true or false, when sufficiently pressed, shows that only assertions are capable of bearing these properties without an apparent conflict with other common practices and beliefs. Propositions, however, are not trouble-free either. Despite the argument given in note 8, ontological issues about propositions are much more problematic than those about sentences. The ontology of sentences is part of the general ontology of types of physical entities, i.e., types whose instances are concrete objects. Propositions do not admit any physical instantiation. Indeed, it is not clear that they admit the type/instance dichotomy in the first place. 8

Sentence-types are sometimes mentioned by advocates of propositions to make a different point. They argue that the uneasiness, which some feel about dealing with abstract entities such as propositions and which leads them to prefer sentences, is largely due to the failure to appreciate the ontological commitment that comes with sentences. For to make sense of treating several token sentences as being the same sentence, we need to consider them as instantiating the same sentence-type. But types, the argument continues, are as abstract as propositions. It should be cautioned that by using the established terminology 'truth values', I do not intend any Fregean connotations (see Section 1.2). The truth value of a sentence (or utterance) is its truth status. For example, 'S has the truth value true' simply means that S is true.

The Tarskian Schema

9

But even if it were reasonable to distinguish between propositiontypes and token propositions, it would remain that both of these "things" are abstract. In addition to their loaded ontology, propositions give rise to other difficulties. For instance, there is the problem of individuating them and there is the difficult task of explicating the relation of expressing, i.e., the relation which holds between a proposition and a sentence (or utterance) expressing that proposition.10 Despite the importance of this debate, it is largely irrelevant to the issues at hand. I am not claiming that the choice of sentences or propositions as the bearers of truth does not influence the resulting semantical theory—indeed, it does in many important ways. But the claim is this. There is nothing essential about the theory of truth presented here that depends on such a choice. In other words, any feature of this theory that is shaped by the choice made is not intrinsic to the theory's approach to the concept of truth and the problem of semantical paradox. The theory could be modified to accommodate propositions instead of sentences as the bearers of truth, and it would remain conceptually the same theory of truth.11 Hence, I choose sentences over propositions simply because sentences are easier to manage due to their familiar ontology and precise structure. 1.2. The Fundamental Intuition: Motivations

As mentioned at the beginning of the previous section, the Tarskian schema represents the most fundamental intuition about the con10

To be fair, I must say it might be objected that a similar task is encountered when utterances are taken to be bearers of truth. When an utterance is established to be (contingently) true, many other utterances (some perhaps instantiating different sentence-types) are also claimed to be true because (intuitively) they all express what is asserted by the original utterance. This suggests that in this case too there is a need to explain what it is for two utterances to express the same thing. 11 An example of some of the modifications required for the accommodation of propositions is the reformulation of the fundamental intuition. One familiar phrasing of the Tarskian schema in terms of propositions is this: the proposition that p is true if and only if p. See Parsons (1974), Barwise and Etchemendy (1987), and Horwich (1990) for similar formulations in terms of propositions.

The Liar Speaks the Truth

10

cept of truth. When we examine the ways in which we ordinarily understand and employ the terms 'true' and 'false', as far as they are used to attribute truth and falsity to sentences (or to what they assert), we find them all based on the intuition represented by this schema. To illustrate this point let us consider two examples. Imagine yourself trying to teach the concept of truth to a child who hasn't yet acquired it.12 You surely wouldn't fuss with issues about the appropriate bearer of truth, or the kind of property truth is, or whether it is a property in the first place. I believe you would consider those issues irrelevant to your task of explaining the notion of truth to the child. Almost certainly you would find yourself telling her instances of the Tarskian schema in terms of sentences, propositions, statements, assertions, beliefs, or any other candidate for the bearer of truth that the child happened to comprehend at the time of your experiment. You might explain to her, "When I say that you are a girl, I say something true, because you are a girl; but when I say that you are a boy, I say something false, because you are not a boy." I fancy that you would continue citing to her other instances of the Tarskian schema until she acquires the concept or becomes uninterested in playing your game. Here is the second example. Suppose in a debate between two gubernatorial candidates the incumbent concluded with this remark: "Everyone agrees that I am the best candidate for this office." Now further suppose that the challenger immediately responded that his opponent's last remark is definitely false. We all could accept the challenger's response as true, and we might reason as follows. Surely the challenger and his supporters (assuming they are rational) do not agree that the incumbent is the superior candidate. Therefore, not everyone agrees that the incumbent is the best candidate for the governor's office. We ought to conclude that the incumbent's remark is false, and since this is what the challenger said, we must accept his response as true. Philosophical reflection joins with ordinary practice in demonstrating how fundamental the intuition encoded by the Tarskian schema is. All theories of truth (or at least all the ones with which I am familiar) accommodate this intuition in some central way. There are two observations to note here. First, every such the12

Compare this example with the story told by Kripke (1975).

The Tarskian Schema

11

ory validates the Tarskian schema or some restricted version of it. More precisely, for every theory of truth, there is a designated collection (or collections) of sentences, such that the theory entails all the biconditionals of the form 'x is true if and only if p', where 'p' is replaced by any sentence in the designated collection and 'x' by some appropriate name or description of that sentence. Second, theories of truth that involve hierarchies of interpretations employ the Tarskian schema as the rule that generates the successive stages of these hierarchies. Kripke's and Gupta's accounts are paradigms of such theories. They also demonstrate how recent theories of truth strive to validate larger collections of Tarskian biconditionals. The various semantical systems which attempt to formalize Gupta's revision theory of truth illustrate, even more forcefully, the two observations mentioned above.13 To illustrate the first observation, we now consider two highly influential accounts of truth: Frege's and Tarski's. The latter is discussed in the following section. Several of Frege's writings (for example, 1892, 1906, and 1918) show serious skepticism about the common view that truth and falsity are properties. He definitely rejected the thesis that truth is, or that it arises from, a certain correspondence between a thought and reality (1897 and 1918).14 Frege's positive account is this: truth and falsity are the referents of sentences that express thoughts, and that "attributing" truth to a thought does not produce a new thought or change the original one in any way (1892, 1902, 1904, 1906, and 1915). To use one of his examples (1892, p. 64), the sentence 'The thought that 5 is a prime number is true' contains the very same thought expressed by the sentence '5 is a prime number'. It is clear that this account entails all the biconditionals of the form 'The thought that p is true if and only if p', where 'p' may be replaced by any sentence that expresses a thought.15 13

These systems are described in the last three chapters and Gupta's theory is discussed in Section 1.3. 14 Frege defined thought as the sense of a sentence (1906, p. 192). He said that the word 'thought' may be understood in the same sense as 'judgment' is used by logicians (1918, p. 292n). 15 This generalization needs to be qualified. For Frege, not every sentence expressing a thought has a truth value, that is, not every such sentence has a

12

The Liar Speaks the Truth

Before considering Tarski's theory of truth, I would like to take advantage of the present context to discuss an issue that has philosophical and historical significance for the problem of truth. It is the issue of whether truth is a property or not. The discussion below is not exactly a digression, since it offers us an opportunity to take a closer look at Frege's views on the nature of truth and to examine two more accounts, Ayer's and Grover's, which take truth and falsity as linguistic operations. Those accounts also serve as further illustrations of our original point, namely that theories of truth validate the Tarskian schema or restricted versions of it. As mentioned above, Frege was quite skeptical about the view that truth is a property. He wrote in "The Thought" (1918): So it seems, then, that nothing is added to the thought by my ascribing to it the property of truth. And yet is it not a great result when the scientist after much hesitation and careful inquiry, can finally say "What I supposed is true"? The meaning of the word "true" seems to be altogether unique. May we not be dealing here with something which cannot, in the ordinary sense, be called a quality at all? In spite of this doubt I want first to express myself in accordance with ordinary usage, as if truth were a quality, until something more to the point is found, (p. 293)

Frege's views on the nature of truth are reported in the references cited above for his positive account. Truth and falsity are objects (he called them the True and the False or collectively the truth values). The relation of a sentence to its truth value is not that of an object to a property but of a sign to its reference (via its sense). Frege argued (1897, pp. 128-29) that any definition of truth implicitly presupposes what is being defined. Every definition of the form that S is true if and only if S meets such-and-such conditions must come back to the question whether it is true that S meets such-and-such conditions.16 He concluded that truth (and falsity) referent. Sentences that express thoughts belonging to fiction or that contain nonreferring terms are neither true nor false. See Frege (1897, pp. 129—30, and 1906, p. 194). (Note that in the latter reference the translators use the word 'meaning' instead of 'referent'.) 16 Although it is very tempting to quarrel with this bizarre argument, I have decided not to do so in order to keep peripheral discussions at a minimum.

The Tarskian Schema

13

cannot be defined, analyzed, or reduced to anything simpler. It follows that the True and the False are simple, irreducible objects. Frege's account of truth did not appeal to the logical positivists. The claim that there are two objects, the True and the False, must have sounded mysteriously and suspiciously metaphysical to the philosophers of the Vienna Circle. Ayer in Language, Truth and Logic (1936, ch. 5) advanced a more deflationary account which was better suited to the positivists' taste. He argued (or more correctly urged) that the words 'true' and 'false' connote nothing. They are simply marks of assertion and denial. Hence, truth and falsity are neither properties nor objects. They are linguistic operations on sentences that produce new sentences which in the case of truth reiterate and in the case of falsity deny the assertions made by the original sentences. Thus, truth cannot be analyzed, not because, as Frege said, it is a simple and primitive object, but rather because there is nothing there to analyze.17 Note that Ayer's view entails all instances of the Tarskian schema without any apparent restrictions. For if 'true' and 'false' are considered merely as syntactical marks of assertion and denial, problematic sentences, such as the notorious Liar, would be either ill-formed or nonsensical. To explain this point, let us consider an example. The Liar sentence in its simplest form is a sentence which says of itself that it is not true. If truth is considered a property, then the phrase 'is true' must be a predicate. In this case, the Liar can be formed by a direct self-reference as follows: let 'c' be a name referring to the sentence 'c is not true'. But if truth is, as negation and disjunction are, a linguistic operation on sentences, then 'true' is only a unary connective, just as 'not' is a unary connective and 'or' a binary connective. In this case, the Liar cannot be formulated, for sentential connectives do not operate on names or definite descriptions of sentences but on the sentences themselves. The claim that 'false' is a mark of denial 17

It is incorrect to assume from the discussion above that Ayer maintained throughout his career most, or even any, of the views stated in Language, Truth and Logic. In an interview broadcast by the BBC in 1978, the interviewer, Bryan Magee, asked Ayer about the defects of Logical Positivism in general and Language, Truth and Logic in particular. Ayer's response was: "Well, I suppose the most important of the defects was that nearly all of it was false" (1978, p. 107).

14

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means that 'It is false that' functions exactly as 'It is not the case that'. Similarly, the claim that 'true' is a mark of assertion means that the sentence 'It is true that p' reiterates the assertion made by the sentence 'p'. In this view, 'true' and 'false' are logically superfluous (Ayer, 1936, p. 88). Hence, it is true that p if and only if p, and it is false that p if and only if not-p. Accounts that take truth and falsity as certain syntactical operations are far from being restricted to members of the Vienna Circle and their followers. For instance, a recent proposal advanced primarily by Dorothy Grover treats 'true' and 'false' as syntactical devices. Like Ayer, she believes that they are neither property-ascribing predicates nor names of abstract objects. But unlike Ayer, she does not think that they are marks of assertion and denial. Grover's proposal is that the words 'true' and 'false' are what she terms prosentences. She defines prosentences as linguistic devices which occupy positions in sentences that declarative sentences can occupy, just as pronouns occupy positions in sentences that singular terms occupy. This proposal, when fully worked out, entails all the biconditionals of the form 'It is true that p if and only if p', where 'p' is replaced by any sentence that either contains no prosentences, or all its prosentences ultimately have antecedents which are themselves sentences containing no prosentences.18 My position is that truth and falsity are properties. Whether they are properties of sentences, propositions, beliefs, or something else need not concern us here. My reason for believing that they are properties is simple and by no means novel (see, for example, Horwich, 1990, pp. 38-41). It seems clear to me that when I say, "The sentence '5 is a prime number' is true," I am talking about a certain sentence having a certain property and not about the number 5 being prime, as Frege claimed. Frege and Ayer would have insisted that 'Snow is white' and "'Snow is white' is true" are making exactly the same assertion, namely that snow is white, but this is hardly an argument. Perhaps there is some merit to their claim, if we rephrase the two sentences as follows: 'Snow is white' and 'It is true that snow is white'. For in this case, our linguistic intuition appears to be less discriminating. However, 18 See Grover, Camp, and Belnap (1975) and Grover (1977 and 1981); also see Pollock (1977) for comments.

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there seems to be no doubt that we intuitively understand 'Snow is white' and "The sentence 'Snow is white' is true" (or "'Snow is white' is true") as having different referential contents: the first is about the natural kind snow and the property of being white, while the second is about the sentence 'Snow is white' and the property of being true. In the face of such strong intuition, the claim that these sentences make the same assertion is, without further argumentation, just that—a claim. It is important to note that the preceding argument (which I shall rehearse two more times in this section) is intended as abductive. The point is that our strong linguistic intuition of understanding 'Snow is white' and "The sentence 'Snow is white' is true" in the manner described above can easily be explained by the hypothesis that truth is a property (of sentences). If one wishes to deny this hypothesis, then he should provide an account explaining away this intuition. Without such an account, it is unreasonable to dismiss a hypothesis that renders a common linguistic phenomenon intelligible in favor of one that leaves the phenomenon mysterious. I must warn, however, that if the argument of the previous paragraph were assumed to be deductive, it would be either invalid or guilty of begging the question. Interpreting the main premise correctly (as saying something about what we take the referential contents of certain sentences to be) produces an invalid argument, while giving the premise a stronger reading (namely that it is saying something about what the referential contents of certain sentences really are) leads to question begging. Grover (1981, pp. 71-73) provided an argument for the thesis that truth and falsity are not properties. Her argument is aimed at undermining the position that 'true' and 'false' are predicates (of sentences). If they were predicates, she argued, then they would partition the domain of all linguistic expressions. But the semantical paradoxes (Liar and company) give reason to believe that no such partition is possible. Hence, there are grounds for believing that 'true' and 'false' are not really predicates after all. But if this is so, then what justification is there for claiming that truth and falsity are properties? I agree that if 'true' and 'false' turn out not to be predicates, then it would be doubtful that truth and falsity are properties. For we are led to believe that they are properties mainly because

16

The Liar Speaks the Truth

we intuitively understand 'true' and 'false' as predicates whose semantical roles typically include attributing truth and falsity to sentences and the assertions they make. But I reject the claim that property-ascribing predicates must partition the domain of all linguistic expressions in the manner that Grover seems to have in mind. The conventional view is that a property-ascribing predicate divides the domain of all linguistic expressions into two mutually exclusive and jointly exhaustive classes. The first class consists of all the expressions which designate objects that have the property named by the predicate. The second class contains those and only those expressions which either connote nothing or denote objects that do not have the property named by the predicate. Moreover, such a division is fixed and unique. This view of course is about the metaphysics of properties. But one might reject this view and argue that properties come with categories. This is another view of the world which presupposes that an object must be of a certain nature in order for it to fall under the applicability of a certain property. For example, numbers are outside the category of the property of redness. A number in this view is neither red, nor is it not red. According to the category story, the kind of partitioning which Grover assumes to be produced by every property is only correct of the category associated with the property. One might also argue that certain property-ascribing predicates are vague and that such predicates do not provide neat partitions. The borders of the two classes carved by a vague predicate are necessarily fuzzy: they may overlap in some cases or leave pools of unclaimed expressions. The paradoxes and other semantically problematic sentences might be taken as providing a reason for believing that 'true' and 'false' are vague predicates.19 I myself do not find the category story credible. I also do not think that vagueness is relevant to a correct solution to the semantical paradoxes. My position is that of Gupta (1989). Truth is 19 Many systems of multiple-valued logic are motivated by considerations about categories or about vagueness. In the recent literature on truth and paradox, several of these logical systems have found fertile grounds for application. See Martin (1967 and 1970) for a category approach to the problem of truth and semantical paradox. And see Kearns (1970) and McGee (1989 and 1991) for solutions to the problem of semantical paradox that take 'true' as a vague predicate.

The Tarakian Schema

17

a property that does not necessarily produce a fixed and unique partition. Rather, it provides a process of partitioning that might yield a unique fixed partition, two or more fixed partitions, or no fixed partition at all. (I shall explain and elaborate on these ideas later.) The point I am trying to make here is this. Grover's claim that the lesson of the semantical paradoxes is that 'true' and 'false' are not property-ascribing predicates cannot be warranted without an adequate defense of the view that properties provide fixed classical partitions of the world. The alternative positions sketched above show that one may consistently maintain that the lesson of the paradoxes is that there are properties such as truth which do not carve the world in a fixed classical way. Two more points ought to be mentioned regarding the claim that truth is a property. First, it may be noted that truth, if it is a property, must be of a peculiar sort. For in most of the cases with which we ordinarily deal, the question of whether a sentence has the property of truth or not is reducible to the question of whether some other object has a certain property or not or whether some objects stand in a certain relation to each other or not. For instance, whether the sentence 'George is curious' has the property of being true is determined by whether George has the property of being curious. This is perfectly all right. Even if truth is a supervenient property, it remains, nevertheless, a property. To look ahead, however, the revision theory of truth supports the view that truth is not supervenient. The second point is this. A thoroughly nominalistic philosophy would deny that the predicates 'is true' and 'is false' denote any properties. But this, of course, is part of a general stand which entails that no predicate denotes any property; for the world contains no abstract entities, such as properties, types, and classes. I trust it is understandable that I do not wish to debate nominalism in this book. But for the record, I state without argumentation my view on this issue. I do not think that nominalism is correct. I believe that any view of the world (human activities included) which is hoped to be adequate, or nearly so, must warrant an ontological commitment to certain abstract entities such as properties and types and to certain abstract structures such as the structure of the whole numbers. Putting nominalistic tendencies aside, there appears to be no

18

The Liar Speaks the Truth

good reason to deny that 'true' and 'false' are property-ascribing predicates, just as 'grammatical' and 'meaningful' are. The phrases 'is true' and 'is false' are usually applied to quotational names and definite descriptions of sentences. This ordinary and frequent usage of 'true' and 'false' ought to be a serious worry for any account that takes truth and falsity as linguistic operations. Grover, Camp, and Belnap (1975, pp. 102-3), for example, expressed a healthy degree of uneasiness about how their prosentential view of 'true' and 'false' may account for this ordinary English usage. Recognizing the seriousness of this difficulty, they went so far as to suggest, without an argument, that quotation in ordinary English should perhaps not be considered as a way of referring to linguistic expressions. Ayer in his essay "Truth" (1963), which contains a more moderate view of truth than the one expressed in Language, Truth and Logic, wrote: I conclude from this that though it is strictly incorrect to refuse to accord the words 'true' and 'false' the status of predicates, yet those who have taken this line are basically in the right. For even when they do function as predicates, the role which these terms essentially fulfil is that of assertion or negation signs. The material content of a sentence s which implicitly or explicitly ascribes truth to a statement p, may differ from the material content of p inasmuch as it may refer to p in a way in which p does not refer to itself; but the information which we may gain from s by way of this reference to p, adds nothing to what we gain from p alone, (pp. 166-67)

It seems obvious that Ayer in this passage has accepted the common and intuitive view that 'true' is a predicate and that the sentences 'S is true' and 5 may have different (material) contents. Ayer's last statement may be interpreted in two ways. If one takes it as claiming that S and 'S is true' make exactly the same assertion (as noted above, this is the view expressed in Language, Truth and Logic), then how is it possible for them to have different contents? On the other hand, if one interprets it as claiming that the truth of a sentence supervenes on what is asserted by the sentence itself, then there is nothing in Ayer's passage that entails or

The Tarskian Schema

19

supports the claim that truth is not a property. It is a strong intuition that 'The sentence 5 is true' is about the sentence S and the property of being true and that the assertion it makes attributes truth to S. Until a convincing case is made to the contrary, there is no justification for doubting that this intuition is legitimate and substantial. I.3. More Motivations: Tarski's Theory of Truth

In his classic paper "The Concept of Truth in Formalized Languages" (1933), Tarski gave negative and positive accounts, each of which had a great impact on logical and philosophical studies of truth and related issues. The central claim of his negative account may be expressed in general terms as follows. Any language which possesses the wherewithal for expressing a "sufficient" amount of its syntax and semantics and which is in harmony with basic logical laws is inconsistent. Tarski argued that in such a language semantical paradoxes can be produced and nurtured to a harmful extent. To sketch the argument, let us consider the extreme case of universal languages. A universal language is roughly a language in which we can speak about anything that is possible to speak about (see Tarski, 1933, p. 164). In particular, a universal language is capable of expressing whatever can be said about its own syntax and semantics. Hence, it must contain names or descriptions of all its sentences, and it must have the means to represent the relation of denoting and the property of being true. It is possible, therefore, to produce in such a language a Liar sentence whose English translation might be 'c is not true', where V denotes the sentence 'c is not true' itself. But given the fundamental intuition about truth and the universality of the language under consideration, it follows that all the Tarskian biconditionals of the sentences of this language are included among its true sentences. Thus, this language has as one of its true sentences the Tarskian biconditional of its Liar. If this Liar has the English translation mentioned above, its Tarskian biconditional would be translated into the English sentence 'c is true if and only if c is not true'. But such a biconditional generates a contradiction: if c is true then it is not, and if it is not true then it is. The last argument, which is known as the Liar

20

The Liar Speaks the Truth

argument, is an English translation of a similar argument in the given language involving its own Liar and truth predicate. The Liar argument sketched above shows that a language need not be universal in order to generate contradictions. As the original claim requires, it must only have the resources to express a "sufficient" amount of its syntax and semantics. It is natural to ask here exactly how much is sufficient. A serious investigation of this question is of greater technical interest than philosophical. What matters philosophically is that the paradoxes do not require an expressively rich and logically powerful linguistic environment in order to flourish. For example, Tarski (1933, pp. 247-65) showed that any mathematical language (i.e., a formal language constructed for the purpose of describing some mathematical structure) with certain arithmetical resources would give rise to semantical paradoxes, if it were also equipped with a predicate whose semantical role is to form a true sentence when, and only when, it is applied to a name (or code) of a true sentence.20 Tarski (1933, p. 165) also argued that any language (whether formal or not) for which the laws of classical logic hold and which meets the following conditions is inconsistent. First, it contains names of all its sentences. Second, all the Tarskian biconditionals of its sentences are themselves true sentences of this language. Third, a sentence whose English translation is "c is identical with 'c is not true'" can be formed as a true sentence of the language (in other words, the language contains a Liar sentence). It is clear that the Liar argument would go through in such a language. In "The Semantic Conception of Truth" (1944, pp. 20-21) Tarski weakened the conditions above by replacing the third condition with the requirement that the language contains its own truth predicate. He called a language which meets the weaker conditions semantically closed. In this case, the claim is that any semantically closed language for which the laws of classical logic hold is inconsistent. However, Gupta (1982, pp. 182-90) proved that these conditions are too weak. He constructed a classical language that is semantically closed in the Tarskian sense yet consistent. In Chapter 2, a simplified version of Gupta's construction is described and his 20 Vann McGee (1985b) proved that the usual language of arithmetic would generate semantical paradoxes, even if it were to have a predicate that satisfies a much weaker requirement than the one stated above.

The Tarskian Schema

21

result is made clear. But for now, I only say that a semantically closed classical language would be inconsistent, if it were further required to be able to express certain syntactical notions such as negation and substitution21 One last point ought to be mentioned in order to complete our summary of Tarski's negative account. One particular consequence of the claim that no universal language is consistent has incited a good deal of discussion. It is the conclusion that no natural language is consistent; for natural languages are widely held to be the paradigm of universal expressiveness. This claim has genuine roots in Tarski's original paper (1933, pp. 164-65). However, in another paper (1944), Tarski cautioned that everyday languages do not have precisely specified structures and that there is thus no clear meaning to the problem of consistency with respect to these languages. He concluded: "We may at best only risk the guess that a language whose structure has been exactly specified and which resembles our every day language as closely as possible would be inconsistent" (p. 21). I leave this point without comments. Further discussions of issues related to the connection between natural language, universality, semantical paradox, and consistency may be found in the works of Herzberger (1967 and 1970), Martin (1976), and Simmons (1990). Tarski's positive account has two characteristic features. First, it involves a hierarchy of languages. Second, it specifies schematically truth conditions for the sentences of a certain type of formal language in terms of satisfaction. The basic unit of a Tarskian hierarchy consists of two levels: an object language and a metalanguage. Both of these languages suffer a severe expressive limitation: they cannot speak directly of themselves.22 In particular, neither is capable, directly or indirectly, of expressing the truth or falsity of its sentences. The truth predicate for the object language is located in the metalanguage, and that for the metalanguage in the meta-metalanguage, and so on. In general, the metalanguage is ex21

See, for example, Gupta (1982, pp. 190-94). I say "directly" because, as Godel (1931) showed, a language with sufficiently rich arithmetical resources can be made to express indirectly its syntax, background logic, and a limited amount of its semantics by constructing in the language itself numerical codes of its expressions and some of what they are intended to denote. 22

The Liar Speaks the Truth

22

pressively richer than the object language. On the meta-level, one can say everything which can be said on the object-level. In addition, one can talk on the meta-level about the object language—its syntax, background logic, and semantics.23 Tarski's definition of truth consists of a recursive list of satisfaction conditions for the formulas of a formal classical language whose interpretation is fixed. I believe that Tarski's original picture is of an object language that consists of certain types of strings whose referents (or perhaps meanings) are "outside" the language (see Tarski, 1933). Another picture painted by Lewis (1975, pp. 3-12) and Soames (1984, pp. 425-28) is of an object language whose interpretation is built into it. In this picture, the language is presented as a collection of types of strings coupled with their meanings (i.e., a collection of ordered pairs), or as a function associating types of strings with their meanings, or as itself being an ordered triple consisting of a collection of types of strings, a collection of objects, and a relation (representing denotation) between the members of the two collections. Whatever picture one may adopt, two things are clear. First, since the object language lacks the resources to describe its own semantics, its fixed, intended interpretation (or the intuitive interpretation, as Tarski called it) must be described on the meta-level. Second, Tarski's satisfaction conditions yield a definition of truth for the object language that is not relativized to an interpretation (or model).24 For in both views, the language is already interpreted.25 23

Kripke (1975, p. 61) noted that a full-scale Tarskian hierarchy comprising finite and transfinite levels has never been seriously investigated. I can at least confirm that I was unable to find such an investigation in print. However, Church (1976) gave a technical account of the finite levels of a Tarskian hierarchy. Quine (1962, pp. 6-10) and Burge (1979, p. 85) offered brief descriptions of a Tarskian hierarchy (finite levels only) of truth predicates. It is worth noting that Burge (1979) advanced a proposal which treats 'true' in ordinary English as an indexical. Thus, a hierarchy of indexed truth predicates emerges within the same language rather than within a hierarchy of languages. My main objection to Surge's account is that (other than being sufficient for evading the paradoxes) there seems to be no linguistic or contextual evidence which supports the indexicality picture of 'true' (see also Gupta, 1982, pp. 202-4). 24 Tarski (1936a) introduced the notion of truth in a model in order to define the concept of logical consequence. 25 It is clear that the concept of truth defined for a language conceived a la

The Tarskian Schema

23

Let us now take a closer look at Tarski's definition of truth. The presentation below is a simplified exposition of Tarski's account as given in "The Concept of Truth in Formalized Languages" (1933). We deal with special cases only, and we don't insist on following Tarski's exact details, but we shall keep within the spirit of Tarski's original work. After all, our goal here is only to illustrate rather than to give a detailed exposition of Tarski's historical work. Although Tarski defined the truth and falsity of all the sentences of the object language in terms of satisfaction, the truth conditions for the atomic and quantifier-free sentences can be stated directly without invoking the technical notion of satisfaction. For example, the sentence 'Earth is a planet' is true if and only if Earth is a planet. In a formal setting we might have the sentence 'a is P'. The condition above would read in this case that 'a is P' is true if and only if a is P. Notice that Tarski did not give an account of the conditions under which a is P, or more generally, the conditions under which an object has a certain property. This is an important point and it constitutes, I believe, a basic Tarskian insight: it is not the business of a theory of truth to specify such conditions. Consider, for example, the sentences 'Earth is a planet' and ' is a rational number'. It is not the duty of a truth theorist to determine or investigate whether Earth is a planet or not or whether is a rational number or not; these are concerns of astronomy and mathematics and not of truth theory. Likewise, it is the business of philosophy of language and metaphysics and not of truth theory proper to explain what it means to say in general that some object possesses or lacks a certain property and under what conditions an "arbitrary" object would have a certain property (is it that the object must partake of a certain form, or instantiate a certain universal, or exhibit some physical structure that enables it to serve a certain function? etc.). A proper concern of a theory of truth, however, is to give the conditions under which a sentence is true. A truth theorist would say, for instance, that the sentence 'a is P' has the property of truth if and only if a is P. Lewis and Soames is a concept of truth in the language rather than of truth in an interpretation, since such a language carries within it its own interpretation. However, Field argued that Lewis' and Soames' conception of a language makes their notion of truth in the language purely mathematical and not essentially different from Tarski's notion of truth in a model (see Field, 1986, pp. 64—67).

24

The Liar Speaks the Truth

As suggested by the examples above, a schematic truth condition for the atomic sentences can be stated as follows: x is true if and only if p where 'p' may be replaced by any atomic sentence of the object language,26 and 'x' by some appropriate name of it (e.g., its quotational name). The truth conditions of the compound sentences that involve only sentential, truth-functional connectives are based on the truth conditions of the atomic sentences and the semantical rules which define the "meanings" of those connectives. For example, '2 is prime and 2 is even' is true if and only if both '2 is prime' is true and '2 is even' is true; given the schema above, it follows that '2 is prime and 2 is even' is true if and only if both 2 is prime and 2 is even. A general argument along similar lines to those illustrated by the previous example would show that such truth conditions entail all the Tarskian biconditionals of the quantifier-free, compound sentences of the object language. The story gets more complicated when the language is equipped with quantifiers. Giving recursive truth conditions for the quantified sentences makes the need for an indirect approach urgent. To illustrate the problem, it suffices to consider one of the simplest possible examples: the sentence ' xPx'. An immediate reaction might be to say that ' xPx' is true if and only if xPx (or using a natural language translation, if and only if everything is P). But Tarski was interested in giving a recursive truth condition, which is roughly a condition based on (or "computed from") the conditions specified for the basic cases (in our situation, these are the atomic sentences). One way of dealing with the problem is to use what logicians call substitutional quantification: ' xPx' is true if and only if the sentences formed by applying the predicate 'P' to every name of the language are all true. In order for this condition to give the right results, every individual must have a name in the language and, of course, all names must have referents. Cardinality considerations alone show that such requirements cannot 26 This presupposes that the object language is part of the metalanguage. Hence, every sentence of the object language is also a sentence of the metalanguage. If the former language is not included in the latter one, then 'p' is to be replaced by any sentence of the metalanguage that translates some sentence of the object language.

The Tarskian Schema

25

always be met, if the languages under investigation are supposed to have fixed vocabularies. There simply might not be enough names in a given language for all the individuals. In order to give recursive truth conditions for the quantified sentences,27 Tarski introduced the technical notion of satisfaction: a relation between sentential functions (i.e., formulas with free variables, such as 'x + y = y + x ' ) and sequences of individuals.28 As an example, consider the sentential function 'x lived longer than y'. The sequence (Beethoven, Mozart) (i.e., the sequence whose only terms are Beethoven and Mozart, in that order) satisfies the sentential function 'x lived longer than y'. The same sentential function, however, is not satisfied by the sequence (Schubert, Mozart). In general, the sequence whose first and second terms are the individuals a and b satisfies the sentential function 'x lived longer than y' if and only if a lived longer than b. The method can be generalized to formulas with any number of free variables. For example, the sequences (3,4,7) and (2,3,9,4) satisfy the sentential functions 'x + y = z' and 'x + y = z — w', respectively. A single individual may be considered as a sequence consisting of one single term. Hence, we may say that 'x is a planet' is satisfied by the object Earth or by the sequence (Earth). Satisfaction provides us with a means of giving a recursive truth condition for the sentence ' xPx': ' xPx' is true if and only if the sentential function 'Px' is satisfied by every object (or by every sequence consisting of one term only).29 Using satisfaction, a schematic truth condition can be defined 27

See Tarski (1933, p. 189). Tarski operated mostly with infinite sequences. He did note, however, that the procedure can be carried out completely with finite sequences of arbitrary lengths (see Tarski, 1933, p. 195n). For the sake of simplicity, we work here with finite sequences only. 29 Most modern texts of mathematical logic define satisfaction for formulas in terms of assignments instead of sequences. An assignment is a function which associates with every variable an individual (usually called its value). For instance, the formula 'x2 = 2' is satisfied by any assignment that gives 'x' the value , and it is not satisfied by any assignment that gives 'x' a rational value. In this approach, ' xPx' is true if and only if 'Px' is satisfied by all assignments. Except for some minor details, the two approaches are identical. See Quine (1970, ch. 3) for a brief, self contained exposition of Tarski's original account, and for a typical modern approach, see Enderton (1972, pp. 79— 86) or Barwise (1977). The papers by Tharp (1971), Wallace (1972), Kripke 28

26

The Liar Speaks the Truth

for the general case of any quantified sentence. The definition follows the main lines sketched in the previous paragraph for the truth condition of the simple sentence ' xPx'. The important point is that such a definition would entail all the Tarskian biconditionals of the quantified sentences. This fact together with what we noted above about the quantifier-free (atomic and compound) sentences make it clear that Tarski's theory of truth entails all the Tarskian biconditionals of the sentences of the object language in question. Indeed, it was Tarski who declared that a definition of truth is materially adequate if and only if it has as consequences all the biconditionals obtained from 'x is true if and only if p' by substituting for 'p' any sentence of the language under consideration and for 'x' some appropriate name or description of that sentence.30 He explained that a materially adequate definition of some notion is a definition that captures the "meaning of the notion as it is known intuitively" (1931, pp. 128-29).

(1976), and Fine and McCarthy (1984) are quite informative, though relatively technical, works that explore issues about substitutional quantification and the possibilities and limitations of defining Tarskian truth without satisfaction. 30 See Tarski (1933, pp. 187-88, 1936b, p. 404, and 1944, pp. 15-17). Readers who might question the textual correctness of stating this requirement as sufficient and necessary are referred to Tarski's paper (1933, p. 247) where he stated his celebrated theorem (Theorem I) about the undefinability of truth in the language of set theory. Tarski's text shows that the reading given above is correct.

1 The TS Conception of Truth

This chapter is devoted to one multilayered argument. The main purpose of the argument is to show that the thesis TS and the initial conditions mentioned on page 3 entail the revision theory of truth and to present a preliminary defense of this theory. The circularity argument (in Section 1.3) unpacks the TS conception: the concept of truth is circular and in a bivalent language containing its own truth predicate, this circularity gives rise to a process of revision that determines the truth status of each sentence of the language. The correctness of the revision theory is demonstrated by its ability to accommodate successfully the diverse patterns of behavior exhibited by the concept of truth. The next three chapters are devoted to that task, but in this chapter we give a preliminary justification for this theory by arguing (in Sections 1.1 and 1.2) that there are grounds for accepting the thesis TS and those initial conditions, which collectively entail the revision theory of truth. 1.1. The Tarskian Schema Defines Truth Completely

I argued in the Introduction that the intuition represented by the Tarskian schema is the most fundamental intuition about the concept of truth. As we have seen, this is manifested by philosophical reflections on this concept as well as by the ways in which we or27

The Liar Speaks the Truth

28

dinarily understand and employ the terms 'true' and 'false' to attribute truth and falsity to sentences (or to what they assert). Now I want to say that this fundamental intuition in some sense characterizes the concept of truth. More precisely, I want to conclude that the Tarskian schema defines the concept of truth completely and correctly, which is the claim made by the thesis TS. Many philosophers and logicians, such as Tarski, Ayer, Gupta, and Horwich, have recognized this schema as defining the concept of truth. Tarski (1944, p. 16) said that every biconditional of the form 'x is true if and only if p', where 'p' is replaced by a sentence and 'x' by a name of it, is "a partial definition of truth, which explains wherein the truth of this individual sentence consists," while the general definition of truth "has to be, in a certain sense, a logical conjunction of all these partial definitions." Ayer (1963, pp. 163-64) explained that although "Tarski's formula is not itself a definition of truth but only a schema," nevertheless this schema "achieves as much as can be expected," for if "we apply it to any given sentence, then provided that we know what statement the sentence is being used to make, the formula will tell us what we mean by saying that the sentence is true." Gupta in "Truth and Paradox" (1982) expressed some reservations about embracing the fundamental intuition depicted by the Tarskian schema as correctly characterizing the concept of truth. He assumed, as is commonly done, that the instances of this schema are material biconditionals. Given this assumption, he argued that "the fundamental intuition cannot be preserved under all conditions" (p. 182),1 for in certain classical languages the Tarskian schema, which formulates the fundamental intuition, has contradictory instances, such as the material biconditional 'c is true if and only if c is not true' (where c is identical with 'c is not true'). However, in "Remarks on Definitions and the Concept of Truth" (1989) Gupta took the instances of the Tarskian schema to be definitional, rather than material, biconditionals.2 Thus the preceding argument no longer applies. Gupta was able, therefore, 1

See also Gupta (1982, pp. 194-95, and 1989, p. 243). To interpret 'x is true if and only if p' as a definitional biconditional is to take the right-hand side as specifying the conditions under which x is true. I hope it is clear that the TS conception of truth presupposes a definitional reading of the Tarskian biconditionals. 2

The TS Conception of Truth

29

to endorse the fundamental intuition as "completely correct" and the instances of the Tarskian schema as "constitutive of the notion of truth" (p. 243). In his recent book Truth (1990), Horwich defended a deflationary conception of truth which he called the minimalist conception. At the outset of the book, he announced that the "triviality" which says "each proposition specifies its own condition for being true" captures the notion of truth entirely. His official statement of the minimal theory of truth is "what is expressed by uncontroversial instances" of the schema "It is true that p if and only if p" (pp. 6-7). Except for taking propositions as the bearers of truth, Horwich's minimalist conception appears to be very similar to the one conveyed by the first part of the thesis TS, namely that the instances of the Tarskian schema exhaust the notion of truth. There is, however, a significant difference: the Tarskian biconditionals are interpreted materially in the minimalist conception and definitionally in the TS conception (see note 2). To give an adequate defense of the thesis TS, we need to argue for both parts of the claim made by this thesis, namely that the Tarskian schema defines the concept of truth completely and that it defines it correctly. An argument (which I find convincing) for the first part is already outlined in the remarks cited above, especially those of Tarski and Ayer. Here is a more elaborate presentation of this argument. Complete definitions of concepts are of several kinds. Some of them are reductionist, that is, they reduce the concepts in question to other concepts, presumably simpler than the ones being defined. For example, the definition 'An even number is a number divisible by 2' reduces the concept of even number to that of divisibility by 2. Another way of giving a complete definition of a concept is by specifying for each object the condition or conditions under which that object is subsumed under the concept. If the concept is that of some property, such as the concept of truth, then such a definition would specify for each object of the appropriate kind the conditions under which this object has the property. The appropriate kind is to be understood as the kind of the bearers of that property. This does not imply that every object of the appropriate kind can bear the property in question, nor does it involve an implicit commitment to categories. For ex-

30

The Liar Speaks the Truth

ample, the claim that sentences are the bearers of truth does not imply that every sentence could be true—indeed a contradiction could not—nor does it imply that the nonsentences are neither true nor not true (whatever that might mean). It does imply, however, that being a sentence is a necessary condition for being true. The last claim entails that the nonsentences are not true.3 The reason for designating an appropriate kind is simply that the objects of this kind are the objects for which the conditions of having that property are significant or interesting. The objects which are not of the appropriate kind have the trivial condition that all of them lack the property in question. This, I believe, is a requirement that should come with any definition of the second type. The Tarskian schema achieves exactly what is described in the preceding paragraph. For each sentence of any given language, the schema produces a biconditional which specifies the condition under which that sentence has the property of truth. For example, the condition under which the sentence 'Earth is a planet' has the property of truth is precisely what the sentence itself asserts, namely that Earth is a planet. This condition is specified by the Tarskian biconditional "The sentence 'Earth is a planet' is true if and only if Earth is a planet." The general conclusion is that for every sentence S, any Tarskian biconditional of 5 specifies that the condition under which 5 has the property of truth is precisely what S itself asserts. Given this general conclusion and the discussion of the preceding paragraph, it follows that the Tarskian schema indeed defines the concept of truth completely. 1.2. The Tarskian Schema Defines Truth Correctly The second part of TS is admittedly more controversial. In fact, among the four philosophers and logicians mentioned above, only Gupta unequivocally endorsed the view that every instance of the Tarskian schema is constitutive of the concept of truth. Horwich 3 Of course, this does not mean that the nonsentences are false. They are neither true nor false. I must emphasize here too that the previous statement does not involve any commitment to categories, for something is false if and only if it is a sentence that is not true and something is true if and only if it is a sentence that is not false.

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31

rejected this view. He said that the instances of this schema must be "restricted in some way so as to avoid paradoxical results," and hence "certain instances of the equivalence schema are not to be included as axioms of the minimal theory" (1990, pp. 41-42). There is textual evidence strongly suggesting that Ayer and Tarski would reject this view too (see, e.g., Ayer's remarks cited above and Tarski's italicized words on page 165 of "The Concept of Truth in Formalized Languages"). The difficulty is that the Tarskian schema has problematic instances. Let us consider two such instances: the Tarskian biconditionals of a Liar and of a Truthteller.4 It is an empirical fact that in a sufficiently expressive language one can form sentences that deny or affirm their own truth. For example, in English it is possible to produce a sentence which by means of self-reference or crossreference denies that it is true. Such a sentence is called a Liar sentence or simply a Liar. The sentence below is a self-referential Liar: The boxed sentence on this page is not true To see a case of Liars generated by cross-reference, consider the following example. Suppose that Smith wrote on occasion t this sole sentence, "The sentence Jones will write on occasion t' is true," and later, on occasion t', Jones wrote only this sentence, "The sentence Smith wrote on occasion t is not true." Both of these utterances are Liars. A Truthteller, on the other hand, is a sentence that by means of self-reference or cross-reference affirms that it is true. Here is a self-referential Truthteller: The italicized sentence on this page is true If the Smith-Jones example were changed so that the sentence Jones wrote on occasion t' was "The sentence Smith wrote on occasion t is true," then both of Smith's and Jones' utterances would be Truthtellers. The Tarskian biconditionals of Liars and of Truthtellers are problematic. The biconditionals 'c is true if and only if c is not true' and 'd is true if and only if d is true' are the instances of the Tarskian schema when it is applied to the Liar 'c is not true' 4

I use 'Truthteller' instead of 'Truth-teller' by analogy with 'Storyteller'.

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and its name V and to the Truthteller 'd is true' and its name 'd'.. Before entering into "deep" philosophical reflection, we all could intuitively judge the first biconditional to be contradictory and the second to be tautologous. Since these biconditionals are generated by two empirical facts, namely that c is identical with 'c is not true' and that d is identical with 'd is true', and by the Tarskian schema, it seems questionable that this schema can be regarded as defining the concept of truth correctly. For if we accept all of its instances to be constitutive of the concept of truth, then we must accept that paradoxicality (as shown by the first biconditional) and arbitrariness (as shown by the second biconditional) are genuine features of this concept.5 Thus, one might conclude, if we believe that the concept of truth is "legitimate," then we should not consider every Tarskian biconditional as constitutive of it, and hence the Tarskian schema must be restricted in some way to prevent it from generating problematic biconditionals. As explained earlier, Tarski's approach to restricting this schema is syntactically based: no Tarskian language is permitted to contain its own truth predicate. This restriction is too severe. Consider the usual language of arithmetic as our object language, and take its metalanguage to be some appropriate extension of it. In this object language we can say, for instance, that 1 + 1 = 2. But as Tarski required, we cannot say (directly) on the object-level that '1 + 1 = 2' is true, for neither the quotational name " '1 + 1 = 2' " nor the predicate 'is true' belongs to the permissible vocabulary of the object language. In the metalanguage, however, we may say that '1+1 = 2' is true and we can assert the biconditional "' 1 +1 = 2' is true if and only if 1 + 1 = 2" which specifies the truth condition of '1 + 1 = 2'. Since the metalanguage is also prohibited from speaking (directly) about its sentences and their truth status, one cannot say in this language that the sentence "'1 + 1 = 2' is 5

The contradictory nature of the Tarskian biconditional of the Liar 'c is not true' shows that the condition under which c is true cannot obtain or fail (either way we land on a contradiction); for this biconditional in effect requires that c is subsumed under the concept of truth if and only if it is excluded from this concept. On the other hand, the truth condition of the Truthteller 'd is true', which is specified by the tautology 'd is true if and only if d is true', can fail or obtain independently of and consistently with all of the nonsemantical facts but that 'd' is identical with 'd is true'.

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true" is true, nor can one express its Tarskian biconditional. To make such assertions we need to move to the meta-metalanguage, and so on. But there is nothing problematic about the Tarskian biconditional of any sentence formed from '1 + 1 = 2' by reiterating the truth predicate and quotation marks some finite number of times. All such sentences are grounded, that is, their truth status depends ultimately on the nonsemantical fact that 1 + 1 = 2.6 There are other syntactical approaches which allow a language to contain the terms 'true' and 'false' but avoid the problematic cases by rendering them ill-formed, nonsensical, or ambiguous. We discussed and alluded to such proposals earlier (e.g., Ayer's, Grover's, and Burge's) and concluded that they lack reasonable philosophical and linguistic justification. Thus, we acknowledge that there is a need to give a coherent definition of the notion of truth for languages that are capable of expressing, at least, the truth or falsity of their sentences. In fact, recent works on truth and semantical paradox have made it clear that it is possible to give a coherent definition of the notion of truth for languages that possess much richer expressiveness than what the minimal requirement above demands (though such languages remain far from being universally expressive). Hence, we restore to the languages under consideration the ability to attribute truth to their sentences. Many recent proposals take a semantical approach to restricting the Tarskian schema by giving up the principle of bivalence.7 This 6

One might object that this argument overlooks the main reason for imposing this kind of expressive limitation, namely that there seems to be no principled way of distinguishing between problematic and unproblematic sentences. But the point of the argument is to suggest that the severity of the expressive limitation is too great to be accepted as sufficiently justified. In fact, much post-Tarskian work in truth theory is motivated by such a concern. 7 The principle of bivalence is the semantical principle which asserts that every sentence is either true or false but not both. This principle is not the same thing as the law of excluded middle. The latter is a logical schema that "generates" logically true sentences (or logical theorems) of the form 'p or notp\ where 'p' may be replaced by any sentence of a given language. Certain logical systems, such as van Fraassen's free logic, validate the law of excluded middle for languages whose semantics, nevertheless, are not bivalent (see van Fraassen, 1966). The instances of the law of excluded middle also might not be statable in some expressively impoverished bivalent languages. However, under reasonable conditions the bivalence of a language must influence, in one way or another, its background logic.

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approach was pioneered by Kripke in his highly influential paper "Outline of a Theory of Truth" (1975).8 I do not intend in this book to give a critical discussion of Kripke's proposal and the many multiple-valued approaches that it inspired.9 These approaches have the virtue of allowing sernantically problematic sentences and their Tarskian biconditionals to be expressible in the language under consideration. But since this language is not bivalent, there are pockets available between the extension of the truth predicate and its "antiextension" (i.e., the class of the "untrue" sentences). These pockets serve two purposes. They allow the theory to declare problematic sentences devoid of any truth status (or of any classical truth status), and in a more sophisticated way, they supply the semantical environment for a notion of logical consequence which lacks certain features (e.g., conditional proof),10 so that the Liar argument and other undesirable arguments can no longer go through. Whatever the employed mechanism might be, the relevant point is that these approaches use nonbivalent semantics to prevent the problematic Tarskian biconditionals from being constitutive of the concept of truth; for they represent no genuine feature of this concept, and hence it must be purified of them and their harmful implications. I want to make one remark here which is similar to the one made when Grover's proposal was discussed in Section 1.2. One needs to give an argument for rejecting bivalence other than that a nonbivalent semantics makes it possible to avoid the paradoxes. For as demonstrated by the various systems of stability semantics that are described in the next three chapters, a coherent bivalent semantics can be developed for languages in which a wide 8 Independently of Kripke, Martin and Woodruff in their short paper "On Presenting 'True-in-L' in L" (1975) arrived at an account which is similar in a central way to that of Kripke. Kripke's paper, however, contains an extended discussion of the inductive construction employed and of its philosophical motivations. The logical apparatus used to arrive at the main result is different in Kripke's work from the one in Martin and Woodruff's. Kripke also offered diagnostic analyses of several semantical phenomena. 9 For an excellent exposition and development of Kripke's approach, see Kremer (1986 and 1988). See also Gupta (1982, pp. 209-12) for a critique of Kripke's theory. 10 Conditional proof is the logical rule that if B is a logical consequence of A, then the conditional 'If A then B' is logically true.

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range of semantically problematic sentences and their Tarskian biconditionals can be expressed. Thus, we restore to our language its bivalent character. Where do we stand now? We refused to restrict the Tarskian schema by either means, syntactically or semantically. We restored to our language its expressive integrity and semantical bivalence. All the Tarskian biconditionals are expressible in the language and are warranted their status as constitutive of the concept of truth. But it seems that we now stand facing the original difficulty: all those problematic instances of the Tarskian schema and their "dangerous" implications. Indeed, that is where we stand and what we do. We face those Tarskian biconditionals and accept their full implications. We give them the stage and allow them to tell their part of the story about the concept of truth. The charge originally made is that the Tarskian biconditionals of Liars and of Truthtellers, if they are constitutive of the concept of truth, imply that paradoxically and arbitrariness are genuine features of this concept; and thus it seems that our insistence on extending legitimacy to all instances of the Tarskian schema has deprived the concept of truth of its own legitimacy. My response is that the first part of this charge is correct. To accept those biconditionals as constitutive of the concept of truth is to accept their implications as descriptive of features of this concept. Hence, we acknowledge that the concept of truth has pathological features such as paradoxicality and arbitrariness. The second part, however, is incorrect. Concepts that exhibit such pathological features need not be illegitimate. In Chapter 4 I describe a coherent, noncreative, 11 and materially adequate semantics of a bivalent first-order language in which all the Tarskian biconditionals of its sentences can be expressed. In that semantical system the various kinds of semantically problematic sentences are given their honest, intuitively expected diagnoses. Unless a concept requires for its legitimacy much more than a coherent, noncreative, and materially adequate analysis, we may conclude that the se11

A noncreative semantics is roughly a semantical system in which no fact is determined, completely or partially, by another fact that is irrelevant to it. For example, the fact that the sentence 'Earth is a planet' is in the extension of the truth predicate must not be determined by whether the Truthteller is also in the extension or not.

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mantical system developed in Chapter 4 establishes our claim that the concept of truth may retain its full legitimacy in spite of its pathological features. 1.3. The Circularity Argument We close this chapter with an exposition and defense of the revision theory of truth. These goals are achieved by what I call the circularity argument. This argument has three stages. The conclusion of the first stage is that the concept of truth is circular. The second stage shows that this circularity, when combined with bivalence, gives rise to a partitioning process which Gupta termed revision. The final stage is concerned with the ways in which the pathological features of the concept of truth influence the revision process, or viewed from a different angle, it is concerned with the ways in which the revision process accommodates the various kinds of problematic sentences in a coherent semantics. We first need to explain, if only crudely, what it means to say that some concept C is circular. In a broad sense, C is circular if there is a nonempty collection of objects such that for each object o in this collection, the conditions under which o is subsumed under C involve ultimately reference to C itself. This broad sense includes the stronger notion of circularity whereby such conditions involve ultimately the condition that o is subsumed under C or that o is excluded from C. For reasons which will become clear in later chapters, I call a noncircular concept grounded. The examples described above of self-referential and crossreferential Liars and Truthtellers show the strong circularity of the concept of truth. This concept is also circular in the broad (nonstrong) sense. Imagine an ideal computer being programmed to enter an infinite loop and to print out indefinitely the sentence 'The next utterance in this list is true'. The conditions under which an utterance in that list is true do not involve "ultimately" any reference to the truth status of the utterance itself, but they involve continuous reference to truth. If the idea of an infinite list of utterances seems implausible, then consider instead this new version of the Smith-Jones example. Suppose that what Smith wrote on occasion t was a self-referential Truthteller and that Jones wrote on

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occasion t': "The sentence Smith wrote on occasion t is true." Although the truth conditions of Jones' utterance do not ultimately involve reference to its own truth status, nevertheless they show that the concept of truth is circular in the broad sense. If we examine the problematic sentences considered anywhere in this book (or for that matter, anywhere in the literature on semantical paradoxes), we discover that their truth conditions are all circular. Hence, we may safely conclude that circularity is a necessary condition for the concept of truth to exhibit pathological behavior. However, it is not sufficient, for this circularity is implied not only by the problematic instances of the Tarskian schema but by perfectly unproblematic ones as well. Consider, for example, either of the generalizations below. NC. No sentence is both true and false. EM. Every sentence is either true or false. The Tarskian biconditional of NC specifies that the truth condition of NC is what NC itself asserts. Now the assertion made by NC involves reference to truth. But this does not show that NC is a circular instance of the concept of truth. We need to show that the truth condition of NC cannot be ultimately reduced to a set of conditions involving no explicit or implicit reference to truth. There are two common views on how "truth-generalizations" such as NC and EM ought to be interpreted. According to the first view, the assertion made by NC is (or is equivalent to) what is asserted by all the sentences of the form 'It is not the case that both p and not-p', where 'p' may be replaced by any sentence. (In this view, NC abbreviates, or represents, an infinite conjunction of all such sentences.) Hence, the truth condition of NC is ultimately given by all what these sentences assert. But if NC itself is a permissible "value" of the variable 'p', then the truth condition of NC ultimately includes what is asserted by NC. Again this assertion involves reference to truth; and if we attempt further reductions, we enter an infinite cycle that always includes the assertion made by NC with its reference to truth. (A similar analysis can be given of the truth condition of EM.) The second view (to which I subscribe) is that generalizations such as NC and EM describe certain characteristics of the extension of the truth predicate (or certain characteristics of the property of truth). NC and EM together

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describe the bivalent characteristic of truth: NC asserts that the extension and antiextension of the truth predicate do not contain any sentences in common, and EM asserts that they collectively do not omit any sentences. If NC and EM make these assertions, then it is clear that their truth conditions involve in an essential manner reference to truth. Both of these views, therefore, imply that NC and EM are circular instances of the concept of truth. But neither NC nor EM is pathological in any reasonable sense of the word. We may conclude that even if the concept of truth were to exhibit no pathological behavior, it would remain, in the presence of sentences such as NC and EM, a circular concept. Hence, if the concept of truth were defined for a language whose expressiveness is sufficiently restricted so that all problematic sentences can no longer be expressed and which includes among its sentences generalizations like NC and EM,12 then we would obtain a counterexample to the claim that being circular is a sufficient condition for a concept to exhibit pathological behavior. The concept of truth in such a language is circular, yet it has no pathological features. We begin the second stage of the circularity argument by asking: given the circularity of the concept of truth, what kind of partition does truth produce? Without bivalence, one can paint the following picture. All the sentences with circular truth conditions are placed in a dump outside the extension and the antiextension of the truth predicate. The remaining (grounded) sentences are divided between the extension and the antiextension in a manner ordained by the nonsernantical facts and the Tarskian biconditionals of these sentences.13 But if the semantics is bivalent, then sentences with circular truth conditions can no longer be discarded; they are either in the extension of the truth predicate or outside it (i.e., in the antiextension, because in bivalent semantics the extension and antiextension complement each other). However, since these sentences have circular truth conditions, the sets of facts which determine their membership status in the extension of the truth predicate must include an initial extension of this predicate. Hence, given a totality of nonsemantical facts together with an ini12

Examples of this type of language are described in Chapter 2. This picture shows that the process of revision does not necessarily extend to nonbivalent semantical systems. 13

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tial extension of the truth predicate, the Tarskian biconditional determine an extension of this predicate which might be different from or identical with the initial extension. If it is different, then given this new extension together with the same totality of nonsemantical facts, a third extension of the truth predicate is determined by the Tarskian biconditionals. Again this third extension might be different from or identical with the preceding extension. The same procedure may be repeated any number of times. This partitioning process is called revision.14 Given an initial extension of the truth predicate, the process of revision yields a hierarchy of extensions of this predicate; let us refer to any such hierarchy as a revision hierarchy. A revision hierarchy might terminate or might not. It terminates if it reaches a, fixed extension of the truth predicate, that is, if it reaches an extension which cannot be altered by any further applications of the revision process. When all possible initial extensions are considered, the revision process yields a large assortment of revision hierarchies. There are several possibilities here: all of these hierarchies terminate, some of them terminate and the others don't, or none of them terminates. Furthermore, two terminating hierarchies might arrive at the same fixed extension or at different ones. In this second stage of the circularity argument we have concluded that revision is characteristic of the concept of truth in bivalent languages because this concept is circular. However, the kinds of hierarchies that revision generates are not determined by the circularity of the concept itself, but rather by the types of circular sentences that can be formed in the language. This is an important point that must be made very clear. The mere fact that there are sentences with circular truth conditions does not tell us much about the revision hierarchies. It does not, for example, imply that there should be a revision hierarchy for each initial extension of the truth predicate or that every hierarchy must have infinitely many levels. What is entailed by this fact is that truth in a bivalent language is not supervenient.15 The membership status 14

Some of the examples discussed in the next chapter explain why it is so called. 15 I use 'Supervenient' here in a fairly strong sense: to say that truth is supervenient in this (strong) sense is to claim that the truth status of each sentence is fully determined by nonsemantical facts. More precisely, truth is

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of a sentence in the extension of the truth predicate is determined by its truth condition. If a sentence has a circular truth condition, then its membership status in the extension of the truth predicate cannot be determined independently of the extension itself (or independently of an initial extension of the truth predicate). The revision process is simply the process of determining for every sentence in a bivalent language its membership status in "the" extension of the truth predicate according to the following rule: the sentence belongs to the extension of the truth predicate if and only if the truth condition specified by the Tarskian biconditional of that sentence obtains. Since the truth conditions of some sentences involve reference to truth in an essential, irreducible manner, these conditions can only obtain or fail in a world that already includes an extension of the truth predicate. Hence, in order for the revision process to determine an extension of the truth predicate, an initial extension of this predicate must be posited. This much follows from circularity and bivalence. However, it does not follow that the revision process must be applied for all possible initial extensions of the truth predicates or that it must be applied infinitely many times. Such features of revision are not due to the circularity of the concept of truth; rather their role is to enable revision to accommodate the various kinds of problematic cases in a coherent bivalent semantics. In the last stage of the circularity argument, we show by means of examples how the revision process is capable of describing and sorting out the complex norms of behavior that the concept of truth exhibits. If the only circular sentences of the given language are sentences like NC and EM (no Liars, Truthtellers, or other pathological sentences), then all revision hierarchies arrive at the same fixed extension of the truth predicate. In fact, a unique fixed extension would still be obtained even if the language were allowed to contain certain sentences that appear pathological. The list below consists of two such sentences. S1. At least one sentence in this list is not true. S2. Every sentence in this list is true. supervenient in this sense only if for all possible worlds W and W' and for every sentence 5, if W and W' share the same totality of nonsemantical facts, then the truth status of S in W is the same as its truth status in W'.

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With some reflection, we discover that there is only one consistent way of assigning truth values to these sentences: S1 is true and S2 is false. Although sentences such as NC, EM, S1, and S2 have circular truth conditions, the truth status of each is determined by the logical nature of the extension of the truth predicate. Since this nature is common to all initial and subsequent extensions of the truth predicate (in our situation, it is a bivalent, classical nature), any sentence whose truth status depends solely on this nature receives the same truth status in all revision hierarchies. If the language described above were further permitted to contain a Truthteller (but no other pathological sentences), all revision hierarchies would still terminate, but there would be two fixed extensions of the truth predicate. All hierarchies starting from initial extensions that declare the Truthteller true (i.e., that contain it) eventually arrive at the same fixed extension that also contains the Truthteller. The rest of the revision hierarchies, which start from initial extensions that omit the Truthteller, arrive at a common fixed extension that also omits the Truthteller. Thus, the revision process captures the intuitive judgment that the truth status of the Truthteller is arbitrary but consistent. However, when a language contains a Liar sentence, its truth predicate can no longer receive fixed extensions. No revision hierarchy in this case terminates. The Liar is tossed back and forth between the extension and its complement from one level to the next in every revision hierarchy. The Liar is paradoxical: its truth status is, either way, contradictory.16 Let us sum up the conclusions of the circularity argument. First, the concept of truth is circular. Second, in bivalent languages this circularity yields a process of revision that employs the Tarskian schema to determine an extension of the truth predicate, once a totality of nonsemantical facts and an initial extension of the truth predicate are posited. Third, the revision process can generate a hierarchy of extensions of the truth predicate for each initial extension of the predicate and these hierarchies are capable of accommodating (and diagnosing) the various kinds of problematic and unproblematic sentences of the languages under consideration. The 16

Chapters 2 through 4 contain extensive analyses of the ways in which the outcome of the revision process is influenced by various kinds of circular sentences.

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conjunction of the first two conclusions of the circularity argument is the central thesis of the revision theory of truth. As this argument shows, the revision theory follows from the TS conception of truth when it is applied to a bivalent language that contains its own truth predicate. If the circularity argument is correct, then we have obtained the following result: a "deflationary" conception of truth gives rise to a "nondeflationary" truth theory. The word 'deflationary' is being used here in two different senses. Let us call a conception of truth deflationary if it does not presuppose about truth more than what is implied by the fundamental intuition. In other words, a deflationary conception of truth is a conception that does not go beyond the Tarskian schema. And let us call a theory of truth deflationary if it entails that truth is supervenient, reducible, or redundant. The two senses of 'deflationary' have often been used interchangeably in describing conceptions, accounts, and theories of truth. The circularity argument shows that these senses are far from being interchangeable. If one takes the fundamental intuition seriously, the metaphysical picture of truth that emerges in a bivalent language is of a nonsupervenient property. The TS conception of truth is highly deflationary (in the first sense). It does not go beyond the Tarskian schema and it does not impose any restrictions on it. Conceptions of truth that restrict the Tarskian schema are less deflationary (in the present sense) than the one defended here. For to restrict this schema, whether by syntactically preventing its instantiation over a range of cases or by semantically blocking the implications of some of its instances, is to assume that the fundamental intuition, which this schema represents, cannot be fully embraced. If this assumption is justified, it must be because the fundamental intuition is incapable of telling the whole story about truth. Therefore to complete the story, one needs to make further presuppositions about the nature of truth, e.g., that truth is vague or that it comes with a category. Some such conceptions, however, may give rise to deflationary theories of truth, which entail that truth is supervenient, reducible, or redundant. There we have the other side of the coin: deflationary truth theories that are based on nondeflationary conceptions of truth. We conclude that the two senses of 'deflationary' are more than just distinguishable: they tend to conflict with each other.

2

Stability Semantics

Stability semantics is a collection of infinitely many formal semantical systems that share a common core: each is a natural candidate for formalizing the revision process described in the previous chapter. In other words, stability semantics is a type of formal semantics whose instances are the semantical systems that are natural candidates for representing formally the revision process. The goal of this chapter is to develop this type of formal semantics and describe some of the important notions connected with it. In the first section we introduce the syntax of the formal languages under consideration. Then, in Section 2.2, we describe the general theory of stability semantics. The third section is devoted to an example that illustrates the central features of this type of formal semantics. In Section 2.4 we describe the rich variety of semantical behavior that is identified and classified by any system of stability semantics.

2.1. Syntax

The formal languages with which we deal in this and subsequent chapters are first-order languages, each of which contains its own truth predicate and quotational names of all its sentences. The vocabulary of any such language, L, consists of a denumerable collection of variables and two countable collections of logical and 43

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nonlogical constants.1 The logical constants are the following: • A unary connective, '—i', representing negation, and four binary connectives, 'A', 'V', '—>•', and '*-»•', representing conjunction, disjunction, material conditional, and material biconditional, respectively; • Universal quantifier, 'V, and existential quantifier, '3'; • Identity predicate, '=', 2 and truth predicate, 'T'. The nonlogical constants of L consist of the following:3 • A countable (possibly empty) collection of predicates; • A countable (possibly empty) collection of nonquotational (ordinary) names; • A denumerable collection of quotational names whose members are all the names of the form , where ' ' is replaced by any sentence of C. The formation rules of C are the familiar rules of first-order languages with identity predicate. However, we need to explain how the quotational names may be formally defined. One way of constructing such names is as follows. Consider a first-order language, Co, whose vocabulary, Voc(Lo), is exactly like that of C except that it contains no quotational names. Let 'So' denote the set of all the sentences of Lo. Now extend Co by adding to its vocabulary the quotational names of all the sentences in So- Call the extended language 'L1'. Thus we have

Voc(d) = Voc(L 0 ) U {' ' : € So} Let 'S1' denote the set of all the sentences of L1. Observe that sentences such as '3xTx' and 'Vx (x = x;)' are members of So (they are Lo-sentences), but sentences such as 'Tr xTx~" and '-\T x(x = x)~" are not in So', they are, however, members of Si. It is clear that SQ is a proper subset of S1 (in symbols, 1

'Denumerable' is used here to mean countably infinite, We use the symbol '=' as our identity predicate on the object-level and on the meta-level. 3 For the sake of simplicity, we assume that C has no function symbols. This assumption does not impose any serious limitations on the formal systems discussed in this book. As is well known, functions can be represented by predicates that satisfy certain existence and uniqueness conditions. 2

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So C S 1 ). Similarly, we extend £1 to £2 by adding to its vocabulary the quotational names of all the sentences in <Si. The set £2, which consists of all the sentences of L2, contains in addition to the members of S1 many new sentences, such as '->TrT"~3:eIVn' and 'Tr->TrVx(x = xyn'. In general, for any natural number n, the language £ n +i is defined to be the first-order language whose vocabulary is that of £„ in addition to the quotational names of all the sentences of £n. Formally,

where Sn is the set of all Ln-sentences. Continuing this construction, we obtain a denurnerable chain of sets of sentences: So C S1 C • • • C Sn C • • • - The language C can now be defined as the first-order language that has the following vocabulary:

where N is the set of natural numbers. Note that S, the set of all L-sentences, is the maximal element of the chain above, i.e.,

There are other ways of defining the syntax of a language such as £. For example, we could have introduced the quotational names not as part of the basic vocabulary of £ but as terms formed by certain formation rules. Such rules employ recursive operations that involve the formation rules of the sentences. I do not intend to discuss here alternative syntactical constructions. I believe that the syntax described above is more transparent than those alternative constructions. Before ending this section, I would like to introduce a few syntactical notions, which are needed in later discussions. An occurrence of a constant or variable in a sentence or formula is called genuine if it is not an occurrence inside a quotational name that is part of the sentence or formula. A constant or variable that has at least one genuine occurrence in a sentence or formula is said to occur genuinely in that sentence or formula. Now we define the

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T-sentences of L as the L-sentences that contain one or more genuine occurrences of the truth predicate.4 Sentences that contain no genuine occurrences of 'T' are said to be T-free. A useful notion is that of a T-form. If <; is any L-name (quotational or nonquotational), its T-forms are the members of the smallest collection, C, that contains T and is closed under the following condition: if belongs to C, then so do T" and -«f>.5 In other words, the T-forms of are the L-sentences obtained by applying the truth predicate 'T' and the negation sign any (finite) number of times and in any order to the sentence T . The pure T-forms of are those T-forms of whose constructions do not involve the negation sign. A pure T-form of is usually abbreviated as Tn where n is the number of times 'T' occurs to the left of the name . For instance, T3r3a; (x = x)"" abbreviates the sentence 'TrTrTr x (x = a;) . As a convention, if is a quotational name of some sentence , its T-forms are referred to as the T-forms of the sentence (rather than of the name The last two notions we define here are needed in Section 3.2.3. If t is a nonquotational name, the number of occurrences of the negation sign in any T-form of is called the negation-complexity of that T-form. Hence, the pure T-forms of any nonquotational name are exactly those T-forms of whose negation-complexity is zero. We finally define a certain binary operation on the T-forms of nonquotational names. Let and be any two T-forms of the nonquotational names and , respectively. The composition of and is the T-form of constructed by replacing in with . For instance, the composition of T and is the sentence T Tr T We use ' ' to denote the composition 2 of 1 and 2- Observe that this operation is associative:if1,2 *Davidson (and others) uses the expression 'T-sentences' to refer to what I call 'Tarskian biconditionals'. See, for example, Davidson (1984, pp. 65—66 and 150). 5 Note that ' ' and ' ' are metalinguistic variables ranging over L-names and L-sentences, respectively. Expressions such as 'T, 'Tr ' , and ' ' are used to denote the sentences obtained by applying the truth predicate 'T' to the name , the truth predicate 'T' to the quotational name of the sentence , and the negation sign ' ' to the sentence , in that order. Hence, we write "The sentence Tr ' instead of "The sentence 'Tr '" In similar fashion, the expression ' ' denotes the quotational name of the sentence ; and thus we write "The name '' instead of "The name ' '."

Stability Semantics

47

and 3 are any T-forms of nonquotational names, then and ( ) are identical. Thus, we simply write
Semantics

In this section we define a type of formal semantics that we call stability semantics. Semantical systems of this type are natural candidates for formalizing the revision process discussed in Chapter 1. Two such systems were advanced by Anil Gupta and Hans Herzberger in 1982.6 Nuel Belnap (1982), while commenting on Gupta's system, introduced a third system of stability semantics.7 In the discussion below we give a general definition of stability semantics of which the systems of Gupta, Herzberger, and Belnap are special cases. In very general terms, a system of stability semantics is a collection of ordinal sequences, which we call stability sequences. Each term of a stability sequence is a classical extension of some partial model of a formal language, L,, whose syntax is of the kind described in the previous section. Let us specify these notions more precisely. We start by denning the relevant kind of partial model ofL. 2.A. DEFINITION. 2( is a base model of £ if and only if 2( = (D, I), where D is the domain of discourse and I is a partial interpretation of L, and 2( meets the following conditions: • S C D, where S is the set of all L-sentences; • I assigns to every name a unique object I( ) in the domain D; • For every quotational name , I( )= • For every n-place predicate that is not 'T', I assigns to Q its extension I( ), which is a collection of ordered n-tuples of objects in the domain D; • I assigns to the identity predicate, '=', its usual extension, which is the set {(a, a) : a e D}; • Every T-free sentence receives in 2( a classical truth value that is determined by Tarski's satisfaction conditions. 6

See Gupta (1982) and Herzberger (1982a and 1982b). Chapter 3 is devoted to a critical examination of those three systems.

7

48

The Liar Speaks the Truth

A model 2( that meets the conditions above is a partial model of £ because it leaves 'T" uninterpreted. 2( is meant to represent the nonsemantical facts of the world, which determine the truth values of the grounded sentences; that is why we refer to 2( as the base model. In order to determine the truth values of all the sentences of £, we need to extend 2( to a full model of £ by assigning some extension to the truth predicate 'T". This leads us to the central notion of stability semantics: the notion of stability sequence (henceforth, s-sequence). For reasons that will become clear, these sequences are natural candidates for formalizing the revision hierarchies described in the previous chapter. An s-sequence on 2( (the base model) is an ordinal sequence of classical extensions of 2( that interpret £ fully. Every s-sequence is defined by three things: an initial stage (an initial extension of the truth predicate), a successor rule (a rule that determines the n + 1st stage, given the nth stage), and a limit rule (a rule that determines the extension of the truth predicate at the limit ordinals, w, w2, . . . , w2, . . . , wiw , and so on). As we shall see, all the s-sequences share the same successor rule. Hence, two such sequences are distinct from each other only if their initial stages or limit rules are different. Here is the formal definition. 2.B. DEFINITION. An s-sequence (denoted by ' % i [ E o , R ] ' ) on 2( with an initial extension E0 of 'T' and a limit rule R is an ordinal sequence of classical models, (2(, Ea), of £. That is, where Ord is the class of all ordinals, and (2(, Ea) is a classical model of £ that is exactly like 2( but it extends 2( by assigning the set Ea, which is a subset of S, to the truth predicate 'T' as its extension. The limit rule 'R, and the sequence {Ea : a 6 Ord} are defined as follows. • R. is a "function" from the class of limit ordinals, Ord1, to the power set, P(S), of S (i.e., the set of all the subsets of S). In symbols,8

R:Ordl

P(S)

8 We shall use, as a convention, 'Ry' instead of 'R (y)', where 7 is any limit ordinal. Observe that Ry is a subset of 5.

Stability Semantics

E0C S.9

For every successor ordinal

49

+ 1,

In other words, the extension of T' at stage of those £-sentences that are true in (2(, For every limit ordinal 7,

+ 1 consists

The members of Uj are the sentences that entered the extension of T' at some previous stages and remained in the extension for all the subsequent stages prior to 7. Formally, Uj is the set

The members of Uj~ are the sentences that were outside the extension of 'T' at some previous stages and remained outside the extension throughout the subsequent stages prior to 7.10 That is, U~ is the following set:

A system of stability semantics is a collection of s-sequences on some base model 2(. Since every s-sequence on 2( is completely determined by its initial extension of T' and the limit rule employed in it, any collection of s-sequences can be fully characterized by an assignment of limit rules to subsets of S. We require that there should be at least one s-sequence for every initial extension of T' 9

Eo might be the empty set, S itself, or any other subset of S. Equivalently, a sentence belongs to T (or U) if and only if there is an ordinal S less than 7 such that is true (or false) in (2(,E a } for every a between and 7. 10

50

The Liar Speaks the Truth

(i.e., for every subset of S). More precisely, R is a limit rule assignment if and only if R assigns at least one limit rule to each subset of S. R may be considered as a collection of ordered pairs such that for each EQ C S, there is a limit rule R, and ( E 0 , R ) £ R. The s-sequence 2( [ E 0 ] on 2( is said to be generated by ( E 0 , R ) . The R-system on 2( is the system of stability semantics generated by the limit rule assignment R, i.e., it is the collection whose members are the s-sequences on 2( that are generated by all the ordered pairs in R. The size of the R-system cannot be smaller than the size of 'P(S), nor can it be larger than the size of R.11 Although the discussion above may appear, with its symbolic and formal terminology, to be rather involved, the ideas expressed there are indeed simple. In the next section, we make these ideas more clear and concrete by giving a detailed example. We spend the remainder of this section explaining the role that Ry plays at the limit stages. The construction as described in Definition 2.B works smoothly for the successor stages. We are given at stage a model (2(, Ep) of £ complete with an extension Ep of the truth predicate 'T'. Since (2(, Ep) is a classical model of L, every sentence of £ is either true or false (but not both) in it. Now applying the Tarskian schema as a defining rule of truth, a sentence is placed in the extension EB+1 of the truth predicate at stage (3 + 1 if and only if tp is true in (2(, EB). Thus, Tr is true in (2(, Ep+1) if and only if if is true in (2(, Ep). This shows how the Tarskian schema is being employed to determine the extension of 'T' at stage b + 1, given the nonsemantical facts (i.e., the base model 2( ) and the extension of T' at the previous stage. As mentioned above, we are actually using this schema as a defining rule of truth; it is our successor rule, which all the s-sequences share. Let us introduce some terminology. If an L-sentence, , is true (or false) in the model (2(, Ea), we say that
Stability Semantics

51

in (2(, Ea).13 Given this terminology, the successor rule can be expressed as follows: for every successor ordinal +!, is declared true in (2(, Ep+1 if and only if is evaluated true in (2(, Ep). At the limit stages, things may get problematic. Suppose that we are at the first limit stage . We cannot use the successor rule to determine Eu, for no stage prior to the w-stage is the previous one. We have, however, a partial solution. We can look at all the previous stages and search for any sentence that has been evaluated true by all the members of some segment prior to w. Such a segment is a sequence whose initial term is (2(, Ep), where is some ordinal less than u, and whose subsequent terms are (2(, Ea) for all a between and . In other words, we look at the cumulative effects of applying the Tarskian schema up to the w-stage. If has "stabilized" as true prior to w, we place it in the extension of (T', Ew. The collection of all such sentences is . On the other hand, if has "stabilized" as false prior to w, we place it outside Eu. The collection of all those sentences is U . The Tarskian schema is still used to determine partially the extension of 'T' at u: " if and only if there is a < such that is evaluated true in (2(, Ea), for every a between and ; • V U if nd only if there is a < such that is evaluated false in (2(, Ea), for every a between and .14 The nonsentences (i.e., the members of D — S) are also unproblematic. They are always outside the extension of 'T". As mentioned in note 3 of Chapter 1, it is important to observe that saying that an object, which is not a sentence, is not true does not entail that it is false. If our language were permitted to have a falsity predicate, as part of its basic vocabulary, its extension would coincide with the complement of the extension of the truth predicate only for the sentences.15 That is to say that for every a;, a; is false if and only if x is a sentence and x is not true. The nonsentences are neither true nor false. 13

Equivalently, is declared true (or declared false) in (2(, Ea) if and only if T" is evaluated true (or evaluated false) in (2(, Ea)-

"Of course, 15

and

- C (S - Ea).

For any two sets X and Y, where X C Y, the complement of X in Y is the set Y — X, which consists of all the members of Y that are not in X. In contexts where the set Y is fixed, the complement of X in Y is simply called the complement of X, and it is usually denoted by 'X '.

52

The Liar Speaks the Truth

The problem arises when we try to deal with sentences that are outside " and ~. These are sentences that continued to change truth values: for every < w, there are two different ordinals a and a' both between and w, such that is true in (2(, Ea) and (f is false in (2(, Eai]. Such a is our problem at w. Here is where we appeal to the limit rule 72. 72 gives us at every limit stage, such as w, a "bootstrapper" (Belnap's terminology). RW is our bootstrapper at w; it is a set of sentences that allows us to make decisions about those sentences which are "unstable" up to w. If we find such a sentence in Rw, we put it in Eu, and if it is not in Rw, we exclude it from Ew. Thus, the extension of 'T' at u is determined by the Tarskian schema and the bootstrapper RW, which is assigned to that stage by the limit rule 72 employed in the s-sequence 2l[E0, R- Now that we have a classical model (2(, Ew) of L, we can continue the process of constructing successive classical models of C by using the Tarskian schema as our successor rule. At the next limit stage, u2, we repeat the same operations used at w in determining Ew, but this time we appeal to the bootstrapper Rw2, which is assigned to this stage by the same limit rule R., in order to determine the extension Ew2- The process continues for every ordinal. One can see why the choice of certain limit rules is what determines the philosophical significance of any system, in contrast to other systems, of stability semantics. As mentioned previously, every semantical system of this kind is totally defined by the limit rule assignment it employs, for all those systems share the same successor rule (the Tarskian schema) in constructing their s-sequences, and all of them allow the initial extension of the truth predicate to be any arbitrary collection of L-sentences (including the empty set). Hence, different systems of stability semantics differ from each other only by the limit rule assignments that generate the s-sequences of these systems. It is not surprising, therefore, that the philosophical correctness of any system of stability semantics is essentially dependent on its choice of limit rules. The assignment of bootstrappers to the limit stages must be motivated by defensible philosophical considerations, or else one should expect artifacts and erroneous consequences. As will be seen in Chapter 3, the original three systems of stability semantics (i.e., Gupta's, Herzberger's, and Belnap's) are troubled by ill chosen limit rules.

Stability Semantics

53

2.3. An Example We now consider an example of a simple language whose semantics is given by a certain system of stability semantics. The purpose of this example is not to study the system employed but to clarify the ideas introduced in Definition 2.B. Let £* be a formal language of the kind described in Section 2.1. The nonquotational names of £* are the decimal numerals, '0', '!', '2', etc., and the names 'a' and 'b'. The predicates of £* are only the logical ones (i.e., '=' and 'T'). The base model of L* is 2(* (= {D*,I*)). The domain D* isNU5*, where N is the set of natural numbers (i.e., {0,1,2,...}) and S* is the set of all £*-sentences. In addition to what is mentioned in Definition 2.A, I* meets the following conditions: I* assigns to every numeral its standard referent (e.g., I* assigns the numbers 0, 1, and 2 to the numerals '0', '1', and '2', respectively); • I* assigns the sentence '-iTo' to the name 'a' and 'Tb' to the name 'b'. Note that '->Ta' is the Liar and 'Tb' is the Truthteller; for the L*-sentences 'a = r-iTa~" and 'b = Tb~" are true in 2(*, and hence '-iTa' says of itself (in 2() that it is not true and 'Tb' that it is true. As a metalinguistic convention, we use ' ' to denote the Liar '-Ta and V to denote the Truthteller 'Tb'.16 Now we need a system of stability semantics to work with. Since our goal in this section is only to illustrate, we choose one of the simplest systems of stability semantics. It is the system of Herzberger (to which we shall return in the next chapter). We refer to this system as the H-system and to its stability sequences as the h-sequences. The limit rule assignment H, which generates the H-system, associates the same limit rule with every subset of S. We denote this unique limit rule by "H As a collection, H consists of all the ordered pairs (E 0 ,H]), where E0 is any subset of S* and H. is Herzberger's (unique) limit rule. H. is a constant limit 16

Similar to the metalinguistic usage explained in note 5, expressions such as "" "", "V, 'T r V, 'TV, and T' refer, respectively, to the quotational names ""-.Ta""and ""Tfe"" and the sentences Tr-iTa"", TrTfe~", and '-iTa AT6'. Thus, we write "The Liar A" instead of "The Liar ' '," "The sentence A A T" instead of "The sentence ' T'," and "The name 'T" instead of "The name ' '."

The Liar Speaks the Truth

54

rule: it assigns the same bootstrapper, which is the empty set, to all limit ordinals. Formally,17 H: Ord1- —> P(S*} such that H-, = 0, for every 7 € Ord l This means that at every limit stage 7 we throw all the -unstable sentences (i.e., the sentences which have not stabilized, as true or false, prior to 7) outside the extension of 'T".18 Every h-sequence is generated by its initial extension E0 and the same limit rule H. Hence, any such sequence on 2(* is completely determined by its initial term (2(*,E 0 ). As usual, we represent any h-sequence on 2(*, with the initial extension E0, as 2(*[E0,7i]. The size of the H-system is \P(S*)\ (which is 2 W ). However, there is a natural equivalence relation, which can be defined on the ssequences of any system of stability semantics, that partitions such a system into a smaller (and in many cases finite) number of equivalence classes of sequences.19 Let us examine closely one example of an h-sequence: 2(*[0, "H]. The first term of 21* [0, H] is (21*, 0), which is a classical model of L* that extends 2(* by assigning the empty extension to 'T'. While many sentences are evaluated true in (2(*,0} (e.g., 'Va;(a; = x)\ x(0 = 1)', '-iT0'), no sentence is declared true in (2(*, ) (i.e., for every sentence , T is evaluated false in (2(*, ) because the extension of 'T" is empty). The second term of 2(*[0,ft] is (%*,E1). The extension EI consists of all the sentences evaluated true in (2(*, ): EI = { e S* : tp is evaluated true in (2(*, E0}} where E0 is the empty set. For example, '0 = 0', '->(! = 0)', ' x; x — x)', T2', and every sentence of the form '-T >~" are members of E1. Since A and T are outside E0, and since 'a' stands for the Liar A and 'b' for the Truthteller r, '-Ta' (which is A) is evaluated true in (2(*,E0) and Tb' (which is T) is evaluated false in (2( ,E0). It follows that A e E1 and T E1. 17

We use "Hy to abbreviate 'H ( )'. "By Definition 2.B, Ey = U U ( -, 19 See Definition 2.D.

). Since

= 0, E^ =

Stability Semantics

55

The third term of 2( [0, H] is (2(*, E2), where E2 is the set consisting of the L*-sentences that are evaluated true in (2(*, E I ) . At this stage more sentences are correctly declared true. For instance, 'T r0 = 0"" does not belong to E0 or to EI, but it does belong to E2. Since A E1 and , T A" is evaluated true and TV evaluated false in (2(*, E 1 I ) . Given that 'a' refers to A and '6' to T, it follows that '- Ta' (which is A) and 'Tb' (which is T) are both evaluated false in (2(*, E1). Said differently, A is declared true but evaluated false in (2(*, E1), while T is declared false and evaluated false in (2(*, E1). However, since A and T are evaluated false in (2(*, E1), they are placed outside E2. The process continues along the finite ordinals with more and more sentences correctly declared true. For example, T3 r0 = 0"1 first enters the extension of 'T' at the 4-stage (i.e., the fifth term, (2(*, E 4 ), because the first term, ( ), is the 0-stage). T continues to be outside the extension of 'T", and A is tossed back and forth between the extension and its complement. At the first limit stage, w, "most" of the sentences have already stabilized as either true or false. Indeed, for this simple language L* and its base model 2(*, if it were not for the Liar, the construction would have reached a fixed point at ui, and as explained in the next paragraph, this entails that the h-sequence 2(*[E0 H] (for any E0 C S*) would have become constant, starting at w, i.e., for every ordinals a and >, (2(* ,Ea) = (2(*, E^}. For example, if 'a' were assigned the sentence '0 = 0', instead of '-iTa', as a referent, no sentence of L* would be paradoxical in any s-sequence regardless of the limit rule R employed and the initial extension E0. Every sentence in this case would stabilize at some finite stage.20 The term 'fixed point' means, in general, a point which is assigned to itself by some operator. Formally, if T is an operator 20 This is true only of simple languages such as L*. Certain sentences of more complex languages, even if they are paradox-free, require much greater ordinals before they achieve stability. For example, suppose that a paradoxfree language contains a predicate, 'P, whose extension in some base model, , consists of the sentence 'x(x = x)' and all its pure T-forms. The sentence 'x? (Par —+ Tar)' first enters the extension of the truth predicate at stage w + 1 of the h-sequence [ ,W]. If this language also contains a predicate, 'Q', whose extension in consists of the sentence ' x(Px —+ Tx)' and all its pure T-forms, the sentence lVx(Qx —> Tx)' first enters the extension of 'T" at stage w2 + 1 of the h-sequence 2[ , H].

The Liar Speaks the Truth

56

(function) on some set X, p is a fixed point of T just in case p X and T(p) = p. The successor rule of every s-sequence may be considered as an operator on P(S). It takes an extension of 'T' (at stage a) as an input and produces a second (or possibly the same) extension of 'T' (at stage a+ 1) as an output. Hence, the successor rule, which is the Tarskian schema, obtains a fixed point wherever 21 an s-sequence reaches a constant Conversely, an s-sequence reaches a constant tail wherever its successor rule obtains a fixed point. The first claim is obvious. The second is not difficult to establish. Here is an informal demonstration of this fact.22 Assume that the successor rule has obtained a fixed point at some stage a in some s-sequence 2l[E 0 ,R.] of any system of stability semantics. This means that applying the successor rule to (2(, Ea) produces the same model again, i.e., (2(, E'a+1) = (2(, Ea). Let 7 be the first limit ordinal that is greater than a. Since the successor rule determines the extension of 'T' at any successor ordinal + 1 solely on the basis of (2( Ep), (2(E ) = (2(, Ea) for every ordinal 8 that is between a and y. Given that T consists of those sentences that entered the extension of 'T' at stages prior to 7 and remained in the extension for all the subsequent stages before 7, and given that there is a constant segment prior to 7 (i.e., the segment between a and 7), it follows that U = E for any S between a and 7. Hence, 7 = U . E is determined without any appeal to any bootstrapper at 7, because the existence of a constant segment prior to 7 entails that every -sentence has already stabilized (either in the extension or outside it) prior to 7. The same kind of argument is applicable to all successor and limit ordinals greater than 7. This shows that if (2(, Ea) is a fixed point, then the tail of 2( [E"0,72.] starting with ( 2 ( E a ) is constant. Fixed points of this kind are called classical fixed points, because the extension of the truth predicate in any such fixed point is classical: the extension of '- T' (i.e., the "antiextension" of 'T" ) is nothing but the complement of the extension of T'. In other words, these fixed points are two-valued interpretations of 21

A tail of a sequence is the rest of its terms after some term. The argument can be made very rigorous by using transfinite induction on to establish that for any ordinal , Ea+t = Ea (see Section 4.1, Lemma 4.C). However, this informal demonstration is sufficient for conveying the key ideas. 22

Stability Semantics

57

the truth predicate. We use the adjective 'classical' to distinguish these fixed points from the ones constructed by theories which employ multiple-valued logics (e.g., Martin and Woodruff, 1975, and Kripke, 1975). One important fact about classical fixed points is that in any such fixed point all the material biconditionals of the form 'T -> come out true. If (2(, Ea} is a classical fixed point, then for every sentence , the material biconditional Tr l < •

(these are the w-unstable sentences). According to the limit rule H, every w-unstable sentence is placed outside Ew. In every h-sequence, Eu consists of all those sentences which have stabilized as true before the w-stage. The process continues with the Tarskian schema as our successor rule and the empty set as our only bootstrapper at all limit stages. No fixed point is obtained in the h-sequence 2 ( * [ , H ] . 23 This fact is rarely the case in theories of truth that employ multiplevalued logics. For further discussion, see Gupta (1982, p. 211). 24 The s-sequence *[<S* ,R] would reach the first fixed point and *[ , R] would reach the second. 25 Compare these observations with the discussion in Section 1.3 about the revision hierarchies.

The Liar Speaks the Truth

58

To study the semantical behavior of any sentence in any stability sequence, we use certain diagrams which we call stability tables. A stability table for a sentence shows how is evaluated and declared in the terms of some s-sequence.26 To illustrate this method, let us consider the semantical behavior of T3 r0 = 0"" in 2(*[ , H]. Table 2.A (at the end of the chapter) shows the semantical behavior of T 3r 0 = 0"1' in the h-sequence 2l*[0,H]. The central two columns display the membership status of 'T3r0 = 0n' in the extension Ea of the truth predicate and in its complement at any stage a (that is, they display its truth declarations). The rightmost column shows its truth values: '1' for true and '0' for false. The leftmost column contains the ordinal levels of successive and limit stages. Table 2.B (also at the end of this chapter) is the stability table for the sentences A, , A A T, and A V . Observe that in those tables a sentence belongs to Ea+i if and only if it receives the truth value 1 at stage a, and it belongs to E if and only if it receives the truth value 0 at stage a. I think there is no need to study this example any further, for the main ideas of Definition 2.B have been sufficiently illustrated. Before ending this section, however, I would like to consider an example that demonstrates Gupta's result mentioned in Section I.3: a language that contains its own truth predicate and names of all its sentences need not be inconsistent, even if it includes among its true sentences the Tarskian (material) biconditionals of all its sentences. Let be a language exactly like L* except that its vocabulary does not contain the names 'a' and 'b', and let 51^ be a base model of L that is exactly like 2(* except, of course, that no object in its domain is named 'a' or '6', for the language does not contain these names. This language, therefore, has no Liar or Truthteller. As mentioned above, the s-sequences of any system of stability semantics on 2( reach, for such a language, a unique fixed point at u. Hence, the classical model (2(, Ew) is a full interpretation of L in which every (material) biconditional of the form 'T ' is true. This shows that L^ is a consistent language. 26

As explained above, every sentence receives at each stage a truth value (whether the sentence is true or false at that stage) and a truth declaration (whether it is inside or outside the extension of 'T' at that stage).

Stability Semantics

59

2.4. The Seven Categories We now return to our general discussion of stability semantics. As in Sections 2.1 and 2.2, £ is a formal language and 21 is its base model. The R-system on 21 is the system of stability semantics generated by the limit rule assignment R. It is the following collection: Each s-sequence 2( [ E o , R ] classifies the sentences of £ into three categories. The first two categories are necessarily nonempty, while the third can be void. They are the categories of the stably true, stably false, and paradoxical sentences in 2([E'o,'K.]. Here is the formal definition. 2.C. DEFINITION. Let 2([.Eo,7R] be any stability sequence. For every £-sentence , • is stably true in 2[E'o,7£] if and only if there is an ordinal 6, such that for every ordinal a > , > is declared true in (*,Ea); • is stably false in 2 [E'o R.] if and only if there is an ordinal , such that for every ordinal a > , is declared false in (*,Ea); • i is paradoxical in 2([E 0 ,R] if and only if it is neither stably true nor stably false in 2([E'o,R], that is, for every ordinal -, there are two ordinals a and a' , such that is declared true in (2(, Ea) and declared false in (21, Ea>). It is clear that we could have stated the definition above using the word 'evaluated' instead of 'declared' — the two statements are equivalent. Note that a sentence is stably true (or stably false) in any s-sequence if and only if its negation is stably false (or stably true) in that sequence. A sentence is said to be stable in 2([E 0 ,R] if it is either stably true or stably false in 2 [ E 0 , R ] . The paradoxical sentences in 2l[_E0 "R,] are exactly those sentences which are unstable in 2(E0,R] It follows that the class of the stably true (or stably false) sentences determines completely the other two classes. Let s t ( 2 ( [ E 0 , R ] ) , s f ( 2 ( [ E 0 , R ] ) , and px(2([E 0 ,'R,]) be the

60

The Liar Speaks the Truth

classes of the stably true, stably false, and paradoxical L-sentences in the s-sequence 2 ( [ E o , R ] . The facts mentioned above may be summarized as follows.

These facts should make clear our next definition. 2.D. DEFINITION. If 2l[E'o,7£] and 2(E0,7£'] are any s-sequences in the R-system, they are said to be semantically equivalent if and only if s t ( 2 ( [ E 0 , R ] ) = s t ( 2 [ E ' 0 , R ' ] ) . Given the facts listed above, two semantically equivalent sequences have identical categories: a sentence is stably true (or stably false, or paradoxical) in one of them if and only if it is stably true (or stably false, or paradoxical) in the other sequence. It is obvious that this relation is an equivalence relation on any system of stability semantics.27 Thus, it partitions such a system into equivalence classes. The s-sequences in each equivalence class are semantically equivalent to each other. These equivalence classes are mutually exclusive and collectively exhaustive of the original system. In our previous example (Section 2.3) about the language L* and its base model 2(*, the h-sequences on 2(* are classified by the relation of semantical equivalence into two classes: those h-sequences that make the Truthteller stably true, and those that make it stably false. The first equivalence class may be represented by the h-sequence 2(*[S* , H] and the second class by the h-sequence 2(*[ ,H]. By quantifying over all the s-sequences of some system of stability semantics, a rich variety of semantical behavior can be identified. The sentences of £ are classified into seven (mutually disjoint) categories. Only the first two are necessarily nonempty. We list these categories in the definition below. 27 Recall that a system of stability semantics is nothing but the class of all the s-sequences generated by some limit rule assignment.

Stability Semantics

61

2.E. DEFINITION. For every L-sentence , the R-system on classifies 9? as belonging to one and only one of the following categories. (We denote the R-system on 2( by (2([R]'.) Cl. Stably True sentences in 2([R]. < is stably true in 2([R] if and only if it is stably true in every s-sequence in the R-system on 21. The subset of S consisting of all the stably true sentences in 2l[R] is denoted by 'st(2l[R])': < *(2l[R]) if and only if

62

The Liar Speaks the Truth

sequence, stably false in the second, and paradoxical in the third. The set of all the tfp-capricious sentences in 2l[R] is denoted by The sentences in the first two categories are said to be invariant in 2l[R] and the set of all such sentences is inv(2l[R]). That is, inv(21[R]) = st(a[R]) U sf(2l[R]) The sentences which are either invariant or paradoxical in 21 [R] are referred to as uniform. 28 They are uniform because each one of them exhibits the same kind of semantical behavior in all the ssequences of the given system. The set of all the uniform sentences in 2l[R] is wn/(2l[R]). The sentences in the last four categories, C4C7, are called capricious, because each such sentence exhibits more than one kind of semantical behavior throughout the R-system on 21 and these different behavior patterns are completely dependent on the arbitrarily chosen initial extension E0. The set of the capricious sentences in 21[R] is cpr(2l[R]). A very important collection is the union of categories C1, C2, and C4. The sentences in this union are the paradox-free sentences of £,. As may be expected, for every sentence in this union, the material biconditional Tr
Stability Semantics

63

our new language L has a Liar '->Ta' and two distinct Truthtellers Tb' and Tc'. We use ' ' to denote the Liar and V and V to denote the two Truthtellers. We work with an arbitrary system of stability semantics; as usual we call it the R-system. The sentences T3 0 = 0', ! xTx', TV T, and -n(T r -I A) are stably true in 21 [R]. The negations of the previous sentences are stably false in 21*[R]. , -. , T , '(0 = 0) A ', and '(1 = 0) V A' are all paradoxical in 21*[R].29 Of the capricious types, and r1 are tf-capricious, r V A is tp-capricious, T AX is fp-capricious, and (T' A) is tfp-capricious in 21 [R]. Concerning the last example, T —>• (r1 A A) is stably true in the s-sequence 21 [0,R.], it is stably false in 21 [{ },7l}, and it is paradoxical in 2t*[{Tr, T'},R,]. Tables 2.C, 2.D, and 2.E, at the end of this chapter, show the semantical behavior of the sentence T (r' A A) in the h-sequences 21 [0,H], 21 [{r},H], and 21[{r,r'},H], respectively.

29 Strictly speaking, the expressions '(0 = 0) A' and '(1 = 0) V ' are not well formed according to the metalinguistic usage described in note 16. But we take some liberty in our notation to avoid unnecessary complications.

TABLE 2.A. Stability Table for the L*-sentences '0 = 0', 'Tr0 = 0~", . h-sequence: 2l*[0,H] a

Eca

Ea

0

0 =0 Tr0 = 0"1

1

T2 r0 = 0n

0

T3 r0 = 0n

0

Tr0 = 0"1

1 1

0 =0 1

2

0

T3 r0 = 0"1

0

1 1

0 =0 Tr0 = <0T T2 0 = 0 1

1

T3 r0 = 0"1

0

1

Tr0 = 0n

T2 r0 = 0n

4

0

T2 r0 = 0"1

0 =0 3

tv;

T3 r

0 = 0"1

1 1 1

1 1 1

0 =0 Tr0 = 0n T2 r0 = 0"1

1

T3 r0 = 0"1

64

TABLE 2.B. Stability Table for the L*-sentences '-.To' (A), 'Tb' (r), '-.To A Tb', and '-.To V T6'. a = r-.Ta'1 and b = rTb\ h-sequence: 21*[0,7i]

a

Ea

7-lC

•Ea

Tb -.TaATfc -iTa V Tb

1 0 0 1

Tb -.To A T6

0 0 0 0

-Ta Tb -.To A Tb -.To V Tb

0 0

-.To Tb -.Ta A Tb -.To V T6

0 0

Tb -VTa A T6

0 0 0 0

-nTa

0

-.To 1

-.To V Tb

2

w

-To w+1 -Ta V Tb

65

tv

1 1

1

1

TABLE 2.C. Stability Table for the L-sentences '- Ta' (A), 'Tb' (T), 'Tc' (r1), and 'Tb (Tc A -.Ta)'. a = r-^Ta b = rTbn, and c = Tc"1. h-sequence: 21 [0,H]

a

Eca

£a

0

-.Ta Tb Tc Tb -> (Tc A -.Ta)

1 0 0 1

Tb Tc

0 0 0

--To

1 Tb -+(TcA -,Ta)

-.Ta Tb Tc

2

Tb -+(TcA -.To)

a;

-Ta Tb

(Tc -.Ta)

66

1 1

0

0

1

1

-Ta Tb Tc

0 0

Tb Tc

0 0 0

Tb -> (Tc A -.Ta)

w +1

tv

1

1

TABLE 2.D. Stability Table for the L*-sentences '-.Ta' (A), 'Tb' (r), Tc' (r'), and 'Tb -•• (Tc A -Ta)', a = r-iTan, 6 = Tb" , and c = rTc h-sequence: 21 [{ },H] a

^or

0

T6

Ea

-,Ta Tc

Tb -v (Tc A -Ta)

1

-.To Tb

1

Tb

Tc

(Tc A -,Ta)

1 1

Tc

0 0

-.Ta

Tb Tb

w+1

0 0

-.Ta Tb -.. (Tc -.Ta)

w

1 1 0 0

0

Tb

2

tv

Tc

(Tc -.Ta)

1 1

0 0

0

-Ta Tb Tc

Tb -+ (Tc A -Ta)

67

1

0 0

TABLE 2.E. Stability Table for the L -sentences '-.Ta' (A), 'TV (T), 'TV ( T), and 'Tb -+ (Tc A -.To)', a = -"- Ta , b = rTb , and c = r Tc. h-sequence: 21*[{T, r'}, H] a

ECa

tv

-.Ta

Tb Tc

0

1

Tb

Tb - (Tc

-.Ta)

-.To Tb Tc (Tc -.Ta)

0 1 1 0

-Ta 2

w

w+1

1 1 1 1

T6 Tc

Tb -+ (Tc A --Ta)

-.Ta

Tfc Tc

Tb -> (Tc A -Ta)

-.Ta Tb Tc

1 1 1 1

1 1 1 1 0

1

1

Tb -> (Tc A -,Ta)

0

68

3

The Original Three Systems

Formalizing the revision theory of truth is not a sheer technical curiosity, rather it is part of our philosophical defense of the theory. One objective of our defense is to show that the theory is consistent; a second objective is to confirm its material adequacy. In Section 2.3 we established the consistency claim by producing a formal representation of the revision process within a mathematical theory, namely the theory of sets, which is evidently, though perhaps not demonstratively, consistent.1 The second objective can be met by developing a formal semantical system that is a "faithful" representation of the revision process, and then showing that it yields the intuitively correct results when it is applied to fairly expressive languages (so that a rich variety of problematic and unproblematic sentences can be formed). The task of producing such a semantical system is carried out in the next chapter. But in order to understand fully how that system succeeds in representing faithfully the revision process, we need to examine how other systems fail. This is our project in this chapter. In the first section we 1 Strictly Speaking, the H-system as described in Section 2.3 is not formalized in standard set theory—it involves quantification over collections of proper classes. In Section 4.1, however, we prove that every system of stability semantics generated by a limit rule assignment, such as H, that admits only constant limit rules is formalizable in set theory. Hence, any such system (on an appropriate base model) shows that the revision theory of truth is consistent.

69

70

The Liar Speaks the Truth

describe the systems of Herzberger, Gupta, and Belnap; we define and explain the limit rule assignments that generate them, and we make a few comparative remarks about their uniform categories. In the next section we show that these three systems are infected with various types of artifacts—they produce erroneous verdicts in a large number of cases. The last section is devoted to a philosophical analysis of the limit rule assignments that generate these three systems. We argue that their limit rule assignments are the source of the trouble: they fail to conform to the conceptual picture of the revision process painted in Section 1.3. Thus we conclude that these systems do not formalize the revision process faithfully and that they are troubled by artifacts exactly because of this failure.2 3.1. Describing the Systems

As explained in the preceding chapter, every system of stability semantics is completely determined by its choice of limit rules. Since a limit rule is an assignment that associates each limit ordinal with a bootstrapper, which is a collection of sentences that is employed to make decisions about the truth declarations of the unstable sentences at that limit stage, every system of stability semantics represents a certain approach to the problem of 7-unstable sentences (where 7 is any limit ordinal). In this section we describe three such approaches; they are those of Herzberger, Gupta, and Belnap.3 We start by defining these systems. 3.A. DEFINITION. Let £ be a formal language whose syntax is of the kind described in Section 2.1, S be the set of all L-sentences, and 21 be a base model of L. • Herzberger's System (the H-system). As noted in Section 2.3, the H-system on 21 is generated by the limit rule assignment H that assigns the limit rule H to every subset of S. H is a constant limit rule that associates the empty 2 The last claim receives its best confirmation in Chapter 4. We show there that the revision system, which is developed to be a faithful representation of the revision process, is free of those and, perhaps, all other artifacts. 3 See Herzberger (1982a and 1982b), Gupta (1982), and Belnap (1982).

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71

bootstrapper with all limit ordinals. Formally, U : Ord1 — > T(S) such that %-f = 0, for every 7 in Ord1; ft) :E0CS} and where '21 [H]' denotes the H-system on the base model 21 and '21 [E 0 ,H)' stands for the s-sequence that is generated by ( E 0 , H ) . We refer to the s-sequences in the H-system as the h-sequences. Gupta's System (the G-system). This system is generated by the limit rule assignment G that assigns to each subset E0 of S a unique and distinct limit rule GE°.4 Hence, Gupta's system employs as many limit rules as there are subsets of S. For every subset EQ, the limit rule GE° , which is associated with E0, is a constant rule: it assigns EQ as a bootstrapper to every limit ordinal 7. The s-sequences in the G-system are called the g-sequences. As usual, we use '21[G]' to denote the G-system on 21 and '21[E 0 ,g E °]' to denote the g-sequence generated by (Eo,GE°). Thus, GE° : Ordi — P(S) such that GE° = E0, for every 7 in Ord1; G = {(E0,GE°) :E0CS} and 21[G] = {21[E0,aEo] : (E0,GE°) € G} Belnap's System (the B-system). This is the "largest" of all systems of stability semantics. It employs all possible limit rules. Belnap's limit rule assignment B is a one-tomany relation; it assigns to every subset of S all limit rules. We use 'B' to denote an arbitrary Belnap rule. Since every limit rule is a Belnap rule, 'B' is simply a variable ranging over the class of all limit rules. Formally,

B = {(E0,B) :E0CS, B: Ord1 -> P(S)} and

4

The superscript 'En' in 'QB°' does not represent exponentiation; it merely indicates that the limit rule QE° is determined by the initial extension EQ.

The Liar Speaks the Truth

72

where 21[B] is the B-system on 21, and 21L[Eo,B] is the ssequence generated by (Eo,B). The s-sequences in the B-system are called the b-sequences.5 Before discussing these systems comparatively and critically, it is important to form some intuitions about their limit rules. As noted in Section 2.3, Herzberger solves the problem of the 7unstable sentences (i.e., the sentences that do not stabilize, as true or false, prior to the limit ordinal 7) by declaring them all false in (21, E ). Gupta, on the other hand, digs in at every limit stage by sticking to Ms initial choice: for every 7-unstable sentence ( , p is declared true (or false) at limit stage 7 if and only if it is so declared initially (i.e., in the initial term (21, Eo})- Belnap's solution is to admit the greatest degree of arbitrariness: at every limit stage 7, the 7-unstable sentences are declared true or false in a totally arbitrary manner. Any collection of L-sentences is an admissible bootstrapper at any limit stage in any b-sequence. Since the B-system consists of all the s-sequences, it immediately follows that every system of stability semantics is a subsystem of it. In other words, if R is any limit rule assignment and 21 is any base model of L, then 21[R] C 21[B], where 21[R] is the R-system on 21. Hence, a sentence that is stably true (or stably false, or paradoxical) in the B-system on 21 is also stably true (or stably false, or paradoxical) in the R-system on 21. Using the notation introduced in Section 2.4,

Any two systems of stability semantics whose uniform categories are related to each other in the manner described by the inclusions above are said to be uniform-category nested (henceforth, uc-nested). Thus, we may restate our first observation as follows: the B-system on any base model is uc-nested in every system of stability semantics on the same base model. In particular, 21[B] is uc-nested in 21 [H] and in 21 [G]. 5

Observe that every s-sequence is a b-sequence.

The Original Three Systems

73

Herzberger's system is not, in general, a subsystem of Gupta's. It can be shown, however, that 21[G] is uc-nested in 21[H].6 The argument is based on a fact mentioned by Herzberger (1982a, pp. 150-53, and 1982b, pp. 194-95). He observed that for any h-sequence, there are infinitely many limit ordinals at which the extension of the truth predicate consists of those sentences that are stably true in that h-sequence. Adopting Herzberger's terminology, we call such ordinals alignment points.7 6

McGee (1985a, pp. 117-20) proved this result (see also McGee, 1991, pp. 130-33). Although his proof deals with countable languages only, it can easily be generalized for languages of any infinite cardinalities. In Section 4.1 we prove a more general result (see Theorem 4.G). 7 He also called them closure points (see Herzberger, 1982a and 1982b). We give an argument in Section 4.1 that establishes the existence of alignment points for every h-sequence. McGee (1991, p. 134) showed that if the language is countable, then the first uncountable ordinal, w\, is an alignment point of every h-sequence for that language (see also McGee, 1985a, pp. 119-20). He further showed that there is a countable language, with sufficiently rich arithmetical resources, such that no ordinal smaller than w1 can be an alignment point of every h-sequence on a certain base model of that language (I982a, prop. 6.4, and 1991, prop. 6.6). We shall prove here the following weaker claim: for every ordinal < w1>, there is a countable language, such that no ordinal less than or equal to 8 can be an alignment point of every h-sequence on a certain base model of that language. The key idea of our proof is fairly simple and it is a modification of the one discussed in note 20 of the preceding chapter. For every infinite ordinal that is smaller than w1, let L be a countable language whose nonlogical predicates are P0 , P1 , ..., PS . (Note that every ordinal smaller than w1 is countable.) Now consider the following collection of sentences: ^ is 'Vx(or = a;)', <>o is ' x(Po:x —* Tx)', < >i is ;(Pi:x —> Tx;)', and in general, 0a is 'Vx(Pax -+ Ta;)' for every a < S. Suppose that the interpretation function / of some base model 21 makes the following assignments: /(P0) = { }, I(Pi) = /(Po)U{o}, and in general, for every successor ordinal B + 1 < S, I(Pb+1) = I(Pp) U {4>p}, and for every limit ordinal -y < , I(P-l) = M /(Po,). For instance, under this interpretation, the extension ofPa is the set {i/1, oi<£ii02}i and the extension of Pw, is {-!/>,$o, 01, • • • } • All of these sentences are stably true in any s-sequence on 21. (The truth conditions of each of these sentences can be reduced to the truth conditions of .) Hence, they are stably true in the h-sequence 21[{^}, H]. With some reflection, we discover that each >a first enters the extension of the truth predicate, in this h-sequence, at stage a + 1. In particular, the sentence 4>i first enters the extension of 1T" at stage 5 + 1- Therefore, no ordinal less than or equal to S can be an alignment point of 2l[{i/>},H]. See Table 3.A at the end of this chapter for the semantical behavior of . in 21{V^i "H.

74

The Liar Speaks the Truth

Suppose that 6 is an alignment point of the h-sequence 21[E H]; hence E = s t ( 2 1 [ E o , H ] ) . Now consider the g-sequence 21[E' 0 ,G E °], where E'0 — Ef. For the sake of simplicity, let us call these sequences H and G, respectively. Since the 0-term of G is identical with the -term of H and since the extension of T' at each successor stage is completely determined by the extension at the previous stage, the n-term of G is identical with the ( + n)-term of H, for every finite ordinal n. At stage + u, the extension of 'T" in H consists of those sentences that have stabilized as true prior to 6 + ui.s Given that the members of E$ are all the stably true sentences in H, it follows that Ef, ES+U. The extension of 'T' in G at stage w, E'u, consists of the sentences that have stabilized as true prior to u> plus all the w-unstable sentences that belong to the initial extension .E^.9 But it is clear that the collection of the stably true sentences at limit stage a + w (where a is either 0 or any limit ordinal) in any s-sequence is fully determined by the underlying base model and the extension of 'T" at a. Since G and H are defined on the same base model 21, and since E'0 is identical with E , the stably true sentences at u in G are exactly those sentences contained in ES+W. Hence E'0, which is the bootstrapper in G, is included in E'w. We conclude that the w-term of G and the ( +u>)-term of H are also identical. This argument can be generalized, with a slight modification, to show that the -tail (i.e., the tail that starts at ) of the h-sequence 21[E0,'H] is identical with the g-sequence 21[E'0G E °]. This entails that 21[Eo,7i] and 21[E' 0 ,G B °] are semantically equivalent (see Definition 2.D). Since 21[-Eo, H] is an arbitrary h-sequence on 21, every h-sequence in 21 [H] is semantically equivalent to some g-sequence in 21[G]. We have shown that 21 [H], though in general is not a subsystem of 21 [G], is embedded, in a certain natural way, in 21[G]. Now, we prove that 21[G] is uc-nested in 21[H]. Suppose that is a stably true sentence in 21[G]. If is not stably true in 21[H], then there is an h-sequence on 21 that classifies as stably false or as paradoxical. But as shown above, every h-sequence is semantically equivalent to some g-sequence; hence there is a g-sequence in 21[G] 8 Recall that Ti. assigns the empty bootstrapper to all limit ordinals. See Definition 3.A. 9 The limit rule QEo assigns the bootstrapper E'0 to every limit ordinal. See Definition 3.A.

The Original Three Systems

75

in which is not stably true. The last claim contradicts our original premise, namely that is stably true in 21[G]. We conclude that must also be stably true in 21[H]. Similar arguments can be used to establish similar conclusions about the other two uniform categories. Thus, 21[G] is uc-nested in 21[H]. The following inclusions between the uniform categories of these three systems summarize the two facts established above.

An immediate consequence of the inclusions above is that if any ^-sentence is classified by one of these three systems as belonging to a certain uniform category then none of the other two systems can classify the same sentence as belonging to a different uniform category.10 For example, if

McGee reported and proved this fact first (see McGee, 1985a, pp. 11617, and 1991, p. 130). His argument, however, does not make use of the fact that these systems are uc-nested; rather it is based on the observation that the s-sequence whose first term is (21, 0) and limit rule is H is common to Herzberger's, Gupta's, and Belnap's systems on 21. (Note that Q — H.) 11 Recall that the seven semantical categories, for any system of stability semantics, are mutually exclusive. See Definition 2.E. 12 As explained at the end of Section 2.3, the s-sequences of any system of stability semantics on 21, which is a base model of the language L, all reach the same fixed point at w. (L does not contain any problematic sentences.) Hence, for this kind of language, the corresponding categories of any two systems of stability semantics are identical. (In fact, the categories C3—C7 of any such system are all empty.)

The Liar Speaks the Truth

76

nonquotational names, 'a' and '6'. Let 21 be a base model of L, and assume that 'a' and 'b refer in 21 to the sentences '-iTa' and -Ta', respectively. Thus, this language contains two distinct Liars; we use ' ' to denote '->Ta' and 'A" to denote 'Tb'. Now, the sentence ' is stably true in 21 [H], but it is tf-capricious in 21 [G].13 Since A and A' are 7-unstable sentences in every s-sequence on 21 (as usual, 7 is any limit ordinal), Herzberger's limit rule ti places them outside the extension of T' at every limit stage. A simple Liar, such as A and ', switches truth values throughout the successor stages of any s-sequence in a very systematic way: . . . , 1, 0, 1, 0, and so on. Hence, after the w-stage, A and A' follow an identical pattern (either both in the extension or both outside it) in every h-sequence in 21[H]. This entails that A and A' are always "in phase" in every h-sequence on 21, and hence A A' is stably true in 21 [H]. Table 3.B at the end of this chapter shows the semantical behavior of A A' in the w-tail of an arbitrary h-sequence on 21. In the G-system there are two possible treatments of A and A' at limit stages: they are given either identical truth declarations or opposite ones. The choice of either policy is totally determined by the initial term (21, E0) of each g-sequence. If the initial term of the g-sequence 21[Eo,G E °] gives A and A' the same truth declaration (i.e., both in E0 or both outside it), then they are so declared at every limit stage of that g-sequence. As in every h-sequence, the two Liars are always in phase in 21[Eo,G E °]; and hence A A' is stably true in that sequence.14 But if the first term of 21[Eo, G B °] gives A and A' different truth declarations, then they will be "out of phase" throughout that sequence, for the limit rule QE° assigns to them opposite truth declarations at every limit ordinal. Hence, A A' is stably false in 21[E"0, G E °}. Table 3.C is the stability table of A A' in the g-sequence 21[{A},
This example is due to Belnap (1982, pp. 110-12). Table 3.B also serves as the stability table, starting at u, of A + A' in any g-sequence on 21 with an initial extension of 'T' that omits both A and A'. 15 The negation of a stably true sentence in any system of stability semantics 14

The Original Three Systems

77

shows that sf(2l[G]) sf(21[H]). The sentences ' and ' are paradoxical in 21 [H], but they are, respectively, tp-capricious and fp-capricious in 21 [G]. Table 3.D shows the semantical behavior of these two sentences in the w-tail of an arbitrary h-sequence on 21, and Table 3.E is their stability table in the g-sequence 21[{A}, (y^]. We conclude that px(21[G]) ^ pa:(21[H]).16 As shown above, the semantical behavior of A > A' in the Gsystern is similar to that of the Truthteller. A <->• A' is paradoxfree (and hence, Tarskian) in 21[G]. This entails that the biconditional TrA An • (A A') is stably true in 21[G] (see Section 17 2.4). This biconditional, however, is not stably true in the Bsystem on 21 because A A' is not paradox-free in 21[B]. In fact, T""A <-> "1 (A A') is tp-capricious in 21[B]. The b-sequence that is discussed in the following paragraph and depicted in Table 3.F classifies this biconditional as paradoxical. No s-sequence can render TrA An (A A') stably false. We can see that this is so by observing that A <--> A' remains in the extension of 'T' or outside it throughout the successor stages of any w-segment of any s-sequence. The fact that T"~A<--->•An <-+ (A<--->A') is stably true in 2l[G] and tp-capricious in 21[B] shows that st(21[B]) st(21[G]). The negation of this biconditional is an example of a sentence that is stably false in the G-system but not in the B-system; hence In order to show that px(21[B]) is a proper subset of pa;(21[G]), we assume that our language further contains a nonlogical predicate (P' whose extension in 21 consists of the sentence A <---> A' and all its pure T-forms, i.e., 'P' has the following extension in 21: {A <-»• A', TrA <---> A, T2 rA +-* A", . . . , Tn <----> A", . . .}, where n is any positive integer. Let 0 be the £-sentence, Vx (Px

(Tx *-> (A<--->A')))

0 asserts that the truth declarations of A<-->+A' and of all its pure T-forms are identical with the truth value of A «-> A'. Since A<-->A' is stably false in that system (see Section 2.4). The negation of a tf-capricious sentence is also tf-capricious in the same system. 16 The B-system on 2t renders A<-->A' and its negation tfp-capricious, and like the G-system, it renders A V A' tp-capricious and ' fp-capricious. 17 Belnap (1982, p. Ill) mentioned this example and credited it to Gupta.

78

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is Tarskian in the G-system on 21, 0 is stably true in that system. Hence, 6 A A is paradoxical in 21[G].18 In the B-system on 21, 0 A A is not paradoxical, rather it is fp-capricious: there is a b-sequence on 21 in which 9 is stably false, and hence is stably false too in that b-sequence.19 Consider the b-sequence 21[0, Ba] whose limit rule Ba is defined by transfinite recursion on a, where a is any nonzero ordinal, as follows: for every limit ordinal wa, 20 • if a = I, then /?£ = {A}; • if a is any successor ordinal greater than 1, i.e., a = /?+ 1 for some nonzero ordinal (3, then

• if a is any limit ordinal, then B%,a — ®In other words, B" is an "alternating" limit rule. It declares A true at u>, false at w2, true at w3, and so on. At the first limit of limits w2, Ba declares A false, and then it repeats the previous pattern: true at u2 +w, false at o>2 +w2, true at w2 +u>3, and so on. This policy is followed systematically by the rule Ba throughout the sequence of limit ordinals. Table 3.F shows the semantical behavior of the sentence 0 in 21[0,Bj. At the 0-stage, A <__> A' and all its pure T-forms are declared false, but A <->• A' is evaluated true because A and A' have the same truth value in the first term (21,0). It takes w steps to transfer A <-> A' and all its pure T-forms to the extension of the truth predicate. Hence, 0 remains false throughout the first w-segment. At the first limit stage w, A <->• A' and its pure Tforms are all declared true; but A<--->A' is evaluated false there 18 It is clear that the conjunction of a stably true sentence and a paradoxical one is paradoxical in every system of stability semantics. 19 McGee (1985a, p. 117, and 1991, p. 131) gave two examples of sentences that are paradoxical in Gupta's system but not in Belnap's. Those examples, especially the second, are similar to the one discussed here. His, however, are formulated in a language that has much more expressive power than our impoverished language. 20 We use juxtaposition of symbols to represent ordinal multiplication. Note that for every limit ordinal -y, there is a unique nonzero ordinal a, such that 7 = wot. (It is obvious that for every nonzero ordinal a, the product wa is a limit ordinal.)

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because the bootstrapper { } forces A and A' to have different truth declarations (and hence, different truth values). Thus, 0 remains false in (21, Eu). Similarly, it takes u steps to transfer A <-->• A' and all its pure T-forms to the complement of the extension of the truth predicate. 0 continues to be false in the segment between u and the next limit ordinal w2. At w2, when 6 is once again ready to become true, A<---->A' gets the truth value true. However, A<-->A' and all its pure T-forms are declared false at that stage. Hence, 0 is false in (21, Eu^) too. The process continues in the same fashion until the sequence reaches u2, which is the first limit of limits. It is clear that A <->• A' and all its pure T-forms are w2unstable. The empty bootstrapper used at this stage places all of these sentences, together with A and A', outside Euy. The relevant situation at w2 is similar to that at 0; hence the same analysis applies, and 6 continues to be false. Since 0 is stably false and A is paradoxical in 21[0,B a ], their conjunction is stably false in 21[0,#"]. Thus, 0AA cannot be paradoxical in 21[B]. We conclude that p x ( 2 1 B ] ) px(21[G}). 3.2. Their Artifacts The three systems described in the preceding section are infected with various types of artifacts. These systems produce erroneous verdicts in a large number of cases. In this section we discuss a representative sample of those artifacts. For the sake of convenience, we assume throughout this section that the language £ has a denumerable collection of nonquotational names.

3.2.1. BlCONDITIONALS OF LlARS

As before, we take A and A' to be the simple Liars '-iTa' and '-TV in that order ('a' refers in 21 to '-.To' and '6' to '-*Tb'.) We showed in the previous section that the sentence A<---->A' is stably true in 21[H], tf-capricious in 21[G], and tfp-capricious in 21[B].21 These verdicts, though intuitively wrong, are not as striking as the rest 21

See Tables 3.B, 3.C, and 3.F at the end of this chapter.

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of the examples in this section. However, we discuss this example first because of its historical interest. Belnap (1982, pp. 110-11) introduced this example to show that systems of stability semantics that employ constant limit rules only, such as the systems of Herzberger and Gupta, render A<--->A' paradox-free, and hence they render the biconditional T <-> A<-->•(A <-+ A') stably true.22 He considered the verdict that A<-->A' is paradox-free to be "nothing but an oddity" of such systems, and the stronger verdict that it is stably true as "dreadful." Belnap is correct in describing these verdicts as odd and dreadful. There is no objective ground for treating two distinct Liars as having truth values that are always in phase or always out of phase. What Belnap failed to see, however, is that such erroneous verdicts are not completely avoided by simply "localizing" them to some b-sequences. These verdicts are wrong not because they are shared by every s-sequence of a certain system, but because there is no logical relation between A and A', and hence there is no justification for any systematic correspondence between their truth values. It is unreasonable, therefore, for any s-sequence to treat A<--->A' as stable. Since every system of stability semantics is a subsystem of Belnap's, the B-system is extremely vulnerable to all kinds of artifacts. The difference is that while other systems may have an artifact that is common to all their s-sequences, the B-system always succeeds in localizing the problem to a nonempty collection of its sequences. On the other hand, Belnap's system can never be free of artifacts, if there is at least one s-sequence that gives an incorrect verdict about some sentence. The sentence Tr <--> A is stably false in every system of stability semantics on 21. Although A and T are both paradoxical, the situation in this case is different from the one in the previous example. There is a certain logical relation between A and T r A n : the assertions made by these sentences contradict each other. A is '-iTa' and TrX is Tr-.TcT'. Since V refers to '-/Ta', the sentences 'Tr-iTa~" and To' are equivalent in 21[R] for any limit rule assignment R (i.e., 'T <--> To1' is stably true in every s-sequence on 21). Hence, TT1 <--> A is equivalent in 21 to Ta <-->VTa'. The 22 A <-->A' is stably true, stably false, or tf-capricious in every system of stability semantics that employs only constant limit rules.

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last sentence asserts that the Liar is inside the extension of the truth predicate if and only if it is in the antiextension. Given the classical nature of the extension, this sentence is obviously false. Now consider the sentences '-iTa' and 'TrTa~". Since the former is not the negation of the latter, one might expect that their material biconditional would not be stably false in every s-sequence on 21. But to discover that '-iTa <--> TTa"" is stably true in some s-sequences of a certain system is quite disturbing. After all, '-iTa' asserts that a is not true, while 'TrTa~" says "It is true that a is true." There are g-sequences (and hence, b-sequences) in which'-.Ta<-->TrTa"" is stably true. 21[{jTa},a{Ta}] is such a sequence. Table 3.G is the stability table for '-.Ta<-->TTef" in the g-sequence 21[{Ta},g{Ta>].23 3.2.2. CONJUNCTIONS AND DISJUNCTIONS OF LIARS The artifacts discussed here are perhaps among the most striking consequences of these systems. In the first section of this chapter, we showed that A A A' is fp-capricious in 21[G] and in 21[B], and A V A' is tp-capricious in both systems. Thus in Gupta's and Belnap's systems on 21, the conjunction and disjunction of two distinct Liars are less paradoxical than each of the Liars taken separately. In the g-sequence 21[{A}, g{ }], depicted in Table 3.E, A and A' are paradoxical, but their conjunction is stably false and their disjunction is stably true. In the world of that g-sequence, one is guilty of paradox if he asserts that A is not true or if he asserts that A' is not true, but if he says "A is not true and A' is not true" then what he says is stably false and if he says "A is not true or A' is not true" then what he says is stably true. This is an intolerable verdict. As noted earlier, there is no logical relation between A and A'. The situation is different when we consider, for instance, AAT r A n . Yes, it is true that in every s-sequence on , azT T are paradoxical but their conjunction is stably false. A AT r A" 1 is equivalent 23 '-Ta<-->TTa ' is paradoxical in the H-system on 21; it is evaluated true at every successor stage and false at every limit stage of any h-sequence in 21[H]. Since the Liar is not Tarskian, the verdict of paradoxicality, I believe, is the correct one in this case.

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in 21 to '-iTa A Ta', and there is a clear logical relation between the conjuncts of '-iTa A Ta': the first is the negation of the second. The stable falsity of this conjunction is simply due to that logical relation between its conjuncts. Similarly, the stable truth of A V T r A n , which is equivalent in 21 to '-iTa V Ta', is due to the logical relation of negation between the disjuncts '->Ta' and Ta'. Although each of A A A' and A V A' receives the right classification, namely paradoxical, in the H-system on 21, a slightly different example shows that Herzberger's system is not free of such intolerable verdicts. In 21[H], the sentences A A T r A" and A V T r A' n receive an incredibly hospitable treatment: they are not capricious but paradox-free, and even better, the first is stably false and the second is stably true in every h-sequence on 21. In other words, the conjunction and disjunction of those two, totally unrelated, paradoxical sentences are classified by 21 [H] as belonging to the category of the invariant sentences, which also contains all the grounded sentences and all the truth-generalizations and their instances. Table 3.H shows the semantical behavior of the sentences '-.Ta A TV and l->Ta V T6', which are equivalent in 21 to A A T r A n and A V T r A'"\ in the w-tail of an arbitrary h-sequence on 21.

3.2.3. TRUTHTELLER CYCLES AND CYCLIC TRUTHTELLERS In Section 1.2, we gave examples of self-referential and crossreferential Liars and Truthtellers. We start this section by defining certain semantical notions that formally represent those types of Truthtellers and Liars. The T-form (p is self-referential (henceforth, SR) in 21 if and only if it is a T-form of some nonquotational name whose referent in 21 is tf> itself. Using the notion of negation-complexity introduced at the end of Section 21, we represent in 21 a self-referential Liar as an SR T-form whose negation-complexity is an odd number, and a selfreferential Truthteller as an SR T-form whose negation-complexity is an even number (including zero). We call the first kind of SR T-form cyclic Liar and the second cyclic Truthteller in 21.24 If the 2 *We did not simply call them "SR Liar" and "SR Truthteller" because some self-referential Liars and Truthtellers may have formal representations in

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nonquotational names 1, is a CR T-form sequence in 21 if and only if there are n distinct nonquotational names, Ci, ft, . . . , and Cn> such that • k is a T-form of ft, for every k 6 {1,2,..., n}, and • in 21, ci = r v>2 n , ft = "Vs"1, • • •, Cn-i = Vn"1, and i,tf>2,..., ifn) be any CR T-form sequence in 21; if the sum of all the negation-complexities of ipi, (p^, ..., and Ts', in that order, then the CR sequence (Tj, —>Ts) is a Liar cycle, and it represents in 21 the group of crossreferential Liars described in the first Smith-Jones example (Section 1.2). If 'j' is assigned in 21 the sentence Ts', instead of '-iTV, as its referent, then the CR sequence (Tj,Ts) is a Truthteller cycle, and it represents in 21 the group of cross-referential Truthtellers described in the second Smith-Jones example (Section 1.2).25 Before discussing the kinds of verdicts that these types of Lsentences receive in the systems of Herzberger, Gupta, and Belnap, 21 that axe not T-forms. For example, the jC-sentences x = c —+ -Tx) and x (x = = —T Tx), where each of the nonquotational names and ' refers in 21 to the sentence in which it occurs, may be considered as formal representations in 21 of a simple self-referential Liar and a simple self-referential Truthteller, respectively. (Indeed, every s-sequence on 21 renders the first sentence paradoxical and the second tf-capricious.) 25 An unfortunate consequence of using sequences, instead of ordinary sets, to formalize groups of cross-referential Liars and Truthtellers is that no such group has a unique formal representation in any base model. For instance, if the CR T-form sequence ( 1, 2, 3) represents in 21 some group of crossreferential Liars or Truthtellers, then so does each of the CR sequences ( 2, 3 > 1) and ( 3, 1, 2)' Using sequences, however, simplifies some technical details.

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there are two facts about CR T-form sequences that merit noting. First, the T-forms in any such sequence are either all paradoxical or all stable in every s-sequence on 21. One can see that this is indeed so by observing that the truth conditions of any two sentences in a CR T-form sequence are interdependent. Hence, if one of the T-forms in a CR sequence is tossed back and forth between the extension of 'T' and its complement in some s-sequence on 21, then this paradoxical T-form drags with it the other T-forms in that CR sequence forcing them to exhibit paradoxical behavior too in the same s-sequence. We restate the first fact as follows: for every ssequence 21 [E0, TZ\ and every CR T-form sequence (< 1, 2, • • • , n) in 21, or is empty Second, each CR T-form sequence can be associated, in a very natural way, with some SR T-form. This association is based on the notion of T-form composition defined in the last paragraph of Section 2.1. Let ( 1, 2, . . ., n) be any CR T-form sequence in 21. Now consider the L-sentence o ( o • • • o n, which is the composition of the T-forms 1, (2, . . ., and n . Suppose that 2 ° • • • ° n is also a T-form of n , and it is also clear that this T-form is not SR in 21 because cn does not refer in 21 to ( 1 o 2 o • • • o n (it refers, instead, to 1). Let 21' be any base model of L, in which the T-form 1 o 2 o • • • o ipn is SR (i.e.,
k+i ° • • • ° fn 6 E'0 if and only if

>* £ EO

then the semantical behavior of (pi o 2 o • • • o n in 21'[E' 0 ,Ti] is exactly the same as the semantical behavior of 1 in 21[E'o,72.].26 26 Indeed, The semantical behavior of o k+l 0° • • • ° n in 21'[EQ, R} is exactly the same as the semantical behavior of k- in 21[Boi7J]) for every k = 1 , 2 , . . . , n. However, if k > 1, the T-form k o k+l o • • • o n is not SR in 21'.

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Hence, the verdict that the first sentence receives in 2 1 [ E ' 0 , R ] and the verdict that the second sentence receives in 21[Er0)R.] are identical. We shall not prove this fact here; its proof is tedious and hardly illuminating. The reader may find it helpful, however, to check a few examples. Table 3.1 is the stability table for the T-forms in the OR sequence (Tc, ->Td, ->Te) in the h-sequence 21[0, H], and Table 3.J shows the semantical behavior of the SRT-form 'T -T - > T e ~ " in the h-sequence 21'[0,H].27 Because of this natural association between CR T-form sequences and SR T-forrns, we restrict our discussion in the remainder of this section to cycles of Liars and Truthtellers. All the relevant conclusions of this discussion can be translated into similar conclusions about cyclic Liars and Truthtellers.28 All systems of stability semantics succeed in producing the intuitively correct verdict about every Liar cycle: all the T-forms in any such cycle in 21 are rendered paradoxical by every s-sequence on 21. Truthteller cycles, on the other hand, have a radically different fortune. The systems of Herzberger, Gupta, and Belnap include "almost all" Truthteller cycles in the tfp-capricious category, rather than in the tf-capricious category. Thus, these Truthtellers are not paradoxfree (and hence, not Tarskian) in these systems. In other words, cross-referential Truthtellers are not treated as Truthtellers by any of these three systems. Let us consider a very simple example. The T-forms in the Truthteller cycle (Tj,Ts), which represents in 21 the second Smith-Jones example, are classified as tfp-capricious in 21[G] and in 21[B]. The g-sequence 21[{Tj},6{Tj'}] renders 'Tj' and 'Ts' paradoxical (see Table 3.K). There are two consistent ways of assigning truth declarations to 'Tj' and 'Ts': either both are declared true or both are declared false. On the other hand, if one of them is declared true and the other false, a contradiction would arise; hence, these truth declarations must be rejected.29 It is incorrect to argue "analogously" that since declaring a simple Truthteller T true makes the 27

Note that the last sentence is the composition of 'Tc', '-iTof, and '-iTe'. An immediate consequence of the definitions of these types of Liars and Truthtellers is that Liar cycles are associated with cyclic Liars and Truthteller cycles with cyclic Truthtellers. 29 Observe that every s-sequence on 21 in which 'Tj' and 'Ts' are initially given identical truth declarations renders them paradox-free. 28

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conjunction of r and a simple Liar A paradoxical, and declaring it false makes that conjunction stably false, the first truth declaration must be rejected as contradictory, and hence T must always be declared false. It is incorrect because one could use the same kind of argument to conclude that r must always be declared true; for such a declaration renders the disjunction of T and A stably true, while declaring T false renders the same disjunction paradoxical. The paradoxicality of r A A and T V A is due to the paradoxicality of the Liar A and not to an inconsistent truth declaration of r. Stabilizing T A A or T V A by declaring T false or true does not stabilize A. The paradoxicality of 'TV and 'Tj' in the g-sequence ^ • [ { T j } , G [ T j } is due to an inconsistent initial truth declaration that the limit rule Q{Tj} fails to correct. Giving consistent truth declarations to 'Tj' and 'Ts makes each of these sentences stable, i.e., it makes them paradox-free. Herzberger's system treats the Truthteller cycle (Tj,Ts) correctly: the sentences 'Tj' and 'TV are tf-capricious in 21[H]. The H-system succeeds in this case not because it is equipped with a limit rule that is designed to correct inconsistent initial truth declarations, but simply because Herzberger's limit rule happens to correct this inconsistency. A different example may show that the limit rule H does not always have the good fortune of correcting inconsistent initial truth declarations. Indeed, there are infinitely many such examples. The T-forms in the Truthteller cycle (Tc, Td, >Te), which was discussed earlier, are all paradoxical in the h-sequence 21 [0, 'H] (see Table 3.I), and they are tfp-capricious in 21[H].30 The cyclic Truthteller 'Tr->Tr-^Te^' is, as shown in 30

There are only two consistent assignments of truth values (or truth declarations) to the sentences in this Truthteller cycle. The first is that 'Tc' and '->Trf' are true and '-.Te' is false. The second is that 'Tc' and '-.Td' are false and '—iTe' is true. Any other truth value assignment produces a contradiction. This Truthteller cycle can be considered as a formal representation, in 21, of the following example of cross-referential Truthtellers. Suppose that Sam, David, and Mary each wrote a single sentence on occasion t. Sam wrote, "The sentence Mary writes on occasion t is true." Mary wrote, "The sentence David writes on occasion t is not true." David wrote, "The sentence Sam writes on occasion < is not true." If Sam's sentence is true, then so is Mary's; and if Mary's sentence is true, then David's must be false. But if David's sentence is true, then Sam's is false, and hence Mary's is false too.

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Table 3.J, also paradoxical in the h-sequence 21'[0,7i], and it is tfp-capricious in the H-system on 21'. 3.2.4. CIRCULAR INVARIANT SENTENCES The various examples of groups of cross-referential Truthtellers discussed in the preceding section confirm the conclusion that for every such group, there are only two consistent assignments of truth values to the sentences in that group. Here is an argument for this conclusion. Since the truth conditions of the sentences in any group of cross-referential Truthtellers are all interdependent, the truth value of any sentence in that group determines the truth values of the other sentences in the same group. But any such sentence is itself a Truthteller, i.e., its truth status is arbitrary but consistent. It follows that there are only two consistent assignments of truth values to the sentences in that group. We may now ask whether there are collections of circular sentences (i.e., sentences with circular truth conditions), such that each sentence can receive exactly one truth value. The answer is yes. We call sentences of this type circular invariant (henceforth, Cl) sentences.31 In this section we discuss three examples of CI sentences, and we show that the systems of Herzberger, Gupta, and Belnap fail to produce the intuitively correct verdicts in the second and third examples. In Section 1.3, we gave a list of two sentences, S1 and S2, and said that there is only one consistent way of assigning truth values to those sentences. Here is the same list: S1. At least one sentence in this list is not true. S2. Every sentence in this list is true. If S2 is true, then SI must be true too; but if S1 is true, then at least one of S1 and S2 must be false; this is a contradiction. On the other hand, if S2 is false, then S1 is true; there is no contradiction 31 This fascinating semantical phenomenon was first observed by Gupta (1982, p. 210). It is also discussed by Barwise and Etchemendy (1987, pp. 2324); they gave it the name "Gupta's Puzzle." Their example, however, is different from the one originally mentioned by Gupta. The examples discussed here are also different from those of Gupta and Barwise and Etchemendy.

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here. We conclude that S1 must be true and S2 false. Let us give a formal representation in 21 of this list. Suppose that 'P' is a nonlogical predicate of the language £ whose extension in 21 consists of the L-sentences: ' x (Px ->Tx)' and 'Va: (Px —>• Tx)\ Indeed, every s-sequence on 21 classifies the first sentence as stably true and the second as stably false. Thus, these two sentences receive the correct verdicts in every system of stability semantics on 21. Table 3.L shows the semantical behavior of'3x (PxA-iTx)' and 'Va; (Px Tx)' in four s-sequences on 21, each of which represents one of the four possible assignments of truth declarations to these sentences. Now we consider a different example. Suppose that Michael and Leora each wrote a single sentence on occasion t. Michael wrote, "The sentence Leora writes on occasion t is true." Leora wrote, "This very sentence is true if and only if the sentence Michael writes on occasion t is true." If Michael's sentence is false, then so is Leora's. But this means that Michael's and Leora's sentences have identical truth values, and hence the biconditional that Leora wrote on occasion t is true after all; this is a contradiction. However, if Michael's sentence is true, then so is Leora's; there is no contradiction here. This argument shows that there is only one consistent way of assigning truth values to these sentences: both are true. If we assume that the nonquotational name m' refers in 21 to the L-sentence 'T1' and '/' to 'Tl<-->Tm', then the collection {Tl, Tl<-->Tm} represents in 21 the previous example. These sentences, given the argument above, are expected to be treated as CI sentences, and to be classified as stably true. However, Herzberger's, Gupta's, and Belnap's systems on 21 render 'TV and 'Tl<-->Tm' tp-capricious. Hence, none of the three systems treats these sentences as CI. As mentioned before, these systems are not equipped with an adequate mechanism for correcting inconsistent initial truth declarations. Table 3.M is the stability table for the sentences 'Tl' and 'Tl<-->Tm' in the h-sequence 21[0,7i], which is also a g-sequence and a b-sequence. Our last example is a variation on the second. Finite collections of CI sentences can be associated with single CI sentences. Since this association is very similar to the one between CR Tform sequences and SR T-forms discussed in the preceding section, we shall not concern ourselves here with producing a general def-

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inition of this relation. We wish only to produce an L-sentence that is, in a certain natural sense, semantically similar to the collection {Tl,Tl<-->Tm}. Observe that the identity 'm = T1" is true in 21. Hence, by substitution, the sentences 'Tl <--> Tm' and 'T1 <--> T21'are equivalent in 21. Now let 21' be any base model of £ in which ' l' refers to the biconditional'Tl<-->T 2 l.' For all s-sequences 21[E 0) R] and 2 1 [ E ' Q , R ] , if the truth declarations that 'T/' and 'Tl <--> Tm' initially receive in 2 1 [ E 0 , R ] are, respectively, identical with the truth declarations that 'T/' and 'Tl<-->f T 2 /' initially receive in 21'[E'd1'R], then the semantical behavior of 'Tl<-->Tm' in the first sequence is exactly the same as the semantical behavior of 'Tl<-->T 2 /' in the second sequence. This fact entails that the biconditional 'Tl<-->T 2 /' is also tp-capricious in each of 2t'[H], 21'[G], and 21'[B]. Table 3.M can be easily modified to serve as the stability table for the sentence 'T/<-->T 2 /' in the h-sequence 21'[0,7i]. I leave this task to the reader. Let us in this paragraph fix 21' as the base model of L. The sentence 'Tl<-->T 2 /' makes an interesting assertion; it says, "I am true if and only if it is true that I am true." In other words, this sentence claims that it is Tarskian, or that it is paradox-free. Indeed, it is a self-referential instance of the Tarskian schema formulated as a material biconditional. An argument similar to the one given above about the Michael-Leora example shows that this sentence must be treated as CI, and that it should be classified as stably true. For if / is false, then each of 'Tl' and 'T2/' are false too; hence the biconditional 'Tl<-->T 2 /' is true. Since / is that biconditional, / is true after all; and thus we obtain a contradiction. However, assigning the truth value true to / does not produce a contradiction. This argument further supports our claim that the verdict produced by 21' [H], 21' [G], and 21'[B] about the sentence 'Tl<-->T 2 /' is indeed erroneous.

3.3. Philosophical Diagnosis

In the introduction to this chapter we said that one objective of our philosophical defense of the revision theory of truth is to confirm its material adequacy. As noted there, our strategy for achieving

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this objective is to develop a formal semantical system that satisfies the following two conditions: it faithfully models the revision theory of truth, and it produces the intuitively correct verdicts in all cases. The various types of examples sampled in Section 3.2 demonstrate that Herzberger's, Gupta's, and Belnap's systems of stability semantics do not satisfy the second condition; hence none of them can fulfill that objective. But there is a more serious worry. If one of these systems turns out to be a faithful representation of the revision theory of truth, then there would be a good reason for believing that the revision theory is not materially adequate. For we can argue that the erroneous verdicts produced by that system must be faults of the philosophical theory of which the system is a formal copy. In this section we put this worry to rest. We argue that none of these systems succeeds in representing faithfully the revision theory of truth. As mentioned in Section 1.3, the central thesis of the revision theory is that the circularity of the concept of truth in bivalent languages gives rise to a revision process in which the semantical status of each sentence is determined by its Tarskian biconditional once a totality of nonsemantical facts and an initial extension of the truth predicate are posited. If the language contains pathological sentences such as Liars and Truthtellers, the revision process yields revision hierarchies, each of which consists of extensions of the truth predicate and is based on an "appropriate" initial extension of this predicate and on the same totality of nonsemantical facts. In such a language, the semantical status of each sentence is fully characterized by the kinds of semantical behavior that the sentence exhibits in all revision hierarchies. According to this conceptual picture, a revision hierarchy has only three essential ingredients: a totality of nonsemantical facts, an initial extension of the truth predicate, and the Tarskian schema. These three ingredients are represented in each s-sequence 21[EQ,'R,] as follows:32 • The base model 21 represents the totality of nonsemantical facts; • The collection EQ represents the initial extension of the truth predicate; 32

See Chapter 2, Definitions 2.A and 2.B.

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• The Tarskian rules, i.e., the successor rule and the rules that determine the sets U and U at every limit stage 7, represent the Tarskian schema. Every revision hierarchy, therefore, can be formalized as some s-sequence. Recall that a system of stability semantics on is nothing but a collection of s-sequences on that base model, such that for every initial extension E0 of 'T', there is an s-sequence in that system whose first term is ( , E 0 ). The last condition is needed to ensure that the system contains a sufficient variety of s-sequences, so that every sentence is given ample opportunity to exhibit all the different kinds of behavior that collectively characterize its semantical status. Thus, each system of stability semantics on formalizes the revision process that is based on the totality of nonsemantical facts represented by , for all the revision hierarchies that are generated by that revision process are formally represented by s-sequences in that system. The important question now is this: if all systems of stability semantics represent formally the revision process, then why do some of them fail to do so faithfully? In order to answer this question, we need to examine carefully the parameters that determine any ssequence. We listed above three such parameters: a base model, an initial extension of 'T', and the Tarskian rules, which correspond to the three factors that generate any revision hierarchy. But there is a fourth parameter, a limit rule. As explained in Chapter 2, a limit rule associates every limit ordinal 7 with a bootstrapper, which is a set of sentences employed to assign truth declarations to the 7unstable sentences, if there are any such sentences at that stage. This fourth parameter appears to have no counterpart in the conceptual picture of the revision process painted above. But since in almost every s-sequence there are pools of unstable sentences at limit stages, some kind of bootstrapping must be performed at these stages. Bootstrapping, however, can be a source of serious trouble: it can influence the semantical behavior of many sentences in ways that are not rooted in the three conceptual components, , -E0, and the Tarskian rules. Some of the verdicts that an s-sequence produces might be nothing but artifacts of the bootstrappers employed in that sequence.33 Thus, the fourth parameter must be 33

The erroneous classifications that

and

receive

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tuned with extreme care, so that the bootstrappers introduced at limit stages do not distort the outcome of the formal process. We call this problem the problem of -unstable sentences. There is one more difficulty that every faithful formalization of the revision process must overcome. The set E0, which represents the initial extension of the truth predicate, might be and usually is excessively arbitrary. As argued in Chapter 1, since there are sentences with circular truth conditions, an initial extension of the truth predicate must be posited in addition to the given totality of nonsemantical facts in order for the revision process to determine the semantical status of every sentence. In positing an initial extension of the truth predicate, a certain degree of arbitrariness may be warranted, for there are certain types of sentences, such as Truthtellers, whose truth status is arbitrary. But every system of stability semantics permits E0 to be any set of sentences, including the empty set. Hence, E0 may carry much more arbitrariness than what is needed for dealing with capricious sentences. A great number of sentences can be given correct or noncontradictory truth declarations at stage 0, before any application of the successor rule takes place. We could have required E0 to contain, for example, all the sentences that are evaluated true in the base model and all their pure T-forms, as well as some true truth-generalizations However, except xsuch (Tx as Tx)'. However,and except for certain types of impoverished languages, it is simply impossible to characterize effectively the seven semantical categories.34 This means that the project of determining which collections of sentences can be selected as "appropriate" initial extensions of the truth predicate is a lost cause. Thus, we have no choice but to let E 0 b e any arbitrary set of sentences. Because of this arbitrariness, most or all of the initial truth declarations in many s-sequences are wrong or even contradictory. If some of these declarations persist, the s-sequences are bound to produce erroneous verdicts. An ssequence must be equipped with an adequate self-correcting mechanism in order to deal successfully with any excessive arbitrariness presented at stage 0. As demonstrated by previous examples,35 in the H-system on are examples of such verdicts. See Sections 3.2.1 and 3.2.2, and Tables 3.B and 3.H. 34 For the definitions of these categories, see Section 2.4, Definition 2.E. 35 See Tables 2.A and 3.L.

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93

the Tarskian rules are capable of correcting many wrong initial truth declarations. Still, a great deal of the initial arbitrariness or its outcome might escape to the limit stages. Thus, the limit rule must supply its share of the correcting process. We call this problem the problem of initial excessive arbitrariness. We have argued that if a system of stability semantics faithfully represents the revision process, then its limit rule assignment must be able to deal successfully with the problem of unstable sentences and with the problem of initial excessive arbitrariness. A brief examination shows that none of the limit rule assignments H, G, and B succeeds in both tasks. Gupta's limit rule assignment, G, successfully deals with the first problem, but it amplifies the second. The policy of using the initial extension E0 as a bootstrapper at every limit stage of the g-sequence [E0, E0] has two consequences. First, it allows the process to assign truth declarations to the 7-unstable sentences without introducing new factors that might influence the outcome of the process beyond the "intent" of , E0, and the Tarskian rules. Second, it provides a secure shelter for the initial arbitrariness that escapes the correcting mechanism of the successor rule, thus allowing whatever bad effects that arbitrariness may have to persist throughout the process. The first consequence shows that the limit rule assignment G presents a simple and effective solution to the problem of 7-unstable sentences. The second consequence, however, makes it clear that G does not only leave the problem of initial excessive arbitrariness unsolved, but further it ensures its presence. All the erroneous verdicts produced by many g-sequences are due to G's totally ineffective approach to that problem; hence they are nothing but artifacts of Gupta's limit rule assignments.36 Belnap's limit rule assignment fails to deal adequately with either problem. Its approach to both problems is "maximal arbitrariness." Belnap (1982) explained that maximal arbitrariness is needed in order to insure that "what survives this arbitrariness really fully depends only on the model with which we began to36 The examples described in Tables 3.K and 3.M are representative of the kinds of errors produced by retaining initial truth declarations that are contradictory. The g-sequence depicted in Table 3.E demonstrates how the formal process can be trapped by arbitrary initial truth declarations that are taken too seriously.

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The Liar Speaks the Truth

gether with the Rule of Reason components of the process of revision" (pp. 106-7).37 The idea is that any sentence that has the same semantical status in every s-sequence on a certain base model must owe its semantical status fully to the base model, the logical nature of the extension of 'T', and the Tarskian rules. Belnap's policy of associating every initial extension of 'T' with every limit rule imposes a stringent purifying process on the pure categories: no sentence can be declared stably true (or stably false, or paradoxical) in [B] unless it is so declared by every s-sequence on .38 This purifying process, however, is too stringent; it removes not only the chaff from the pure categories but some of the wheat as well. The conjunction of two distinct Liars is a case in point (see Section 3.2.2 and Table 3.E). On the other hand, Belnap's policy is too relaxed in its criteria for membership in the various capricious categories. Simply put, this policy, by allowing every s-sequence to participate, turns the capricious categories into a mess. Almost all the erroneous verdicts mentioned in Section 3.2 are indications of the chaotic status of these categories in the B-system. The limit rule assignment B is an ineffective approach to the problem of 7-unstable sentences, because the influence that bootstrappers might have on the outcome of the process cannot be eliminated by giving every limit rule a full consideration. B also fails in dealing with the problem of initial excessive arbitrariness because the problem does not simply disappear by ignoring the capricious categories. To sum up, Belnap's limit rule assignment, by admitting all s-sequences into the system, admits with them all the artifacts that these sequences may have. It is hard to see what kind of solution to either problem is advanced by Herzberger's limit rule assignment. I believe that this limit rule assignment does not even attempt to approach those two problems. We should be surprised if the policy of declaring false all the 7-unstable sentences at every limit stage 7 turns out to have 37 The Rule of Reason components are the successor rule and the rules that determine the sets U and U at every limit stage , i.e., they are the Tarskian rules. 38 Recall that the B-system is uc-nested in every system of stability semantics on the same base model. This means that being of a certain uniform type in [B] is sufficient for being of the same uniform type in any other system on

The Original Three Systems

95

some defensible philosophical basis.39 However, let us entertain one possibility. Perhaps, Herzberger's policy can be justified on the grounds that the task of is to produce in the h-sequence [E0, ] approximations of the extension of the metapredicate 'is stably true in [E0, H]'. There are three reasons why we should reject this assumption. First, this justification is backward. Whether a sentence has the property of stable truth in an s-sequence is determined by the overall semantical behavior of the sentence in that s-sequence. But such behavior may be shaped by the kinds of truth declarations assigned to the unstable sentences at limit stages. This means that the category of stably true sentences in any s-sequence may be partly determined by the limit rule employed in that s-sequence. Hence, it is hardly convincing to claim that . produces approximations of the category of stably true sentences, where these "approximations" themselves can be among the factors that determine that category. Second, if approximates the extension of 'is stably true in this s-sequence' by declaring all the unstable sentences false, then analogously the limit rule y that declares all the 7-unstable sentences true must produce approximations of the extension of 'is stably false in this s-sequence'.40 Since the categories of stably true and stably false sentences in every s-sequence codetermine each other, one would expect the y-sequence [E 0 ,Y] to be semantically equivalent to the h-sequence [E0, .]. But this is not true in general. The sentence T VT is paradoxical in every h-sequence on and stably true in every y-sequence on .41 39 It is incorrect to interpret this sentence as claiming that Herzberger's system lacks philosophical significance. Herzberger (1982a, p. 135, and 1982b, p. 479) suggested that his system serves as a consistent reconciliation of Kripke's construction and classical logic. He is right, the H-system serves this purpose successfully. Although every system of stability semantics also serves this purpose, the H-system. does so very elegantly, for the limit rule is definitely among the simplest limit rules. Furthermore, as argued at the beginning of this section, all systems of stability semantics are formal representations of the revision process. This consideration alone ensures that no system of stability semantics lacks philosophical significance. 40 The limit rule y is a constant limit rule that assigns the bootstrapper S (the set of all -sentences) to every limit, ordinal 7. 41 See Herzberger (1982a, p. 147) for interesting remarks about the semantical behavior of the sentence T VT in any h-sequence.

The Liar Speaks the Truth

96

Third, as explained in Section 3.1, every h-sequence has alignment points; at these points the extension of 'T' coincides with the set of stably true sentences in that h-sequence. Let 6 be the first alignment point of an arbitrary h-sequence [E0, ]. Now if the task of is to produce approximations of the extension of 'is stably true in this s-sequence', then it should be expected that the extension of 'T' at every limit stage higher than 6 would be identical with the extension at , for completes its task once it produces the perfect approximation. This expectation is not met. In almost all cases, [E0, ] goes into a loop of extensions deviating from the perfect approximation E for a while and then returning to it periodically. For example, the sentence T VT is not in the extension E but it is in the extension at the next limit stage, i.e., it is in E +W . The limit rule assignment H leaves the problem of 7-unstable sentences and the problem of initial excessive arbitrariness unsolved. Assigning the empty bootstrapper to all limit stages is bound to distort the outcome of the formal process in many hsequences. The erroneous verdicts that the sentences ', T ", and T receive in every h-sequence on are clear examples of such distortion.42 H has a very poor correcting mechanism. For example, it permits many contradictory initial truth declarations to persist. The Truthtellers shown in Tables 3.I and 3.J demonstrate this fact. These erroneous verdicts are artifacts of Herzberger's limit rule assignment. All the artifacts mentioned in Section 3.2 are classification errors. We have discussed many examples of sentences that are semantically misclassified by one or more of these three systems. For instance, the sentences 'Tj' and 'Ts', where 'j' refers in to 'Ts and V to 'Tj', should be treated as Truthtellers, and hence they should be classified as tf-capricious. But as shown in Section 3.2.3, the G-system and B-system on classify these sentences as tfp-capricious (see Table 3.K). Let us call the artifacts of this type "CRASS errors" ('CRASS' for 'Classifying with Rampant Arbitrary Stability Sequences'). There is a second type of artifact that we haven't discussed yet. We shall call the artifacts of this type "CHOP 42

See Sections 3.2.1 and 3.2.2, and Tables 3.B and 3.H.

The Original Three Systems

97

errors" ('CHOP'for'Change in Haste of Original Purpose').43 Here is an example of a CHOP error. Let be the -sentence ' (Px Tx)'. Suppose that the extension of 'P' in the base model consists of the sentence 'g = g' and all its pure T-forms, where 'g' is some nonquotational name in . asserts that the sentence 'g = g ' and all its pure T-forms are in the extension of 'T'. Since 'g = g' is a logical truth, we should expect that is classified as stably true. This expectation is indeed fulfilled by every s-sequence on . Hence, is stably true in [B]. Now consider the sentence 'T ', and suppose that 'g' refers in to 'T ' itself. Informally, this sentence says: "My truth declaration is identical with the truth value of ." The following intuitive reasoning shows that 'Tg ' must be treated as a Truthteller, i.e., it should be classified as tf-capricious. If 'Tg ' is declared true, then its truth declaration is identical with the truth value of a; hence it is true. On the other hand, if 'Tg ' is declared false, then its truth declaration is different from the truth value of ; hence it is false. Thus, this sentence exhibits the same semantical behavior as a Truthteller: its truth conditions are circular and arbitrary. To make 'Tg ' true or to make it false, we only need to declare it so initially. Indeed, every system of stability semantics on classifies 'Tg ' as tf-capricious. Thus, no s-sequence on produces a CRASS error in determining the semantical status of this sentence. However, something bizarre can happen. Consider the h-sequence (and hence, the b-sequence) [{Tg }, H]. In this sequence 'Tg ' is declared true initially. Our "intent" is to make 'Tg ' stably true. It shouldn't matter that , 'g = g', and the pure T-forms of 'g = g' are all declared false at stage 0 of that h-sequence; for these erroneous truth declarations are part of the initial excessive arbitrariness that the Tarskian rules can correct. Contrary to our original declaration and intent, however, 'Tg ' is rendered stably false by [{Tg },H].44 Similarly, the limit rule y, which constantly declares all the 7-unstable sentences true, makes 43 These two colorful acronyms are products of the witty imagination of Janet Kelley. 44 Table 3.N at the end of the chapter shows the semantical behavior of [{Tg 'Tg ' in

98

The Liar Speaks the Truth

'Tg ' stably true, if the initial extension is empty (i.e., if 'Tg ' is initially declared false). This example shows that the bootstrappers assigned by the limit rules H and y can distort the outcome of the formal process by ignoring an arbitrary initial truth declaration that is seriously intended. The initial arbitrariness is not totally superfluous; a portion of it is conceptually warranted. As argued previously, circularity and arbitrariness are genuine features of the concept of truth. Hence, an initial arbitrary truth declaration (e.g., declaring the sentence 'Tg ' true) may represent a genuine choice, and if so, it must be respected and maintained. CHOP errors are produced by limit rules, such as H and Y, that hastily alter some of the choices made at the initial stage. Thus, these errors are artifacts of limit rules that fail to deal successfully with the problem of 7unstable sentences. Gupta's system is not troubled by any CHOP errors because its limit rule assignment has an effective approach to this problem: it never alters the initial choices. The systems of Herzberger and Belnap, however, are infected with both kinds of artifacts, CRASS and CHOP.

TABLE 3. A. Stability Table for the L sentences ' x (x = x)' ( ), ' x(Pnx Tx)' ( n) for n 0, and ' x(P x Tx)' ( ). /(P0) = { }, /(Pn) = { , 0, ..., n], and I(P ) = { , o, 1,... }. h-sequence:

[{}, H]

0

1 1 0

1

0 1 1 1 0

1 1 1

2

0

1 1 1 1 1 1 1

+1

1

99

TABLE 3.B. Stability Table for the L-sentences ' Ta' ( ), ' Tb' ( '), and ' Ta Tb'. a = r Ta and b = r Tb. h-sequence:

[E 0 ,H], starting at

a

1 1 1

+l

0 0 1

+2

1 1 1

2

1 1 1

0

2+l

0

1

100

TABLE 3.C. Stability Table for the L-sentences ' T a ' ( ), 'Tb' and '-Ta Tb'. a = r Ta and b = r Tb g-sequence:

[{ },

{

}

] tv

a

0

0 1 0

1

1 0 0

0

1

2

0

0 1 0

1

+1

0 0

101

( '),

TABLE 3-D. Stability Table for the L-sentences 'Ta' ( ), 'Tb' 'Ta V Tb', and ' Ta Tb'. a = Ta and b = r Tb. h-sequence:

[E0,H]

( '),

starting at

a

1 1 1 1

0 0 0 0

+1

+2

1 1 1 1

2

1 1 1 1

0 0 0 0

2+l

102

TABLE 3.E. Stability Table for the L-sentences ' Ta' ( ), 'Tb' ( '), 'Ta V Tb', and Ta Tb'. a = r Ta and b = r Tb. g-sequence: [{ },

{ }

]

tv

0 1 1 0

0

1

1

0

1

0

0

1

2

1 0

0

1 1

0

1 0

+1

1

0

103

TABLE 3.F. Stability Table for the L-sentences ' Ta' ( ), ' Tb' ( '), ', Tn for n > 0, and ' x (Px (Tx ( ')))' ( ). r a = Ta and b = rTb . The extension of 'P' consists of ' and all its pure T-forms. (b-sequence: [ ,B a ])

0

1 1 1

0 0 0

1

0 0 1 1 0 0_ 0

1

0

1 1

1 1 1 0 0 0

0 0

1 1

0

0

0

1 0

1 1 0

1

0 0 0

+1

1

0

0

104

TABLE 3.G.
Stability Table for the L-sentences ' T a ' ( ), 'Ta', Ta T Ta . a = r Ta . g-sequence:

[{Ta}, g{Ta}]

1 0 1 1

0

0

1

1

0

1 1

2

0

1 1 1

0

1 1

0

1

+1

0

1 1 0

+2

1 1

105

TABLE 3.H. Stability Table for the L-sentences ' Ta' ( ), 'Tb' 'Tb', ' Ta Tb, and ' Ta V Tb'. a = r Ta and b = r Tb. h-sequence:

[E 0 ,H], starting at

1 1 0 0 1 0 0

1

+1

0

1 1

1

+2

0 0

1

1 1

2

0 0

1 0

0

1

2+l

0

1

106

( '),

TABLE 3.1. Stability Table for the L-sentences 'Tc', c = r Td , d = r Te , and e = Tc . h-sequence:

Td', and ' Te'.

[ , H]

1 1 0

0

1 0

1

1

0 0 0

2

1 1

3

0

1 1 0

1

+1

0

1

107

TABLE 3.J. Stability Table for the L-sentences 'T -T '-
Te,

'[ ,H]

1 1 0

0

1 0

1

1

0 0 0

2

1 1

3

0

1 1 0

1

+1

0

1

108

TABLE 3.K. Stability Table for the L-sentences 'Tj' and 'Ts'. j and s = Tj. g-sequence:

=Ts

[{Tj}, g{Tj}]

0

0 1

1

1 0

2

0

1

0

1

+1

0

1

109

TABLE 3.L. Stability Table for the L sentences ' x (Px Tx)' ( and ' x (Px Tx)' ( )- The extension of 'P' consisits of and s-sequences:

1 0

0

a

tv

0

1 0

1

1 0 tv

a

1 0

0

1

1

0

a

tv

0

0

1 1

1

0

1

2

0

110

)

TABLE 3.M. Stability Table for the L-sentences 'Tl, 'Tm', and 'Tl h-sequence:

[ , H]

0

0 0 1

1

1 0 0 0

1

2

0

0 0

3

1

0 0

1 1

+1

0 0

111

TABLE 3.N. Stability Table for the L-sentences ' x (Px Tx)' ( ) and Tg . g = Tg and the extension of 'P' consists of 'g = g' and all its pure T-forms. (h-sequence: [{Tg }, H])

1

0

0 0

1

0

1 1

1 0 0

1 1 1

2 0

1 0

I 1 1 0 0

1 1

+1

1 0 0

112

4

The Revision System

The last phase of our defense of the revision theory of truth is to confirm its material adequacy. We said in the preceding chapter that we shall achieve this goal by developing a formal semantical system that correctly models the revision theory of truth, and then showing that it delivers the intuitively correct verdicts in a representative sample of cases. In this chapter, we make good our promise: we advance a system of stability semantics whose limit rule assignment is designed to deal successfully with the problem of -unstable sentences and the problem of initial excessive arbitrariness. We call this system the revision system. In Section 4.2 we describe the limit rule assignment that generates the revision system and explain its approach to these two problems. We show there that this system does not produce any of the CRASS or CHOP errors discussed in Chapter 3. Given the type of limit rule that the system employs, it is very likely that this system is free from all artifacts. Hence, we conclude that the revision system of stability semantics confirms the material adequacy of the revision theory of truth. In Section 4.3, we carry this confirmation one step further. We define a notion of logical consequence that is appropriate for any formal language whose semantics is of the type described in Section 4.2, and we show that this logical consequence relation validates all modes of deductive reasoning that ought to be preserved. We call the logical system that is based on this notion of logical consequence stability logic. In Section 4.1 we establish a few tech113

114

The Liar Speaks the Truth

nical results about constant limit rules. Some of those results are needed in Section 4.2 in order to define the limit rule assignment that generates the revision system. 4.1. Remarks on Constant Limit Rules Our project in this section is to prove two pivotal theorems, which we call the Loop Theorem and the Alignment Point Theorem.1 The first asserts that for every s-sequence on any base model of a language whose size is 77, where 77 is any infinite cardinal, if the limit rule employed in that sequence is constant, then there is a limit ordinal 8 whose size is smaller than or equal to 2n and at which the s-sequence becomes periodic, i.e., it enters a loop and remains in it permanently.2 The second says that if K is the least cardinal that is larger than 2n (i.e., K = (2 n ) + ), then every sentence is stably true, stably false, or unstable at K if and only if it is, respectively, stably true, stably false, or paradoxical in that s-sequence. We start by defining our notation and stating our presuppositions. As usual, L is a formal language of the type defined in Section 2.1, is any base model of L, and [ E 0 , R ] is the s-sequence generated by ( E 0 , R ) , where E0 is a subset of S (the set of all Lsentences) and R is a limit rule. In order to ensure that all the definitions and results mentioned in this and the following section are applicable to languages of any infinite cardinalities, we shall make no use of the fact that L is countable. As above, we take K to be the least cardinal that is larger than 2 s ; thus K = (2 s )+ = P(S) +. 'K ' denotes the set of all limit ordinals that are less than K. As a reminder, we use 'Ord' to denote the class of all ordinals and 'Ord to denote the class of all limit ordinals. 1

The discussion in this section is fairly technical. Readers who do not wish to go through the mathematical material are advised to read only the following: the introductory remarks through the paragraph that precedes Lemma 4.B, the paragraph that precedes Theorem 4.A, and the statements of Theorems 4.C-4.G. 2 The periodicity of the s-sequences whose limit rules are constant was first observed by Herzberger (1982a, pp. 148-53). (See also Gupta, 1982, pp. 22122, and Belnap, 1982, p. 107.) For proofs and details, Herzberger referred the reader to an unpublished manuscript of his: "Notes on Periodicity," May 1980. I have no knowledge of the contents of that manuscript.

The Revision System

115

Many basic facts about the ordinals and their arithmetic are presupposed throughout this section and the following one.3 Here are some of these facts. Every infinite cardinal is a limit ordinal. Hence, S , P(S) , and K are limit ordinals, is the first limit ordinal and the smallest infinite cardinal. For every a and (3 in Ord, < if and only if a . If X is a set of ordinals, then U X (the union of all the members of X) is also an ordinal, and it is the least ordinal that is greater than or equal to a for all € X. UX may also be written as U €X . Every limit ordinal can be expressed uniquely as the ordinal product of and some nonzero ordinal, i.e., for each Ord , there is a unique Ord such that > 0 and = a. Every successor ordinal can be expressed uniquely as + n, where Ord and n is a nonzero finite ordinal. Indeed, for all a and in Ord, if > 0, then there are two unique ordinals and " such that =

' + ", where 0

"<

This fact is the ordinal generalization of Euclid's Division Algorithm: for all natural numbers n and d, if d > 0, then there are two unique natural numbers q and r such that n = dq + r, where 0

r
The following lemma is an immediate consequence of Definition 2.B (Section 2.2). 'U ' denotes the set of all 7-unstable sentences. ' abbreviates 'if and only if and ' ' abbreviates 'only if'. 4.A. LEMMA. In every s-sequence, if 7 is any limit stage of that sequence, then (1) for each < and each L-sentence , U there is an ordinal such that < and ) E ; <

(2)

3

U UUand

U U.

,

U

U,

and

and

See Suppes (1972) for an excellent introduction to ordinal arithmetic. For a more advanced treatment of this subject, see Kunen (1980).

116

The Liar Speaks the Truth

In Section 3.1 we introduced the notion of alignment point. We said there that an alignment point of an h-sequence is a limit ordinal at which the extension of T' coincides with the category of the stably true sentences in that h-sequence. Definition 4. A generalizes this notion to an arbitrary s-sequence. 4.A. DEFINITION. An alignment point of an s-sequence is a limit ordinal 7 at which U =st([E0,R])

[E 0 ,R]

It is clear that the definition given in Section 3.1 is a special case of Definition 4.A, for the h-sequences employ the empty bootstrapper at all limit stages. The corollary below follows from the previous definition, Lemma 4. A, and the facts reported in Section 2.1 after Definition 2.C. 4. A. COROLLARY. If is an alignment point of U = sf( [E 0 ,R]) and U = px( [E0,R]).

[E 0 ,R], then

In the remainder of this section we consider only those ssequences that are generated by constant limit rules. C is a constant limit rule if and only if C assigns the same bootstrapper to all limit ordinals, i.e., C: Ord P(S), and there is an X C S such that for every 7 Ord , C = X. The limit rules H, GE° , and y are examples of constant limit rules. It is obvious that there are as many constant limit rules as there are subsets of S. We use 'C' as a variable ranging over the class of all constant limit rules. Any s-sequence that is generated by ( E 0 , C ) is called a c-sequence and denoted by [ E 0 , C ] ' , The next three results describe basic features of all c-sequences. 4.B. LEMMA. For every [ E 0 , C ] and all limit ordinals U = U , then E = Eri.

and , if

PROOF: Since U = U , U = U (see Section 2.2). Given that C = C, E = U T U (C - U ), and En = U T U (Cn - U ), it immediately follows that E = E . 4.C. LEMMA. For every [ E 0 , C ] and all ordinals E = E , then E + = E + .

, , and

, if

The Revision System

117

PROOF: Use transfinite induction on . The zero-case is trivial. Let = + 1 for some Ord, and assume that E + ; = E + . Hence, ( , Ea+ ) = ( , E + ). But for every successor ordinal + 1, E +1 = { S : is true in ( , E )}. Therefore, E + +1 = E + +1 Now we consider the limit-case. Let = for some 7 Ord , and assume that E + = -E + for every < . By Lemma 4.A(1), there is a

' such that

there is a

such that

From Lemma 4.B, it follows that Ea+ = E + 4.B. COROLLARY. For every UT = UT then

.

[ E 0 , C ] and all 7 and

' in Ord , if

for every for every PROOF: (1) follows from Lemmas 4.B and 4.C. The argument for (2) is similar to the limit-case argument given in the proof of Lemma 4.C (the inductive hypothesis is guaranteed by Part 1). D Our goal now is to prove the Loop Theorem. Since the proof is relatively long, we shall break it into smaller units. 4.D. LEMMA. For every U T +p , then

[E0,C] and all 6 and p in Ord , if UT =

(1) UT = UT+pn for every n < w>; (2) UT UT+pw . PROOF: We prove (1) by induction on n. The zero-case is trivial. Assume that UT = UT +pm for some m < . UT +p(m+1) From the inductive hypothesis and Corollary 4.B(2),

The Liar Speaks the Truth

118 it follows that lore,

But ThereWe conclude that for every n < ,

Now we prove Part (2). By Lemma 4.A(1), there is an n <

there is an m <

From Part (1), it follows that

such that

such that

UT

.

4.E. LEMMA. For every [E 0 ,C] and all and in Ord , if UT = UT+p - U T + pw , then UT E + for any < p . PROOF: We prove the lemma by showing that if E + for some < pw, then UT . Thus, let E + . Since < p , there exist an n < and an a < p such that — pn + . By Lemma 4.D(1), U = UT . Hence by Corollary 4.B(1), E + = E + n+ . It follows that for some < , E + . From the preceding lemma, Part (1), UT = UT+pm for all m < . By applying Corollary 4.B(1) again, we obtain that E +a = E +pm+a for all m < . Thus, there is an a < p such that for every m < , E+ m+ . This entails that UT+ . But, UT = UT+ . T Therefore, U . 4.F. LEMMA. For every [E0, C] and all 6 and UT +P = UT +P , then UT = UT+p for each

in Ord, if UT = Ord.

PROOF: The proof is by transfinite induction on a. The zero-case is trivial. For the successor-case, assume that = + 1 and that UT +p = UT Therefore by Corollary 4.B(2), Now let a = 7 for some 7 6 Ord , and assume that UT — UT+p

The Revision System for every 4.A(1),

<

. First, we show that

there is a

there is an

<

<

UT+p

119 UT.

By Lemma

such that

such that

Since < and Ord , + 1 < . Hence by the inductive hypothesis, UT. Finally, we show that UT UT+ . The inductive hypothesis asserts that UT = UT + for all < j. By Corollary 4.B(2), for every 77 < 7 and every < , E + = E + + • But Lemma 4.E ensures that UT E + for each < . Hence, UT E + + , for all < and < p . It follows from Lemma 4.A(l) that UT UT+ , . We conclude that 4.C. COROLLARY. For every c-sequence [ E 0 , C\ and for all limit ordinals and p, if UT = U T +p = U T +p then UT E + for any a Ord. PROOF: Write a as + , where Ord and 0 < . From the preceding Lemma, it follows that UT = UT+pw . By Corollary 4.B(2), E + = E + + , where 0 < >. We conclude, by Lemma 4.E, that UT C E + + . The following theorem describes an important feature of all csequences. Intuitively speaking, it says that if a c-sequence enters a loop at a certain limit stage and after cycling in that loop for times it fails to exit from it, then the c-sequence will cycle in that loop forever.4 We call such a loop the characteristic loop of that c-sequence. 4

The semantical behavior of the sentence T V T - in the h-sequence [ , H] provides an illustration of a process that enters a loop at and cycles times and then exists this loop at w2. (T V T - is stably true at every n, where 1 n < , and it is unstable at 2 .)

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4.A. THEOREM. For every [E 0 ,C] and all and UT = UT+ = UT+ , then UT = U T +pa for each

in Ord , if Ord.

PROOF: Write as + n, where Ord and n < . We get: UT + ( +n) = UT +p + P n -From Lemma 4.F, Corollary 4.B(2), and Lemma 4.D(1), it follows that UT+pw +pn = U T +pn = UT. The proofs of the next two theorems invoke a certain claim about infinite cardinals: every infinite successor cardinal is regular. An infinite cardinal is regular if and only if for every < and every f: , U < f( ) < Equivalently, is regular if and only if there is no family of sets such that |F| < , |X| < for each X in F, and | UF| . The last definition entails that the first uncountable ordinal, is regular if and only if it is not 1, the countable union of countable sets. It is known, however, that without using the Axiom of Choice one cannot establish that 1 or 2 is regular. The general claim that every infinite successor cardinal is regular is proved by invoking the Axiom of Choice.5 Thus, the arguments that we give for Theorems 4.B and 4.C below presuppose that axiom.6 4.B. THEOREM. For every ( [ E 0 , C ] , there is an K such that for each K ,if > , then UT = UT+ for some K . PROOF: We prove the theorem by reductio ad absurdum. Thus, we take [E0,C] to be some c-sequence and we assume that for every K , there is an K such that > and UT UT + for all K . Let [ ] be the least ordinal in K such that [ ] > and UT UT+ for every K . The reductio hypothesis ensures that [ ] is well defined for each in K . By transfinite recursion, we define f( ) for each < K as follows:

5 For more information, see Kunen (1980, pp. 30-33) and Jech (1978, pp. 2728 and 39-40). 6 I was unable to prove those theorems without invoking the claim that « is regular.

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By transfinite induction, we show that f( ) K for every a 6 K. The zero-case and the successor-case are immediate consequences of the definition of [ ]. It is the limit-case where we need K to be regular. Let = for some K , and assume that f( ) K for each < ; it is clear that [f( )] k . Since K is an infinite successor cardinal (k = (2s ) + ), it is regular. Therefore, U < f( ) is also in k ,i.e., f( ) K - We conclude that f() K for every a € K. Hence, f: K K. Now consider the set {UTf(a): K}. It is not difficult to show that if , then UTf() UTf( ) . The proof depends on the fact that for all and in k , if > [ ], then UT UT .7 It follows that UTf(

)

:

k}

has k distinct members. But this is

impossible, for there are only 2 subsets of 5 and K is larger than 2 . Hence, we reject the reductio hypothesis and conclude that there exists an K such that for each k , if > , then UT - UT, for some K.

4.C. THEOREM (THE Loop THEOREM). Every c-sequence on enters its characteristic loop at a limit stage prior to K. In other words, for Tevery [ E 0 , C ] , there exist a 6 and a in K such that T = T U

=

U

U

PROOF: By Theorem 4.B, there is an K such that for every K, if > , then U T = U T +v for some K . Let 0 be the least such . For every > 0, take ( ) to be the least ordinal in K - such that UT = UT +( ) . Now define g( ), for all K, by transfinite recursion as follows:

(1)(2)

(3)

First, we show that g( ) K for each a 6 K. As usual, we use transfinite induction on a. The zero-case and the successor-case 7 Observe that if k and > [ ], then there is some K such that = [ ] + . The proof is a straightforward inductive argument. Take a and to be two distinct ordinals in K and assume, without loss of generality, that > . Thus, = + for some nonzero ordinal K. Now use transfinite induction on to show that UTf(B+ ) > UTf(B). for each K.

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are straightforward. For the limit-case, we invoke the claim that K is regular. Hence, if K and g( ) K for each < , then 8 U < g ( ) k . We conclude that g: K K. Second, we prove that for all and in K, if , then UTg() UTg() .Since every ordinal that is higher than a can be written as + for some Ord, we use transfinite induction on T T to show that U g() U g() for all K. The zero-case is trivial. Now take to be + 1 for some K, and assume thatUTg() UTg( + )• By definition, g( + + 1) = g( + ) + (g( + ))W . From the definition of (g( + )) and Lemma 4.D(2), it follows that UTg( + ) UTg( + +1)9 The inductive hypothesis and our last conclusion entail that UTg( ) UTg( + +1) . For the limit-case, = for some K, and assume that UTg() UTg( < . By Lemma 4.A(1), UTg( + )

let all

+

)

for

for some for some Since < , conclude that

+ 1 < UTg( ).

; and by the inductive hypothesis, we Hence the set {UTg( ): K} is linearly

ordered by ' '. But there are at most 2 s members in this set. Therefore, there exist an a and a such that < < K and

UTg()=UTg() .It follows that UTg() = UTg() = UTg() for every

between and . Hence, there is an K such that UTg( ) = UTg( +1). Let o be the least such a. Take 6 to be g( 0) and to be (g( o ) ) . We conclude, by the definitions of (g( o)) and g( 0 + 1), that 4.D. COROLLARY. For every c-sequence [E0,C], there exist a and a in K such that UT = UT +p for all Ord. PROOF: The corollary follows immediately from Theorems 4.C and 4.A. 8

As mentioned in the proof of the preceding theorem, K is regular because it is an infinite successor cardinal. 9 Recall that by the definition of

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The Alignment Point Theorem is an obvious consequence of the Loop Theorem. We first establish a link between the notions of characteristic loop and alignment point. 4.D. THEOREM. For every [ E 0 , C ] and all and in Ord , if UT = UT+p = UT+P then for each ordinal a, + pa is an alignment point of [E0, C]. PROOF: By Corollary 4.C, UT E +a for every ordinal a. Hence, UT st( [E0,C]). We now show that st [ E 0 , C ] ) U T . By Lemma 4.A(1),10 for some for some

But from Theorem 4. A, it follows that for every ordinal a, UT = UT +P . Therefore, UT . We conclude that st( [E 0 ,C]) = T U +p for al1 Ord. 4.E. THEOREM (THE ALIGNMENT POINT THEOREM), k is an alignment point of every c-sequence on any base model of C. PROOF: Let [E 0 ,C] be any c-sequence. The Loop Theorem ensures that there exist a 8 and a in K such that UT = UT+p . UT +p • Hence by Theorem 4.D, for every Ord, + is an alignment point of [E 0 ,C]. Since and are smaller than K, K = + K. Therefore, AC is an alignment point of [E0, C] 4.E. COROLLARY. For every if there is an ordinal

[E0, C],

such that

< K and

st( .[E0,C]) if and only E .

PROOF: The corollary is a trivial consequence of Definition 2.C (Chapter 2), Theorem 4.E, and Corollary 4.C. 10

Lemma 4.A(l) with a slight modification would make a parallel claim about st( [ E 0 , C ] ) .

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There are three applications of the Alignment Point Theorem that are of particular interest to us. The first two are discussed below and the third one in the next section. 4.F. THEOREM. Every system of stability semantics whose limit rule assignment admits only constant limit rules is formalizable in ZFC (Zermelo-Fraenkel set theory with the Axiom of Choice). PROOF: Let C be any limit rule assignment that associates the subsets of 5 with constant limit rules. We use ' [C]' to denote the system of stability semantics generated by C, and ' [E 0 ,C]
, E ) : ( , E ) is the

[E0, C] and 0 <

-term of

< K}

The Alignment Point Theorem entails that [E0, C] and [E0, C]
,

[G], is uc-nested in every

PROOF: Let [E 0 ,C] be any s-sequence in [C], and let EK be the extension of 'T' at the k-stage of [E 0 ,C]. Consider the gsequence [E'0,GE°] whose initial extension E'0 is identical with 11 See Definition 2.D (Chapter 2). The definition can easily be adjusted to cover the case under consideration. 12 Note that the size of [C]
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EK . By an argument similar to the one sketched in Section 3.1, where we showed that the G-system is uc-nested in the H-system, we can establish that the K-tail of [E 0 ,C] is identical with the g-sequence [E0,gE0].14 Hence, [E 0 ,C] and [E0,gE0] are semantically equivalent. It follows that every s-sequence in [C] is semantically equivalent to a g-sequence in [G]. As argued in Section 3.1, this entails that [G] is uc-nested in [C]. At first encounter, Theorem 4.G might seem surprising. With some reflection, however, one can see that it is an expected result. By its definition, Belnap's limit rule assignment B associates every subset of S with all limit rules. Hence every s-sequence is a bsequence and every system of stability semantics is a subsystem of Belnap's. In this sense, the B-system is the largest of all systems of stability semantics. As mentioned in Section 3.1, an immediate consequence of this fact is that the B-system is uc-nested in (and hence, uc-compatible with) every other system on the same base model. Gupta's limit rule assignment G does not associate every subset of S with every constant limit rule, and hence not every c-sequence is a g-sequence. However, G admits all constant limit rules, for the limit rule C that assigns the bootstrapper X to all limit ordinals is the same as Gupta's rule Gx. This means that for every constant bootstrapping policy, there is a g-sequence that follows this policy.15 It should be expected, therefore, that each csequence is semantically equivalent to some g-sequence. Thus, any system that only admits constant limit rules is, in a certain sense, embedded in the G-system on the same base model. It follows that Gupta's system on is uc-nested in (and hence, uc-compatible with) every [C]. We close this section with one last remark. If G* is the limit rule assignment that associates every subset of S with all constant limit rules, then one might expect that [G*] and [G] are "semantically equivalent." In one sense they are, and in another they are not. Their corresponding semantical categories are identical (e.g., si( [G]) = st( [G*])). However, some of their semantical 14

See also the proof of Lemma 4.C. For Belnap, a bootstrapping policy also involves the assignment of an initial extension E0 of 'T'. We use 'bootstrapping policy' to mean limit rule or restriction of a limit rule to a sequence of limit ordinals. 15

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features are different. The G*-system produces CHOP errors, while the G-system is free from this type of artifact.16

4.2. The Limit Rule Assignment V

As explained in Section 3.3, in order for an s-sequence to be a faithful representation of some revision hierarchy, its limit rule must satisfy the following requirements: Rl. It does not prematurely alter any truth declaration made at the initial stage or at a limit stage, so that every truth declaration that constitutes a genuine initial choice or a correction of a previous error is allowed to persist, and hence, to realize its full impact on the outcome of the process. R2. It nullifies all effects and corrects all errors that are produced by an excessively arbitrary initial extension or a bootstrapper. Our goal now is to define a limit rule assignment that associates every subset of S with limit rules that satisfy the requirements above. 4.B. DEFINITION. Let S be the set of all £-sentences the least cardinal that is larger than P(S) (i.e., K = The limit rule assignment V associates every subset E0 every limit rule V (V: Ord P(S)) that meets the conditions:

and K be P(S) +). of S with following

Nl. For every limit ordinal , if < K, then V = E 0 ', N2. For every limit ordinal , if k, then V = VK , where is the unique ordinal such that K < K( + I);17 N3. For all X, Y, and Z in P ( S ) and each £ in Ord, there is 16

Observe that the h-sequence [{Tg } , H ] , which was discussed in Section 3.3, is a sequence in [G*], but it is not a g-sequence. See Table 3.N. 17 Recall that for each Ord, there are two unique ordinals ' and " such that = k ' + ", where 0 < " < K. Hence for each Ord, there is a unique such that k ' < k( ' + 1).

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an ordinal

, and for all

such that K

(

X, y, Z,

if if if

127

,

is even is odd =

Condition Nl says that the initial extension £0 is the bootstrapper assigned to and to every limit ordinal between and K. Condition N2 requires that a bootstrapper introduced at limit stage must be employed at all limit stages between and + K. Hence, these conditions ensure that V satisfies the requirement Rl, for the Loop Theorem and the Alignment Point Theorem entail that any bootstrapper maintained in a segment of length K must achieve its full impact on the process in that segment. Condition N3 is designed to equip V with the kind of correcting mechanism that R2 requires. Let us explain how N3 fulfills this function. The formal process that V transcribes views the class of all ordinals as being composed of K segments arranged in a manner that is similar to (and indeed, is induced by) the well-ordering on Ord. Thus every k-segment is either the initial segment, a successor segment, or a limit segment. As mentioned in the preceding paragraph, N2 requires every K-segment to employ a constant bootstrapping policy, and Nl reserves the initial K-segment for the bootstrapper E0. Condition N3 assures that every bootstrapper and every combination of two or three bootstrappers have free and equal access to the process. Hence, it prevents any systematic bootstrapping policy from dominating the whole process. The condition achieves this goal by requiring that after any stage each bootstrapper and each combination of two or three bootstrappers must take control of at least one -sequence of successive Ksegments, and of their limit segment as well.18 Thus, every such bootstrapping policy is given unlimited access to the process: it enters after each stage, exerting its influence and reversing some of the residual effects of previous bootstrapping policies. Every constant limit rule is an example of a systematic bootstrapping policy. All s-sequences that are generated by constant 18 By taking all, two, or none of X, Y, and Z to be identical, we can see that the existence of each of these bootstrapping policies follows from N3.

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limit rules are, in general, infected with artifacts. For example, every such s-sequence renders ' Tarskian, and hence it renders T ) stably true.19 Indeed, these types of artifacts and similar ones are epidemic in systematic bootstrapping policies. Here is a telling example. Let = |P(S)| and K = +. For every E0 S, we take W to be a one-to-one correspondence between and P ( S ) such that W(0) = E0. We now define the , limit rule W as follows: Ql. For every limit ordinal , if < K, then W = W(0); Q2. For every limit ordinal , if K, then W = WK , where is the unique ordinal such that K < K( + 1); Q3. For every ordinal 1, WK = W( ), where is the unique ordinal such that 0 < n and = 1 + + for some Ord. We shall refer to W as a list of all bootstrappers and to W as the limit rule induced by W. Ql and Q2 are the same conditions as Nl and N2. Like N3, the condition Q3 entails that after each stage of the process, every subset of S must be used as the bootstrapper in a k-segment. But unlike N3, Q3 requires the list W to dominate the process. In fact, Q3 is inconsistent with N3: every limit rule that satisfies the former condition cannot satisfy the latter. The bootstrapping policy that Q3 gives rise to is a very systematic policy: it employs the bootstrapper W(0) in the initial K-segment, W(1) in the second, W(3) in the third, and so on until the list W is exhausted, and then it repeats the same process indefinitely using the same list W. The condition Q3 equips W with a powerful correcting mechanism. Indeed, all the erroneous verdicts discussed in Sections 3.2 and 3.3 are corrected by W. This type of limit rule, however, is not free from artifacts. Let 9 be the sentence ' and be ( T ) V (T T2 ). Consider the limit rule W induced by a list W that satisfies these two conditions: W(0) = E0 = { , T , T2 }, and W(r) E0, for every limit ordinal < Take [ E 0 , W ] to be the s-sequence generated by (E0,W). The , sentences , T , T2 T , and T T2 are all 19

See Section 3.1 and Tables 3.B and 3.C.

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paradoxical in [E0; W], but is stably true in this s-sequence. This verdict is an artifact of the systematic bootstrapping policy that W represents. Of course, there is no justification for this verdict; is simply a paradoxical sentence that is forced into the category of stable truths by a persistent systematic treatment of , T , andT 2 . Before explaining how the correcting mechanism furnished by N3 deals with these and other types of artifacts, we give the definition of the revision system of stability semantics. 4.C. DEFINITION. Let be any base model of L. The revision system on (denoted by ' [V]') is the system of stability semantics that is generated by the limit rule assignment V. We call the s-sequences in this system the v-sequences, and we use ' [E0, V]' to denote the v-sequence generated by (E 0 , V). As usual, we also refer to the revision system as the V-system. All the sentences mentioned in Sections 3.2.1 and 3.2.2 are classified as paradoxical in [V]. Let us discuss a representative example. Consider the sentences

and

Every v-sequence on renders these sentences paradoxical. In order to see that this is so, take X to be the empty set, Y to be { }, and Z to be { , }. For any [E0,V], conditions N2 and N3 entail that after each ordinal , there is an -sequence of K-segments such that X is the bootstrapper in every k-segment that is an evennumbered term of that -sequence, Y is the bootstrapper in the odd-numbered terms, and Z is the bootstrapper in the k-segment that is the limit of that -sequence. To simplify matters, we introduce some notation. Let 'K ' denote any such sequence that comes after £, and 'K ' denote the n-term of K . 'K ,' stands for the K-segment that is the limit segment of K . By definition, every K (where 0 ) is a K-segment of the v-sequence [E0, V]. N2 requires that every K employs a constant bootstrapping policy. Thus, we use 'br(K )' to represent the bootstrapper employed

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The Liar Speaks the Truth

in K . We also take K ( ) to be the -term of Kg and E[K ( )] to be the extension of 'T' in K ( ).20 Using this notation, we restate our last conclusion as follows: for every [E0, V] and every Ord, there is a K such that X — br(K ), if n is even y = br(K ), if n is odd

Z = br(K ) The discussion in Section 3.1 and Tables 3.B and 3.C make it clear that for each n < , the sentences , T , and T2 are stably true in K 2n and stably false in K 2n+1 . Hence, these sentences are unstable in the sequence K . Given that Z = br(K ) = { , }, it follows that A and 0 are in E[K (0)] but ', T , and T2 are outside E[K (0)]. This shows that T , T T2 , and are evaluated false in K ,(0). These sentences, however, are stably true in K for all n < .21 We conclude that , ', , T , T2 , T , T T2 , and V are all paradoxical in [E0, V]. Hence, they are paradoxical in [V]. All the cyclic Truthtellers and the T-forms in the Truthteller cycles that were discussed in Section 3.2.3 are treated in the revision system as Truthtellers: they are all -capricious in [V]. The circular invariant sentences mentioned in Section 3.2.4 are all correctly classified as invariant. The sentences 'Tl and 'Tl Tm' (where 'm' refers in to 'Tl' and 'l' to 'Tl Tm') are both stably true in [V]. The sentences 'Tl T2 l' (where 'l' refers in 2 ' to 'Tl T l') is stably true in '[V]. When an initial extension E0 presents some contradictory truth declarations that can be corrected in one way only, every [ E 0 , V ] corrects these initial declarations in the same way. However, if there are two ways in which such truth declarations can be corrected, then some of the v-sequences starting from ( , E0) correct these declarations in one way and the rest in the other. As an example, consider the Truthteller cycle (Tj,Ts) described in Section 3.2.3.22 If E0 = {Tj}, 20 Note that K ( ) = ( ,E ), where = k + K + , K , and 0 < K. (See Definition 4.B.) Thus, E[K ( )] is simply E . 21

Interestingly enough,

is stably true in K , while 2

is evaluated false in K (0) and T T 22 Since (Tj,Ts) is a cycle, 'j' refers in

T

O , and T

is evaluated false in K (l). to 'Ts' and 'S' to 'Tj'.

,

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then E0 presents a contradictory truth declaration. This error can be corrected in two ways: placing 'Ts' in the extension of 'T' together with 'Tj', or placing 'Tj' outside the extension together with 'Ts'. Thus, some of the v-sequences starting from ( , {Tj}) render 'Tj' and 'Ts' stably true and the rest of them render these sentences stably false. The sentence 'Tg ', whose semantical behavior in the hsequence [{Tg } , H ] was discussed at the end of Section 3.3,23 is stably true in every v-sequence in which it is initially declared true, and it is stably false in every v-sequence in which it is initially declared false. Thus, no v-sequence makes a CHOP error in determining the truth status of 'Tg '. We can be certain that the V-system is free from all CHOP errors. This type of artifact can only be produced by a bootstrapping policy that hastily alters truth declarations assigned at the initial stage or at some limit stages. The conditions Nl and N2 of Definition 4.B ensure that all truth declarations are given sufficient consideration. I am not aware of any CRASS errors in the revision system. I believe that this system is free from all types of artifacts. It is very hard to imagine that some limit rule satisfying the conditions Nl, N2, and N3 would generate an erroneous verdict or permit one to persist.

4.3. Stability Logic In this section we carry out our final project: to define a notion of logical consequence that is appropriate for revision semantics. More precisely, our project is to explain what it means to say that is a, logical consequence of , where is any L-sentence, is any set of L-sentences, and £ is any uninterpreted formal language whose syntax is of the type defined in Section 2.1 and whose possible interpretations are all the V-systems on its base models.24 Throughout this section, we use the metavariable ' ' to range over the class of all base models of L, 'V over the class of all limit rules admitted by V, ' ' and ' ' over S (the set of all Lsentences), and ' ' and 'Eo' over P(S). We also use the standard 23

See also Table 3.N. In other words, for every I, I is a possible interpretation of £ if and only if there is a base model of L such that I is [V]. 24

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132

notation ' =' to denote the relation of logical consequence. As in Section 4.1, ' ' abbreviates 'if and only if and ' ' abbreviates 'only if. Let us first consider the simpler case of a (nonempty) finite . Since each finite collection of premises can be represented by the conjunction of its members, we first attempt to define the relation of logical consequence between single sentences. We start by advancing four proposals. for every for every and for every

st(

for all

, E0, and

for all

, E0, and

(E0,V]) and for all

sf( '[E 0 ,V'])

and

sfl

Each of these proposals has its rationale. Definitions (la) and (2a) are motivated by the following consideration: since every ( , E0) of any v-sequence is a classical model of L, it seems natural to define ' ' for revision semantics in a manner similar to the way in which it is traditionally defined in classical logic. The motivation behind definitions (lb) and (2b) is this: given that 'stably true' is a complex predicate, and hence, that being stably false is not simply failing to be stably true, it might be more appropriate to define ' ' for revision semantics in a way similar to that in which it is usually defined in multiple-valued logics. The difference between group (1) and group (2) is that in the latter the possible interpretations of L are decomposed into their basic units: the v-sequences. There are two problems common to these four proposals. First, each of them fails to distinguish between certain semantical categories despite the fact that revision semantics draws clear boundaries between these categories. According to definition (la), capricious, paradoxical, and logically false sentences entail all sentences. For example, the following claims are true according to (1a). PI. For all

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P2. For all P3. For all Definition (lb) invalidates P1 and P2. However, according to (1b), capricious and paradoxical sentences entail each other. For example, a Truthteller entails a Liar and vice versa, i.e.,

P4. P5. Definition (2a) invalidates P1 and P4 but it implies P2 and P5. (2a) treats paradox on a par with contradiction: each entails all sentences. Definition (2b) makes ' ' more sensitive to semantical categories than any of the previous definitions. Still, according to (2b), paradoxical sentences entail each other indiscriminately:

P6. P7. The second problem is that each of these definitions invalidates certain important forms of deductive reasoning. Here is a sample of logical forms that are rendered invalid by some or all of them. P8 and P9 below follow from each of these definitions.

P8. P9.

but

Therefore, Conditional Argument and Constructive Dilemma fail for (la), (lb), (2a), and (2b). Definitions (la) and (2a) also invalidate reductio ad absurdum, for from either one it follows,

and and

P10. Pll.

and

Definitions (lb) and (2b) render Conjunction Argument invalid:

but A careful examination of the previous proposals reveals a certain feature that all of them share. The meta-level conditional ' ' used in each of them to define ' ' has material truth conditions. The object-level conditional ' ', however, has stability truth conditions. It is this feature, perhaps, that explains why none of these proposals succeeds in defining a notion of logical

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consequence that is congruous to revision semantics.25 Let us put this speculation to the test. Here is our plan. First, we define a relation of logical consequence between single sentences in a way that forces the meta-level conditional ' ' to have stability truth conditions. Second, we generalize the definition to an arbitrary set of L-sentences. Third, we study some of the important features that this relation of logical consequence has in order to confirm its material adequacy. In the remainder of this section, the proofs of all the corollaries are omitted, for all of those proofs are straightforward arguments from the relevant definitions. 4.D. DEFINITION. For all and , is stably true in [V] for every .

if and only if

In the corollary below, ' [E0, V] ' denotes the of the v-sequence [E0, V]. 4.F. COROLLARY. For all equivalent,

and

-term, ( , E ),

, the following statements are

(1)

(2) For every [E0, V], is stably true in [E0, V]. (3) For every [E0,V], there is an ordinal such that for each is evaluated true in [ E 0 , V ] is evaluated true in [E0, V] . (4) is stably false in every [V],

Part (3) makes it clear that the meta-level conditional ' ' that is implicit in Definition 4.D has stability truth conditions. The last statement in the preceding corollary shows a certain parallel between this notion of logical consequence and its classical counterpart: in classical logic, 'is a logical consequence of ' if and only if is false in all interpretations of { ', '}. Part (4) also suggests a way of generalizing Definition 4.D. We first introduce certain terminology. At least one thing is certain: Conditional Argument must fail under these circumstances. It merits noting that each of the logics of truth advanced by Kremer (1986 and 1988) and by McGee (1985a and 1991) invalidates Conditional Argument.

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4.E. DEFINITION. For all ,

135

, E0, and V,

is satisfied in [E 0 ,V] (or [E0, V] satisfies ) if and only if there exists an ordinal such that for every and every , is evaluated true in [E0, V]a; (2) is refuted in [E0, V] (or [E0, V] refutes ) if and only if there exists an ordinal such that for every , there is a , and is evaluated false in [E0, V]; (3) is satisfied in [V] (or [V] satisfies ) if and only if every v-sequence in [V] satisfies ; (4) is refuted in [V] (or [V] refutes ) if and only if every v-sequence in [V] refutes .

(1)

4. G. COROLLARY. For all finite , (1) if is empty, then it is satisfied in every [V]; (2) if is not empty, then for each , is satisfied [V] A is stably true in [V], and is refuted in [V] A is stably false in [V].26 4.F. DEFINITION. For all and is refuted in every [V].27

,

U{

if and only if

}

The notion of logical consequence defined above is congruous to the semantical theory on which it is based, ' ' is very sensitive to the differences between the semantical categories. Furthermore, it validates all classical forms of deductive reasoning. In the corollary below we sample some of the important features that ' ' has. Following the usual convention, we write ' ' instead of ' ', ' ' instead of '{ } ', and ' , ' instead of ' U 4.H. COROLLARY. Let any L-sentences. (1) If (2) (3) 26

and

be any subsets of , and

and L is stably

is a nonempty finite set, then is stably true in every is stably false in every

[V] [V]

. for

all .

' ' denotes the conjunction of all the sentences in . Notice the parallel to classical logic: ' is a logical consequence of ' if and only if ' { } 'is unsatisfiable. 27

The Liar Speaks the Truth

136

(4)

for each

(5) If (6) If

, ,

(7) ,

,then E for some , then for some , then

.

; and if .

(8) For any L-name , ; and if is distinct from , then . (9) If and ' are any L-names and occurs in , then { = ', } , where is tie sentence obtained from by replacing one or more of the occurrences of in with '. (10) For every L-name , , where [ ] is the sentence obtained from by replacing all the free occurrences of 'x' in with . (11) If and is a nonquotational L-name that occurs genuinely in but not in any member of , then [], where [x ] is the formula obtained from the sentence by replacing all the genuine occurrences of in with 'x'.28 Part (1) shows that Definition 4.D is a special case of Definition 4.F. Part (3) indicates that ' ' does not confuse paradoxical sentences with contradictions.29 Parts (4) through (11) show that ' ' validates all classical rules of inference, for each of the classical logic connectives and quantifiers can be expressed in terms of ' ', ', and ' '. The next corollary represents a feature of ' that is of particular significance to the problem of semantical paradox. We first introduce certain terminology. 4.G. DEFINITION. For all and , is -Tarskian if and only if is Tarskian in all the v-sequences that do not refute .30 28

Recall that an occurrence of a constant in a sentence is genuine if it is not inside any quotational name (see Section 2.1). 29 See P2 above. 30 Observe that according to this definition if is refuted in each [.E0, V], then every L-sentence is -Tarskian.

The Revision System

137

4.I. COROLLARY. For all and , if is T-Tarskian, then and for all nonzero natural numbers n and m. Given Corollary 4.H(7), the first part of the preceding corollary entails that for every that is -Tarskian, , T and ,T . This means that ' ' validates the rules of T-introduction and T-elimination for all those sentences that are paradox-free relative to . ' ' invalidates these rules for sentences that exhibit paradoxical behavior in some v-sequences that do not refute . For example.

By invalidating these rules for such sentences, ' ' blocks all Liar and Liar-like arguments. Hence, it prevents the semantical paradoxes from generating contradictions. Finally, we prove a negative result. We show that ' ' is not compact, i.e., there exist a and a such that is a logical consequence of , but it is not a logical consequence of any finite subset of . 4.H. THEOREM. The relation of logical consequence introduced in Definition 4.F is not compact. PROOF: We give two counterexamples.31 (1) Let = {c : S}. The members of are all T-free sentences. Thus, they receive truth values in the base models of L. Now for any , if assigns the truth value 'true' to all the sentences in , then V must refer in to a nonsentence. Since the nonsentences are always outside the extension of 'T', ' Tc' is evaluated true in . It follows that Tc. However, if 0 is any finite subset of , then 0 Tc, for we can construct a base model in 31

The first counterexample is modeled alter one used by Kremer (1986).

138

The Liar Speaks the Truth

which 'c' refers to a true sentence and all of 0'S members are evaluated true. (2) Let ( 0, 1, 2, • • •) be a denumerable sequence of distinct nonquotational L-names. Take £ to be the following subset of S: { n= Tn+1 : n N}. For any base model in which all of 's members are evaluated true, T n is tfcapricious in [V] for each n N; for there are only two consistent ways of assigning truth values to the members of £: either they are all true or they are all false. Hence, |= T 0 T2 O. But there is no finite subset 0 of £ such that 0 T0 T2 0- Any finite subset 0 of £ must have an "open end." Thus, we can construct a model in which all the members of 0 are true and 0 refers to a paradoxical sentence. We call the logic whose logical consequence relation is as defined in Definition 4.F stability logic. Theorem 4.H shows that stability logic is incomplete, i.e., it cannot have a complete proof theory, as long as proofs are considered to be finite derivations. However, we raise two open questions. (1) If we permit the proof theory to include an infinite rule (i.e., a rule that requires infinitely many premises), can we prove a completeness theorem for stability logic, given an appropriate choice of deductive rules? (2) Is stability logic weakly complete? In other words, if we consider only finite collections of premises, can we prove a completeness theorem in this case? The answers to these questions, I believe, depend on whether it is possible or not to characterize recursively the property of being -Tarskian, where is a decidable set of L-sentences. Despite its incompleteness, stability logic is an adequate logic of truth. It respects the (semantical) modes of argument that are invoked in intuitive reasoning involving the use of the truth predicate. It blocks those arguments that engender paradox by invalidating the only reasonable suspect: the unrestricted application of the T-introduction and T-elimination rules. I find quite strange the suggestion that the culprit behind the Liar argument is Conditional Proof or some other classical schema. Is it not more reasonable and natural to suspect the new recipe that has been used

The Revision System

139

with the introduction of the additional ingredient 'true'? Stability logic is incomplete because it doesn't compromise. It maintains the expressive integrity of the language L and its logical power. We might be able to purchase completeness if we were willing to prevent many logical consequences of from being considered as such. We could have imposed an expressive limitation on £ so that many of 's consequences are no longer expressible in the language, or we could have reduced its logical power to prevent from semantically implying those consequences that challenge the proof theory of the underlying logic. We chose not to follow any such strategy, and thus we had to let go of completeness. Throughout this work this has been our policy: to let the Tarskian biconditionals tell their story about the concept of truth and to accept what they tell as true until something to the contrary is found. But what we found is remarkable: a consistent and materially adequate theory of truth. We had to give up, however, certain things on the way. The metaphysical picture of truth as a supervenient property is replaced with a portrait of a circular irreducible property. The tradition of having a complete logic is left unfulfilled. It is a philosopher's decision whether the revision theory of truth with its unmatched record of consistency and adequacy is worthy of this price or not. I believe it is (see the title of this book).

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Bibliography

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Horwich, P. [1990]. Truth, Basil Blackwell, Oxford. Jech, T. [1978]. Set Theory, Academic Press, New York. Kearns, J. T. [1970]. "Some Remarks Prompted by van Fraassen's Paper," in R. L. Martin (ed.), The Paradox of the Liar, second edn., Ridgeview, Reseda, 1978, pp. 47-58. First edition published in 1970. Kremer, M. [1986]. "Logic and Truth," Ph.D. dissertation, University of Pittsburgh, Pittsburgh. Kremer, M. [1988]. "Kripke and the Logic of Truth," Journal of Philosophical Logic 17: 225-278. Kripke, S. [1975]. "Outline of a Theory of Truth," in R. L. Martin (ed.), Recent Essays on Truth and the Liar Paradox, Oxford University Press, Oxford, 1984, pp. 53-81. Originally printed in The Journal of Philosophy 72 (1975): 690-716. Kripke, S. [1976]. "Is There a Problem about Substitutional Quantification?" in G. Evans and J. McDowell (eds.), Truth and Meaning: Essays in Semantics, Oxford University Press, Oxford, pp. 325-419. Kunen, K. [1980]. Set Theory: An Introduction to Independence Proofs, Elsevier, Amsterdam. Lewis, D. [1975]. "Languages and Language," in K. Gunderson (ed.), Minnesota Studies in the Philosophy of Science, vol. 7, University of Minnesota Press, Minneapolis, pp. 3-35. Magee, B. [1978]. Men of Ideas: Some Creators of Contemporary Philosophy, Oxford University Press, Oxford. Martin, R. L. [1967]. "Toward a Solution to the Liar Paradox," The Philosophical Review 76: 279-311. Martin, R. L. [1970]. "A Category Solution to the Liar," in R. L. Martin (ed.), The Paradox of the Liar, second edn., Ridgeview, Reseda, 1978, pp. 91-112. First edition published in 1970. Martin, R. L. [1976]. "Are Natural Languages Universal?" Synthese 32: 271-291. Martin, R. L., and P. W. Woodruff [1975]. "On Representing 'True-in-L' in L," in R. L. Martin (ed.), Recent Essays on Truth and the Liar Paradox, Oxford University Press, Oxford, 1984, pp. 47-51. Originally printed in Philosophia 5 (1975): 213-217.

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McGee, V. [1985a]. "Truth and Necessity in Partially Interpreted Languages," Ph.D. dissertation, University of California, Berkeley. McGee, V. [1985b]. "How Truthlike Can a Predicate Be? A Negative Result," Journal of Philosophical Logic 14: 399-410. McGee, V. [1989]. "Applying Kripke's Theory of Truth," The Journal of Philosophy 86: 530-539. McGee, V. [1991]. Truth, Vagueness, and Paradox, Hackett, Indianapolis. Parsons, C. [1974]. "The Liar Paradox," in R. L. Martin (ed.), Recent Essays on Truth and the Liar Paradox, Oxford University Press, Oxford, 1984, pp. 9-45. Originally printed in the Journal of Philosophical Logic 3 (1974): 381-412. Pollock, J. L. [1977]. "The Liar Strikes Back," The Journal of Philosophy 74: 604-606. Putnam, H. [1978]. Meaning and the Moral Sciences, Routledge & Kegan Paul, London. Quine, W. V. [1962]. "The Ways of Paradox," in The Ways of Paradox and Other Essays, revised and enlarged edn., Harvard University Press, Cambridge, Massachusetts, 1976, pp. 1-18. Originally printed in Scientific American 206 (Apr. 1962): 84-96. Quine, W. V. [1970]. Philosophy of Logic, second edn., Harvard University Press, Cambridge, Massachusetts, 1986. First edition published in 1970. Quine, W. V. [1974]. "Truth and Disquotation," in The Ways of Paradox and Other Essays, revised and enlarged edn., Harvard University Press, Cambridge, Massachusetts, 1976, pp. 308-321. Originally printed in the Proceedings of Symposia in Pure Mathematics 25 (1974): 373-384. Simmons, K. [1990]. "The Diagonal Argument and the Liar," Journal of Philosophical Logic 19: 277-303. Soames, S. [1984]. "What Is a Theory of Truth?" The Journal of Philosophy 81: 411-429. Suppes, P. [1972]. Axiomatic Set Theory, Dover, New York. Tarski, A. [1931]. "On Definable Sets of Real Numbers," in Logic, Semantics, Metamathematics, second edn., translated by J. H. Woodger, edited by J. Corcoran, Hackett, Indianapolis, 1983, pp. 110-142. First edition published in 1956 by Oxford

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Symbols

T A T £ Voc(L) S P(S) 1

1

Eo R H GE° B C V R H G B V [E 0 ,R]

Truth predicate, 44 Liar, 53 Truthteller, 53 Formal language, 43-44 The vocabulary of L, 45 The set of all L-sentences, 45 The power set of S, 48 Quotational names, 44-45 The composition of 1 and 1, 46-47 Base model of C, 47 Initial extension of T, 48 Limit rule, 48 Herzberger's limit rule, 53-54, 70-71 Gupta's limit rule, 71 Belnap's limit rule, 71-72 Constant limit rule, 116 Revision limit rule, 126-27 Limit rule assignment, 50 Herzberger's limit rule assignment, 53, 71 Gupta's limit rule assignment, 71 Belnap's limit rule assignment, 71-72 The revision limit rule assignment, 126-27 The s-sequence on generated by ( E 0 , R ) , 48, 50 147

Symbols

148 [E 0 ,H] [E 0 ,G E 0 ] [E0,B] [E0,C] [E0,V] [R]

H]

[G] [B] [V] St(

[E0,R])

sf([E0,R]) px( [E 0 ,R])

st( [R]) sf( [R]) px( [R]) tf( [R]) tp( [R]) fp( [R])

tfp( [R]) cpr( [R]) inv( [R]) unf( [R]) trs( [R]) 7 Ord Ord K K

uT U U

The The The The The The The The The The The

h-sequence on generated by ( E 0 , H ) , 71 g-sequence on generated by (E 0 ,G E o ), 71 b-sequence on generated by ( E 0 , B ) , 72 c-sequence on generated by ( E 0 , C ) , 116 v-sequence on generated by (E0, V), 129 R-system on , 61 H-system on , 71 G-system on , 71 B-system on , 71-72 V-system on , 129 set of all stably true sentences in [E0, R]

59-60

The set of all stably false sentences in [ E 0 , R ] , 59-60 The set of all paradoxical sentences in [ E 0 , R ] 59-60 The set of all stably true sentences in [R], 61 The set of all stably false sentences in [R], 61 The set of all paradoxical sentences in [R], 61 The set of all tf-capricious sentences in [R], 61 The set of all tp-capricious sentences in [R], 61 The set of all fp-capricious sentences in [R], 61 The set of all tfp-capricious sentences in [R], 62 The set of all capricious sentences in [R], 62 The set of all invariant sentences in [R], 62 The set of all uniform sentences in [R], 62 The set of all Tarskian sentences in [R], 62 Limit ordinal, 49 The class of all ordinals, 48 The class of all limit ordinals, 48 The least cardinal larger than p(S) , 114 The set of all limit ordinals less than K, 114 The set of all stably true sentences at 7, 49 The set of all stably false sentences at 7, 49 The set of all 7-unstable sentences, 115

Index

Ackrill, J. L., 6n Alignment points, 73-74, 96, 116, 123 Alignment Point Theorem, 114, 123-24, 127 Alternating limit rule, 78 Arbitrariness, 32, 35 initial excessive, 93-94, 96-98, 113 Aristotle, 6, 141 Artifacts, 5, 70, 79-81, 91, 93-94, 96, 98, 113, 128, 131 Austin, J. L., 7, 141 Axiom of Choice, 120, 124 Ayer, A. J., 4, 12-14, 18, 28-29, 31, 33, 141

Belnap's limit rule assignment, 71, 93-94, 125, 147 Belnap's system, 5, 47, 52, 7071, 75n, 78n, 80-81, 85, 8788, 90, 98, 125. See also B-system Bivalence, 33-36, 38, 40 and the law of excluded middle, 33n Bivalent language, 3-4, 27, 33n, 34-36, 38-42, 90 Bootstrapper, 52, 70, 74, 91-92, 126 Bootstrapping, 91, 125, 127-28 Burge, T., 22n, 33, 141 c-sequences, 116, 119, 121-23, 125, 148 Cairns, H., 6n, 143 Capricious sentences, 61-62, 92, 148 fp-, tf-, tfp-, tp-, 61-62 Cartwright, R., 7, 141 Categories, 16, 29, 30n semantical, 5, 59-62, 75n, 92, 124, 132, 135 Characteristic loop, 119, 121, 123 CHOP errors, 96-98, 113, 126, 131

b-sequences, 72, 80, 125, 148 B-system, 71-72, 80, 125, 148. See also Belnap's system Barnes, J., 6n, 141 Barwise, J., 7, 25n, 87n, 141 Bearers of truth, 7-9, 30 Belnap, N. D., Jr., 5, 14n, 18, 47, 52, 70-72, 75n, 76n, 77n, 78n, 80-81, 83, 85, 87-88, 90, 93-94, 98, 114n, 125, 141, 147 149

150

Church, A., 22n, 142 CI sentences. See Circular invariant sentences Circular concept, 36, 38 Circular invariant sentences, 8788, 130 Circularity argument, 27, 36, 3842 Circularity of the concept of truth, 4, 27, 36-41, 90, 98, 139 Closure points, 73n. See also Alignment points Composition of T-forms, 46, 147 Constant limit rule, 53-54, 69n, 80, 114, 116, 124-25, 12728, 147 Constant tail of an s-sequence, 56-57 Correspondence notion of truth, 7, 11 CR sequences. See Crossreferential T-form sequences CRASS errors, 96-98, 113, 131 Cross-referential T-form sequences, 83-85, 88 Davidson, D., 46n, 142 Deflationary conception (theory) of truth, 3, 42 Disquotation, 7 Enderton, H. B., 25n, 142 Etchemendy, J., 7, 87n, 141 Field, H., 7n, 23n, 142 Fine, K., 26n,142 Fixed extension, 5, 39-41 Fixed point, 55-58, 75n classical, 56-57 Frege, G., 4, 11-14, 142-43

Index Fundamental intuition about truth, 4, 6-7, 9, 19, 27-29, 42 formulations of, 6—7 g-sequences, 71, 74, 93, 124-25, 148 G-system, 71, 125-26, 148. See also Gupta's system G6del, K., 21n, 143 Grounded sentences, 33, 38, 48, 82 Grover, D. L., 12, 14-18, 33-34, 143 Gupta, A., 4n, 5, 7n, 11, 16, 20, 21n, 22n, 28, 30, 34n, 36, 47, 52, 57n, 58, 70-73, 75n, 77n, 78n, 80-81, 83, 85, 87-88, 90, 93, 98, 114, 124-25, 143, 147 Gupta's limit rule assignment, 93, 125, 147 Gupta's system, 4n, 5, 47, 52, 70-71, 73, 75n, 78n, 80-81, 85, 87-88, 90, 98, 124-25. See also G-system h-sequences, 53-54, 71, 73-74, 95-96, 116, 148 H-system, 53-54, 69n, 70-71, 81-82, 86-87, 92n, 95n 125, 148. See also Herzberger's system Hamilton, E., 6n, 143 Herzberger, H. G., 5, 21, 47, 5253, 62n, 70, 72-73, 75n, 76, 80, 82-83, 85-88, 90, 94-96, 98, 114n, 143, 147 Herzberger's limit rule assignment, 94, 96, 147 Herzberger's system, 5, 47, 5253, 70, 73, 75n, 80, 82-83,

Inde 85-88, 90, 95n, 98. See also H-system Horwich, P., 7, 9n, 14, 28-30, 144 Invariant sentences, 62, 82, 148. See also Circular invariant sentences Jech, T., 121n, 144 Jowett, B., 6n Kearns, J. T., 16n, 144 Kelley, J., 97n Kremer, M., 34n, 134n, 137n, 144 Kripke, S., 10n, 11, 22n, 25n, 34, 57, 95n, 144 Kunen, K., 115n, 120n, 144 Lewis, D., 22, 23n, 144 Liar, 3, 13, 15, 19-20, 31, 32n, 35, 40-41, 53, 55, 57-58, 63, 76, 79-82, 85n, 86, 90, 94, 133, 147 cross-referential, 31, 36, 82-83 cyclic, 82-83, 85 self-referential, 13, 31, 36, 82, 83n Liar argument, 19-20, 34, 137-38 Liar cycles, 83, 85 Limit rule, 48-50, 52, 55, 70, 91, 93, 147 Limit rule assignment, 50, 52, 59, 60n, 72, 93, 147 Logical consequence, 6, 22n, 34, 113, 132-35, 137-38 Loop Theorem, 114, 117, 121-24, 127 Magee, B., 13n, 144 Martin, R. L., 16n, 21, 34, 57, 144

151 Materially adequate definition, 26 McCarthy, T., 26n, 142 McGee, V., 16n, 20n, 73n, 75n, 78n, 134n, 145 Minimalist conception (theory) of truth, 29 Model, base, 47-50, 57, 72, 90-94, 147 classical, 48, 50, 132 partial, 47-48 Negation-complexity, 46, 82-83 Nominalism, 17 Noncreative semantics, 35 Nonsemantical facts, 4, 32, 3839, 41, 48, 50, 90-92 Nonsentences, 30, 51, 137 Paradox-free sentences, 55, 62, 137 Paradoxes, 15-17, 19-20, 34, 133, 136-38 Paradoxical sentences, 57, 59-62, 148 Paradoxicality, 32, 35 Parsons, C., 9n, 145 Plato, 6, 143 Pollock, J. L., 14n, 145 Power set, 48, 147 Proper classes, 69n, 124 Prosentences, 14 Putnam, H., 7n, 145 Quine, W. V., 7n, 22n, 25n, 145 Quotational names, 18, 43-45, 47, 147 R-system, 50, 59-63, 72, 148 Regular cardinal, 120-22 Revision hierarchies, 39-41, 48, 57n, 90-91, 126

Index

152

Revision process, 4-5, 27, 36, 38n, 39-41, 43, 47, 69-70, 90-94, 95n Revision semantics, 131-32, 134 Revision system, 5, 70n, 113-14, 129, 131. See also V-system Revision theory of truth, 4-6, 11, 17, 27, 36, 42, 69, 89-90, 113, 139 consistency of, 3, 5, 69, 139 material adequacy of, 3, 5-6, 35, 89-90, 113, 139 Ross, W. D., 6n s-sequences, 48-50, 52, 54, 5657, 59-60, 72, 90-91, 11516, 125-26, 147. See also Stability sequences Satisfaction, 21-23, 25, 47 Self-referential T-forms, 82, 8485, 88 Semantically closed language, 20-21 Semantically equivalent sequences, 60, 74, 95, 12425 Set theory, 26n, 69n Zermelo-Fraenkel, 124 Simmons, K., 21, 145 Soames, S., 22, 23n, 145 SR T-forms. See Self-referential T-forms Stability logic, 6, 113, 131, 13839 Stability semantics (systems of), 5, 34, 43, 47-50, 52-54, 59, 60n, 69n, 70, 72, 80, 85, 91,

95n, 125 Stability sequences, 47-48, 5354. See also s-sequences Stability tables, 58, 64-68, 99112

Stable sentences, 59 Stably true (false) sentences, 5961, 76n, 95, 114, 132, 148 Successor rule, 48, 50-52, 56-57, 91-93, 94n Supervenient property, 17, 39, 42, 139 T-elimination (-introduction), 137-38 T-forms (pure), 46-47 T-free sentences, 46-47 T-sentences, 46 Tarski, A., 4, 12, 19-26, 28-29, 31-32, 47, 145-46 Tarskian biconditionals, 3-4, 1920, 24, 26, 30-35, 37-40, 46n, 90 definitional vs. material, 28-29 Tarskian hierarchy, 21, 22n Tarskian rules, 91, 93-94, 97 Tarskian schema, 3-4, 6-7, 9-13, 27-33, 35, 37, 41-42, 50-52, 56-57, 89-91 Tarskian sentences, 62, 148 F-, 136-38 Tharp, L. H., 25n, 146 Truthteller, 31, 35-36, 40-41, 53, 57-58, 60, 82, 85, 87, 90, 92, 133, 147 cross-referential, 31, 36, 82-83, 85, 86n, 87 cyclic, 82-83, 85-86, 130 self-referential, 31, 36, 82, 83n Truthteller cycles, 82-83, 85-86, 130 TS conception of truth, 3-4, 2730, 42 uc-compatible systems. See Uniform-category compatible systems

Index uc-nested systems. See Uniformcategory nested systems Uniform-category compatible systems, 75, 125 Uniform-category nested systems, 72-75, 94n, 124-25 Uniform sentences, 62, 148 Universal language, 19-21 Unstable sentences, 52, 54, 59, 70, 72, 91, 95, 114-15, 148 the problem of -, 70, 72, 9296, 98, 113

153

v-sequences, 129-32, 148 V-system, 129, 131, 148. See also Revision system van Fraassen, B. C., 33n, 146 Wallace, J., 25n, 146 Woodruff, P. W., 34n, 57, 144 y-sequence, 95

ZFC, 124