etchemendy, the concept of logical consequence

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THE CONCEPT OF LOGICAL CONSEQUENCE

JOHN ETCHEMENDY

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THE DAVID HUME SERIESPHILOSOPHY AND COGNITIVE SCIENCE REISSUES

CSLI PUBLICATIONS


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Copyright © 1999

CSU Publications

Center for the Study of Language and Information

Leland Stanford Junior University

Printed in the United States

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Library of Congress Cataloging-in-Publication Data

Etchemendy, John, 1952-

The concept of logical consequence / John Etchemendy.

p. cm.

Originally published: Cambridge, Mass.: Harvard University Press, 1990.

Includes bibliographical references and index.

I S B N 1-57586-194-1 (pbk.: alk. paper)

1. Logic, Symbolic and mathematical 1. Title.[B C 135.E 83 1999]

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For Nancy and Max


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Acknowledgments

I owe many thanks to many people. For their help and encourage-ment, without which I may never have finished the book, and theircriticism, without which I would certainly have finished too soon, I

would like to thank Ian Hacking, Calvin Normore, Ned Block, GregO’Hair, Richard Cartwright, Leora Weitzman, and, in particular,John Perry, Genoveva Marti, and Paddy Blanchette. For their pa-tience, I thank my family, and especially my wife, Nancy. And for allof the above and more, I thank my friend and colleague Jon Barwise.Finally, I am indebted to the Mrs. Giles Whiting Foundation and tothe Center for the Study of Language and Information for supportwhile working on various stages of this book.


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Contents

1 Introduction 1

2 Representational Semantics 12

3 Tarski on Logical T ru th 27

4 Inte rpre tational Semantics 51

5 Interpre ting Quantifiers 65

6 Modality and Consequence 80

7 The Reduction Principle 95

8 Substantive Generalizations 107

9 The Myth of the Logical Constant 125

10 Logic from the Metatheory 136 11 Completeness and Soundness 144

12 Conclusion 156

Notes 161

Bibliography 171

Index 173


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Introduction

1

The highest compliment that can be paid the author of a piece ofconceptual analysis comes not when his suggested definition surviveswhatever criticism may be leveled against it, or when the analysis isacclaimed unassailable. The highest compliment comes when the suggested definition is no longer seen as the result of conceptual analysis—when the need for analysis is forgotten, and the definition istreated as common knowledge. Tarski’s account of the concepts oflogical truth and logical consequence has earned him this compliment.

Anyone whose study of logic has gone beyond the most rudimentarystages is familiar with the standard, model-theoretic definitions of thelogical properties. According to these definitions, a sentence islogically true if it is true in all models; an argument is logically valid, itsconclusion a consequence of its premises, if the conclusion is true inevery model in which all the premises are true. These definitions,along with the additional machinery needed to understand them, areset forth in every introductory textbook in mathematical logic.1 Inthese texts we are taught how to delineate a class of models for a simplelanguage and how to provide a recursive definition of truth in amodel—in short, how to construct a simple model-theoretic semantics. Once this semantic theory is in place, the model-theoretic definitionsof the logical properties can be applied.

This method of defining logical truth and logical validity is generally traced to Tarski’s 1936 article, “On the Concept of Logical Consequence.”2 In this article Tarski sets out to give a precise and generalaccount of what he calls the “intuitive” consequence relation and the

corresponding property of logical truth. The definitions that result aremeant to be applicable to any language whose truth predicate can be


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2 Introduction

defined, and to remain, as Tarski puts it, “close in essentials” to thecommon, everyday concepts.

Tarski devotes most of his attention in this brief, twelve-page articleto shortcomings of other attempts to define the consequence relation,in particular attempts to characterize it syntactically, by means offormal systems of deduction. His own, semantic account, sketched in amere four pages, is devoted in part to the exposition of some ancillarynotions treated at length in his earlier monograph on truth. The mainthrust of the article is not to discuss details of the semantic account ofconsequence, or even to give a simple example of its application, butrather to urge that “in considerations of a general theoretical naturethe proper concept of consequence must be placed in the foreground”(1956, p. 413).

Tarski begins his article by emphasizing the importance of the intu

itive notion of consequence to the discipline of logic. He dryly notesthat the introduction of this concept into the field “was not a matter ofarbitrary decision on the part of this or that investigator” (1956, p. 409). The point is that when we give a precise account of this notion,we are not arbitrarily defining a new concept whose properties we thenset out to study—as we are when we introduce, say, the concept of agroup, or that of a real closed field. It is for this reason that Tarskitakes as his goal an account of consequence that remains faithful to theordinary, intuitive concept from which we borrow the name. It is for

this reason that the task becomes, in large part, one of conceptualanalysis.

Tarski’s account of the logical properties is widely regarded as successful in this respect, as capturing, in mathematically tractable form,the “proper” concepts of logical truth and logical consequence. We cansee this not only from explicit acknowledgments of its success by many philosophers and logicians, but also from the treatment given it bythose not interested in conceptual analysis as such. Perhaps the moststriking indication is the different status afforded syntactic characterizations of consequence, formal systems of deduction.

It has long been acknowledged that the purely syntactic approachdoes not yield a general analysis of the ordinary notion of consequence, and in principle cannot. The reason for this is simple. It isobvious, for starters, that the intuitive notion of consequence cannot

be captured by any single deductive system. For one thing, such asystem will be tied to a specific set of rules and a specific language,while the ordinary notion is not so restricted. Thus, by “consequence”

we clearly do not mean derivability in this or that deductive scheme.But neither do we mean derivability in some deductive system orother, for any sentence is derivable from any other in some such system.


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Introduction 3

So at best we might mean by “consequence” derivability in some sound deductive system. But the notion of soundness brings us straight backto the intuitive notion of consequence: a deductive system is sound if itallows us to prove only genuinely valid arguments, those whose conclusions follow logically from their premises.

We recognize that a syntactic definition does not capture the ordinary notion of consequence, and we recognize this even though wemay be convinced, for one reason or another, that a given deductivesystem is adequate for a given language—that is, even if we believe thatall valid arguments, and only valid arguments, are provable within thesystem. This recognition is at a conceptual level, but its main impact isat the extensional. The upshot is that systems of deduction requireexternal proofs of their extensional adequacy (or inadequacy, as thecase may be). To be sure, with careful selection of our rules of proof, it

is fairly easy to guarantee that only valid arguments are provable in agiven system. But our assurance that all valid arguments are provablein the system—if such an assurance is to be had—must come fromsomewhere other than the deductive system itself. We need outsideevidence that our system is “complete,” evidence we would not requireif the system straightforwardly captured, in mathematically tractableform, the ordinary concept of consequence.

To appreciate how different our attitude is toward the model-theoretic account of consequence, consider the significance we read

into Gödel’s completeness theorem. It is now common to state thistheorem in the following form, where 5 is any sentence in a first-orderlanguage and K is an arbitrary set of such sentences:

If K (= 5 then K \-S.

Here, the relation indicated by “ f=” is the model-theoretically definedconsequence relation, while “ |- ” indicates a syntactic or proof-theoretically defined consequence relation. This theorem, plus its con

verse, the soundness theorem,

If K |- 5 then K \=S,

shows that the model-theoretic and proof-theoretic definitions of consequence coincide, that they apply to the same pairs (K , S) in thefirst-order language. But we think of these results as having an intuitive significance that goes beyond the mere coincidence of two alternative characterizations of the consequence relation. Specifically, we

think of them as demonstrating the extensional adequacy of the deductive system in question. They are thought to show that the system issound, that it will not allow the derivation of conclusions that are not


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4 Introduction

genuine consequences of their premises, and that it is complete, that itallows the derivation of all the consequences of any given set of sentences in the language.

What is revealing is that the significance we read into these results isasymmetric, even though their form alone would not seem to warrant

it. After all, for any given language there will be a wealth of theoremsdisplaying the same general pattern:

If K |-i 5 then K \-2S,If K \-2 S then K |~i 5.

But if, for example, both h i and |~2 are syntactically defined consequence relations, perhaps involving variant proof regimes, we wouldhardly take these results as showing the adequacy, the soundness andcompleteness, of one regime rather than the other. In such a case we

would take the theorems as showing nothing more than the coextensiveness of the two characterizations. To think they demonstrate,say, the extensional adequacy of |~2» would obviously presupposeadditional theorems showing the completeness and soundness of h i •In this case, the pair of results would be viewed as entirely symmetric.

The felt asymmetry in our original two theorems stems from ourassumption that the model-theoretic definition of consequence, unlikesyntactic definitions, involves a more or less direct analysis of theconsequence relation, and so its extensional adequacy, its “complete

ness” and “soundness,” is guaranteed on an intuitive or conceptuallevel, not by means of additional theorems. If it were not for thisassumption, we would feel equal need for external evidence that themodel-theoretic characterization of consequence is extensionally correct, that it applies to all valid arguments, and only valid arguments, ofthe language in question.

How do we know that our semantic definition of consequence isextensionally correct? How do we know it does not declare somelogically valid arguments invalid, or declare some invalid argumentslogically valid? Many readers will find this question quite odd. But it isnot odd in the same way as the quesdon “How do we know that allstructures satisfying the group axioms are really groups?” This secondquesdon is simply confused: the notion of a group is arbitrarily defined to mean those structures satisfying our characterization. But asTarski points out, the situadon is quite different with the concept oflogical consequence. Here the correctness of our model-theoretic definition is not determined by arbitrary fiat; on the contrary, whether

the definition is right or wrong will depend on how closely it corresponds to the pretheoretic notion it is meant to characterize. Thatthe first question now strikes us as odd just indicates how deeply


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Introduction 5

ingrained is our assumption that the standard, semantic definitioncaptures, or comes close to capturing, the genuine notion of consequence.

The situation here might be illuminated by analogy with some basicresults in recursion theory. Recursion theory, like logic proper, was

originally driven by an interest in a rather imprecise and intuitivenotion. Here the notion was that of an effectively computable function, a function whose values could in principle be calculated by algorithmic means—that is, using fixed instructions requiring no insightor creativity. During the 1930s, many mathematically precise characterizations of the class of computable functions were proposed, byChurch, Gödel, Turing, and others, and various important resultsconcerning the precisely defined classes were proved. Among themwas the striking result that, although the precise characterizations

proceeded in widely divergent ways, they were nonetheless coextensive; they carved out exacdy the same class of functions. This resultwas taken as evidence that this class of functions, however specified,formed a natural and important collection. But did it also show thatthe specified class was exacdy the class of intuitively computable functions? The answer, of course, is no. For if none of the precise characterizations individually captured the intuitive notion of computability,the question of whether they coincide exactly with this concept hardlyfollowed from their convergence. The coincidence of the various de

finitions provided some indirect evidence, as did the fact that noobviously algorithmic function could be found that fell outside thedefined class. But these do not amount to a mathematical demonstration. Because of this, logicians take great care to distinguish the variousmathematical results in recursion theory from the claim that all intuitively computable functions fall into the precisely delineated class.This claim is usually called Church’s thesis, and although it is almostuniversally accepted, it is not considered amenable to mathematical proof.

This situation is parallel to the one that confronted early, formallogicians. Much of their work was driven by an interest in the intuitivenotions of logical truth and logical consequence, but the only preciseaccess to these notions was through specific, proof-theoretic characterizations, specific deductive systems. These syntactic characterizations, however, clearly did not capture the intuitive notion; theywere not straightforward analyses. Because of this, the claim that a particular proof regime, say for some first-order language, coincides

with the language’s genuine consequence relation, seemed at best toadmit of indirect evidence. The coincidence of various different systems of proof provided some support, as did our ability to construct


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6 Introduction

formal derivations of many specific instances of valid reasoning. But asHilbert once put it, evidence accrued only “through experiment,” notthrough mathematical proof.3To emphasize the parallel with recursion theory, we might call this claim—the claim that all and onlylogically valid arguments of a given language are provable within a

given deductive system— Hilbert's thesis.

Now, what ever happened to this latter thesis? Why has Church’sthesis been given such a prominent position in logical pedagogy, whileits counterpart has not? Both involve the relationship between a mathematically precise definition and one of the central, albeit intuitive,notions of our discipline. The difference is that in the latter case, thethesis has been replaced by theorems: the soundness and completenesstheorems are thought to provide a mathematical proof of Hilbert’sthesis for first-order languages, a proof that the syntactic characteri

zations of consequence do in fact coincide with the genuine consequence relation for these languages. And of course it is such a proof,on the assumption that the model-theoretic definition captures thegenuine concept of consequence. It is such a proof, on the assumptionthat Tarski’s analysis is right.

It is precisely this assumption that I question in this book. Briefly put, my claim is that Tarski’s analysis is wrong, that his account oflogical truth and logical consequence does not capture, or even comeclose to capturing, any pretheoretic conception of the logical proper

ties. The thrust of my argument is primarily at the conceptual level, but again the main impact is at the extensional. Applying the model-theoretic account of consequence, I claim, is no more reliable a technique for ferreting out the genuinely valid arguments of a languagethan is applying a purely syntactic definition. Neither technique isguaranteed to yield an extensionally correct specification of the language’s consequence relation. Needless to say, this conclusion requiresthat we reassess the intuitive significance of Gödel’s completeness theorem, as well as the import of the failure of analogous results when wemove, for example, to second-order logic.

The intuitive concept of consequence, the notion of one sentencefollowing logically from others, is without doubt the most centralconcept in logic. It is what has driven the study of logic for more thantwo thousand years. On the other hand, the remarkable achievementsin logic during the past century have been the direct result of themathematization of the field. The infusion of mathematically precisedefinitions and techniques has turned a field dominated by homely

admonitions into one capable of suppordng significant and illuminating theorems. My aim in this book is to attack a common misunderstanding of one widely used mathematical technique, not to ad-


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8 Introduction

ideas and techniques. For one thing, as Tarski points out, the definitions he gives presuppose “methods which have been developed [only]in recent years.” Specifically, they involve techniques for defining thenotions of satisfaction and truth, concepts that had been left at anintuitive level by all earlier authors. Second, and more important, is

Tarski’s attempt to present and motivate the definitions in a com pletely general setting. It is easy to underestimate the importance ofthis contribution. But clearly, the ordinary notions of logical truth andlogical consequence are not restricted to a specific language or smallcollection of languages, and so our definition of a single language’sconsequence relation, or of its set of logical truths, must flow fromsome more general analysis of these concepts. Finally, unlike his immediate predecessors, Tarski extends his account to the notion oflogical consequence as well as logical truth.

For the purposes of this book, I simply assume that the model-theoretic definitions originated with Tarski’s analysis. The historicalquestion of who should receive primary credit for the definitions is acomplicated one, both for the reasons sketched here and for anotherimportant reason that will emerge in Chapter 5. It turns out thatcertain paradigmatic instances of the model-theoretic definitions involve a subtle but significant departure from Tarski’s analysis, one thathas gone completely unnoticed. But to explain that departure at this

point would be premature.

The Plan of This Book

This book consists of a single, extended argument. The conclusion ofthe argument is that the standard, semantic account of logical consequence is mistaken. What I mean by this is, first of all, that when weapply the account to arbitrary languages—even perfecdy familiar,well-behaved ones—it will regularly and predictably define a relationat variance with the genuine consequence relation for the language inquestion. The definition will both undergenerate and overgenerate: it willdeclare certain arguments invalid that are actually valid, and declareothers valid that in fact are not.

This is not to say that every application of Tarski’s account is exten-sionally incorrect. Indeed, I will eventually argue that with suitablyweak languages (and with certain qualifications that I explain later) thedefinition does get the extension right. But even in these cases we mustseek external guarantees of that fact. This is the second point, and

though a bit more subtle, it is at least as important as the first. The point is that the semantic account shares with syntactic accounts thefollowing limitation: there is no way to tell from the definition alone or


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Introduction 9

from characteristics of the language whether the extension of theaccount is correct. Clearly, no amount of pondering a syntactic systemof deduction can assure us of its extensional adequacy; for that, wemust turn to indirect evidence, whether in the form of theorems or,failing these, evidence of a more “experimental” sort. I claim that

exacdy the same holds true of any application of the model-theoreticaccount of consequence.As I said, this book consists of one, rather long argument. Most of

the argument deals with various intuitive or conceptual considerations bearing on the adequacy of Tarski’s account. The reason for thisemphasis is simple. I think the basic problem with Tarski’s account is insome sense obvious, once certain confusions and misunderstandingsare cleared away. But there are several of these confusions, and each ofthem lends a certain plausibility to the analysis. Together, they give

rise to a remarkably persuasive illusion, an illusion that the account (asTarski puts it) captures the “essential” features of the ordinary conceptof consequence.

Of course, if this were really the case, if the account simply translated our intuitive concept into mathematically tractable form, wewould have an ironclad guarantee of its extensional adequacy whenapplied to arbitrary languages. The situation would then be analogousto, say, our inductive definition of N, the set of natural numbers.According to this definition, N is the smallest set that contains 0 and is

closed under the successor operation.5 Now, it is perfectly clear thatthis definition is not identical to the intuitive notion it supplants. Thus,it employs a variety of set-theoretic concepts that are not, by anystretch of the imagination, part of our ordinary understanding of thenatural numbers. Conversely, certain things that are arguably centralto our intuitive concept (say, the concrete process of counting) are at

best dimly reflected in the inductive definition. But the definitionobviously captures the essential feature of the intuitive notion, and soits extensional adequacy is apparent from the definition itself. We donot, so to speak, have to try it out to see that it really works.

Most people react to the model-theoretic account of consequence inthe same way they react to the inductive definition of N. Neither isgiven extensive justification since neither seems to need it. I claim thatthis reaction is, in the former case, mistaken. But it is not, unfortunately, a simple mistake—or, for that matter, a single one. For thisreason, much of this book is devoted to explaining the variety ofconfusions and misunderstandings that have made Tarski’s analysis

seem so convincing. Until these are finally laid to rest, purely extensional evidence against Tarski’s account, evidence that I think we havelong had, will continue to be explained away.


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io Introduction

I try to treat these misunderstandings one by one, in what I hope isan orderly, comprehensible way. Unfortunately, treating them one ata time—the only way I see to do it—has certain drawbacks. For onething, not everyone will share a given misunderstanding, and so anindividual reader may find certain parts of the book obvious, while

another might find those same points illuminating and others not. Forexample, the first few chapters are addressed to a confusion extremelycommon among those who enter logic through philosophy or linguistics, but almost nonexistent among those who enter through mainstream mathematics. Here, I can only ask the reader’s patience. If Iappear, at points, to be addressing the wrong issue, and perhapsignoring entirely some key insight that justifies the account, I hope thereader will nonetheless persevere.

This gives rise to a second problem—namely, that different parts of

the book are really addressed to somewhat different audiences. Sincethese audiences will have different technical backgrounds (not to mention different interests and concerns), I have tried not to assume muchcommon ground, at least in covering the key points of my argument.The model-theoretic account of consequence has had a tremendousinfluence on all logic-related disciplines, from philosophy and linguistics to mathematics and computer science. Thus, I have tried to makethe book understandable to anyone who has had a first course inmathematical logic. I hope it does not seem tedious to those who have

had more.My main criticism of Tarski's account is contained in Chapters 7

through 10. There, I explain two things. First, I explain what I take to be the central defect in the account, the reason it will, in general, beextensionally incorrect. Second, I describe what I believe is the mainsource of the account’s remarkable persuasiveness. The chapters leading up to this are devoted to untangling some of the more straightforward confusions that surround the analysis, and to giving a clearexplanation of Tarski’s original definition and of its relation to themodel-theoretic treatment with which we are now familiar.

In order to understand Tarski’s account it is essential to distinguishit from what I call representational semantics. Representational semanticsis a perfectly legitimate approach to semantics, but (as will becomeclear) it bears no relation whatsoever to Tarski’s account of the logical properties. Unfortunately, Tarski’s analysis is frequently conflatedwith representational semantics. For this reason I will begin, in Chapter 2, by discussing this alternative approach to semantics, so that it can

be usefully contrasted with Tarski’s account rather than vaguely confused with it. Chapters 3 through 5 are devoted to a careful expositionof Tarski’s original definitions and their relation to the standard,


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Introduction 11

model-theoretic account. Then, in Chapter 6, I consider and rejectTarski’s own positive arguments in support of his analysis.

In Chapter 11, I try to reconcile the lessons learned in Chapters 7through 10 with widespread intuitions about completeness andsoundness theorems. There, I modify an argument of Kreisel’s in

order to see how, and in what precise sense, we can verify the exten-sional adequacy of certain applications of the model-theoretic definitions.

One final point before beginning. Through large stretches of this book I focus, for simplicity, on the notion of logical truth. Logicaltruth, since it is a property of single sentences, is often far easier todiscuss than logical consequence, which is a relation between a collection of sentences (say, premises of an argument) and another sentence(the conclusion). For example, it is much easier first to look at the

details of Tarski’s account as they bear on the concept of logical truth,and then to explain briefly the more general account of consequence,than it is to tackle the consequence relation head on.

This greatly facilitates the exposition, but it could also be misleading.We must not lose sight of the fact that the concept of consequence is farmore important than that of logical truth, both intuitively and technically. On their own, logical truths are of very little interest—recall thatthese are sentences we often describe as trivial, devoid of information,true by virtue of meaning, and so forth. Where the notion of logical

truth gains its importance is as the limiting case of the consequencerelation: these are sentences that follow logically from any set of sentences whatsoever. The crucial notion, ultimately, is that of one sentence following logically from others. Logic is not the study of a bodyof trivial truths; it is the study of the relation that makes deductivereasoning possible.


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Representational Semantics

2

To understand Tarski’s account of the logical properties, we need todistinguish clearly between it and representational semantics. But to dothat, we need a fairly clear idea of what the latter approach to semantics is all about. A good place to begin is with a simple puzzle suggested

by Donald Davidson. In a well-known article in which he defends hisown approach to semantics, Davidson draws a broad distinction between “theories that characterize or define a relativized concept oftruth” and his own call for a “theory of absolute truth” (1973, p. 79).

Davidson points out that as we ordinarily understand it, truth is a property of sentences, a property whose holding or failing to hold isexpressed by a monadic predicate. In this respect, truth sets itself apartfrom many other concepts that we consider peculiarly semantic. Thus,denotation is a relation between a singular term and an object denoted,satisfaction a relation between an open sentence and the things it “holdstrue of,” and so forth. But truth, perhaps the preeminent semanticconcept, does not relate a sentence to something else; it simply appliesor fails to apply, so to speak, absolutely.1

Davidson goes on to note that at least on a superficial level, muchcontemporary work in semantics seems to belie this simple point.Much effort is devoted to the investigation of what Davidson sees asirreducibly relational notions, notions like “truth in a model, truth in aninterpretation, valuation or possible world.” These technical concepts,which Davidson subsumes under the generic term “truth in a model,”hold or fail to hold between sentences and objects of some other sort:generically, “models.” Because of this, Davidson argues, such theories

of relative truth do not have as consequences the so-called T-sentencesdistinctive of the theory of absolute truth. The T-sentence


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Representational Semantics 13

‘Snow is white’ is true if and only if snow is white

does not, as Davidson puts it, “fall out of” a theory that simply tells uswhich models ‘Snow is white’ is true in. And for this reason, theories ofrelative truth “do not necessarily have the same sort of interest as atheory [of absolute truth]” (1973, p. 79). A theory that yields T-

sentences provides, first and foremost, an explication of absolutetruth—that is, of truth as we, ordinarily understand it; theories of relativetruth must, at least on the surface, be seen as providing explications ofsomething else.

I am not concerned here with the merits or demerits of competingsemantic programs, and in particular I will not spend time consideringDavidson’s own approach. But it is worthwhile taking seriouslyDavidson’s simple, initial point: truth is, after all, a property; truth in amodel, a relation. What bearing can a characterization of such a relational concept have on our ordinary “monadic” concept of truth? Ifthere is no close tie between the two, as Davidson occasionally implies,then why is the relation of “truth in a model” given a name that soundsso misleading?2

We can look at Davidson’s puzzle this way. A theory of relative truth provides us with a characterization of “x is true iny.” Yet it is commonto think of such theories as telling us something about truth, as havingat least intuitive or informal consequences involving the ordinary mo

nadic predicate “x is true.” Davidson, of course, is particularly interested in the so-called T-sentences, but the same point might be madeabout any claims involving “absolute” truth. That point is this. Before atheory of relative truth can be judged to have consequences, formal orotherwise, involving the standard monadic concept, we must give someexplanation of exactly how the defined “x is true in y" is related to thealready understood “x is true.” Somehow, we must explain how we areto move from our theory about the relation to claims involving the

property. If we can give no such explanation, then the simple, prima facie

evidence is that our theory of relative truth has no bearing on theconcept of truth as we ordinarily understand it. But that, of course, isabsurd.

Truth as Specification

We often find it advantageous to explain a monadic concept in termsof a relational one. So, for example, we may find the explication of “x is

a brother” far more tractable if we first set out to analyze “x is a brotherof y.” The former then reduces to an existential generalization of thelatter: brotherhood is just brother-of-someone-hood. There are similar cases


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14 Representational Semantics

in which we gain access to the monadic concept through a universal generalization of the relational; thus with comparatives and superlatives—say, taller than and tallest. But clearly the monadic concept oftruth, the concept we ordinarily employ, is no generalization of any ofthe various relational concepts. A sentence can be true in some model,

yet not be true; a sentence can be true, yet not be true in all models.If the monadic concept of truth is not a generalization, universal orexistential, of the concept of truth in a model, then the natural alternative is to think of the former as a specification of the latter. In otherwords, perhaps the monadic concept emerges from the relational byfixing on a specific instance of the nonsentential parameter, the “y” in“x is true iny.” Being true simpliciter would then be viewed as equivalentto being true in some particular model, and getting from a theory ofrelational truth to a theory of absolute truth would be a matter of

indicating which specific model was the “right” model. Our conceptualanalogy might then run: “x is true in y” stands to “x is true” as “x is a

brother of y” stands to “x isFred’s brother.”In broad outline, this is clearly the intended relation between theo

ries of relative truth and the ordinary, monadic concept of truth. In asense it is the relational concept that is a generalization of the monadic concept; what justifies the appearance of the word “true” in theories ofrelative truth is that the relation studied comes from abstracting or“unfixing” an implicitly fixed parameter embedded in the ordinary

notion of truth. Theories of relative truth try to characterize “x is truein y,” while theories of absolute truth aim to characterize, so to speak,“x is true in Fred."

Of course, this still does not tell us who or what Fred is. We have notdetermined what sort of “hidden parameter” our models are meant tofill, or what makes one model the “right” one, the model that binds theordinary concept of truth to the more general concept of truth in. I willdevote several chapters of this book to exploring one possible answer

to this question, the answer presupposed by the model-theoretic definitions of the logical properties. But there is another very naturalanswer, one assumed in what I have called representational semantics.Briefly, this answer is that Fred is the accurate model, the one thatrepresents the world as it really is.

Truth in a Row

Consider the simplest and most familiar theory of relative truth, a

theory we are taught during the first few days of any inaugural coursein logic. This is the theory of truth in a row, the “theory” that enables usto construct truth tables.


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To fill out a truth table for a simple sentence of English, we have toacquire two principal skills. In the first place, we must master the proper technique for constructing the reference column of the truthtable, a column headed by a horizontal list of the atomic components ofthe sentence in question. This technique generally involves some sim-

ple, extendable pattern of writing the words“t r u e ”

and“f a l s e ”

inhorizontal rows beneath our list of atomic sentences, a pattern guaran-teed to capture all the required permutations for a given number ofsuch components. Thus, depending on the atomic sentences con-tained in the target sentence 5, each of the following would serve as proper reference columns:

Snow is white S

TRUEFALSE

Snow is white Roses are red S

TRUE TRUE

TRUE FALSE

FALSE TRUE

FALSE FALSE

Snow is white Roses are red Violets are blue S

TRUE TRUE TRUE

TRUE TRUE FALSE

TRUE FALSE TRUE

TRUE FALSE FALSE

FALSE TRUE TRUE

FALSE TRUE FALSE

FALSE FALSE TRUE

FALSE FALSE FALSE

Our reference column—everything to the left of the double lines— provides us with the rows that our target sentence is to be true or falsein. The ultimate goal is to write the words “t r u e ” or “f a l s e ” in eachrow below S; “t r u e ” if S is true in that row, “f a l s e ” if S is not true inthat row. But to do that, of course, no standard pattern of the sort used

in constructing the reference column will suffice, will ensure that weenter the correct value in each row. Rather, we need a radically differ-ent technique, a technique that involves the repealed application of


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certain recursive tables. The following are two sample recursive tables;the ‘not’ table:

1 6 Representational Semantics

p not p

TRUE FALSE

FALSE TRUE

and the ‘or’ table:

P


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“monadic” value. But this was to be expected, since our theory is atmost a theory of relative truth.3 It does, however, tell us exactly whichrows our sentence is true in; specifically, it tells us that the sentence istrue in every row save the third. But what bearing does this informa-tion have on the genuine, “monadic” truth value of our sentence?

At the close of the last section we noted that truth simpliciter wasmeant to be a specific instance of “relative” truth. Translating to present terminology, the truth of a sentence should boil down to itstruth in some specific row. And since we know that the current sentenceis actually true, we can rule out the third row without further ado; thatrow is surely not “Fred.” On the contrary, as any student of introduc-tory logic could quickly tell us, our target sentence is true simpliciter

because it holds true in the first row of the present table. Here, at least,it is the first row that binds relative truth to truth.

But what makes the first row the right row? This may seem like a sillyquestion; after all, ‘Snow is white’ and ‘Roses are red’ are both true—that is, genuinely true—and the first row is the only row in which thesesentences both come out true. But notice that in offering this reply, wehave simply put off solving Davidson’s puzzle. There is no questionthat ‘Snow is white’ is true in the first row of this table; for that, we neednot even apply our recursive techniques. Yet it is equally clear, even onthe level of atomic sentences, that being true in a row is quite differentfrom being “absolutely” true; evidence for that will be found in any of

the remaining rows of our table.

Language and the World

Davidson’s puzzle reappears at the very bottom level of our theory oftruth in a row, with the atomic sentences that acquire their values in thereference columns of our tables. If truth is to be truth in some specificrow, then clearly the first row of our sample table must be the “right”one. But it is equally clear that this observation does not provide anyaccount of the link between our theory of relative truth and the ordi-nary, monadic concept from which we pirate the name. To providesuch an account we must explain how the first row, so to speak, comes to be the right row. Furthermore, our explanation cannot simply reduceto the plea that if we picked any other row, various sentences would be“true in” the “right” row and yet not be true simpliciter. Such a responsewould leave our theory of relative truth entirely suspended in air.

If we could not pinpoint some implicit parameter in our ordinary

notion of truth, some parameter whose potential effect on the “abso-lute” truth values of our sentences is mimicked by the effect of changesfrom row to row in the theory of relative truth, then Davidson would


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1 8 Representational Semantics

be completely justified in claiming that the defined “x is true in y” isirreducibly relational. And consequently he would be justified inclaiming that, for this reason, our theories of relative truth cannot bethought to illuminate the notion of truth as we ordinarily understandit. But this conclusion would obviously be wrong. It is perfectly clear

that truth tables tell us something about truth, about ordinary monadic truth, and that the relation of “truth in a row” was not just conjured up by some logician or semanticist with no concern at all for its tie to theordinary concept.

But Davidson’s puzzle is not unsolvable. The problem is not findingan appropriate parameter in our ordinary notion of truth, but ratherchoosing between two obvious alternatives. Consider the move fromthe first row of our sample truth table to the third. Here the relevantchange in our reference column is the value assigned to the atomic

sentence ‘Snow is white.’ The effect of this move is that the resultingvalue of our target sentence turns from true to false. Now the questionis simply what change would have a similar effect on the “absolute”truth value of ‘Snow is white,’ and a similar effect on the “absolute”value of our target sentence.

There are only two parameters to which the sentence ‘Snow is white’owes its truth: broadly speaking, the language and the world. It is due tothe language that the sentence means what it means, that it makes theclaim it does. But it is due to the world that snow is white. Appropriate

changes on either side would have made our atomic sentence false.Thus, had the language been somewhat different, this sentence wouldhave been false in spite of the whiteness of snow—say, if ‘white’ hadmeant hot. On the other hand, had the world been different, thissentence might have been false in spite of its meaning —say, had snow

been red.We can interpret the move from row to row in our truth table in

either of these two ways. In the first place, we can view our theory oftruth in a row as explicating the relation “x is true in L” for a limited,though nontrivial range of languages L. From this perspective, wewould assume that any extralinguistic fact that might influence thetruth value of sentences—say, the color of snow or roses—is heldfixed; our concern is not with changes in the world. Viewed this way,the first row of our sample table is “right” simply because English, theimplicitly specified parameter in “x is true,” happens to be one of thelanguages that expresses true propositions by both ‘Snow is white’ and‘Roses are red.’ Thus, the third row would have been “right” had

we been speaking a language exactly like English save that “white”meant hot.


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Representational Semantics 1g

If we adopt the alternative perspective, then the first row is still“right,” but for entirely different reasons. Here we view our theory as,throughout, a theory of truth for English, or for some fragmentthereof. Our aim is to explicate the relation “x is true in W,” where “W”ranges over various intuitively possible configurations of the world,

the world our language describes. Thus, the first row of our table is“right” just because snow really is white and roses are indeed red. Fromthis perspective, the move to the third row involves no change inmeaning; that row would have been “right” simply had snow not beenthe color it is.

We commonly think of truth tables as capable of supporting certaincounterfactual claims about the (absolute) truth values of their tar-get sentences. We imagine these claims to be supported because ourtheory assigns values to these sentences even in rows that are not

“right,” rows in which the atomic sentences are not assigned theiractual values. So, for example, the third row of our sample tablesupports a claim of the form:

The sentence ‘Snow is white or roses are not red’ would have been false had . . .

Obviously, the appropriate completions of this counterfactual willvary depending on which parameter we view as changing in the movefrom row to row—that is, depending on what we take to be the relation

between “truth in a row” and the monadic truth predicate appearingin the claim. In effect, our theory will support those completions thatwe consider elucidations of “had the third row been the ‘right’ row.”Thus, if we view our parameter to be the language, we might offer thecompleted counterfactual:

The sentence ‘Snow is white or roses are not red’ would have been false had ‘white’ meant hot.

While if we view the parameter to be the world, we would likely produce:

The sentence ‘Snow is white or roses are not red’ would have been false had snow not been white.

As these sample counterfactuals show, the significance we read intoour truth tables depends critically on which perspective we assume, onthe nature of the parameter that corresponds to the rows our sentencesare true in. Of course, since both points of view are possible here, we

might justify either of the above counterfactuals by referring to thethird row of our sample truth table. Or, to simplify matters, we might


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even merge both of our claims into a single counterfactual:

The sentence ‘Snow is white or roses are not red’ would have been false had ‘Snow is white’ been false.

But the fact that we can do this does not mean the resulting claim is

somehow justified by the abstract theory, quite independent of anyaccount we might give of the relation between “x is true in y” and “x istrue.” Or, to put it another way, the fact that our theory of “truth in arow” seems doubly illuminating because it admits of either perspectiveshould not lull us into thinking that it retains its illumination independent of these perspectives. Rather, as Davidson’s puzzle nicely pointsout, the purely abstract characterization of relative truth, of “x is trueiny,” supports no claim whatsoever about absolute truth, about truth aswe ordinarily understand it.

A Representational Semantics

When we view a particular theory of relative truth as explicating “x istrue in W,” we see it as providing an account of how the world wields itsinfluence on the truth values of sentences within a fixed language. Ifcharacterizing this influence is the aim of our relativized theory oftruth, then I will say we are engaged in representational semantics. The

reason I use this somewhat unusual term is simple. Our theory provides an account of a relation, “x is true iny,” and what the theory takesto satisfy the “y” position are, for all intents, just ordinary objects ofsome sort or other—chunks of the actual world. Thus, in our theory of“truth in a row,” the “y” term was filled by rows, rows that were fixed bythe reference column of our truth table. Other representational theories might define a relation between sentences and abstract, set-theoretic objects, maybe functions of some sort. But obviously these inno case actually are the “possible configurations of the world” that they

are meant to represent. Rows of a truth table are just blotches of ink,and functions are set-theoretic constructs; the world, thankfully, isneither of these.

The point is a simple one, but all too easily overlooked. When weviewed our theory of “truth in a row” as explicating “x is true in W thefact that the target sentence came out false in the third row of the tablewas taken to indicate that the sentence would have been false in a worldin which roses were red but snow not white. But the third row itself, theink marks on paper, is not a world in which roses are red but snow not

white. It is just a handy surrogate, used for purposes of our theory.From this “representational” standpoint, our truth table gives us valuable information about truth, but certainly not about how truth would


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be affected by changes in row. Rather, it tells us how truth would beaffected by changes in the world, by changes that are represented ordepicted by changes in row.

The techniques used in constructing truth tables are not generallythought to constitute a full-fledged semantic theory for any language

or language fragment. More than anything else, this is due to certaintraditions of fairly recent vintage concerning the accepted format ofsuch theories. Still, it may seem perverse to view our theory of “truth ina row” as a representational semantics, insofar as it may seem perverseto view it as a semantics at all. But this can easily be remedied.

Suppose we are interested in the fragment of English containing theatomic sentences ‘Snow is white,’ ‘Roses are red,’ and ‘Violets are blue,’ plus whatever complex sentences can be formed from these using asign for negation, ‘not,’ and a sign for disjunction, ‘or.’ I will assume

that we have a precise syntactic theory for our language, one thatenables us to form the negation of any sentence and the disjunction ofany two.4 A standard representational semantics for this simple language might proceed in the following way. First we define a class ofmodels that will represent all possible configurations of the worldrelevant to the truth values of our sentences. Thanks to the simplicityof our language, this purpose can be served by the class of functionsthat assign a truth value, either true or false, to each of our three atomicsentences. Thus, our class of models consists of eight functions, one

that assigns true to each sentence (representing worlds in which snow iswhite, roses are red, and violets blue), one that assigns false to each(representing worlds in which snow is not white, roses not red, andviolets not blue), and so forth.

Our next step is to provide a recursive definition of S is true in f forarbitrary sentences S and models/. Since we will take this relation as anindirect characterization of “x is true in W,” our aim will be to ensurethat any given sentence of our language is true in exactly those models

which represent worlds that would indeed have made the sentencetrue. So if a model depicts a world in which snow is not white, ourdefinition should guarantee that ‘Snow is white’ comes out false in thatmodel. Here we assume, of course, that the sentence ‘Snow is white’means what it actually means; the sentence is ours, even though theworld depicted by the model is not.

The definition proceeds in the obvious way, by recursion on the setof sentences in our language:

• If S is an atomic sentence, then it is true in a model/just in case/assigns it the value true.

• If .S' is the negation of .S', then it is true in a model/just in case S' isnot true in /.


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• If S is the disjunction of S' and S", then it is true in a model /ju st incase either S' is true in f or S" is true in/

For the most part, what we have done here just involves a recasting ofour theory of “truth in a row.” But there are two changes worthmentioning. In the earlier theory, we constructed reference columns

for each sentence encountered, the number of rows being determined by the atomic components of the target sentence. In the new theory,our models take over part of the burden shouldered by the referencecolumns, since they provide the objects our sentences are true or falsein. Indeed, they do so with somewhat more aplomb, allowing us to usethe same models for any sentence in our fragment. Thus, we havemanaged, in the new theory, to introduce a standard collection ofobjects, each of which fully determines the apportionment of truthvalues throughout the entire language.5

Now, although it could easily escape notice, the reference columnsof our earlier theory actually did a bit more than our models. Thereference columns both delineated the needed rows and simultaneouslyspecified the values of our atomic sentences in those rows. In contrast,whether an atomic sentence comes out true in a given model is deter-mined not by the model itself but by the base clause of our recursivedefinition, the clause beginning “if S is an atomic sentence . . . ” Thefact that we took models to be functions that yield the values true and

false is entirely a mnemonic convenience in the new theory; any twoobjects would have worked as well—for example, the numbers zeroand one. Indeed, if we had used zero and one, the substantial contribu-tion made by the base clause of our definition would have been high-lighted: without the base clause, we would not know whether a modelthat assigns zero to ‘Snow is white’ represents a world in which snow is white, or one in which it is not. To provide similar freedom in thereference columns of our truth tables, say, the freedom to use “+” and

rather than “t r u e ” and “f a l s e ,”we would have to supplement our

recursive tables with base tables to complete the definition of truth in arow. Such tables would look something like this:

Snow is white ‘Snow is white’

+ TRUE — FALSE

Thus, our new semantic theory, unlike the earlier truth tables, ex- plicitly distinguishes the definition of “x is true in y” from the de-lineation of the class of objects that sentences of the language are to be“true in.”


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Representational Semantics 2 3

Representational Guidelines

The basic motivation underlying a representational semantics, an indirect characterization of “x is true in W,” is fairly clear. The approach provides a natural framework in which to couch a theory of meaning,or at any rate a theory of those aspects of meaning relevant to the truth

values of sentences, both the values they actually have and the valuesthey would have, were the world differently arranged. Needless to say,the simple representational semantics of the last section can at best beconsidered a partial theory of meaning for the relevant fragment,since it offers no detailed account of the semantic functioning of thethree atomic sentences. In giving the semantics, we simply assumedthat ‘Snow is white’ somehow comes to mean what it does, and for thisreason is true in exactly those worlds in which snow is white. A moredetailed semantics would presumably say something on this score aswell.

Of course, the fact that the motivation is clear does not mean the taskof devising a representational semantics for any interesting language iseither easy or philosophically unproblematic. But these difficulties arenot, at present, our concern. For Tarski’s analysis of the logical properties does not involve giving a characterization of “x is true in W”;in effect, it involves a characterization of “x is true in L,” for a specifiedrange of languages L. As we will see, Tarski’s is a remarkably different

goal from that presupposed by the representational approach to semantics, in spite of the fact that one and the same account of “x is truein / ’ may occasionally admit of both construals. Failing to recognizethis difference, many philosophers have assumed that Tarski, in defining the logical properties, had in mind something akin to representational semantics, a characterization of “x is true in W," for all “possi

ble worlds” W. For example, we find David Kaplan extolling theinsight of “Tarski’s reduction of possible worlds to models,” a reduction Kaplan claims to be “implicit in” the analysis of the logical proper

ties developed in Tarski’s article.6 But this, as we will see, is just aconfusion, one of several that lend undeserved credence to Tarski’sanalysis.

Let me conclude this chapter by emphasizing the guidelines that willseem natural if our aim in constructing a model-theoretic semantics isto give a characterization of “x is true in W." First, there is the obviousthough rather vague criterion we use in judging the adequacy of ourclass of models. In a representational semantics the class of modelsshould contain representatives of all and only intuitively possible con

figurations of the world. This was accomplished in the semantics of thelast section by employing a rather crude but effective system of repre-sentation. Our collection of models imposed, so to speak, a complete


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partition on the class of possible worlds, a partition whose boundarieswere determined by the color of snow, roses, and violets in thoseworlds. Had we excluded any one of our eight functions, the remaining class of models would have been inadequate in this respect, leavingno representative for certain perfectly conceivable worlds. On the

other hand, had our atomic sentences been ‘Snow is white,’ ‘Snow isred,’ and ‘Snow is blue,’ then we would have been justified in limitingthe class of models to those functions that assign false to at least two ofour atomic sentences. The remainder would not represent genuine possibilities.

Once we have specified the class of models, our definition of truth ina model is guided by straightforward semantic intuitions, intuitionsabout the influence of the world on the truth values of sentences in ourlanguage. Our criterion here is simple: a sentence is to be true in

a model if and only if it would have been true had the model been accurate—that is, had the world actually been as depicted by that model.Obviously, the possibility of success on this score is not independent ofthe objects we have chosen to include in our class of models. In particular, it is this ultimate goal that determines the amount of detail we needto incorporate into our models, how crude a system of representationwe can get by with. So, for example, with our sample fragment wecould not have used functions that assigned truth values only to ‘Snowis white’ and ‘Roses are red.’ Although these models would indeed

have given us a complete partition of possible worlds, the partitionwould not have been fine-grained enough to allow us to carry out oursemantic task: the accuracy of any of these models would have beenconsistent with either the truth or falsehood of ‘Violets are blue.’ Andof course with more complicated languages, say, languages containingquantifiers, our technique of constructing representations will have toallow for a considerably more detailed depiction of the world.

Now, the final points to notice about representational semanticsconcern the sentences that turn up true in all models. It is an immediate and trivial consequence of the two criteria I have just described thatsentences which are true in all models should be exactly those that arenecessarily true. If a sentence is not necessarily true, yet comes out truein all models, then we have either omitted representations for some

possible configurations of the world, namely those that would havemade the sentence false, or our definition of truth in a model has goneastray, having declared the sentence true in at least one model thatdepicts a world in which it would actually have been false. Just so, a

sentence that is necessarily true can only come out false in a model if wehave gotten its semantics wrong or if the model fails to depict a genuine possibility.


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Representational Semantics 2 5

Clearly, all and only necessary truths will come out true in all modelsof an adequate representational semantics. And so if logical truths arethought to be necessarily true, these will of course be among those truein every model. Similarly, if one sentence comes out true in everymodel in which a second sentence is true, then the truth of the first

must be a necessary consequence of the second. That is, it must beimpossible for the first to be false while the second is true, at least if oursemantics really satisfies the representational guidelines.

Equally trivial is the observation that analytic truths, sentences thatare true solely by virtue of the fixed semantic characteristics of thelanguage, will come out true in all models. If a sentence is not true in allmodels, then its truth is clearly dependent on contingent features ofthe world, and so cannot be chalked up to meaning alone. Thus,insofar as logical truths are analytic, true in virtue of meaning, these

must again be among the sentences that are true in every model of anadequate semantics, one that satisfies the stated criteria.7

These are all immediate consequences of the simple representa-tional guidelines sketched above. But in spite of these consequences, itwould clearly be wrong to view representational semantics as giving usan adequate analysis of the notion of logical truth. For one thing, ifthere are necessary truths that are not logically true, say, mathematicalclaims, then these will also come out true in all models of a representa-tional semantics. But more important, even if we are prepared to

identify necessary truth and logical truth—an identification most people would balk at—it is still clear that representational semantics af-fords no net increase in the precision or mathematical tractability ofthis notion. Any obscurity attaching to the bare concept of necessarytruth will reemerge when we try to decide whether our semanticsreally satisfies the representational guidelines—in particular, when weask whether our models represent all and only genuinely possibleconfigurations of the world.

The value of representational semantics does not lie in an analysis ofthe notions of logical truth and logical consequence, or in the analysisof necessary or analytic truth. Rather, what this approach gives us is a perspicuous framework for characterizing the semantic rules that gov-ern our use of the language under investigation. It should be seen as amethod of approaching the empirical study of language, rather thanan attempt to analyze any of the concepts employed in that task.Certainly, all necessary truths of a language—of whatever ilk—shouldcome out true in every model of a representational semantics. If they

do not, this just shows that our semantics for the language is somehowdefective, perhaps that we are wrong about the meanings of certainexpressions. But this is only a test of the adequacy of the semantics, not


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a sign that we also have an analysis of necessary truth. The latter notionis simply presupposed by this approach to semantics. This is not anobjectionable presupposition, by any means, so long as our goal is toilluminate the semantic rules of the language and not the notion ofnecessary truth.

I have sketched some simple and general criteria that guide theconstruction of a representational semantics, a theory of “x is true inW,” for variable W. As I explain in Chapter 4, Tarski’s analysis of thelogical properties gives rise to an alternative approach to semantics,one whose aim is to characterize the relation “x is true in L,” for somerange of languages L. The intuitive importance of such a theory, andthe general guidelines appropriate to it, are not nearly so apparent asthose of representational semantics. To get a clear idea of these guide-lines, and to see how they differ from those I have just sketched, we

need to take a close look at Tarski’s account of logical truth and logicalconsequence.


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Tarski on Logical Truth

3

My remark that Tarski’s account involves the notion of “x is true in L”for variable L would seem odd to anyone familiar with his originalanalysis but unfamiliar with modern presentations of it. There is nomention in Tarski’s article of any “range of languages,” or of anynotion of relative truth, of “truth in.” The remark is appropriate only,so to speak, in hindsight, as the natural way of viewing the model-theoretic definitions that emerge from Tarski’s account. In Chapter 4,

I explain how making a few minor (though somewhat confusing)changes in Tarski’s original account yields a recognizable model-theoretic semantics. But to see exactly how the resulting semanticsdiffers from a representational semantics, it is important to start fromthe beginning, with a clear understanding of Tarski’s original definitions and their underlying motivation.

I approach Tarski’s account of logical truth and logical consequenceindirectly, by considering first a simpler account developed by Bolzanonearly a century earlier.1The two accounts are remarkably similar;

indeed, Tarski initially entertains what is, for all intents, precisely thesame definition as Bolzano’s, but modifies it for reasons I will eventually explain. But in spite of the striking similarity in the two accounts,Tarski was unaware of Bolzano’s work until several years after theinitial publication of his article. The key difference between the twoaccounts is simply that Bolzano employs substitution where Tarski usesthe more technical, and for the purposes more adequate, notion ofsatisfaction.


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2 8 Tarski on Logical Truth

Bolzano on Logical Truth

We normally think of logical truth as a single property that holds orfails to hold of sentences within a language. Both Bolzano and Tarskiadopt a slightly different approach, in effect treating logical truth as arelation that holds between sentences and sets of atomic expressions in

the language, or alternatively, as a collection of properties that can beobtained from this relation by fixing its second argument.2On eitherBolzano’s or Tarski’s account, there will be sentences that are logicallytrue with respect to one set of atomic expressions, but not logically truewith respect to another. The logical truth of such sentences depends,as Bolzano puts it, on which expressions we take to be variable andwhich we take to be fixed. To use Tarski’s phrase, it depends on whichexpressions we treat as logical constants.

According to Bolzano, what is distinctive about logical truths is thatthey remain true when we exchange some subset of their componentexpressions for any other expressions of similar type.3Bolzano notes,for example, that the sentence

If Caius was a man then Caius was mortal

remains true regardless of the subject term we put in the two positionscurrently occupied by ‘Caius.’ On the other hand, the sentence thatresults from inserting the term ‘omniscient’ in the position occupied by

‘mortal’ is false. Thus, Bolzano concludes, this sentence is logically truewhen we allow only the first sort of exchange, though it is not logicallytrue when we also allow substitutions for the expression ‘mortal.’ Wecannot say the sentence is or is not logically true simpliciter, since thiswill depend, as Bolzano sees it, on which sorts of substitutions we permit.

Following Bolzano, I shall call the terms we allow to vary variable terms and those we keep fixed fixed terms. Assuming that all grammati-cally correct sentences are either true or false, we can take expressions

to be of “similar type” just in case they are members of the samegrammatical category. We can then describe Bolzano’s account oflogical truth as follows. A sentence 5 is logically true with respect to a set ^offixed terms just in case 5 is true and every sentence S' that results frommaking permissible substitutions for expressions in 5 is also true. Asubstitution of a for b in S is permissible ifa and b are expressions of thesame grammatical category, if all of the occurrences of b are uniformlyreplaced by a, and if expression b contains no member of 5, the set offixed terms.

Consider an example. The following sentence is true:

Snow is white or snow is not white.


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Tarski on Logical Truth 29

Also true is the sentence that results from substituting ‘grass’ for‘snow,’

Grass is white or grass is not white,

and the sentence that results (ignoring the awkward placement of

‘not’) from the uniform replacement of ‘is white’ by ‘is green’:Snow is green or snow is not green.

Even simultaneous substitution of ‘grass’ and ‘is green’ produces thetrue sentence

Grass is green or grass is not green.

It seems reasonable to assume that the truth of this sentence survivesany grammatically appropriate substitution for the expressions ‘snow’

and ‘is white.’4 In which case, the sentence ‘Snow is white or snow is notwhite’ is logically true with respect to any set $ that contains the terms‘or’ and ‘not.’

According to Bolzano’s account, though, this sentence is not logically true with respect to every selection of fixed terms. So for instanceif $ contains just the three expressions ‘not,’ ‘snow,’ and ‘is white,’ thatis, if the expression ‘or’ is considered a variable term, then the sentencecan easily be turned into a false one. Thus, the false sentence

Snow is white and snow is not whiteresults from the substitution of the expression ‘and’ for ‘or,’ a substitution permitted on this selection of Similarly if we take as our onlyfixed terms ‘or’ and ‘is white,’ we can presumably get the false sentence

Grass is white or grass is necessarily white

by making grammatically appropriate substitutions for the two remaining variable terms. On the other hand, ‘Snow is white or snow isnot white’does seem to be logically true with respect to the set containing ‘snow,’‘is white,’ and ‘or.’ Regardless of what we put in for ‘not,’ theresulting sentence will, by all appearances, be true.

The result of Bolzano’s substitutional test for logical truth dependscrucially on the set of terms we decide to hold fixed. Bolzano was wellaware of, and indeed welcomed, this dependence, chalking it up to thefact that different terms have different logics. Thus, the sentence

If Tom knew Carolyn to be a dean then Tom believed Carolynto be a dean

is logically true when we hold fixed the three expressions ‘if-then,’‘knew,’and ‘believed’; substituting at will for ‘Tom,’ ‘Carolyn,’ and ‘to


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jjo Tarski on Logical Truth

be a dean’ never yields a false sentence. On the other hand, when weconsider ‘knew’ to be a variable term, we get substitution instances like

If Tom wanted Carolyn to be a dean then Tom believed Carolyn to be a dean.

One of these instances will no doubt be false, if not this particularinstance (Tom may be prone to wishful thinking) then one that resultsfrom further substitutions for the other variable terms. We might takethis to indicate that our sentence is a truth of, say, epistemic logic, butnot a truth of, say, mere doxastic logic.

For any language there will be as many versions of logical truth, asmany “logics,” as there are subsets of the atomic expressions of thelanguage. This is just to view Bolzano’s account as providing, insteadof a relation between sentences and sets of expressions, the collection

of properties that can be obtained from that relation by holding con-stant one of its arguments, the set $ of fixed terms. If we setde on theempty set, if we hold no expressions fixed, then in general no sentencewill qualify as logically true. At the other end of the scale, allowing allatomic expressions into we find that logical truth merely reduces totruth. Thus, the sentence ‘Snow is white’ is logically true if we fix both‘snow’ and ‘is white.’ This, simply because it is true; if all of a sentence’scomponent expressions are in $, there are no permissible substitutioninstances to worry about.

The Violation of Persistence

On all of these points, Tarski’s conception of logical truth coincideswith Bolzano’s. Tarski argues, though, that the substitutional test de-scribed above should not be considered a sufficient condition for logicaltruth, but only a necessary condition. As I have characterized Bolzano’sdefinition, it has an obvious drawback: logical truth depends not onlyon our selection of but on the expressive resources of the languageas well.5 This is where Tarski and Bolzano part company.

Suppose we were applying Bolzano’s definition to a very simplelanguage, one containing two names, say, ‘George Washington’ and‘Abe Lincoln’; two predicates, ‘was president’ and ‘had a beard’; andsome truth functional operators, say, ‘or’ and ‘not.’ Now, when weconsider the two names to be our only variable terms, the sentence‘Abe Lincoln was president’ passes Bolzano’s test for logical truth,

though the sentence ‘Abe Lincoln had a beard’ does not. Both of theseare in fact true sentences. But in the first case, when we substitute theonly other available name we get a true sentence, ‘George Washington


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was president/ while in the second case, the same substitution pro-duces a false one, ‘George Washington had a beard/

Of course, the difference here is just a quirk of our language. Theworld has plenty of people who have never been president. If ourmeager language had a name for just one of them, say Ben Franklin,

the sentence ‘Abe Lincoln was president’ would suffer the same fate as‘Abe Lincoln had a beard’: neither would be logically true on theimagined selection of fixed terms.

This example shows that Bolzano’s substitutional test is liable to giveresults that depend on purely accidental features of the language.With our current choice of $, the sentence ‘Abe Lincoln was president’has only two substitution instances, one that results from the trivialsubstitution of ‘Abe Lincoln’ for itself, the other resulting from thesubstitution of ‘George Washington’ for ‘Abe Lincoln.’ But this seems

artificially restrictive in light of the fact that, had we simply increasedour list of names, the test would obviously have produced oppositeresults. Thus it happens that ‘Ben Franklin was president’ does notresult from making a permissible substitution in ‘Abe Lincoln was

president/ ‘Ben Franklin’ not being an expression of the language. But‘Ben Franklin’ could have been introduced into an existing category,could have been given an appropriate interpretation, and therebywould have provided us with a false substitution instance of the sen-tence at issue. In that case ‘Abe Lincoln was president’ would not have

come out logically true.We should characterize this problem more precisely. What under-

lies our intuition here is perhaps best isolated by considering contractions rather than expansions of the language, by considering the con-verse of the problematic case we have encountered. It seems clear thaton our ordinary conception, logical truth has at least the following property: if a sentence S is not a logical truth of a given language, thenneither should it become a logical truth simply by virtue of the deletionof expressions not occurring in S. After all, nothing directly relevant tothis sentence, to its meaningfulness or its truth, has been changed. If‘Abe Lincoln was president’ is not logically true, it should not becomeso merely through the deletion of an otherwise irrelevant name, ‘BenFranklin,’ from the language.

If the property of not being logically true should persist throughcontractions of the language, the property of being logically true should persist through expansions. This desideratum, which I will call therequirement of persistence, presumably remains binding regardless of

how we specify our set $ of fixed terms. That is, the property of being logically true with respect to a given $ should persist through simple expansions of the language.


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3u Tarski on Logical Truth

As we have seen, Bolzano’s definition of logical truth fails to meetthe requirement of persistence. Tarski’s account aims to avoid thisdefect by appealing to the notion of the satisfaction of a sentential function where Bolzano relies on the considerably simpler though less powerfulnotions of truth and substitution.

Sentential Functions

We can think of a sentence as the limiting case of a sentential function, where this latter notion permits variables of appropriate type to takethe place of ordinary expressions.6 So, for example, if V is a variable ofappropriate type, the linguistic object ‘x was president’ will be called asentential function; it is exactly like the sentence ‘Abe Lincoln was president’ save that a variable has been inserted in the position here

occupied by the name ‘Abe Lincoln.’ Sentential functions may containmore than one variable, indeed more than one type of variable; thus (x g’ might be the sentential function that results from allowing ‘g’ to takethe place of‘was president’ in lx was president.’ I will say that sentencesare just sentential functions that contain no variables.7

The notion of a variable should not be confused with that of a variable term. A variable term is an ordinary expression of the language, onethat differs from a fixed term only for the immediate purposes of ourtest for logical truth. Thus, in the last section we chose $ to include ‘was

president’ and to exclude ‘Abe Lincoln’; the former was therebydubbed a fixed term, the latter a variable term. But neither is a variable. Hence, regardless of our selection of ‘Abe Lincoln was president’ isa sentence—that is, a sentential function that contains no variables.

To simplify the transition from Bolzano’s definition of logical truthto Tarski’s more complicated account, it will help to introduce thenotion of a sentential function into the former. We can think ofBolzano’s test for logical truth proceeding in the following way. Firstwe introduce a stock of variables for each grammatical category. Nextwe replace each variable term in sentence 5 with a variable of appropriate type, ensuring that multiple occurrences of a term receive thesame variable, and distinct terms, distinct variables.

The result of this operation is a sentential function S' containingonly expressions that occur in the chosen set of fixed terms. We nowconsider the collection of substitution instances of 5'—that is, thecollection of sentences that result from S' by placing expressionsdrawn from appropriate categories back in the variable positions. If

every member of this set is true, then 5 is judged logically true withrespect to the current selection of fixed terms; if one or more is false,then 5 is not logically true with respect to that selection.


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According to the present account, the violation of persistence observed in the last section arises from the limited stock of names available to insert for V in the sentential function ‘x was president.’ Tarski’saccount of logical truth allows us to go beyond the actually availablesubstitution instances of this sentential function. The key concept is, of

course, satisfaction. Using it, Tarski bestows some measure of persistence on logical truth.

From Substitution to Satisfaction

It is impossible to give a general definition of satisfaction applicableto all languages; this for various reasons, not the least of which are theso-called semantic paradoxes. But in simple cases the concept is prettyintuitive. So, for instance, satisfaction is the relation that holds betweenAbe Lincoln, the person, and the sentential function x was president,’

but that fails to hold between Ben Franklin, the person, and this samesentential function—in the first case because Lincoln was president, inthe second because Franklin was not.

Let us try to capture this intuitive description in a somewhat moreformal setting. For the moment we will confine our attention to sentential functions which, like ‘x was president,’ contain a single variablestanding in a position ordinarily occupied by a name. It will be convenient to assume that our metalanguage contains the object language

and hence, in particular, that any sentential function of the objectlanguage is also a sentential function of the metalanguage.Let “. .. x . . .” be a schematic placeholder for an arbitrary senten

tial function of the sort described—that is, a sentential function containing (perhaps multiple) occurrences of a single name variable. Wewill use . . . ” ’ as a schematic placeholder for a name (in themetalanguage) of that same sentential function, “n” as a placeholderfor any name, and " .. . n . . . ” as a placeholder for the sentence thatresults from replacing all occurrences of “x” in the sentential function

\ . . x . . .’ with the name that replaces “n.” Using these notationalconventions, we can offer a schema, analogous to Tarski’s T-schema,that partially captures the concept of satisfaction:8

(1) n satisfies \ . . x . . .’ if and only i f . . . n . . .

This schema, and the various constraints placed on its instantiation,are stated in the metametalanguage. But like Tarski’s celebrated T-schema for characterizing the notion of truth, all instances are sentences of the metalanguage. Thus, we find among the instances

(1.1) Abe Lincoln satisfies ‘x was president’ if and only if AbeLincoln was president


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34 Tarski on Logical Truth

and

(1.2) Ben’Franklin satisfies ‘x was president’ if and only if BenFranklin was president.

These instances sustain our intuitive remark that satisfaction is a rela

tion that holds between Lincoln and ‘x was president’ because Lincoln was president, while it fails to hold between Franklin and ‘x was president’ since Franklin was not president.

Like Tarski’s T-schema, (1) is important not because its instances provide a definition of satisfaction, but because they provide a fairly precise measure of the success of any attempted definition. Schema (1)gives us a clear idea of what a relation, so to speak, must look like beforeit deserves to be called satisfaction. We will return to this topic in a latersection; for now, let us remark on the obvious bearing of our schema

on substitution.On the assumption that our metalanguage contains the object lan

guage, any object language name will be a permissible replacementfor “n.” Furthermore, the sentence that results from inserting thisname into the sentential function—that is, the sentence that replaces“.. . n . . . ” in our schema—will also be a sentence of the objectlanguage. Let us introduce “ ‘. .. n .. .’ ” as a placeholder for a nameof this sentence. We can now offer a second schema:

(2) n sati

etchemendy, the concept of logical consequence

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