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THE CONCEPT OF LOGICAL CONSEQUENCEJOHN ETCHEMENDYTHE DAVID HUME SERIESPHILOSOPHY AND COGNITIVE SCIENCE REISSUESCSLI PUBLICATIONSCopyright (0 1999 CSU Publications Center for the Study of Language and Information Leland Stanford Junior University Printed in the United States 03 02 01 00 99 12345Library of Congress Cataloging-in-Publication DataEtchemendy, John, 1952- The concept of logical consequence / John Etchemendy. p. cm.Originally published: Cambridge, Mass.: Harvard University Press, 1990. Includes bibliographical references and index.ISBN 1-57586-194-1 (pbk.: alk. paper)1. Logic, Symbolic and mathematical 1. Title.[BC135.E83 1999] i6o-dc2i 99-12538 CIP00 The acid-free paper used in this book meets the minimum requirements of the American National Standard for Information Sciences - Permanence of Paper for Printed Library Materials, ansi Z39.48-1984.The David Hume Series of Philosophy and Cognitive Science Reissues consists of previously published works that are important and useful to scholars and students working in the area of cognitive science. The aim of the series is to keep these indispensable works in print in affordable paperback editions.In addition to this series, CSLI Publications also publishes lecture notes, mono

graphs, working papers, and conference proceedings. Our aim is to make new results, ideas, and approaches available as quickly as possible. Please visit our web site at http://csli-publications.stanford.edu/ for comments on this and other titles, as well as for changes and corrections by the author and publisher.For Nancy and MaxAcknowledgmentsI owe many thanks to many people. For their help and encourage-ment, without which I may never have finished the book, and their criticism, without whichI would certainly have finished too soon, I would like to thank Ian Hacking, Calvin Normore, Ned Block, Greg O 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 all of the above and more, I thank my friend and colleague Jon Barwise. Finally, I am indebted to th

e Mrs. Giles Whiting Foundation and to the Center for the Study of Language andInformation for support while working on various stages of this book.Contents1 Introduction 12 Representational Semantics 123 Tarski on Logical Truth 274 Interpretational Semantics 515 Interpreting Quantifiers 656 Modality and Consequence 807 The Reduction Principle 958 Substantive Generalizations 1079 The Myth of the Logical Constant 12510 Logic from the Metatheory 136

11 Completeness and Soundness 14412 Conclusion 156Notes 161Bibliography 171 Index 1731IntroductionThe highest compliment that can be paid the author of a piece of conceptual analysis comes not when his suggested definition survives whatever criticism may beleveled against it, or when the analysis is acclaimed unassailable. The highestcompliment comes when the sug-gested definition is no longer seen as the result



of conceptual analy-sis when the need for analysis is forgotten, and the definition is treated as common knowledge. Tarski s account of the concepts of logical truth and logical consequence has earned him this compliment.Anyone whose study of logic has gone beyond the most rudimentary stages is familiar with the standard, model-theoretic definitions of the logical properties. According to these definitions, a sentence is logically true if it is true in allmodels; an argument is logically valid, its conclusion a consequence of its premises, if the conclusion is true in every model in which all the premises are true. These definitions, along with the additional machinery needed to understand them, are set forth in every introductory textbook in mathematical logic.1 In these texts we are taught how to delineate a class of models for a simple languageand how to provide a recursive definition of truth in a model in short, how to construct a simple model-theoretic semantics. Once this semantic theory is in place, the model-theoretic definitions of the logical properties can be applied.This method of defining logical truth and logical validity is gener-ally tracedto Tarski s 1936 article, On the Concept of Logical Conse¬quence. 2 In this article Tarski sets out to give a precise and general account of what he calls the intuitive consequence relation and the corresponding property of logical truth. The definitions that result are meant to be applicable to any language whose truth predicate can be defined, and to remain, as Tarski puts it, close in essentials to the common, everyday concepts.Tarski devotes most of his attention in this brief, twelve-page article to shortcomings of other attempts to define the consequence relation, in particular attempts to characterize it syntactically, by means of formal systems of deduction.

His own, semantic account, sketched in a mere four pages, is devoted in part tothe exposition of some ancillary notions treated at length in his earlier monograph on truth. The main thrust of the article is not to discuss details of the semantic account of consequence, or even to give a simple example of its application, but rather to urge that in considerations of a general theoretical nature the 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 notes that the introduction of this concept into the field was not a matter of arbitrary 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 then set out to study as we are when we introduce, say, the concept of a group, or that of a real closed field. It is for this reason that Tarski

takes as his goal an account of consequence that remains faithful to the ordinary, intuitive concept from which we borrow the name. It is for this reason thatthe task becomes, in large part, one of conceptual analysis.Tarski s account of the logical properties is widely regarded as suc-cessful in this respect, as capturing, in mathematically tractable form, the proper concepts of logical truth and logical consequence. We can see this not only from explicit acknowledgments of its success by many philosophers and logicians, but also fromthe treatment given it by those not interested in conceptual analysis as such. Perhaps the most striking indication is the different status afforded syntactic characteri¬zations of consequence, formal systems of deduction.It has long been acknowledged that the purely syntactic approach does not yielda general analysis of the ordinary notion of conse-quence, and in principle cannot. The reason for this is simple. It is obvious, for starters, that the intuiti

ve notion of consequence cannot be captured by any single deductive system. Forone thing, such a system will be tied to a specific set of rules and a specificlanguage, while the ordinary notion is not so restricted. Thus, by consequence weclearly do not mean derivability in this or that deductive scheme. But neither do we mean derivability in some deductive system or other, for any sentence is derivable from any other in some such system.So at best we might mean by consequence derivability in some sound deductive system. But the notion of soundness brings us straight back to the intuitive notion of consequence: a deductive system is sound if it allows us to prove only genuinely valid arguments, those whose con¬clusions follow logically from their premises.



We recognize that a syntactic definition does not capture the or-dinary notion of consequence, and we recognize this even though we may be convinced, for one reason or another, that a given deductive system is adequate for a given language that is, even if we believe that all valid arguments, and only valid arguments, are provable within the system. This recognition is at a conceptual level, but its main impact is at the extensional. The upshot is that systems of deduction require external proofs of their extensional adequacy (or inadequacy, as the case 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 a given system. But our assurance that all valid arguments are provable in the system if such an assurance is to be had must come from somewhere other than the deductive system itself. We need outside evidence that our system is complete, evidence we would not require if the system straightforwardly captured, in mathematically tractable form, the ordinary concept of consequence.To appreciate how different our attitude is toward the model- theoretic accountof consequence, consider the significance we read into Godel s completeness theorem. It is now common to state this theorem in the following form, where 5 is anysentence in a first-order language and K is an arbitrary set of such sentences:If K |= 5 then K \-S.Here, the relation indicated by |= is the model-theoretically defined consequence 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 con-sequence c

oincide, that they apply to the same pairs (K, S) in the first-order language. But we think of these results as having an intu¬itive significance that goes beyond the mere coincidence of two alter¬native characterizations of the consequence relation. Specifically, we think of them as demonstrating the extensional adequacyof the de¬ductive system in question. They are thought to show that the system issound, that it will not allow the derivation of conclusions that are not genuine consequences of their premises, and that it is complete, that it allows the derivation of all the consequences of any given set of sen¬tences in the language.What is revealing is that the significance we read into these results is asymmetric, even though their form alone would not seem to warrant it. After all, for any given language there will be a wealth of theorems displaying the same general pattern:If K |-i 5 then K |-2 5,

If K |-2 5 then K |~i 5.But if, for example, both |~i and (~2 are syntactically defined conse¬quence relations, perhaps involving variant proof regimes, we wouldhardly take these results as showing the adequacy, the soundness and completeness, of one regime rather than the other. In such a case we would take the theorems as showing nothing more than the coex-tensiveness of the two characterizations. To think they demonstrate, say, the extensional adequacy of |~2> would obviously presuppose additional theorems showing the completeness and soundness of |-1 In this case, the pair of results would be viewed as entirely symmetric.The felt asymmetry in our original two theorems stems from our assumption that the model-theoretic definition of consequence, unlike syntactic definitions, involves a more or less direct analysis of the consequence relation, and so its extensional adequacy, its complete-ness and soundness, is guaranteed on an intuitive or

conceptual level, not by means of additional theorems. If it were not for this assumption, we would feel equal need for external evidence that the model-theoretic characterization of consequence is extensionally cor-rect, that it applies to all valid arguments, and only valid arguments, of the language in question.How do we know that our semantic definition of consequence is extensionally correct? How do we know it does not declare some logically valid arguments invalid,or declare some invalid arguments logically valid? Many readers will find this question quite odd. But it is not odd in the same way as the question How do we know that all structures satisfying the group axioms are really groups? This second question is simply confused: the notion of a group is arbitrarily de¬fined to mea



n those structures satisfying our characterization. But as Tarski points out, the situation is quite different with the concept of logical consequence. Here the correctness of our model-theoretic de¬finition is not determined by arbitrary fiat; on the contrary, whether the definition is right or wrong will depend on howclosely it cor¬responds to the pretheoretic notion it is meant to characterize. That the first question now strikes us as odd just indicates how deeply ingrainedis our assumption that the standard, semantic definition captures, or comes close to capturing, the genuine notion of conse¬quence.The situation here might be illuminated by analogy with some basic results in recursion theory. Recursion theory, like logic proper, was originally driven by an interest in a rather imprecise and intuitive notion. Here the notion was that of an effectively computable func¬tion, a function whose values could in principlebe calculated by al-gorithmic means that is, using fixed instructions requiring no insight or creativity. During the 1930s, many mathematically precise charac¬terizations of the class of computable functions were proposed, by Church, Gõdel, Turing, and others, and various important results concerning the precisely defined classes were proved. Among them was the striking result that, although the precise characterizations proceeded in widely divergent ways, they were nonetheless coex-tensive; they carved out exactly the same class of functions. This result wastaken as evidence that this class of functions, however specified, formed a natural and important collection. But did it also show that the specified class wasexactly the class of intuitively computable func¬tions? The answer, of course, isno. For if none of the precise charac¬terizations individually captured the intuitive notion of computability, the question of whether they coincide exactly with

this concept hardly followed from their convergence. The coincidence of the various de¬finitions provided some indirect evidence, as did the fact that no obviously algorithmic function could be found that fell outside the defined class. But these do not amount to a mathematical demonstra¬tion. Because of this, logicians take great care to distinguish the various mathematical results in recursion theory from the claim that all intu¬itively computable functions fall into the precisely delineated class. This claim is usually called Church's thesis, and although it is almost universally accepted, it is not considered amenable to mathematicalproof.This situation is parallel to the one that confronted early, formal logicians. Much of their work was driven by an interest in the intuitive notions of logicaltruth and logical consequence, but the only precise access to these notions wasthrough specific, proof-theoretic charac-terizations, specific deductive systems

. These syntactic characteri-zations, however, clearly did not capture the intuitive notion; they were not straightforward analyses. Because of this, the claimthat a particular proof regime, say for some first-order language, coincides with the language s genuine consequence relation, seemed at best to admit of indirect evidence. The coincidence of various different sys-tems of proof provided somesupport, as did our ability to construct formal derivations of many specific instances of valid reasoning. But as Hilbert once put it, evidence accrued only through experiment, not through mathematical proof.3 To emphasize the parallel with recur¬sion theory, we might call this claim the claim that all and only logically 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 s thesis been givensuch a prominent position in logical pedagogy, while its counterpart has not? Bo

th involve the relationship between a math¬ematically precise definition and one of the central, albeit intuitive, notions of our discipline. The difference is that in the latter case, the thesis has been replaced by theorems: the soundness and completeness theorems 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 conse¬quence relation for theselanguages. And of course it is such a proof, on the assumption that the model-theoretic definition captures the genuine concept of consequence. It is such a proof, on the assumption that Tarski s analysis is right.It is precisely this assumption that I question in this book. Briefly put, my cl



aim is that Tarski s analysis is wrong, that his account of logical truth and logical consequence does not capture, or even come close 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 tech-nique for ferreting out the genuinely valid arguments of a language than is applying a purely syntactic definition. Neither technique is guaranteed to yield an extensionally correct specification of the lan-guage s consequence relation. Needless to say, this conclusion requires that we reassess the intuitive significance of Godel s completeness theo¬rem, as well as the import of the failure of analogous results when we move, for example, to second-order logic.The intuitive concept of consequence, the notion of one sentence following logically from others, is without doubt the most central concept in logic. It is what has driven the study of logic for more than two thousand years. On the other hand, the remarkable achievements in logic during the past century have been the direct result of the mathematization of the field. The infusion of mathematically precise definitions and techniques has turned a field dominated by homely admonitions into one capable of supporting significant and illumina-ting theorems. My aim in this book is to attack a common misun-derstanding of one widely used mathematical technique, not to ad-vocate a return to homely admonitions, or even to suggest that we abandon the particular technique. The fact that neither the model- theoretic nor the proof-theoretic account of consequence alone cap-tures the genuine notion does not mean they are useless for studying this very same concept. Direct analysis is just one way to gain access to an important, intuitive co

ncept; lessons from elsewhere in mathemat¬ics should convince us of that.Some HistoryThough my concern in this book is not historical, a few preliminary words should be said about the complicated heritage of the model- theoretic definitions of the logical properties. As I mentioned, these definitions are generally creditedto Tarski s 1936 article, and for the purposes of this book, there is no need to question this attribution. What is clearly right about it is that Tarski s articlecontains the only serious attempt to state, in its most general form, the analysis underly¬ing the standard definitions, and to put forward a detailed philo¬sophical justification for that analysis. It is, so to speak, the philosophi¬cal locus of the model-theoretic definitions.From a historical point of view, though, attributing the definitions to Tarski alone oversimplifies the situation a great deal.4 For one thing, most of the main

features of the analysis were anticipated, in various different ways, by earlier authors, including Bolzano (1837), Padoa (1901), Bernays (1922), Hilbert and Ackermann (1928), and Gõdel (1929). Of all of these, Bolzano s discussion is by far the most exten¬sive; in Chapter 3, I will briefly describe his account and motivate certain features of Tarski s analysis by comparing it with Bolzano s. Padoa, unlike Bolzano, does not offer an analysis of logical truth and logical consequence, but gives a general statement of the familiar, model-theoretic technique for establishing a sentence s logical inde¬pendence from a given set of axioms, a techniquethat presupposes one direction of the definition of consequence. Bernays, Hilbert and Ackermann, and Gõdel all present, with varying degrees of clarity, a model-theoretic definition of logical truth, though none of them tries to justify it, or offers the corresponding definition of logical conse¬quence.When Tarski proposed his analysis in 1936, he was fully aware of these predecess

ors, with the notable exception of Bolzano. In his article, Tarski emphasizes that his treatment of the logical properties makes no very high claim to complete originality, and that the ideas involved . . . will certainly seem to be something well known (1956, p. 414). Still, the article is not just a codification of commonly accepted ideas and techniques. For one thing, as Tarski points out, the defini¬tions he gives presuppose methods which have been developed [only] in recent years. Specifically, they involve techniques for defining the notions of satisfaction and truth, concepts that had been left at an intuitive level by all earlier authors. Second, and more important, is Tarski s attempt to present and motivate thedefinitions in a com¬pletely general setting. It is easy to underestimate the impo



rtance of this contribution. But clearly, the ordinary notions of logical truthand logical consequence are not restricted to a specific language or small collection of languages, and so our definition of a single language s consequence relation, or of its set of logical truths, must flow from some more general analysisof these concepts. Finally, unlike his im¬mediate predecessors, Tarski extends his account to the notion of logical 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 historical question of who should receive primary credit for the definitions is a complicated one, both for the reasons sketched here and for another important reason that will emerge in Chapter 5.It turns out that certain paradigmatic instances of the model-theoretic definitions in-volve a subtle but significant departure from Tarski s analysis, one that has gone completely unnoticed. But to explain that departure at this point wouldbe premature.The Plan of This BookThis book consists of a single, extended argument. The conclusion of the argument is that the standard, semantic account of logical conse-quence is mistaken. What I mean by this is, first of all, that when we apply the account to arbitrarylanguages even perfectly familiar, well-behaved ones it will regularly and predictably define a relation at variance with the genuine consequence relation for the language in question. The definition will both undergenerate and overgenerate: it will declare certain arguments invalid that are actually valid, and declare others valid that in fact are not.This is not to say that every application of Tarski s account is exten¬sionally inco

rrect. Indeed, I will eventually argue that with suitably weak languages (and with certain qualifications that I explain later) the definition does get the extension right. But even in these cases we must seek 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 the following limitation: there is no way to tell from the definition alone or from characteristics of the language whether the extension of the account is correct. Clearly, no amount of pondering a syntactic system of deduction can assure us of its extensional adequacy; for that, we must turn to indirect evidence, whether in the form of theorems or, failing these, evidence of a more experimental sort. I claim that exactly the same holds true of any application of the model-theoretic account 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 this emphasis is simple. I think the basicproblem with Tarski s account is in some sense obvious, once certain confusions and misunderstandings are cleared away. But there are several of these confusions, and each of them lends a certain plausibility to the analysis. Together, they give rise to a remarkably persuasive illusion, an illusion that the account (as Tarski puts it) captures the essential features of the ordinary concept of consequence.Of course, if this were really the case, if the account simply trans-lated our intuitive concept into mathematically tractable form, we would have an ironclad guarantee of its extensional adequacy when applied to arbitrary languages. The situation would then be analogous to, say, our inductive definition of N, the setof natural numbers. According to this definition, N is the smallest set that con

tains 0 and is closed under the successor operation.5 Now, it is perfectly clear that this definition is not identical to the intuitive notion it supplants. Thus, it employs a variety of set-theoretic concepts that are not, by any stretch of the imagination, part of our ordinary understanding of the natural numbers. Conversely, certain things that are arguably central to our intuitive concept (say, the concrete process of counting) are at best dimly reflected in the inductive definition. But the definition obviously captures the essential feature of theintuitive notion, and so its extensional adequacy is apparent from the definition itself. We do not, so to speak, have to try it out to see that it really works.



Most people react to the model-theoretic account of consequence in the same waythey react to the inductive definition of N. Neither is given extensive justification since neither seems to need it. I claim that this reaction is, in the former case, mistaken. But it is not, unfor-tunately, a simple mistake or, for that matter, a single one. For this reason, much of this book is devoted to explainingthe variety of confusions and misunderstandings that have made Tarski s analysis seem so convincing. Until these are finally laid to rest, purely exten-sional evidence against Tarski s account, evidence that I think we have long had, will continue to be explained away.I try to treat these misunderstandings one by one, in what I hope is an orderly, comprehensible way. Unfortunately, treating them one at a time the only way I see to do it has certain drawbacks. For one thing, not everyone will share a given misunderstanding, and so an individual reader may find certain parts of the book obvious, while another might find those same points illuminating and others not.For example, the first few chapters are addressed to a confusion extremely common among those who enter logic through philosophy or linguis¬tics, but almost nonexistent among those who enter through main¬stream mathematics. Here, I can only ask the reader s patience. If I appear, at points, to be addressing the wrong issue,and perhaps ignoring entirely some key insight that justifies the account, I hope the reader will nonetheless persevere.This gives rise to a second problem namely, that different parts of the book are really addressed to somewhat different audiences. Since these audiences will have different technical backgrounds (not to men¬tion different interests and concerns), I have tried not to assume much common ground, at least in covering the key p

oints of my argument. The model-theoretic account of consequence has had a tremendous influence on all logic-related disciplines, from philosophy and linguis¬tics to mathematics and computer science. Thus, I have tried to make the book understandable to anyone who has had a first course in mathematical 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, be extensionally incorrect. Second, I describe what I believe is the main source of the account s remarkable persuasiveness. The chapters lead¬ing up to this are devoted to untangling some of the more straightfor¬ward confusions that surround the analysis, and to giving a clear explanation of Tarski s original definition and of its relation to the model-theoretic treatment with which we are now familiar.

In order to understand Tarski s account it is essential to distinguish it from what I call representational semantics. Representational semantics is a perfectly legitimate approach to semantics, but (as will become clear) it bears no relation whatsoever to Tarski s account of the logical properties. Unfortunately, Tarski s analysis is frequently conflated with representational semantics. For this reason I will begin, in Chap¬ter 2, by discussing this alternative approach to semantics, so that it can be usefully contrasted with Tarski s account rather than vaguelycon¬fused with it. Chapters 3 through 5 are devoted to a careful exposition of Tarski s original definitions and their relation to the standard,model-theoretic account. Then, in Chapter 6, I consider and reject Tarski s own positive arguments in support of his analysis.In Chapter 11, I try to reconcile the lessons learned in Chapters 7 through 10 with widespread intuitions about completeness and soundness theorems. There, I mo

dify 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 defini-tions.One final point before beginning. Through large stretches of this book I focus,for simplicity, on the notion of logical truth. Logical truth, since it is a property of single sentences, is often far easier to discuss than logical consequence, which is a relation between a collec-tion of sentences (say, premises of anargument) and another sentence (the conclusion). For example, it is much easierfirst 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 far more important than that of logical truth, both intuitively and techni¬cally. On their own, logical truths are of very little interest recall that these are sentences we oftendescribe 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 consequence relation: these are sentences that follow logically fromany set of sen¬tences whatsoever. The crucial notion, ultimately, is that of one sen¬tence following logically from others. Logic is not the study of a body of trivial truths; it is the study of the relation that makes deductive reasoning possible.2Representational SemanticsTo understand Tarski s account of the logical properties, we need to distinguish clearly between it and representational semantics. But to do that, we need a fairly clear idea of what the latter approach to seman¬tics is all about. A good place to begin is with a simple puzzle suggested by Donald Davidson. In a well-knownarticle in which he defends his own approach to semantics, Davidson draws a broad distinction be¬tween theories that characterize or define a relativized concept of truth 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 is expressed by a monadic predicate.In this respect, truth sets itself apart from many other concepts that we consid

er 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 holds true of, and so forth. But truth, perhaps the preeminent semantic concept, does not relate a sentence to something else; it simply applies or fails to apply, so to speak, absolutely.1Davidson goes on to note that at least on a superficial level, much contemporary work in semantics seems to belie this simple point. Much effort is devoted to the investigation of what Davidson sees as irreducibly relational notions, notions like truth in a model, truth in an interpretation, 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-sentences distinctive of the th

eory of absolute truth. The T-sentenceSnow is white is true if and only if snow is whitedoes not, as Davidson puts it, fall out of a theory that simply tells us which models Snow is white is true in. And for this reason, theories of relative truth do not necessarily have the same sort of interest as a theory [of absolute truth] (1973, p. 79). A theory that yields T- sentences provides, first and foremost, an explication of absolute truth that is, of truth as we ordinarily understand it; theories of relative truth must, at least on the surface, be seen as providing explications of something else.I am not concerned here with the merits or demerits of competing semantic programs, and in particular I will not spend time considering Davidson s own approach. But it is worthwhile taking seriously Davidson s simple, initial point: truth is, after all, a property; truth in a model, a relation. What bearing can a character

ization of such a rela¬tional concept have on our ordinary monadic concept of truth? If there 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 sounds so misleading?2We can look at Davidson s puzzle this way. A theory of relative truth provides uswith a characterization of x is true in);. Yet it is common to think of such theories as telling us something about truth, as having at least intuitive or informal consequences involving the ordinary mo¬nadic predicate x is true. Davidson, of course, is particularly inter¬ested in the so-called T-sentences, but the same pointmight be made about any claims involving absolute truth. That point is this. Before a theory of relative truth can be judged to have consequences, formal or other



wise, involving the standard monadic concept, we must give some explanation of exactly how the defined x is true in y is related to the already understood x is true. Somehow, we must explain how we are to move from our theory about the relation to claims involving the property. If we can give no such explanation, then thesimple, prima facie evidence is that our theory of relative truth has no bearing on the concept of truth as we ordinarily understand it. But that, of course, is absurd.Truth as SpecificationWe often find it advantageous to explain a monadic concept in terms of 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 brother of 31. The former then reduces to an existential generalization of the latter: brotherhood is just brother-of-someone-hood. There are similar cases in which we gain access to the monadicconcept through a universal generalization of the relational; thus with comparatives and super¬latives say, taller than and tallest. But clearly the monadic concept of truth, the concept we ordinarily employ, is no generalization of any of thevarious 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 or existential, of the concept of truth in a model, then the natural alterna-tive is to think of the former as a specification of the latter. In other words, perhaps the monadic concept emerges from the relational by fixing on a specific instance of the nonsentential parameter, the y in x is true in;y. Being true simpliciter would then be viewed as equivalent to being true in some particular model, and getting fro

m a theory of relational truth to a theory of absolute truth would be a matter of indicating which specific model was the right model. Our conceptual analogy might then run: x is true in y stands to x is true as x is a brother of y stands toed'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 a sense it is the relational concept that is a generalization of the monadic concept; what justifiesthe appearance of the word true in theories of relative 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 true in 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 not determin

ed what sort of hidden parameter our models are meant to fill, or what makes one model the right one, the model that binds the ordinary concept of truth to the more general concept of truth in. I will devote several chapters of this book to exploring one possible answer to this question, the answer presupposed by the model-theoretic defi¬nitions of the logical properties. But there is another very natural answer, one assumed in what I have called representational semantics. Briefly, this answer is that Fred is the accurate model, the one that represents the world as it really is.Truth in a RowConsider the simplest and most familiar theory of relative truth, a theory we are taught during the first few days of any inaugural course in logic. This is the theory of truth in a row, the theory that enables us to construct truth tables.To fill out a truth table for a simple sentence of English, we have to acquire t

wo principal skills. In the first place, we must master the proper technique for constructing the reference column of the truth table, a column headed by a horizontal list of the atomic components of the sentence in question. This technique generally involves some sim¬ple, extendable pattern of writing the words TRUE and FLSE in horizontal rows beneath our list of atomic sentences, a pattern guaran¬teedto capture all the required permutations for a given number of such components.Thus, depending on the atomic sentences con¬tained in the target sentence 5, eachof the following would serve as proper reference columns:Snow is white 5TRUE



FALSE

Snow is white Roses are red 5TRUE TRUETRUE FALSEFALSE TRUEFALSE FALSE

Snow is white Roses are red Violets are blue 5TRUE TRUE TRUETRUE TRUE FALSETRUE FALSE TRUETRUE FALSE FALSEFALSE TRUE TRUEFALSE TRUE FALSEFALSE FALSE TRUEFALSE 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 false in. The ultimate goal isto write the words TRUE or FALSE in each row below S; TRUE if 5 is true in that row,LSE if 5 is not true in that row. But to do that, of course, no standard patternof the sort used in constructing the reference column will suffice, will ensurethat we enter the correct value in each row. Rather, we need a radically differ¬en

t technique, a technique that involves the repeated application of certain recursive tables. The following are two sample recursive tables; the not table:p notpTRUE FALSEFALSE TRUE

and the or table:P <1 porqTRUE TRUE TRUETRUE FALSE TRUEFALSE TRUE TRUEFALSE FALSE FALSE

These recursive tables are meant to tell us when a complex sentence is to be considered true in a row, on the assumption that we have already determined whether its immediate constituents are true in that row. Equipped with the values of the constituents, we need only match them to the appropriate row of the appropriate recursive table and read to the right. Often the recursive tables will also have been applied in order to determine the values of the relevant constituents, and,in turn, of their relevant constituents. Indeed, there is no upper bound on thenumber of times a recursive table may have to be applied within a single row before the final value of the target sentence is reached.Once we are adept at these techniques we can easily produce tables in which ourtarget sentence is assigned a definite value in each row. Thus, taking the target sentence to be Snow is white or roses are not red (and abbreviating our reference column somewhat), we get the following simple table:

S W RR Snow is white or roses are not redT T TRUET F TRUEF T FALSEF F TRUE

Now, consider exactly what this table tells us. First of all, it clearly does not tell us the actual truth value of our target sentence that is, its

monadic value. But this was to be expected, since our theory is at most a theory of relative truth.3 It does, however, tell us exactly which rows our sentence is



true in; specifically, it tells us that the sentence is true 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 was meant to be a specific instance of relative truth. Translating to present terminology, the truth of a sentence should boil down to its truth in some specific row. And sincewe know that the current sentence is actually true, we can rule out the third row without further ado; that row 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 silly question; after all, Snow is white and Roses are red are both true that is, genuinely truehe first row is the only row in which these sentences both come out true. But notice that in offering this reply, we have simply put off solving Davidson s puzzle. There is no question that Snow is white is true in the first row of this table;for that, we need not even apply our recursive techniques. Yet it is equally clear, even on the level of atomic sentences, that being true in a row is quite different from being absolutely true; evidence for that will be found in any of the remaining rows of our table.Language and the WorldDavidson s puzzle reappears at the very bottom level of our theory of truth in a row, with the atomic sentences that acquire their values in the reference columns of our tables. If truth is to be truth in some specific row, then clearly the f

irst row of our sample table must be the right one. But it is equally clear that this observation does not provide any account of the link between our theory of relative truth and the ordi¬nary, monadic concept from which we pirate the name. To provide such an account we must explain how the first row, so to speak, comes to be the right row. Furthermore, our explanation cannot simply reduce to the plea that if we picked any other row, various sentences would be true in the right rowand yet not be true simpliciter. Such a response would 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 changes from row to row in tlie theory of relative truth, then Davidson would be completely justified in claiming that the defined x is true in y is irreducibly relational. And consequently he would be justif

ied in claiming that, for this reason, our theories of relative truth cannot bethought to illuminate the notion of truth as we ordinarily understand it. But this conclusion would obviously be wrong. It is perfectly clear that truth tablestell 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 the ordinary concept.But Davidson s puzzle is not unsolvable. The problem is not finding an appropriate parameter in our ordinary notion of truth, but rather choosing between two obvious alternatives. Consider the move from the first row of our sample truth table to the third. Here the relevant change in our reference column is the value assigned to the atomic sentence Snow is white. The effect of this move is that the resulting value 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 Sn

ow 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 to the language that the sentence means what it means, that it makes the claim 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 would have been false in spite of the whiteness of snow say, if white had meant hot. On the other hand, had the world been different, this sentence 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 of truth in a row as explicating the relation x is true in L for a limited, though nontrivial range of languages L. From this perspective, we would assume that any extralinguistic fact that might influence the truth value of sentences say, the color of snow or roses is held fixed; our concern is not with changes in the world. Viewed this way, the first row of our sample table is right simply because English, the implicitly specified parameter in x is true, happens to be one of the languages that expresses true propositions by both Snow is white and Roses are red. Thus, the third row would havebeen right had we been speaking a language exactly like English save that white meant hot.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 fragment thereof. 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. From this perspective, the move to the third row involves no change in meaning; that row would have been right simply had snow not been the color it is.We commonly think of truth tables as capable of supporting certain counterfactual claims about the (absolute) truth values of their tar¬get sentences. We imaginethese claims to be supported because our theory assigns values to these sentences even in rows that are not right, rows in which the atomic sentences are not assigned their actual values. So, for example, the third row of our sample table supports 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 will vary depending on which parameter we view as changing in the move from row to row that is, depending on what we take to be the relation between truth in a row and the monadic truth predicate appearing in the claim. In effect, our theory will support those completions that we consider elucidations of had the third row been the right row. Tus, if we view our parameter to be the language, we might offer the completed 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 into our truth tables depends critically on which perspective we assume, on the nature of the parameter that corresponds to the rows our sentences are true in. Of course, sinceboth points of view are possible here, we might justify either of the above counterfactuals by referring to the third row of our sample truth table. Or, to simplify matters, we might even merge both of our claims into a single counterfactual:The sentence Snow is white or roses are not red would havebeen 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 any account we might give of the relation between x is true in / and x is true. Or, to put it another way, thefact that our theory of truth in a row seems doubly illuminating because it admits

of either perspective should not lull us into thinking that it retains its illumination indepen¬dent of these perspectives. Rather, as Davidson s puzzle nicely points out, the purely abstract characterization of relative truth, of x is true iny, supports no claim whatsoever about absolute truth, about truth as we ordinarily understand it.A Representational SemanticsWhen we view a particular theory of relative truth as explicating x is true in W, we see it as providing an account of how the world wields its influence on the truth values of sentences within a fixed language. If characterizing this influence is the aim of our relativized theory of truth, then I will say we are engaged



in representational semantics. The reason I use this somewhat unusual term is simple. Our theory pro¬vides an account of a relation, x is true in;y, and what the theory takes to satisfy the position are, for all intents, just ordinary objectsof some sort or other chunks of the actual world. Thus, in our theory of truth in a row, the term was filled by rows, rows that were fixed by the reference column of our truth table. Other representational theo¬ries might define a relation between sentences and abstract, set- theoretic objects, maybe functions of some sort.But obviously these in no case actually are the possible configurations of the world that they are meant to represent. Rows of a truth table are just blotches ofink, and functions are set-theoretic constructs; the world, thankfully, is neither of these.The point is a simple one, but all too easily overlooked. When we viewed our theory of truth in a row as explicating x is true in W, the fact that the target sentence came out false in the third row of the table was taken to indicate that the sentence would have been false in a world in which roses were red but snow not white. But the third row itself, the ink 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 valu¬able information about truth, but certainly not about how truth would be affected by changes in row. Rather, it tells us how truth would be affected by changes in the world, by changes that are represented or depicted by changes in row.The techniques used in constructing truth tables are not generally thought to constitute a full-fledged semantic theory for any language or language fragment. More than anything else, this is due to certain traditions of fairly recent vinta

ge concerning the accepted format of such theories. Still, it may seem perverseto view our theory of truth in a row as a representational semantics, insofar as it may seem perverse to view it as a semantics at all. But this can easily be remedied.Suppose we are interested in the fragment of English containing the atomic sentences Snow is white, Roses are red, and Violets are blue, plus whatever complex sentences can be formed from these using a sign for negation, not, and a sign for disjunction, or. I will assume that we have a precise syntactic theory for our language,one that enables us to form the negation of any sentence and the disjunction ofany two.4 A standard representational semantics for this simple lan¬guage might proceed in the following way. First we define a class of models that will represent all possible configurations of the world relevant to the truth values of our sentences. Thanks to the simplicity of our language, this purpose can be served b

y the class of functions that assign a truth value, either true or false, to each of our three atomic sentences. Thus, our class of models consists of eight functions, one that assigns true to each sentence (representing worlds in which snow is white, roses are red, and violets blue), one that assigns false to each (representing worlds in which snow is not white, roses not red, and violets not blue), and so forth.Our next step is to provide a recursive definition of 5 is true in f for arbitrary sentences 5 and models/. Since we will take this relation as an indirect characterization of x is true in W, our aim will be to ensure that any given sentenceof our language is true in exactly those models which represent worlds that would indeed have made the sentence true. So if a model depicts a world in which snow is not white, our definition should guarantee that Snow is white comes out false in that model. Here we assume, of course, that the sentence Snow is white means w

hat it actually means; the sentence is ours, even though the world depicted by the model is not.The definition proceeds in the obvious way, by recursion on the set of sentences in our language:

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

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

If 5 is the disjunction of S' and S", then it is true in a model/just in case either S' is true in f or S" is true in/.



For the most part, what we have done here just involves a recasting of our theory of truth in a row. But there are two changes worth mentioning. 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 reference columns, since they provide the objects our sentences are true or false in. Indeed, they do so with somewhat more aplomb, allowing us to use the same models for any sentence in our fragment. Thus, we have managed, in the new theory, to introduce a standard collection of objects, each of which fully determines the apportionment of truth values throughout the entire language.5Now, although it could easily escape notice, the reference columns of our earlier theory actually did a bit more than our models. The reference columns both delineated the needed rows and simultaneously specified 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 recursive definition, the clause beginning if 5 is an atomic sentence . .The fact that we took models to be functions that yield the values true and false is entirely a mnemonic convenience in the new theory; any two objects would have worked as well for example, the numbers zero and 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 model that 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 the reference columns of our truth tables, say, the freedom to use + and rather than TRUE and FALSE, we

d have to supplement our recursive tables with base tables to complete the definition of truth in a row. 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. Representational GuidelinesThe basic motivation underlying a representational semantics, an indi¬rect 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 as

pects of meaning relevant to the truth values of sentences, both the values they actually have and the values they would have, were the world differently arranged. Needless to say, the simple representational semantics of the last section can at best be considered a partial theory of meaning for the relevant fragment,since it offers no detailed account of the semantic functioning of the three atomic sentences. In giving the semantics, we simply assumed that Snow is white somehow comes to mean what it does, and for this reason is true in exactly those worlds in which snow is white. A more detailed semantics would presumably say something on this score as well.Of course, the fact that the motivation is clear does not mean the task of devising a representational semantics for any interesting language is either easy orphilosophically unproblematic. But these difficulties are not, 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 xis true in L, for a specified range of languages L. As we will see, Tarski s is a remarkably different goal from that presupposed by the representational approachto se¬mantics, in spite of the fact that one and the same account of x is true in y may occasionally admit of both construais. Failing to recognize this difference, many philosophers have assumed that Tarski, in de¬fining the logical properties,had in mind something akin to represen¬tational semantics, a characterization of xis true in W, for all possi¬ble worlds W. For example, we find David Kaplan extolling the insight of Tarski s reduction of possible worlds to models, a reduc¬tion Kaplan claims to be implicit in the analysis of the logical proper-ties developed in Tarsk



i s article.6 But this, as we will see, is just a confusion, one of several that lend undeserved credence to Tarski s analysis.Let me conclude this chapter by emphasizing the guidelines that will seem natural if our aim in constructing a model-theoretic semantics is to give a characterization of x is true in W. First, there is the obvious though rather vague criterion we use in judging the adequacy of our class of models. In a representational semantics the class of models should contain representatives of all and only intuitively possible con¬figurations of the world. This was accomplished in the semantics of the last sect ion by employing a rather crude but effective system of repre¬sentation. Our collection of models imposed, so to speak, a completepartition on the class of possible worlds, a partition whose boundaries were determined by the color of snow, roses, and violets in those worlds. Had we excluded any one of our eight functions, the remain¬ing class of models would have been inadequate in this respect, leaving no representative for certain perfectly conceivable worlds. On the other hand, had our atomic sentences been Snow is white, Snow is red, and Snow is blue, then we would have been justified in limiting the classof models to those functions that assign false to at least two of our atomic sentences. The remainder would not represent genuine possibilities.Once we have specified the class of models, our definition of truth in a model is guided by straightforward semantic intuitions, intuitions about the influenceof the world on the truth values of sentences in our language. 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 accu¬rate that is, had the world actually been as depicted by that model. Obviously, the possibility of success on this score is not inde

pendent of the objects we have chosen to include in our class of models. In particu¬lar, it is this ultimate goal that determines the amount of detail we need toincorporate into our models, how crude a system of representation we can get bywith. So, for example, with our sample fragment we could not have used functions that assigned truth values only to Snow is white and Roses are red. Although thesemodels would indeed have given us a complete partition of possible worlds, the partition would not have been fine-grained enough to allow us to carry out our semantic task: the accuracy of any of these models would have been consistent with either the truth or falsehood of Violets are blue. And of course with more complicated languages, say, languages containing quantifiers, our technique of constructing representations will have to allow for a considerably more detailed depiction of the world.Now, the final points to notice about representational semantics concern the sen

tences that turn up true in all models. It is an immedi-ate and trivial consequence of the two criteria I have just described that sentences which are true in all models should be exactly those that are necessarily true. If a sentence is not necessarily true, yet comes out true in all models, then we have either omitted representations for some possible configurations of the world, namely those that would have made the sentence false, or our definition of truth in a model has gone astray, having declared the sentence true in at least one model that depicts a world in which it would actually have been false. Just so, a sentence thatis necessarily true can only come out false in a model if we have gotten its semantics wrong or if the model fails to depict a genu¬ine possibility.Clearly, all and only necessary truths will come out true in all models of an adequate representational semantics. And so if logical truths are thought to be necessarily true, these will of course be among those true in every model. Similar

ly, if one sentence comes out true in every model in which a second sentence istrue, then the truth of the first must be a necessary consequence of the second. That is, it must be impossible for the first to be false while the second is true, at least if our semantics really satisfies the representational guidelines.Equally trivial is the observation that analytic truths, sentences that are true solely by virtue of the fixed semantic characteristics of the language, will come out true in all models. If a sentence is not true in all models, then its truth is clearly dependent on contingent features of the 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 eve



ry model of an adequate semantics, one that satisfies the stated criteria.7These are all immediate consequences of the simple representa-tional guidelinessketched above. But in spite of these consequences, it would clearly be wrong to view representational semantics as giving us an adequate analysis of the notion of logical truth. For one thing, if there are necessary truths that are not logically true, say, mathematical claims, 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 peo¬ple would balk at it is still clear that representational semantics af¬fords no net increase in the precision or mathematical tractability of this notion. Any obscurity attaching to the bare concept of necessary truth will reemerge when we try to decide whether our semantics really satisfies the representational guidelines in particular, when we ask whether our models represent all and only genuinely possibleconfigurations of the world.The value of representational semantics does not lie in an analysis of the notions of logical truth and logical consequence, or in the analysis of necessary oranalytic 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 a method of approaching the empirical study of language, rather than an attempt to analyze any of the concepts employed in that task. Certainly, all necessary truths of a language of whatever ilk should come out true in every model of a representational semantics. If they do not, this just shows that our semantics for the language is somehow defec tive, perhaps that we are wrong about the meanings of certain expressions. But this is only a test o

f the adequacy of the semantics, nota sign that we also have an analysis of necessary truth. The latter notion is simply presupposed by this approach to semantics. This is not an objectionable presupposition, by any means, so long as our goal is to illuminate the semantic rules of the language and not the notion of necessary truth.I have sketched some simple and general criteria that guide the construction ofa representational semantics, a theory of x is true in W, for variable W. As I explain in Chapter 4, Tarski s analysis of the logical properties gives rise to an alternative approach to semantics, one whose aim is to characterize the relation xis true in L, for some range of languages L. The intuitive importance of such a theory, and the general guidelines appropriate to it, are not nearly so apparentas those 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 logical consequence.3Tarski on Logical TruthMy 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 original analysis but unfamiliar with modern presentations of it. There is no mention in Tarski s article of any rangeof languages, or of any notion 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 semantics differs from a representational semantics, it is important to start from the beginning, with a clear understanding of Tarski s original defi

ni¬tions and their underlying motivation.I approach Tarski s account of logical truth and logical consequence indirectly, by considering first a simpler account developed by Bolzano nearly a century earlier.1 The two accounts are remarkably similar; indeed, Tarski initially entertains what is, for all intents, precisely the same definition as Bolzano s, but modifies it for reasons I will eventu¬ally explain. But in spite of the striking similarity in the two accounts, Tarski was unaware of Bolzano s work until several years after the initial publication of his article. The key difference between the two accounts is simply that Bolzano employs substitution where Tarski uses the more technical, and for the purposes more adequate, notion of satisfaction.



Bolzano on Logical TruthWe normally think of logical truth as a single property that holds or fails to hold of sentences within a language. Both Bolzano and Tarski adopt a slightly different approach, in effect treating logical truth as a relation that holds between sentences and sets of atomic expressions in the language, or alternatively, as a collection of properties that can be obtained from this relation by fixing its second argument.2 On either Bolzano s or Tarski s account, there will be sentences that are logically true with respect to one set of atomic expressions, but not logically true with respect to another. The logical truth of such sentences depends, as Bolzano puts it, on which expressions we take to be variable and whichwe take to be fixed. To use Tarski s phrase, it depends on which expressions we treat as logical constants.According to Bolzano, what is distinctive about logical truths is that they remain true when we exchange some subset of their component expressions for any other expressions of similar type.3 Bolzano notes, for example, that the sentenceIf Caius was a man then Caius was mortalremains true regardless of the subject term we put in the two positions currently occupied by Caius. On the other hand, the sentence that results from inserting the term omniscient in the position occupied by mortal is false. Thus, Bolzano concludes, this sentence is logically true when we allow only the first sort of exchange, though it is not logically true when we also allow substitutions for the expression mortal. We cannot say the sentence is or is not logically true simpliciter, since this will 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 sentencesare either true or false, we can take expressions to be of similar type just in case they are members of the same grammatical category. We can then describe Bolzano s account of logical 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 from making permissible substitutions for expressions in 5 is also true.A substitution of a for b in 5 is permissible if a and b are expressions of thesame grammatical category, if all of the occurrences of b are uniformly replaced by a, and if expression b contains no member of the set of fixed terms.Consider an example. The following sentence is true:Snow is white or snow is not white.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 the true sentenceGrass is green or grass is not green.It seems reasonable to assume that the truth of this sentence survives any grammatically appropriate substitution for the expressions snow and is white. 4 In which case, the sentence Snow is white or snow is not white 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 logi¬cally true with respect to every selection of fixed terms. So for instance if $ contains just thethree expressions not, snow, and is white, that is, if the expression or is consid

variable term, then the sentence can easily be turned into a false one. Thus, the false sentenceSnow is white and snow is not whiteresults from the substitution of the expression and for or, a substi-tution permitted on this selection of Similarly if we take as our only fixed terms or and is white, we can presumably get the false sentenceGrass is white or grass is necessarily whiteby making grammatically appropriate substitutions for the two re-maining variable terms. On the other hand, Snow is white or snow is not white does seem to be logically true with respect to the set contain¬ing snow, is white, and or. Regardless of



at we put in for not, the resulting sentence will, by all appearances, be true.The result of Bolzano s substitutional test for logical truth depends crucially on the set of terms we decide to hold fixed. Bolzano was well aware of, and indeed welcomed, this dependence, chalking it up to the fact that different terms have different logics. Thus, the sentenceIf Tom knew Carolyn to be a dean then Tom believed Carolyn to be a deanis logically true when we hold fixed the three expressions if-then, knew, and believd ; substituting at will for Tom, Carolyn, and tobe a dean never yields a false sentence. On the other hand, when we consider knew to be a variable term, we get substitution instances likeIf Tom wanted Carolyn to be a dean then Tom believedCarolyn to be a dean.One of these instances will no doubt be false, if not this particular instance (Tom may be prone to wishful thinking) then one that results from further substitutions for the other variable terms. We might take this to indicate that our sentence is a truth of, say, epistemic logic, but not a truth of, say, mere doxastic logic.For any language there will be as many versions of logical truth, as many logics, as there are subsets of the atomic expressions of the language. This is just toview Bolzano s account as providing, instead of a relation between sentences and sets of expressions, the collection of properties that can be obtained from thatrelation by holding con-stant one of its arguments, the set $ of fixed terms. If we settle on the empty set, if we hold no expressions fixed, then in general no sentence will qualify as logically true. At the other end of the scale, allowin

g all atomic expressions into $, we find that logical truth merely reduces to truth. Thus, the sentence Snow is white is logically true if we fix both snow and is ite. This, simply because it is true; if all of a sentence s component expressionsare in there are no permissible substitution instances to worry about.The Violation of PersistenceOn all of these points, Tarski s conception of logical truth coincides with Bolzano s. Tarski argues, though, that the substitutional test de-scribed above should not be considered a sufficient condition for logical truth, but only a necessarycondition. As I have characterized Bolzano s definition, it has an obvious drawback: logical truth depends not only on our selection of $, but on the expressive resources of the language as well.5 This is where Tarski and Bolzano part company.Suppose we were applying Bolzano s definition to a very simple language, one conta

ining two names, say, George Washington and Abe Lincoln ; two predicates, was president and had a beard ; and some truth functional operators, say, or and not. Now, whenonsider 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 Lincolnhad a beard does not. Both of these are in fact true sentences. But in the firstcase, when we substitute the only other available name we get a true sentence, George Washingtonwas president, while in the second case, the same substitution pro-duces a falseone, George Washington had a beard.Of course, the difference here is just a quirk of our language. The world has plenty of people who have never been president. If our meager language had a namefor 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 tru

e on the imagined selection of fixed terms.This example shows that Bolzano s substitutional test is liable to give results 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 trivial substitution of Abe Lincoln for itself, the otherresulting from the substitution of George Washington for Abe Lincoln. But this seems artificially restrictive in light of the fact that, had we simply increased our list of names, the test would obviously have produced opposite results. Thus it happens that Ben Franklin was president does not result 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 thereby would 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 contrac-tions rather than expansions of the language, by considering the con-verse of the problematic case we have encountered. It seems clear that on our ordinary conception, logical truth has at least the following property: if a sentence 5 is not a logical truth of a given language, then neither should it become a logical truth simply by virtue ofthe deletion of expressions not occurring in S. After all, nothing directly relevant to this sentence, to its meaningfulness or its truth, has been changed. If Abe Lincoln was president is not logically true, it should not become so merely through the deletion of an otherwise irrelevant name, Ben Franklin, from the language.If the property of not being logically true should persist through contractionsof the language, the property of being logically true should persist through expansions. This desideratum, which I will call the requirement 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.As we have seen, Bolzano s definition of logical truth fails to meet the requirement of persistence. Tarski s account aims to avoid this defect by appealing to thenotion of the satisfaction of a sententialfunction where Bolzano relies on the c

onsiderably simpler though less powerful notions of truth and substitution.Sentential FunctionsWe can think of a sentence as the limiting case of a sentential function, wherethis latter notion permits variables of appropriate type to take the place of ordinary expressions.6 So, for example, if V is a variable of appropriate type, the linguistic object x was president will be called a sentential 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 contain more than one variable, indeed more than one type of variable; thus x g might be the sentential function that results from allowing g to take the place of was president in x was president. I will say that sentences are just sentential functions that contain no variables.7The notion of a variable should not be confused with that of a variable term. A

variable term is an ordinary expression of the language, one that differs from a fixed term only for the immediate purposes of our test for logical truth. Thus, in the last section we chose ^ to include was president and to exclude Abe Lincoln the former was thereby dubbed a fixed term, the latter a variable term. But neither is a variable. Hence, regardless of our selection of Abe Lincoln was president is a sentence that is, a sentential function that contains no variables.To simplify the transition from Bolzano s definition of logical truth to Tarski s more complicated account, it will help to introduce the notion of a sentential function into the former. We can think of Bolzano s test for logical truth proceeding in the following way. First we introduce a stock of variables for each grammatical category. Next we replace each variable term in sentence 5 with a variable of appro¬priate type, ensuring that multiple occurrences of a term receive the same variable, and distinct terms, distinct variables.

The result of this operation is a sentential function S' containing only expressions that occur in the chosen set of fixed terms. We now consider the collection of substitution instances of S' that is, the collection of sentences that resultfrom S' by placing expressions drawn from appropriate categories back in the variable positions. If every member of this set is true, then 5 is judged logically true with respect to the current selection of fixed terms; if one or more is false, then 5 is not logically true with respect to that selection.According to the present account, the violation of persistence ob-served in thelast section arises from the limited stock of names avail-able to insert for x inthe sentential function x was president. Tarski s account of logical truth allows us



to go beyond the actually available substitution instances of this sentential function. The key concept is, of course, satisfaction. Using it, Tarski bestows some measure of persis¬tence on logical truth.From Substitution to SatisfactionIt is impossible to give a general definition of satisfaction applicable to alllanguages; this for various reasons, not the least of which are the so-called semantic paradoxes. But in simple cases the concept is pretty intuitive. So, for instance, satisfaction is the relation that holds between Abe Lincoln, the person, and the sentential function x was president, but that fails to hold between BenFranklin, the person, and this same sentential function in the first case becauseLincoln was president, in the second because Franklin was not.Let us try to capture this intuitive description in a somewhat more formal setting. For the moment we will confine our attention to senten¬tial functions which, like x was president, contain a single variable standing in a position ordinarily occupied by a name. It will be conve¬nient to assume that our metalanguage contains the object language and hence, in particular, that any sentential function of the object language is also a sentential function of the metalanguage.Let .. x . . . be a schematic placeholder for an arbitrary senten¬tial function ofthe sort described that is, a sentential function con-taining (perhaps multiple) occurrences of a single name variable. We will use \ . . x . . . as a schematic placeholder for a name (in the metalanguage) of that same sentential function, w asa placeholder for any name, and . . . n . . . as a placeholder for the sentence that results from replacing all occurrences of x in the sentential function .. . x ... with the name that replaces w. Using these notational conventions, 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 if. . . n . . .This schema, and the various constraints placed on its instantiation, are stated in the me/ametalanguage. But like Tarski s celebrated T- schema for characterizing the notion of truth, all instances are sen¬tences of the metalanguage. Thus, wefind among the instances(1.1) Abe Lincoln satisfies x was president if and only if Abe Linc oln was presidentand(1.2) Ben Franklin satisfies x was president if and only if Ben Franklin 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 presi¬dent 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 thesuccess of any attempted definition. Schema (1) gives us a clear idea of what arelation, so to speak, must look like before it deserves to be called satisfaction. We will return to this topic in a later section; 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 replacement for w. Furthermore, the sentence that results from inserting this name into the sentential function that is, the sentence that replaces .. . n ... in our schema will also be a sentence of the object

language. Let us introduce . . . n . . . as a placeholder for a name of this sentence. We can now offer a second schema:(2) n satisfies . . . x .. . if and only if . .. n . . . is true in L.This schema is a direct consequence of (1) and Tarski s T-schema.9 The only additional constraint we must place on the instantiation of (2) is that n be replaced by a name actually appearing in the vocabulary of the object language L. For otherwise . .. n . . . would not be a sentence of L, and hence never true in L.Consider again the language that caused problems for Bolzano s account. Since Abe Lincoln is a name in this language, we are allowed the following instantiation of(2):



(2.1) Abe Lincoln satisfies x was president if and only if Abe Lincoln was president is true in L.However, since Ben Franklin is not a name occurring in L, but only a name in our metalanguage, the restriction placed on schema (2) prevents us from taking the further step to(2.2) Ben Franklin satisfies x was president if and only if Ben Franklin was president is true in L.When Bolzano s test for logical truth turned in positive results for Abe Lincoln was president (holding fixed was president ), we la-mented the fact that there was a simple expansion of the language that would provide a false substitution instance for the function x was president. Our two schemata allow us to clarify this hazyintuition. Franklin was never president, and so by (1.2) he does not satisfy the function x was president. This latter fact, along with the presence of schema (2), supports the counterfactual claim that Ben Franklin was president would have been false had Ben Franklin been an object language name with the same meaning it enjoys in the metalanguage. For then we could have carried out the forbidden instantiation of (2) to (2.2).Of course, this all suggests a simple way to circumvent miscarriages of the substitutional test, a way to meet the requirement of persistence while still retaining the spirit of Bolzano s account. The idea is to rule out the logical truth of Abe Lincoln was president simply by virtue of the fact that there is some perhapsunnamed object that fails to satisfy x was president. Then no expansion of L which merely includes a name of this object can affect the logical status of our original sentence. That, in short, is Tarski s strategy for getting around the shortco

mings of the original, substitutional definition. But to make good on this idea, we first have to generalize the notion of satisfaction in two ways, one simpleand one not so simple.Multiple VariablesThe simple generalization is aimed at handling sentential functions with more than one variable. Thus, when we want to test Abe Lincoln was president or George Washington was not president for logical truth (with names the only variable terms), we first convert this to the sentential function x was president or y was notpresident. Any per-missible substitution will here result in a true sentence, since both available names name presidents. But it also happens that any single object we choose will either satisfy x was president or satisfy 'y was not president. Yet there are obvious expansions of the language that would give us false substitution instances of this function, witness Ben Franklin was president or Thomas J

efferson was not president. What we need is an account of the satisfaction relation that captures this intuition, one that allows us to say that Franklin and Jefferson, as a pair and in that order, fail to satisfy x was president or;y was not president.We will say that sentential functions are satisfied by sequences, where a sequence is any function that assigns an object to each of the variables introduced for the purpose of testing logical truth.10 Thus, no se¬quence that assigns Ben Franklin to x and Thomas Jefferson to y will satisfy x was president or y was not president ; on the other hand,sequences that assign a president to V or a nonpresident to y will indeed satisfythis sentential function.In the spirit of our earlier discussion, we can think of sequences as providinga technique for simultaneously entertaining a collection of possible expressions f

or substitution into our sentential function, one for each variable. Rather than consider a general schema, which would be premature at this point, we can see this by employing a sample instantiation:( .1) Sequence/satisfies x was president or;y was not president ifand only if/(V) was president or f( y ) was not president.In ( .1) f names a sequence and /( x ) and '/('/) are complex names of objects, the obje that result from applying sequence / to, respectively, variables x and 'y. xx If language L also contained the names /(V) and /( >T though of course it does not then wd have in addition:( .2) Sequence/satisfies x was president or;y was not president if and only if /(V) w



as president or f( y ) was not president is true in L.In this eventuality the sentence mentioned in the second half of ( .2), /(V) was president or/( y ) was not president, would be a permissible substitution instance of the sentential function x was president or;y was not president. Further, if/assigns Franklin to x and Jefferson to y the substitution instance would be false. Which is to say, Bolzano s substitutional test would have produced negative results had the object language contained a few more aptly chosen names, perhaps Ben Franklin and Thomas Jefferson, perhaps /( x ) and f('y ).This technique works for sentential functions with arbitrarily many variables standing in place of names. Consequently, we can now use the notion of satisfaction to define logical truth with respect to certain choices of Suppose $ containsall the atomic expressions of a lan-guage except perhaps one or more names. In other words, let us assume that any atomic expression which is not a name is a fixed term. Let S' be any sentential function that results from the sentence 5 after we replace all variable terms with variables, ensuring of course that the same variable is used for all occurrences of a given variable term, and that distinct variable terms receive distinct variables. Then we can say that 5 is logically true with respect to ^ just in case S' is satisfied by all sequences.This definition meets the requirement of persistence in the follow¬ing way: If a sentence is logically true (with names the only variable terms), then it will remain logically true even if the lan^ua^e is ex¬panded to include additional names. For regardless of what object the name names, that individual has already been found to satisfy the sentential function in question. In this sense, satisfaction puts at our disposal all possible names tha

t might be incorporated into the lan-guage.On Generalizing SatisfactionAccounting for logical truth in terms of satisfaction avoids certain problems in the substitutional approach, but it encounters some new ones as well. So far the account is not nearly so general as Bolzano s, which allowed atomic expressionsof any grammatical category to be considered variable terms. Thus, if we choose^ to contain Abe Lincoln but not to contain was president, Bolzano s test for the logical truth of Abe Lincoln was president simply involves substituting vari¬ous predicate expressions into the sentential function Abe Lincoln g. This operation is no more problematic than inserting names in the sentential function x was president.We have generalized the notion of satisfaction to the point where we can handlesentential functions with multiple variables, so long as the variables stand proxy for names. Now the not so simple generaliza-tion mentioned earlier must be face

d: explaining the satisfaction of sentential functions that contain variables of arbitrary grammatical type.First a word of motivation. It should be clear that the same intuitions that led us to forsake substitution for satisfaction are at stake even when the variable terms selected include expressions other than names. For example, suppose our language allows only the expres¬sions was president and had a beard to be inserted into the sentential functions Abe Lincoln g and George Washington g. The first of these has all true substitution instances, whereas the second has a true in¬stance, George Washington was president, and a false instance, George Washington had a beard. Consequently, the sentence Abe Lincoln was president is logically true with respect to the fixed terms Abe Lincoln and George Washington, while the sentence George Waington was president is not. Here again something seems amiss, something precisely parallel to the problem solved earlier by invoking satisfaction. There are obv

ious possible expansions of the object language that contain false substitution instances of Abe Lincoln g, for instance any language with the predicate wore a powdered wig interpreted, of course, as it is in English.Bol/.aiio s substitutional account violates the requirement of persis¬tente regardless of what we take to be variable terms: contractions orexpansions of the language can always affect the substitution class of a particular type of variable, whether it stands in place! of a name, a predicate, a sentential connective, or something else. Our simple ac-count of satisfaction allows us to avoid this problem so long as the variable terms are all names; the movefrom substitution to satisfaction thereby bestows some measure of persistence on



logical truth. But since the danger of artificially restricting our substitution class is en¬tirely general, not limited to names, we need to offer a generalized account of satisfaction to handle arbitrary choices of $.So much for motivation; now for the problems. First recall schema (1), which seemed to capture the intuitive characteristic of satisfaction that makes it a natural extension of substitution:(1) n satisfies . . x . . . if and only if.. . n .. .Recall that we require the sentential function to have a single variable in name position, and that w must be replaced with a name (which at least occurs in the metalanguage) of some object or other. A parallel schema for sentential functions with a single predicate variable might look like this:(3) p satisfies . g . . . if and only if. .. p . . .Our first sign of trouble comes when we try to instantiate (3). In instantiating (1) we inserted a name for both occurrences of w ; thus, with (3) we might try replacing p with a predicate:(3.1) Was president satisfies Abe Lincoln g if and only if Abe Lincoln was president.But unlike instantiations of (1), (3.1) is not even a grammatical sen-tence. This may be easy to overlook, especially if we confuse it with the perfectly grammatical sentence(3.2) Was president satisfies Abe Lincoln g if and only if Abe Lincoln was president.But (3.2), although grammatically correct, is not what we are after. Satisfaction is explicitly intended not to be a relation between a linguis¬tic entity (here a

predicate) and a sentential function.12 Alternatively we might try(3.3) Having been president satisfies Abe Lincoln g if and only if Abe Lincoln having been president.Or perhaps(3.4) The set of former presidents satisfies Abe Lincoln g* if and only if AbeLincoln the set of former presidents.Both of these instantiations start out fine, but quickly degenerate into nonsense. The first begins with the name of a property, the property Abe Lincoln has just in case he was once president. The second begins with the name of a set, theset that contains all the individuals, including Abe Lincoln, who once were president. But neither of these names can comfortably occupy the predicate positionin which it later finds itself.When dealing with sentential functions containing variables other than those sta

nding in place of names, we obviously need a more complex schema than (1). Thisis clear from the purely grammatical troubles spawned by (3). But the real problem is not simply finding the right phrasing for a schema, phrasing that produces a collection of tolerably grammatical sentences of the metalanguage. Rather, the problem lies in knowing what exactly we are looking for.Semantic Presuppositions of PersistenceOur ultimate aim is for satisfaction to take the place of substitution in our definition of logical truth, to take its place even when the expres-sions substituted are not names. But satisfaction is a relation, and all relations hold or fail to hold between objects of one sort or other. In our search for a schema withgrammatically proper instances, this was reflected in the fact that the term satisfies must be sandwiched be¬tween two names, which of course is not the case in (3.1). In (3.3) and

(3.4) , on the other hand, we have taken heed that satisfaction is a relationbetween objects, that satisfies must be flanked by names. But it then becomes obscure precisely why such a relation would be thought of as a simple replacement for substitution; the obvious demon¬stration of its relevance, inserting a name of the first object into the sentential function that constitutes the second object, again produces only an ungrammatical string of signs.One thing this exercise demonstrates is that satisfaction is not as innocent anextension of substitution as it might at first seem. Let me explain. Bolzano s test for logical truth requires a division of expres-sions into grammatical categories, basically into groupings whose members can be freely exchanged within sente



nces without risking ungrammatically. Such exchanges often produce sentences that dif¬fer in truth value. This shows that although there may be some similar¬ity running through each category, something that accounts for the endurance of grammaticality through such exchanges, there are also differences. In particular, there are semantic differences, differences in the way members of the same category contribute to the truth value of sentences in which they occur. Thus, we know that the grammatically similar ex¬pressions Abe Lincoln and George Washington do something differ¬ent when they appear in the sentences Abe Lincoln had a beard and George Washington had a beard, simply because the first is true and the second false. Justso we know that the grammatically similar was president and had a beard must contribute differently to truth values of sentences in which they occur, as must the grammatically similar or and and. In each case the evidence is simple and incontrovertible: the presence of pairs of sentences diverging only in the occurrence of these expressions and, of course, in truth value.When we maintain the purely substitutional approach there is no need to provideany account of how the various expressions we classify as grammatically similarcontribute to the truth value of their contain¬ing sentences; we are officially interested only in the end result of that contribution, in the truth or falsity of the sentence. In this way the substitutional definition allows us to keep our semantic theory to a minimum.13 However, as soon as we try to extend the substitutional approach using satisfaction, we are forced to hazard at least a simple theory about the semantic functioning of expressions within a given grammatical category, a theory of how they each contribute, and differ in their contribution,to the truth values of sentences in which they occur.

This change in perspective was easily disguised in the case of names, thanks inlarge part to the apparent simplicity of schema (1). In the last section we described satisfaction as a simple technique for taking into account possible expansions of the list of expressions which, in our object language, fall into the category of names. But we then assumed, without making our assumption explicit, that the range of possible names (for example, Ben Franklin ) that could be incorporated into the object language was determined by the range of objects (for exam¬ple,Ben Franklin) that could be picked out or denoted by a name.But this assumption requires that we see the category of names as held togetherby more than just the grammatical similarity of its mem¬bers. We could certainly introduce an expression say, Nix that we allow to occur in all and only positions thatalso admit Abe Lincoln, but whose contribution to the truth value of a sentence cannot be explained by appeal to the fact that an object named by Nix satisfies a gi

ven sentential function. Perhaps every sentence containing Nix, including Nix was president or Nix was not president, is simply false, with complete disregard for what else might be going on in the sen¬tence. No purely grammatical grounds for rejecting this possible expansion of the object language spring to mind none, at any rate, that do not also threaten the inclusion of Ben Franklin. But the possibility of such a bizarre name does not seem to call into question the logical truth of AbeLincoln was president or Abe Lincoln was not president. Nix would be a name only in grammar.When we used satisfaction to extend Bolzano s account, we assumed that our grammatical category was also a semantic category, that the expansion of the category was constrained not only by the requirement of grammatical interchangeability, but also by the requirement that each member of the category display some common semantic feature. It seems clear that the names Abe Lincoln and George Washington bot

h pick out or name, or denote, or refer to individuals. Further¬more, the fact that these expressions pick out different individuals can alone account for any divergence in truth value among sentences in which they occur, at least in the simple languages we have considered so far. It was this that made it so natural to turnfrom names to objects, to individuals that could have been named by expressionsin the lan¬guage. It seemed obvious that for each such individual our substitution class now taken to be a semantic category could have been appro¬priately extended. On the other hand, the possible expansion of our substitution class to include anexpression that behaves like Nix is ruled out by the move to satisfaction. This hardly seems an objection¬able bias.



Well-Behaved Expansion and Satisfaction DomainsLet us now return to the problem of generalizing the notion of satisfac¬tion to arbitrary sentential functions. Satisfaction must still be a rela¬tion between objects of some sort and sentential functions (which are also, of course, a type of object). The difficulty we encountered with schema (3) arises because we are nowdealing with expressions not naturally thought of as names, whose contribution to the truth value of a sentence is not easily reduced to the simple naming of an individ¬ual. Consequently, it is not obvious how to extend satisfaction to the newbreed of sentential function. In particular, it is not obvious what sort of object, if any, might stand in the satisfaction relation to these sentential functions.Let us call the class of individuals, things that could have been picked out bynames, the name domain of the satisfaction relation. Intuitively, this is the collection of objects that stand in the satisfaction relation to some sentential function displaying a single name variable. Our prob¬lem is now to specify the predicate domain of the satisfaction relation, the class of objects that can satisfy sentential functions which contain a single predicate variable. But more important, if our account of logical truth is to achieve a generality that approachesthat of Bolzano s, we need a fairly clear idea of what should guide us in choosing a satisfac¬tion domain regardless of the type of variable appearing in the senten¬tial function.The most recent considerations suggest the required perspective on satisfaction. Our aim is to provide a notion of logical truth that persists through expansions of the language. But the considerations of the last section make it clear that

our concern with persistence does not extend to all conceivable expansions of the language, to all conceivable altera¬tions in Bolzano s substitution classes. After all, any grammatical cate¬gory could be expanded to include a semantically ill-behaved expres¬sion like Nix. Thus, for any sentence that is logically true and contains at least one variable term, there will be a possible expansion of the language in which it fails Bolzano s test, in which it is not logically true with respect to the same choice of constant terms.But such possibilities are not the sort that led us to impose the requirement of persistence. Rather, we were concerned with perfectly well-behaved expansions of the language, with the introduction of expressions whose semantic behavior seemed no more different from that of present members of the substitution class than the behavior of the present members differed from one to another. The possibility of adding a name like Ben Franklin is quite another thing from the possibility

of adding a name like Nix.Abe Lincoln and George Washington both stand in a particular relation to two members of the name domain of the satisfaction rela-tion, Lincoln and Washington; these individuals are named by the ex-pressions. The remainder of the domain comprises all individuals that, intuitively, could have stood in that same relation to other expressions, and hence to expressions that contribute to the truth value of a sen¬tence in a fashion similar to Abe Lincoln and George Washington. Thus, if a given sentential function is satisfied by all of these individ¬uals, no semanticallywell-behaved expansion of the category of names will provide a false substitution instance of that sentential function.The other expressions of our language was president, or, and so on do not name objects;only names, so to speak, name.14 But it seems equally clear that the category of, say, predicates admits of seman¬tically well-behaved expansions, just like the c

ategory of names. Obvi¬ously the inclusion of wore a powdered wig should be permitted, while the inclusion of nixes, the predicate analogue of Nix, should not. The onlyquestion is whether the notion of satisfaction offers a technique for clarifying this intuition, for distinguishing appropriate from inappropriate expansions of the category of predicates.With a simple language like the one we have defined, it clearly does. In fact there are several ways to circumscribe the new domain. Perhaps the most intuitiveis to take the predicate domain of the satisfactionrelation to contain properties for example, having been president, hav¬ing had a beard, having worn a powdered wig, and so forth. This does not commit us to the cla



im that predicates name properties, only to the claim that expansions of the category of predicates are constrained by the availability of properties.15 The underlying idea is that the pre¬dicates in our language contribute to the truth value of sentences by asserting, of some object, that it possesses a particular property. Thus, for any given property, we could appropriately expand the category of predicates to include one which asserts possession of that property, just as the category of names can be expanded, for any given individual, to include one which names that individual.We can now see exactly why schema (3) caused so many problems not encountered with schema (1). When we are dealing with sentential functions containing a single predicate variable, we will need a schema of the following sort:(3') P satisfies . . . g . . . if and only if.. . p .. .One of the conditions governing the instantiation of (3') will have torun as follows: p must be replaced by an expression that assertspossession of the property named by the expression that replaces P. Thus, the following would be a proper instantiation of (3'):(3.5) Having been president satisfies Abe Lincoln g* if and only if Abe Lincoln was president.Now consider the following alteration of schema (1):(1') N satisfies . .. x . . if and only if. . . n .. .This restatement is precisely parallel to (3'), and would require a similar condition to govern instantiations of AT and w. However, since names do not assert possession of properties, but rather name individuals, the condition would now run: n must be replaced by an expression that names the individual named by the express

ion that replaces N. But of course every name names the individual named by itself So our restriction can be built directly into the schema by changing N to w which ocourse yields (1) and demanding that both occurrences of w simply be replaced by a single name.We are prevented from making a similar simplification of (3') since no predicate asserts possession of a property named by itself, and likewise, no name asserts possession of a property named by itself. This merely because predicates do not name, and names do not assert possession of, properties. But this does not indicate that satisfaction is any less natural an extension of substitution in the case of predicates than it is in the case of names. It indicates only that namesand pre¬dicates contribute differently, in both the object language and the metalanguage, to the truth values of sentences in which they occur. But in both casesthe move to satisfaction represents an attempt to isolate that contribution and

to extrapolate the way in which further expressions of a similar type might function.I remarked that with our simple language there are several ways to delineate the predicate domain of the satisfaction relation.16 I sug-gested populating this domain with properties, since the resulting account of the semantic functioning of predicates seems intuitively appealing. Intuition aside, this move is not generally adopted. Rather, it is standard to take this domain to consist of sets ofindividuals drawn from the name domain. We can then think of predicates as asserting, of some individual, that it is a member of a given set. Once again we need not claim that predicates name sets, only that expansion of the category of predicates is constrained by set availability.The present language gives us no reason to prefer one of these options over theother; there are no sentences in which the contribu-tion of predicates is not eq

ually well explained either as the assertion of set membership or as the assertion of property possession. But this is not to say that the two are equivalent. There may on the one hand be sets that do not correspond to any property, or on the other hand multiple properties shared by all (and only) members of a single set. So in either case the possible expansions of the category of predicates will be differently circumscribed.There are two remaining categories whose satisfaction domains we must specify: the category containing or (which I will call a sentential connective) and the category containing not (which I will call a sentential operator). For simplicity, wecan take sentential functions with a single connective variable to be satisfied



by binary truth functions, and those with a single operator variable to be satisfied by unary truth functions. Again, there is no need to say that sentential connectives and operators name truth functions, only that there is a fixed relation that holds between each of them and some member of the appropriate satisfac¬tion domain. I will say connectives and operators express truth func¬tions. Thus, taking c to be a connective variable, the sentential func¬tion Abe Lincoln was presidentc George Washington had a beard is satisfied by the truth function expressed by or, though by neither the truth function expressed by and nor the truth function expressed by nor.Clearly, our choice of domains here severely restricts the possible expansions of these two categories. According to the present account, there are only sixteen possible sentential connectives and four possible sentential operators. The category of operators could not, for exam- pie, be expanded to include necessarily as it is ordinarily understood. Although this term may be grammatically similar to not, its contribu¬tion to the truth value of sentences in which it occurs cannot be reduced to the expression of a unary truth function. Our decision thus treats necessarily with the same disdain earlier afforded Nix ; both are, from the present perspective, semantically ill-behaved.We can give schemata parallel to (1') and (3') that characterize satisfaction for sentential functions with single connective or operator variables. Thus, for the former we will have:(4) B satisfies . . . c . . . if and only if. . . b . . .Here we require that the expression replacing b must express the binary truth function named by the expression that replaces

Finally, for sentential functions with single operator variables, we will use the following schema:(5) U satisfies . . . 0 . . if and only if. . . u . . . ,requiring that the expression replacing u must express the unary truth function named by the expression replacing U.A Persistent Account of Logical TruthOur technique for extending the notion of satisfaction to sentential functions with an arbitrary number of variables is again to employ sequences. But now our sentential functions may also contain variables of arbitrary type. Say that a sequence is any function that assigns to each variable an object from the appropriate satisfaction domain. Let S(x, g, c, 0) be a schematic placeholder for any sentential function all of whose name variables are among xi, ... , x*; all of whosepredicate variables are among g\, ... ,gh\ all of whose connective variables are

among c\, ... ,Ck\ and all of whose operator variables are among oi, ... , Ok-Let \S(x, g, c, 0) stand for a name of that sentential function, and finally, let(x/w, glp, c/l, o/m) be the result of uni¬formly replacing variable x,- with expression niy gi withp,, c, with biy and Oi with Ui (for 0 < i < A), wherever they occur in that sentential function. For any given sequence/, we require that w, name the individual/(x,), that pi assert possession of the property/(gv), that bi express the binary truth function /(c,-), and that w, express the unary truth function/(o,). We then have:(6) Sequence/satisfies S(x, g, c, o) if and only ifS(x/n, g/fi, ell, o/u).If a given sequence/assigns Ben Franklin to xi, the property of having worn a powdered wig to and the truth function expressed by and to ci, then the following are sample instantiations of (6):

(6.1) Sequence / satisfies xi g\ if and only if Ben Franklin wore a powdered wig.(6.2) Sequence/ satisfies xi was president c\ Abe Lincoln g{ if and only if Ben Franklin was president and Abe Lincoln wore a powdered wig.When, for a given sequence/, we also have available object language expressionswi, . . . , w*; pi, .. . ,p*; b\, .. . , bk\ and u\,... , m*which meet the above conditions on naming, assertion, and expres-sion, we can offer the following analogue of schema (2):(7) Sequence/satisfies \S(x, g, c, o) if and only if\S(x/w, g/p, c/l, o/u)' is true in L.



Thus if/is a sequence that assigns George Washington to xi and the property of having been president to gi, we get:(7.1) Sequence/satisfies xi g{ if and only if George Washington was president istrue in L.Together (6) and (7), like (1) and (2) before them, demonstrate the connection between satisfaction and substitution. Satisfaction is just an extension though not as simple an extension as it first appeared of substitution. It allows us to extend our various substitution classes to include expressions from any semantically well-behaved expansion of the language. An expansion is well-behaved just in case any new mem¬ber of a given category of expressions stands in the specified relation to an object in the appropriate satisfaction domain. In the case of our present language we allow new names if they name individuals, new predicates if they assert possession of properties, new connectives and operators if they express appropriate truth functions. Our upcoming definition of logical truth will thus meet the requirement of persis¬tence, with the implicit qualification we have all along been assuming', logical truth will be persistent through semantically well-behaved expansions of the language.Before applying the generalized notion of satisfaction to the defini-tion of logical truth, it should again be emphasized that we have not given a definition of satisfaction, either of the general notion, which resists definition in principle, or even of satisfaction for sentential functions of our current object language. Instances of schema (6) can be taken only as adequacy conditions that constrain the formal defini-tion of satisfaction for the present language. A formal definition of satisfacti

on for arbitrary sentential functions of the language would proceed by a simplerecursion on the set of sentential functions.Once we have access to a definition of satisfaction for arbitrary sentential functions of a particular language, we can give the following definition of logical truth. Let S' be any sentential function that results from uniformly replacingall atomic expressions in 5, other than mem¬bers of $, with variables of appropriate type. Then we will say that 5 is logically true with respect to $ just in case S' is satisfied by all sequences. This is Tarski s definition of logical truth.Logical ConsequenceHow should we define logical consequence? One route that might seem attractive is a simple reduction of this notion to that of logical truth. Certainly, if a sentence 5 is a logical consequence of a set of sentences K = {K\, . . . , Kr), then the single conditional sentence whose antecedent is the conjunction of the me

mbers of K, and whose consequent is S, must be logically true. That is, 5 will be a logical consequence of K if and only if the sentenceIf K\ and . . . and Kn, then 5is logically true. Given our definition of logical truth, it might seem naturalto rally this observation into an account of the consequence relation.There are three problems with this idea, not overwhelming, but still significant. First, we would have to assume that the language we are dealing with containsthe expressions and and if .. . then, or the equivalent, and this assumption would restrict the applicability of the account.17 Second, we would have to assume that these expressions are always included in $, and this again restricts the generality of the suggested definition.18 Finally, and most important, the reduction will work only if K is finite, or, alternatively, if the language allows infinitely long sentences. For otherwise we could never form the antecedent of our condi

tional sentence.For these reasons, neither Bolzano nor Tarski tries to reduce the notion of logical consequence to that of logical truth. But their defini-tions of consequenceare, not surprisingly, quite similar, Tarski s being a simple emendation of Bolzano s. We can describe them quite suc¬cinctly.Say that an inference or argument of a language L is any ordered pair {K, S) inwhich S is a sentence of L, and K a set of sentences of L. An expression will he .said to occur in an argument (K, S) if it occurs either in 5 or in some member of K; we will call the argument truth preserving just in case either 5 is trueor some member of K is false. So, for instance, any argument whose conclusion (t



hat is, S) is the English sentence Abe Lincoln was president will be truth preserving, as will any argument with the sentence George Washington had a beard among its premises (that is, in K). This simply because the first sen¬tence is true and the second false.According to Bolzano, an argument (K, S) is logically valid with re-spect to a selection of fixed terms just in case it is truth preserving and every argument (K't S') that results from making one or more permissible substitutions for expressions occurring in (K, S) is also truth preserving. A permissible substitutionis defined in the obvious way: all members of $ must be left untouched, and thereplacement of variable expressions must be uniform throughout the argument. A sentence 5 is a logical consequence, with respect to $, of a set of sentences K just in case the argument (K, S) is logically valid with respect to $.As Bolzano defines it, the logical consequence relation, like the property of logical truth, depends crucially on our selection of fixed terms. Just as every true sentence can be rendered logically true by including all its atomic expressions in $, so too every truth-preserving argument becomes logically valid when wefix all of its component terms. Obviously any argument that concludes with the true sentence Abe Lincoln was president will be logically valid when $ contains each expression appearing in the argument; such arguments will in fact be logically valid on any selection of ^ that includes both Abe Lincoln and was president. Since all permitted substitution instances share the same conclusion, the continued truth of that sentence ensures the truth preservation of those instances. On theother hand, if $ is empty, if we hold no terms fixed, then in general the only valid arguments will be those in which the same sentence appears both as premise

and conclusion that is, where 5 is a member of K.According to the present account, the logical consequence relation is not persistent. Whether 5 is a logical consequence of K does not depend only on our choice of fixed terms; it can also be affected by the size of the substitution classes for the variable terms. In particular, the relation will not persist through semantically well-behaved expansions of the language, although our choice of $ remains constant. Thus, when we fix the atomic predicates in our previous language, the sen¬tence Abe Lincoln was president is a logical consequence of any set containing the sentence Abe Lincoln had a beard ; this due to our omission of names for bearded nonpresidents. Had we merely in¬cluded the expression Robert E. Lee, interpreted, as in English, to denote the Confederate gentleman, Lincoln s presidency would not have turned up a consequence of his having a beard.Tarski s definition employs satisfaction, thereby ensuring that the consequence re

lation will persist through semantically well-behaved expansions of the language. Let us take an inferential function (or, better, an argument form) to be any ordered pair whose first member is a set of sentential functions and whose second member is a single sen¬tential function. Thus, an argument is just an argument form in which no (free) variables occur. We will say that an argument form (K', S') is satisfaction preserving on sequence f ]\\st in case/either satisfies S' ordoes not satisfy some member of K'.Suppose now that (K',S') is an argument form that results from uniformly replacing all atomic expressions in argument (K, S), other than members of $, with variables of appropriate type. Then we will say that (K, S) is logically valid withrespect to $ just in case (K', S') is satisfaction preserving on all sequences.Finally, sentence 5 is a logical consequence, with respect to $, of set K if the corresponding argument (K, S) is logically valid with respect to $. This is Tar

ski s definition of logical consequence.By replacing truth preservation with satisfaction preservation, we avoid the violation of persistence noted two paragraphs back. Once we have specified the class of well-behaved expansions of the language that is, once we have chosen satisfaction domains and defined the satisfaction relation for arbitrary sentential functions we are assured that any argument judged logically valid will remain so throughout those expanded versions of the language. In this sense Tarski s defini¬tion of logical consequence, like that of logical truth, successfully meets the demand for persistence.Recapitulation



Tarski s goal is to provide an analysis of the notions of logical truth and logical validity, to provide definitions that are, as he puts it, close in essentials to the common concepts. To this end, he develops an account that refines the substitutional definitions first proposed by Bolzano. He notes that the substitutionaltests must be demoted from the status of necessary and sufficient conditions tomere necessary conditions; to achieve persistence, the limitations encountered with actual substitution classes must be overcome.The idea behind Tarski s solution is simple. If a given sentential function is satisfied by all sequences, then naturally all its permissible substitution instances will be true.19 But of course the converse of this docs not always hold: 11 sentential function may survive the substitu¬tional test, though not be satisfied by certain sequences. Thus, no sentence (or argument) can pass Tarski s more stringent test without passing Bolzano s as well, and where the tests produce differentre¬sults, the problem will invariably lie in the limited resources available in one or more of the original substitution classes. So any divergence marks a potential failure of persistence for Bolzano s account.Now Tarski s solution, though simple in conception, may not be so simple in execution. The new complexity is an immediate consequence of the concern over persistence: the goal of achieving a persistent account of the logical properties makesno sense except in the context of a theory of (or assumptions about) how existing members of a category contribute, and how potential members could contribute,to the truth values of sentences in which they occur. The required ac¬count of satisfaction must provide such a theory, both to give precise (and plausible) sense to the demand for persistence, and of course to give us resources with which to

meet that demand.In arriving at a definition of satisfaction for a sufficiently broad class of sentential functions, we attribute to each expression classed as a variable term a specific semantic function naming an individual, asserting possession of a property, expressing a truth function, and so forth. By populating a satisfaction domain with the appropriate type of object individuals, properties, truth functions we take a stand on how the existing category might be expanded: we condone new members so long as their semantic contribution, their contribution to the truth value of sentences, can be charted in a fashion similar to that of the present members. Thus, any expression that names an individ-ual is treated as a potential member of the category containing Abe Lincoln, any expression that asserts possessionof a property may belong to the category containing was president, and any expression that expresses either a unary or a binary truth function is admitted into th

e category containing either not or or. We will eventually see how certain other semantic categories, specifically quantifiers, can be han¬dled within this same framework.Once we have an account of satisfaction, Tarski s definitions run as follows: 5 is logically true if and only if S' is satisfied by all sequences (where S' results from 5 by replacing all atomic expressions, except those in $, by variables).Similarly, 5 is a logical consequence of K if and only if every sequence eithersatisfies S' or fails to satisfy some member of K'. Note that, given this latter definition, logical truth can be seen as a reduced form of logical consequence: 5 will be logically true just in case it is a consequence of the empty set, or, alternatively, if it is a consequence of any set of sentences whatsoever.4Interpretational Semantics

In Chapter 1, I remarked that the standard, model-theoretic defini¬tion of consequence is an outgrowth of Tarski s account. I will begin this chapter by explaininghow, upon making some minor ad¬justments, the direct application of Tarski s account gives way to a recognizable model-theoretic semantics that is, to the characteri¬zation of a relation, "x is true in y, holding between sentences and models. However, the conception of model-theoretic semantics that emerges is strikingly different from that presupposed in the represen¬tational approach sketched in Chapter 2. For reasons that will become obvious, I adopt the term interpretational semantics for the Tarskian conception of model-theoretic semantics.Interpretational and representational semantics occasionally inter¬sect. That is,



we sometimes find that one and the same model-theoretic semantics can be viewedfrom either the interpretational or the repre¬sentational perspective. I will discuss a couple of points of intersection: one is the simple semantics devised forthe language of Chapter 2, the other a slightly more intricate semantics for the language of Chapter 3. But in spite of the occasional intersection, interpretational and repre¬sentational semantics are radically different approaches to semantics, approaches whose adequacy must be judged by completely different standards. In Chapter 2, I sketched the standards applied to a repre¬sentational semantics;as we will see, the counterparts for interpreta¬tional semantics are simply the criteria already discussed for delineat¬ing satisfaction domains and for defining satisfaction. For in an interpretational semantics our class of models is determined by the chosen satisfaction domains; our definition of truth in a model is a simple variant of satisfaction.Distinguished Sentential FunctionsHow do we get to model-theoretic semantics from Tarski s account of the logical properties? The steps are by and large just minor modifica¬tions of the definitionsdescribed in the last chapter. Unfortunately the end result of these modifications is a certain blurring of the careful distinction Tarski draws between sentences and sentential functions, between the ordinary expressions of the language and the variables we introduced for defining the logical properties.In considering the following changes, it will be convenient to assume we are interested in logical truth and logical consequence only with respect to a. particular selection ^ of fixed terms. When speaking of our sample language from Chapter 3,1 assume that ^ is the set containing the atomic expressions or and not. In this

way we avoid repeated mention of ^ and can simply speak of the fixed terms andthe variable terms of the language. But it is important to keep in mind the (hence¬forth implicit) relativization to our choice of In particular, it is crucial to remember that variable terms are not variables. Variable terms are ordinary atomic expressions of the language, differing from fixed terms only in their omission fromRecall that in testing a sentence 5 for logical truth, we first convert 5 to a sentential functional S'. S' results from uniformly replacing vari¬able terms withvariables of appropriate type. It does not matter which variables we choose in constructing S' so long as distinct variable terms receive distinct variables, and multiple occurrences of a single variable term are converted to multiple occurrences of a single variable. So, for example, the sentence Abe Lincoln was president corresponds quite indiscriminately to the sentential functions xi gi, xi g2, lX

g2> and so forth.The first modification we will make ensures that a specific sentential function5* corresponds to each sentence 5 in the language. To do this we need only assign a specific variable, once and for all, to each variable term different variables, of course, to different terms. Thus, we might take xi to be the variable corresponding to Abe Lincoln, X2 to be the variable corresponding to George Washington, he variable corresponding to was president, g2 the variable corresponding to had a beard. We need not choose special variables for or and not, the remaining atomic expressions, since at present they are both mem¬bers of ft.We can now take the distinguished sentential function 5* correspond¬ing to 5 to be the result of replacing each variable term in 5 with its assigned variable. So, for example, xi g2 or X2 gi is the distinguished sentential function corresponding to Abe Lincoln had a beard or George Washington was president.

This change allows us to simplify our sequences considerably. Cur-rently a sequence assigns objects from appropriate satisfaction do-mains to many variables that never appear in the new, distinguished sentential functions. But it can hardly make any difference what is assigned to, say, predicate variable when we know that any distin-guished sentential function 5* contains occurrences of at most lg\ and g2- So without modifying our account of satisfaction, we can simply take the domain of a sequence to be limited to the chosen variables that is, to the variables assigned to specific variable terms in the lan-guage. For the present language, such a limited sequence/* will be any function that assigns members of thename domain to x^ and x2, and members of the predicate domain to lg{ and lg2. Clearl



these simpler sequences suffice for our current needs.Tarski s test for logical truth can now be characterized in the follow¬ing way: we convert a sentence 5 to the distinguished sentential func¬tion 5* that results from replacing each variable term with its assigned variable. We then run through our new, pruned down sequences to see whether they all satisfy 5*. If so, 5 is logically true; if not, not. The test for logical validity proceeds similarly. First we replace an argu¬ment (K, S) with its distinguished argument form (K*, S*) thatis, the result of replacing all variable expressions occurring in (K, S) with their chosen variables. We then check to see that (K*, S*) is satisfaction preserving on all limited sequences/*. If so, 5 is a logical consequence of K; if not,not.D-Sequences and D-SatisfactionWe are now halfway to model-theoretic semantics. The remaining change is equally slight, though potentially more confusing. Since we have set up a one-to-one correspondence between variable terms and variables, and between sentences and sentential functions, there is a way to achieve the same results as our present tests without bothering to detour through variables and sentential functions. The new method will yield a recognizable model-theoretic semantics for our language.First we must introduce a new type of sequence, one whose domain consists of the variable terms of the language rather than the chosen variables. Let us say that a direct or d-sequence is any function that assigns to each variable term an object from the appropriate satisfaction do¬main. Thus, a d-sequence will assign Ben Franklin directly to the ex¬pression Abe Lincoln, whereas a limited sequence assigns Franklin to '*1, the variable chosen to correspond to Abe Lincoln.

For any d-sequence f, let /* be the corresponding limited se¬quence that is, the function that assigns the same object to a chosen variable (for example, xi, lg\) as/assigns to the corresponding vari¬able term ( Abe Lincoln, was president ). We can now introduce a relation, parallel to satisfaction, which holds between d-sequences and sentences. Specifically, say that a d-sequence f d-satisfies sentence 5 if and only if the corresponding limited sequence/* satisfies the distin-guished sentential function 5*.Although d-satisfaction is defined in terms of satisfaction, it is im-portant not to confuse the two notions. For one thing, if we briefly reflect on schema (6) of the last chapter, it will be clear that sentences, sentential functions with no variables, are only trivially satisfied or not satisfied by sequences. A true sentence is satisfied by all sequences, while no sequence satisfies a false sentence. Thus, for any limited sequence/* we have the following instantiation of

(6):(6.2) Sequence/* satisfies Abe Lincoln was president if and only if Abe Lincolnwas president.Since Lincoln was president, every sequence satisfies Abe Lincoln was president ; had he not been, no sequence would.Now suppose that / is a d-sequence that assigns Franklin to Abe Lincoln and the property of having worn a powdered wig to was president. If /* is the correspondinglimited sequence, we will have the following instantiation of (6):Sequence/* satisfies xi g{ if and only if Ben Franklin wore a powdered wig.Since xi gV is the distinguished sentential function corresponding to Abe Lincolnwas president, our definition of d-satisfaction gives us:D-sequence/d-satisfies Abe Lincoln was president if and only if sequence/* satisfies x! g|.

And from these we get:(6.2D) D-sequence/d-satisfies Abe Lincoln was president if and only if Ben Franklin wore a powdered wig.The comparison of (6.2) and (6.2D) points up the difference be¬tween satisfactionand d-satisfaction. In (6.2) the makeup of sequence /* is quite immaterial, since the sentential function Abe Lincoln was president has no variables; it is simply a true sentence. But it is clear from the derivation of (6.2D) that d-sequences do not trivially d-satisfy true sentences, nor will they trivially fail to d-satisfy false sentences.In effect, d-satisfaction tells us whether a sentence would have been true had i



ts variable terms been interpreted in accord with the assign-ments of the d-sequence. In fact Abe Lincoln was president is a true sentence. But had Abe Lincoln namd Ben Franklin and had was president meant wore a powdered wig, then this sentence would have been true just in case Ben Franklin wore a powdered wig. Since Franklin did not, as a matter of fact, wear powdered wigs, this sentence would have been false on the interpretation suggested by d-sequence fWe have now, of course, arrived at model-theoretic semantics, though our ungainly terminology could stand some revision. But be-fore making the final, terminological change, let us note how Tarski s definitions of logical truth and logical consequence survive the altera-tions already in place. It is a trivial consequence of the former defini-tion and our present account of d-satisfaction that a sentence is logi-cally true just in case it is d-satisfied by every d-sequence. Just so, an argument is logically valid, its conclusion a logical consequence of its premises, if and only if it is d-satisfaction preserving on all d-sequences. There is now no need to move to sentential functions or argument forms to apply Tarski s definitions.Our final terminological change will be this: replace d-sequence with model, and the phrase is d-satisfied by with is true in. Thus, a sentence will be logically true if and only if true in all models, and an argument logically valid just in case it is truth preserving in all models.1Semantically Well-Behaved ReinterpretationConsider for a moment the nature of d-sequences, or of models, as we are presently calling them. In Chapter 3, we saw how the technique of satisfaction is meant to extend Bolzano s substitutional tests for logical truth and logical validity,

our stock of sequences allowing considera¬tion of all semantically well-behaved expansions of the various substi¬tution classes. The technique of d-satisfaction, truth in models, em¬bodies precisely the same extension of the substitutional account, though the style of the tests is slightly modified. In particular, no syntactic manipulations of the sentences or arguments being tested, no exchanges of variables for variable terms, are now required.We can think of the new technique in various ways. For example, we can obviously consider it a simple abbreviation, somewhat confusing perhaps, of Tarski s original method. As such, we must imagine the variable terms of the language doing double duty: on the one hand, they act as ordinary expressions of the language, taking part in genu-ine sentences whose logical properties we hope to reveal. But when it comes time to test for logical truth and logical validity, the variable terms also act as variables, their replacement by actual variables now rendered s

uperfluous thanks to the slight technical modifications de-scribed in the last section. If expressions like Abe Lincoln are just considered odd-looking variables, then d-sequences are simply se-quences and d-satisfaction simply satisfaction.There is another view of the model-theoretic technique that is con-siderably more natural, and equally faithful to the basic idea of Tarski s definitions. As I suggested above, we can think of a d- sequence as providing a possible reinterpretation of the variable terms in our language, of the atomic expressions not currently being held fixed. From this perspective, our model theory provides a characterization of x is true in L for a limited range of languages L. Thus, the class of d-sequences, or models, does not encompass all conceivable reinterpre¬tations of the variable terms, but instead encompasses all semantically well-behaved reinterpretations. No model suggests that Abe Lincoln might have contributed to the truth value of sentences in a manner akin to Nix. Rather, since Abe Lincoln presently c

ontributes by naming an individual, the permissible reinterpretations of this expres¬sion are taken to be constrained by the availability of nameable individuals, by the name domain of the satisfaction relation. Similarly, permissible reinterpretations of predicates are limited by the predicate domain. And of course had or and not been left out of the set $ of fixed terms, their range of interpretation would be constrained by the connective and operator domains, respectively.Among these interpretations we will find what is called the intended interpretation. For our language, the intended interpretation is the model that assigns Abe Lincoln to Abe Lincoln, having been presi-dent to was president, and so forth. This is simply the trivial reinter¬pretation of the variable terms, the interpretation i



n which all expres¬sions of the language mean what they actually mean. If our class of models omitted this assignment, we could not be sure that a logically truesentence was not actually false that is, false when the variable expressions are interpreted in the normal way. Similarly, if the in¬tended interpretation were notincluded in the test, we would have no general assurance that logically valid arguments in fact preserve truth.Obviously it makes no real difference whether we see models as interpretations of our language, or whether we simply view variable terms as variables of convenience, with models cast as ordinary assignments to these not-so-ordinary variables. The difference is purely heuristic. Either way, I will call the present conception of model-theoretic semantics the Tarskian or interpretational view. Accord¬ing to it, our models are meant to range over all semantically well- behaved interpretations of some subset of the expressions in the lan¬guage. Let us now contrastthe interpretational perspective with the representational view described in Chapter 2.Samples of the Contrasting ViewsThe best way to emphasize the contrast between the interpretational and representational views is to consider specific examples. In Chapter 2, I sketched a simple representational semantics for a language con¬taining five atomic expressions:three were sentences ( Snow is white, Roses are red, and Violets are blue ), one a connctive ( or ), and one an operator ( not ). Obviously we could devise a more finely grained grammatical analysis of this simple language, but it will be an instruc¬tive exercise to devise a Tarskian semantics while retaining the coarser parsing.Let us begin the old way, introducing variables for each type of atomic expressi

on. For sentences we will use pi, p2> ; for connec¬tives 'cu C2, ;for operators 01, 02. The next step is tospecify satisfaction domains for the various types of variables. As we saw in the Chapter 3, this requires that we hazard a simple theory of how existing members of a category contribute, and differ in their contribution, to the truth values of sentences in which they occur. We can again take connectives and operatorsto express appropriate truth functions, and construct the respective satisfaction domains accord¬ingly. Thus, it remains to settle on the sentence domain of the satisfaction relation.As before, we will opt for the simplest plausible satisfaction domain. In the present language, we can explain the semantic contribution of any embedded sentence to its embedding sentence in one of two ways: either the component sentence says something true or it says something false. Thus, we can take the sentence dom

ain to consist of the two truth values, true and false. Again, there is no reason to say sentences name truth values, any more than predicates name properties.I will simply say they have truth values; Snow is white has the value true because it says something true specifically, that snow is white.Sequences will, of course, be functions that assign truth values to sentence variables, binary truth functions to connective variables, and unary truth functions to operator variables. The analogue of schema (6) for the present language will then run as follows:(8) Sequence/satisfies S(p, c, o) if and only if S(p/s, ell, o/u).Recall that S(p, c, 0) is to be replaced by the name of an arbitrary sentential function of the language, and S(p/s, c/l, o/u) is to be replaced by a sentence thatresults from inserting appropriate expres¬sions of the metalanguage for variablesof the sentential function. The appropriateness of the replacement expressions w

ill now be deter¬mined by the following instantiation conditions: sentence s, must have the truth value /(/>;); connective ft, must express the binary truth function f (Ci); and operator w, must express the unary truth function f (Oi). Thus,if/ assigns false to pi and the truth function expressed by and to 'cu we would have the following instantiation of (8):(8.1) Sequence/ satisfies Snow is white ci pi if and only if snow iswhite and George Washington had a beard.Here again, the relationship between satisfaction and the substi-tution of possible expressions is obvious: George Washington had a beard is not a sentence of the present language, nor is and a connec¬tive of the language. But had they been, the se



ntence Snow is white and George Washington had a beard would have been a false substi¬tution instance of the sentential function Snow is white ci piNaturally, in constructing (8.1) we could have chosen sentences of the metalanguage other than George Washington had a beard, so long as the chosen sentence saidsomething false that is, actually had the truth value false. The situation is similar with and : any meta¬language connective that expresses the same truth function would have met the requisite instantiation condition; thus, moreover might have been used in lieu of and.Let us now convert to d-sequences and d-satisfaction. Suppose we are again interested only in logical truth and logical validity with respect to the expressions or and not. In other words, our three atomic sentences will be considered the onlyvariable terms. For any sentence 5 we will let S* be the sentential function that results from uniformly replacing Snow is white with pi, Roses are red with ands are blue with />3. A d-sequence (model) will be any function whose domain is theset of atomic sentences and which assigns a truth value, a member of the sentence domain of the satisfaction relation, to each of those sentences. D-satisfaction (truth in a model) will of course be defined as before: d-sequence / d-satisfies sentence 5 if and only if the corresponding limited sequence /* satisfies the distin-guished sentential function 5*.Notice that there are precisely eight d-sequences, or models. Fur-ther, these models happen to be exactly the same eight functions we introduced for our representational semantics in Chapter 2: there is a model that assigns true to all three atomic sentences, one that assigns false to all three, and various models that assign the remaining combi¬nations of values.

But of course, according to the Tarskian view these models are not meant to represent possible configurations of the world, as the repre-sentational view wouldhave it; rather, they are meant to canvass semantically well-behaved reinterpretations of the atomic sentences of the language. Thus, at present the sentence Snow is white says some¬thing true. Yet it could have been provided with another interpre¬tation, an interpretation in which it said something false say, that Washingtonhad a beard. The d-sequences that assign false to this sen¬tence are meant to take account of these possible reinterpretations. Complex sentences which are d-satisfied by every d-sequence that is, sentences which are true in all models are just those whose truth would survive any semantically well-behaved reinterpretation of the atomic sentences of the language.We have finally arrived at the alternate view of truth tables described n Chapter 2. And we can emphasize the difference between our interpretational and repres

entational semantics much as we did the difference between the two perspectivestaken on our theory of truth in a row. According to both views, our model theory supports certain counterfactual claims about the truth values of sentences in the lan¬guage. But the counterfactual claims that emerge are strikingly differ¬ent. Thus, from the representational perspective our semantic theory supports the claimthat the sentence Snow is white or snow is not white would have been true even ifsnow had not been white; that contin¬gency is, after all, depicted by various of our models. However, from the interpretational perspective no claim is made about what would have happened to the truth value of our sentence had snow not been white. Rather, our theory supports the quite different claim that this sentence would still be true even if the component expression Snow is white were somehow reinterpreted, perhaps given the interpretation pres¬ently enjoyed by the English sentence George Washington had a beard.

Now consider the model theory constructed for our second lan-guage. There, too,we held fixed the interpretations of or and not, and so our models consisted of functions that assign individuals to Abe Lincoln and George Washington, and properties to was presi-dent and had a beard. We have already considered at length the interpretational perspective on these models; let us now look at them briefly from the contrasting representational perspective. For here again we can view our models in either way.To provide a representational account of these models, we begin by assuming that all the expressions of our language have their ordinary interpretation, regardless of the assignments made by a model. The purpose of assigning various objects



to various expressions is to con-struct representations of alternative configurations of the world. The individual assigned to Abe Lincoln in a given model represents Lincoln in that model, the property assigned to was president repre¬sents the property of having been president. If the individual has the property, the model depicts a world in which Lincoln was president; if the individual does not, the model depicts one in which Lincoln was not president. The fact that the individual may happen to be Ben Franklin (or perhaps an abstract object likethe number one), and the property that of having worn a wig (or perhaps that ofbeing an even number), has no bearing on our interpretation of Abe Lincoln or was presi¬dent. On the contrary, the interpretation of these expressions, their actual interpretation, is our key to understanding what the model represents, what configuration of the world it depicts.Again the difference emerges in the counterfactuals our theory supports. The sentence Abe Lincoln was president is not true in any model that assigns Franklin to Abe Lincoln and the property of having worn a powdered wig to was president. According to the Tarskian view, this supports a counterfactual claim about how the truth value of this sentence would have changed had Abe Lincoln named Ben Franklin and had was president meant wore a powdered wig. From the representational perspective, it supports a claim about how the truth value of this sentence would have changed had Lincoln not been president. Here Franklin is just a convenient stand-in, the property of wearing a wig a handy prop. The same representational roles could have been played equally well by innumerable other objects and properties, and in each case the moral would have been the same: the sentence Abe Lincoln waspresident would have been false had Lincoln not been president.

The Failure of IntersectionWe have here two very different conceptions of model-theoretic se-mantics. According to the representational view, the models ap-pearing in our semantics are simple depictions of possible configura-tions of the nonlinguistic world, the worldour language talks about. A sentence is true in a given model just in case it would have been true if the world had been as depicted by the model. Conse-quently if, judging by some intuitive metaphysics, all possible configu¬rations of the world receive some manner of depiction, then sentences that come out true in all models are true regardless of how the world might be, perhaps they are true simplydue to the way the language works. Of course, should some possibilities be omitted, inadvertently or otherwise, these results will hold only modulo the metaphysical assumptions embodied in our semantics. This is arguably the case with the model theory for our second language; the semantic theory does not tell us, for i

nstance, how the truth value of sentences would have been affected had Lincoln not existed. Perhaps there are other possi¬bilities our theory fails to cover.According to the second conception, the Tarskian view, each model provides a possible interpretation of certain expressions appearing in the language, those not included in the set ^ of fixed terms. A sentence is true in a given model if, so to speak, what it would have said about the world on the suggested interpretation is, in fact, the case. Thus, sentences that come out true in all models aretrue regardless of how we interpret a subset of their component expressions. Here, too, the regardless must be qualified: the result holds only modulo our cir-cumscription of the class of semantically well-behaved reinterpre-tations of the variable terms. It is assumed that Abe Lincoln would not have functioned like Nix, oreven like the considerably less bizarre Pegasus. The semantic theory does not tell us how the truth values of our sentences would react to such reinterpretations.

With the semantic theories considered in the last section, the two conceptions seem aptly described as differences in perspective: to move from one to the other requires nothing more than a subtle shift in gestalt. But it would be a serious mistake to imagine that this will always be the case. Indeed in our two simpleexamples we have just been lucky; we have just hit upon a fortuitous intersection of the two ap¬proaches.Clearly, not every model-theoretic semantics allowed from the inter¬pretational perspective can also be viewed representationally. In the case of our sample languages, this becomes apparent when we con¬sider different theories that emerge from different selections of the set of fixed terms. With other choices of ^ we encount



er one of two problems: either the resulting class of models, when seen representa-tionally, omits depictions of genuinely possible configurations of the world, or there is simply no way to view the class of models as represen¬tations.We would have run into the first problem had Snow is white been included in On this choice of fixed terms our models would consist of functions that assign truthvalues to Roses are red and Violets are blue. These models can still be taken representationally, but as such they contain an obvious omission: we have no models that depict worlds in which snow is not white. A similar problem would arise withour second language were we to include, say, Abe Lincoln and was president in Amongthe resulting class of models we would still find depictions of worlds in whichWashington was not president (namely, any sequence that assigns a nonpresident to George Washington ) and worlds in which Lincoln had no beard (namely, any sequence that assigns a property that Lincoln does not possess to had a beard ), but we would have no models representing worlds in which Lincoln was not president.The second problem would have arisen, with either language, had we excluded or or not from *$. Consider, for instance, the d-sequences we get for our second language when or is considered an additional variable term. These consist of functions that assign individ¬uals to our two names, properties to our predicates, and a binary truth function to our sole connective. If we try to view such models representationally, we must somehow imagine that or receives its ordinary interpretationand that our assignment of various truth functions to this expression is just atechnique for representing possible configura¬tions of the nonlinguistic world. Butthere is no plausible way of understanding, representationally, models in which or is assigned, say, the truth function ordinarily expressed by and. This is not to

say that such models depict extremely bizarre possible worlds, worlds we have difficulty conceiving. There is just no representational counter¬part to such a Tarskian semantics.Consider a more familiar example. Suppose L is the quantifier-free fragment of the language of elementary number theory. Thus, L contains such sentences as 2 + 2 = 4 and either 7x8 = 49 or 7x8 = 56. A standard interpretational semantics will hold fixed the meanings of the identity predicate and the connectives, but will reinterpret the numerals ( 0/ 1/ 2/ etc.) and function symbols ( +, x, etc.) of the languae. Thus, one model might assign the empty set to 2/ the set containing the emptyset to 4, and set union to +. In this model that is, according to this interpretation+ 2 = 4 comes out false, since the union of the empty set with itself is the empty set, not the set containing the empty set. Such a semantics makes perfect sense from the interpretational standpoint, but obviously cannot be viewed repre-sen

tationally. There is no way to construe the model described as somehow representing a possible world in which two plus two does not equal four. That way madness lies: 2 + 2 = 4 might well have said something false, perhaps something about the union of sets. But what it says that is, what it actually says is necessarily true.In all of these cases, the theories described meet the standards of interpretational semantics, but make no sense if we apply the stan-dards of representational semantics. And it is not hard to find exam-ples of the opposite sort as well: theories that meet the requirements of representational semantics but that violate the Tarskian conception. Consider a simple example. Clearly, a perfectly acceptable representa¬tional semantics for our second language could get by with far fewer models than are needed for a plausible interpretational semantics. Many of the models inherited from the interpretational semantics are representationally isomorphic. That is, although two models might assign different individuals and p

roperties to the names and predicates of our language, this does not mean they depict different configurations of the world. All that matters to the depiction is whether the individ¬uals assigned to the names have or do not have the properties assigned to the predicates. For this language a representational semantics could get by with a small number of nonisomorphic models: sixteen, to be exact. Viewed interpretationally, any such move would constitute an unmotivated restriction of the class of semantically well-behaved reinterpretations of the language, an unjustified limitation on the name and predicate domains of the satisfaction relation. Such a se¬mantics would thus be ruled out by the interpretational guidelines.



To take a more interesting example, recall from Chapter 2 our discussion of a representational semantics for a language whose atomic sentences are Snow is white, Snow is red, and Snow is green. There we suggested models that assign truth values to these sentences, but with the added proviso that we exclude any model that assigns true to more than one atomic sentence. This gives us four models rather than the original eight, the limitation being motivated by the obvious fact that the remaining models would not depict genuine possibilities. Now notice that the resulting semantics would be ruled inadequate from the interpretational standpoint. By including d-sequences that assign true to Snow is red, we acknowledge that this sentence, though false, could be assigned a different meaning, perhaps thatLincoln had a beard, and thereby say something true. Similarly for Snow is green :it could be reinterpreted to mean, say, that Lincoln was president. But if these sentences can be assigned such interpretations individually, it must surely bepossible to so interpret them simultaneously. Ruling out models that assign both of these interpretations at once is no more justified than ruling out d-sequences that both assign Ben Franklin to Abe Lincoln and Thomas Jefferson to George Washington, even though we allow these same interpretations individually. Thus, thisrestriction from eight to four models, though easily motivated from the representational standpoint, would make little sense in interpreta-tional semantics.Clearly, representational and interpretational semantics are entirely differententerprises, governed by entirely different standards. They are not simply two perspectives from which we can view an arbitrary semantics. The two approaches do happen to come together at certain fortuitous points, in simple theories that do not explicitly violate either standard. Such was the case with the examples di

scussed in the last section: the same class of models and the same definition of truth in a model were equally suited to either a representational or an interpre-tational semantics, to either an explication of x is true in W" or an explication of x is true in L.Now, there is little significance in the fact that the two approaches occasionally intersect, or that they do so where they do. The fact that the same functions can sometimes be used as models for either type of semantics is hardly more surprising than that a brick can both break a window and hold up a bookshelf. But what is important to note here is that the intersection of these approaches is not trivial. Not trivial, but only in this somewhat trivial sense: it does not always happen. If we had merely described two perspectives, two ways of viewing one and the same endeavor, then every interpretational semantics would have a repre

sentational counterpart, and every representational semantics could also be seen interpretationally. The difference would just de¬pend on how we screw up our eyes while watching the move from model to model.5Interpreting QuantifiersBefore going on we should consider one final example, an example in which the same model theory appears at first glance to satisfy both the aims of interpretational semantics and the aims of representational semantics. In the languages we have considered so far, quantifiers have been conspicuously absent. Yet the standard model theory for first-order quantified languages seems an obvious case in which inter¬pretational and representational semantics intersect or so we might assume.Actually, the situation is not so simple, and thus this final example goes beyon

d mere illustration. The motivation underlying the tradi-tional technique of defining a first-order model seems quite straight-forward when we imagine ourselves offering a representational semantics. But it turns out that those same models, considered interpretationally, embody a significant departure from Tarski s analysis of the logical properties. The departure stems from the introduction of what I call cross-term restrictions on the permissible interpretations of expressions.Cross-term RestrictionsSuppose our second language were supplemented with the expression something, the unrestricted (or trivially restricted) existential quanti¬fier. The standard semant



ics for the resulting language would have us build models in the following way:first we choose an arbitrary set called the universe or domain of the model; second we choose a function that assigns ail objec t from that set to each name inthe language, and a .subset of that set to each predicate.1 Truth in a model isdefined recursively, the clause governing the newly introduced quantifier en¬suring that Something was president is true just in case some member of the universe falls in the set assigned to the predicate was presi¬dent. 2These models have a simple and natural motivation from the repre¬sentational viewpoint. As always, the representational semantics draws no distinction between fixed and variable terms: again, the purpose of assigning objects of various sortsto the names and predicates is purely representational, a technique precisely parallel to our earlier account. The new element in our models, the universe set,provides some added detail to our representation: it allows us to depict worldswith various populations, and with various distributions of properties among those populations. In particular we can represent worlds in which, say, someone has been president, though that someone is neither Washington nor Lincoln. And once again, according to the present system of representation, Washington and Lincoln are always depicted as existing, their proxies always chosen from among the universe set.Suppose we try now to construct a complementary, interpretational account of these models. We must first see if we can trace the boundary between fixed and variable terms implicit in the semantics. Clearly, since or and not receive the same treatment

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