Truth (Princeton Foundations of Contemporary Philosophy)

Truth

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P RINCETON FOuN daTIONs OF CONTEMPORaRY PhIlOsOPhY

Scott Soames, Series Editor Philosophical Logic by John P. Burgess Philosophy of Language by Scott Soames Philosophy of Law by Andrei Marmor

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trUtH Alexis G. Burgess & John P. Burgess

P r i n c eto n U n i v e r S i ty P r eS S P r i n c eto n A n d ox fo r d

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Copyright © 2011 by Princeton University Press Published by Princeton University Press, 41 William Street, Princeton, New Jersey 08540 In the United Kingdom: Princeton University Press, 6 Oxford Street, Woodstock, Oxfordshire OX20 1TW press.princeton.edu All Rights Reserved Library of Congress Cataloging-in-Publication Data Burgess, Alexis, 1980–  Truth / Alexis G. Burgess and John P. Burgess.    p. cm. — (Princeton foundations of contemporary philosophy)  Includes bibliographical references and index.  ISBN 978-0-691-14401-6 (hardcover : alk. paper) 1. Truth. I. Burgess, John P., 1948– II. Title.  BD171.B85 2010  121—dc22 2010041439 British Library Cataloging-­in-­Publication Data is available This book has been composed in Archer and Minion Pro Printed on acid-­free paper. ∞ Printed in the United States of America 10 9 8 7 6 5 4 3 2 1

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To Anna and Aigli

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Contents

Starred (*) technical sections optional Preface Acknowledgments

xi xiii

CHAPTER ONE Introduction 1.1 Traditional Theories 1.2 Contemporary Theories 1.3 Paradoxes 1.4 Plan 1.5 Sentences 1.6 Propositions

1 2 4 5 7 10 12

CHAPTER TWO Tarski 2.1 “Semantic” Truth 2.2 Object Language vs Metalanguage 2.3 Recursive Definition 2.4* Direct Definition 2.5* Self-­Reference 2.6* Model Theory

16 16 18 22 24 28 29

CHAPTER THREE Deflationism 3.1 Redundancy 3.2 Other Radical Theories 3.3 Disquotation 3.4 Other Moderate Theories 3.5 Sloganeering 3.6 Reference

33 34 38 41 44 47 49

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Contents

viii

CHAPTER FOUR Indeterminacy 4.1 Presupposition 4.2 Vagueness 4.3 Denial, Disqualification, Deviance 4.4 Doublespeak, Dependency, Defeatism 4.5 Relativity 4.6 Local vs Global

52 53 54 55 59 61 65

CHAPTER FIVE Realism 5.1 Realism vs Deflationism 5.2 Correspondence Theories 5.3 Truthmaker Theories 5.4 Physicalism 5.5 Utility 5.6 Normativity

68 68 70 72 74 77 79

CHAPTER SIX Antirealism 6.1 Meaning and Truth 6.2 Davidsonianism 6.3 Dummettianism vs Davidsonianism 6.4 Dummettianism vs Deflationism 6.5 Holism 6.6 Pluralism

83 84 87 90 93 96 97

CHAPTER SEVEN Kripke 7.1 Kripke vs Tarski 7.2 The Minimum Fixed Point 7.3 Ungroundedness 7.4* The Transfinite Construction 7.5* Revision 7.6* Axiomatics

102 103 105 107 109 112 113

CHAPTER EIGHT Insolubility? 8.1 Paradoxical Reasoning 8.2 “Revenge”

116 116 118

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Contents



8.3 8.4 8.5 8.6

Logical “Solutions” “Paraconsistency” Contextualist “Solutions” Inconsistency Theories

Further Reading Bibliography Index

120 123 124 127 135 143 153

ix

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Preface

This volume is, in accordance with the aims of the series in which it appears, a somewhat opinionated introductory survey of its subject, at a level suitable for advanced undergraduate or beginning graduate students of philosophy, or the general reader with some philosophical background. The subject is truth, and more specifically, what it is for something to be true, with special emphasis on the question whether there is anything interesting all truths have in common, besides their all being true. We set aside the postmodernist answer, “All truths are enforced by the hegemonic structures of society,” on the grounds that it confuses something’s passing for true in a given society with its actually being true, overlooking the question, “What is something that passes for true in a given society passing for?” Questions about what passes for true but isn’t, and how it is able to do so, are needless to say of great importance, but writers addressing them have not been lacking. We also leave to others the question of “the value of truth,” at least insofar as it is a question of intrinsic value, not just practical utility. The important issues that have been extensively discussed under this heading seem to us really about the intrinsic value, not so much of truth itself, as of something else related to it: of discovering the truth, or of speaking the truth. That one should value knowledge and honesty, rather than contenting oneself with what Harry Frankfurt calls bullshit, we take for granted without comment, and we take for granted that the reader takes it for granted, too. The issues about truth that remain, the ones we do address, have in recent years been made the subject of dozens of monographs and anthologies, as well as scores of journal articles not yet anthologized. Any survey of this large volume of material will inevitably give more attention to some parts of it than others, with more than a few subtopics getting merely a passing mention (plus references to further literature for the interested reader). Though other authors would doubtless make different judgments about

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Preface

xii

what to emphasize and what not, we do not consider our judgment to be at all eccentric. We do give especially detailed attention to the pros and cons of so-­called deflationism, and its claim that there is nothing interesting all truths have in common; but then it is hard to think of an issue that has been more intensively debated in the recent specialist literature on truth. Though not eccentric, our emphases are not entirely conventional, either. Though for expository purposes we first treat the question of the nature of truth in isolation from consideration of such paradoxical utterances as “What I am now saying is false,” our real opinion is that the two topics are inseparable. One can go a long way toward the goal of an adequate theory of truth while ignoring the bearing of the paradoxes, but not all the way, and so we have devoted our last two chapters to the paradoxes and attempted solutions. The least conventional aspect of our choice of topics is perhaps our giving attention to a position often treated as unmentionable, the defeatist view that the paradoxes admit no solution. The paradoxes nonetheless remain for us a subordinate issue. The literature on them fairly quickly gets involved in technicalities, but our discussion goes only as far as is possible without requiring any deep knowledge of technical notions. Moreover, the more technical parts of the material we do include have been placed in sections starred as optional reading, whose omission should not seriously impede understanding of the remainder of the book. About the division of labor in producing this volume the following may be said. We have drawn on previous work by both of us, especially AGB’s doctoral dissertation on fictionalism about truth and public lectures of JPB on Tarski and Kripke. We began with a literature search, making a list of topics needing coverage, with AGB and JPB taking responsibility for the more philosophical and the more technical material, respectively. AGB then undertook extensive drafting, and JPB intensive rewriting to condense to fit the publisher’s word limits. As a result of this way of working, JPB is mainly to be blamed for any faults in the condensed style, though AGB did undertake a final editing. In matters of substance rather than style, we stand together.

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Acknowledgments

Thanks to Scott Soames, for organizing the present series and inviting us to contribute to it. Thanks as well to the staff at Princeton University Press, especially Ian Malcolm, Heath Renfroe, and Leslie Grundfest; to our copyeditor, Jodi Beder; and to the two readers who provided comments on the manuscript.

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chapter one

Introduction

Inquiry, it is said, aims at the truth. Yet it’s doubtful there is any such thing as the truth. So it might be better to say that inquiry aims at truths, and better still to say that different inquiries from archeology to zoology aim at different truths from archeological to zoological. Such inquiries have had many successes, but in many cases inquiries are still underway, and success has not yet been achieved. Thus some truths are known, others unknown. But what, if anything, do the different truths, known and unknown, about different topics have in common, to make them all truths? If we knew the answer to this question we’d at least have a better understanding of the nature of inquiry, and perhaps even a better chance of finding what we’re looking for when we inquire. But with so many kinds of truths, the project of coming up with a unified conception of what truths are might seem hopeless. Perhaps that is why other inquiries leave it to philosophy. This little book is about recent philosophical inquiries into what it is for a thing to be true. There are other philosophical questions about truth—Is truth of value in itself or only as a means to other ends? How much sense can be made of the idea of one untruth’s being closer to truth than another?—but there are so many such questions that it would take a book longer than this even just to introduce them all. The question on which we focus—“What is it for a thing to be true?”—has a certain priority simply because some sort of answer to it has to be presupposed by any serious attempt to answer almost any of the others.

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Chapter One 1 .1  Traditional

Theories

If the best-­known saying about truth is Pilate’s question, “What is truth?” (John 18:38), probably the best-­known saying about truth by a philosopher is Aristotle’s assertion (Metaphysics, 1011b25): (1) 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. This saying, like others of Aristotle’s, has been interpreted in more than one way; and it is not the only thing Aristotle said about truth, either. The definition modern philosophy inherited from medieval Aristotelianism ran along the following lines: (2)  Truth is agreement of thought with its object.

2

Most early modern philosophers from Descartes to Kant professed to accept something like (2) as a definition of truth. Some, however, even as they did so, complained, in a terminology itself inherited from medieval Aristotelianism, that (2) was only a “nominal” definition, revealing the meaning of the word “truth,” and not a “real” definition, revealing the essence of the thing, Truth. The supposed contrast between real and nominal definition was in deep disrepute through much of the last century, but in recent decades its reputation has recovered somewhat, and we will see in later chapters that there is a fundamental division today among writers on truth over the question whether, once it has been explained what it means to call something true, there remain any further questions about what it is to be true. Be that as it may, (2) was as diversely interpreted as it was widely endorsed. By a century or so ago at least three interpretations had emerged, differing over the location of the objects of thought—in an external world, in the mind along with thought itself, or in the interaction between the two—and therewith over the nature of agreement. In surveys of philosophical thought about truth one typically encounters early on a list of “theories” of truth represented by slogans loosely based on things that were said in a three-­cornered debate over truth about a century ago, in which the realist insurgent Bertrand Russell attacked the

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Introduction

dominant British idealism and American pragmatism of the day, as represented by the now-­forgotten H. H. Joachim and the ever-­ famous William James. The slogans are biconditional in form, involving “if and only if ” (henceforth abbreviated “iff ”). They read as follows: (3) Realist or correspondence theory: A belief is true iff it corresponds to reality. (4) Idealist or coherence theory: A belief is true iff it coheres with other ideas. (5) Pragmatist or utility theory: A belief is true iff it is useful in practice. By bringing in the notions of reality and idea and practice, whose homes are metaphysics and epistemology and ethics, such views tend to suggest truth is a metaphysical or epistemological or ethical notion. Both (4) and (5) invite immediate objections: May not a paranoid’s delusions of persecution be frighteningly coherent? May not a patient’s faith that a mere placebo is a wonder drug be therapeutically useful? Russell was quick to claim in opposition to Joachim that multiple systems of beliefs may be internally consistent, though incompatible with each other. Nietzsche had already suggested well before James that false beliefs may be not merely useful but indispensable for life. These objections are so obvious that the reader will likely guess that Joachim and James must have held more interesting views than a simplistic reading of the coherence and utility slogans would suggest. Indeed, idealists understood that multiple belief systems, including crazy ones, might be classifiable as coherent if one meant by coherence just bare logical consistency; but they meant something more. Likewise, pragmatists recognized that there might be counterexamples to the principle of the utility of truth if one understood it as a purported exceptionless universal law; but they understood it as something less. Both groups also made cogent criticisms of crude “copy” versions of the correspondence theory. And the coherence and utility views put two questions on the agenda for any inquiry into the nature of truth: to explain why

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Chapter One

consistency is at least a necessary condition for truth, and why, as a general rule subject to particular exceptions, true beliefs tend to be more useful than false ones. Still, Joachim leaves quite obscure what beyond mere consistency is required for a system of beliefs to constitute a “significant whole,” and thus be coherent in his sense; and James does not satisfactorily explain how utility, or any feature that only claimed to hold “in the long run and for the most part,” could be definitive of truth. 1 .2

4

 Contemporary Theories

Any historical treatment of our subject would have a great deal more to say about the figures already mentioned and several others. Gottlob Frege, the great precursor of the analytic tradition in philosophy, held that truth could not be defined as correspondence or in any other way. G. E. Moore, cofounder with Russell of that tradition, held like Russell that truth is correspondence, and unlike Russell that correspondence is unanalyzable. C. S. Peirce, the philosopher and logician from whom James took the word “pragmatism,” defined truth roughly as what would come to be believed if inquiry were pursued to its ideal limit; John Dewey, a younger pragmatist whom James frequently cited, eventually concluded that one should simply avoid talk of truth in favor of talk of “warranted assertability.” This book, however, in accordance with the aims of the series in which it appears, must be concerned mainly with the status of the question among philosophers in the analytic tradition in the early twenty-­first century, and so, after taking note of the debates of the first years of the twentieth, must leave those earlier debates behind. Henceforth the coherence and utility theories will be mentioned only as occasional foils for views having a significant number of present-­day defenders. But note the plural: “views having a significant number of present-­day defenders.” The analytic tradition has become the mainstream in Anglophone philosophy, wholly supplanting idealism and largely absorbing pragmatism, but in achieving this

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Introduction

status it has ceased to represent, if it ever did, a uniform doctrine. Its founders’ realist or correspondence view of truth is by no means universally accepted, and the reader should not infer from our indication that (4) and (5) are no longer widely defended that there is now a consensus in favor of (3). The problem with (3) is not that what it tells us seems obviously wrong, as with (4) or (5), but rather that it tells us so very little, pending specification of what its key terms (“reality” and “correspondence”) are supposed to mean, and that every attempt to say something more specific has proved highly contentious. The rival theories that attract philosophers today are not, however, those that attracted philosophers a century ago. Today the kind of idealism that predominated a century ago is dead, its heir being an idealism that dares not speak its name, and calls itself “antirealism.” Antirealism holds a distinctive view of the nature of truth, but it resembles the traditional idealist Joachim’s view less than it resembles Peirce’s. Pragmatism survives, but some of its most noted recent adherents have been, like many nonpragmatists, attracted to a view of the nature of truth, called deflationism, that attributes no interesting common property to all truths. In this respect deflationism is unlike the view of James; it derives, rather, from F. P. Ramsey, the most talented British philosopher of the generation after Russell and Moore. So in place of the traditional three-­cornered realist-­idealist-­ pragmatist debate, one has today a three-­cornered realist-­ antirealist-­deflationist debate, complicated by each of the three positions coming in several variant versions and by the presence also of several less popular views on the scene. That debate will be the primary topic of this book. 1 .3  Paradoxes

Russell’s work in logic became very influential among philosophers in the decades following the First World War, especially among the logical positivists, the dominant school of the period. But many positivists’ views on truth were less like Russell’s than like Dewey’s, in that they tended to hold that the concept of truth

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Chapter One

6

has no role to play in a scientifically oriented philosophy. Ironically, one of Russell’s own discoveries contributed to the spread of this tendency, which was also found among logicians and mathematicians. In the background was a crisis in the then-­novel branch of mathematics known as set theory, where antinomies or paradoxes had emerged. Some of these (Burali-­Forti’s, Cantor’s) were quite technical, but Russell discovered one that is easily stated. Consider the set R of all sets that are not elements of themselves. Then R is an element of R, which is to say, is one of the sets that is not an element of itself, iff R is not an element of R—a contradiction! Russell’s paradox reminded many of a fact long known but little cited in the realist-­idealist-­pragmatist debate, that the notion of truth, too, is subject to paradoxes. In particular it reminded many of a paradox attributed to ­Aristotle’s contemporary Eubulides, called the pseudomenos or liar. Suppose I say, “What I am now saying is not true.” Then it seems that what I am saying is true iff what I am saying is not true—another contradiction! Medieval logicians had added similar examples, under the label insolubles: Suppose Socrates says, “What Plato is saying is true,” while Plato says, “What Socrates is saying is false.” Modern mathematicians and logicians and philosophers now added more examples, finding that truth is but one of a family of related notions, all of which involve what seem to be similar contradictions. One member of this family of notions—called alethic, from the Greek for “truth”—is that of a predicate or an adjective being true of something. This notion gives rise to an exact parallel to Russell’s paradox of the set of all sets that are not elements of themselves, namely, Grelling’s paradox of the adjective that is true of all adjectives that are not true of themselves. “English” is English, “short” is short, and “polysyllabic” is polysyllabic. Hence these three are autological or true of themselves. By contrast, “French” is not French, “long” is not long, and “monosyllabic” is not monosyllabic. So these three are heterological, or not true of themselves. What about “heterological”? Another member of the family of alethic notions is that of definability, where an object is definable iff it is the one and only

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Introduction

thing of which some predicate is true. This notion likewise gives rise to several paradoxes, of which Berry’s, the easiest to state, goes somewhat as follows: “The smallest natural number not definable in English in twenty-­six syllables or fewer” defines a natural number in English in twenty-­six syllables. Other definability paradoxes (Richard’s, König’s) were more technical, and became entangled with debates over the status of set theory. Many mathematicians were only too happy to join those philosophers who classed alethic notions as unacceptably “psychological” if not damnably “metaphysical” in character. However, one important mathematician and logician, Alfred Tarski, and following him quite a few philosophers, attempted to rehabilitate the notion of truth by “solving” or “resolving” or “dissolving” or at least blocking the paradoxes. Debate over the solution (or insolubility) of the paradoxes and debate over the nature (or lack of nature) of truth proceeded separately over most of the last century, but it has become increasingly clear in recent years that the two questions cannot really be kept apart. (For instance, some think deflationism makes it harder, while others think it makes it easier, to deal with the paradoxes.) Accordingly, the paradoxes will be a secondary topic of this book. 1 .4  Plan

It is with Tarski, probably still today the writer most often cited in discussions of truth by philosophers in the analytic tradition, that we begin our survey of theories of truth. To accommodate readers differing in their degree of background and interest in technical matters, we divide our account of Tarski’s work in two. The first half of chapter 2 gives a nontechnical account of the most often discussed aspects of Tarski’s views, which should be enough to enable the reader to follow allusions to those views later in the book. More technical material is confined to the starred sections making up the second half of the chapter, which readers who so choose may postpone or omit. We then turn to the deflationism-­realism-­antirealism debates, taking deflationism first. Tarski gave a central role in the theory of

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Chapter One

8

truth to the principle that saying something and saying it is true are equivalent. Common to all forms of deflationism is the claim that, roughly speaking, this equivalence principle is all there is to the theory of truth. Chapter 3 is devoted to the exposition of various deflationistic positions from Ramsey’s time to the present; a half-­dozen variants, some more radical, some more moderate, are described. The present authors are sympathetic to the general idea behind deflationism, but not fully satisfied with any of the existing versions. It is our dissatisfaction with existing versions that will be most evident in this chapter, while our sympathy with the general idea will become more evident when we turn in later chapters to deflationism’s “inflationist” (realist and antirealist) rivals and critics. One popular objection to the equivalence principle, and hence to any theory of truth, such as deflationism, that embraces it, goes as follows. If you have never practiced cannibalism, then neither “Yes” nor “No” is an appropriate answer to “Have you stopped eating people?” since the question seems to presuppose that you at least used to eat people. This suggests that “You have stopped eating people” is neither true nor false. But if it is not true (as well as not false), then to say that it is true is to say of what is not that it is, which is false, and we have a case where saying something is not equivalent to saying that it is true, because the latter is false while the former is not false (though not true, either). Similar purported counterexamples turn on the phenomena of vagueness and relativity, which we lump together with presupposition in chapter 4 under the bland label “indeterminacy.” We survey various lines of defense deflationists have taken against purported indeterminacy counterexamples, without pretending to achieve a full resolution of the issues. Paradoxical examples like the liar are often viewed as further cases of indeterminacy, and the discussion of presupposition and vagueness is in some respects a warm-­up for tackling the paradoxes later in the book, though it will be seen when we come to them that the paradoxes involve an additional twist. Chapter 5 is devoted to views we classify as “realist,” namely, views taking truth to involve standing in some appropriate relation to some portion(s) or aspect(s) of reality. Russell and Moore were realists about truth in this sense at key stages in their careers,

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Introduction

though each changed his mind about truth more than once—our attaching just one view to Russell’s name in §1.1 and to Moore’s in §1.2 was a caricature—and though their versions of realism were distinct and incompatible. Among the heirs of Russell and Moore there is even more disagreement than there was between those pioneers themselves. There is disagreement both over the nature of the “appropriate relation” involved (which some do and some don’t call “correspondence”) and over the “portion(s) or aspect(s)” of reality involved (which some do and some don’t call “truthmakers”). Moreover, there is a division between those satisfied with a fairly abstract and metaphysical account of these matters, and those who see a need for a more concrete and physical account. Along with the different realist views we consider also a realist objection to deflationism alleging that the latter cannot explain why true beliefs are useful, and an objection to realism and deflationism alike alleging that the notion of truth has an evaluative role that both groups wrongly neglect. Chapter 6 is devoted mainly to those who call themselves “antirealists.” They reject both deflationism and what we call “realism,” though they do not much discuss either. What they do discuss at length, and most emphatically reject, is something else that they call “realism,” which amounts to what others call truth-­ conditional semantics, to which they oppose something called verification-­conditional semantics. (Explaining the tangled usages of “realism” will be one of our tasks in this chapter.) Along with antirealism we take note in the same chapter of a more recent position, pluralism, which holds that a realist view of truth may be more appropriate for some “domains of discourse” and an antirealist for others. The discussion of pluralism concludes our survey of contemporary theories of the nature of truth, insofar as those can be discussed without bringing in issues about the paradoxes. We begin our examination of views on liar-­style paradoxes in chapter 7, with an account of the work of Saul Kripke, whose “Outline of a Theory of Truth” (1975) has probably been the most influential work on its topic since Tarski’s “The Concept of Truth in Formalized Languages” (1935). Our Kripke chapter is organized like our Tarski chapter, with a nontechnical account, containing what is needed to follow discussions in our next and final chapter,

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Chapter One

coming in the first half, followed in the second half by sections starred as optional reading, containing more technical material. Tarski held that it is impossible to avoid paradox unless one distinguishes the language for which one is formulating a theory of truth from the language in which one is formulating that theory. Kripke in the end seems to concede, however reluctantly, that he has been unable to avoid something like Tarski’s split or the “ghost” of it. It is on this point that several subsequent writers have sought to improve on Tarski and Kripke alike. Several proposals are considered in chapter 8, especially views advocating deviation from classical logic and views emphasizing the role of context in communication. Also considered is the defeatist view that no proposed solution to the paradoxes can ever be wholly successful, because the intuitive notion of truth ultimately is simply incoherent. Finally, a connection between this issue of the solvability or unsolvability of the paradoxes and the issues between deflationism and inflationism is briefly sketched. The book then ends, not with a final verdict on the issues, but with suggestions for further reading. Before launching into our survey we must address a question that we have sidestepped so far, but can hardly hope to continue evading when we get down to closer consideration of the views of specific authors. The question is this: What kinds or sorts of things or items are true, or as is said, are bearers of truth? Presumably the same kinds of things are false as are true, so what we are really asking is: What kinds of things bear truth or falsehood, or as is said, bear truth values? Or if (as is surely the case) more than one kind of thing can do so, which are the fundamental truthbearers? An early confrontation with this question is unavoidable, since the writers whose work we survey often have quite strong opinions on the matter, so that the position adopted on it can affect the whole character of an author’s account of truth. 1 .5  Sentences

10

One answer quickly suggests itself. Some truths have been written down. (The preceding remark provides an example.) Other truths

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Introduction

have only been spoken. (Readers will have to provide their own examples.) When truths are written or spoken it is sentences that are written or spoken, and so it may seem that it is sentences that are the truths, the bearers of truth. The same conclusion can also be reached in a different way. Consider the following dialogue: (6) X:  Where there’s a will, there’s a won’t. Y:  That’s true. Z:  What’s true? I didn’t hear. Y may answer Z with a direct quotation, thus: (7)  Y: X said, “Where there’s a will there’s a won’t,” and that’s true. It seems that the quotation of a sentence designates a sentence, the one quoted. If so, it seems to be a sentence that Y is calling true in (6). But what are sentences? The distinctions that have to be made in response to this question, though they may at first seem pedantic, have proved fundamental to the study of language. Sticking with speech rather than writing for the moment, suppose each of ten greeters on a reception line says successively to each of ten guests, “I’m glad to see you.” Is that a hundred sentences or one sentence a hundred times? The usual answer is that it is one sentence type and a hundred sentence tokens. Does our language have one sentence type “The post office is near the bank” with two meanings, or two with the same pronunciation? Using Greek-­derived words for pronunciation and meaning, we may say there is one phonological type but two semantic types. A phonological type is a sound pattern, a semantic type a sound pattern plus a meaning. Once made, the distinction phonological vs semantic can be seen to apply to tokens as well: Producing a phonological token amounts to emitting sounds, as both parrots and people can do, but only the people and not the parrots can thereby speak meaningfully, which is what producing a semantic token amounts to. Writing (or recorded as opposed to live speech) complicates the story. If the greeters have laryngitis and each greeter writes on a card, “I’m glad to see you,” and successively shows it to each

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Chapter One

guest, then there is one type, ten inscriptions of the type, and one hundred presentations of the inscriptions to a potential reader. Usually the word “token” is used with writing for what there are ten of, but to maintain parallelism with speech it might better be used for what there are one hundred of. As with speech one distinguishes phonological from semantic, so with writing one must distinguish orthographic from semantic, since two meanings may share a spelling. Ambiguity can prevent an orthographic or phonological type from having a fixed truth value: “A bank is an especially dangerous place to be during a flood” is true if riverbanks are meant but false if moneybanks are meant. An even more common phenomenon than ambiguity is so-­called indexicality, the kind of dependence on features of context (such as who is speaking to whom) that is signaled by what are called indexicals (such as “I” and “you”). It can prevent even a semantic type from having a fixed truth value: “I’m glad to see you” is true when said by a greeter sincerely glad to see the guest, and false when said by one merely being polite. In general, a sentence can be the bearer of a fixed truth value only if we understand “sentence” in the sense of semantic token. Where indexicality is absent, and any possible semantic token would have the same truth value, there will be no harm if we say that the semantic type has that truth value in a secondary sense, and if ambiguity is absent as well, that the orthographic or phonological type has that truth value in a tertiary sense. Thus some types may be recognized as derivative truthbearers even if semantic tokens are recognized as the primary truthbearers. There is, however, a rival proposal. 1 .6  Propositions

Returning to the dialogue (6), Y may answer Z with an indirect rather than a direct quotation, thus:

12

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(7)  Y: X said that where there’s a will there’s a won’t, and that’s true.

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Introduction

What is called a that-­clause, the word “that” with a sentence coming after it, is often held to designate a proposition, the proposition expressed by that sentence. If so, it is a proposition that Y is calling true in (7), and that we are calling true when we use the form of expression “That     is true” or its stylistic variant “It is true that     .” Many take propositions to be the primary bearers of truth, with sentences being called “true” only in a derivative sense, a sentence being true if it expresses a true proposition. According to this propositionalist view, what we took earlier to be reasons for giving first place to semantic sentence tokens as bearers of truth are better taken as reasons for giving them first place as ­expressers of propositions (but only second place, after propositions, as bearers of truth). Some proponents of the rival sententialist view, while still insisting on sentences as the primary truthbearers, allow that a proposition may be called true in a derivative sense if it is or could be expressed by a true sentence. Many sententialists, however, are suspicious of the whole idea of propositions, not least on account of the frequency of disagreements about them among propositionalists themselves. Propositionalists generally agree that, just as different tokens of the same sentence type may express different propositions, so inversely the same proposition may be expressed by tokens of different sentence types. If Jack says to Jill, “I am younger than you are,” and Jill says back to Jack, “You are younger than I am,” they have expressed the same proposition, that he is younger than she is. Similarly if Jack says to Jill in English, “I love you,” and then in French «Je t’aime.» But here agreement ends. A sentence like “Jack fell down” has a certain grammatical structure. Does the proposition it expresses have a structure as well? One of us may say “Jack loves Jill,” calling them by name, and another of us, “He loves her,” pointing first to the one, then to the other. Have we expressed the same proposition? To each question, some say yes and some say no. And then there are embarrassing questions asked by linguists. If “that the earth moves” designates the same thing that “the proposition that the earth moves” denotes, why is it that we can say

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Chapter One

(8a) Copernicus hypothesized that the earth moves, while the Inquisition anathematized the proposition that the earth moves. but not (8b) Copernicus hypothesized the proposition that the earth moves, while the Inquisition anathematized that the earth moves.

14

(Evidently designating a proposition by a that-­clause is not quite the same as denoting one by a noun phrase beginning with “the proposition that.”) But despite doubts and difficulties, it seems that propositionalists outnumber sententialists. It also seems that, though many other kinds of things are spoken of as true and false—from assertions and beliefs and conjectures and declarations to remarks and statements and thoughts and utterances—there are at present no serious candidates for the role of fundamental truthbearers beyond propositions and sentences (the latter in the sense of semantic tokens). In particular, while the traditional theories (3)–(5) were formulated in terms of beliefs, few today regard beliefs as the primary bearers of truth and falsehood, for the following sort of reason, among others: It is possible to believe a disjunction (for instance, that it was either Professor Plum or Colonel Mustard who killed Mr. Boddy) without believing either disjunct, and such a belief may be true; but a disjunction cannot be true unless one disjunct is; so there must be truths that are not believed. Similar considerations apply to assertions. In any case, the very notions of “assertion” and “belief ” are ambiguous, since we must distinguish the act or state or event or process or whatever of asserting or believing from the content thereof, from what is asserted or believed. The most common view today seems to be that the content is a proposition, and that if the act or state or event or process can be called “true” at all, it is only in a derivative sense. Items of yet other sorts are also spoken of as true or false. One speaks, for instance, of false teeth or false friends. But here we are clearly dealing with different senses of the key words “true” and “false,” roughly synonymous with “genuine” and “fake,” as can be

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Introduction

seen from the fact that while a false sentence or proposition is as much a sentence or proposition as a true one, false teeth or friends are no teeth or friends at all. (What is false about them is the proposition that they are teeth or friends.) In sum, other truthbearers beyond sentences and propositions can be and will be more or less ignored for the remainder of this book. But the division between sententialists and propositionalists, which cuts across the division of theorists into realists and antirealists and deflationists, must be borne in mind as we turn to the examination of the views of particular authors, beginning with Tarski.

15

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chapter two

Tarski

Tarski was, by his own account, primarily a mathematician, though also “perhaps a philosopher of a sort.” He foresaw important applications for a notion of truth in mathematics, but also saw that mathematicians were suspicious of that notion, and rightly so given the state of understanding of it circa 1930. In a series of papers in Polish, German, French, and English from the 1930s on he attempted to rehabilitate the notion for use in mathematics, and his efforts had by the 1950s resulted in the creation of a branch of mathematical logic known as model theory. This fact alone makes Tarski’s work an enduring achievement, even if its philosophical bearing remains in dispute. What follows is a simplified account (telescoping earlier and later developments) of the most basic features of Tarski’s influential work. 2 .1  “Semantic”

Truth

Tarski held that almost all words of ordinary language, “true” included, are ambiguous, and his first task was to distinguish the sense of “true” that concerned him, and to give it a label. To give a somewhat facetious example, a follower of Williams James might hold that “Snow is white” is true on the grounds that it is useful for people to believe so, while a follower of William Jennings Bryan might hold that “Snow is white” is true on the grounds that the Bible often uses the phrase “white as snow.” Tarski considered himself a follower of Aristotle, because he held that “Snow is white” is true on the grounds that snow is white (and that “to say of what is that it is, is true”).

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Tarski

He gave his notion of truth the label “semantic” truth, which could mislead readers familiar only with the present-­day usage of “semantics” for the branch of linguistics concerned with meaning. Tarski was following a usage current among positivistically inclined philosophers in the 1930s, on which “semantic” was one of a trio of terms. A notion was called syntactic if it pertains to relations among words, semantic if it pertains to relations between words and things, and pragmatic if it pertains to relations among words and things and people. Thus a notion involving only words and compilations of words (such as the Bible) would count as syntactic, while a notion that brought in people (as believers, for instance) would count as pragmatic. By contrast, Tarski’s notion of truth was “semantic” simply because his principle (1)  “Snow is white” is true iff snow is white. involves words and a thing, snow, but not people. No more should be read into the “semantic” label than that. Anything of the same form as (1) is called a T‑biconditional. The general pattern (2)  “    ” is true iff     . where the same sentence goes into each blank, is called the T‑scheme. Tarski is famous for the emphasis he places on the T‑scheme, but against a background of classical logic, which Tarski assumes without question, the scheme is equivalent to a pair of rules, T‑introduction and T‑elimination: (3a)  from “    ” to infer “ ‘    ’ is true” (3b)  from “ ‘    ’ is true” to infer “    ” Tarski’s whole discussion could be recast in terms of these rules. Nonetheless we will stick to formulations in terms of the scheme in what follows. Another notion Tarski counted as “semantic” was that of the satisfaction of a condition by a thing or things. Parallel to (1) Tarski held, for instance, (4a)  “x is white” is satisfied by snow iff snow is white.

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Chapter Two

(4b) “x is the same color as y” is satisfied by snow and milk iff snow is the same color as milk. Yet another notion Tarski counted as “semantic” was that of denotation, as illustrated by (5)  “snow” denotes snow. Tarski’s ambition was to rehabilitate for use in rigorous mathematics the whole list of what he called “semantic” notions. (Note, however that these are more or less what we called “alethic” notions in §1.3, and meaning is not on Tarski’s list, though it would be at the head of any list of “semantic” notions in present-­day linguistics.) 2 .2

18

 Object Language vs Metalanguage

Before going further we need to look at how quotation marks are being used in (1). The sequence of letters es, en, oh, double-­yu, and so on, is an orthographic type that is used in English to express the truth that snow is white, but that could be used in some other language to express the falsehood that grass is red. Setting aside complications owing to ambiguities within a single language, there are two ways we could acknowledge this fact in a reading of (1). One way would be to take the quotation marks with what they enclose to designate the semantic type consisting of the orthographic type together with its English meaning. The other way would be to take it to denote the orthographic type, but take “true” to be elliptical for “true in English.” The latter is Tarski’s way. For him quotation marks are used to designate orthographic types, and what he is concerned with under the label “truth” is truth-­in-­English, or truth-­in-­L for some other language L. Tarski calls the language L for which truth is being discussed the “object language,” and the language L* in which truth is being discussed the “metalanguage.” So far we have been writing as if both languages L and L* were just English. But Tarski insists that paradox will be inevitable if one takes both L and L* to be the whole of some natural language, including the word “true” itself. For then one can write down something like this:

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Tarski

(6)  The sentence numbered (6) is not true. One then has (7a) The sentence numbered (6) is “The sentence numbered (6) is not true.” which together with the T‑biconditional (7b) “The sentence numbered (6) is not true” is true iff the sentence numbered (6) is not true. yields the contradiction (7c) The sentence numbered (6) is true iff the sentence numbered (6) is not true. If, however, one distinguishes the object language L from the metalanguage L*, and takes “true sentence of L” to belong only to the latter and not the former, then while one can still write down this: (6)  The sentence numbered (6) is not a true sentence of L. it will be only a sentence of L*, and there will be no paradox in the conclusion that it is neither a sentence of L that is true nor a sentence of L that is not, since it will not be a sentence of L at all. Thus for Tarski, though it is sentences of one language L that go into the blanks in (2), what results from filling the blanks is a sentence of another language L*. Three points should be noted about the relationship between L and L* as described so far. First, since for any instance of (2) the same sentence of L that is mentioned on the left side is used on the right side as a clause in a biconditional sentence of L*, the metalanguage must contain the object language. Second, in order to be able to mention sentences of L, the metalanguage must contain the device of quotation. Third, the metalanguage must contain something more besides, namely, the truth predicate “is true-­in-­L.” But the first two requirements can be relaxed. First, L* need not literally contain L, so long as one can translate L into L*. In subsequent mathematical applications of Tarski’s work, often L and L* are each a fragment of some natural

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Chapter Two

language, today most often English, but with L being abbreviated and written in symbols rather than words, while L* is not. Thus items of nonlogical vocabulary that may be written “less than” or “perpendicular” in L* might be abbreviated to “<” or “” in L, while the logical vocabulary of negation, conjunction, disjunction, conditional, biconditional, and universal and existential quantification that in L* would be written in words “not, and, or, if, iff, all, some” would in L be written in symbols “~, , , , , , .” In such a case, L is not literally contained in L* and one does needs to translate L into L*, even if the “translation” involved amounts to no more than undoing abbreviations. Second, L* must be able to mention sentences of L somehow, but it need not be by quoting them. Thus one might designate “Snow is white” as the sentence consisting of the 19th letter of the alphabet followed by the 14th followed by the 15th, and so on, or the sentence “~x~(x = x)” as the negation of the existential quantification with respect to the variable ex of the negation of the sentence consisting of the identity symbol flanked by two occurrences of the variable ex. Tarski speaks of designating a sentence of L by a “structural description,” a label broad enough to cover any of these possibilities. With the first requirement relaxed the T‑scheme becomes (2)  “    ” is true in L iff     . where an object language sentence goes into the first blank, and its translation into the metalanguage into the second blank. With the second relaxed it becomes (2)     

20

is true in L iff     .

where a “structural description” of a sentence of the object language goes into in the first blank, and its translation into the metalanguage into the second blank. (2) is Tarski’s ultimate “official” version of the T‑scheme. Tarski calls a proposed definition of the truth predicate formally correct if it is mathematically rigorous, and materially adequate if it implies every instance of the T‑scheme for the object language. Tarski’s aim is to make the notion of truth respectable by giving a formally correct, materially adequate definition.

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Tarski

Tarski’s aim is achieved by using some set-­theoretic apparatus available in the metalanguage but not the object language. Invocation of set-­theoretic apparatus makes it utterly unlikely that his mathematical definition could be achieving what a lexicographer’s definition aims to do, namely, to give an account of what ordinary people have meant all along by the word. It is no part of Tarski’s aim to do so. To put the matter another way, traditional logic distinguished a predicate’s extension, the things to which the predicate correctly applies, from its intension, the information about a thing conveyed by applying the predicate to it; and Tarski is only aiming at a definition that will have the right extension, without worrying about the intension. For some object languages he finds two or more formally correct, materially adequate definitions, differing in meaning or intension from each other. The smaller the object language, the more likely it is that one can find two definitions thus coextensive but nonsynonymous. With such a pair, though each definition would tell us something all truths have in common, at most one could be telling us what “truth” means. Yet in such a case Tarski is equally prepared to adopt either definition. But why bother with definition at all? In mathematics, new terms are indeed often introduced by defining them in terms of old ones and deriving basic results about them by proof from older results plus the new definitions. But new terms sometimes are (and as Aristotle already pointed out, on pain of circularity or infinite regress, sometimes must be) introduced simply as undefined primitives, with basic facts about them assumed as unproved axioms. Why not, then, simply introduce the truth predicate as a primitive, and the T‑biconditionals as axioms? Tarski finds a philosophical reason for not taking the notion of truth as a primitive in a view called “physicalism” current in his day among the positivistically inclined. According to this view, the only scientifically admissible primitives are logical and physical. (Arithmetical and geometrical notions were assimilated to logical and physical, respectively.) The traditional literature on truth, however, certainly makes the notion seem, not logical or physical, but psychological or metaphysical.

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Chapter Two 2 .3  Recursive

Definition

Tarski also has another reason why, even if one did take truth as a primitive, one shouldn’t take the T‑biconditionals as one’s axioms for it. That procedure won’t suffice for establishing various theorems about truth. One of the first theorems one might want to establish is that consistency is a necessary condition for truth, that deductions starting from true premises can never lead to contradictory conclusions. To establish this one might hope to start from the analyses that logicians such as Frege and Russell had given of deduction, showing that every deduction can be broken down into a series of steps proceeding according to very simple rules, such as inference to a disjunction from either of its disjuncts, or from a conjunction to either of its conjuncts. The program would then be to show, on the one hand, that such rules are truth-­preserving, always carrying one from true premises to true conclusions, and on the other hand, that contradictory conclusions, one the negation of the other, cannot both be true. To carry out this program, one would need such principles as the following: (8a) A disjunction is true iff at least one of its disjuncts is true. (8b)  A conjunction is true iff both of its conjuncts are true. (8c)  A negation is true iff what it negates is not true. Now T‑biconditionals, if taken as axioms, would enable one to deduce each instance of any of these principles. For example, from the T‑biconditionals (9a)  “Coal is black” is true iff coal is black. (9b)  “Blood is green” is true iff blood is green. (9c) “Coal is black or blood is green” is true iff coal is black or blood is green. one can deduce this instance of (8a): 22

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(10) “Coal is black or blood is green” is true iff “Coal is black” is true or “Blood is green” is true.

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Tarski

But there is no hope of getting any universal principle on the order of (8abc) just from the T-­biconditionals. One can, however, for a suitably restricted object language, hope to do the reverse, and obtain all T‑biconditionals from a shortish list of principles including (8abc) and a few more. To avoid technicalities, let us consider just a toy example. Consider a language L whose only nonlogical vocabulary consists of the numerals “0” and “1.” The only atomic, or logically simple, sentences of the language are equations “m = n” between numerals. The only sentences of the language are those that can be obtained from the atomic sentences by repeated use of negation, conjunction, and disjunction (writing “m ≠ n” for the negation of “m = n”). We can supplement the principles (8abc) by a few more principles bringing in the auxiliary alethic notion of denotation, thus: (8d) An equation is true iff the denotations of the numerals on the two sides are identical. (8e)  The denotation of the numeral “0” is the number zero. (8f)  The denotation of the numeral “1” is the number one. Then (8abcdef) together give what is called a recursive definition of the truth predicate, enabling us to deduce a T‑biconditional for every one of the (infinitely many) sentences of the language, as required by Tarski’s criterion of material adequacy. We can deduce the T‑biconditional for any sentence in a canonical way, following the structure of the sentence itself. If our target sentence is “0 = 1 or 1 ≠ 1,” we note it is the disjunction of one atomic sentence with the negation of another, and so start with the atomic sentences and work our way up, thus: (11a)  “0 = 1” is true iff zero is identical with one. (11b)  “1 = 1” is true iff one is identical with one. (11c)  “1 ≠ 1” is true iff one is not identical with one. (11d) “0 = 1 or 1 ≠ 1” is true iff zero is identical with one or one is not identical with one. Given that (8abcdef) provide a recursive definition of truth, if one were to take the truth predicate as a primitive governed

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Chapter Two

by certain axioms, (8abcdef) would be the axioms to take, rather than the infinite list of instances of the T‑scheme. But only a direct definition, which is to say, something of the form “A sentence is true iff . . . ,” with no occurrence of the truth predicate on the right side, would permit us to regard the truth predicate as merely an abbreviation for some compound of notions previously understood. And “physicalist” scruples lead Tarski to demand such a direct definition. Fortunately for him, the nineteenth-­century algebraist Richard Dedekind had decades earlier devised a method by which a recursive definition can always be converted into a direct definition, making use of set-­theoretic apparatus (specifically, the mathematical notion of function). But one can understand the allusions to Tarski in our discussion of the deflationism-­realism-­antirealism debate without having to know any of the technical details of Dedekind’s method and Tarski’s exploitation of it, or of the various mathematical uses to which the notion of truth, once defined, was put by Tarski and later model theorists. The reader who wishes may proceed at this point straight to the next chapter and post-­Tarskian philosophical debates. (Some of the more technical material that occupies the remaining sections of this chapter would be needed to follow the correspondingly technical portions of our discussion of Kripke’s work on the paradoxes in later sections of chapter 7, but those sections may themselves be treated as optional reading.) 2 .4 * Direct

24

Definition

To illustrate some of the finer points, we need a couple of examples more substantial than the toy language of the previous section. The first will be the formal language of arithmetic. This language contains variables v1 and v2 and v3 and so on for natural numbers. (We do not write variables “v1” and “v2” and “v3” and so on because the custom when writing in unformalized English about formal languages is to let expressions of the latter name themselves, omitting quotation marks.) The language contains

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Tarski

besides the numerals or constants 0 and 1 also operators + and •. This is all the nonlogical vocabulary. We call something an atomic term if it is either a variable or a constant. Then more terms such as (v1 + 1)  (1 + 1) are built up from atomic terms using the operators and parentheses for punctuation. A term is called open if it is or contains a variable or variables, and closed otherwise. It does not make sense to ask for the numerical value of an open term such as v1 + 1, since that would depend on the value of the variable; only closed terms such as 1  + 1 can be said to have a numerical value or denotation. Tarski’s first task is to give a rigorous definition of this auxiliary notion of the denotation |t| of a closed term t. A recursive definition sufficient to determine, step by step, the denotations of more and more complex closed terms, is provided by the following clauses: (12a)  |0| is zero. (12b)  |1| is one. (12c)  |s + t| is the sum of |s| and |t|. (12d)  |s  t| is the product of |s| and |t|. The first step in applying Dedekind’s method to convert the recursive definition into a direct one is to prove an existence and uniqueness lemma, telling us that there is one and only one function f taking closed terms as inputs and giving natural numbers as outputs, that satisfies the following conditions: (13a)  f(0) is zero. (13b)  f(1) is one. (13c)  f(s + t) is the sum of f(s) and f(t). (13d)  f(s  t) is the product of f(s) and f(t). All the hard work goes into the proof of this lemma, which we omit. When the proof is complete, |t| may be defined to be f(t) where f is the function of the lemma, and (12abcd) follow immediately as theorems.

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Chapter Two

The atomic formulas of the language consist of two terms flanking the identity symbol. Then other formulas are built up from atomic formulas by the logical operations. An occurrence of a variable in a formula is called bound if it is “caught” by a quantifier, and free otherwise. For example in (14)  v2(v1 = v2  v2) which intuitively says that v1 is a perfect square, v1 is free and v2 is bound. A formula is called closed if all occurrences of variables in it are bound, and otherwise is called open. Open formulas may also be called conditions, and closed formulas sentences. It does not make sense to ask whether a condition such as (14) is true, since that would depend on the value of the free variable. Tarski’s next task is to give a rigorous definition of the notion of truth for sentences. A recursive definition sufficient to determine, step by step, the truth values of more and more complex sentences is given by the following clauses: (15a)  s = t is true iff |s| is identical with |t|. (15b)  ~A is true iff A is not true. (15c)  A  B is true iff A is true and B is true. (15d)  A  B is true iff A is true or B is true. (15e)  vA(v) is true iff for every closed term t, A(t) is true. (15f)  vA(v) is true iff for some closed term t, A(t) is true. In (15ef), A(t) represents the result of substituting the term t for every free occurrence of the variable v in A(v). The lemma needed for conversion to a direct definition by Dedekind’s method is that there is one and only one function g taking closed formulas as inputs and giving as output either the number one (representing truth) or the number zero (representing falsehood), and satisfying the conditions (16a)  g(s = t) is one iff |s| is identical with |t|. 26

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(16b)  g(~A) is one iff g(A) is not one.

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Tarski

and four more related to (15cdef) as (16ab) are related to (15ab). Then A is defined to be true iff g(A) is one, where g is the function of the lemma, and (15abcdef) become theorems given this definition. The definition becomes more complicated for an object language with variables ranging over objects not all of which are denoted by closed terms of the language. Such is the case with the formal language of geometry, which has variables for points of the Euclidean plane, but no constants or operators. The only items of nonlogical vocabulary are two predicates B and C, for betweenness and congruence, abbreviating the following: (17a)     lies on the line segment between     and     . (17b)     is as exactly as far from     from     .

as    

is

There are three kinds of atomic formulas, from which to build up other formulas: B followed by three variables, C followed by four variables, and = flanked by a variable on each side. For geometry we cannot begin a recursive definition of truth for closed formulas with a clause like (15a) about truth for closed atomic formulas, since there are no closed atomic formulas. Nor can we handle quantification by clauses like (15ef), since there are no closed terms. Tarski’s solution to these difficulties is to bring in the auxiliary alethic notion of satisfaction. He gives a recursive definition (convertible to a direct definition by Dedekind’s method) of what it is for an open formula to be satisfied by an assignment of a sequence of points to its free variables. For example, the atomic formula Cv1v2v1v3 is satisfied by the assignment of p, q, r to v1, v2, v3 iff p is exactly as far from q as it is from r. For another example, the existentially quantified open formula v1Cv1v2v1v3 is satisfied by the assignment of q, r to v2, v3 iff for some point p the formula Cv1v2v1v3 is satisfied by the assignment of p, q, r to v1, v2, v3. Truth for closed formulas is simply the special case of satisfaction by the “empty sequence” of points. We will not give the details, since Tarski’s approach or some close variant can be found expounded in all good logic textbooks today.

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Chapter Two 2 .5 * Self-­Reference

The language of geometry is a case where there can be found coextensive but nonsynonymous truth definitions. For a major result of Tarski is that for the language of geometry truth under his “semantic” definition coincides with the “syntactic” notion of deducibility from a list of axioms for Euclidean geometry. By contrast with this result, Kurt Gödel’s famous incompleteness theorems tell us that there is no system of axioms from which all truths of arithmetic can be deduced. This difference between geometry and arithmetic is closely connected with the fact that we have in arithmetic, though not geometry, a way of referring in the formal language to expressions of the formal language, comparable to the ability of natural language to refer to its own expressions by quoting them. To be sure, arithmetic is about numbers, not expressions; but a code number #E may be assigned to every expression E, as in effect happens today when expressions are transmitted electronically by encoding them as sequences of zeros and ones, the binary digits of a number. The formal language L of arithmetic has numerals 0 and 1 for zero and one and closed terms 1 + 1 and (1 + 1) + 1 and so on that can in effect serve as numerals 2 and 3 and so on for other numbers. The numeral e for the code number e = #E of an expression E amounts to something like a quotation of the expression, and we may suggestively write "E" for it. Many important notions pertaining to expressions of the language can then be expressed in the language. One notion we can express is (18) x is the code number of an equation whose two sides have the same denotation. For a fixed system of axioms, giving us a fixed notion of provability, or deducibility from the axioms, another notion we can express is (19)  x is the code number of a provable sentence. 28

Sentences can be constructed that in effect refer to themselves. (So can pairs of sentences that refer to each other, and sequences

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Tarski

of sentences in which each refers to the next.) If “x is the code number of a sentence with property Π” is expressible by a formula P(x) of L, a sentence Σ of L can be constructed that in effect says of itself that it has the property Π. What Σ literally says is that for any number x, if S(x), then P(x), where S(x) is a formula that provably holds of one and only one number, namely, the code number of the sentence Σ itself. Thus the sense in which Σ says of itself that it has the property Π is that the following is a theorem: (20)  Σ  P("Σ") Such self-­referential sentences are the analogues of natural language sentences of the type (21)  The sentence numbered (21) has the property Π. Gödel exploited self-­reference to construct a sentence that says of itself that it is not provable, which sentence he showed is indeed not provable, hence is true: an unprovable truth. Tarski exploited self-­reference to show that no formula T(x) of the language L of arithmetic can express “x is the code number of a true sentence of L,” because for any formula T(x) one can construct a sentence having Σ  ~T("Σ") rather than Σ  T("Σ"). The details are given in textbooks of intermediate-­level logic. 2 .6 * Model

Theory

Languages like those of formal arithmetic and formal geometry are called first-­order languages. They are the main object of study in model theory. To specify such a language one must do two things. On the “syntactic” side, one must specify a vocabulary of nonlogical symbols. One must indicate what constants, operators, and predicates occur in the language (as for arithmetic we have two constants 0 and 1 and two two-­place operators + and ∙, while for geometry we have one three-­place predicate B and one four-­place predicate C). Application of the usual logical apparatus to a vocabulary of nonlogical symbols yields an “uninterpreted language.” On the “semantic” side, one must specify an interpretation. What this means is that one must indicate what domain (as

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numbers for arithmetic, or points for geometry) the variables are supposed to range over. Also, one must associate objects of appropriate kinds to each item of nonlogical vocabulary. That means that for each constant, one must say what element of the domain it denotes (as 0 denotes zero). For each operator, one must say what function on the domain it indicates (as + indicates addition). For each predicate, one must specify its extension, that is, one must say which sequences of elements of the domain satisfy it (as triples of points with the first between the other two satisfy B). Providing an interpretation turns an “uninterpreted language” into an “interpreted language.” Note that an “interpretation” in the technical sense just indicated need not assign any meaning to the nonlogical vocabulary. Predicates need to be assigned an extension, but not an intension. In describing the language of geometry, for instance, we read C as “    is exactly as far from     as     is from     ,” and used this reading to specify which quadruples satisfy an atomic formula beginning with C. But all that really matters for Tarski’s definition of truth is the set of quadruples. This we could have arrived at a different way, by reading C as “    is exactly as far from     as     is from     and some swans are black.” For with this alternate reading, though the intension would be different, still (since some swans are black) the extension would be the same. For a language containing, say, an operator & meaning “Euclid knew that . . . ,” the difference between the two readings of C might make a difference. Indeed, black swans being native to Australia, not Egypt, Euclid knew the first but not the second of the following: (22a) For any two points, the first is exactly as far from the second as the second is from the first. (22b) For any two points, the first is exactly as far from the second as the second is from the first, and some swans are black. 30

Hence &v1v2Cv1v2v2v1 would be true with our original swanless reading of C, but not with the alternate swanful reading. But

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such operators as & are not present in first-­order languages. Their presence would obstruct Tarski’s whole procedure, which depends on the truth values of compounds depending only on the truth values of their components, as the truth value of ~A depends only on the truth value of A. With an operator like & in the language (and with Euclid having been less than omniscient), this feature would be lost, since &A would be true for some true A and not for others. If we change not only the intension but the extension of our basic predicates, truth values can change even in first-­order languages. We can obtain from the geometric language another geometric language by keeping the vocabulary but changing the interpretation. Instead of taking the variables to range over points of the Euclidean plane or pairs of real numbers (x, y), take them to range over points in the interior of the unit circle, or pairs of real numbers (x, y) with x2 + y2 < 1. Instead of reading the nonlogical predicate B as lying between on a straight line, read it as lying between on an arc of a circle orthogonal to the unit circle. Instead of reading the nonlogical predicate C as equal in distance according to the usual Cartesian distance formula, read it as equal in distance according to the alternative Poincaré distance formula. Then where the sentences that came out true in the old language were the theorems of Euclidean geometry, the sentences that come out true in the new language will be the theorems of what is known as hyperbolic geometry. (Such is the so-­called Poincaré disk model of non-­Euclidean geometry, famously illustrated by Escher’s Circle Limit graphics; the technical details are immaterial for present purposes.) Model theory generally treats truth for whole batches of different interpreted languages having the same nonlogical vocabulary, or what is the same thing, truth for an uninterpreted language relative to whole batches of different interpretations. Now though the two geometries in our example do not have the same set of theorems, they do have some theorems in common. These include all logical truths, such as the law v1(v1  =  v1) that everything is self-­identical. Tarski suggested that logical truths may indeed be characterized as precisely those sentences that remain true for all interpretations. This characterization is adopted in logic

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textbooks today. Indeed, most of the basic material on truth and logical truth in modern logic texts derives from Tarski. (A few basic results do antedate Tarski, notably the Löwenheim-­Skolem and Gödel completeness theorems. Such results were originally stated in terms of a notion of truth not reduced to more ordinary mathematical notions, leaving the impression that “metamathematics” was a subject outside mathematics. Tarski’s definition brought it inside.) The subject of model theory today extends far beyond such basic material. It has grown into a flourishing enterprise with applications across mathematics and allied fields. Tarski’s work has an ongoing influence in philosophy as well, and a greater one than will be evident just from the scattered mentions of his name in the chapters to follow.

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Deflationism

In turning from Tarski’s work to the deflationist-­realist-­ antirealist debate we are turning from work largely motivated by concerns about the paradoxes to work more directly motivated by questions about the nature of truth. About this work there is a great deal to be said even ignoring the bearing of the paradoxes, so let us set them aside for the time being, and return to the question with which we began this book, “What do the different truths about different topics all have in common, to make them all truths?” The answers offered by traditional views tended to make truth a notion of metaphysics or epistemology or ethics. That tendency, and all the traditional views, are opposed by the cluster of positions commonly labeled “deflationist.” The question “What do the different views called ‘deflationist’ all have in common, to make them all deflationist?” itself admits no easy answer. Deflationists are, however, typically committed to three theses about the phrase “is true,” usually called the natural language truth predicate. (That label also covers the phrase’s synonyms “holds” and “is so” and “is the case,” along with corresponding expressions in other languages.) First, applying the truth predicate to something is equivalent to just saying it. One version of this equivalence principle is embodied in Tarski’s T‑scheme, but there are others. Different deflationists, besides holding different views on whether the “something” in question should be taken to be a sentence or a proposition, give different accounts of what the “equivalence” here amounts to. Second, the equivalence principle is a sufficient account of the meaning of the truth predicate. There is nothing more to understanding the truth predicate than recognizing the equivalence

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principle, and that by itself ultimately suffices to account for our usage of the predicate and its utility. Different deflationists give different accounts of what the “recognition” here amounts to. Third, an account of the meaning of “true” is a sufficient account of the nature of truth. There is nothing to be said about what it is for something to be true once one has said what it means to call something true. Commitment to this last thesis is implicit in the practice of the typical deflationist, who begins by promising an account of the nature of truth (often quoting Pilate’s question), but in the end offers only an account of the meaning of “true.” Explicit enunciation of the principle is less common. But the different varieties of deflationism may be united more by family resemblance than shared essence, and our initial characterization doubtless accommodates some varieties less comfortably than others. What follows is a quick survey of the main types of deflationism, offered in hopes that as it proceeds the theme behind the variations will gradually become more apparent. The main division among deflationists is between two groups we will call radicals and moderates. The radicals maintain that what is conventionally called the truth predicate is not truly a predicate, or is one “only grammatically and not logically.” They implicitly or explicitly contradict Tarski’s view that if two speakers successively say (1)  The love of money is the root of all evil. (2)  It is true that the love of money is the root of all evil. then the second speaker has made a statement about the first speaker’s statement. 3 .1  Redundancy

34

The prehistory of deflationism vs inflationism can be traced back to the tension between the two Aristotelian formulations “to say of what is that it is” and “agreement of thought with its object,” but what is usually accounted the earliest version of deflationism was the redundancy theory of Ramsey. His view is radical. He holds that (1) and (2) are equivalent in the very strong sense of

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expressing exactly the same proposition. The difference between them is merely stylistic, and the truth predicate in (2) is redundant. Note, however, that if someone has just said (1), then the effect of repeating (1) as if taking no notice of its having just been said and the effect of saying (2) can be quite different. (The effects would be closer if we inserted “as was just said” after “that” in (2).) To say that the difference between (1) and (2) is stylistic is not to say that it is unimportant. The truth predicate appears in many forms other than the simple, present-­tense affirmative found in (2). Consider for instance the following: (3a) It is not true that the love of money is the root of all evil. (3b) Is it true that the love of money is the root of all evil? (3c) It used to be true that the love of money was the root of all evil. (3d) If it is true that the love of money is the root of all evil, then it is easier for a camel to go through the eye of a needle, than for a rich man to enter into the kingdom of God. If the truth predicate is to be truly redundant, it must be eliminable from such formulations as well. How is it to be eliminated? Well, presumably if (1) and (2) express the same proposition as each other, then their negations express the same proposition as each other, the corresponding interrogatives express the same question, and so on. Thus Ramsey’s account provides for the equivalence of (3abcd) to “The love of money is not the root of all evil” and “Is the love of money the root of all evil?” and so on. Also we may ask how the truth predicate is to be eliminated from items like the following: (4)  That’s true. (5a)  That’s not true.

(5b)  Is that true?

when said in response to another’s assertion of (1). But on almost any theory these are presumably equivalent to (2) and (3ab). The

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same brevity as is achieved by (5ab) can be achieved without “true” as well, thus: (6a)  It isn’t.

(6b)  Is it?

Here the anaphoric pronoun “it” and pro-­verb “is” have as antecedents the subject “the love . . .” and predicate “is the root . . .” of (1). Dispensing with “true” is more difficult in the case of what is called blind affirmation or denial or querying, when “true” is applied to a proposition that is not exhibited by displaying a sentence of our language expressing it, but specified in some other way, thus: (7a)  Radix malorum est cupiditas is true. (7b)  If 1 Timothy 6:10 is true, then Mark 10:25 is true. These may, however, be construed as equivalent to generalizations: (8a)  Whatever Radix malorum est cupiditas says, it is true. (8b) Whatever 1 Timothy 6:10 and Mark 10:25 say, if the former is true, so is the latter. though the conversion reads a bit awkwardly in the case of questions. But how is “true” to be dispensed with in these or other generalizations? Consider, say, (9)  If Moses says something, then it is true. There are various ways one might claim to be able to express (9) without the use of “true.” It is sometimes said that (9) amounts to the huge conjunction of the ascriptions of truth to all propositions expressed by Moses. But this is not so, since one cannot infer that huge conjunction from (9) or vice versa without knowing exactly what things Moses did and didn’t say. It may be said that what (9) directly expresses can be indirectly suggested by a list of a few hypothetical examples on the order of 36

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(10a)  If Moses says that Cain slew Abel, then he did.

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(10b)  If Moses says that Lilith slew Eve, then she did. (10c) If Moses says that there were giants in the earth, then there were. (10d) If Moses says that there were angels on the moon, then there were. plus an “et cetera.” But indirect suggestion is not direct expression. It is sometimes said that (9) amounts to the huge conjunction of all conditionals of type (10abcd) for all sentences of English. But this cannot be so if, as it seems, one can understand (9) without understanding all sentences of English. It may be said that much at least of the content of (9) can be expressed using anaphoric expressions as in (11) If Moses says that someone did something, then he or she did. But (11) does not express the whole content of (9). It covers (10ab) but not (10cd), since “someone did something” and “he or she did” cover only (propositions expressed by) subject-­predicate sentences with singular, personal subjects and active, past-­tense predicates. Ramsey would represent (9) symbolically, and then apply the supposed identity between a proposition and the proposition that it is true to eliminate the truth predicate, thus: (12a)  p(Moses says p  p is true) (12b)  p(Moses says p  p) But putting (12b) back into words produces nonsense: (13)  If Moses says something, then it. The problem is grammatical. “For any” demands a noun like “proposition” as complement, but “if . . . then” demands a sentence like “it is true.” Ramsey thought this a defect of English and other natural languages rather than of his notation, but the long and short of it is that Ramsey does not show that “true” is redundant in English. He only shows that it would become redundant if English were supplemented by his grammar-­defying quantifiers.

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That will be seriously significant only if Ramsey can claim that, though he was speaking English and using the word “true” long before he invented his kind of quantifiers, nonetheless he has an understanding of those quantifiers that is independent of any prior understanding of “true.” Such a claim is easy to make, but difficult to prove; it may be equally difficult to disprove, but the burden of proof is arguably on its proponents, not its opponents. Another way to render “true” redundant would be to add to our language prosentences, anaphoric expressions standing to sentences as anaphoric pronouns stand to nouns—in effect, expressions like “he or she did,” but without the limitation of covering only sentences of a certain specific form. Arthur Prior actually coined expressions to serve as prosentences, thus: (14)  If Moses says that somewhether, then thether. (Compare: If Moses says to go somewhither, then go thither.) These expressions even permit a definition of truth: The proposition that somewhether is true iff thether. But again there would be a question whether one can have an understanding of the new “somewhether” and “thether” independent of any understanding of the old “true.” The closest approximation to “somewhether/ thether” in English is perhaps “things are some/that way,” using which the Priorese definition of truth becomes: The proposition that things are some way is true iff things are that way. The definition only works, however, on the understanding that any proposition whatsoever, even one about how things used to be or might have been or ought to be, counts as being about “how things are.” 3 .2

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 Other Radical Theories

Historically, the next version of deflationism was the performative theory of P. F. Strawson, a leader of the ordinary language school, which dominated British philosophy in the decades immediately after World War II. It is a professed development of Ramsey’s views, and an explicit rejection of Tarski’s. On Strawson’s view, (2) does not, pace Tarski, express a proposition about the proposition expressed by (1), nor does it, pace Ramsey, express the same

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proposition as (1). Rather, it does not express a proposition at all. To utter (2) is not to say something about (1), but to do something to (1), namely, to endorse it. In jargon, it is not constative, but performative. (Other ordinary language philosophers said about “good” what Strawson said about “true,” that to say “That’s good” about something is not to say anything about it, but to do something to it, namely, to commend it.) Strawson gives little attention to examples beyond the simplest. By dropping Ramsey’s view that (2) expresses the same proposition as (1), he loses the redundancy view’s ability to account for examples like (3abcd). Needless to say, these cannot be regarded as instances of endorsement, though it is still open to Strawson to say that they are cases of doing something to, rather than saying something about: repudiating in the case of (3a), querying in the case of (3b), and so on. Perhaps with (9) one may speak of “blind” and “blanket” endorsement. But the need to find different kinds of “performances” (endorsement, repudiation, querying, blanket endorsement) to go with the different logical types of sentences involving “true” (affirmative, negative, interrogative, universal) is an unattractive feature of the theory, and performativism seems incapable of dealing with examples that are really logically complex. Consider, for instance, the following, in which truth is attributed not just generally but so to speak fractionally: (15) I don’t know what percentage of the things Berlusconi said in his last speech were true, but I suspect it’s well under fifty. What are we doing to the things Berlusconi said when we assert (15)? (Change “said in his last speech” to “did during his last term in office” and “true” to “good” to see the corresponding problem with performative theories of “good.”) For Strawson, (2) endorses rather than saying anything about (1), but one may well wonder why one cannot endorse a proposition by saying something about it. (Isn’t that how one endorses most other kinds of things?) In jargon, one may well wonder whether the constative and performative are exclusive. J. L. Austin, who invented the jargon, eventually came to conclude that

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40

they are not. In debate with Strawson, he defended a version of the correspondence theory (to be described in §5.2). Departing somewhat from historical sequence, the later view closest to Ramsey’s has been the prosententialism of Dorothy Grover and associates at the University of Pittsburgh, which dates from the 1970s. Where Prior proposed adding prosentences to English, Grover maintains that English already has them: We don’t need “thether,” because, despite the way it is written, “it is true” or “that’s true” is such a prosentence already. On this view, applying the truth predicate to a given sentence is equivalent to repeating the sentence itself in the same sense in which using a pronoun whose antecedent is a given noun is equivalent to repeating the noun itself. Grover can handle examples like (3abcd) simply by elaborating on the (postulated) analogy between prosentences and pronouns. When the pronoun “he” has “Whistler” as antecedent, saying the inflected and compounded form “he and his mother” of the pronoun is equivalent to saying the correspondingly inflected and compounded form “Whistler and Whistler’s mother” of the noun. So likewise, according to prosententialism, if “That’s true” has as antecedent “Beer is sold in grocery stores,” then saying “That used to be true, but no longer” is equivalent to saying “Beer used to be sold in grocery stores, but no longer.” Grover’s account is still dependent, however, on the not especially plausible construal of “blind” assertions like (7ab) as generalizations like (8ab). And prosententialism shares the fundamentally implausible claim of earlier radical theories that “is true” is not only not a genuine predicate in the fullest sense, but not even a separately significant unit, anymore than “soever” in “whatsoever” or “tofore” in “heretofore.” These features are avoided by Robert Brandom’s neoprosententialism. According to Brandom, “is true,” though not a predicate, is a separately significant unit of a special kind: a prosentence-­ forming operator. Attached to a phrase denoting a sentence, such as “Proverbs 16:18,” it forms, not a disguised generalization, but an anaphoric expression, “Proverbs 16:18 is true,” expressing the same proposition as the sentence denoted by the original phrase (namely, that pride goeth before a fall).

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Neoprosententialism, however, besides attributing to prosentences a kind of behavior without obvious precedent in the behavior of pronouns, seems to have an implausible implication of its own, namely that (16) The most famous conjecture in number theory is true. has the same content as Fermat’s theorem, which seems wrong, since (16) says something about fame, but Fermat’s theorem does not. Further bells and whistles can be added, but nonetheless (neo)prosententialism remains a minority view among deflationists. 3 .3  Disquotation

Historically, the next important view after Strawson’s was the disquotationalism of W. V. Quine, the most influential American philosopher of the 1950s and 1960s. Quine’s view counts as “moderate” in the sense that he took the apparent grammar of truth-­talk at face value. On his view, quotation turns a sentence into something noun-­like, while adding the truth predicate gives back a sentence, and one equivalent to the original, so that we understand what it is to say a sentence is true as well or as poorly as we understand the sentence itself. (If blind people cannot fully understand what sighted people mean by “Snow is white,” the same will be the case for “ ‘Snow is white’ is true.”) The back-­and-­forth is needed to express generalizations because quantifier expressions in English demand noun-­like expressions as complements. The indispensable role of the truth predicate in expressing generalizations that otherwise could only be suggested by lists of examples plus an “and so on”—compare (9) and (10abcd)—accounts for the appearance of “true” in theses of metaphysics, epistemology, and ethics (among other areas), without our having to suppose that the concept of truth itself is metaphysical or epistemological or ethical. Quine distinguishes immanent from transcendent use of the truth predicate, its application “at home” to sentences of our own language, and its application “abroad” to foreign languages. Initially, “at home” has to be understood quite narrowly. Quine was

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a militant antipropositionalist, and on Quine’s account the truth predicate specifically undoes direct, rather than indirect, quotation. Quine thus gives priority to the first over the second of the following: (17a)  “    ” is true iff     . (17b)  It is true that     iff     . The result is that initially the truth predicate applies only to our own sentences, here and now. For where indexicality is present, (17a) holds no further than that. To apply the truth predicate to the sentences of other speakers of our language, or even our own at other times and places, we must first transpose, in precisely the manner needed to turn direct into indirect quotation: (18a) wrong:  If Pope Benedict says to Tony Blair, “We disagree with you about gay rights,” then what he says to him is true iff we (the authors) disagree with you (the reader) about gay rights. (18b) right:  . . . iff he (the pontiff) disagrees with him (the ex-­premier) about gay rights. More importantly, to apply the truth predicate to speakers of other languages we must first translate, thus: (19) wrong:  If Carmen sings «L’amour est un oiseau rebelle, que nul ne peut apprivoiser», then what she sings is true iff l’amour est un oiseau rebelle, que nul ne peut apprivoiser. (19b) right:  . . . iff love is a rebellious bird that none can tame.

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How to transpose—to change pronouns and so forth in going from direct to indirect quotation—is something we know as part of our knowledge of our native language. How to translate is not; but with a language like French, where there is an established bilingual community, its customs and conventions establish a standard for correct translation if nothing else does.

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Quine notoriously held that with radical translation, as between our language and that of some group with which there has been no previous contact, there is no right and wrong translation, but only better or worse; and that there might be equally good translations of the same foreign sentence, one turning it into a truth, the other into a falsehood; and that there might even be languages that cannot be translated into ours at all. If all human languages on some deep level share some universal but species-­ specific features, the languages of intelligent extraterrestrials might be of this kind. The application of the truth predicate to foreign sentences we perhaps cannot translate determinately or at all would be a theoretical extrapolation underdetermined by evidence or even by all nonlinguistic facts, according to Quine, and he does not operate with it. Quine’s account only implies that if X says what we would say by saying “    ,” then what X says is true iff     . If X says “    ” and there is no “    ” such that what X says by saying the former is just what we would say by saying the latter, then Quine’s account does not imply anything about the conditions under which what X says is true. The view that the truth predicate with which ordinary English speakers actually operate is a theoretical extrapolation, to untranslatable utterances of others, of what started out as an immanent disquotation operator would no longer be deflationist in Quine’s sense. Such a view, however, would still remain light-­years away from any traditional rival views, and might be called quasi-­deflationist. In addition to this much-­discussed aspect of Quine’s view, there is a problem of formulation less often noted: How exactly are we to understand the “equivalence” between saying something and saying that it is true? Quine is much less explicit than Ramsey on this point. To claim interchangeability in absolutely all contexts would be to claim too much. For apart from the matter of stylistics, there is a problem about examples such as the following: (20a)  Fido believes that his master is at the door. (20b) Fido believes that it is true that his master is at the door.

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These do not seem equivalent. For surely a dog, though it may have the belief ascribed in (20a), does not have the concepts needed to have the belief ascribed in (20b), any more than (to borrow Wittgenstein’s example) it has those needed to believe that his master will come the day after tomorrow. Quine may hold that all T‑biconditionals are true, and that all T‑introductions and T‑eliminations are truth-­preserving, but claiming as much will not do as an explanation of equivalence. For the procedure of explaining truth in terms of a notion of “equivalence” itself explained in terms of truth or truth-­preservation is unacceptably circular: An explanation of what truth amounts to had better not itself use the T‑word. (Besides, if all one knows about X is that all biconditionals of the form “ ‘    ’ is X iff     ” are X, then for all one knows, X might be truth or might be falsehood.) 3 .4  Other

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Moderate Theories

The most-­discussed version of deflationism in recent years has been that advocated by Paul Horwich under the label minimalism. Horwich’s view differs from Quine’s by being propositionalist, and by offering a definite account of “equivalence.” According to minimalism, for one to understand the truth predicate is for one to have the disposition to accept any T‑biconditional proposition, or more precisely, anything that one recognizes as such. Against a background of classical logic, this is more or less the same as having the disposition to infer the conclusion proposition from the premise proposition in any T‑introduction or T‑elimination, or more precisely, anything one recognizes as such. The requirement of recognition is imposed in an attempt to get around the problem of mistakes that we correct when they are pointed out to us, and especially the problem of instances no one is disposed to accept because they are simply too long and complicated to take in. (There remain further objections, beyond these problems, to any identification of knowledge of meaning with a disposition, but the matter is too deep and elusive to be gone into here.)

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Horwich emphasizes that his theory does not leave understanding the notion of truth dependent on understanding some A‑notion, acceptance or assertion, or some I‑notion, inference or implication. To be sure, one can only think a biconditional is acceptable if one has the concept of acceptance, but one can accept a biconditional without having the thought that it is acceptable. To suppose otherwise, that before accepting something one must accept that it is acceptable, involves an infinite regress. This claimed advantage of freedom from entanglements comes at a cost. Minimalism is open to a version of Tarski’s objection to the procedure of simply taking T‑biconditionals as ­axioms, namely, that such a conception of truth leaves us unable to deduce certain general laws. Consider these successively more general formulations: (21a) If it is true that the tomato on your plate is ripe and juicy, then it is true that the tomato on your plate is ripe. (21b) For any tomato, if it is true that it is ripe and juicy, then it is true that it is ripe. (21c) For any conjunction, if it is true, then its first conjunct is true. The natural ways to argue for these would be as follows: (22a) Suppose it is true that the tomato on your plate is ripe and juicy. It follows (by T‑elimination) that it is ripe and juicy, hence (by the logic of “and”) that it is ripe, hence (by T‑introduction) that it is true that it is ripe. (22b) Consider any tomato, call it Tom, and suppose it is true that it is ripe and juicy. From its being true that Tom is ripe and juicy, it follows that Tom is ripe and juicy, hence that Tom is ripe, hence that it is true that Tom is ripe, that is, that the given tomato is ripe. (22c) Consider any conjunction, say the conjunction that things are thus and things are so, and suppose it is true.

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From its being true that things are thus and things are so, it follows that things are thus and things are so, hence that things are thus, hence that it is true that things are thus, that is, that the first conjunct is true.

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Here (22a) requires only acceptance of T‑biconditionals for specific sentences, expressing specific propositions. Not so with (22bc). For “Tom” does not really denote any specific tomato, and “Tom is ripe” does not really express any specific proposition. That goes double for “things are thus,” which is really only a placeholder for a proposition-­expressing sentence. It is not plausible to claim (21b) or (21c) is arrived at inductively from examples like (21a); but arriving at either deductively requires “schematic reasoning,” as the type of reasoning in (22bc) is called. What seems needed is a disposition to infer a conclusion from a premise by T‑introduction and T‑elimination (or, what is more or less equivalent, to accept a T‑biconditional) even in the case of placeholders. That and more also would presumably follow if understanding the truth predicate were identified, not with a disposition to accept each T‑biconditional, but with a recognition that all T‑­biconditionals are acceptable or assertable, or that the conclusion is inferable from or implied by the premise in all ­T‑introductions and T‑eliminations. Such an identification makes acquisition of the T‑notion depend on prior possession of an A‑notion or I‑­notion, and so has the disadvantage of leaving one unable to explain A‑notions or I‑notions in terms of the T‑notion, lest a chicken-­and-­egg problem arise: One cannot explain assertion as utterance purporting to be true, or explain implication in terms of truth-­preservation. If one is comfortable with talk of propositions and properties, the view being contemplated can be formulated as saying that truth is the property of propositions for which its ascription to a given proposition and the proposition itself always imply each other, or the property for whose possession by a given proposition the proposition itself gives the necessary and sufficient condition. Such is Colin McGinn’s self-­effacement theory. It is not always recognized as a variety of deflationism. Thus there is at present not only no consensus among deflationists as to what is

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the optimal formulation of deflationism, but not even any consensus as to which views count as genuinely deflationist. Nonetheless, the idea that something fitting our initial characterization of deflationism must be right remains widespread. 3 .5  Sloganeering

In recent years this idea has often been accompanied by rhetoric of a kind one does not find in Ramsey or Quine—it may derive ultimately from the pragmatist philosopher or antiphilosopher Richard Rorty—involving such slogans as “Truth is not a property” or “Truth is not a substantive property.” The thought perhaps is something like the following. The equivalence principle suffices to explain the usage and usefulness of “true” (notably, for avoiding repetition and for formulating generalizations). We do not have to posit that the use of “true” is to report the detection of the presence of some interesting property all truths share. This being so, it would be an improbable coincidence if there were any such interesting property. Despite the recent popularity of such slogans, however, there is no hope that any slogan of the kind could provide a characterization of deflationism acceptable to all deflationists. The reason why not is as follows. Colloquially we may always expand “so-­and-­so does such-­and-­such” to “so-­and-­so has the property of doing such-­and-­such.” Such linguistic padding is called pleonasm, and some philosophers consider this colloquial, pleonastic notion of property the only legitimate one. Others, however, think the only legitimate notion of property is some more substantive one, on which perhaps a natural predicate like “is a human being” would pick out a property, while an artificial one like “is either female or else both over six feet tall and also left-­handed” would not. Yet others take “property” to be ambiguous between a pleonastic and a substantive sense, both legitimate. Yet others are uncomfortable with any talk of properties. These divisions exist among deflationists as much as among other philosophers, and cut across the distinction between moderates who do and radicals who do not grant that “is true” is a predicate.

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Now a radical who admitted only the pleonastic notion of property might agree on the slogan “Truth is not a property,” as might a moderate who admitted only the substantive notion. But a radical who admitted only the substantive notion would find the slogan an understatement, since merely to say that “is true” does not indicate a substantive property is not to say that it is not a predicate. A moderate who admitted only the pleonastic notion would find the slogan plain wrong, since if “is true” is a predicate, it ipso facto indicates a pleonastic property. Those with other combinations of views might find the slogan ambiguous or meaningless. Similar problems plague any slogan formulated in terms of properties. We think our third thesis, that there is nothing to be said about what it is to be true, once one has said what it means to call something true, does a little better. The thought is that whether or not an account of meaning leaves room for a further question about being depends on the nature of the account of meaning. For example, if the right definition of “hot” is something like “having whatever physical property it is that causes heat sensations,” then even if one fully understands what it is to be a physical property or a cause or a heat sensation, still there would be room for a question about what heat is, namely, the question “Which physical property is it that causes heat sensations? The presence of caloric fluid? Rapid random motion of molecules? Or what?” But if the right definition of “hot” is something like “having a temperature in °F significantly higher than 150,” then if one fully understands what it is for an object to have a certain temperature in °F, and what it is for a number to be significantly higher than 150, then there seems to be no room for any further question about what heat is. For deflationists, the equivalence principle tells the whole story about the meaning of “true,” and that story, though it is not strictly speaking a definition, seems to be much more like the second than the first candidate definition of “hot,” and to be not at all the sort of account of meaning that leaves room for a further question about being. Another slogan sometimes met with as a capsule summary of deflationism is “There are no substantive questions about truth.” This may easily be made to seem absurd. For suppose a

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Frenchman, Monsieur Orgon, emits certain sounds, and a bystander knowing his circumstances and his language tells us he has said something true. Then surely we may ask why, and surely our question is both substantive and about truth. The deflationist has an answer to this objection, but before stating it, a word about quotation marks is called for. Inverted commas may be used to designate at least eight different kinds of things: phonological or orthographic tokens or types, or semantic tokens or types from our own or a foreign language. In writing about Tarski we followed his usage, taking quotation to designate an orthographic type. In stating the deflationist answer to the objection about Monsieur Orgon, a different usage will be in order. In discussing this example it will be convenient to designate phonological entities by using italics and approximate phonetic spelling, and to use ordinary English-­style quotation marks to designate expressions of English with their ordinary English meanings, and French-­style quotations marks to designate expressions of French with their ordinary French meanings. The deflationist account of the supposed example of a substantive question about truth is then that the answer to the question comes in three parts: (i) by emitting sounds very roughly representable as mah mehr eh fahshay, Monsieur Orgon is uttering the French sentence «Ma mère est fâchée», which translates as “My mother is angry,” which transposes to “His mother is angry”; (ii) “His mother is angry” is true iff his mother is angry; (iii) his mother is angry. Deflationists claim that (i) and (iii) are substantive but about meaning and emotion, respectively, not truth, while (ii) is about truth, but is a triviality. The original question thus can be substantive and about truth without, so to speak, being substantive about truth, much as (to borrow Plato’s example) a dog can be yours, and a father, without being your father. 3 .6  Reference

Most deflationists about truth are likewise deflationists about other alethic notions. The most important of these is probably the one we called denotation in our discussion of Tarski, and

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that is elsewhere called reference. (We note that, though we and many others use these terms interchangeably, they are sometimes distinguished. Some may use denoting for what certain expressions do, and referring for what people who use those expressions thereby do. Others, holding that the relation of name to named and the relation of description to described are importantly different, may reserve referring for the former, while allowing denoting to cover both.) The deflationist view is that, call the relation what you will, there are no substantive questions about it, any more than about truth. Why does Monsieur Orgon, by emitting the sounds mah mehr, refer to Madame Prunelle? Well, (i) by emitting the sounds he utters «ma mère», which translates as “my mother,” which transposes to “his mother”; (ii) “his mother” refers to an individual iff that individual is his mother; and (iii) Madame Prunelle is his mother. Everything substantive belongs to (i) semantics or (iii) genealogy; only a triviality belongs to (ii) the theory of reference. More specifically, prosententialists about truth tend to be pronomialists about reference, minimalists about truth tend to be minimalists about reference, disquotationalists about truth tend to be disquotationalists about reference, and so on. The referential analogue of (17a) reads as follows: (23a) “    ” refers to an individual iff he/she/it is     . Curiously, a referential analogue of (17b), undoing indirect rather than direct quotation, is harder to formulate. Some write (23b) An individual is the one referred to as     she/it is     .

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iff he/

but this is not entirely satisfying, since in ordinary language the form of words “referred to as” seems most often used to distance the speaker from a certain way of referring to an item, as if short for “referred to by some other people as.” For instance, a political reporter who says “The person referred to as The Dear Leader is Kim Jong-­Il” probably does not herself regularly refer to the ruler of North Korea as The Dear Leader, unless perhaps ironically.

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Deflationism

While deflationists about truth may also be deflationists about reference, it must not be automatically assumed, just because some doctrine has come to be labeled “deflationism about X” in the literature, that deflationists about alethic notions will be committed to it. Deflationism about truth is subject to enough criticisms of its own, without saddling deflationists about truth with any and every doctrine to which the “deflationist” label has been applied. The label has become rather fashionable of late, so there are quite a few such doctrines, not all of which are even consistent with each other. Criticism of any one form of deflationism about truth is implicit in advocacy of any other, and we have already seen many issues about whether this or that version can account for this or that use of the truth predicate (to endorse or repudiate or query, to do so blindly or generally or fractionally, and in inflected form and transcendently and for placeholders). Deflationist sloganeering has provoked criticism going far beyond such issues, so that deflationism finds itself embattled on many fronts. But there is at least this much to be said for it: that its opponents are by no means agreed what is supposed to be wrong with it.

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Indeterminacy

The equivalence principle (“Saying something is true is equivalent to just saying it”) has been less controversial than deflationism’s other theses, but it has been challenged by examples of what we will call indeterminacy. These are cases where we have a question “Are things thus?” to which neither the affirmative nor the negative answer seems appropriate, and not just because of ignorance on our part. If the question “Are things thus?” cannot be answered “Yes” or “No,” then it seems the corresponding declarative “Things are thus” cannot be called “true” or “false,” and so it seems we have a counterexample to the bivalence principle, according to which every proposition is either true or false. And any threat to the bivalence principle is a threat to the equivalence principle. For if we were to assume (A) things are thus, then applying the equivalence principle, in the form of the rule of T‑introduction, it would follow that (B) it is true that things are thus. But we are supposed to be dealing with a counterexample to bivalence, hence a case where (~B) it is not true that things are thus (and also not false). It follows that (~A) things are not thus (here using the rule of classical logic that allows us, having seen a conclusion B to follow from an assumption A, to infer the negation ~A of the latter given the negation ~B of the former). From this intermediate conclusion that things are not thus, it then follows by T‑introduction that it is true that things are not thus, which is to say, that it is false that things are thus. But again we are supposed to be dealing with a case where it is not false that things are thus (and also not true). And so we have a contradiction. Thus it is that indeterminacy threatens the equivalence principle, a central thesis of deflationism, and so centrally threatens deflationism. This threat does not depend on any distinctively realist

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or antirealist assumptions. On the contrary, since the equivalence principle is at least a peripheral thesis of most forms of inflationism, indeterminacy at least peripherally threatens them, too (though inflationists, whose theories are generally more elaborate than deflationist theories, may claim to have more resources available for warding off such threats). In this chapter we will concentrate mainly on two alleged classes of examples, involving phenomena of presupposition and vagueness, on each of which there is a vast literature in linguistics and philosophy of language. We first briefly describe the two phenomena, and then enumerate various conceivable lines of response to the problems they raise. Finally we examine a purported third type of case, involving a purported special kind of relativity. 4 .1  Presupposition

The fallacy of many questions was illustrated in antiquity by the example “Have you stopped beating your father?” This question is said to have been addressed by one Alexinus to one Menedemus, son of Cleisthenes, who refused to give a yes-­or-­no answer. It really should be two questions, “Did you ever beat your father? If so, have you stopped?” where if the answer to the first is negative, the second does not arise. Either answer, yes or no, to the original question “Have you stopped?” seems to involve an admission of father-­beating, and thus to be inappropriate. Eubulides, to whom the pseudomenos (liar) is attributed, is also credited with another paradox, the keratines (horned one), turning on this phenomenon: What you have not lost, you have; you have not lost horns; therefore, you have (cuckold’s) horns. Modern linguistics and philosophy of language discuss the same phenomenon under the rubric “presupposition.” Thus one says that (1) The phlogiston has been removed from the air under the bell jar. presupposes that there was phlogiston in the air under the bell jar to be removed, and hence that there is such stuff as phlogiston.

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Since, contrary to eighteenth-­century chemical opinion, there is not—what early chemists described as the presence of phlogiston modern chemists would describe as the absence of oxygen, and vice versa—we have a case of presupposition failure. Being presupposed by a sentence is supposed to differ from being entailed by it, or implied by its meaning, by the fact that the negation of a sentence is supposed to have the same presuppositions as the original, as is the corresponding interrogative. By contrast, a conclusion is never entailed both by an hypothesis and by its negation except in the case where the conclusion is logically valid or analytic. “Phlogiston exists” assuredly is not logically valid or analytic, but it is said to be presupposed both by (1) and by its negation, and by the mere question “Has the phlogiston been removed from the air under the bell jar?” That is why a negative may seem as inappropriate as an affirmative in answer to the interrogative, giving an example of indeterminacy. 4 .2

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 Vagueness

A stock example of vague predicates is provided by color words. On a simplified model of language-­learning, we learn a word like “red” by being given certain paradigms, or samples of red things, and certain foils, or samples of things that are not red but orange or brown or purple or pink, plus certain principles for projecting beyond such examples. On this model, we may be left with certain borderline cases. Suppose, for instance, a large number of color chips are placed in a hopper, one is drawn at random but kept covered, and players may bet on whether it is red or not, after which it will be revealed. Then it may happen that after some have guessed “Yes, it is red” and others “No, it is not red,” the chosen chip is uncovered and turns out to be exactly halfway between the most orangish of the red paradigms and the most reddish of the orange foils. In this case it may seem that the only appropriate thing to do is to cancel all bets. If one takes the relevant projection principle to be that anything intermediate in color between two red things counts as red, and anything intermediate in color between two nonred things counts as nonred, then the case of the problem chip seems to be

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underdetermined, and an example of a truth-­value gap, something neither true nor false. If instead one takes the relevant projection principle to be that anything very similar in color to a red thing counts as red, and anything very similar in color to a nonred thing counts as nonred, then the case of the problem chip seems to be overdetermined, and an example of a truth-­value glut, something both true and false. For a paradigmatically red and a paradigmatically orange chip can be connected by a series of intermediates each scarcely (if at all) distinguishable from the next. Numbering the chips from zero (pure red) to a thousand (pure orange), we seem able to argue: Chip #0 is red, so chip #1 is red, so chip #2 is red, and so on, until we reach the conclusion that our problem chip #500 is red; and inversely that chip #1000 is nonred, hence chip #999 is nonred, hence chip #998 is nonred, and so on until the chosen chip #500 is nonred. This puzzle is a version of a pair of paradoxes attributed to the ubiquitous Eubulides. According to the sorites (heap) paradox, since adding one grain of sand won’t turn a nonheap to a heap, adding a million one by one won’t do so either. According to the phalakros (bald one) paradox, since plucking one hair won’t turn a nonbald man bald, neither will plucking a million one at a time. The phenomenon of vagueness is not limited to “red” and “heap” and “bald,” but is all-­pervasive. Frege sometimes wrote as if he took this to show that natural languages are defective, and should ideally be replaced by some perfectly precise artificial language. But actually the vagueness of natural language is more virtue than vice, since if we were forced to say only what could be said perfectly precisely—to describe colors by exact numerical wavelengths, for example—we would hardly be able to say anything at all. The vagueness of ordinary discourse is inseparable from its utility. It does, however, pose problems about what to say in borderline cases, of which the threat to the equivalence principle is only one. 4 .3  Denial,

Disqualification, Deviance

Denial. We turn next to consideration of possible defenses against the threat posed by borderline cases, and the similar threat posed

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by presupposition failure. One defense would be simply to deny that “You have stopped” and “Chip #500 is red” or other sentences like them provide examples neither true nor false. This would require denying the reality of the linguistic phenomena—failed presupposition and borderline cases—supposedly illustrated by the examples. One might claim, for instance, that so-­called presuppositions are simply a subclass of entailments. On such a view, “Have you stopped?” is not a case where neither a yes nor a no answer is appropriate. The answer to the question is a plain “No,” because having stopped entails having started, and you never started. Such denialism is implausible to the degree that the evidence leading linguists and philosophers of language to posit a category of presupposition is convincing. The question “Is chip #500 red?” may remain one we are undeniably unable to answer, but it has been claimed that this is just because of ignorance on our part, that in such a case it still is either true or false to call the chip “red,” even though nothing about our usage of “red” enables us to know which. One version of this epistemicist view holds that somewhere between the most orangish paradigm and the most reddish foil there is a “natural joint,” visible perhaps to God but not to us, and that though our usage does not track this joint exactly, owing to the joint’s naturalness a kind of “magnetism” attracts our word “red” to it, so that without any help from us the meaning of the word aligns perfectly with the joint, thereby making chip #500 count as red or count as nonred, depending on which side of the joint it lies on. This view may seem fantastic, but that does not distinguish it from many other views in the vagueness literature. Disqualification. Another defense would be to claim that though the sentences “You have stopped” and “Chip #500 is red” are neither true nor false, which is to say, neither express true propositions nor express false propositions, that is not because they express propositions that are neither true nor false, but simply because they do not express propositions at all. That would disqualify them as counterexamples to bivalence or for that matter any other thesis about propositions. Now some sentences indeed do not express propositions. Interrogatives and imperatives are uncontroversial examples. Even

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sententialists who avoid the notion of proposition need something like a distinction between proposition-­expressing sentences and others, and have introduced a bit of jargon, “truth-­apt,” to mark this distinction. The defense against the threat to the equivalence principle being contemplated is the easy one of just claiming that the problem sentences are like interrogatives and imperatives in not being proposition-­expressing or truth-­apt. This line is very often taken, independently of any concern about debates over the nature of truth, with failed presuppositions, and is sometimes taken with borderline cases as well. However, it faces a serious objection. It is widely held that people’s actions are to be explained in terms of their beliefs (and their desires), and also widely held that what one believes when one believes is a proposition. Given these widely held views, if someone when asked “Why did you act as you did?” replies with evident sincerity, “Because things were thus and so, or so I believed,” then there is a presumption that the person did believe that things were thus and so, and that there is a proposition to the effect that things are thus and so for the person to believe. Citation in explanation of action thus provides a presumptive sufficient condition for the existence of a proposition, and our examples seem to pass this test. Asked why he put the unconscious mouse under the bell jar to revive it, an eighteenth-­ century chemist might well have answered, “Because I was assured the air under it had been dephlogisticated.” Asked why she staked her money on red in the color-­chips game, a player might well say, “Because there’d been a run of red, and I had a hunch this one would turn out to be red, too.” Deviance. Another defense would be to concede that there are counterexamples to bivalence, but deny that these give counterexamples to the equivalence principle. Since the principle of bivalence follows from the equivalence principle given classical logic, to take this line one would have to reject classical logic. Some deviant logic would then presumably be needed, but a number of them have been worked out in the literature on indeterminacy. A proposal called the Kleene strong trivalent logic was developed to deal with cases of presupposition failure. This logic tells how the status of a compound as true or false or neither is supposed to be determined systematically by the status of its

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components. For example, it counts a disjunction as true iff either disjunct is true, and false iff both disjuncts are false, and neither true nor false (a status variously called neuter or indeterminate or gappy) in all other cases. This logic of truth-­value gaps has a mirror-­image logic of truth-­value gluts. To deal with vagueness, some have proposed fuzzy logics, according to which, contrary to all the theories of truth considered in this book, there are infinitely many truth values or degrees of truth, from zero or pure falsehood to one or pure truth, through every real (or surreal) number between; “Chip #500 is red” would presumably be a half-­truth. A different proposal called the Van Fraassen supervaluation logic was also developed to deal with vagueness. The thought behind it is that the very circumstance that there is nothing to make “red” the right thing to call the problem chip is enough to make “orange” a right thing to call it, and vice versa. One needn’t adopt the attitude that everything not obligatory is forbidden. There may be no natural line between red and orange, but there are several places where it would be permissible to draw an artificial line, and “precisify” the concept of red. Paradigm chips among others are determinately red, or red for every permissible precisification, while foil chips among others are determinately nonred, or nonred for every permissible precisification, but Chip #500 and others like it are neither determinately red nor determinately nonred. Something is counted supervaluationally true (or false) iff it is true (respectively false) for any permissible precisification. Thus “Such-­and-­such a chip is red” is true (or false) iff the chip in question is determinately red (or determinately nonred), but is neither true nor false in borderline cases, which count as red on some precisfications and nonred on others. Before the reader becomes too enthusiastic about nonclassical logics, it must be noted that the logics we have been discussing were not originally motivated by a desire to save the equivalence principle, and whatever their other merits, as originally designed they fail to save the equivalence principle simply because as originally designed they do not say anything at all about sentences involving the truth predicate. But we will meet some of these logics again when we turn later to consideration of “solutions” to the alethic paradoxes.

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Indeterminacy 4 .4  Doublespeak,

Dependency, Defeatism

Doublespeak. Another defense might begin by positing that the word “not” has, besides its ordinary meaning in expressing logical negation, another, special meaning expressing rejection of the saying of something rather than negation of the thing said. It could then be claimed that though one is tempted to say of such an example as “You have stopped” or “Chip #500 is red” that it is not true and also not false, perhaps the two occurrences of “not” here are expressing the special meaning and not the ordinary one. In that case, by saying the examples are “not” true and “not” false, one is not really contradicting the principle of bivalence, which would require one to say “not true and not false” in the ordinary, not the special sense. This suggestion looks like a clear violation of the methodological maxim known as Occam’s Eraser (“Senses are not to be multiplied beyond necessity”), an analogue of the better-­known Occam’s Razor (“Entities are not to be multiplied beyond necessity”). Positing an ambiguity in the word “not” looks like proposing a linguistic thesis merely to get out of a philosophical difficulty. But it turns out that linguists have, for their own purely linguistic reasons, been moved to posit a “metalinguistic” negation distinct from ordinary negation, whose use indeed only commits us to its being in some way inappropriate to say something, and not to the logical negation of the thing said. The distinction is supposed to be illustrated by such pairs as the following: (2a) In that country, they don’t eat tomatoes, they suspect they are poisonous. (2b) In this country, we don’t eat tomahtoes, we eat tomaytoes. (3a)  He’s not intelligent, his I.Q. is 80. (3b)  She’s not intelligent, she’s a world-­class genius. Linguistic evidence for the reality of the distinction includes the fact that (2b) would be pronounced with a distinctive stress (indicated by our italics) not present in (2a), and the fact that in (3b)

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“not intelligent” cannot be contracted to “unintelligent” as it can be in (3a). The two-­negations theorist is in effect answering the question “Have you stopped?” by saying “Neither yes nor no would be an appropriate answer,” and would answer the follow-­up question “Well, would it be true to say that you have stopped?” in exactly the same way. Giving the same answer to both questions is, of course, just what one would expect of a defender of the equivalence principle. If there is any problem here it is that both answers are what lawyers call nonresponsive. The question whether “You have stopped” is true or false or neither or both does not go away simply because some theorist declines to answer it. In short, the two-­negations approach looks less like a solution than an evasion. Dependency. A somewhat less unresponsive answer than a mere “I don’t say yes and I don’t say no” would be “It all depends.” It may be suggested that in our examples of indeterminacy, the problematic question might appropriately be answered “Yes” in some contexts and “No” in others, and that it is only on account of its context-­dependence that it cannot be answered either way in isolation. For instance, with the color chips, it might be suggested that it is natural to call the borderline chip “red” in some contexts (say where one is working with pure red, orangish red, borderline, and pure orange, but no reddish orange) and to call it “orange” in others (say where one is working with pure orange, reddish orange, borderline, and pure red, with no orangish red). Less plausibly perhaps, it might be suggested that “Have you stopped?” is in some contexts most naturally taken as “Do you agree that if you ever did, you have since stopped?” and answered “Yes” (understanding the conditional as in mathematics, so that it counts as true if its antecedent is false), while in other contexts it would be most naturally taken as “Do you agree that you used to, but have now stopped?” and answered “No.” The view that the paradox of the horned one involves some kind of context-­shift between one premise and the other may not be especially common or inviting, but a view that context-­shift is crucial to the paradoxes of the heap and the bald one is fairly widely held. More generally speaking, “contextualism” of one sort

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or another is an approach to the problems of vagueness that is of growing popularity among philosophers of language, though like almost any philosophical ’ism it also has its detractors. Defeatism. Another course would be to admit defeat, to surrender the equivalence principle and try to reconstruct one’s theory of truth so as to retain its most important features while giving up whatever features committed it to the now-­abandoned position that the equivalence principle holds as an exceptionless law. For a deflationist, reconstruction would presumably involve (i) coming up with some amended equivalence principle; (ii) revising the thesis that the equivalence principle is all there is to the meaning of the truth predicate so as to refer to the equivalence principle as amended; (iii) retaining the thesis that an account of the meaning of the truth predicate is all that is required of a theory of truth. If presupposition failure were the only problem to worry about, an amended equivalence principle might say that saying something and saying it is true are equivalent except that presuppositions of the former become entailments of the latter. It would be left to the theory of meaning, not of truth, to identify what the presuppositions of a given sentence are. This amended equivalence principle implies that if the presupposition of the question whether p is that q, then it is true that p iff q and p, and false that p iff q but not p, and neither iff not q. This proposed adjustment to the deflationist position in order to accommodate presupposition failure is painless enough that one might hope that the problem of borderline cases could be handled by assimilating it to presupposition failure. Perhaps “Chip #500 is red” can be construed as having the failed presupposition that chip #500 is either noticeably closer in color to one of the paradigms than to any of the foils, or else noticeably closer to one of the foils than to any of the paradigms. 4 .5  Relativity

Our discussion thus far, needless to say, does not settle the question of the bearing of indeterminacy phenomena on the theory of truth; nor can the extensive literature on the issue be said to

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resolve it decisively. A question mark, or rather, a cluster of them, hangs over the subject. Hartry Field, one of the most prolific writers on deflationism, urges its sympathizers to adopt it only as a methodological stance, something to be held onto provisionally until forced to give it up. The same would go for most of deflationism’s rivals, to the degree that they, too, are committed to the equivalence principle. Our discussion has been not only inevitably incomplete in its treatment of the vast topics of presupposition and vagueness, but also incomplete in that it addressed only those two types of alleged indeterminacy. We should, and in this section do, address at least one further type, a certain kind of alleged relativity that has inspired the development by John MacFarlane and others of a recently much-­discussed view called by its sympathizers “truth relativism,” and by the less sympathetic “new-­age relativism.” The kind of relativity in question is claimed to affect judgments of probability, of morality, and of a number of other matters. We will concentrate on moral relativity, but let us first, by way of contrast, consider the less problematic case of legal relativity. Consider the following exchange: (4)  Abortion is legally prohibited. (5)  That’s true in Chile, but not in China. Everyone accepts legal relativism, the view that legality is relative to legal systems, which vary with time and place. (4) is in the present tense, indicating the pertinent time is the present, but it gives no indication of the pertinent place, leaving an opening for (5). But do we really want to say that there is a single item, the proposition that abortion is legally permissible, that is true in some places and not others? The more usual view is that what (5) conveys is not that, but something like the following: (5) It’s true that abortion is legally prohibited in Chile, but not that abortion is legally prohibited in China. 62

On this view, considered in isolation (4) fails to express a specific proposition, while uttered in a context where Chilean (respectively, Chinese) law is understood to be at issue, it expresses a true

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(respectively, false) one. If two legal experts have been discussing the legal situation in different countries for so long that they have lost track of which country is currently under discussion, and one who thinks they are still on Chile says (4), while one who thinks they have moved down the alphabet to China says (4)  Abortion is legally permitted. then the two have miscommunicated rather than disagreed. Considered out of context, the form of words “The proposition that abortion is legally prohibited” fails to denote anything, while taken in a context where some specific legal system is tacitly understood to be intended, it denotes the proposition that abortion is legally prohibited under that system. As with “I’m glad to meet you,” the context of utterance must supply something not supplied by the semantic sentence type to get us from uttering the sentence to expressing a specific proposition. There is nothing visible or audible comparable to the indexical pronouns “I” and “you” to mark this context-­dependence, but philosophers speak of hidden indexicality in (4) or (4). Contrast this situation with a situation where a religious fundamentalist and a radical feminist say, respectively, (6a)  Abortion is morally prohibited. (6b)  Abortion is morally permissible. Of course the proponent of (6a) is judging by religious fundamentalist standards, and is aware of doing so, while the proponent of (6b) is judging instead by radical feminist standards, and again is aware of doing so. But each thinks that he or she is doing something more than just judging by certain moral standards or a certain moral system: Each thinks that his or her moral system is right while the other’s is wrong. (Or at least, this is certainly the attitude of the religious fundamentalist and probably the attitude of the radical feminist.) In this case there seems to be genuine disagreement, not miscommunication, and so it seems there is a single proposition that abortion is morally prohibited that the two are disagreeing about, which the form of words “Abortion is morally prohibited” suffices to express, without any help from any context of utterance.

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The opposite attitudes of belief and disbelief on the part of the fundamentalist and the feminist towards one and the same proposition may explain their opposite actions, say of demonstrating and counterdemonstrating, on one and the same occasion. But does this proposition have a truth value, or does it exemplify a further type of indeterminacy, beyond failed presupposition and borderline cases? For moral absolutists, who take there to be a single objectively correct moral standard or system, the proposition expressed by (6a) is either true or false, according as that objectively correct moral standard or system does or does not prohibit abortion. For moral relativists, who hold the opposite view, the proposition expressed by (6a) may exhibit a truth-­value gap. For one version of moral relativism, the error theory, ordinary people today still tend to be implicit moral absolutists, as five hundred years ago ordinary people tended to be implicit geocentrists; but today as five hundred years ago, ordinary people are just making a mistake, since absolute morality is as nonexistent as phlogiston. For the error theorist, moral relativity provides examples of indeterminacy, but examples of a familiar kind, raising no new issues: essentially, cases of presupposition failure. But many philosophers have shown themselves equally reluctant to endorse moral absolutism or to accuse ordinary people of making some systematic error akin to belief in phlogiston. Consequently a large literature on the supposed tension between deflationism and “nonfactualism,” especially about morality, has developed. So-­called truth relativism is one fairly recent and rather influential position on this issue. Often, as in the legal example, something that can only be supplied by the context of utterance is needed to get us from a semantic sentence token to a proposition. According to “truth relativism,” sometimes something that can only be supplied by a “context of assessment” is needed to get us from a proposition to a truth value. The anthropological or sociological claim (7) Abortion is prohibited according to religious fundamentalist morality, but not according to radical feminist morality. 64

should be no more controversial than the claim

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(8) Abortion is prohibited under Chilean law, but not under Chinese law. But “truth relativism” in effect construes the uncontroversial (7) as supporting a more controversial theoretical claim: (9) The proposition that abortion is morally prohibited is true in a religious fundamentalist context of assessment, but not in a radical feminist context of assessment. Whether there is any need or use for the apparatus of contexts of assessment is at present still hotly debated, which is to say that the reality of the phenomenon of “truth relativity” is hotly debated. So much so, indeed, that comparatively little attention has been given to the follow-­up question whether, if there really is such a phenomenon, it poses a significantly different and perhaps more serious threat to the equivalence principle than do the phenomena of presupposition and vagueness. At present “truth relativism” is another question mark in the cluster of question marks hanging over the center of deflationism (and over the periphery at least of most of its rivals). 4 .6  Local vs

Global

The views that truths about law and morality that we have just been discussing, to the effect that they are both in different ways relative, are examples of “local” relativisms, often contrasted with “global” relativism, the view that all truth is relative. Global relativism is sometimes considered another theory of truth of the same sort as the correspondence, coherence, utility, deflationist, and so on, but it does not seem to be a theory that any significant number of analytic philosophers have ever been seriously tempted to adopt. Our brief discussion of it will therefore be something in the nature of a digression. Global relativism is often claimed to be self-­defeating. One argument claims that if the thesis of global relativism were true, then according to itself it would only be true relative to something, so presumably false relative to something else. This objection is not compelling. After all, though all legality is relative,

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breathing happens to be lawful under every law code (though perhaps heavily taxed under some). Might it not be that though all truth is relative, the thesis that this is so is true relative to every whatever-­it-­is that truth is supposed to be relative to? A perhaps better argument is due to Paul Boghossian. Suppose all truths were only true relative to something or other. Then the truth that grass is green, for example, would only be true relative to something. But if it is true that grass is green relative to something, then that too would only be true relative to something: It would only be true relative to something that it is true relative to something that grass is green. But if this is true, then . . . . This regress shows that the only genuine truth in the vicinity is the “infinite” proposition that we can only express in English using ellipsis: “It is true relative to something that it is true relative to something that . . . it is true relative to something that grass is green.” Surely the idea that these sorts of propositions are the only truths is absurd. Boghossian has attempted to parlay this argument against global relativism into a challenge to various local relativisms as well. He assumes that every local relativist owes us an account of the things to which he or she takes truth in the pertinent locality to be relative—let us call them “systems,” as in the legal case, just to have a word for them, without reading too much into the word—and then poses a dilemma. Are these systems themselves true or false absolutely or only relatively? On the one hand, if systems are only relatively true, a regress argument like that against global relativism looms: Each system is true relative only to some system itself true only relative to some system, and so on. And saying that each is true relative to itself only trades regress for circularity, leaving systems equally baseless. On the other hand, if one system is absolutely true and the others absolutely false, then we have after all a notion of absolute truth for the local area where truth was supposed to be merely relative, namely, truth relative to the system that is absolutely true. As we have crudely formulated it, the argument proves too much: A general refutation of all local relativisms is impossible, since whatever may be the status of moral relativism, legal relativism is a truism. The best response of the local relativist is

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presumably to claim that systems just aren’t the sorts of things that can be evaluated for truth or falsehood. Legal and moral systems, for instance, invite being thought of as systems of imperatives, not declaratives. A charge of “baselessness” may still be raised, but depending on the “locality” in question, the local relativist may not be afraid of it. (Boghossian’s special target is local relativism about “epistemic justification,” where a charge of baselessness may be less easy to shrug off than in some other areas.) Whatever may be the case with other local relativisms, there is one that seems to raise special problems because it threatens to collapse into global relativism. The local relativism in question is relativism about judgments, not of legality or morality, but of truth. Relativism about this kind of judgment is what “truth relativism” would naturally be taken to mean, were that label not already in use for MacFarlane-­type views, but the point is that, call it what you will, it threatens to collapse into global relativism: If all the truths about truth are only true relative to something, then it seems all truths are only true relative to something. For consider an arbitrary truth, say the truth that grass is green. This truth is not about truth, in any straightforward sense, but it seems equivalent to the truth that it is true that grass is green, and this second truth is about truth. If truths about truth are only relatively true, this second truth is only relatively true, and then presumably the first truth, that grass is green, is also only relatively true—though obviously this little argument assumes the equivalence principle, which as we have seen has been subject to various objections. But it is time to draw this digression to a close, and turn to deflationism’s inflationist rivals.

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CHAPTER FIVE

Realism

Even in traditional philosophy, the label “realism” had multiple uses whose connections with each other were anything but clear. There was, for instance, realism about universals (properties and relations), as opposed to conceptualism and nominalism, but also realism about material objects, as opposed to idealism. In present-­day analytic philosophy the terminological situation is so bad as to have led one important contributor to our subject to say that a philosopher who announces without further explanation that she is a realist has accomplished no more than to clear her throat. With other terms it is reasonable to demand that authors use them in their accepted senses. Since “realism” has no accepted sense, all that can reasonably be demanded is that authors explain the sense in which they propose to use it. We are not here concerned with realism about universals or about material objects, but only about truth. What we mean by “realism” about truth has been indicated in §1.4: A realist is someone who holds that truth involves an appropriate relation between a truthbearer and some portion(s) or aspect(s) of reality. We will successively consider a variety of theories that are realist in this sense, beginning with correspondence theories and proceeding to others more exotic. We also consider objections different kinds of realists raise against rival views or themselves face. 5 .1  Realism vs

Deflationism

Traditional realists would insist, against traditional idealists and pragmatists, that whether the proposition that snow is white is true depends on whether snow really is white, and not on whether

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the thought that it is fits comfortably with our other ideas or is convenient to adopt in practice. Deflationists cannot be accused of holding that the truth of the proposition depends on our comfort or convenience, but some realists complain that deflationism cannot account for a clear commonsense intuition according to which it is snow’s being white that makes the proposition true— and not vice versa. For according to deflationism, to say that snow is white and to say that the proposition is true are equivalent by virtue of meaning, and such equivalence is a symmetric relation, whereas dependence is not. The deflationist seems to have no more justification for giving the answer “Because snow is white” to the question “Why is the proposition that snow is white true?” than for giving the answer “Because the proposition that snow is white is true” to the question “Why is snow white?” Deflationists answer that even where a biconditional holds by definition, there generally remains the asymmetry that the term whose definition is involved occurs on one side but not the other, and that is enough to explain and justify a corresponding asymmetry in colloquial language. Thus we may say that if a woman’s husband is dead, that makes her (count as) a widow, but not that if she is a widow, that makes her husband (count as) dead. In a similar sense, we may say that if snow is white, that makes the proposition that snow is white (count as) true, but not that if the proposition is true, that makes snow (count as) white. Inflationists may insist that asymmetry between definiens and definiendum is not a genuine explanatory asymmetry in any philosophically weighty sense, but deflationists can reply that while there may be a commonsense intuition that snow’s being white in some sense “makes” the proposition that snow is white true, it is not clear that it is a philosophically weighty sense. It is worth mentioning that Googling turns up many examples of questions of the form “Why is it true that p?” in areas from pop psychology (“Why is it true that matters of the heart and wallet are the areas where we are most susceptible to self-­deception?”) to differential geometry (“Why is it true that if the charts of a smooth structure overlap in only one connected component then the manifold is orientable?”) and that the answer offered by online experts is never the simple “Because p.” The construction

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“Why is it true that . . .” or “Why is it that . . .” seems in practice to be used mainly as a convenient device for turning an assertion into a why-­question, without having to insert the auxiliary verb “do” and so forth—an observation entirely in harmony with deflationism. 5 .2

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 Correspondence Theories

Russell and Moore were each for a time tempted by the view that facts are just true propositions. That view, sometimes dignified with the title of the identity theory, leaves no room for the criticism of deflationism just discussed, that facts make propositions true in some serious sense of “making,” and it was soon enough abandoned for the view that a true proposition corresponds to a fact. Variant formulations of the correspondence theory may take the truthbearer to be, not a proposition, but a sentence or a belief, and may take that to which a truthbearer corresponds to be, not a fact but a state of affairs or situation (though “states of affairs” may be the same as “facts” not conceived of as just true propositions). But the most important differences among correspondence theorists are over the nature of correspondence. The main division, which goes back to Russell and Moore, is between congruence and correlation theorists. A congruence theorist holds that a truthbearer and what it corresponds to are both structured complexes, and that when one corresponds to the other, there is likeness of structure, and correspondence of components to components. A version of this view taking sentences as truthbearers and facts or states of affairs as what they correspond to might hold that the true sentence “Snow is white” has two components, the subject “snow” and the predicate “is white,” while the fact or state of affairs of snow’s being white has two components, the substance snow and the property whiteness. The sentence is true because it corresponds to the fact or state of affairs, with the subject corresponding to the substance it denotes and the predicate corresponding to the property it connotes. Such a view suggests analyzing truth in terms of denotation and connotation.

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Congruence theorists subdivide over the requirement of “likeness” of structure, with some demanding identity of structure and others only similarity. The two groups are labeled with Greek-­ derived terms borrowed from mathematics, being called isomorphism and homomorphism theorists. But the differences between the two are subtleties we will not pursue. A correlation theorist simply holds that a true truthbearer as a whole corresponds to a fact or state of affairs or situation as a whole, with no further analysis. The main complaint about correlation theories is that they add nothing but rhetoric to the deflationist account on which it is true that things are some way iff things are that way. Austin’s variant does posit descriptive conventions relating a sentence type to a general kind of situation and demonstrative conventions relating a sentence token to a specific historic situation, with a token being true iff its specific situation is of its type’s general kind. That’s telling us something, but it’s not telling us much. There are serious worries about congruence theories, however. A first worry, often cited by correlation theorists, concerns atomic sentences, ones that like “Snow is white” are not logically compounded out of simpler ones. The problem is that though it may be clear what the components, snow and whiteness, of the corresponding fact are supposed to be, it is not so clear what is supposed to unite them, to bind them together to produce the fact. Some say that snow instantiates whiteness, but that only raises further questions. For saying that instantiation glues snow to whiteness seems to make instantiation itself a component of the fact; that then seems to call for some kind of meta-­instantiation to glue snow and whiteness to instantiation; and that then seems to call for some further adhesive; and so on. A regress threatens. (Many see a similar difficulty on the side of truthbearers if these are taken to be propositions.) A second worry is that for many logically compound truths, including heterogeneous disjunctions such as “Snow is white or grass is green,” and especially negative existentials such as “Hippogriffs do not exist,” it is not clear what the corresponding fact is supposed to be, or even that there is one. We may say colloquially that the fact that hippogriffs do not exist explains

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why hippogriff-­hunters always come home empty-­handed, and Ramsey explained such colloquial speech in terms of a deflationist or lightweight theory of facts, paralleling his theory of truth. But metaphysically inclined theorists have a weightier conception of facts, as structured complexes of objects and properties or whatever. It is not clear what structured complex if any could plausibly be identified with the heavy-­duty fact that hippogriffs do not exist. Are we to recognize the whole wide world as an object, and freedom from hippogriffs as a property it instantiates? That sounds so far-­fetched that many metaphysically inclined theorists have been reluctant to recognize negative existential facts. Some hesitate to recognize even disjunctive facts. 5 .3  Truthmaker

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Theories

Reluctance to posit such facts eventually led Russell to the view (also seen in early Wittgenstein) that the truth of compound propositions derives from facts only indirectly, by way of their logical relations to atomic components whose truth derives directly from correspondence with facts. This logical atomism represents a serious departure from the correspondence theory. A not dissimilar departure is found in the writings of many metaphysically inclined theorists today, who put less emphasis on correspondence, and more on each truth having some thing(s) that make(s) it true, in some philosophically weighty sense of making. Whereas the word “correspondence” may suggest a one-­ one relation, many today allow that a single truthmaker may make more than one truthbearer true, and a single truthbearer may be made true by more than one truthmaker. The fact that snow is white may make the two propositions true, that snow is white and that either snow is white or grass is green, which latter may also be made true by the fact that grass is green. As the example suggests, truthmaker theory does away with any need for disjunctive facts, such as a fact that snow is white or grass is green. But there is still the problem of negative existentials. The late David Lewis, for instance, though sympathetic to motivations

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underlying the positing of truthmakers, suggested that only affirmatives have them, while taking negatives to be made true, not by the presence of a truthmaker, but by the absence of any falsehoodmaker. The truthmaker principle asserts that for any two possible world-­states, if something would have been true if the world had been in one of them that would not have been true if the world had been in the other, then something would have existed in the former case that would not have existed in the latter. Lewis would amend this by adding “or something would not have existed in the former case that would have existed in the latter,” thus obtaining the (weaker) principle of supervenience of truth on being. Debates continue over simple cases as well, not only about what unites the components of the fact or state of affairs, but also over whether it is facts or states of affairs that are the truthmakers at all. Some say that it is not the fact or state of affairs of Frosty the Snowman’s being white that makes the proposition that Frosty the Snowman is white true, but rather a feature of Frosty, Frosty’s whiteness. Various unattractive labels (“tropes,” “abstract particulars,” “qualitons”) for such features compete with each other in the literature. Terminological differences combine with substantive disagreements to make truthmaker theory one of the murkier areas of metaphysics. Fortunately we are excused from having to penetrate into the murk by Lewis’s observation that truthmaker metaphysics may not really have very much to do with truth. For the metaphysicians’ view is not that, even if snow is white, that is not sufficient to make the proposition that snow is white true. Their view is not that, without the existence of the fact of snow’s being white (or “trope” or “abstract particular” or “qualiton,” snow’s whiteness), snow could still somehow or other manage to be white, but the proposition that snow is white couldn’t manage to be true. Rather, their view is that snow couldn’t be white without the fact (or feature) existing. “Things being some way necessitates the being of some thing,” might be their slogan. This is perhaps not so catchy a slogan as “No truth without a truthmaker,” but in the catchier formulation “truth” may be serving as little more than a device of generalization of the kind deflationism emphasizes.

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Chapter Five 5 .4  Physicalism

Molière mocked physicians of his day who “explained” why opium is capable of making people sleepy by positing that it has a sleepy-­ making capability (virtus dormitiva). The logical positivists, who (like Nietzsche before them) liked to cite that example of “meaningless metaphysics,” would have taken an attitude like Molière’s towards metaphysicians debating which of the two it is, snow’s being white or snow’s whiteness, that makes it true that snow is white. For the positivists, the very notion of truth was suspect because it seemed metaphysical. Tarski convinced many positivists that his definition showed that acceptance of a notion of truth was compatible with physicalism as the positivists understood it. Decades later Hartry Field questioned whether the definition really establishes compatibility with physicalism (as he understands it). What did Field want that Tarski did not supply? Tarski’s recursive definition of truth depends on a list of base clauses like the ones for the toy language considered in §2.3 that tell us “0” denotes zero and “1” denotes one. (If there were more constants in the language, there would be a longer list.) Field in effect wanted a single, uniform, general, physical account of the relation (1) Symbol     denotes object     used by person     .

in the language

and ultimately a single, uniform, general, physical account of the relation (2) Symbol-­sequence     is true in the language used by person     .

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This is a very ambitious goal indeed. Any account of (2) in physical terms would stand to metaphysical accounts of truth (such as “homorphism with a fact” or “correlation with a situation”) much as present-­day physiological explanations of the action of opiates on the nervous system stand to the “explanations” offered by Molière’s contemporaries. It would make identity, isomorphism, homomorphism, correlation, atomist, truthmaker, supervenience, and any other metaphysical theories all seem like empty verbiage by comparison. An account of (2) in physical

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terms would also accomplish tasks that deflationism insists explicitly (and that metaphysical theories seem to assume implicitly) belong to the theory of meaning rather than the theory of truth, beginning with the task of determining whether or not emitting given sounds or displaying given marks amounts to utterance of a proposition-­expressing or “truth-­apt” sentence at all. It would refute deflationism by showing that there is enormously more to be said about what truth is than what the equivalence principle tells us. But why would anyone have imagined that a task so ambitious could be accomplished? Two factors were at work. On the one hand, Tarski’s work (which Field commended for going as far as it did and condemned for not going further) seemed to show a good deal about how to get to (2) from (1). On the other hand, work of Saul Kripke on proper names (along with related work by him and by Hilary Putnam on natural-­kind terms), which commentators often labeled a “causal theory of reference,” seemed to show a good deal about how to provide a causal, natural, physical account of (1). The hope seemed to be that a cross between Tarski’s theory of truth and Kripke’s causal theory might breed a causal theory of truth. Critics, notably Scott Soames, pointed out that Tarski did not really supply as much help in getting from (1) to (2) as it may initially have appeared. For the further clauses Tarski adds to the ones about “0” denoting zero and “1” denoting one in order to complete his recursive definition assume that one knows the meaning of negation, conjunction, disjunction, and so forth. Tarski makes no pretence of providing a physicalist account of “symbol     expresses disjunction in the language used by person      ,” for instance. There seems also to have been a misunderstanding about the aims and claims of the so-­called causal theory of reference. Kripke’s aim was to show how a proper name or natural-­kind term could refer to an individual or kind even though not associated with any description thereof. Grossly simplified, his account, which he called “the chain of communication picture,” ran as follows. Jay may describe an individual or kind and introduce a name by stipulating that it is to refer thereto, as in “Let ‘Aldebaran’

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denote that orangish star [pointing],” or “Let ‘tigers’ denote that striped kind of thing [pointing].” So long as in subsequent uses of the word he intends—and the intention need not be a conscious thought—to refer to whatever he has been using it to refer to, it will continue to do so, even after he has forgotten what description he used and everything else about how he originally introduced it. Kay may hear Jay use the word and decide to use it herself to refer to whatever Jay uses it to refer to. So long as in subsequent uses of the name she intends it to refer to whatever she has been using it to refer to, it will continue to do so, even after she has forgotten when, where, why, how, and from whom she originally picked it up. Then Ella may pick it up from Kay, Emma from Ella, and so on. It was a mistake for commentators to call this a “causal theory of reference.” There is no requirement of causal contact between the first introducer of the name and the object named, which might be a number rather than a star. And not only does the chain of communication picture not reduce the notion of reference to causal notions, it does not offer any reductive analysis of reference at all. Reread the summary account of the picture in the preceding paragraph, and you will see that it simply takes for granted a notion of intention to refer. Kripke explicitly and emphatically disavowed any aim to provide a reductive analysis. His work may raise doubts about the deflationist claim that there are no substantive questions about reference, simply by being an example of a substantive “theory” about reference, but if his work makes trouble for deflationism, it offers very little help to physicalism. Once it is realized how very little Tarski and Kripke offer in the direction of a physicalist account of truth, discouragement may set in. (Field himself eventually abandoned physicalism for a qualified defense of disquotationalism.) Discouragement is reinforced by certain nagging questions. The question how a naturalistic, causal theory could apply to truth in mathematics or ethics might be dismissed by physicalists hardheaded enough to question whether there is any truth in mathematics and ethics, but how could a general, uniform account hope to cover even the physical sciences, considering the great variety of objects, from quarks to quasicrystals to quasars, with which such sciences are concerned,

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and the diversity of our relations to them? The more fully one understands what the physicalist project would have to accomplish, the more it comes to seem that the project is unrealistic—in the everyday, nonphilosophical sense of “realistic.” It is a pipe dream. 5 .5  Utility

If physicalism survives today, it is less as an active program than as an inchoate feeling that a physicalist account is needed, even if no one will for the foreseeable future be in a position to provide one, for certain explanatory purposes, and especially for explaining the utility of truth. Traditional pragmatists explained the utility of truth by defining truth as utility, but with that no longer considered a live option, it may be thought that a genuine explanation would require a naturalistic, causal account of truth: that a metaphysical theory could only provide a pseudo-­explanation (of the virtus dormitiva type), and a deflationist theory not even that. But deflationists claim they can provide an explanation of the utility of truth, after all. Consider first a special case. Imagine an agent with a choice among several courses of action that may lead to various possible outcomes among which the agent has preferences. Call beliefs of the form (3) Acting thus-­and-­so will lead to the outcome most preferred. directly action-­guiding. Counting a belief as useful to an agent just in case it is useful for obtaining the agent’s preferred outcome, and supposing that the agent does not exhibit weakness of the will, so that if the agent believes (3) then the agent will act thus-­ and-­so, it follows tautologically that it will be useful for the agent to believe (3) just in case acting thus-­and-­so will lead to the outcome most preferred—or in other words, just in case (3) is true, as deflationists understand truth. Thus it is explained why having true beliefs is useful, and similarly having false ones is harmful, in the special case when the beliefs are directly action-­guiding. And now for the general case. Since directly action-­guiding beliefs will generally be inferred as

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conclusions in some manner from other beliefs taken as premises, it will be indirectly useful if the other beliefs involved are true and the manner of inferring conclusions from premises involved is truth-­preserving. This explanation leaves room for exceptions, as any explanation should, since the utility of true beliefs is a general tendency, not an exceptionless law. True beliefs can be useless if they do not serve as premises for inferring action-­guiding conclusions. True beliefs about trivia of antiquarian scholarship or popular culture may be useless except to contestants on quiz shows. False beliefs can be useful if, taken as premises, they yield true action-­guiding conclusions. A false belief that there is a lion behind the door may be as useful as a true belief that there is a tiger in persuading one to leave the door closed. Successful applications of scientific theories now superseded provide other examples. And there are other types of exceptions. (True beliefs can be harmful if, taken as premises together with some false beliefs, they lead to false action-­guiding conclusions that would not have been inferred from the false premises alone. Weakness of the will can render true beliefs useless and false ones harmless.) What now is there left to explain about the general tendency of true belief to be useful, for which a physicalist or other inflationary theory of truth would be needed? Well, if there is nothing more to be said about the general tendency, there still seems to be more to say about particular cases. The issue may be easier to illustrate in the case not of truth for a particular part of language, but of correctness for a particular kind of nonverbal representation. Parallel to the deflationist account of truth, “Things are as they are said to be,” there is a deflationist account of correctness, “Things are as they are represented as being.” For some particular kinds of representation, however, there seems to something more to be said about what correctness is than just this, and this “something more” may play an important role in explaining the utility of the representation. Consider, for instance, subway diagrams. When such a diagram is correct, a certain physical relation holds between it and the subway system, which may be roughly described as follows: For every dot on the diagram there is a unique station in the system whose walls bear a label of the same shape as (but of a much

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larger size than) the label beside the dot in the diagram; and for every station there is a unique dot thus related to it; and two dots are connected by a line on the diagram iff the stations thus related to them are connected by a subway track. This physical relation can be used to explain why a rider who uses the diagram can successfully navigate the subway system. To this the deflationist replies that of course if the diagram is correct, in the deflationist sense that things are as it represents them as being, then this physical relation will hold, for that is how the diagram represents things as being, as anyone knows who knows how to read a subway diagram. With road maps, which unlike subway diagrams represent geometrical features, distances and directions, and not just topological features, order and connections, there will also be a physical relation, but it will be a different one. With celestial globes, which represent the relative directions of stars but not their distances from the earth, there will be yet another relation. In each case, one who understands how the representation is representing things as being will understand that the appropriate relation will hold if the representation is correct in the deflationist sense that things are as it represents them. What the physicalist seems to want is a physical, or physiological, account of the rider’s or motorist’s understanding of subway diagrams or roadmaps. And there may be some scientific, or scientistic, sense of “explanation” in which nothing less than this could provide a genuine explanation of the rider’s success in navigating the system relying on the diagram, or the motorist’s success in navigating the countryside relying on the map. But the deflationist will insist that the account that is being called for is an account of meaning, in a broad sense applicable to nonverbal as well as linguistic representations, and not of correctness, about which the formula “Things are as they are represented as being” says all there is to be said. 5 .6  Normativity

Deflationists often describe truth as a “logical” or “quasilogical” notion. Realists take it to be a metaphysical or physical one. A

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common feature of logical, metaphysical, and physical notions alike is that they are neutral or value-­free. To the extent that truth is useful one may expect it will be valued, but for deflationism and realism alike, “true” is not itself an evaluative term. One common criticism of deflationism and realism, going back to Michael Dummett, the founder of so-­called antirealism, is precisely that “true” is an evaluative or “normative” term. An assertion that is untrue is open to criticism on that account, or in jargon (4)  Truth is a norm of assertion. According to a favorite analogy of Dummett, truth is to making an assertion what winning is to playing a game. The deflationist account of truth is analogous to the following analysis of winning: (5) Winning a game consists in whatever the rules defining the game say constitutes winning. This may get the extension of “win” right, but without the principle (6)  Winning is the aim of playing.

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the point of the win/lose distinction is unaccounted for. Analogously, Dummett claims, without the principle (4) the point of the truth/false distinction is unaccounted for. And any theory, deflationist or other, that leaves the point of the distinction out must be considered inadequate. One line of response grants the critic that truth is a norm of assertion, but insists that something’s being taken as a norm by some practice is an extrinsic fact about it that does not suffice to make it intrinsically normative. Being 94 feet long remains a neutral, value-­free notion even though it is taken as the norm for courts in professional basketball. For truth to be intrinsically normative, (4) must not merely hold but do so as part of the very meaning of the truth predicate. But perhaps (4) is better regarded as part of the meaning not of “true” but of “assert,” distinguishing assertion from uttering during an elocution competition, a report of another’s speech, a storytelling session, or whatever. If someone is uttering declarative sentences without even pretending to be aiming at uttering true

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ones, must she be making assertions but flouting a norm of that practice, or may she not be doing something other than making assertions, perhaps narrating fiction? Likewise it is not obvious that (6) must be regarded as part of the meaning of “win” rather than of “play.” If someone is moving checkers on a checkerboard otherwise in accordance with the rules of checkers, but making no effort to capture all the opponent’s pieces, must he be playing checkers but violating a norm of that practice, or may he not be doing something other than playing checkers, perhaps demonstrating how the pieces move, or reproducing the moves of some famous game, or playing giveaway checkers? Such a response should be available to most versions of deflationism or realism. It will presumably not, however, be available to a version of deflationism that makes the notion of truth depend on that of assertion (by identifying understanding of the truth predicate with recognition that all T‑biconditionals are assertable). But perhaps (4) can be regarded as part of the meaning neither of “true” nor of “assert,” even if by the time both words are learned a child will have learned that there is such a rule as (7)  Don’t assert what isn’t true. and even though (4) is no more than a jargon-­ridden way of saying that there is such a rule. How could this come about? A hypothetical scenario might run as follows. Soon after its first words a child begins to hear prohibitions of the form “Don’t say . . . ,” and not much later the child learns there are exceptions, thus: (8) Don’t say “The duck is in the muck” if the duck is not in the muck, or “The goose is on the loose” if the goose is not on the loose, or anything like that. (9) But saying when reciting a rhyme, or making believe, or telling what someone else said, or something like that, doesn’t count. It is obvious why there should be such a rule as (8): We want to be able to infer from the child’s saying “Things are thus” that things are thus. It is obvious also why there should be such exceptions as

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(9). (There may be none for the rule “Don’t say the word Daddy said when he hit his thumb with the hammer.”) The child later on learns to use “true” as a device for making generalizations that otherwise could only be suggested by a few examples plus “or anything like that,” as in (8). Slightly earlier or later, the child learns to use “asserting” for “saying” minus the open-­ended list of kinds of exceptions mentioned in (9) (and “­uttering” for “saying” without exception). Once both words have been learned, the child will be able state the rule in the grown-­up way (7), but the rule need not be considered part of the meaning of the vocabulary needed to state it. Such a response may have at least some initial plausibility, but other objections remain to be considered, arising from the theory of meaning.

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Antirealism

As mentioned at the beginning of the preceding chapter, in traditional philosophy there were several debates pitting a group called “realists” against a group called something else—a different something else in each debate. The realists were those who maintained the “real” or “objective” or “independent” existence of something—a different something in each debate. Early in the last century there was a debate between two groups of mathematicians: classical mathematicians (now the immense majority) and mathematical intuitionists (now a tiny minority). The former accepted certain existence proofs that the latter rejected, and so the debate was usually perceived as being about the existence and nature of mathematical objects, and sometimes perceived as a revival or reincarnation of a traditional debate between “realists” and “conceptualists” over the existence and nature of universals. Inspection shows that the mathematicians’ differences over which existence proofs are acceptable derive from more fundamental differences over which logical inferences are acceptable. Classical mathematicians accepted and mathematical intuitionists rejected the principle of excluded middle, according to which assertion of A or not A is always warranted. Such differences themselves derive from more fundamental differences over the meanings of “not” and “or” and other logical vocabulary. For several decades now Michael Dummett has been urging that such differences over the meanings of specific words should be seen as derivative from a more fundamental difference over the general form an account of meaning should take. He has urged that not just the mathematical debate, but a whole range of debates over various forms of realism, should be reconfigured as a debate over the proper form for an account of meaning. The

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realist side would be advocates of an approach, to be described shortly, called truth-­conditional semantics, while the antirealist side, which Dummett himself generally favors, would advocate a rival approach called verification-­conditional semantics. When we spoke of “realism” in the preceding chapter, we meant a view directly about truth, to the effect that it consists in an appropriate relation to some portion(s) or aspect(s) of reality. By contrast, when Dummett speaks of “realism” he means a view about meaning (and its relation to truth) that he thinks those traditionally called “realists” about various other matters (beginning, but only beginning, with mathematics) are ultimately committed to, whether or not they acknowledge as much (as not all do). Dummettian antirealism is an influential view about truth (and its relation to meaning), but what it is anti, namely, truth-­ conditional semantics, is not what we called “realism” about truth in the preceding chapter. (Dummett opposes that, too, but it isn’t his main concern.) In this chapter, therefore, we will generally ignore the kinds of views discussed in the preceding one, to concentrate on the relations among the trio of views, truth-­conditional semantics and verification-­conditional semantics and deflationism (fairly often, however, remarking that something we are saying about deflationism applies much more widely). It will be a tangled tale. The concerns and considerations that motivate the three views are so different that the trio more often seem to be addressing different questions than taking differing positions on a single issue. At the end of the chapter we will bring in a fourth, compromise view, called pluralism. 6 .1  Meaning and

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Truth

What does meaning have to do with truth? Even deflationists want to say (as we saw in §5.1) that in some sense the state of the world “makes” the proposition that snow is white true. Does it also make the sentence “Snow is white” true? Only so long as we think of the sentence as being not just certain sounds or shapes, but certain sounds or shapes with a certain meaning. This thought

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may be put by saying that the meaning of a sentence together with the state of the world determine whether the sentence is true. In a slogan: (1) Meaning and world together determine truth value. Natural though it may seem, the naive formulation (1) calls for critical scrutiny. For any complete account of “the state of the world” or “how things are” would have to include an account of how things are as regards the meanings of sentences, and for that matter, their truth values. The sentence meaning what it does is part of the state of the world, and separate mention of it is otiose, unless we restrict the scope of “the state of the world” to pertinent aspects. Should we, then, understand “the state of the world” to mean “the state of the world as regards nonlinguistic matters”? That will not do, because some sentences are about linguistic matters, and the state of the world as regards linguistic matters will be pertinent to their truth. Rather it is the meaning of the sentence that determines which aspects of the state of the world (usually nonlinguistic aspects, sometimes linguistic aspects) are pertinent. The meaning of the sentence determines what conditions the state of the world must fulfill in order for the sentence to count as true. In a slogan: (2) Meaning determines truth conditions: A sentence’s truth conditions are consequences of its meaning. But there are objections to this slogan, too. For one, where indexicality is present, the meaning of the sentence type is not enough to determine under what conditions a sentence token counts as true; context must provide something more. For another, according to the chain of communication picture (sketched in §5.4), “tigers” may refer to tigers only owing to something “outside the head,” the history of the usage of the term, even though “inside the head” there may be no more meaning attached to the term than a broad sortal classification as “a kind of animal.” If so, then the meaning of “Tigers are striped” will in a sense be insufficient to determine its truth conditions (since from a complete description of the world’s fauna, but identified only by their scientific names, plus the information that “tiger” is the common

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name for an animal species, one could not determine whether tigers are striped). We mention objections of these kinds, however, only to ask the reader to set them aside and not be distracted by them. Our interest here will be in dissent from (2) in a different direction, represented by the counterslogan: (3) A sentence’s truth conditions constitute, or at least are constituents of, its meaning. Whereas (2) suggests meaning comes first, and determines truth conditions, (3) suggests that truth conditions are already a part of meaning, or perhaps even the whole of it. We call (3) the principle of truth-­conditional semantics, and its apparent corollary (4) Knowledge of meaning consists, at least in part, in knowledge of truth conditions. the principle of hardcore truth-­conditional semantics. By suggesting that any question that involves meaning ipso facto involves truth, (3) and (4) threaten the deflationist slogan “There are no substantive questions about truth,” since (as we saw in §3.5) the defense of that slogan involves claiming that various undeniably substantive questions about meaning are not about truth. More directly and seriously threatening to deflationism, and to many other theories of truth besides, are these further apparent corollaries: (5) Knowledge of the meaning of any sentence requires possession of the concept of truth. (6)  The concept of truth cannot be defined or analyzed.

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Just as one cannot know that the conditions under which it is legal to do something are such-­and-­such unless one has the concept of legality, so one cannot know that the conditions under which it is true to say something are such-­and-­such unless one has the concept of truth. Hence the step from (4) to (5). If even toddlers who have not yet learned the word “true” would have to possess the concept of truth (presumably on the same unconscious level where speakers possess various syntactic and phonological concepts for which many never learn the words at all) before they could learn

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the meaning of anything, then truth is a concept so basic that there can hardly be any more basic concepts in terms of which it might be analyzed or defined. Hence the step from (5) to (6). The thesis (6), called truth primitivism, was upheld (for reasons of his own) by Frege, but is at least implicitly rejected by every other theory of truth mentioned in this book, since all in one way or another do attempt some sort of analysis. Thus (3) threatens, via (6), not only deflationism but almost every other theory of truth. (We say “threatens” rather than “contradicts” as acknowledgment that the steps from (3) to (6) are not logically watertight.) But why should one accept any version of truth-­conditional semantics? Where does truth-­conditional semantics come from? The short answer is: from Donald Davidson’s reaction to Tarski. 6 .2

 Davidsonianism

Tarski was mainly concerned to define truth for object languages useful in pure mathematics. Mathematical language lacks many features found in nonmathematical language, and this circumstance made Tarski’s task easier. For instance, indexicality is absent from pure mathematics, and that is why Tarski was able to get away with working with types rather than tokens. Also, in mathematical language the logical forms of sentences are usually more or less obvious, which made it easier for Tarski to find the clauses in his recursive truth definition. Where the logical form of a construction is unobvious, so is the form the clause for that construction should take in a recursive truth definition. To give one example, consider singular negative existentials, such as (7)  Pegasus does not exist. This is a common enough construction outside mathematics (since only in mathematics is one required, before introducing a name, to prove the existence of its bearer). Two competing accounts assign (7) logical forms symbolizable as follows: (8a) Np, where p is a constant denoting Pegasus and N a predicate for nonexistence.

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(8b) ~xPx, where P is a predicate for being Pegasus. And there are other proposals. How a Tarski-­style truth definition should treat (7) depends on which account of its logical form is right. Thus corresponding to (8a) or (8b) one might have clauses like these: (9a) (7) is true iff the thing denoted by “Pegasus” satisfies “x is nonexistent.” (9b) (7) is true iff no thing satisfies “x is Pegasus.” Corresponding to some other proposal one would have something else. Distinctions of tense and mood, propositional attitude operators (“So-­and-­so knows/believes that . . .”), and many more constructions found in nonmathematical language also have unobvious logical forms. To extend Tarski’s definition to any large fragment of extramathematical language is therefore not easy. No one has done more to promote the program of doing so than Davidson, though Davidson’s perspective was the opposite of Tarski’s. For Tarski, truth was a problematic notion, for which a definition was sought, while the various notions of the object language were taken for granted. For Davidson, it is truth that is more or less taken for granted, and the clauses of the recursive definition give information about the logical form and other aspects of the meaning of constructions in the object language. One feature of Tarski’s procedure that impressed Davidson was the phenomenon we encountered in connection with a toy language in §2.3, where we saw that from the (finitely many) clauses of Tarski’s recursive definition of truth one can derive in a canonical way, for any of the (infinitely many) sentences of the object language, a biconditional of form (10)  “    ” is true in L iff     ­ .

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with the given object language sentence on the left and on the right a sentence of the metalanguage that is a translation of it. Davidson sought an extension of this result beyond toy cases to natural language. In this connection, he notoriously said that giving such a theory of truth for a language is a way of giving a theory of meaning for it.

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This dictum has been variously interpreted, elaborated, and amended by Davidson’s disciples and Davidson himself at different stages in the development of his ideas. As a result, there are many quite different views today that call themselves “Davidsonian.” Now just as, given Davidsonianism’s Tarskian origins, and ­Davidson’s own militant antipropositionalism, one would not expect such views to challenge sententialism, so also, given those origins, and Davidson’s heavy emphasis on the T‑biconditionals (10), one would not expect them to challenge the equivalence principle, deflationism’s first thesis. Some forms of “Davidsonianism,” however, may not challenge any principle of deflationism— or of any other theory of truth, for that matter. For in some the role of truth in the theory of meaning has become so very indirect that it is no longer clear that it even matters whether it is truth rather than something else that is playing the role. Such versions may not even endorse the slogan (3) and so may not even count as truth-­conditional semantics as we are using that label here, however they label themselves. Other forms of Davidsonianism may endorse (3) without (4), or (4) without (5), or (5) without (6). Davidson himself, however, did emphatically endorse the thesis (6) of primitivism, the most threatening item on our list. There is another line of thought suggesting a close connection between meaning and truth conditions, based on a naive picture of what goes on in the situation where a child is beginning to learn its first language. Davidson, it may be remarked, despite his sometimes arguing in favor of his approach on the ground that it is necessary to explain the learnability of language, was more concerned with the situation where speakers of a first language are trying to interpret a second, and many later Davidsonians seem even less concerned than Davidson himself with learning a first language. (Notoriously, at one well-­known hotbed of Davidsonianism some even deny that we ever do learn a first language. On their view, we do not learn, but rather are born knowing, a language of thought; what is commonly mistaken for learning a first language is really learning to interpret a second language in this innate language; and courses mistitled “English as a Second Language” are really courses on English as a third language.) So the line of thought we have in mind is largely independent of developments in high theory deriving from Davidson.

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The naive picture is simply that the child receives parental approval for itself saying “The cat is on the mat” when Felix is over here and not elsewhere, and likewise “The dog is in the bog” when Fido is over there and nowhere else, and so on, and comes to associate with these sentences—along with others formed by recombining components—the conditions under which approval has been experienced and/or is to be expected, which is to say, the conditions under which the sentences in question are true. The nebulous conclusion here, that knowledge of meaning involves “associating” truth conditions with sentences, stops well short of hardcore truth-­conditional semantics, let alone truth primitivism. But any argument against truth-­conditionalism broad enough to rule out the nebulous, naive version will a fortiori suffice to rule out the hardcore version, and so rescue any threatened theories of truth. And though rescuing deflationism was the furthest thing from his thoughts, a quite broad purported refutation of truth-­conditionalism is precisely what is offered us by—to get back to him at last—Dummett. 6 .3  Dummettianism vs

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Davidsonianism

Dummett is centrally concerned with what knowledge of meaning consists in, and so is concerned to refute truth-­conditional semantics only in versions that claim to tell us something about that—essentially, what we have called hardcore versions. For Dummett, knowledge of the meaning of the most elementary parts of a first language is crucial, because such knowledge must be tacit, whereas knowledge of the meaning for more advanced parts may be verbalizable, expressible using vocabulary from the more elementary parts, as knowledge of meaning for a second language may be expressible using a first language. For Dummett, the verbalizable/tacit distinction is central, for assimilating it to the theoretical/practical distinction, he concludes that knowledge of meaning is ultimately not a matter of knowing-­that, but of knowing-­how. It is not a matter of having a certain thought, but of having a certain ability. Which ability? According to the verification-­conditional semantics that Dummett opposes to truth-­conditional semantics,

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learning the meaning of a sentence is acquiring an ability to recognize whether or not something constitutes a verification (or warrant for assertion) thereof. Thus to know the meaning of (11)  The cat is on the mat. is to be able to recognize whether or not a given experience constitutes an observation of the cat’s being on the mat. Note that the ability in question is not an ability to tell whether the cat is on the mat, or whether it is true that the cat is on the mat, since we have no such ability: If someone asks whether the cat is on the mat, we will be able to tell if we happen to be in the room with the cat and the mat, and unable to tell if we are far away; but we grasp the meaning of the question equally well wherever we may be. Knowledge of meaning as a recognitional ability is supposed to extend beyond such elementary examples. Knowing the meaning of, say, Goldbach’s conjecture (12) Every even number greater than four is a sum of two primes. is supposed to consist in being able to recognize whether or not a given argument constitutes a proof of (12). What would count as a proof, by the way, what logical steps are permitted in a proof, itself depends on the meaning of the logical vocabulary. Here verification-­conditional semantics, like truth-­ conditional semantics, assumes that the meanings of compound sentences are determined by the meanings of their components and the mode of composition, but replaces the truth-­conditional account of each mode of composition by a verification-­conditional account. In the case of disjunction, for example, the Tarski-­style condition (13a) “    or     ” is true iff one of the pair “    ” and “    ” is true. is replaced by the condition (13b) Something is a verification of “    or     ” iff it is a verification of one of the pair “    ” and “    .”

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The difference between classical and intuitionistic mathematics (over excluded middle, for instance) is supposed to trace back ultimately to such differences as that between (13a) and (13b), and Dummett argues for intuitionist as against classical mathematics by arguing for making verification (or warranted assertability), rather than truth, central to meaning. His main argument starts from the premise (14) Ascription of tacit knowledge makes sense only if one can say what would constitute manifestation of that knowledge.

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Dummett then claims that verification-­conditional theories can meet the challenge of identifying what constitutes manifestation of tacit knowledge of meaning in a way truth-­conditional theories cannot. The positive side of Dummett’s claim is that if knowledge of meaning is an ability to recognize when a sentence has been verified (or when assertion of it is warranted), there is no problem about saying what constitutes a manifestation of that ability. It is manifested when speakers show themselves willing (unwilling) to assert or assent to the sentence when presented with something that does (doesn’t) constitute a verification of it. The negative side of Dummett’s claim is that there is no plausible candidate for what manifestation of tacit knowledge of truth conditions would be, given that we have no general ability to recognize truth. The premise (14) of Dummett’s manifestation argument sounds similar to the rhetoric of the behaviorists who dominated psychology and linguistics around the middle of the last century, as do other of his formulations. Since psychologists and linguists today regard behaviorism as outmoded and discredited, this makes Dummett easy for Davidsonians to ignore or dismiss. But let us think of Dummett as attempting, not to give a knock-­down argument, but to raise an embarrassing question: What could grasp of truth conditions—never mind the behavioristic demand for manifestation of that grasp—consist in? In the case of (11), there may be a “mental representation” or “picture in the head” of the cat on the mat. But imagining what it would look like for the cat to be on the mat (for (11) to be true) is indistinguishable from imagining what it would be like to see

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the cat on the mat (to verify (11)). Thus having the mental picture could with no less and arguably more justice be called a “grasping a verification condition” rather than “grasping a truth condition.” And in any case, if we switch from examples about cats and mats to examples about, say, catalysis and matriculation, let alone Goldbach’s conjecture, there will be no relevant picture in the head. 6 .4  Dummettianism vs

Deflationism

There is no obvious reason why either a truth-­conditional or a verification-­conditional theorist should deny that the meaning of the word “true” is given by some version of the equivalence principle. Corresponding to the contrast between (13a) and (13b) we would have a contrast between the following: (15a)  “It is true that     ” is true iff    

is true.

(15b) Something is a verification of “It is true that     ” iff it is a verification of “    .” But a truth-­conditional semanticist who is committed to (5) will have to say that to learn the meaning of the word “true” is not to acquire a new concept, but merely to learn the label for a concept one already had on some level, as Monsieur Jourdain did when he learned the word “prose.” Verification-­conditional semanticists are not committed to (5), and so lack this reason to deny that the concept of truth is acquired when the word “true” is learned, suggesting that they may be able to go a step further in agreement with deflationism than truth-­conditional semanticists can. (In this connection note the contrast that, while “true” is mentioned on the left side in both (15a) and (15b), it is used only in (15a) and not in (15b).) Yet there seems to remain a conflict between Dummettianism and deflationism insofar as the latter is committed to there being no interesting property coextensive with truth across the whole range of truths, while the former advances the slogan “There is no verification-­transcendent truth,” apparently committing Dummettians to the thesis that truth is coextensive with verifiability. One may question, however, how serious the conflict is.

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On the one hand, is Dummettianism after all seriously committed to this verifiability thesis? It is not obvious how to get to the verifiability thesis from verification-­conditional semantics. For instance, how exactly are we supposed to get from the first to the second of the following? (16) To know what it means to say that there is intelligent life in other galaxies is to be able to recognize whether or not something constitutes a verification that there is. (17) If there is intelligent life in other galaxies, then it is at least in principle possible to verify that there is. One route to (17) exploits a verification-­conditional account of the conditional “if . . . then. . . .” Corresponding to the contrast between (13a) and (13b) we would have a contrast between the following: (18a) “If     , then     ” is true iff, if “    ” is true, then “    ” is true. (18b) Something is a verification of “if     , then     ” iff it is a verification that a specified method will convert any verification of “    ” into a verification of “    .” If (18b) gives the meaning of the conditional construction, then to verify the particular conditional (17) all we need to do is specify a method that verifiably will convert any verification of (i) “There is intelligent life in other galaxies” into a verification of (ii) “It is verifiable that there is intelligent life in other galaxies.” That seems easy. The method is just this: According to (16), we have an ability to recognize a verification of (i) when presented with one; so given a verification of (i), just apply that recognitional ability to it, to see that (i) has been verified and hence a fortiori is verifiable; for to see this is to verify (ii). There is an air of hocus-­pocus, however, to such argumentation, and some uneasiness that it may prove too much. For we don’t want a proof that 94

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(17*) If there is intelligent life in other galaxies, then it has actually been verified that there is.

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Antirealism

The situation is complicated by the existence of an argument, Fitch’s paradox of knowability, purporting to show that the validity of the principle “If A then it is verifiable that A” would imply the validity of the principle “If A then it has been verified that A.” While there is almost universal agreement that there is a fallacy somewhere in the Fitch argumentation, there is no agreement as to just where. But the substantial literature on this topic cannot be gone into here. On the other hand, how seriously does the verifiability thesis conflict with deflationism? Deflationism predicts that substantive theses of metaphysics, epistemology, semantics, and so on that involve the truth predicate in their formulation will turn out on examination merely to be using the truth predicate to achieve formulation as a single generalization of what otherwise could be conveyed only by examples. We have seen in §5.3 that Lewis argues something like this point in the case of the claim “Every truth has a truthmaker.” Soames has suggested something similar in the case of the claim “Every truth is verifiable.” Perhaps the verifiability thesis, properly understood, no more clashes with deflationism than does the truthmaker principle, properly understood. For the argument is not that, whether or not (17) holds, still (19) If it is true that there is intelligent life in other galaxies, then it is at least in principle verifiable that there is. must hold simply because “true” just means “verifiable.” Rather, the argument is that (19) holds because (17) holds (and then, because there is nothing special about this one example, the further generalization holds that anything true is verifiable). This argument only requires the equivalence principle to get from (17) to (19), and so, it seems, does not require an inflationist notion of truth. Insofar as deflationism implies that the concept of truth just isn’t as important a notion as it has often been taken to be, any victory for deflationism must in a sense be pyrrhic. In the contest for recognition as the true theory of truth, if deflationism wins, then the very fact that it has won in a sense suggests that the gold medal may be only gilt. Lewis and Soames between them suggest that much that avowed realists and antirealists care about could still be argued about even if deflationism were conceded to be the correct account of truth. Such a concession might not have to be

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viewed as surrender on the points that are really important to the realist or antirealist. But such irenic reflections have not so far noticeably reduced realist and antirealist hostility to deflationism. 6 .5  Holism

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The twists and turns of the dialectic include one last radical reversal: Dummett argues that truth-­conditionalism could not give the correct account of meaning for any possible language; but Dummett does not and cannot claim that it is verification-­ conditionalism instead that gives the correct account of our actual language. For as Dummett certainly holds, meaning is what guides use, and the intuitionist/verificationist meaning (13b) is not what is guiding our use of disjunction, for instance. If it were, forensic scientists would not say things like “The DNA evidence shows the culprit was one of the identical twins X and Y, but we have no way of determining which.” Nor would mathematicians accept intuitionistically unacceptable existence proofs, as they systematically do. In the face of this systematic usage, it would make no more sense to claim that (13b) gives the correct account of what English speakers really mean by “or” than to claim that black is what English speakers really mean by “white.” Verification-­conditionalism is a nonstarter as a description of what we have. It is, rather, a prescription to replace it with something radically different. And while Dummett does not exactly highlight this point, he does not conceal it, either. But if this is so, and if also Dummett is right in his criticism of truth-­conditional semantics, then it must be some third alternative that gives the correct account of the meaning of our actual language. Dummett does not endorse any specific such third account, but he does seriously discuss one, the picture of the language of science that he associates with the name of Quine, and that is sometimes called “holism.” On this view, meaning is not given by conditions for truth or verification, at least not once one gets beyond the elementary part of language that is used for recording and predicting empirical observations. The meanings of theoretical terms are constituted in

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a different way, by clusters of basic laws posited to hold for them. These laws are used together with some empirical observations to predict other empirical observations. By the success or failure of such predictions we decide whether to retain the basic laws and the theoretical terms they involve, or revise them, or reject them altogether, as phlogiston has been rejected. The predictions of empirical observations are derived using logic, raising the question of what constitutes the meaning of logical vocabulary. The alternative Dummett considers to truth conditions or verification conditions, as in (13ab) or (18ab), is that meaning is given directly by introduction and/or elimination rules. For disjunction and the conditional, these would read as follows: (20a)  to infer “    or     ” from “    ” (20b)  to infer “   

or     ” from “    ”

(20c) having seen “    ” to follow from “    ” and from “    ,” to infer “    ” from “    or     ” (21a) having seen “    ” to follow from “    ,” to infer “if     , then     ” (21b) from “if     , then     ­” and “    ” to infer “    ” The holist picture may be especially congenial to deflationists, who suggest the meaning of “true” is constituted by T‑biconditionals or by T‑introduction and T‑elimination; but this is not the place to elaborate a picture Dummett himself only adumbrates. It is enough that we have sketched one alternative beyond Davidsonianism “realism” and Dummettianism “antirealism”; there are others. Indeed, the vast literature inspired by Dummett is full of other “arealisms” and “irrealisms” and “quasirealisms.” 6 .6  Pluralism

Though Dummett may concentrate on mathematics as an example, he repeatedly indicates that the considerations he is

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advancing are supposed to apply generally. By contrast, a number of recent writers, beginning with Crispin Wright (long a sympathetic though not uncritical commentator on Dummettian antirealism) and continuing with Michael Lynch, have been prepared to contemplate the possibility that truth may amount to something like verifiability for some topics—or for some “domains of discourse”—but not for others. This view is now called (alethic) pluralism. (Wright initially, and unfortunately, gave it the same label, “minimalism,” as Horwich’s very different view.) The view has points both of agreement and of disagreement with deflationism, as with (various forms of) inflationism. It is easiest to state the point of agreement with deflationism: There is no one, single, interesting, substantive property coextensive with truth across all domains of discourse. The points of disagreement are several. We remarked in §2.2 that for small enough fragments of language, truth of sentences may be coextensive with something else interesting (adding in §2.5 that for Euclidean geometry, truth is coextensive with provability, and noting in §5.5 that correctness for subway diagrams is coextensive with a certain physical relation). Certainly for any fragment to which Tarski’s method can be applied, there will always be a set-­theoretic property coextensive with truth, namely, Tarski-­truth. Perhaps a holist of the kind considered in the preceding section will want to combine deflationism with the view that truth for observation sentences is coextensive with verifiability or even some physical property. The pluralist, however, suggests that it is not just for small fragments, or for especially simple sentences, but for large blocks of language that truth is coextensive with some more substantive property: maybe one property in ethics, and another in mathematics, and yet another in science. This is already something that most deflationists (sensitive as they are to the diversity of the things we talk about even within, say, such a medium-­sized chunk as physical science) will be reluctant to accept. And they will be even more reluctant to accept the pluralist’s further claim that, for this or that large block, the substantive property truth is coextensive with is just the sort of thing that this or that inflationist theory has traditionally taken truth in general to be: perhaps

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something like verifiability in one domain, and coherence in another, and correspondence in yet another. On top of this, the pluralist wants to claim that truth is not just coextensive for this or that large block with this or that traditional inflationist notion, but that for each large block truth literally is (or in more obscure metaphysical jargon, is “constituted” or “realized” by) that substantive property. Here pluralism directly contradicts the deflationist thesis that once one has explained what it means to call something true by citing the equivalence principle, there is no room for further questions about what it is for something to be true. Though there are philosophers who maintain that we may always posit that “there is more to a property than our concept of it” (an extremely dubious claim if intended to apply even to mere pleonastic properties), we have suggested on the contrary that whether, having been told what “    ” means, there is room for a further question about what     is, depends on the nature of the meaning. On this view, if “pain” means a certain distinctive kind of sensation known by acquaintance, then there is no room for a further question about what pain is. If, by contrast, “pain” means, as some forms of materialism maintain, “whatever natural phenomenon it is that plays such-­and-­such a causal role” (where the role in question may involve, say, being characteristically caused by injury and characteristically causing aversive behavior), then there is room for the question “And what natural phenomenon is that?” and for such an answer as “Firing of C‑fibers,” and also perhaps for such an answer as “One thing for earthlings, another for Martians.” Lynch explicitly compares his pluralism to this form of materialism, generally called functionalism, by way of explaining how it is supposed to be possible for truth to be (in his almost mystical formulation) “both one and many.” Pluralists oppose the deflationist account of the meaning of the truth predicate, in favor of an account on which that meaning is something like “whatever substantive property it is that plays such-­ and-­such a role” or “having some substantive property or other that plays such-­and-­such a role,” with the understanding that different properties may play the role in different domains. The characterization of the “truth role” that pluralists offer does include

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(perhaps a bit grudgingly) the equivalence principle, but Wright and Lynch both go on to include more, beginning with being the norm of assertion, which we have seen (in §5.6) deflationists may suggest is part of the meaning of “assertion” rather than of “true.” There is then a further longish list of “platitudes” or “truisms” any deflationist is bound to claim are part of the meaning of other notions such as “honest” or “sincere,” rather than of “true.” One worry about alethic pluralism is the threat of pluralism creep. Pluralism about truth is less innocuous than some other forms of pluralism just because the notion of truth is routinely used in the explication of other concepts of philosophical interest (though according to deflationists the truth predicate may be serving only to enable us to state in a single generalization what otherwise could only be suggested by examples). If the notion of truth is implicated in the analyses of honesty or sincerity, assertion, knowledge, implication, and so on, then presumably if there are many “truth properties” or things that truth is, there will for each of these other items be many things that it is, too. In particular, alethic pluralists are often naturally led to logical pluralism, the view that different logics are appropriate for different domains of discourse: perhaps classical logic for a domain where truth is correspondence, but the weaker intuitionistic logic for one where it is something like verifiability or coherence. Pluralism seems to raise drastically the chances that arguments involving inferences across domains where truth is differently “constituted” or “realized” may be fallacious because of equivocation. Consider, for instance, the Puritan syllogism: (22a)  Whatever causes pleasure without hard work is a vice. (22b)  Whatever is a vice is ultimately deleterious to health. (22c) Ergo, whatever causes pleasure without hard work is ultimately deleterious to health.

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If truth for the premises (22a) and (22b), which involve evaluative notions, is a matter of something like verifiability or coherence, but truth about descriptive matters, as in the conclusion (22c), is a matter of correspondence, what grounds have we for thinking that the syllogism is truth-­preserving? More fundamentally,

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if truth for “Driving Hummers is expensive” is correspondence, while truth for “Driving Hummers is reprehensible” is something else, what is truth for their conjunction or disjunction supposed to be? There have been several rounds of debate between pluralists and antipluralists about this problem of mixed discourse. There has even, perhaps, been too much attention to this one apparent weakness of pluralism, and too little to others. Even if pluralists only claimed that truth is coextensive with correspondence in some domains, and something like verifiability or coherence in others, there would still be (i) a need to specify the boundaries of these domains and what truth is supposed to be coextensive with in each, and (ii) for each substantive property (correspondence or coherence or verifiability or whatever) with which truth is supposed to be coextensive in some domain, a need to answer standard objections to the thesis identifying truth with that property, insofar as these still apply when the association is restricted to that domain and downgraded from identity to coextensiveness. Existing pluralist accounts generally do little towards satisfying needs (i) and (ii). Even sympathizers have to admit that existing pluralist discussions are, despite containing many suggestive ideas, still “programmatic.” The less sympathetic might say “sketchy.” Perhaps this is inevitable given that, at this writing, pluralism is the nouvelle vague in the theory of truth, still under development. (The antithetical view, advocated by Mark Richard, that truth simply isn’t the right dimension of evaluation in many domains, is newer, but not yet a wave.) With pluralism we come to the end of our survey of contemporary theories of truth, insofar as the issue of the nature of truth can be discussed without taking into account the paradoxes. It is time to take them into account.

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chapter seven

Kripke

Tarski was the most prominent advocate of the inconsistency theory of truth, notoriously maintaining that the intuitive notion of truth is self-­contradictory. (Holding, like many positivistically inclined philosophers of his day, that languages come equipped with “meaning postulates” or “semantic rules,” he expressed his view that those governing “true” permit the deduction of contradictions by saying that natural languages like English are inconsistent, a claim that some later commentators, in real or feigned ignorance of the historical background, have professed to find unintelligible.) His aim was to define rigorously a restricted substitute for the intuitive notion of truth, safely usable for mathematical purposes, and to demonstrate its utility. Many later writers have, by contrast, written as if they took it to be the job of a philosophical account of the paradoxes to vindicate the intuitive notion of truth, and show that despite all apparent paradoxes the notion involves no real antinomies. In what is surely the most influential work since Tarski on the paradoxes, Saul Kripke presents a mathematically rigorous, paradox-­free treatment of truth for certain formal languages, different in spirit from Tarski’s. He adds hints about how his formal construction might model some features of natural language, but his hints steer a path between an inconsistency view and a vindicationist one. Reserving discussion of Kripke’s philosophical position for the next chapter, we describe in informal terms some main features of his construction in the first half of this one, adding in the second half, whose sections are starred as optional, technical details about the constructions of Kripke and some of his successors.

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Kripke 7 .1  Kripke vs

Tarski

Tarski’s reconstructed notion of truth is severely restricted. He starts with an object language L0 not mentioning truth at all, and defines a truth predicate T0 for L0 in a metalanguage L1. His method could be applied again to define a truth predicate T1 for L1 in a metametalanguage L2, and so on, to create a hierarchy of truth predicates Tn for a hierarchy of languages Ln, but we never get a language L containing its own truth predicate T. The natural-­language analogue of Tarski’s hierarchical approach would be a language with no unrestricted truth predicate “is true,” but only a series of restricted ones: “is true0” or “is a true sentence not mentioning truth,” then “is true1” or “is a true sentence mentioning at most truth0,” then “is true2” or “is a true sentence mentioning at most truth0 and truth1,” and so on. (And similarly for “false,” with falsehood being identified, as always, with truth of the negation.) Tarski never advocated that we should add such subscripts to natural language; nonetheless, Kripke finds it worthwhile to rehearse why the use of audible or visible subscripts in speech or writing would be unworkable, and would leave us far short of what we can say, usually without encountering any contradictions, in natural language. Kripke’s work dates from the Watergate era, and he considers things John Dean and Richard Nixon might have wanted to say about each other’s veracity in Watergate-­related matters. Suppose Dean wants to say, on inductive grounds, (1) Nothing Nixon said about Watergate up to the time of his resignation is true. Dean would have to put a subscript on his “true,” and the subscript would have to be higher than the subscript on “true” in anything Watergate-­related that Nixon said up to the time of his resignation. Finding an appropriate subscript would be infeasible in practice, unless Dean had a complete recording of everything pertinent Nixon said during the relevant time period. But as is well known, the Nixon tapes contain an eighteen-­minute gap.

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In some related cases subscripting would not even be possible in principle, let alone feasible in practice. In Kripke’s most famous example, Dean wants to say (2a)  Most of Nixon’s Watergate-­related statements are false. while Nixon wants to say (2b)  Most of Dean’s Watergate-­related statements are true. On the one hand, the subscript in (2a) would have to be higher than any subscript in any of Nixon’s Watergate-­related statements, including (2b), while on the other hand, the subscript in (2b) would have to be higher than any subscript in any of Dean’s Watergate-­related statements, including (2a); and this cannot be. Yet intuitively (2a) and (2b), understood as in ordinary, unsubscripted English, might both well be true. If at least 75% of Nixon’s Watergate-­related statements did not use “true” at all, and at least 75% of those were false, then since 75% of 75% is more than 50%, intuitively (2a) would be true regardless of the status of (2b). And similarly with the roles of Nixon and Dean reversed. The impossibility of subscripting does point to a risk of paradox, however, which might be realized in exceptionally unfortunate circumstances. If Nixon and Dean each made, apart from (2ab), an even number of Watergate-­related statements, exactly half true and half false, then their saying (2ab) would produce a situation like that in the medieval example of Socrates and Plato mentioned in §1.3. As Kripke notes, the case is similar to one of the oldest of all examples, where Epimenides the Cretan says (3)  Everything said by a Cretan is false.

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Intuitively, this could easily be false. To make it so, it is enough for even one thing said by a Cretan to be true. There is a risk of paradox, but it is realized only in case everything else said by Epimenides or any other Cretan is false. Kripke’s point is that Tarskian subscripting or more generally any syntactic criterion sufficient to exclude all paradoxical cases will have to exclude many harmless and even useful cases as well. Syntax can only show whether there is a risk of paradox, so a

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syntactic criterion will have to ban all risky cases, many of which will not be paradoxical in actual circumstances. Even if, like Tarski, one is only interested in producing a replacement for an intuitive notion of truth deemed inconsistent, one might hope to be able to produce a more flexible one than Tarski’s. This is in the first instance what Kripke aims to do. He aims to develop rigorously a suggestion floated in some of the previous literature to the effect that one can have a truth predicate that is self-­applicable, provided that one allows truth-­value gaps. 7 .2

 The Minimum Fixed Point

Kripke’s idea is that sentences—for simplicity Kripke, like Tarski, works with “true” as a predicate of orthographic sentence types— should not be assigned levels on syntactic grounds, but should be allow to “find their own level,” with those that fail to find one exhibiting a truth-­value gap. Sentences that make no mention of truth at all would be at level zero, and for these the equivalence principle is enough to determine which of them are to count as true. Thus (4a)  Snow is white. is true at level zero. But since (4a) is true, so also is (5a) Either snow is white or most of Nixon’s Watergate-­ related statements are false. and this regardless of the status of (2a), assuming as Kripke does that the truth of the first disjunct is sufficient for the truth of a disjunction, regardless of the status of the second disjunct. Thus Kripke’s level zero contains more than Tarski’s syntactically defined level zero. By contrast with (5a), the following (6) Either snow is black or most of Nixon’s Watergate-­ related statements are false. would not be true at level zero, since its truth value (if any) would depend on the truth value (if any) of something itself mentioning truth, namely, the second disjunct (2a).

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At level one come sentences not themselves at level zero whose truth value is determined by the truth values of sentences of level zero, for instance these two: (4b)  “Snow is white” is true. (5b) Either “Snow is white” is true or most of Nixon’s Watergate-­related statements are false. At level two come sentences not themselves at level one or lower whose truth value is determined by the truth values of sentences of level one or lower, such as (4c)  “ ‘Snow is white’ is true” is true. Kripke’s hierarchy, unlike Tarski’s, extends beyond all finite levels. For instance, let us call the infinite sequence whose first three items are (4abc) the snow sequence. Then for Kripke the following would be true at the first level beyond all finite ones: (7)  Every sentence in the snow sequence is true.

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But as one proceeds through higher and higher levels, one eventually comes to what Kripke calls the minimum fixed point, the first level where no new sentences get classified as true that were not already so classified at some earlier level. At the minimum fixed point, whenever “    ” has been classified as true, so has “ ‘    ’ is true,” and vice versa. As is said, one has closure under the rules of T‑introduction and T‑elimination. To make these ideas rigorous, one thing that will be needed is a way of counting levels beyond all finite ones. That is provided by the notion of transfinite ordinal number from set theory, but we do not want to enter into such technicalities at this point. Another thing that will be needed is a rigorous account of when the truth values of certain components are sufficient to determine a truth value for a compound. For instance, in the examples (5ab) we assumed that truth of one disjunct is enough to make a disjunction true. What one needs is precisely a logic of truth-­value gaps. We have already encountered the names of a couple of such logics (in §4.3) in connection with presupposition and vagueness. Kripke uses one of them, the Kleene strong trivalent approach

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(mentioning another, the Van Fraassen supervaluation approach, as a possible alternative). 7 .3  Ungroundedness

Various pathological sentences will never obtain truth values at any level. Such, for example, are the liar or falsehood-­teller and the truth-­teller: (8)  (8) is false.

(9)  (9) is true.

Why such examples receive no truth value is easy enough to see. In order for (8) to become true (respectively, false) at a given level, it already would have to have become false (respectively, true) at some earlier level. In order for (9) to become true (respectively, false) at a given level, it would already have to have become true (respectively, false) at some earlier level. Therefore there can be no first level where either gets a truth value, and hence there is no level at all at which either does. Similarly with (10a)  (10b) is true.

(10b)  (10a) is false.

In general, for a sentence mentioning truth to get a truth value, enough sentences it mentions will have to get a truth value earlier. And if any of these mentions truth, for it to get a truth value, enough sentences it mentions will have to get a truth value even earlier. The truth value of one sentence may depend on the truth value of another sentence, which may in turn depend on the truth value of yet another sentence, and so on. In pathological cases, a sentence may depend on itself, as with (8) or (9), or two sentences may depend on each other, as with (10ab), or there may be a vicious circle of three or more, or an infinite regress. (Examples of this last kind are implicit in Kripke’s technical work, and an explicit nontechnical example was given by Steve Yablo: Think of an infinite sequence of sentences in which each says “All later sentences in this sequence are false.”) Kripke’s construction makes rigorous sense of the notion of an “ungrounded” sentence, one for which the unpacking procedure never hits bottom because a vicious circle or infinite regress of dependence is encountered:

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The ungrounded sentences are ones that receive no truth value at the minimum fixed point. Not all ungrounded sentences are equal, however. If we started by arbitrarily declaring (9) true (or false), we could still carry out Kripke’s procedure, and would obtain a fixed point different from the minimum one, where we still had closure under T‑­introduction and T‑elimination, but (9) was true (false). By contrast, if we tried to declare (8) true (false) at the beginning, we would have to declare it false (true) at the next stage, and the resulting contradiction would obstruct the procedure. Falsehood-­ tellers have no truth value at any fixed point, while truth-­tellers are true at some, false at others, and without truth value at yet others, including the minimum fixed point. There are even subtler distinctions to be made, illustrated by (11)  Either (9) is true or (9) is false. (12)  Either (12) is true or (12) is false.

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Each of these can be true at some fixed point, can be false at no fixed point, and is without truth value at the minimum fixed point. But (11) can only be made true by making true something that could have been made false (either (9) or its negation), while (12) can be made true without making true anything that could have been made false. Kripke calls examples like (12) intrinsically true, and shows that there is a maximum intrinsic fixed point, where all and only the intrinsically true sentences are made true. But we have come about as far as one can go without assuming a little more technical background (including material from the later, more technical sections of chapter 2). Without assuming such background it is hardly possible to bring out Kripke’s main achievement, which is that of making all the ideas we have been ever-­more-­sketchily sketching rigorous and precise, or to say much of anything about the main rival approach, revision theory, that was developed in the wake of Kripke’s work. However, understanding of these matters is not really required to follow the more purely philosophical developments in the next chapter, and so the more technical discussion in the remaining sections of this chapter may be treated as optional reading.

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Kripke 7 .4 * The

Transfinite Construction

A test case is provided by the language L of arithmetic, as in §§2.4 and 2.5, and its expansion L adding a predicate T. For an interpretation of L, Tarski would require an assignment of an extension for T, which since the variables range over natural numbers would just be some set of natural numbers. Such a set can conveniently be represented by the function T that takes the value one or zero for a given number as argument according as that number is or isn’t in the set. Tarski’s definitions give a denotation |t| to each closed term t of the language, and a truth value to all sentences of L, in such a way that the following composition laws hold: (13a) t0 = t1 is true (false) iff |t0| and |t1| are the same (different). (13b)  T(t) is true (false) iff T(|t|) is one (zero). (13c)  ~A is true (false) iff A is false (true). (13d) A0  A1 is true (false) iff each Ai is true (at least one Ai is false). (13e) A0  A1 is true (false) iff at least one Ai is true (each Ai is false). (13f) xA(x) is true (false) iff A(t) is true (false) for each (some) term t. (13g) xA(x) is true (false) iff A(t) is true (false) for some (each) term t. Sentences A of L can be assigned code numbers #A, each of which can in turn be assigned an appropriate closed term as a numeral denoting it, and the numeral for its code number can serve as a kind of quotation of the sentence. Accordingly we write "A" for it. With this notation, if T is assigned as its extension the set of all code numbers of Tarski-­true sentences of L, then the following will hold in restricted form, applicable only to sentences A not involving T: (14a)  If T("A") is true (false), then A is true (false). (14b)  If A is true (false), then T("A") is true (false).

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However, T("A") will not be true for any sentence A involving T. Kripke aims to prove that truth values can be assigned to some though not all sentences of L in such as way that (13) holds and (14ab) hold unrestrictedly. An interpretation, we have seen, can be represented by a valuation or function T assigning each natural number a value one or zero, indicating that the predicate T holds of some things and fails of the rest. A partial interpretation can be represented by a partial valuation, or function T assigning some natural numbers values one or zero, indicating that the predicate T holds of some and fails of some others, while leaving the rest unclassified. The composition laws (13), given an interpretation, determine an assignment of truth values to all of the sentences of L, and equally, given a partial interpretation, determine an assignment of truth values to some of the sentences of L. For the partial case, the composition laws (13) encapsulate Kleene’s strong trivalent logic. Let us write T[A] for the truth value of A given the partial interpretation T. Then the truth laws (14ab) we seek can be restated as follows: (15a)  If T(#A) is one (zero), then T[A] is truth (falsehood). (15b)  If T[A] is truth (falsehood), then T(#A) is one (zero). Kripke aims to prove the existence of a T for which (15ab) hold. We call a partial interpretation T coherent if (15a) holds. Note that the empty valuation T0, with T0(n) undefined for all n, trivially fulfills this condition, though it fails badly to fulfill condition (15b). Towards getting a T for which (15b) as well as (15a) holds, Kripke defines the jump T* to be given by (16) T*(#A) is one (zero) iff T[A] is truth (falsehood).

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(We may suppose the coding has been so arranged that every number is the code for some sentence, so that (16) suffices to determine the value of T* for all numbers.) The goal of finding a T for which (15a) and (15b) both hold can be restated as that of finding a T for which T* = T. Such a T is a fixed point of the jump operation, and Kripke’s aim is to obtain a fixed point in this rigorously defined sense. For valuations T and U, we say U extends T iff they are identical or differ only by U(n) being defined for some

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n for which T(n) is undefined. Coherence can be restated as the requirement that the jump T* extends T. Kripke actually aims to show, not merely that there exists a fixed point, but that every coherent partial valuation has an extension that is a fixed point. To do so, he uses a general theorem from the mathematical theory of inductive definitions, together with the following observation. If U extends T, then the partial truth-­valuation obtained by applying the composition rules (13) to U will extend that obtained from T. If U(#A) is defined while T(#A) is not, then we will get from U a truth value for the atomic sentence T("A") while we did not get one from T, and then other sentences will get truth values that did not have them; but for no sentence that already had a truth value will that value be altered or lost. It follows that if U extends T then U* extends T*, a property called the monotonicity of the jump operation. There is a general theorem to the effect that for any monotonic operation, every coherent T can be extended to a fixed point, and there is moreover a minimum fixed point, of which all others are extensions. Kripke does not just appeal to this general theorem, but indicates a proof for the case of his jump, making use of Cantor’s countable ordinals, an extension of the sequence of natural numbers 0, 1, 2, . . . into the transfinite. Cantor’s two principles are: (17a) For every ordinal α there are ordinals greater than α and a least among these, the successor of α. (17b) For every increasing sequence α0 < α1 < α2 < . . . of ordinals there are ordinals greater than all of them and a least among these, their limit. Thus the natural numbers 0 < 1 < 2 < . . . themselves have a limit called ω, which has a successor called ω + 1, which has a successor called ω + 2, and the sequence ω < ω + 1 < ω + 2 . . . has a limit called ω + ω or ω  2, and similarly we get an ω  3 and so on, and ω < ω  2 < ω  3 . . . has a limit called ω  ω or ω2, and similarly we get an ω3 and so on, and ω < ω2 < ω3 . . . has a limit ωω, and that’s just the beginning. Kripke assigns each countable ordinal α a coherent partial valuation T in such a way that if α < β then Tβ extends Tα. We start with the empty valuation T0, already noted to be coherent.

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Monotonicity implies that the jump of a coherent partial valuation is coherent. (If T is coherent, that means T* extends T; monotonicity then implies that T** extends T*, making T* coherent.) So at successors we can let Tα+1 be the jump of Tα. Monotonicity also implies that if we have a sequence of coherent partial valuations Un with each extending the one before, then their union (which gives a number a value one or zero iff some Un and hence every Um with n ≤ m does so) is coherent. (Since the union U extends each Un, monotonicity implies that U* extends Un*, while the coherence of Un means that Un* extends Un; thus U* extends each Un and therefore their union U, making U coherent.) So at limits we can let Tβ be the union of the Tα for α < β. Let A0 be the sentence 0 = 0, and for each n let An+1 be T("An"). It is not hard to see that T0(#A0) is undefined but T1(#A0) = 1, T1(#A1) is undefined but T2(#A1) = 1, and so on. (These An are a formal analogue of the snow sequence (4abc) and so on.) So we keep getting new truths as we go through the natural numbers. It can be shown that we keep getting new truths beyond that—but not forever. For each A for which there exists any ordinal α for which Tα(#A) is defined, there is a least such α, which we may call αA. If we let βn be the largest of the αA for sentences A with code number #A < n, then the ordinals βn form a nondecreasing sequence, and have a limit γ. By the time we get to Tγ, everything that is ever going to get a value has got it already, so there can be nothing that will get a value for the first time with Tγ+1. Thus Tγ = Tγ+1 and we have a fixed point, the minimum fixed point. If we start with a valuation assigning 1 only to the code number of a truth-­teller (9), a little thought shows that its jump will also value that code number 1, so that we have coherence. Starting from such a valuation rather than the empty valuation, we get a fixed point in which the truth-­teller is true, and similarly we can get one where it is false. 7 .5 * Revision

112

Now for the rival revision theory. The definition (16) of the jump makes perfect sense even for a total valuation T. But no total valuation is a fixed point. Rather the jump T* will disagree with the original T over the status of, for instance, any sentence B of type

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(8), any B equivalent to ~T("B"). For such a B, if T(#B) = 1, then T[T("B")] = true, so T[~T("B")] = false, so T[B] = false, so T*(#B) = 0; and similarly if T(#B) = 0, then T*(#B) = 1. If we start with some total valuation U0 and consider U1 = U0* and U2 = U1* and so on, the truth value of such a paradoxical sentence will flip-­flop as we go through the sequence of valuations. The jump represents a revision, not an extension. If we want to consider a stage ω, by that stage there will be some numbers that from some point on have always received the same value. For instance, for a sentence A of L that is true (false), Um(#A) will be one (zero) for all m ≥ 1, while Um(#T("A")) will be one (zero) for all m ≥ 2, and so on. If we want to introduce a Uω, we will want it to give the value one (zero) to those numbers that have persistently been valued one (zero) in this way. But what of other numbers, whose value has been flip-­flopping? Different revision theorists, Hans Herzberger and Anil Gupta and Nuel Belnap, take different approaches. The first would assign all such numbers zero, the second would assign them all whatever value they were assigned back at the beginning, while the third would allow a new arbitrary assignment of ones and zeros to them. On any of the three approaches, the process may be iterated indefinitely. Any sentence that eventually reaches a point after which it is always true (false) may be called stably true (false). No fixed point will ever be reached, though there will be an ordinal by which the process is essentially complete, after which no stably true or stably false sentence ever changes truth value. Even for the simplest of the several versions of revision theory, this ordinal is much larger than the ordinal connected with Kripke’s theory—a reflection of the greater complexity of the revision-­theoretic approach. Still more elaborate constructions, combining elements of Kripkean and elements of the revision-­theoretic approaches, have been undertaken by Hartry Field in order to incorporate a conditional-­like operator into the language. 7 .6 * Axiomatics

Logicians have ways of classifying the complexity of sets of numbers. For instance, the set of code numbers of Tarski-­true

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sentences of L is more complex than any set that is arithmetically definable, any for which membership is expressible by a formula of L. But the set of Tarski-­truths is in some sense just beyond arithmetically definable sets in complexity. Perhaps unsurprisingly, the sets associated with Kripke’s theory turn out to be more complex than those associated with Tarski’s theory, and the sets associated with revision theory to be more complex still. Logicians also have ways of classifying the logical strength of different theories. For instance, there is a natural set of axioms for arithmetic, formulated in the language L, called the first-­order Peano axioms P. By Gödel’s incompleteness theorems, neither it nor any other system can prove all truths expressible in L, and among the truths P cannot prove is (a coded version of) the statement that P is consistent. There is also a natural set of axioms, which we may call P, for arithmetic plus Tarski-­truth for arithmetic sentences, which can be formulated in the language L, adding to the apparatus of P, as additional axioms, formalized versions of (13), and of (14) restricted to sentences not involving T. In P one can prove the consistency of P (essentially by a formalized version of the argument that each axiom is true, and that deduction preserves truth). Because P can prove the consistency of P and not vice versa, logicians count P as being of greater “consistency strength.” One would expect the natural system of axioms for arithmetic plus Kripke-­truth to be higher up on the scale of consistency strength than arithmetic plus Tarski-­truth, and the natural system of axioms for arithmetic plus revisionist-­truth to be still higher. But actually, it is not entirely clear what “the natural system of axioms” for Kripke-­truth is, and it is entirely unclear what “the natural system of axioms” for revisionist-­truth is. One branch of axiomatic truth theory begins with work of Solomon Feferman that was specifically directed towards axiomatizing Kripke’s theory, and determining its place on the logicians’ scale of strength. Workers in this area have still not achieved agreement on which candidate is “the” natural axiomatization, but have succeeded in finding the places of the various candidates on the scale. Another branch of axiomatic truth theory began with joint work of Harvey Friedman and Michael Sheard that was directed

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towards a complete determination of which combinations of items from a list of candidate truth principles are consistent, whether or not the combination corresponded to any philosopher’s conception. The principles considered include composition laws, introduction or elimination rules, and the like. Friedman also began work on determining the strengths of the consistent combinations. So far the branch of the theory concerned with consistent combinations has worked mainly against a background of classical logic. Though from several points of view partial logic might be a more natural choice, it is in many ways technically less tractable. In sum, despite a fair amount of work, many questions remain open. Before leaving the topic of axiomatic truth theories, mention should be made of an objection sometimes raised against deflationism, based on the fact that a theory like P is stronger than one like P. How “thin” or “anemic” can a property be, the critic asks, if the natural axioms for it enable us to prove things about other matters (such as consistency) that we couldn’t prove before? While such a rhetorical question may be an understandable reaction to some deflationist sloganeering, if the basic idea of deflationism is that there is no more to the meaning of the truth predicate than a few simple rules for its use, then issues about the strength of formal theories are ultimately irrelevant. For it is well known that simple rules can be of great logical strength. If there is a threat to deflationism, it is perhaps in the complexity, rather than the strength, of the rules of axiomatic theories of truth. For if one had to think of the meaning of the truth predicate as somehow constituted by the principles of one of the axiomatic truth theories in the literature, one certainly could not say that the meaning is constituted by a few, simple rules.

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Insolubility?

Those engaged in mainly technical work of the kind considered in the preceding chapter do not generally discuss at any length whether their constructions are to be regarded as models describing our intuitive notion of truth, showing it to be consistent, or as prescriptions for modification of an intuitive notion of truth that is inconsistent. Those writing on the more purely philosophical side of the question of the paradoxes of truth do have to take a stand, and among them consistency theorists far outnumber inconsistency theorists. In surveying the more purely philosophical side of the subject in this chapter we will accordingly take consistency theories first, before turning to inconsistency theories, and then the relation between the consistency vs inconsistency debate and the realist vs antirealist vs deflationist debate. 8 .1  Paradoxical

Reasoning

It will be well to have before us a particular paradoxical derivation. The one we wish to consider uses the equivalence principle not in the form of the T‑biconditionals (which immediately raise questions about what kind of conditional is involved), but of rules of T‑introduction and T‑elimination. Writing T for “it is true that,” these and their negative counterparts read as follows: (1a)  from A to infer TA

(1b)  from TA to infer A

(1c)  from ~A to infer ~TA

(1d)  from ~TA to infer ~A

Paradoxes result on applying such rules to such examples as (2)  (2) is false.

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If we write p for (2) and F for “it is false that,” with falsehood as usual identified with truth of the negation, then passage back and forth among p and Fp and T~p is in effect just application of the rule of reiteration: (3)  from A to infer A The only other classical logical principle needed to arrive at the contradictory conclusions p and ~p is refutation by reductio: (4)  having seen B and ~B each to follow from A, to infer ~A The further elaboration of the argument to get from negation consistency, or the deducibility of a pair of contradictory conclusions p and ~p, to absolute inconsistency, or the deducibility of any arbitrary conclusion q whatsoever, even “Santa Claus exists,” two more principles are needed, disjunction introduction and disjunctive syllogism: (5)  from A to infer A  B (6)  from A  B and ~A to infer B The elaborated liar reasoning may be represented symbolically as follows, indenting steps that are merely an hypothesis and what follows under it, and leaving unindented those asserted categorically:  p  (7b)  Fp  (7c)  T~p  (7d) ~p  (7e) ~p

Hypothesis

(7f) T~p

from (e) by T‑introduction

(7g) Fp

from (f) by reiteration

(7h) p

from (g) by reiteration

(7i) p  q

from (h) by ‑introduction

(7a)

from (a) by reiteration from (b) by reiteration from (c) by T‑elimination from (a)–(d) by reductio

(7j) q from (e) and (i) by disjunctive syllogism

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Two remarks will be in order. First, note that T‑elimination was used at step (7d) hypothetically, to deduce a consequence from something merely assumed, while T‑introduction was used at step (7f) only categorically, to deduce a consequence from something outright asserted. The reasoning could be rearranged to use T‑introduction hypothetically and T‑elimination categorically, but one or the other of the T‑rules must be used hypothetically at some point. Second, though negation is essentially involved in the derivation (7), paradox arises also without negation, using the conditional. The introduction and elimination rules for the conditional (stated in §6.5) and the rules of T‑introduction and T‑elimination together lead to absolute inconsistency: Curry’s paradox derives the conclusion that Santa Claus exists by considering not “This sentence is false” but “If this sentence is true, then Santa Claus exists.” (Since it would take us too far afield to go further into the matter here, we must leave it as an exercise to the reader to figure out how.) The same half-­dozen strategies we saw used in §§4.3 and 4.4 to address problems of indeterminacy have been used to address the paradoxes. Considerations of space limit us to considering various strategies only in their simplest, most naive forms. Actual proposals in the literature—for citations see the “Further Reading” section following this chapter—are generally more sophisticated, with complications intended to forestall various objections. 8 .2

118

 “Revenge”

Denial is represented by the claim that the equivalence principle is not after all a part or consequence of the intuitive notion of truth. Thus some claim that the principle “It is true that things are some way iff they are that way” is only a “habitual” generalization, on the order of “Birds fly,” not a genuinely universal generalization. This is usually maintained without offering any systematic account of the scope and limits and nature of the exceptions, except the indication that they are supposed to include liar-­type examples. Assurances are made that we can really understand how it could be that either “(2) is not true” is not true, though (2) is not true,

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or that (2) is true, though “(2) is not true” is true. But those who find Moore’s paradox (“It’s raining outside, but I don’t believe it is”) troubling will find this suggestion at least as much so. Disqualification is represented by the claim that (2) does not express a proposition, perhaps the commonest initial reaction. This line is open to the same kind of objection that disqualificationism faced in connection with presupposition, namely, that supposed non-­proposition-­expressing sentences are found cited in explanation of action. As an eighteenth-­century chemist, unaware of the failure of the presupposition that phlogiston exists, may explain an action by saying “The phlogiston has been removed from the air under the bell jar,” so a tourist ignorant of the paradoxical features of Epimenides’ warning may cite that warning in explanation of why he skipped Crete on his Mediterranean tour, or a voter ignorant of the paradoxical features of Dean’s accusation may cite that accusation in explanation of why she voted against Nixon. But there is a far more serious problem, without parallel for presupposition, often referred to in the literature by a title like that of a B-­movie in the horror genre: “The Revenge of the Liar.” The problem is that, given a purported solution to the liar paradox, it is generally possible to formulate, using the jargon of the solution, a new strengthened liar intuitively just as paradoxical as the original liar. If the solution to the ordinary liar is to claim that “This sentence is false” is in some sense defective, then the strengthened liar can be taken to be just “This sentence is false or defective.” Specifically, if the “solution” to the falsehood-­teller, “This sentence is false,” or (8a)  (8a) expresses a false proposition. is to say that (8a) does not express a proposition, then the strengthened liar is just “This sentence either expresses a false proposition or no proposition at all,” which amounts to the untruth-­teller (8b)  (8b) does not express a true proposition. or “This sentence is not true.” (We leave it to the reader to think through how (8b) leads to paradox by a slight variant of the reasoning by which (8a) led to paradox.)

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The would-­be vindicator of the intuitive notion of truth who says of (8b) what was said of (8a), that it does not express a proposition, faces the reply, “So a fortiori (8b) does not express a true proposition, and since that is just what it says, it is true.” The would-­be vindicator may then insist, “No, (8b) does not say that; (8b) does not ‘say’ anything; (8b) does not express a proposition. So a fortiori (8b) does not express a true proposition, and contrary to what you say, (8b) is not true.” But then the would-­be vindicator will be in the position of having said something (namely, (8b), which is to say, “(8b) does not express a true proposition”), and also having said that it is not true. The problem, sometimes called that of ineffability, is that the would-­be vindicator’s theory cannot be enunciated, at least not without saying something that, according to that theory itself, is untrue. 8 .3  Logical

120

“Solutions”

Deviance is represented by proposals to adopt a logic of truth-­ value gaps, such as the Kleene strong trivalent logic, also called partial logic. (Adopting the intuitionistic logic advocated by antirealists would be no help with the paradoxes, since the principles of classical logic that lead to trouble in the liar and Curry cases are all accepted in intuitionistic logic, too.) According to partial logic, the truth values of compounds are determined from those of their components according to the adjoining tables, wherein 1 represents truth, 0 falsehood, and ? truth-­valuelessness or gappiness. (The tables restate in symbols what those who have read §2.4 have seen stated there in words.) In this logic, as the reader can work out from the tables, if B is gappy, so is A = B  ~B. But B and ~B each follow by a truth-­ preserving rule (to infer a conjunct from a conjunction) from A, even though ~A is not true, and in this sense reductio (4) fails, and the paradoxical reasoning is blocked at step (7e). But the equivalence principle in the form of the rules (1abcd) is saved, if one takes TA to have always the same truth value (1 or 0 or ?) as A. That is, in effect, what the Kripke construction of the preceding chapter shows.

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Kleene Strong Trivalent Truthtables

~

Ù 1

?

0

Ú 1

?

0

1 ? 0

0 ? 1

1 ? 0

? ? 0

0 0 0

1 ? 0

1 ? ?

1 ? 0

1 ? 0

1 1 1

This “solution,” however, faces the problem of ineffability. For the would-­be vindicator wants to say that (2) is neither true nor false, whereas ~Tp  ~T~p comes out not true but gappy if p is gappy, so the enunciation of the view involves saying something that according to the view is not true. Kripke is perfectly aware of the problem. For him, the fixed-­ point construction may be a model of natural language, but if so it is a model of natural language only at a “prereflective” stage. His own description of that model, in which he says that liar examples are neither true nor false, is from the perspective of a later and higher “reflective” stage of development. One conspicuous difference between the two stages is in their logics. The internal logic, the logic involved in the model, is a nonclassical one, the Kleene strong trivalent logic. The external logic, the logic of Kripke’s proofs about the model, is as Kripke emphasizes classical, as in any other contribution to orthodox mathematics. Kripke writes: “If we think of the minimal fixed point . . . as giving a model of natural language, then the sense in which we can say, in natural language, that a Liar sentence is not true must be thought of as associated with some later stage in the development of natural language, one in which speakers reflect on the generation process leading to the minimal fixed point. It is not itself a part of that process. . . . The ghost of the Tarski hierarchy is still with us.” An analogy offered by Soames can help to clarify this aspect of Kripke’s view. Imagine someone pointing to a number of persons, none taller than four-­foot-­six, and saying that these paradigms, along with anyone shorter than any of them, are all smidgets, and then pointing to a number of other persons, none shorter than five-­foot-­six, and saying that these foils, along with anyone taller

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than any of them, are nonsmidgets. These specifications leave Mr. Smallman, who is exactly five feet tall, unclassified. For Soames, the (partial) specifications Kripke gives for what is to count as “true” (which those who have read §7.4 have seen set down as (13) and (14) of that section) are comparable to the partial specifications for the term “smidget.” Liar and other pathological sentences are analogues of Mr. Smallman. Presumably the pre‑ or unreflective speaker will simply go forward with what he has been told, classifying lots of children and a few adults as smidgets, and a few children and lots of adults as nonsmidgets, and saying nothing about Mr. Smallman, of whose existence he may be unaware. The reflective speaker, by contrast, will ponder what she has been told by way of explanation of the term “smidget,” notice that she has been given no basis to call Mr. Smallman a smidget or a nonsmidget, and perhaps be tempted to conclude that he is not a smidget, though not a nonsmidget, either. The sense in which she says that Mr. Smallman is not a smidget is associated with “a later stage in the development of the language, in which one reflects on the process.” The long and short of it is that Kripke, with his prereflective/reflective distinction, can make and does make no claim of achieving a total vindication of the intuitive notion of truth. Doublespeak is represented by the proposal to distinguish, in hopes of improving on Kripke and exorcizing Tarski’s ghost, not Kripke’s two stages in the development of the notion of truth, but two speech acts, strong affirmation and weak affirmation. The former is ordinary assertion, or putting forward as true, while the latter is putting forward as unfalse. The gappiness of liar examples, ~Tp  ~T~p, can then be at least weakly affirmed, and to that extent the ineffability problem can be overcome, and perhaps a vindication claimed. Let us look a little closer at this proposal. If one distinguishes strong and weak affirmation, one must then double various other notions. One must distinguish strong rejection (ordinary denial or assertion of the negation), from weak rejection, the weak affirmation of the negation. (Weak rejection somewhat resembles the metalinguistic negation considered in §4.4.) Likewise one must distinguish strong entailment (ordinary implication or following

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in a truth-­preserving manner) from weak entailment (or following in an unfalsehood-­preserving manner). As strong entailment preserves strong affirmability, so weak entailment preserves weak affirmability. (The logics of strong and weak entailment differ; for instance, disjunctive syllogism preserves strong affirmability but does not preserve weak affirmability in the case where A is truth-­valueless but B is false.) While we are at it, we may as well define strong veracity to be ordinary truth, and weak veracity to be unfalsehood. Then we can say that, weak or strong, affirmation is putting forward as veracious, rejection is affirmation of the negation, and entailment is veracity‑ and affirmability-­preserving. In the jargon of the literature on many-­valued logics, the strong and weak versions of trivalent logic differ in that, while they have the same truthtables, the former takes only 1 while the latter takes both 1 and ? as “designated values.” So far, so good, but there is a problem. With the intuitionistic logic advocated by antirealists, while not all classical forms of mathematical reasoning are acceptable, a great many still are, and a substantial body of intuitionistic mathematics has been built up. In particular, when intuitionists prove formal results about their logic, they are careful always to reason intuitionistically, so that their results will be theorems of this intuitionistic mathematics. By contrast, as Solomon Feferman has said, with gap logic “nothing like sustained ordinary reasoning can be carried on.” The lack of a useful conditional-­type operator is one handicap, but only one. So it is no surprise to find that advocates of gap logic, in proving results about gap logic, never restrict themselves to gap-­ logically acceptable forms of inference. This is the real “Revenge of the Liar,” raising doubts whether gap theorists believe their own theory seriously enough to let it affect their practice. 8 .4  “Paraconsistency”

Another type of “solution” advocates recognizing truth-­value gluts rather than gaps. Partisans of this approach reject disjunctive syllogism (6), and block the paradoxical reasoning at its last step (7j). They allow negation inconsistency but not absolute

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inconsistency, a combination known as paraconsistency, and speak of the rival gap approach under the contrasting label paracompleteness, a term not traditionally used by partisans of the gap approach themselves. But glut theorists, too, face an ineffability problem. It is not that they cannot enunciate their theory without saying something that, according to that theory itself, is untrue (as well as unfalse), but that they cannot enunciate it without saying something that is false (as well as true); and so they, too, are led to resort to a kind of doublespeak. The most important fact about the glut theory is one well known to specialists but rarely mentioned in the literature, namely, that it is identical to the gap theory except for terminology: One group pronounces the 1 and 0 and ? of the trivalent tables as “true” and false” and “gappy,” while the other pronounces them as “unfalse” and “untrue” and “glutted”; one group applies the ordinary terms truth and assertion and denial and implication to the strong versions of veracity and affirmation and rejection and entailment, while the other applies them to the weak versions. If glut theories are perceived as radical, exciting, and transgressive, while gap theories are perceived as moderate, dull, and conventional, that is a testimony to the power of advertising, for there is no more real difference between the two than between a used and a “pre-­owned” car. This is all that really needs to be said here about “paraconsistent” vs “paracomplete” approaches. An option genuinely distinct from both would be presented by a quadrivalent or four-­valued logic with both gaps and gluts (where it might be held, for instance, that the truth-­teller is gappy, while the falsehood-­teller is glutted). Such an option would still face, perhaps even more acutely, the problem of a split between the “external logic” and the “internal logic.” 8 .5  Contextualist

124

“Solutions”

Dependency is represented by “solutions” that retain classical logic, but reject the formalization (7) as involving a fallacy of equivocation. For (7) represents all tokens of “(2) is false” by the

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same letter p, whereas (it is claimed) different tokens, because they occur in different contexts, express different propositions, and can have different truth values. One type of contextualist theory claims that the extension of the truth predicate can be context-­dependent in this way. In its simplest version, it is just the view that each occurrence of the truth predicate bears an inaudible, invisible Tarski-­style subscript. (The problem of practical feasibility mentioned in §7.1 for audible, visible subscripts does not arise, since “context” is supposed to supply the subscripts, whether or not the speaker or writer can.) How is the imputed subscript to be determined? The most simplistic proposal would be just this, that in any stretch of discourse, one starts with subscript zero, and raises the subscript only when the principle of charity requires one to do so in order to be able to construe some argumentative step as cogent. Of the various rules of inference we have considered, the only one that might require such subscript-­raising is T‑introduction (1a) (along with its negative counterpart (1c)), which on the kind of view we are contemplating is only appropriate with the proviso that the subscript on T must be higher than any in A. Thus in the deduction (7) the subscript would have to be raised at step (7f). Restating the deduction in words rather than symbols (again indenting hypothetical but not categorical steps), it then becomes the following:  (9a) is false0.  (9b)  It is false0 that (9a) is false0.  (9c)  It is true0 that (9a) is not false0.  (9d)   (9a) is not false0. (9e) (9a) is not false0. (9a)

(9f) It is true1 that (9a) is not false0. (9g) It is false1 that (9a) is false0. (9h) (9a) is false1. (9i) (9a) is false1 or Santa Claus exists. (9j) Santa Claus exists.

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Read without the subscripts, each step considered in isolation is at least initially and superficially plausible, but according to the present analysis, the last is fallacious. It appears to be an instance of the valid (6), but owing to the difference in subscripts between (9i) and (9e), it is really an instance of the invalid (6)  from A  B and ~A to infer B This seems very tidy, but it is hard to see how to apply the policy of giving lower subscripts to earlier-­occurring tokens in a case where two tokens occur simultaneously, as in the old Socrates-­ Plato example, or the postcard paradox, where one is presented with a postcard with “What is written on the other side of this card is true” on one side, and “What is written on the other side of this card is false” on the other side. Worse, strengthened liars threaten: (10a) What I am now saying is of a type that has no true tokens. (10b)  What I am now saying is not true on any level. The proposal under consideration seems to require that it is impossible to quantify over all tokens or contexts or levels or subscripts. But why should this be impossible, and why should one believe that (10ab) don’t accomplish it? A variant version of this strategy distinguishes levels of propositions rather than of truth. Since any hierarchy of propositions brings with it a hierarchy of truth predicates (with truthi being truth for propositionsi) this is not so much an alternative to as an elaboration of the proposal we have been considering. An example like (2) is construed in an elaborated fashion as involving an implicit quantification over propositions, somewhat as follows: (11) There is a proposition that is expressed by (11), and any such proposition is false.

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So the specific form of context-­dependence claimed to be involved is variation from context to context of the domain over which some quantifier ranges. This specific phenomenon, quantifier-­domain variance, is widely recognized by linguists and philosophers of language, usually illustrated by such examples as the following:

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(12)  All the beer is in the refrigerator. where it is not reasonable to understand the quantifier as ranging over all the beer in the whole wide world, but rather over all the beer in the house, or all the beer for the party, or some such, depending on the context in which (12) is uttered. Note, however, that the speaker in (12) is not intending to make any absolutely universal statement about all beer, while the solution to the liar paradox being contemplated requires, in order to avoid strengthened liars, that anyone who speaks of “all propositions,” even if she is trying with all her might to say something about absolutely all propositions, must fail in the attempt. This is something quite different. Moreover, this proposed solution invites an obvious ineffability objection. The would-­be vindicator of the intuitive notion of truth seems to need to maintain (13) All propositions that appear to be about all propositions are not really so, but only about some restricted range of propositions. And (13) is in an obvious way self-­stultifying: If it is not really about all propositions, then it falls short of its purpose, but if it is really about all propositions, then it is a counterexample to itself. This is an embarrassment faced long ago by Russell, who in his own solution to the Russell paradox, his theory of types, wanted to say something very like (13). Russell invented a doctrine, adopted by proponents of the kind of contextualism under consideration, of a special kind of unnegatable systematically ambiguous or schematic utterance, through which (it is claimed) the thing the theorist wants to say, though it cannot be said, yet can be shown. Critics don’t see it. 8 .6  Inconsistency

Theories

Defeatism seems the natural inductive conclusion from the observation that one “solution” after another to the liar paradox fails to convince. Nonetheless, avowed inconsistency theorists are scarce, though the authors of this book are among them. The more usual reaction to the failure of old “solutions” is to propose some new one.

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While for consistency theorists the question is how to defend our intuitive notion of truth, for inconsistency theorists it is whether that notion should, in view of its inconsistency, be renounced or revised or retained. In a specialist area that puts a premium on rigor (think: pure mathematics), retention is not an option. The notion must be renounced at least until someone (think: Tarski) devises a consistent revision. If someone else (think: Kripke) later devises another, there is no reason not to adopt as many as are available and useful, so long as one distinguishes them, using different ones for different purposes. For the general public, by contrast, the liar paradox poses no threat. It is something discovered in historical times by a philosopher and passed on by books to later philosophers, but most people never encounter it. Hence for philosophers to urge the general public to renounce or revise their notion of truth would be futile. It will be retained no matter what philosophers say, and in speaking with their neighbors philosophers will be obliged to speak as their neighbors do, whatever they themselves think. But what should philosophers think? Even among the small number of inconsistency theorists at least three different attitudes can be discerned. A first answer is suggested by Matti Eklund’s endorsement of the principle that when nothing can meet all the conditions built into the meaning of some term, then the distinction the term in fact marks (or should be understood as marking) is the one that comes closest to doing so. On such a view, though it was not intended that the equivalence principle should hold anything less than absolutely universally, and though each step of the liar paradox reasoning is indeed intuitive, the equivalence principle in fact holds only “habitually,” and contrary to intuition one of the steps of the liar paradox reasoning fails. Assuming classical logic is retained, it must be either the T‑elimination step (7d) or the T‑introduction step (7f). To determine which of the two it is, one would have to determine whether the true/untrue distinction that comes closest to meeting the conditions built into the meaning of the truth predicate is one that marks the liar paradox as true or one that marks it as untrue. To make this determination will be no easy task. If we understand “coming as close as possible” as “retaining as many

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T‑­biconditionals as possible,” we have to confront the fact that there is not just one but many a maximal consistent set of T‑­biconditionals (a consistent set that becomes inconsistent if even one more T‑biconditional is added). Should we understand something to be marked true only if it is so marked whichever of these sets one chooses? Or should we understand the true/ untrue distinction as ambiguous or otherwise indeterminate? The issues are further complicated by Vann McGee’s observation that any arbitrary conclusion q whatsoever is equivalent to some T‑­biconditional Tp  p. (Just choose a p that says Tp  q.) It might be thought that the solution here would to be to establish some priority among T‑biconditionals, holding some more worthy of retention than others. But there can be no basis for doing so if the equivalence principle is the whole of the meaning or intension of “true,” and if extension is determined solely by goodness of fit with intension. For the equivalence principle is simply a universal generalization, and as such does not discriminate among its instances. Perhaps, then, we should take extension to be determined not by fit with intension as such, but by fit with intension and the use we make of it. This would justify giving higher priority to retaining useful over useless or harmful T‑biconditionals. The inconsistency theorist would then be like a Tarskian trying to replace an inconsistent notion of truth with a useful substitute, but with this difference, that once the useful substitute is found, the inconsistency theorist will claim, not indeed that this is what we intended all along, but at least that this is the extension we had succeeded in capturing all along. A skeptic may wonder how the inconsistency theorist can be justified in making such a self-­congratulatory claim—to call it an application of the “principle of charity” would be to misapply the maxim “charity begins at home”—but a prior question is why one would even want to. A second answer to the question what attitude the philosopher should take, involving no such claim, is explored by A. G. Burgess in his doctoral dissertation, the option of adopting the same “fictionalist” stance towards truth that so many philosophers have advocated adopting towards so many other things (from mathematics to modality to morals). Central to fictionalism is a

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distinction between what one says about the “fiction” (what philosophers say to each other in the philosophy seminar room) and what one says within it (what philosophers say to members of the general public on the street). About the “fiction of truth” one might say (14a) The liar paradox shows that the notion of truth is ultimately incoherent. as one might say (14b) Sherlock Holmes is not a real person but a character invented by Arthur Conan Doyle. Within the fiction one might say (15a) To assert that some proposition is true is equivalent to just asserting that proposition. as one might say (15b) Sherlock Holmes shares rooms with Doctor Watson in Baker Street.

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One maintains an attitude of genuine belief towards what one says about the fiction, and to what one says within it if prefaced by the disclaimer “According to the pertinent fiction.” Towards what one says within the fiction, taken without the disclaimer, one only maintains a pretence of belief. Philosophical discussions of fictions in the literal sense of novels and short stories and the like—Kendall Walton’s discussion of prop-­oriented make-­believe is especially often cited—can be drawn on for principles about when one can and can’t fairly say “According to such-­and-­such a fiction, p.” If one takes a fictionalist view of truth, one will doubtless want to take a similar view of other alethic notions such as satisfaction, and also (since the extension of a predicate is just the set of things that satisfy it) of the notion of extension. So from a fictionalist point of view, it is just a mistake to want to ask what the extension of the truth predicate really is. Extensions are fictional, not real. What becomes of the liar paradox reasoning on this proposal? Speaking about the fiction, what one says is precisely (14a).

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Speaking within the fiction, how could things fairly be said to stand “according to the fiction of truth” with respect to the liar paradox reasoning? The analogy with fiction in the literal sense of imaginative literature suggests comparison with a story about a mathematician who finds a counterexample to Fermat’s theorem. How could things fairly be said to stand “according to the story” with respect to Wiles’s proof of that theorem? Presumably in both cases one should say that according to the fiction there can be no genuine proof of a certain kind, so any purported such proof must contain a fallacy somewhere. But there is no one step one can point to and say, “Here is where, according to the fiction, the fallacy lies.” The fictionalist’s feigned belief would thus be much like the real belief of those would-­be consistency theorists who are dissatisfied with all existing consistency theories: There must be a fallacy somewhere in the liar paradox reasoning, though we can’t (yet) say where. But probably the fictionalist philosopher will never be called upon by any member of the general public to express a belief about the liar paradox reasoning, since even today the proverbial “man in the street” will most likely never have heard of it. Be all that as it may, the within/about distinction works enough like Tarski’s object/meta‑ distinction to prevent strengthened liars from being a problem (though this conclusion does depend on assuming that the “fiction of truth” isn’t like one of those tricky fictions that refers to itself as fictitious). A third answer to the question what attitude the philosopher should take is that we should proceed as much as possible in the same unself-­conscious way as those who have never encountered the liar, not keeping our fingers crossed as we speak, or saying “but all that is just according to a certain fiction” sotto voce after every remark to a neighbor. The fact that we have encountered the liar paradox reasoning and recognized the inconsistency of our intuitive notion of truth just means that if we ever do stumble upon similar reasoning in daily life we will be able to recognize it for what it is, and have the good sense not to “follow the argument wherever it leads,” since we will remember it leads us someplace we don’t want to go. By way of justification for adopting such a nonchalant attitude towards contradiction, it may be noted that there is illustrious

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precedent for something like it. Mathematicians from the time of Leibniz noted that manipulations of infinite series can lead to paradox, thus: (16) 0 = 0 + 0 + 0 . . . = (1 – 1) + (1 – 1) + (1 – 1) . . . = 1 – [(1 – 1) + (1 – 1) + (1 – 1) . . .] = 1 – (0 + 0 + 0 . . .) = 1–0=1

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They nonetheless went right on using series, without any clear or distinct notion of the scope and limits of “safe” and “dangerous” manipulations, and if they stumbled into uses leading to paradox, they just ignored them. When mathematicians became differentiated into pure and applied, and pure mathematicians found reason to insist on strict rigor, they banned all series except those for which they could give a rigorous definition of sum (convergent series); but even then applied mathematicians and physicists went right on using series to which that definition did not apply (divergent series) for half a century before rigorous definitions of sum were found for these as well. If there was nothing unreasonable in the attitude of Leibniz and his successors—and in view of the great successes of mathematics and physics during the period it would be hard to argue that there was—why should there be thought to be anything unreasonable in retention, at any rate outside of fully rigorous pure mathematics, of a notion of truth found to be inconsistent in cases that in practice are seldom encountered and easily recognized when they are? All theories of the nature of truth, deflationist or inflationist, are controversial, and that is even more emphatically so for all approaches to the paradoxes. There is an understandable tendency for a philosopher already committed to a controversial position on one topic to want to avoid further controversies on others. Yet we would like to urge that a theory of the nature of truth can only really be satisfying if it can be matched with a satisfactory account of the paradoxes of truth. It is widely held that since deflationism holds truth to have no nature, and only by appeal to some substantive nature of truth could one hope to solve the paradoxes, deflationism will have a harder time than its rivals dealing with them. This opinion,

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however, assumes that the paradoxes are to be dealt with by solving them, rather than by declaring them insoluble. We would like to suggest, on the contrary, that at present deflationism is ahead of its rivals precisely because it is well matched with an approach to the paradoxes, namely, the inconsistency theory, while other theories of the nature of truth remain without an obvious, natural partner. The background assumptions about meaning behind an integrated deflationist/defeatist theory might run as follows. Meaning can be given by rules (as on a holist account of the kind considered in §6.5), for there can be no objection to positing rules in semantics, given that linguists have been freely positing them in syntax and phonology ever since the downfall of behaviorism. Rules, however, can be inconsistent, and there is even a result in mathematical logic (Church’s theorem) to the effect that there is no mechanical test for inconsistency of rules, making it unlikely we have any filter preventing us from ever internalizing inconsistencies. If we sometimes do so, we’ll need some override mechanism to halt the operation of rules in emergencies, as a computer may stop and return a message of “too much recursion” if it finds itself going round and round the same loop; and though internalizing rules may or may not be a matter of acquiring dispositions, it is certainly not a matter of acquiring irresistible compulsions, so there is no conceptual obstacle to positing such an emergency override mechanism. Such a general mechanism for suspending rules would make building separate safeguards into the rules for each separate term superfluous. Its operation would be in evidence whenever we felt the tug of the rules but resisted it, whenever we found a step intuitive, and yet refrained from taking it. Given such background assumptions, the specific claim about the truth predicate might be that its meaning is given by T‑­introduction and T‑elimination rules, with no subtle and sophisticated restrictions to evade paradox, and these rules are simply inconsistent. The fact that each step in the liar paradox reasoning is felt to be intuitive, though we refrain from inferring arbitrary conclusions, would be construed as a case of the postulated general fail-­safe mechanism at work.

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Whether or not such ideas are ultimately tenable, whether or not such an attempt to integrate the deflationist and inconsistency theories can ultimately succeed, we would suggest that the next step in the development of the theory of truth must be the closer integration of views about its nature with views about its paradoxes, if not in the way just sketched, then in some other.

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Further Reading

The topic of truth leads off in multiple directions into others. Let us list some sources where the reader interested in one or another of these directions can pursue it further, taking the various topics in order by the chapter in this work to which they are most pertinent. Introduction. We have said that a satisfactory account of the nature of truth must incorporate an account of the paradoxes. One might add that it really ought also to incorporate accounts of approximate truth and of the value of truth, a pair of topics we set aside at the beginning of our survey as beyond its scope. The interested reader can make a start on the study of the first of the pair with Oddie (2008), which like other entries in the reference work of which it is a part offers an ample bibliography. The sort of thing analytic philosophers have found to say (mainly in opposition to postmodernism) about the second of the pair can be seen from Blackburn (2005), Lynch (2005), and most concisely Frankfurt (2008). As to the issues we have not set aside, among single-­ author surveys with emphases different from ours we have found the article Glanzberg (2004) and the book Kirkham (1992) most useful. (We have borrowed the terminology congruence vs correlation from the latter, for instance.) Künne (2003) and Schmidt (1995) may also be mentioned. The reader with historical interests will find Candlish and Damnjanovic (2007) very useful. Such readers will ultimately also want to look at the actual formulations of Joachim, James, Russell, and other historical figures themselves. The Oxford Readings anthology Blackburn and Simmons (1999) conveniently makes available a number of historic texts going back to Frege, including selections from the three authors just named, the contributions of Austin and Strawson to their famous Aristotelian Society symposium, the passages in Ramsey that became the wellspring of deflationism, and more—as well as concise statements of their views by authors still actively contributing today. The rival anthology

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of Lynch (2001) contains similar material, along with material from the coherentist and pragmatist traditions (ranging from Blanshard to Putnam and Rorty). The reader whose interest is really piqued by coherentism may consult the survey Young (2008), or for a somewhat idiosyncratic book-­length treatment, Walker (1989). The Lynch volume even includes token representation of Continental views, though unfortunately Nietzsche (1873/1954) is missing, along with anything from Jürgen Habermas. For the ins and outs of propositionalism vs sententialism, see Soames’s volume Philosophy of Language in the present series. Tarski. The monumental monograph of Tarski (1935/1956) is tough going for any reader, but the volume cited for the English translations also contains translations of some of Tarski’s semipopular expositions of the same ideas, complementing our discussion in chapter 2. For the impact of Tarski on logic, see any contemporary textbook, for instance, Boolos, Burgess, and Jeffrey (2007). See Patterson (2008) for a wide range of views on Tarski, with emphasis on his impact on philosophy. Deflationism. The Lynch anthology contains material from Quine (1990) and Field (1994) on disquotationalism, but the fullest collection of material on that topic is to be found in Armour-­Garb and Beall (2005), an anthology entirely devoted to the pros and cons of deflationism in the various guises we discussed in chapter 3 and others besides. There one can find the Pittsburgh philosophers Brandom (neoprosentialism) and Gupta (antideflationism) debating each other, for instance. But for the somewhat idiosyncratic “thick deflationism” of McGinn one will have to turn to another useful collection, Schantz (2001), which consists entirely of essays written specially for that volume; while for the objection that there is a need for a “transcendent” application of the truth predicate one should see Shapiro (2003). If all this is not enough, Grover (1992) and Horwich (1990) have provided book-­length expositions of their views, with responses to critics. The former incorporates the minor classic Grover et al. (1972), the latter amounts to a little catechism of minimalism. J. P. Burgess (2002) defends an idiosyncratic view on the optimal formulation of deflationism that we have not insisted upon here. For an overview with different emphases from ours, see Stoljar and Damnjanovic (2009).

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Indeterminacy. The philosophical literature on the topics we introduce in chapter 4 is enormous, though unfortunately the connections with the literature on the nature of truth are often made only in scattered passages. For a logical approach to vagueness we barely mentioned in passing, see Hajek (2009), which provides basic information and references on fuzzy logic and the view that truth comes in numerical degrees. Fuzzy logic is criticized by Haack (1996), which also treats other nonclassical logics, while Smith (2008) defends an elaborate numerical-­degrees theory. There is a recent whole anthology Kölbel and García-­Carpintero (2008) devoted to MacFarlane-­style relativism, including a contribution by MacFarlane himself, while Field (2001) has published a collection of his lengthy papers on truth and “nonfactualism.” See also Boghossian (2006) for a fuller enunciation of the arguments we sketchily indicated in our discussion of global vs local. Realism. The spread of views called “realism” in the literature is even wider than what we have indicated in chapter 5. Like deflationist theories, realist theories have their own anthology, Greenough and Lynch (2006), though the editors’ conception of “realism” differs from that adopted in this book. Hill (2002) sees less incompatibility between deflationism and realism than most others, and Alston (1996) offers a quasideflationist version of realism. As to unblushingly metaphysical theories, see David (2009) for a survey of correspondence views. For the identity theory, which we barely mention, see the article Candlish (2008), or for advocacy of a particular version, the book Dodd (1981). See Armstrong (2004) for the fullest statement of the views of the preeminent truthmaker theorist, addressing in particular issues about negative and general truths. Merricks (2007) surveys the metaphysical controversies over truthmakers, supervenience, and more, from a critical point of view. For an even more critical point of view, see Soames (2008). And then there is the unforgettable “fuggedaboutit” paper Lewis (2001). And as to physicalist theories, Field (1972), castigating Tarski for his insufficient “physicalism,” is classic, as for that matter is the reply of Soames (1984) (which also includes a brief version of the argument that the question whether all truth is verifiable isn’t

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really about truth). A more-­or-­less physicalist position is taken in Devitt (1997), to be compared with the same author’s earlier work on causal theories of reference (1981). Related works of Putnam (1978) and Leeds (1978) concern, among other things, the utility issue. Price (1989) is a large-­scale study by a writer whose position on truth is best known for his emphasis on normativity. Antirealism. Practically everything Dummett has written— and he has written a lot, from Dummett (1959), the locus classicus for the normativity objection to deflationism, onwards—is related to his antirealism. The less technical first half of Dummett (1973) is perhaps the place to start tackling that author’s works. J. P. Burgess (1984) is one of many, many replies this work has provoked. The background literature on Davidsonianism, often Dummett’s ultimate target even if Dummett seldom mentions Donald Davidson by name, is again enormous, though works explicitly connecting it to debates over the nature of truth are less common than one might expect. The Soames volume in this series contains extended discussion of the ups and downs of the Davidson program. The statement of Davidson (1996) himself, endorsing primitivism, is of course a must-­read. Bar-­On et al. (2001) is also important. See Lepore and Ludwig (2007) for, among many other things, a version of “Davidsonianism” that, if it does not exactly minimize the role of truth in truth-­conditional semantics, anyhow maximizes the indirectness of that role. For pluralism see Wright (1992), a work containing much more than the ideas we have sketched and criticized, and Lynch (2009), a work heavily influenced thereby, and much concerned to address the objection about mixed inferences, for which the locus classicus is Tappolet (1997). Among even newer views than pluralism, about which no very large literature has yet gathered, but which can be expected to generate discussion, Richard (2008) will repay study, as may Millgram (2009). Kripke. There is no substitute for tackling Kripke (1975), directly­—and one can say the same thing for Kripke (1972), though it has played only a walk-­on part in our discussion. How far Kripke transformed discussion of the paradoxes becomes clear on comparing the pre‑ and post-­Kripke anthologies Martin (1970) and Martin (1984). Some guidance beyond what we were

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able to provide in chapter 7 may be desirable. Soames (1999) is especially recommended, and not just for its treatment of Kripke. Martin and Woodruff (1975) is the most important precursor of Kripke’s theory of truth on its technical side. Visser (1989) is, for those with the requisite background in logic, one of the earliest and still one of the best comparative guides to Kripke and his revisionist rivals. As to Kripke’s successors, shorter and longer statements of such revisionist positions can be found in the various works of Herzberger, Gupta, and Belnap listed in our bibliography below; for further advocacy see Yaqub (1993), and for a more neutral survey, Kremer (2009). We have written as if there were a clear separation between technical additions and amendments to Kripke and philosophical discussions of the paradoxes, but a look at the actual literature will show that many works straddle the border. That is clearly the case with Barwise and Etchemendy (1987), which consists partly of a formal reconstruction of a Kripke-­type theory using Peter Aczel’s theory of non-­well-­founded sets, but also of an attempted solution to the paradoxes drawing (remote) inspiration from Austin. Likewise with McGee (1991), which gives a treatment of the paradoxes of truth and of vagueness making heavy use of a notion of determinateness, besides including substantial exposition and evaluation of Kripke’s theory. (A little gem by the same author, McGee (1992), shows that the policy “Retain as many instances of the T‑scheme as possible without inconsistency” leads to no definite result.) And then there is Field (2008), presenting, along with a good deal of informative discussion of earlier proposals, an elaborate construction claimed to be “revenge-­free.” That claim seems to have been a main impetus for putting together the anthology Beall (2007). Among critiques of Field there, Rayo and Welch (2007) may be the most useful. Only the reader well trained in logic will want to tackle the work on complexity of J. P. Burgess (1987/1988) and Welch (2009) (to cite only the most recent contribution of that prolific author). For axiomatic theories, the forthcoming studies by Halbach (2011) and Horsten (2011) will be indispensable, and perhaps more accessible than the seminal papers of Feferman (1991) and Friedman and Sheard (1987). Versions of the curious objection

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to deflationism based on the “nonconservativeness” of truth theories can be found in Shapiro (1998) and Ketland (1999); such criticisms are one of the topics discussed by Horsten, who is specifically concerned with axiomatic theories and deflationism. Insolubility. The gap theory as expounded in chapter 8 is, of course, in a sense best represented by Kripke himself. The underlying logic is discussed in Blamey (1986). The two-­speech-­act—one might call it the “forked-­tongue”—approach appears in early and pure form in Tappenden (1993). Maudlin (2004) makes do with a single speech act, but one for which truth is not always the norm. His work is unusual for that of a would-­be paradox-­solver in admitting defeat, though Gaifman (1992), the purest approach based on type/token distinctions, also admits there are “black holes.” The “paraconsistent” or “dialethist” approach is advocated by Priest (2006), pontifex maximus of the cult of contradiction. Two reviews, the sympathetic Field (2006) and unsympathetic Scharp (2007), discuss some of the problems with the position. The equivalence of “paracomplete” and “paraconsistent” trivalent approaches is spelled out in Beall and Ripley (2004). A quadrivalent approach is advocated in Beall (2005), a contribution to a collection, Beall and Armour-­Garb (2005), devoted to attempts to uphold deflationism and solve the paradoxes (rather than declare them insoluble, as the present authors would). Many of the contributions to that volume can be seen as responses to the call, echoed here, for closer integration between work on the nature of truth and work on the paradoxes. For an extended version of that call, see Beall and Glanzberg (2008), and for an extended answer (quite different from ours), see Beall (2009), which combines dialethism (trivalent, not quadrivalent) with deflationism. Parsons (1974) is a seminal work in the contextualist tradition, striking in that it actually antedates Kripke’s influential work. Burge (1979) presents a proposed solution to the paradoxes based on variation in the extension of the truth predicate from context to context. A rival contextualist approach perhaps closer to Parsons is worked out in detail by Glanzberg (2004). Simmons (1993) presents a distinctive version of contextualism, called the “singularity” approach, loosely inspired by remarks of Gödel. There are also weirder “solutions” to the paradoxes than any we have

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discussed. Notably, Rahman and coeditors (2008) collect papers on medieval views, and modern ones loosely inspired thereby, that involve the thesis that the same sentence, in the same context, can “say more than one thing.” Chihara (1979) was the first important restatement of the inconsistency view after Tarski, and has heavily influenced both the present authors, as has Barker (1999), a detailed defense of the inconsistency theory in the face of post-­Kripkean proposals. Eklund (2002) is probably the best-­known version of the inconsistency theory as of this writing. A. G. Burgess (2007) advocates a fictionalist approach, but we have not insisted on it here. Doubtless we have overlooked some important work, and certainly also by the time this survey appears the literature will have grown further. The one point on which virtually all contributors to the literature on truth, despite their many disagreements, will agree is that no one has yet arrived at the full and final truth about it.

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Bibliography

Alston, William (1996) A Realist Conception of Truth (Ithaca: Cornell University Press). Armour-­Garb, Bradley, and J. C. Beall (2005) (eds.) Deflationary Truth (Chicago: Open Court). Armstong, David M. (2004) Truth and Truthmakers (Cambridge: Cambridge University Press). Austin, J. L. (1950) “Truth,” Aristotelian Society Supplement 24: 111–129, reprinted in Blackburn and Simmons (1999), 149–161; also in Lynch (2001), 25–40. Barker, John (1999) The Inconsistency Theory of Truth, doctoral dissertation, Princeton University. Bar-­On, Dorit, William Lycan, and Claire Horisk (2001) “Deflationism and Truth-­Conditional Theories of Meaning,” Philosophical Studies 28: 1–28. Barwise, K. Jon, and John Etchemendy (1987) The Liar: An Essay on Truth and Circularity (Oxford: Oxford University Press). Beall, J. C. (2005) “Transparent Disquotationalism,” in Beall and Armour-­Garb (2005), 7–22. (2007) (ed.) The Revenge of the Liar: New Essays on the Paradox (Oxford: Oxford University Press). (2009) Spandrels of Truth (Oxford: Clarendon Press). Beall, J. C., and Bradley Armour-­Garb (2005) (eds.) Deflationism and Paradox (Oxford: Clarendon Press). Beall, J. C., and Michael Glanzberg (2008) “Where the Paths Meet: Remarks on Truth and Paradox,” in French and Wettstein (2008), 169–198. Beall, J. C., and David Ripley (2004) “Analetheism and Dialetheism,” Analysis 64:30–35.

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Index

A-­notions, 45–46 abbreviation, 20 abstract particulars, see features Aczel, Peter, 139 alethic notions, 6, 18 Alexinus of Elis, 53 Alston, William, 137 antirealism, 5, 7, 9, 53, 116, 120, 123; see also Dummett, Michael approximate truth, 135 Aristotle and Aristotelianism, 2, 6, 16, 21, 34 arithmetic, 24, 28–30, 109 Armour-­Garb, Bradley, 136, 140 Armstrong, David, 137 assertion, 14, 80–82; see also A‑notions atomic sentences and formuals, 23, 26 Austin, J. L., 39–40, 71, 135, 139 axiomatic theories of truth, 21, 114–115, 139–140 bald one (phalakros) paradox, 55 Barker, John, 141 Bar-­On, Dorit, 138 Barwise, Jon, 139 Beall, J. C., 136, 139, 140 beliefs, 14, 77–78; see also disqualification strategy Belnap, Nuel D., Jr., 113, 139 Berry’s paradox, 7 biconditionals, 3 bivalence, 52 Blackburn, Simon, 135 Blamey, Stephen, 140 Blanshard, Brand, 135 blind assertion, 36, 40 Boghossian, Paul, 66–67, 137 bound vs free variables, 26 Brandom, Robert, 40–41, 136 Bryan, William Jennings, 16

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Burali-­Forti paradox, 6 Burge, Tyler, 140 Burgess, A. G., 129, 141 Burgess, J. P., 136, 138, 139 Candlish, Stewart, 135, 137 Cantor, Georg, 111; his paradox, 6 categorical vs hypothetical use of rules, 118 causal theories, 75–76, 137 Chihara, Charles, 141 Church’s theorem, 133 code numbers, 28 coextensiveness, 21, 28 coherence theory, 3–4, 136 coherent partial valuation, 110 color, see vagueness complexity, 114, 139 composition laws, 109, 115 conceptualism, 68, 83 conditionals, 94, 97 congruence theories, 70–72, 135 conjunction, 22, 45–46 consistency strength, 114 context of assessment, see truth relativism contextualism, 60–61, 125–127 contradiction, see inconsistency correctness of nonlinguistic representations, 78–79 correlation theories, 70–71 correspondence theories, 3–5, 9, 70–72, 135, 137 Curry’s paradox, 118, 120 Damnjanovic, Nic, 135, 136 David, Marian, 137 Davidson, Donald, and Davidsonianism, 87–89, 97, 138 Dedekind, Richard, 24–27

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Index defeatism, 10, 61, 127 definition and definability, 6–7; see also recursive vs direct definition deflationism vs inflationism, xii, 5, 7–9, 33–53, 62, 65, 69, 75–79, 93,95–96, 98, 100, 116, 132–134, 136 denial strategy, 55–56, 118–119 denotation, 18, 23, 25, 30, 49–50, 70, 109; see also reference dependency strategy, 60–61, 124–125 Descartes, René, 2, 31 descriptive vs demonstative ­conventions, 71 determinateness, 58 deviance strategy, 57–58, 120–121 Devitt, Michael, 138 Dewey, John, 4–5 dialethism, see paraconsistency direct definition, see recursive vs direct definition disjunction, 22, 91, 97 disjunction introduction and disjunctive syllogism, rules of, 117 disqualification strategy, 56–57, 119–120 disquotationalism, 41–44, 50 Dodd, Julian, 137 domain of quantification, 29; variance of, 126 doublespeak strategy, 59–60, 122–123 Dummett, Michael, and Dummettianism, 80, 83–84, 90–98, 138

154

Eklund, Matti, 128, 141 elimination rules, see introduction and elimination rules endorsement, speech act of, 39 equivalence principle, 8, 33–34, 43–44, 48, 52–53, 55, 61, 93, 100 Escher, M. C., 31 Euclidean vs non-­Euclidean geometry, 31 extension, 128–129; vs intension, 21 external vs internal logic, 121, 124 Epimenides paradox, 104 epistemicism, 56 equivalence principle, 8

Burgess_FINAL.indb 154

Escher, M. C., 31 Etchemendy, John, 139 Eubulides of Miletus, 6, 53, 55 Euclid of Alexandria, 30–31 Euclidean vs (hyperbolic) non-­ Euclidean geometry, 31 excluded middle, see intuitionism facts and states of affairs, 70–73 fallacy of many questions, 53 falsehood 10, 52 falsehood-­teller vs untruth-­teller, 119 features, 73–74 Feferman, Solomon, 114, 123, 139 Field, Hartry, 62, 74–75, 76, 113, 136, 137, 139, 140 fictionalism, 129–131 Fitch’s paradox of knowability, 95 fixed points, 108, 110, 111; maximum intrinsic, 108; minimum, 106–107, 111–112 formal correctness, 20–21 Frankfurt, Harry, xi, 135 free vs bound variables, 26 Frege, Gottlob, 4, 22, 55, 87, 135 Friedman, Harvey, 114–115, 139 functionalism, 99 fuzzy logic, 137 Gaifman, Haim, 140 games, 80–81 gaps and gluts, 55, 58, 105, 123–124; see also trivalent logic Garcia-­Carpintero, Manuel, 137 geometry, 27, 31 Glanzberg, Michael, 135, 140 Gödel, Kurt, 28–29, 32, 114, 140; his completeness theorem, 32; his incompleteness theorems, 28, 114 Greenough, Patrick, 137 Grelling (or heterological) paradox, 6 Grover, Dorothy, 40, 136 Gupta, Anil, 113, 136, 139 Haack, Susan, 137 Habermas, Jürgen, 136

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Index Hajek, Peter, 137 Halbach, Volker, 139 heap (sorites) paradox, 55 Herzberger, Hans, 113, 139 heterological (or Grelling) paradox, 6 Hill, Christopher, 137 holism, 96–97 homorphism vs isomorphism, 71 horned one (keratines) paradox, 53 Horsten, Leon, 139–140 Horwich, Paul, 44–45, 136 hypothetical vs categorical use of rules, 118 I-­notions, 45–46 idealism, 3–4, 5, 68 identity theory, 70, 137 “iff ” abbreviation, 2 immanence vs transcendence, 41–42, 136 inconsistency, absolute vs negation, 117, 124 inconsistency theories 102, 116, 127–134 indeterminacy, 8, 52–65, 137 indexicals and indexicality, 12, 63 ineffability, 120–121 inference, see I‑notions, rules of inference insolubles, 6 instantiation, 71 internal vs external logic, 121, 124 interpretation, 29–30 intrinsic truth, 108 introduction and elimination rules, 97; see also T-­introduction, T‑elimination intuitionism, 83, 92, 120, 123 James, William, 3–5, 16 Joachim, H. H., 3–5 jump operation, 110 Kant, Immanuel, 2 Ketland, Jeffrey, 139 Kirkham, Richard, 135 Kleene, S. C., see trivalent logic

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knowledge: of truth-­conditions, 86; of meaning, 90; tacit vs verbalizable, 90, 92; see also manifestation Kölbel, Max, 137 König’s paradox, 7 Kremer, Philip, 139 Kripke, Saul, 9–10, 24, 75–76, 102–112, 114, 120–122, 128, 138–139, 140 language: first order, 29; interpreted vs uniterpreted, 29–30; object vs meta‑, 19–20; of thought, 89; see also arithmetic, geometry Leeds, Stephen, 138 Leibniz, 132 Lepore, Ernest, 138 Lewis, David, 72–73, 95, 137 liar (pseudomenos) paradox 6, 53, 107; see also paradoxes linguists and linguistics, 13–14, 17; see also metalinguistic negation logic, see deviance strategy logical atomism, 72 logical pluralism, 100 logical positivism, 5, 21, 74 logical truths, 31–32 Löwenheim-­Skolem theorem, 32 Ludwig, Kirk, 138 Lynch, Michael, 98–100, 135–138 MacFarlane, John, 62, 67, 137 manifestation argument, 92–93 Martin, Robert, 138–139 material adequacy, 20–21 Maudlin, Timothy, 140 McGee, Vann, 129, 139 McGinn, Colin, 46, 136 meaning, 30, 49–50, 84–85; see also semantics Menedemus son of Cleisthenes, 53 Merricks, Trenton, 137 metalanguage vs object language, 19–20 metalinguistic negation, 59–60, 122 metaphysical theories of truth, 74–75; see also correspondence theories, truthmaker theories

155

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Index Millgram, Elijah, 138 minimalism, 44–45, 50, 98 model theory, 16, 29–32 Molière (J.-­B. Poquelin), 74 monotonicity, 111 Moore, G. E., 4–5, 8–9, 70 negation, 22; metalinguistic, 59–60; see also trivalent logic negative existentials, 71–72, 87–88 Nietzsche, Friedrich, 3, 74, 136 Nixon-­Dean example, 103–104 nominal vs real definition, 2 normativity, 80–82, 138 numerals, 28 object language vs metalanguage, 18–20 Occam’s Razor and Eraser, 59 Oddie, Graham, 2008 open vs closed terms and formulas, 24, 26 ordinals, 106, 111 orthographic types or tokens, 12, 18

156

paraconsistency vs paracompleteness, 124, 140 paradigms and foils, 54–55 paradoxes, xii, 5–7, 8, 9–10, 18–19, 33, 116–118; bald one, 55; Berry’s, 7; Curry’s, 118; heap, 55; heterological, 6; horned one, 53; liar, 6, 53; postcard, 126; Russell’s 5–6; Socrates-­Plato, 6 Parsons, Charles, 140 partial valuations and interpretations, 110 Patterson, 208 Peirce, C. S., 4–5 performative vs constative speech acts, 39–40 phlogiston, see presupposition phonological types or tokens, 11–12 physicalism, 21, 24, 74–77, 137–138 Pilate, Pontius, 2, 34 platitudes or truisms and the truth role, 100

Burgess_FINAL.indb 156

pluralism: alethic, 9, 98–101; logical, 100 Poincaré, Henri, 31 postcard paradox, 126 postmodernism, xi, 135 pragmatism, 3–4, 47, 68, 77, 136 prereflective vs reflective stages, 121–122 presupposition and presupposition failure, 8, 53–54, see also strategies of response Price, Huw, 138 Priest, Graham, 140 Prior, Arthur N., 38, 40 properties, pleonastic vs substantive, 47–48, 98–99 prosentences, 38 prosententialism and neoprosententialism, 40–41, 146 propositions, 13–15; see also disqualification strategy propositionalism vs sententialism, 13–15, 33, 42–44, 136 pseudomenos (liar) paradox 6, 53, 107 Puritan syllogism, 100 Putnam, Hilary, 75, 136, 138 quadrivalent logic, 124 quantification, grammar-­defying, 37–38 quasi-­deflationism, 43 Quine, Willard Van Orman, 41–44, 47, 96, 136 quotation, 10, 19–20; direct vs indirect, 13, 44; marks of, 49 radical vs moderate deflationism, 34 Rahman, Shahid, 141 Ramsey, Frank P., 5, 8, 34–38, 39, 43, 47, 72, 135 Rayo, Augustin, 139 real vs nominal definition, 2 realism, 3–5, 7–9, 52–53, 68–82, 116; usage of the term, 8–9, 68, 83–84, 137 recursive vs direct definition, 23–26, 87–88 redundancy theory, 34–38

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Index reference, 50–51; see also causal theories, denotation refutation by reductio, rule of, 117 reiteration, rule of, 117 rejection, strong vs weak, 122–123 relativity and relativism, 8, 61–67; legal, 62–63; local vs global, 65–67; moral, 63–65; see also truth relativism revenge, 119–120, 123 revision theories, 108, 112–113, 139 Richard, Mark, 101, 138 Richard’s paradox, 7 Ripley, David, 140 Rorty, Richard, 47, 136 rules of inference, 117–118; see also introduction and elimination rules Russell, Bertrand, 2–5, 8–9, 22, 70, 127; his paradox, 5–6, 127 satisfaction, 17–18, 27 Schantz, Richard, 136 Scharp, Kevin, 140 schematic reasoning, 46 self-­effacement, 46 self-­reference in formal languages, 28–29 semantic types or tokens, 11–12 semantics, 17–18, 29; see also truth-­ conditions, verification-­conditions sentences, 10–12; of a formal language, 26; see also propositionalism vs sententialism set theory and set-­theoretic notions, 6–7, 21, 24 Shapiro, Stewart, 136, 139 Sheard, Michael, 115, 139 Simmons, Keith, 135, 140 situations, 70–71 smidgets, 121–122 Smith, Nick, 137 Soames, Scott, 75, 95, 121, 137–139 Socrates-­Plato paradox, 6, 104, 126 sorites (heap) paradox, 55 speech acts, see doublespeak, performative vs constative

Burgess_FINAL.indb 157

stable truth and falsehood, 113 Stoljar, Daniel, 136 strategies of response, see: defeatism, denial, dependency, deviance disqualification, doublespeak Strawson, Peter F., 38–40, 41, 135 strong vs weak affirmation, entailment, rejection, and veracity, 122–123 strong trivalent logic, see trivalent logic structual description, 20 subscripts, 103–104, 125–126 supervaluations, 58, 107 supervenience of truth on being, 73, 137 syntax and syntactic notions, 17, 29, 104–105 T-­biconditionals and T‑scheme, 17, 20, 33, 44, 97 T‑introduction and T-­elimination, 17, 44, 97, 106, 117–118 tacit vs verbalizable knowledge, 90, 92 Tappenden, James, 140 Tappolet, Christine, 138 Tarski, Alfred, 7, 9–10, 11, 15, 16–33, 34, 38, 45, 74–75, 76, 87–88, 98, 102, 128, 136, 137, 141 Tarski hierarchy, 103–106; ghost of, 121–122; see also subscripts that-­clauses, 13–14 thick deflationism, 136 three-­valued logic, see trivalent logic transcendence vs immanence, 41–42, 136 translation and transposition, 19–20, 42, 48–49 trivalent logic (strong), 57–58, 106, 110, 120–121 tropes, see features truisms or platitudes defining the truth role, 100 truth-­conditions and truth-­ conditional semantics, 9, 85–87; hardcore, 86, 90

157

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Index truth predicate, 33, 40–42 truth preservation, 22, 44, 100 truth primitivism, 87, 89 truth relativism, 62, 64–65, 137 truth values, 10; see also gaps and gluts truthbearers, 10–15, 70–72; fundamental or primary vs derivative, 12, 14; see also propositionalism vs sententialism truthmakers, 9, 69–70, 72–73, 137 truth-­teller, 107 types vs tokens, 11–12 types, theory of, 127 under‑ and overdetermination, 54 ungroundedness, 107–108 universals, 68, 83 use vs mention, 19, 93 utility, 3–4, 77–79, 138 untruth-­teller vs falsehood-­teller, 119 vagueness, 8, 54–55, see also strategies of response

valuations, 110 value of truth, xi, 1, 135; see also utility Van Fraassen, Bas, see super­valuations variables, 24, 26 verification-­conditions and verification-­conditional semantics, 9, 90–95 virtus dormitiva, 74, 77 Visser, Albert, 139 Walker, Ralph, 136 Walton, Kendall, 130 warranted assertability, 4, 91 Welch, Philip, 139 winning a game, 80–81 Wittgenstein, Ludwig, 44, 72 Woodruff, Peter, 139 Wright, Crispin, 98, 100, 138 Yablo, Steve, 107 Yaqub, Alladin, 139 Young, James, 136

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Burgess_FINAL.indb 158

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