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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 friendand colleague Jon Barwise. Finally, I am indebted to the Mrs. Giles Whiting Foundationand to the Center for the Study of Language and Information for support while workingon various stages of this book.

Contents

1 Introduction 1

2 Representational Semantics 12

3 Tarski on Logical Truth 27

4 Interpretational Semantics 51

5 Interpreting Quantifiers 65

6 Modality and Consequence 80

7 The Reduction Principle 95

8 Substantive Generalizations 107

9 The Myth of the Logical Constant 125

10 Logic from the Metatheory 136

11 Completeness and Soundness 144

12 Conclusion 156

Notes 161

Bibliography 171 Index 173

1

Introduction

The highest compliment that can be paid the author of a piece of conceptual analysiscomes not when his suggested definition survives whatever criticism may be leveledagainst it, or when the analysis is acclaimed unassailable. The highest complimentcomes when the sug-gested definition is no longer seen as the result of conceptualanaly-sis — when the need for analysis is forgotten, and the definition is treated ascommon 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 tothese definitions, a sentence is logically true if it is true in all models; an argument is



logically valid, its conclusion a consequence of its premises, if the conclusion is true inevery model in which all the premises are true. These definitions, along with theadditional machinery needed to understand them, are set forth in every introductorytextbook in mathematical logic.1 In these texts we are taught how to delineate a class ofmodels for a simple language and how to provide a recursive definition of truth in amodel — in short, how to construct a simple model-theoretic semantics. Once thissemantic 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 traced to Tarski‟s

1936 article, “On the Concept of Logical Conse¬quence.”2 In this article Tarski sets outto 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 aremeant 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 ofother attempts to define the consequence relation, in particular attempts to characterizeit syntactically, by means of formal systems of deduction. His own, semantic account,sketched in a mere four pages, is devoted in part to the exposition of some ancillarynotions treated at length in his earlier monograph on truth. The main thrust of the articleis not to discuss details of the semantic account of consequence, or even to give asimple 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 ofconsequence to the discipline of logic. He dryly notes that the introduction of thisconcept into the field “was not a matter of arbitrary decision on the part of this or thatinvestigator” (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 tostudy — as we are when we introduce, say, the concept of a group, or that of a real closedfield. It is for this reason that Tarski takes as his goal an account of consequence thatremains faithful to the ordinary, intuitive concept from which we borrow the name. It is

for this reason that the task becomes, in large part, one of conceptual analysis.

Tarski‟s account of the logical properties is widely regarded as suc-cessful in thisrespect, as capturing, in mathematically tractable form, the “proper” concepts of logical

truth and logical consequence. We can see this not only from explicit acknowledgmentsof its success by many philosophers and logicians, but also from the treatment given it

by those not interested in conceptual analysis as such. Perhaps the most strikingindication 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 yield ageneral 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 intuitive notion ofconsequence cannot be captured by any single deductive system. For one thing, such asystem will be tied to a specific set of rules and a specific language, while the ordinarynotion is not so restricted. Thus, by “consequence” we clearly do not mean derivability

in this or that deductive scheme. But neither do we mean derivability in some deductivesystem 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 ofconsequence: a deductive system is sound if it allows us to prove only genuinely validarguments, those whose con¬clusions follow logically from their premises.

We recognize that a syntactic definition does not capture the or-dinary notion ofconsequence, and we recognize this even though we may be convinced, for one reasonor another, that a given deductive system is adequate for a given language — that is, evenif we believe that all valid arguments, and only valid arguments, are provable within thesystem. This recognition is at a conceptual level, but its main impact is at theextensional. The upshot is that systems of deduction require external proofs of theirextensional adequacy (or inadequacy, as the case may be). To be sure, with carefulselection 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 inthe system — if such an assurance is to be had — must come from somewhere other thanthe 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 account ofconsequence, 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 any sentence in afirst-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 consequencerelation. 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 coincide,that they apply to the same pairs (K, S) in the first-order language. But we think of theseresults as having an intu¬itive significance that goes beyond the mere coincidence oftwo alter¬native characterizations of the consequence relation. Specifically, we think ofthem as demonstrating the extensional adequacy of the de¬ductive system in question.They are thought to show that the system is sound, that it will not allow the derivationof 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 ofsen¬tences in the language.

What is revealing is that the significance we read into these results is asymmetric, eventhough 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¬quencerelations, perhaps involving variant proof regimes, we would

hardly take these results as showing the adequacy, the soundness and completeness, ofone 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 theydemonstrate, say, the extensional adequacy of |~2> would obviously presupposeadditional 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 themodel-theoretic definition of consequence, unlike syntactic definitions, involves a moreor 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 equalneed for external evidence that the model-theoretic characterization of consequence isextensionally 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 declaresome 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 mean those structuressatisfying our characterization. But as Tarski points out, the situation is quite differentwith the concept of logical consequence. Here the correctness of our model-theoreticde¬finition is not determined by arbitrary fiat; on the contrary, whether the definition isright or wrong will depend on how closely it cor¬responds to the pretheoretic notion itis meant to characterize. That the first question now strikes us as odd just indicates howdeeply ingrained is our assumption that the standard, semantic definition captures, orcomes 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 arather imprecise and intuitive notion. Here the notion was that of an effectively





the assumption that the model-theoretic definition captures the genuine concept ofconsequence. 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 claim is thatTarski‟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 thelogical proper-ties. The thrust of my argument is primarily at the conceptual level, butagain the main impact is at the extensional. Applying the model- theoretic account ofconsequence, I claim, is no more reliable a tech-nique for ferreting out the genuinelyvalid arguments of a language than is applying a purely syntactic definition. Neithertechnique is guaranteed to yield an extensionally correct specification of the lan-guage‟s

consequence relation. Needless to say, this conclusion requires that we reassess theintuitive 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 logicallyfrom others, is without doubt the most central concept in logic. It is what has driven thestudy of logic for more than two thousand years. On the other hand, the remarkableachievements in logic during the past century have been the direct result of themathematization of the field. The infusion of mathematically precise definitions andtechniques has turned a field dominated by homely admonitions into one capable ofsupporting significant and illumina-ting theorems. My aim in this book is to attack acommon 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-theoreticaccount of consequence alone cap-tures the genuine notion does not mean they areuseless for studying this very same concept. Direct analysis is just one way to gainaccess to an important, intuitive concept; lessons from elsewhere in mathemat¬icsshould convince us of that.

Some History

Though my concern in this book is not historical, a few preliminary words should besaid about the complicated heritage of the model- theoretic definitions of the logical

properties. As I mentioned, these definitions are generally credited to Tarski‟s 1936article, 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 article contains 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 aloneoversimplifies the situation a great deal.4 For one thing, most of the main features of theanalysis were anticipated, in various different ways, by earlier authors, including

Bolzano (1837), Padoa (1901), Bernays (1922), Hilbert and Ackermann (1928), andGõdel (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 thefamiliar, model-theoretic technique f or establishing a sentence‟s logical inde¬pendence

from a given set of axioms, a technique that presupposes one direction of the definitionof consequence. Bernays, Hilbert and Ackermann, and Gõdel all present, with varyingdegrees of clarity, a model-theoretic definition of logical truth, though none of themtries 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 predecessors,with the notable exception of Bolzano. In his article, Tarski emphasizes that histreatment 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 leftat an intuitive level by all earlier authors. Second, and more important, is Tarski‟s

attempt to present and motivate the definitions in a com¬pletely general setting. It iseasy to underestimate the importance of this contribution. But clearly, the ordinarynotions of logical truth and logical consequence are not restricted to a specific languageor 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 generalanalysis of these concepts. Finally, unlike his im¬mediate predecessors, Tarski extendshis 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 definitionsoriginated 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 foranother important reason that will emerge in Chapter 5. It turns out that certain

paradigmatic instances of the model-theoretic definitions in-volve a subtle butsignificant departure from Tarski‟s analysis, one that has gone completely unnoticed.

But to explain that departure at this point would be premature.

The Plan of This Book

This book consists of a single, extended argument. The conclusion of the argument isthat 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 arbitrary languages — even perfectly familiar, well-behaved ones — it will regularly and predictably define a relationat variance with the genuine consequence relation for the language in question. Thedefinition will both undergenerate and overgenerate: it will declare certain argumentsinvalid 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 incorrect.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 inthese cases we must seek external guarantees of that fact. This is the second point, andthough a bit more subtle, it is at least as important as the first. The point is that thesemantic account shares with syntactic accounts the following limitation: there is noway to tell from the definition alone or from characteristics of the language whether theextension of the account is correct. Clearly, no amount of pondering a syntactic systemof deduction can assure us of its extensional adequacy; for that, we must turn to indirectevidence, 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 dealswith various intuitive or conceptual considerations bearing on the adequacy of Tarski‟s

account. The reason for this emphasis is simple. I think the basic problem with Tarski‟s

account is in some sense obvious, once certain confusions and misunderstandings arecleared 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 intuitiveconcept into mathematically tractable form, we would have an ironclad guarantee of itsextensional adequacy when applied to arbitrary languages. The situation would then beanalogous to, say, our inductive definition of N, the set of natural numbers. According

to this definition, N is the smallest set that contains 0 and is closed under the successoroperation.5 Now, it is perfectly clear that this definition is not identical to the intuitivenotion it supplants. Thus, it employs a variety of set-theoretic concepts that are not, byany stretch of the imagination, part of our ordinary understanding of the naturalnumbers. Conversely, certain things that are arguably central to our intuitive concept(say, the concrete process of counting) are at best dimly reflected in the inductivedefinition. But the definition obviously captures the essential feature of the intuitivenotion, 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 way theyreact to the inductive definition of N. Neither is given extensive justification sinceneither seems to need it. I claim that this reaction is, in the former case, mistaken. But itis not, unfor-tunately, a simple mistake — or, for that matter, a single one. For thisreason, much of this book is devoted to explaining the variety of confusions andmisunderstandings 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 givenmisunderstanding, and so an individual reader may find certain parts of the bookobvious, while another might find those same points illuminating and others not. Forexample, the first few chapters are addressed to a confusion extremely common amongthose who enter logic through philosophy or linguis¬tics, but almost nonexistent amongthose 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 ignoringentirely 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 reallyaddressed to somewhat different audiences. Since these audiences will have differenttechnical backgrounds (not to men¬tion different interests and concerns), I have triednot to assume much common ground, at least in covering the key points of my

argument. The model-theoretic account of consequence has had a tremendous influenceon all logic-related disciplines, from philosophy and linguis¬tics to mathematics andcomputer science. Thus, I have tried to make the book understandable to anyone whohas had a first course in mathematical logic. I hope it does not seem tedious to thosewho 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, thereason it will, in general, be extensionally incorrect. Second, I describe what I believe isthe 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 thatsurround 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 legitimateapproach to semantics, but (as will become clear) it bears no relation whatsoever toTarski‟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, bydiscussing this alternative approach to semantics, so that it can be usefully contrasted

with Tarski‟s account rather than vaguely con¬fused with it. Chapters 3 through 5 aredevoted 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 withwidespread intuitions about completeness and soundness theorems. There, I modify anargument 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, forsimplicity, on the notion of logical truth. Logical truth, since it is a property of singlesentences, is often far easier to discuss than logical consequence, which is a relation

between a collec-tion of sentences (say, premises of an argument) and another sentence(the conclusion). For example, it is much easier first to look at the details of Tarski‟s

account as they bear on the concept of logical truth, and then to explain briefly the moregeneral 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 losesight of the fact that the concept of consequence is far more important than that oflogical truth, both intuitively and techni¬cally. On their own, logical truths are of verylittle interest — recall that these are sentences we often describe as trivial, devoid ofinformation, true by virtue of meaning, and so forth. Where the notion of logical truthgains its importance is as the limiting case of the consequence relation: these are

sentences that follow logically from any set of sen¬tences whatsoever. The crucialnotion, ultimately, is that of one sen¬tence following logically from others. Logic is notthe study of a body of trivial truths; it is the study of the relation that makes deductivereasoning possible.

2

Representational Semantics

To 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 asimple puzzle suggested by Donald Davidson. In a well-known article in which hedefends 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 tohold is expressed by a monadic predicate. In this respect, truth sets itself apart frommany other concepts that we consider peculiarly semantic. Thus, denotation is a relation

between a singular term and an object denoted, satisfaction a relation between an open

sentence and the things it “holds true of,” and so forth. But truth, perhaps the preeminent semantic concept, does not relate a sentence to something else; it simplyapplies or fails to apply, so to speak, absolutely.1

Davidson goes on to note that at least on a superficial level, much contemporary workin semantics seems to belie this simple point. Much effort is devoted to the investigationof 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 theso-called T-sentences distinctive of the theory of absolute truth. The T-sentence



„Snow is white‟ is true if and only if snow is white

does not, as Davidson puts it, “fall out of” a theory that simply tells 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 absolutetruth — that is, of truth as we ordinarily understand it; theories of relative truth must, atleast 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 characterization of such arela¬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?2

We can look at Davidson‟s puzzle this way. A theory of relative truth provides us with acharacterization 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 consequencesinvolving the ordinary mo¬nadic predicate “x is true.” Davidson, of course, is

particularly inter¬ested in the so-called T-sentences, but the same point might be madeabout any claims involving “absolute” truth. That point is this. Before a theory of

relative truth can be judged to have consequences, formal or otherwise, 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 the simple, prima facie evidence isthat our theory of relative truth has no bearing on the concept of truth as we ordinarilyunderstand it. But that, of course, is absurd.

Truth as Specification

We 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 wefirst set out to analyze “x is a brother of 31.” The former then reduces to an existentialgeneralization of the latter: brotherhood is just brother-of-someone-hood. There aresimilar cases in which we gain access to the monadic concept through a universalgeneralization of the relational; thus with comparatives and super¬latives — say, tallerthan and tallest. But clearly the monadic concept of truth, the concept we ordinarilyemploy, is no generalization of any of the various relational concepts. A sentence can betrue 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 aspecification of the latter. In other words, perhaps the monadic concept emerges from





Snow is white Roses are red 5

TRUE TRUE

TRUE FALSE

FALSE TRUE

FALSE FALSE

Snow is white Roses are red Violets are blue 5

TRUE TRUE TRUE

TRUE TRUE FALSE

TRUE FALSE TRUE

TRUE FALSE FALSE

FALSE TRUE TRUE

FALSE TRUE FALSE

FALSE FALSE TRUE

FALSE FALSE FALSE

Our reference column — everything to the left of the double lines — provides us with therows that our target sentence is to be true or false in. The ultimate goal is to write thewords “TRUE” or “FALSE” in each row below S; “TRUE” if 5 is true in that row,“FALSE” if 5 is not true in that row. But to do that, of course, no standard pattern of the

sort used in constructing the reference column will suffice, will ensure that we enter thecorrect value in each row. Rather, we need a radically differ¬ent technique, a techniquethat involves the repeated application of certain recursive tables. The following are twosample recursive tables; the „not‟ table:

p notp

TRUE FALSE

FALSE TRUE

and the „or‟ table:

P <1 porq



TRUE TRUE TRUE

TRUE FALSE TRUE

FALSE TRUE TRUE

FALSE 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 theconstituents, we need only match them to the appropriate row of the appropriaterecursive table and read to the right. Often the recursive tables will also have beenapplied in order to determine the values of the relevant constituents, and, in turn, of

their relevant constituents. Indeed, there is no upper bound on the number of times arecursive table may have to be applied within a single row before the final value of thetarget sentence is reached.

Once we are adept at these techniques we can easily produce tables in which our targetsentence 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 red‟

T T TRUE

T F TRUE

F T FALSE

F F TRUE

Now, consider exactly what this table tells us. First of all, it clearly does not tell us theactual 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 specificinstance of “relative” truth. Translating to present terminology, the truth of a sentenceshould boil down to its truth in some specific row. And since we know that the current



sentence is actually true, we can rule out the third row without further ado; that row issurely not “Fred.” On the contrary, as any student of introduc¬tory logic could quicklytell 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; afterall, „Snow is white‟ and „Roses are red‟ are both true— that is, genuinely true — and thefirst row is the only row in which these sentences both come out true. But notice that inoffering 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 atomicsentences, 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 World

Davidson‟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 ourtables. If truth is to be truth in some specific row, then clearly the first row of oursample 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 theordi¬nary, monadic concept from which we pirate the name. To provide such anaccount 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” row and yet not be truesimpliciter. Such a response would leave our theory of relative truth entirely suspendedin 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, thenDavidson would be completely justified in claiming that the defined “x is true in y” is

irreducibly relational. And consequently he would be justified in claiming that, for thisreason, our theories of relative truth cannot be thought to illuminate the notion of truth

as we ordinarily understand it. But this conclusion would obviously be wrong. It is perfectly clear that truth tables tell us something about truth, about ordinary monadictruth, and that the relation of “truth in a row” was not just conjured up by some logicianor 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 obviousalternatives. 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 atomicsentence „Snow is white.‟ The effect of this move is that the resulting value of our target

sentence turns from true to false. Now the question is simply what change would have a



similar effect on the “absolute” truth value of „Snow is white,‟ and a similar effect on

the “absolute” value of our target sentence.

There are only two parameters to which the sentence „Snow is white‟ owes its truth:

broadly speaking, the language and the world. It is due 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 snowis white. Appropriate changes on either side would have made our atomic sentencefalse. Thus, had the language been somewhat different, this sentence would have beenfalse 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 itsmeaning — say, had snow been red.

We can interpret the move from row to row in our truth table in either of these twoways. In the first place, we can view our theory of truth in a row as explicating therelation “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 truthvalue of sentences — say, the color of snow or roses — is held fixed; our concern is notwith 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 have been “right” had we been speaking alanguage 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 iswhite and roses are indeed red. From this perspective, the move to the third rowinvolves 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 counterfactualclaims about the (absolute) truth values of their tar¬get sentences. We imagine these

claims to be supported because our theory assigns values to these sentences even inrows that are not “right,” rows in which the atomic sentences are not assigned theiractual values. So, for example, the third row of our sample table supports a claim of theform:

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 onwhich parameter we view as changing in the move from row to row — that is, dependingon 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.” Thus, 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 tablesdepends critically on which perspective we assume, on the nature of the parameter thatcorresponds to the rows our sentences are true in. Of course, since both points of vieware 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 bothof our claims into a single counterfactual:

The sentence „Snow is white or roses are not red‟ would have

been false had „Snow is white‟ been false.

But the fact that we can do this does not mean the resulting claim is somehow justified by the abstract theory, quite independent of any account we might give of the relation between “x is true in /‟ and “x is true.” Or, to put it another way, the fact that our theoryof “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 abstractcharacterization of relative truth, of “x is true in y,” supports no claim whatsoever about

absolute truth, about truth as we ordinarily understand it.

A Representational Semantics

When 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 valuesof 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 anaccount of a relation, “x is true in;y,” and what the theory takes to satisfy the position

are, for all intents, just ordinary objects of some sort or other — chunks of the actualworld. 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¬riesmight define a relation between sentences and abstract, set- theoretic objects, maybefunctions 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 of ink, and functions are set-theoretic constructs; the world, thankfully, isneither of these.



The point is a simple one, but all too easily overlooked. When we viewed our theory of“truth in a row” as explicating “x is true in W,” the fact that the target sentence cameout 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 rowitself, 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 ushow truth would be affected by changes in the world, by changes that are represented ordepicted by changes in row.

The techniques used in constructing truth tables are not generally thought to constitute afull-fledged semantic theory for any language or language fragment. More thananything else, this is due to certain traditions of fairly recent vintage concerning the

accepted format of such theories. Still, it may seem perverse to view our theory of “truthin a row” as a representational semantics, insofar as it may seem perverse to view it as asemantics 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 of anytwo.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 possibleconfigurations of the world relevant to the truth values of our sentences. Thanks to thesimplicity of our language, this purpose can be served by the class of functions thatassign a truth value, either true or false, to each of our three atomic sentences. Thus, ourclass 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 thatassigns false to each (representing worlds in which snow is not white, roses not red, andviolets not blue), and so forth.

Our next step is to provide a recursive definition of 5 is true in f for arbitrary sentences

5 and models/. Since we will take this relation as an indirect characterization of “x istrue in W,” our aim will be to ensure that any given sentence of our language is true inexactly those models which represent worlds that would indeed have made the sentencetrue. So if a model depicts a world in which snow is not white, our definition shouldguarantee that „Snow is white‟ comes out false in that model. Here we assume, ofcourse, that the sentence „Snow is white‟ means what 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 it thevalue true.

• If .S' is the negation of S', then it is true in a model/just in case S' is not true 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 beingdetermined by the atomic components of the target sentence. In the new theory, ourmodels 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 somewhatmore 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 entirelanguage.5

Now, although it could easily escape notice, the reference columns of our earlier theoryactually did a bit more than our models. The reference columns both delineated theneeded rows and simultaneously specified the values of our atomic sentences in thoserows. In contrast, whether an atomic sentence comes out true in a given model isdeter¬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 thenew theory; any two objects would have worked as well — for example, the numberszero 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 baseclause, 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 similarfreedom in the reference columns of our truth tables, say, the freedom to use “ + ” and

rather than “TRUE” and “FALSE,” we would 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 thatsentences of the language are to be “true in.”



Representational Guidelines

The basic motivation underlying a representational semantics, an indi¬rectcharacterization of “x is true in W,” is fairly clear. The approach provides a naturalframework in which to couch a theory of meaning, or at any rate a theory of those

aspects of meaning relevant to the truth values of sentences, both the values theyactually 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 beconsidered a partial theory of meaning for the relevant fragment, since it offers nodetailed account of the semantic functioning of the three atomic sentences. In giving thesemantics, 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 moredetailed 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 arepresentational semantics for any interesting language is either easy or philosophicallyunproblematic. 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 “x is 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 approach to se¬mantics, in spite of the fact that oneand 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, inde¬fining the logical properties, had in mind something akin to represen¬tational

semantics, a characterization of “x is true in W,” for all “possi¬ble worlds” W. Forexample, 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 Tarski‟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 ouraim 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 theadequacy 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 theworld. This was accomplished in the semantics of the last sect ion by employing arather crude but effective system of repre¬sentation. Our collection of models imposed,so to speak, a complete

partition 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 ofour eight functions, the remain¬ing class of models would have been inadequate in thisrespect, 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 class of models to those functions that



assign false to at least two of our atomic sentences. The remainder would not representgenuine 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 influence of 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 beenaccu¬rate — that is, had the world actually been as depicted by that model. Obviously,the possibility of success on this score is not independent of the objects we have chosento include in our class of models. In particu¬lar, it is this ultimate goal that determinesthe amount of detail we need to incorporate into our models, how crude a system ofrepresentation we can get by with. So, for example, with our sample fragment we couldnot have used functions that assigned truth values only to „Snow is white‟ and „Roses

are red.‟ Although these models would indeed have given us a complete partition of

possible worlds, the partition would not have been fine-grained enough to allow us tocarry out our semantic task: the accuracy of any of these models would have beenconsistent with either the truth or falsehood of „Violets are blue.‟ And of course with

more complicated languages, say, languages containing quantifiers, our technique ofconstructing representations will have to allow for a considerably more detaileddepiction of the world.

Now, the final points to notice about representational semantics concern the sentencesthat turn up true in all models. It is an immedi-ate and trivial consequence of the twocriteria 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 comesout true in all models, then we have either omitted representations for some possibleconfigurations of the world, namely those that would have made the sentence false, orour definition of truth in a model has gone astray, having declared the sentence true in atleast one model that depicts a world in which it would actually have been false. Just so,a sentence that is necessarily true can only come out false in a model if we have gottenits 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 adequaterepresentational semantics. And so if logical truths are thought to be necessarily true,

these will of course be among those true in every model. Similarly, if one sentencecomes out true in every model in which a second sentence is true, then the truth of thefirst must be a necessary consequence of the second. That is, it must be impossible forthe first to be false while the second is true, at least if our semantics really satisfies therepresentational guidelines.

Equally trivial is the observation that analytic truths, sentences that are true solely byvirtue of the fixed semantic characteristics of the language, will come out true in allmodels. 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 every model of an adequate semantics, one thatsatisfies the stated criteria.7

These 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 logicaltruth. 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 arepresenta¬tional semantics. But more important, even if we are prepared to identifynecessary truth and logical truth — an identification most peo¬ple would balk at — it isstill clear that representational semantics af¬fords no net increase in the precision ormathematical tractability of this notion. Any obscurity attaching to the bare concept ofnecessary truth will reemerge when we try to decide whether our semantics reallysatisfies the representational guidelines — in particular, when we ask whether our models

represent all and only genuinely possible configurations of the world.The value of representational semantics does not lie in an analysis of the notions oflogical truth and logical consequence, or in the analysis of necessary or analytic truth.Rather, what this approach gives us is a perspicuous framework for characterizing thesemantic rules that gov¬ern our use of the language under investigation. It should beseen as a method of approaching the empirical study of language, rather than an attemptto analyze any of the concepts employed in that task. Certainly, all necessary truths of alanguage — of whatever ilk — should come out true in every model of a representationalsemantics. 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 certainexpressions. But this is only a test of the adequacy of the semantics, not

a 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 andnot the notion of necessary truth.

I have sketched some simple and general criteria that guide the construction of arepresentational 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 alternativeapproach to semantics, one whose aim is to characterize the relation “x is true in L,” for

some range of languages L. The intuitive importance of such a theory, and the generalguidelines appropriate to it, are not nearly so apparent as those of representationalsemantics. To get a clear idea of these guide-lines, and to see how they differ from thoseI have just sketched, we need to take a close look at Tarski‟s account of logical truth and

logical consequence.

3

Tarski on Logical Truth



My remark that Tarski‟s account involves the notion of “x is true in L” for variable L

would seem odd to anyone familiar with his original analysis but unfamiliar withmodern presentations of it. There is no mention in Tarski‟s article of any “range of

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- theoreticdefinitions that emerge from Tarski‟s account. In Chapter 4, I explain how making a

few minor (though somewhat confusing) changes in Tarski‟s original account yields a

recognizable model- theoretic semantics. But to see exactly how the resulting semanticsdiffers from a representational semantics, it is important to start from the beginning,with a clear understanding of Tarski‟s original defini¬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 allintents, 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, Tarskiwas unaware of Bolzano‟s work until several years after the initial publication of hisarticle. The key difference between the two accounts is simply that Bolzano employssubstitution where Tarski uses the more technical, and for the purposes more adequate,notion of satisfaction.

Bolzano on Logical Truth

We normally think of logical truth as a single property that holds or fails to hold ofsentences within a language. Both Bolzano and Tarski adopt a slightly differentapproach, in effect treating logical truth as a relation that holds between sentences andsets of atomic expressions in the language, or alternatively, as a collection of propertiesthat can be obtained from this relation by fixing its second argument.2 On eitherBolzano‟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 expressionswe take to be variable and which we 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 truewhen we exchange some subset of their component expressions for any otherexpressions of similar type.3 Bolzano notes, for example, that the sentence

If Caius was a man then Caius was mortal

remains true regardless of the subject term we put in the two positions currentlyoccupied 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 notlogically 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 Bolzanosees it, on which sorts of substitutions we permit.

Following Bolzano, I shall call the terms we allow to vary variable terms and those wekeep fixed fixed terms. Assuming that all grammati¬cally correct sentences are either

true or false, we can take expressions to be of “similar type” just in case they aremembers 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 $ offixedterms just in case 5 is true and every sentence S' that results from making permissiblesubstitutions for expressions in 5 is also true. A substitution of a for b in 5 is permissibleif a and b are expressions of the same grammatical category, if all of the occurrences of

b are uniformly replaced by a, and if expression b contains no member of the set offixed 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 sentence

Grass is green or grass is not green.

It seems reasonable to assume that the truth of this sentence survives any grammaticallyappropriate 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 the threeexpressions „not,‟ „snow,‟ and „is white,‟ that is, if the expression „or‟ is considered a

variable term, then the sentence can easily be turned into a false one. Thus, the falsesentence

Snow is white and snow is not white

results from the substitution of the expression „and‟ for „or,‟ a substi-tution permitted onthis selection of Similarly if we take as our only fixed terms „or‟ and „is white,‟ we can

presumably get the false sentence

Grass is white or grass is necessarily white



by making grammatically appropriate substitutions for the two re-maining variableterms. On the other hand, „Snow is white or snow is not white‟ does seem to belogically true with respect to the set contain¬ing „snow,‟ „is white,‟ and „or.‟ Regardless

of what 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 setof terms we decide to hold fixed. Bolzano was well aware of, and indeed welcomed, thisdependence, chalking it up to the fact that different terms have different logics. Thus,the sentence

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

is logically true when we hold fixed the three expressions „if -then,‟ „knew,‟ and

„believed‟; substituting at will for „Tom,‟ „Carolyn,‟ and „to

be a dean‟ never yields a false sentence. On the other hand, when we consider „knew‟ to be a variable term, we get substitution instances like

If Tom wanted Carolyn to be a dean then Tom believed

Carolyn to be a dean.

One of these instances will no doubt be false, if not this particular instance (Tom may be prone to wishful thinking) then one that results from further substitutions for theother 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 to viewBolzano‟s account as providing, instead of a relation between sentences and sets of

expressions, the collection of properties that can be obtained from that relation byholding con-stant one of its arguments, the set $ of fixed terms. If we settle on theempty set, if we hold no expressions fixed, then in general no sentence will qualify aslogically true. At the other end of the scale, allowing all atomic expressions into $, wefind that logical truth merely reduces to truth. Thus, the sentence „Snow is white‟ is

logically true if we fix both „snow‟ and „is white.‟ This, simply because it is true; if all

of a sentence‟s component expressions are in there are no permissible substitutioninstances to worry about.

The Violation of Persistence

On all of these points, Tarski‟s conception of logical truth coincides with Bolzano‟s.

Tarski argues, though, that the substitutional test de-scribed above should not beconsidered a sufficient condition for logical truth, but only a necessary condition. As Ihave 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

containing 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, when we consider the two names to be our only variable terms, the sentence

„Abe Lincoln was president‟ passes Bolzano‟s test for logical truth, though the sentence„Abe Lincoln had a beard‟ does not. Both of these are in fact true sentences. But in the

first case, when we substitute the only other available name we get a true sentence,„George Washington

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

Of course, the difference here is just a quirk of our language. The world has plenty of people who have never been president. If our meager language had a name for just oneof them, say Ben Frank lin, the sentence „Abe Lincoln was president‟ would suffer the

same fate as „Abe Lincoln had a beard‟: neither would be logically true on the imagined

selection of fixed terms.

This example shows that Bolzano‟s substitutional test is liable to give results thatdepend on purely accidental features of the language. With our current choice of thesentence „Abe Lincoln was president‟ has only two substitution instances, one that

results from the trivial substitution of „Abe Lincoln‟ for itself, the other resulting fromthe 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 Franklinwas 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 givenan appropriate interpretation, and thereby would have provided us with a falsesubstitution 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 hereis 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. Itseems 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 of the deletion of expressions not occurring inS. 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 contractions of the

language, the property of being logically true should persist through expansions. Thisdesideratum, 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 expansionsof 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 the notion of thesatisfaction of a sententialfunction where Bolzano relies on the considerably simplerthough less powerful notions of truth and substitution.

Sentential Functions

We can think of a sentence as the limiting case of a sentential function, where this latternotion permits variables of appropriate type to take the place of ordinary expressions.6So, for example, if V is a variable of appro priate 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 hereoccupied by the name „Abe Lincoln.‟ Sentential functions may contain more than onevariable, 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 novariables.7

The notion of a variable should not be confused with that of a variable term. A variableterm is an ordinary expression of the language, one that differs from a fixed term onlyfor 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 therebydubbed 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, asentential 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 theformer. 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, ensuringthat multiple occurrences of a term receive the same variable, and distinct terms, distinctvariables.

The result of this operation is a sentential function S' containing only expressions thatoccur in the chosen set of fixed terms. We now consider the collection of substitutioninstances of S' — that is, the collection of sentences that result from S' by placingexpressions drawn from appropriate categories back in the variable positions. If everymember of this set is true, then 5 is judged logically true with respect to the currentselection of fixed terms; if one or more is false, then 5 is not logically true with respect

to that selection.





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 the success of anyattempted definition. Schema (1) gives us a clear idea of what a relation, so to speak,must look like before it deserves to be called satisfaction. We will return to this topic ina later section; for now, let us remark on the obvious bearing of our schema onsubstitution.

On the assumption that our metalanguage contains the object lan-guage, any objectlanguage name will be a permissible replacement for “w.” Furthermore, the sentence

that results from inserting this name into the sentential function — that is, the sentencethat 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 cannow 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 additionalconstraint 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 „AbeLincoln‟ 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 furtherstep 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 simpleexpansion of the language that would provide a false substitution instance for thefunction „x was president.‟ Our two schemata allow us to clarify this hazy intuition.

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 thecounterfactual 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 thesubstitutional 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 perhaps unnamed 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, isTarski‟s strategy for getting around the shortcomings of the original, substitutional

definition. But to make good on this idea, we first have to generalize the notion ofsatisfaction in two ways, one simple and one not so simple.

Multiple Variables

The simple generalization is aimed at handling sentential functions with more than onevariable. 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 firstconvert this to the sentential function „x was president or y was not president.‟ 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 Jefferson was not president.‟ What we need is an

account of the satisfaction relation that captures this intuition, one that allows us to saythat 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 anyfunction that assigns an object to each of the variables introduced for the purpose oftesting logical truth.10 Thus, no se¬quence that assigns Ben Franklin to „x‟and Thomas

Jef ferson 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 satisfy this

sentential function.In the spirit of our earlier discussion, we can think of sequences as providing atechnique for simultaneously entertaining a collection of “possible expressions” for

substitution into our sentential function, one for each variable. Rather than consider ageneral schema, which would be premature at this point, we can see this by employing asample instantiation:

( — .1) Sequence/satisfies „x was president or;y was not president‟ if

and 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, theobjects 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 we would have in addition:

( —.2) Sequence/satisfies „x was president or;y was not president‟ if and only if „/(V)was 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 thesentential 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 objectlanguage 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 definelogical truth with respect to certain choices of Suppose $ contains all the atomicexpressions of a lan-guage except perhaps one or more names. In other words, let usassume that any atomic expression which is not a name is a fixed term. Let S' be anysentential function that results from the sentence 5 after we replace all variable termswith variables, ensuring of course that the same variable is used for all occurrences of agiven 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 allsequences.

This definition meets the requirement of persistence in the follow¬ing way: If asentence is logically true (with names the only variable terms), then it will remainlogically true even if the lanûaê is ex¬

panded to include additional names. For regardless of what object the name names, thatindividual has already been found to satisfy the sentential function in question. In thissense, satisfaction puts at our disposal all possible names that might be incorporated

into the lan-guage.On Generalizing Satisfaction

Accounting for logical truth in terms of satisfaction avoids certain problems in thesubstitutional approach, but it encounters some new ones as well. So far the account isnot nearly so general as Bolzano‟s, which allowed atomic expressions of 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.‟





Our first sign of trouble comes when we try to instantiate (3). In instantiating (1) weinserted 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 beeasy 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 isexplicitly 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 Lincolnhaving been president.

Or perhaps

(3.4) The set of former presidents satisfies „Abe Lincoln g* if and only if Abe Lincoln

the set of former presidents.

Both of these instantiations start out fine, but quickly degenerate into nonsense. Thefirst 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, the set that contains allthe individuals, including Abe Lincoln, who once were president. But neither of thesenames can comfortably occupy the predicate position in which it later finds itself.

When dealing with sentential functions containing variables other than those standing in place of names, we obviously need a more complex schema than (1). This is clear fromthe purely grammatical troubles spawned by (3). But the real problem is not simplyfinding the right phrasing for a schema, phrasing that produces a collection of tolerablygrammatical sentences of the metalanguage. Rather, the problem lies in knowing whatexactly we are looking for.

Semantic Presuppositions of Persistence

Our ultimate aim is for satisfaction to take the place of substitution in our definition oflogical truth, to take its place even when the expres-sions substituted are not names. Butsatisfaction is a relation, and all relations hold or fail to hold between objects of one sortor other. In our search for a schema with grammatically proper instances, this wasreflected 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 relation betweenobjects, that “satisfies” must be flanked by names. But it then becomes obscure





„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 given sentential

function. Perhaps every sentence containing „Nix,‟ including „Nix was president or Nixwas 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 alsothreaten the inclusion of „Ben Franklin.‟ But the possibility of such a bizarre “name”

does not seem to call into question the logical truth of „Abe Lincoln 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 categorywas 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‟ both pickout — or name, or denote, or refer to — individuals. Further¬more, the fact that theseexpressions pick out different individuals can alone account for any divergence in truthvalue among sentences in which they occur, at least in the simple languages we haveconsidered so far. It was this that made it so natural to turn from names to objects, toindividuals that could have been named by expressions in the lan¬guage. It seemedobvious that for each such individual our substitution class — now taken to be a semanticcategory — could have been appro¬priately extended. On the other hand, the possibleex pansion of our substitution class to include an expression that behaves like „Nix‟ is

ruled out by the move to satisfaction. This hardly seems an objection¬able bias.

Well-Behaved Expansion and Satisfaction Domains

Let us now return to the problem of generalizing the notion of satisfac¬tion to arbitrarysentential functions. Satisfaction must still be a rela¬tion between objects of some sortand sentential functions (which are also, of course, a type of object). The difficulty weencountered with schema (3) arises because we are now dealing with expressions notnaturally thought of as names, whose contribution to the truth value of a sentence is noteasily reduced to the simple “naming” of an individ¬ual. Consequently, it is not obvious

how to extend satisfaction to the new breed of sentential function. In particular, it is not

obvious what sort of object, if any, might stand in the satisfaction relation to thesesentential functions.

Let us call the class of individuals, things that could have been picked out by names, thename domain of the satisfaction relation. Intuitively, this is the collection of objects thatstand in the satisfaction relation to some sentential function displaying a single namevariable. Our prob¬lem is now to specify the predicate domain of the satisfactionrelation, 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 approaches that 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 appearingin the senten¬tial function.

The most recent considerations suggest the required perspective on satisfaction. Ouraim 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 allconceivable alter a¬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 thelanguage, with the introduction of expressions whose semantic behavior seemed nomore different from that of present members of the substitution class than the behaviorof the present members differed from one to another. The possibility of adding a namelike „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 otherexpressions, and hence to expressions that contribute to the truth value of a sen¬tence ina fashion similar to „Abe Lincoln‟ and „George Washington.‟ Thus, if a given sentential

function is satisfied by all of these individ¬uals, no semantically well-behavedexpansion of the category of names will provide a false substitution instance of thatsentential function.

The other expressions of our language —„was president,‟ „or,‟ and so on— do not nameobjects; only names, so to speak, name.14 But it seems equally clear that the categoryof, say, predicates admits of seman¬tically well-behaved expansions, just like the

category 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 only question is whether the notion of satisfaction offers a technique for clarifyingthis intuition, for distinguishing appropriate from inappropriate expansions of thecategory of predicates.

With a simple language like the one we have defined, it clearly does. In fact there areseveral ways to circumscribe the new domain. Perhaps the most intuitive is to take the

predicate domain of the satisfaction

relation 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 claim



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 thatthe 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, wecould 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 anygiven individual, to include one which names that individual.

We can now see exactly why schema (3) caused so many problems not encounteredwith 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 to

run as follows: “p” must be replaced by an expression that asserts

possession 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 togovern 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 expression thatreplaces “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 ofcourse yields (1) —and demanding that both occurrences of “w” simply be re placed by asingle 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 donot assert possession of, properties. But this does not indicate that satisfaction is anyless natural an extension of substitution in the case of predicates than it is in the case ofnames. It indicates only that names and pre¬dicates contribute differently, in both theobject language and the metalanguage, to the truth values of sentences in which theyoccur. But in both cases the move to satisfaction represents an attempt to isolate thatcontribution 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 domainwith properties, since the resulting account of the semantic functioning of predicatesseems intuitively appealing. Intuition aside, this move is not generally adopted. Rather,it is standard to take this domain to consist of sets of individuals drawn from the namedomain. We can then think of predicates as asserting, of some individual, that it is amember of a given set. Once again we need not claim that predicates name sets, onlythat 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 the other;there are no sentences in which the contribu-tion of predicates is not equally wellexplained 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 possibleexpansions of the category of predicates will be differently circumscribed.

There are two remaining categories whose satisfaction domains we must specify: thecategory containing „or‟ (which I will call a sentential connective) and the category

containing „not‟ (which I will call a sentential operator). For simplicity, we can takesentential functions with a single connective variable to be satisfied by binary truthfunctions, and those with a single operator variable to be satisfied by unary truthfunctions. Again, there is no need to say that sentential connectives and operators nametruth 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 andoperators express truth func¬tions. Thus, taking “c” to be a connective variable, the

sentential func¬tion „Abe Lincoln was president c 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 thesetwo categories. According to the present account, there are only sixteen possiblesentential connectives and four possible sentential operators. The category of operatorscould 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¬tionto the truth value of sentences in which it occurs cannot be reduced to the expression ofa 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 sententialfunctions with single connective or operator variables. Thus, for the former we willhave:

(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 followingschema:

(5) U satisfies „. . . 0 . . if and only if. . . u . . . ,

requir ing that the expression replacing “u” must express the unary truth function named

by the expression replacing “U. ”

A Persistent Account of Logical Truth

Our technique for extending the notion of satisfaction to sentential functions with anarbitrary number of variables is again to employ sequences. But now our sententialfunctions may also contain variables of arbitrary type. Say that a sequence is any

function that assigns to each variable an object from the appropriate satisfactiondomain. 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 whose predicate variables are amongg\, ... ,gh\ all of whose connective variables are among c\, ... ,Ck\ and all of whoseoperator variables are among oi, ... , Ok- Let “\S(x, g, c, 0)‟ ” stand for a name of that

sentential function, and finally, let “S(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/, werequire 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 truthfunction/(o,). We then have:

(6) Sequence/satisfies „S(x, g, c, o)‟ if and only if

S(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 BenFranklin was president and Abe Lincoln wore a powdered wig.

When, for a given sequence/, we also have available object language expressions wi, . . ., w*; pi, .. . ,p*; b\, .. . , bk\ and u\,... , m*

which meet the above conditions on naming, assertion, and expres-sion, we can offerthe 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 betweensatisfaction and substitution. Satisfaction is just an extension — though not as simple anextension as it first appeared — of substitution. It allows us to extend our varioussubstitution classes to include expressions from any semantically well-behavedexpansion of the language. An expansion is well-behaved just in case any newmem¬ber of a given category of expressions stands in the specified relation to an objectin 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. Ourupcoming definition of logical truth will thus meet the requirement of persis¬tence, withthe 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, eitherof the general notion, which resists definition in principle, or even of satisfaction forsentential 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 satisfaction forarbitrary sentential functions of the language would proceed by a simple recursion onthe 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 anysentential function that results from uniformly replacing all 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 isTarski‟s definition of logical truth.

Logical Consequence

How should we define logical consequence? One route that might seem attractive is asimple reduction of this notion to that of logical truth. Certainly, if a sentence 5 is alogical consequence of a set of sentences K = {K\, . . . , Kr), then the single conditionalsentence whose antecedent is the conjunction of the members of K, and whoseconsequent is S, must be logically true. That is, 5 will be a logical consequence of K if

and only if the sentence



If K\ and . . . and Kn, then 5

is logically true. Given our definition of logical truth, it might seem natural to rally thisobservation 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 contains theexpressions „and‟ and „if .. . then,‟ or the equivalent, and this assumption would restrictthe applicability of the account.17 Second, we would have to assume that theseexpressions are always included in $, and this again restricts the generality of thesuggested definition.18 Finally, and most important, the reduction will work only if K isfinite, or, alternatively, if the language allows infinitely long sentences. For otherwisewe could never form the antecedent of our conditional 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 consequence are, notsurprisingly, 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) in which Sis a sentence of L, and K a set of sentences of L. An expression will he .said to occur inan argument (K, S) if it occurs either in 5 or in some member of K; we will call theargument truth preserving just in case either 5 is true or some member of K is false. So,for instance, any argument whose conclusion (that 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 selectionof fixed terms just in case it is truth preserving and every argument (K't S') that resultsfrom making one or more permissible substitutions for expressions occurring in (K, S)is also truth preserving. A permissible substitution is defined in the obvious way: allmembers of $ must be left untouched, and the replacement 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 validwith respect to $.

As Bolzano defines it, the logical consequence relation, like the property of logicaltruth, 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 everytruth-preserving argument becomes logically valid when we fix all of its componentterms. 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 substitutioninstances share the same conclusion, the continued truth of that sentence ensures the



truth preservation of those instances. On the other hand, if $ is empty, if we hold noterms fixed, then in general the only valid arguments will be those in which the samesentence 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 fixedterms; it can also be affected by the size of the substitution classes for the variableterms. In particular, the relation will not persist through semantically well-behavedexpansions of the language, although our choice of $ remains constant. Thus, when wefix 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 relation

will persist through semantically well-behaved expansions of the language. Let us takean inferential function (or, better, an argument form) to be any ordered pair whose firstmember is a set of sentential functions and whose second member is a single sen¬tentialfunction. 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 ]\\stin case/either satisfies S' or does not satisfy some member of K'.

Suppose now that (K',S') is an argument form that results from uniformly replacing allatomic expressions in argument (K, S), other than members of $, with variables ofappropriate type. Then we will say that (K, S) is logically valid with respect to $ just incase (K', S') is satisfaction preserving on all sequences. Finally, sentence 5 is a logicalconsequence, with respect to $, of set K if the corresponding argument (K, S) islogically valid with respect to $. This is Tarski‟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 areassured that any argument judged logically valid will remain so throughout thoseexpanded 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 substitutional tests must be demoted fromthe status of necessary and sufficient conditions to mere necessary conditions; to



achieve persistence, the limitations encountered with actual substitution classes must beovercome.

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 survivethe 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 different re¬sults, the problem will invariably lie in thelimited resources available in one or more of the original substitution classes. So anydivergence 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: thegoal of achieving a persistent account of the logical properties makes no sense except inthe context of a theory of (or assumptions about) how existing members of a categorycontribute, and how potential members could contribute, to the truth values of sentencesin 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 togive us resources with which to meet that demand.

In arriving at a definition of satisfaction for a sufficiently broad class of sententialfunctions, we attribute to each expression classed as a variable term a specific semanticfunction — naming an individual, asserting possession of a property, expressing a truth

function, and so forth. By populating a satisfaction domain with the appropriate type ofobject — individuals, properties, truth functions — we take a stand on how the existingcategory might be expanded: we condone new members so long as their semanticcontribution, their contribution to the truth value of sentences, can be charted in afashion similar to that of the present members. Thus, any expression that names anindivid-ual is treated as a potential member of the category containing „Abe Lincoln,‟

any expression that asserts possession of a property may belong to the categorycontaining „was president,‟ and any expression that expresses either a unary or a binarytruth function is admitted into the 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 byreplacing all atomic expressions, except those in $, by variables). Similarly, 5 is alogical consequence of K if and only if every sequence either satisfies S' or fails tosatisfy some member of K'. Note that, given this latter definition, logical truth can beseen as a reduced form of logical consequence: 5 will be logically true just in case it is aconsequence of the empty set, or, alternatively, if it is a consequence of any set of

sentences whatsoever.4



Interpretational Semantics

In Chapter 1, I remarked that the standard, model-theoretic defini¬tion of consequenceis an outgrowth of Tarski‟s account. I will begin this chapter by explaining how, 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 arelation, "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 thatwill become obvious, I adopt the term interpretational semantics for the Tarskianconception of model-theoretic semantics.

Interpretational and representational semantics occasionally inter¬sect. That is, wesometimes find that one and the same model-theoretic semantics can be viewed fromeither the interpretational or the repre¬sentational perspective. I will discuss a couple of

points of intersection: one is the simple semantics devised for the language of Chapter2, the other a slightly more intricate semantics for the language of Chapter 3. But inspite of the occasional intersection, interpretational and repre¬sentational semantics areradically different approaches to semantics, approaches whose adequacy must be judged

by completely different standards. In Chapter 2, I sketched the standards applied to arepre¬sentational semantics; as we will see, the counterparts for interpreta¬tionalsemantics are simply the criteria already discussed for delineat¬ing satisfaction domainsand for defining satisfaction. For in an interpretational semantics our class of models isdetermined by the chosen satisfaction domains; our definition of truth in a model is a

simple variant of satisfaction.

Distinguished Sentential Functions

How 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 acertain blurring of the careful distinction Tarski draws between sentences and sententialfunctions, between the ordinary expressions of the language and the variables weintroduced for defining the logical properties.

In considering the following changes, it will be convenient to assume we are interestedin logical truth and logical consequence only with respect to a. particular selection ^ offixed terms. When speaking of our sample language from Chapter 3,1 assume that ^ isthe set containing the atomic expressions „or‟ and „not.‟ In this way we avoid re peatedmention of ^ and can simply speak of the fixed terms and the variable terms of thelanguage. But it is important to keep in mind the (hence¬forth implicit) relativization toour choice of In particular, it is crucial to remember that variable terms are notvariables. Variable terms are ordinary atomic expressions of the language, differingfrom fixed terms only in their omission from



Recall that in testing a sentence 5 for logical truth, we first convert 5 to a sententialfunctional S'. S' results from uniformly replacing vari¬able terms with variables ofappropriate type. It does not matter which variables we choose in constructing S' so longas distinct variable terms receive distinct variables, and multiple occurrences of a singlevariable term are converted to multiple occurrences of a single variable. So, forexample, the sentence „Abe Lincoln was president‟ corresponds quite indiscriminately

to the sentential functions „xi gi,‟ „xi g2,‟ lX2 g2> and so forth.

The first modification we will make ensures that a specific sentential function 5*corresponds to each sentence 5 in the language. To do this we need only assign aspecific 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,‟ 'g\ the variablecorresponding 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 theresult 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 sequenceassigns objects from appropriate satisfaction do-mains to many variables that never

appear in the new, distinguished sentential functions. But it can hardly make anydifference 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 variableterms in the lan-guage. For the present language, such a limited sequence/* will be anyfunction that assigns members of the name domain to „x^‟ and „x2,‟ and members of the

predicate domain to lg{ and lg2.‟ Clearly these simpler sequences suffice for our current

needs.

Tarski‟s test for logical truth can now be characterized in the follow¬ing way: weconvert a sentence 5 to the distinguished sentential func¬tion 5* that results fromreplacing 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; ifnot, not. The test for logical validity proceeds similarly. First we replace an argu¬ment(K, S) with its distinguished argument form (K*, S*) — that is, the result of replacing allvariable expressions occurring in (K, S) with their chosen variables. We then check tosee that (K*, S*) is satisfaction preserving on all limited sequences/*. If so, 5 is alogical consequence of K; if not, not.

D-Sequences and D-Satisfaction



We are now halfway to model-theoretic semantics. The remaining change is equallyslight, though potentially more confusing. Since we have set up a one-to-onecorrespondence between variable terms and variables, and between sentences andsentential functions, there is a way to achieve the same results as our present testswithout bothering to detour through variables and sentential functions. The new methodwill yield a recognizable model-theoretic semantics for our language.

First we must introduce a new type of sequence, one whose domain consists of thevariable terms of the language rather than the chosen variables. Let us say that a director d-sequence is any function that assigns to each variable term an object from theappropriate satisfaction do¬main. Thus, a d-sequence will assign Ben Franklin directlyto 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, thefunction 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 andsentences. Specifically, say that a d-sequence f d-satisfies sentence 5 if and only if thecorresponding limited sequence/* satisfies the distin-guished sentential function 5*.

Although d-satisfaction is defined in terms of satisfaction, it is im-portant not to confusethe two notions. For one thing, if we briefly reflect on schema (6) of the last chapter, itwill 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 havethe following instantiation of (6):

(6.2) Sequence/* satisfies „Abe Lincoln was president‟ if and only if Abe Lincoln was 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 powder ed 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 Lincoln was

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 satisfaction andd-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 truesentence. 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 its variableterms 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‟ named 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 amatter of fact, wear powdered wigs, this sentence would have been false on theinterpretation suggested by d-sequence f

We have now, of course, arrived at model-theoretic semantics, though our ungainlyterminology could stand some revision. But be-fore making the final, terminologicalchange, let us note how Tarski‟s definitions of logical truth and logical consequencesurvive 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 incase it is d-satisfied by every d-sequence. Just so, an argument is logically valid, itsconclusion 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 orargument 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.1

Semantically Well-Behaved Reinterpretation

Consider 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 extendBolzano‟s substitutional tests for logical truth and logical validity, our stock ofsequences allowing considera¬tion of all semantically well-behaved expansions of thevarious substi¬tution classes. The technique of d-satisfaction, truth in models,em¬bodies precisely the same extension of the substitutional account, though the styleof the tests is slightly modified. In particular, no syntactic manipulations of thesentences or arguments being tested, no exchanges of variables for variable terms, arenow required.

We can think of the new technique in various ways. For example, we can obviouslyconsider it a simple abbreviation, somewhat confusing perhaps, of Tarski‟s original



method. As such, we must imagine the variable terms of the language doing doubleduty: on the one hand, they act as ordinary expressions of the language, taking part ingenu-ine sentences whose logical properties we hope to reveal. But when it comes timeto test for logical truth and logical validity, the variable terms also act as variables, theirreplacement by actual variables now rendered superfluous thanks to the slight technicalmodifications 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 morenatural, 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 thevariable 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 notencompass all conceivable reinterpre¬tations of the variable terms, but insteadencompasses 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 contributes by naming an individual, the

permissible reinterpretations of this expres¬sion are taken to be constrained by theavailability of “nameable” individuals, by the name domain of the satisfaction relation.Similarly, permissible reinterpretations of predicates are limited by the predicatedomain. 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. Forour 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 in which all

expres¬sions of the language mean what they actually mean. If our class of modelsomitted this assignment, we could not be sure that a logically true sentence was notactually false — that is, false when the variable expressions are interpreted in the normal

way. Similarly, if the in¬tended interpretation were not included in the test, we wouldhave no general assurance that logically valid arguments in fact preserve truth.

Obviously it makes no real difference whether we see models as interpretations of ourlanguage, or whether we simply view variable terms as variables “of convenience,” with

models cast as ordinary assignments to these not-so-ordinary variables. The differenceis purely heuristic. Either way, I will call the present conception of model-theoreticsemantics the Tarskian or interpretational view. Accord¬ing to it, our models are meantto range over all semantically well- behaved interpretations of some subset of theexpressions in the lan¬guage. Let us now contrast the interpretational perspective with

the representational view described in Chapter 2.



Samples of the Contrasting Views

The best way to emphasize the contrast between the interpretational andrepresentational views is to consider specific examples. In Chapter 2, I sketched asimple representational semantics for a language con¬taining five atomic expressions:

three were sentences („Snow is white,‟ „Roses are red,‟ and „Violets are blue‟), one aconnective („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¬tiveexercise to devise a Tarskian semantics while retaining the coarser parsing.

Let us begin the old way, introducing variables for each type of atomic expression. Forsentences we will use „pi,‟ „p2>‟ • • • ; for connec¬tives 'cu „C2,‟ ; and for operators „01,‟ „02.‟ • • • The next step is to

specify 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 acategory contribute, and differ in their contribution, to the truth values of sentences inwhich they occur. We can again take connectives and operators to express appropriatetruth functions, and construct the respective satisfaction domains accord¬ingly. Thus, itremains to settle on the sentence domain of the satisfaction relation.

As before, we will opt for the simplest plausible satisfaction domain. In the presentlanguage, we can explain the semantic contribution of any embedded sentence to itsembedding sentence in one of two ways: either the component sentence says somethingtrue or it says something false. Thus, we can take the sentence domain to consist of the

two truth values, true and false. Again, there is no reason to say sentences name truthvalues, any more than predicates name properties. I will simply say they have truthvalues; „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 operatorvariables. 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 ar bitrary sententialfunction of the language, and “S(p/s, c/l, o/u)” is to be replaced by a sentence that

results from inserting appropriate expres¬sions of the metalanguage for variables of thesentential function. The appropriateness of the replacement expressions will now bedeter¬mined by the following instantiation conditions: sentence s, must have the truthvalue /(/>;); 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 is





semantics much as we did the difference between the two perspectives taken on ourtheory 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 thecounterfactual claims that emerge are strikingly differ¬ent. Thus, from therepresentational perspective our semantic theory supports the claim that the sentence„Snow is white or snow is not white‟ would have been true even if snow had not been

white; that contin¬gency is, after all, depicted by various of our models. However, fromthe interpretational perspective no claim is made about what would have happened tothe truth value of our sentence had snow not been white. Rather, our theory supports thequite different claim that this sentence would still be true even if the componentexpression „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, weheld 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 thecontrasting representational perspective. For here again we can view our models ineither way.

To provide a representational account of these models, we begin by assuming that allthe expressions of our language have their ordinary interpretation, regardless of the

assignments made by a model. The purpose of assigning various objects to variousexpressions 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 modeldepicts a world in which Lincoln was president; if the individual does not, the modeldepicts one in which Lincoln was not president. The fact that the individual may happento be Ben Franklin (or perhaps an abstract object like the number one), and the propertythat of having worn a wig (or perhaps that of being an even number), has no bearing on

our interpretation of „Abe Lincoln‟ or „was presi¬dent.‟ On the contrary, theinterpretation of these expressions, their actual interpretation, is our key tounderstanding 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 ofthis sentence would have changed had „Abe Lincoln‟ named Ben Franklin and had „was

president‟ meant wore a powdered wig. From the representational perspective, itsupports 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 ofwearing a wig a handy prop. The same representational roles could have been playedequally well by innumerable other objects and properties, and in each case the moralwould have been the same: the sentence „Abe Lincoln was president‟ would have been

false had Lincoln not been president.

The Failure of Intersection

We have here two very different conceptions of model-theoretic se-mantics. Accordingto the representational view, the models ap-pearing in our semantics are simpledepictions of possible configura-tions of the “nonlinguistic” world, the world our

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 bysome intuitive metaphysics, all possible configu¬rations of the world receive somemanner of depiction, then sentences that come out true in all models are true regardlessof how the world might be, perhaps they are true simply due to the way the languageworks. Of course, should some possibilities be omitted, inadvertently or otherwise,these results will hold only modulo the metaphysical assumptions embodied in oursemantics. This is arguably the case with the model theory for our second language; thesemantic theory does not tell us, for instance, how the truth value of sentences wouldhave been affected had Lincoln not existed. Perhaps there are other possi¬bilities ourtheory 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 theset ^ of fixed terms. A sentence is true in a given model if, so to speak, what it wouldhave said about the world on the suggested interpretation is, in fact, the case. Thus,sentences that come out true in all models are true regardless of how we interpret asubset 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-behavedreinterpre-tations of the variable terms. It is assumed that „Abe Lincoln‟ would not have

functioned like „Nix,‟ or even like the considerably less bizarre „Pegasus.‟ The semantic

theory does not tell us how the truth values of our sentences would react to suchreinterpretations.

With the semantic theories considered in the last section, the two conceptions seemaptly described as differences in perspective: to move from one to the other requiresnothing more than a subtle shift in gestalt. But it would be a serious mistake to imaginethat this will always be the case. Indeed in our two simple examples we have just beenlucky; 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 encounterone 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 noway 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 truth values to

„Roses are red‟ and „Violets are blue.‟ These models can still be takenrepresentationally, but as such they contain an obvious omission: we have no modelsthat depict worlds in which snow is not white. A similar problem would arise with oursecond language were we to include, say, „Abe Lincoln‟ and „was president‟ in Among

the 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, anysequence 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 languagewhen „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 truthfunction to our sole connective. If we try to view such models representationally, wemust somehow imagine that „or‟ receives its ordinary interpretation and that our

assignment of various truth functions to this expression is just a technique forrepresenting possible configura¬tions of the “nonlinguistic world.” But there 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 modelsdepict 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 thelanguage 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 thenumerals („0/ „1/ „2/ etc.) and function symbols („+,‟ „x,‟ etc.) of the language. Thus,

one model might assign the empty set to „2/ the set containing the empty set to „4,‟ and

set union to „+.‟ In this model— that is, according to this interpretation —„2 + 2 = 4‟comes out false, since the union of the empty set with itself is the empty set, not the setcontaining the empty set. Such a semantics makes perfect sense from theinterpretational standpoint, but obviously cannot be viewed repre-sentationally. There isno 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 itsays — that is, what it actually says — is necessarily true.

In all of these cases, the theories described meet the standards of interpretationalsemantics, 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 therequirements of representational semantics but that violate the Tarskian conception.Consider a simple example. Clearly, a perfectly acceptable representa¬tional semanticsfor our second language could get by with far fewer models than are needed for a

plausible interpretational semantics. Many of the models inherited from theinterpretational semantics are representationally isomorphic. That is, although twomodels might assign different individuals and properties to the names and predicates ofour language, this does not mean they depict different configurations of the world. Allthat matters to the depiction is whether the individ¬uals assigned to the names have ordo not have the properties assigned to the predicates. For this language arepresentational semantics could get by with a small number of nonisomorphic models:sixteen, to be exact. Viewed interpretationally, any such move would constitute anunmotivated restriction of the class of semantically well-behaved reinterpretations of thelanguage, an unjustified limitation on the name and predicate domains of the

satisfaction relation. Such a se¬mantics would thus be ruled out by the interpretationalguidelines.

To take a more interesting example, recall from Chapter 2 our discussion of arepresentational 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 assignstrue to more than one atomic sentence. This gives us four models rather than theoriginal eight, the limitation being motivated by the obvious fact that the remainingmodels would not depict genuine possibilities. Now notice that the resulting semanticswould 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 that Lincoln had a beard, andthereby 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 suchinterpretations individually, it must surely be possible to so interpret themsimultaneously. Ruling out models that assign both of these interpretations at once is nomore 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, this restriction from eight to four models,though easily motivated from the representational standpoint, would make little sense ininterpreta-tional semantics.

Clearly, representational and interpretational semantics are entirely different enterprises,governed by entirely different standards. They are not simply two perspectives fromwhich we can view an arbitrary semantics. The two approaches do happen to cometogether at certain fortuitous points, in simple theories that do not explicitly violateeither standard. Such was the case with the examples discussed in the last section: thesame 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 occasionallyintersect, 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 thanthat a brick can both break a window and hold up a bookshelf. But what is important tonote here is that the intersection of these approaches is not trivial. Not trivial, but onlyin this somewhat trivial sense: it does not always happen. If we had merely describedtwo perspectives, two ways of viewing one and the same endeavor, then everyinterpretational semantics would have a representational counterpart, and everyrepresentational 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 tomodel.

5

Interpreting Quantifiers

Before going on we should consider one final example, an example in which the samemodel theory appears at first glance to satisfy both the aims of interpretationalsemantics and the aims of representational semantics. In the languages we haveconsidered so far, quantifiers have been conspicuously absent. Yet the standard modeltheory for first-order quantified languages seems an obvious case in whichinter¬pretational and representational semantics intersect — or so we might assume.

Actually, the situation is not so simple, and thus this final example goes beyond mereillustration. The motivation underlying the tradi-tional technique of defining a first-order model seems quite straight-forward when we imagine ourselves offering arepresentational semantics. But it turns out that those same models, considered inter-

pretationally, embody a significant departure from Tarski‟s analysis of the logical

properties. The departure stems from the introduction of what I call cross-termrestrictions on the permissible interpretations of expressions.

Cross-term Restrictions

Suppose our second language were supplemented with the expression „something,‟ the

unrestricted (or trivially restricted) existential quanti¬fier. The standard semantics forthe resulting language would have us build models in the following way: first wechoose an arbitrary set called the universe or domain of the model; second we choose afunction that assigns ail objec t from that set to each name in the language, and a .subsetof that set to each predicate.1 Truth in a model is defined recursively, the clausegoverning 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.‟2



These models have a simple and natural motivation from the repre¬sentationalviewpoint. As always, the representational semantics draws no distinction betweenfixed and variable terms: again, the purpose of assigning objects of various sorts to thenames and predicates is purely representational, a technique precisely parallel to ourearlier account. The new element in our models, the universe set, provides some addeddetail to our representation: it allows us to depict worlds with various populations, andwith various distributions of properties among those populations. In particular we canrepresent worlds in which, say, someone has been president, though that someone isneither Washington nor Lincoln. And once again, according to the present system ofrepresentation, Washington and Lincoln are always depicted as existing, their proxiesalways chosen from among the universe set.

Suppose we try now to construct a complementary, interpretational account of thesemodels. 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 treatmentas before, they have been given the status of fixed terms. Similarly, the names and

pre¬dicates are considered variable expressions, since they receive differ¬entinterpretations in different models. The problem is to decide on the status of„something,‟ the newly introduced quantifier. There are several accounts we might givehere; I will describe the two most natural. In the end both of these come to much thesame thing; in the end neither is satisfactory.

As a first shot we might judge the expression „something‟ to have the status of avariable term, with the range of permissible reinterpre-tations limited to variously

restricted existential quantifiers. Reverting to our pre-model-theoretic terminology, wecan take a sentential func¬tion containing the variable „£‟ in place of „something‟ to be

satisfied by an arbitrary set. The underlying idea here is this: for each such set there is a possible expansion of the language which contains an expres¬sion that existentiallyquantifies over that particular set. For example, our present language contains neitherthe expression „someone‟ nor the expression „somedog.‟ But the satisfaction domain for

this class of expressions will include the set of humans and the set of dogs. Thus, weneed only define satisfaction in such a way that for some sequences/ we will have:

/ satisfies „E was president‟ if and only if someone was presi-dent

h satisfies „E was president‟ if and only if somedog was presi-dent.

If / assigns the set of humans to then the expression „someone‟ quantifies existentially over the set named by “/(„£‟)”; while if h assigns the set of dogs to the same semantic

relation holds between „somedog‟ and h(*E‟).5

Here we have taken existential quantifier to constitute a single seman¬tic category, the possible members of which differ only in their (per¬haps implicit) restrictions. Theexistential quantifier domain of the satisfaction relation thus consists of these sundry

restriction sets. In choosing a domain for our model — that is, an assignment to„some¬thing‟ by our d-sequence — we are simply selecting one possible inter¬pretation



from that semantic category.4 Hence, „something‟ might have meant someone or

somedog, but not everything, everyone, eachdog, thedog, and so on. Though perfectlyunderstandable, these latter inter¬pretations have been ruled semantically ill-behaved:so interpreted, „something‟ would no longer contribute as an existential quantifier to the

truth values of sentences in which it occurs.

The second approach is similar. Suppose we parse our language so that „some thing‟ is

seen as a complex expression formed by joining the determiner „some‟ with the

common noun „thing.‟ We might then consider „some‟ to be a fixed term and „thing‟ to

be the expression subject to reinterpretation. According to this view, our semanticstreats three expressions as members of —„some,‟ „or,‟ and „not‟— and the remainder asvariable expressions. Our language happens to have only one common noun, but thereare obviously semantically well-behaved expansions which contain others. Indeed foreach set — whether the set of humans or the set of dogs — our language might have

included a common noun that extends to all and only the members of that set. Hence, toensure a persistent yield of logical truths we can take the common noun domain of thesatisfaction relation to consist of sets. That is to say, we can allow our models, our d-sequences, to variously interpret „thing‟ as meaning human, dog, and so forth.

According to this approach, our “universe” set is meant to provide one such

interpreta¬tion for the common noun.

The difference between this and the earlier view is primarily one of taste. Both accountsallow us to construe „something was president‟ as meaning some dog was president,

while neither permits the construal everything was president. The first account

discriminates between these by invoking a category of existential quantifiers, the second by holding iixed the determiner „some.‟ The first sees the interpretation everything as

semantically ill-behaved, the second as disregarding our selection of fixed terms.5

I said earlier that neither of these accounts is entirely satisfactory. The problem is this.Recall that our models are constructed by first specifying a “universe” set and then

choosing appropriate assign-ments, objects or sets, for the names and predicates in ourlanguage. As before, an assignment to „Abe Lincoln‟ must fall within the name domain

of the satisfaction relation; an assignment to „was president,‟ within the predicate

domain. We have now described two ways of viewing our selection of a universe set

consistent with the aims of interpretational semantics. On the one hand we see ourselvesas-signing to „something‟ a member of the satisfaction domain set aside for existential

quantifiers (the class of all possible quantifier restriction sets); on the other we seeourselves assigning to „thing‟ a member of the common noun domain of the satisfaction

relation (the class of all possible common noun extensions).

According to either of these accounts, each model is simply a d-sequence, anassignment of some object within the appropriate satis-faction domain to each variableterm. But notice that we demand more of a model than that it be an acceptable d-

sequence. We allow models that assign Ben Franklin to „Abe Lincoln,‟ and also modelsthat assign Fido to „Abe Lincoln.‟ Both are considered permissible reinterpre¬tations of



this variable term. Further, we allow models that assign the set of humans to„something‟ (or „thing‟), and models that assign the set of dogs to „something‟ (of

„thing‟). Again these both generate well- behaved reinterpretations of a variable term.But we do not permit a model simultaneously to assign Ben Franklin to „Abe Lincoln‟

and the set of dogs to „something‟; nor do we include models that assign Fido to „Abe

Lincoln‟ and the set of humans to „something.‟ For recall that the object assigned to

„Abe Lincoln‟ must fall within the universe set of our model, the set assigned to

„something.‟ Clearly there are plenty of d-sequences in which this will not happen. Butwe are no longer admitting all d-sequences into the class of models. So far this changein strategy seems quite unwarranted.

Demanding that our interpretation of „Abe Lincoln‟ not only fall within the name

domain of the satisfaction relation but also that it somehow be constrained by theinterpretation of „something‟ (or „thing‟) imposes a cross-term restriction on the class of

models. In the present semantics, certain d-sequences are excluded from the class ofmodels not because they suggest a semantically ill-behaved interpreta¬tion of anyindividual expression, but because the interpretations of different expressions fail tostand in some fixed relation to one another. Specifically, if/is an arbitrary d-sequencefor the current language, it will be disqualified as a model should either /(„Abe

Lincoln‟) or /(„George Washington‟) not be members of /(„something‟), or should

/(„was president‟) or/(„had a beard‟) not be subsets of/(„something‟).

So far we have seen no motivation for imposing cross-term restric-tions in aninterpretational semantics. At first glance we have no better reason for requiring that

/(„Abe Lincoln‟) be a member of /(„some¬thing‟) than we have for demandingthat/(„Abe Lincoln‟) be a member of/(„had a beard‟) or that /(„had a beard‟) be a subsetof /(„was presi¬dent‟). Any of these restrictions will naturally alter the yield of our

semantics, will affect which sentences and arguments qualify as logi¬cally true orlogically valid. As we will see, for this very reason any such restriction violates theintegrity of Tarski‟s account of the logical properties.

Substitution, Persistence, and Cross-term Restrictions

Recall the motivation underlying Tarski‟s definitions of logical truth and logical

validity. First is the substantial point of agreement with Bolzano‟s substitutionaltreatment of the logical properties: according to Tarski, meeting the substitutional testsis a necessary condition for a sentence to be logically true or an argument to be logicallyvalid (all, of course, with respect to the chosen ^). A proper definition of satisfac¬tionwill ensure that this necessary condition is met. The second point motivates the move tosatisfaction: passing the substitutional test is not «1 sufficient condition, according toTarski, since the logical properties must be persistent. That is, an adequate account oflogical truth and logical validity must show that these properties persist through well-

behaved expansions of the language. Or, turning persistence around, sentences and

arguments that do not qualify as logically true or logi¬cally valid should never come toqualify merely through a purge of otherwise irrelevant expressions from the language.



In the last section I remarked that the introduction of cross-term restrictions requires asignificant de parture from Tarski‟s original account. We can now pinpoint that

departure: if we do not admit all d-sequences into the class of models, and if thisrestriction materially alters the yield of our semantics, then we cannot both demand thatthe logical properties be persistent and demand that all logical truths and logically validarguments meet the substitutional tests.

The reason is simple. A restriction of the class of models can affect the output of oursemantics only by increasing the set of logical truths or logically valid arguments. Andit can do so only if we have excluded at least one d-sequence that provides a well-

behaved interpretation on which some logical truth is false (is not d-satisfied) or somelogically

valid argument does not preserve truth (is not d-satisfaction preserv¬ing). In which casewe can expand the language to include expressions whose actual interpretations areexactly those specified by the offend¬ing d-sequence. But then in the expandedlanguage the original sen¬tence or argument will fail the substitutional test with exactlythe same choice of fixed terms. Either persistence has been abandoned (a sen¬tence,say, is judged logically true in the original fragment but not in the newly expandedlanguage) or else the substitutional test has been rendered violable (the sentence is

judged logically true in the ex¬panded language in spite of its false substitutioninstances).

Consider our current semantics. So long as we maintain the cross- term restriction, the

following argument is valid —holding fixed „some,‟ „or,‟ and „not‟: (A) Abe Lincoln was president.

So, something was president.

But note that it is crucial here that we exclude from our class of models the d-sequencein which „thing is assigned the set of dogs and the remaining expressions receive theirintended interpretations. For oth¬erwise the argument would not preserve truth in everymodel.

Now suppose our language were expanded to contain the common noun „dog.‟ Byincluding the set of dogs in the appropriate satisfaction domain, we have explicitlyapproved this expansion as semantically well- behaved. Yet as soon as we introduce thisexpression into the langu¬age, there will be a permissible substitution instance of thesame argu¬ment —still holding fixed „some,‟ „or,‟ and „not‟— which fails to be truth

preserving:

(A') Abe Lincoln was president.

So, some dog was president.



We must now choose either to allow (A) to remain valid in the new language, despitenon-truth-preserving substitution instances like (A'), or to declare (A) invalid, eventhough it was judged valid in the preceding fragment. One way we give up substitution;the other per-sistence.

For better or worse, an interpretational semantics that avails itself of cross-termrestrictions cannot avoid straying from Tarsk i‟s original conception of the logical

properties, assuming of course that the re¬strictions actually alter the output of thesemantics. In the next section I will consider a slightly different account of the presentsemantics, one that attempts to minimize, or at least disguise, the change in underlyingconception. But first let me emphasize the generality of the problem just described.

Cross-term restrictions impose constraints on the simultaneous in-terpretation of two ormore expressions. There are, of course, in-numerable such constraints we might imagineimposing. For example, we might require that /(„Abe Lincoln‟) be a member of /(„was

presi¬dent‟), or that /(„was president‟) be a subset of/(„had a beard‟). The first of these

would restrict the simultaneous interpretation of expressions from two differentsemantic categories, as we do when we constrain the interpretation of „something‟ and

„Abe Lincoln,‟ while the second involves our interpretation of two expressions within

the same seman¬tic category.

Now the ultimate effect of any cross-term restriction is the same: it excludes certain d-sequences from the class of models. And in so doing the restriction will, except in trivialcases, expand both the set of sentences that come out true in all models and the set of

arguments that preserve truth in all models. For this reason the use of any cross- termrestriction will require that we abandon one of Tarski‟s de-siderata: either we allow thatlogical truths (or logically valid argu-ments) can occasionally be turned false (non-truth-

preserving) by substituting for variable terms, or we admit that the logical properties arenot persistent through well-behaved expansions of the language.

So far I have emphasized one common use of cross-term restric-tions: placingconstraints on the simultaneous interpretation of a quantifier and a name or predicate.But there is no significant differ-ence between this sort of restriction and the use of so-called meaning postulates. The technique of excluding any model that falsifies a par-

ticular meaning postulate is simply a roundabout way of imposing cross-termrestrictions, of limiting the class of models to d-sequences in which the interpretationsof two of more variable expressions stand in some fixed relation to one another.Commonly these expressions will fall within the same semantic category, but there is noreason the same technique might not be used to legislate restrictions across categories aswell. So, for example, we might exclude any model in which „Abe Lincoln was

president‟ is false, thus indirectly imposing the first re¬striction mentioned two

paragraphs back. Of course, not all cross-term restrictions can be imposed indirectly,through an appeal to meaning postulates. To use an obvious example, no set of

sentences in any first-order variant of our current language would guarantee that /(„was president‟) has the same cardinality as /(„had a beard‟).



The important point, though, is not how cross-term restrictions are imposed, whetherthrough meaning postulates or through direct con-straints on d-sequences. Rather, it isthat any such restriction, however enforced, stands in equal violation of Tarski‟s

original account of the logical properties. I suspect this violation underlies manytraditional

objections to the use of meaning postulates in an interpretational semantics, in particularto the use of postulates that constrain the simultaneous interpretation of two predicates.But to my knowledge, no similar objections have been voiced concerning the cross-termre-strictions built into the standard semantics for quantified languages.

No doubt one reason the use of meaning postulates has been con-sidered objectionablewhile the standard semantics has not is simply oversight. For one thing, with meaning

postulates the violation of Tarski‟s analysis is often quite noticeable, since the sorts of

postulates generally used give rise to immediate failures of the substitutional test. Forexample, if our language contained the predicate „was an elected official‟ and we

required that /(„was president‟) be a subset of /(„was an elected official‟), then the

inference from „George Washington was president‟ to „George Washington was anelected official‟ would come out valid. But here we get a non-truth-preservingsubstitution instance without expanding the language: we need only substitute „had a

beard‟ for „was an elected official.‟

It should be clear, though, that this immediate violation arises only because the relevantsubstitution class contains more than one mem¬ber, and of course not all of these

members are subject to identical restrictions: we could hardly place the same restrictionon „had a beard‟ as we have placed on „was an elected official,‟ for this would rule out

the intended interpretation of the language. The reason we had to expand our languagein order to find a non-truth-preserving instance of (A) was simply that the presentlanguage offers no nontrivial substitution instances: there is only one expression in therelevant substitution class (the class of common nouns or the class of existentialquantifiers, depending on our parsing). Had we started out in the expanded version andimposed our restrictions, violations of substitution like (A') would arise immediately.

The main objection to using meaning postulates in an analysis of the logical properties,

though, does not involve such failures of substi¬tution. Rather, it is the apparentcircularity that their use injects into the analysis. Suppose we considered the inferencefrom „George Washington was president‟ to „George Washington was an elected

official‟ to be intuitively valid, and wanted it to be judged so by our interpretational

semantics. Of course, there are many interpretations of the predicates that make the firstof these true and the second false, but if we allow appeal to meaning postulates, we caneasily exclude these: we need only throw out any interpretations that falsify the meaning

postulate „All presidents are elected officials.‟ Then, of course, this postulate will come

out logically true — that is, true in all the remaining interpretations — and the

corresponding inference will come out logically valid.



The problem, though, is that our decision to take this sentence as a meaning postulate isnot guided by anything over and above our intuitions about which inferences should bevalid and which sentences should be logically true. Thus, we risk reducing our generalaccount of logical truth to something like this: a sentence is logically true if it is true inall interpretations that do not falsify any sentences that seem logically true. Unless wehave an independent account of when a sentence qualifies as a meaning postulate, onethat does not simply appeal to the logical properties we are trying to account for, theiruse in a general definition of those properties is simply circular.

All of this applies equally to the use of cross-term restrictions. Tarski has given us arelatively straightforward technique for specifying the class of d-sequences orinterpretations for a given choice of variable terms. If we must then impose cross-termrestrictions to “fine tune” the semantics, but can motivate the restrictions only by noting

that the unmodified semantics gets things wrong, then our analysis is in serious trouble.

Interpreting Cross-term Restrictions

Tarski‟s analysis is supposed to provide the theoretical underpinning of the

interpretational approach: constructing such a semantics is thought to be an applicationof Tarski‟s general account of the logical properties to a particular language. In

subsequent chapters I consider general questions about the adequacy of Tarski‟s

definitions; at present our problem is more immediate. It is clear that there are variousinterpretational semantics for our quantified language — that is, various theoriesconsistent with Tarski‟s original definitions. The two accounts sketched earlier in the

chapter are examples; others would result from different selections of fixed terms ordifferent demarcations of satisfaction domains. Yet these seem consistent with Tarski‟s

definitions only if we admit all d-sequences into our class of models. But when we do,various intuitively valid arguments — such as argument (A) — do not preserve truth inevery model. Hence, they do not qualify as logically valid according to an unmodifiedTarskian semantics.

Now if our only concern were to give a Tarskian semantics that judged (A) valid, itwould be quite easy to solve the problem. One way would be by brute force: we couldsimply include all of the constituent expressions in ft. But this would produce an equally

counterintuitive yield of “logically valid” arguments; in particular, any argument with„Something was president‟ as conclusion would come out valid, this being, 011 thatselection of ft, a logical truth. Or we could try a more ddicatc approach, say, holdingfixed the interpretation of„something‟ (that is, of both „some‟ and „thing‟) but not of the

names or predicates. But this selection also runs into problems — for example, if ourlan¬guage contains identity. Thus, we do not want „There are at least two things‟—

symbolically, „3x3;y(x 4- y)‟—to come out logically true, much less „There are at least

two billion things.‟ Yet these will be deemed logical truths if we allow no variation inthe interpretation of the existential quantifier.6

The problem is that with quantified languages, there is no single selection of fixed termsthat gives exactly the right judgments about validity, at least with an unmodified



Tarskian semantics. Yet we have seen that there is a simple modification of theunrestricted semantics that does seem to yield an intuitively plausible collection oflogical truths and logically valid arguments. It is the modification incorpo-rated into thestandard semantics for such languages: impose cross- term restrictions on thesimultaneous interpretation of various atomic expressions. But it looks like we canemploy such restrictions only by abandoning the very conception of the logical

properties that under¬lies the interpretational perspective.

What then should we say about the standard semantics for quanti-fied languages? Thissort of semantics is almost universally thought of as firmly grounded in Tarski‟s

analysis. But our recent considerations suggest otherwise. When applied to our samplelanguage, the result seems acceptable as a representational semantics, but unacceptableas an interpretational semantics. This seems surprising enough. But the situation is evenworse when we consider the usual semantics for, say, the language of first-order number

theory. For there, the semantics clearly cannot be construed representationally, forreasons sketched in Chapter 4, but neither does it conform to the interpretationalguide¬lines, thanks to the use of cross-term restrictions. Can it be that ourunderstanding of the standard, first-order semantics is so completely undermined by theintroduction of cross-term restrictions?

There are three options open to us. First of all, we can simply accept the surprisingconclusion of the recent analysis. We would then have to count our sample, first-ordersemantics among those in which the representational and interpretational approachesfail to intersect, and banish the semantics for first-order number theory to a limbo some-

where between the two approaches. Second, we might try to revise Tarski‟s generalaccount so that the occasional use of cross-term re-strictions is vindicated, is shownconsistent with some modified interpre¬tational analysis of the logical properties.Finally, we might argue that the recent considerations are somehow faulty, that in factthe standard semantics is perfectly consistent with Tarski‟s original definitions.

I will not consider the first two alternatives in any detail. The first option is, obviously,the option of last resort. If we cannot devise interpretational semantics that produce

plausible results for simple first-order languages, then we have uncovered a serious, ifnot devas-tating, defect in Tarski‟s general account. At present this option must remain

on the sidelines; I will return to it later.

The main problem with the second option is simple: the requisite modification ofTarski‟s analysis is not at all apparent. The most direct way to incorporate cross-termrestrictions is obviously circular. We can hardly say that a sentence is logically true in L

just in case it is true in all interpretations that remain after imposing those cross-termrestric¬tions needed to produce the proper collection of logical truths for L.

Of course, the circularity of this definition could be disguised in various ways. Forexample, we might point out that Tarski‟s original account proceeds by first specifying

a class of languages that are seman¬tically similar to L — the fixed terms are interpretedidentically, the interpretations of variable terms fall within the same satisfaction



do¬mains, and the underlying definition of satisfaction is the same. It then offers thefollowing definition: a sentences is logically true inLjust in case for any language L' thatis semantically similar to L, S is true in L'. Consequently, we might claim that cross-term restrictions serve simply to tighten the “semantic similarity” relation. Here, of

course, the entire burden is transferred to our account of the new “semantic similarity”

relation. If all we can say about languages in which /(„Abe Lincoln‟) is not a member of

/(„something‟), or in which /(„was president‟) is not a subset of /(„was an elected

official‟), is that certain intuitively valid arguments of L fail to preserve truth (and sosuch languages must be semantically dissimilar to L), then our definition is just ascircular as the trivial one given above, though it might initially appear less so.

Other than incorporating cross-term restrictions in a blatantly circu¬lar fashion, noobvious modifications of Tarski‟s account yield defini¬tions consistent with, on the one

hand, the use of cross-term restric¬tions and, on the other, the basic intuitions

underlying the interpretational approach. To be sure, we can hardly rule out the possibility that such a modified account can be developed. But until we have areasonably clear account to look at, little more can be said about the second option.

A few words should be said about the third alternative. It might be argued that althoughcross-term restrictions are indeed inconsistent with Tarski‟s original definitions, the

need for them arises only when we construe the universe or domain of a model asinterpreting an individual expression in the language (whether the quantifier „some-thing‟ or the common noun „thing‟). For then the requirement that /(„Abe Lincoln‟) be

chosen from the universe set appears as a direct restriction on the joint interpretations of

„something‟ and „Abe Lincoln,‟ and hence no more acceptable than, say, invoking themean-ing postulate „All presidents are elected officials,‟ or demanding that /(„was a

president‟) be a subset of /(„was an elected official‟).

But there are other ways of describing the standard semantics. In particular, when wechoose the universe of a model, we commonly see ourselves as interpreting an implicit

parameter of the language, the language‟s domain of discourse. Now, we might claim

that such implicit parameters place constraints on the interpretation of severalexpres¬sions in the language, but that they do not directly provide the inter¬pretation ofany. In this way, it might be argued, the standard seman¬tics can avoid the illicit appeal

to cross-term restrictions. After all, demanding that our interpretation of„Abe Lincoln‟fall within the set chosen as the domain of discourse does not restrict the jointinterpre¬tation of two expressions, but rather the joint interpretation of an indi¬vidualexpression and a parameter of some entirely different sort.

But what exactly have we bought by adopting this alternative de-scription? First of all,it is clear that our semantics still treats „some-thing‟ (or „thing‟) as a variable term: its

contribution to the truth values of sentences still differs as radically from model tomodel as that of „Abe Lincoln‟ or „was president.‟ Consequently, if we were to list all

the various constraints imposed by our new “implicit parameter,” we would have toinclude among them the following three: „Abe Lincoln‟ can name any individual we



please, so long as that individual is a member of the domain of discourse; „was

president‟ can have any exten¬sion we please, so long as that set is a subset of the

domain of discourse; and finally „something‟ can be restricted however we please, so

long as the restriction set is identical to the domain of discourse.

None of those constraints directly involves the interpretation of more than oneexpression. But it should be clear that by arranging our constraints like spokes aroundan implicit parameter, we do not avoid cross-term restrictions but simply honor themwith a name. Obviously, any cross-term restriction can be imposed in a similarlyroundabout fashion. For example, rather than requiring that /(„was president‟) be a

subset of /(„was an elected official‟), we might instead posit a genus domain, an implicit

parameter that constrains both the interpretation of„was president‟ and the interpretation

of„was an elected official,‟ but that is not thought of as directly interpreting either. We

could then allow models to assign „was president‟ any extension at all, so long as that

set is a subset of the genus domain, and similarly permit „was an elected official‟ tohave any extension we please, so long as that set is identical to this same parameter.

Appealing to implicit parameters may provide an alternative tec h-

nique for imposing cross-term restrictions, a technique that differs somewhat fromeither the direct imposition of those restrictions or the less direct appeal to meaning

postulates. But it does not address the fundamental conflict between the use of suchrestrictions and Tarski‟s general account of the logical properties. The notion of a

domain of discourse does not solve the problem, but simply disguises it.

For now, we seem to be left with the first option. The standard model theory for first-order languages seems to violate the very guide¬lines that underlie the interpretationalapproach to semantics. Of course at present, this is only a tentative and provisionalconclusion. It is always possible that Tarski‟s account of the logical properties can be

suitably revised so that the standard cross-term restrictions used in first-order modeltheory turn out to be consistent with it. I will not be concerned with this possibility anyfurther, since the real problem with Tarski‟s analysis applies equally whether or not we

employ cross-term restric¬tions. But this will be the topic of Chapters 7 through 9.

RecapitulationThe superficial similarities between representational and interpreta-tional semantics areobvious but misleading. In fact, these two “views” of model-theoretic semantics arecompletely different approaches to charting the semantic properties of a language. Thisdifference comes out most clearly in the radically different standards that must be usedin judging the adequacy of the two main features of the theory: the class of models andthe definition of truth in a model. The class of models is adequate for a representationalsemantics if it contains a representative for each genuinely possible configuration of the“em¬pirical” or “nonlinguistic” world. To make this judgment we must naturally

presuppose some technique of representation — we must un¬derstand what our modelsmean — as well as various intuitions about what is and is not a genuine possibility. With



interpretational seman¬tics, the class of models (d-sequences) is determined by thesatisfaction domains assigned to each category of expression — more accurately, by thedomains assigned to those categories containing members not included in ft, the set offixed terms. Thus, the class of models in an interpretational semantics is to be judgedaccording to the criteria for delineating satisfaction domains. Such a domain mustcontain an ob¬ject for each existing member of the given semantic category, as well asobjects for other potential members of the same category — intuitively, expressions thatwould contribute similarly to the truth values of sen¬tences in which they occurred.

Standards forjudging the relation of truth in a model differ accord-ingly. Withinterpretational semantics we must ask whether sentences declared true in a particularmodel would indeed have been true under the suggested interpretation of the variableexpressions. This is simply the intuitive content of the satisfaction schemata of Chapter3, the relation of truth in a model being our trivial modification of the earlier

satisfaction-by-a-sequence. Representational semantics, on the other hand, requires thata sentence be declared true in a given model if and only if it would have been true hadthe model been accurate — that is, had the world actually been as depicted by thatmodel.

Once we take these differences seriously, once we realize that the criteria that apply to atheory of “x is true in W” are entirely differ ent from those that apply to a Tarskiantheory of “x is true in L,” then it should seem a remarkable fact when one and the same

“model- theoretic semantics” admits of both readings. But such points of inter¬section

do occasionally occur. Thus, with the simple semantic theories devised for our two

nonquantified languages, the same class of models and the same definition of truth in amodel met the separate demands imposed by the two approaches. With these semantictheories there is a straightforward sense in which the importance of the perspective weadopt is minimized. Here, as with the “theory” of truth tables discussed in Chapter 2,

the perspective makes little difference simply because both are readily available.

But clearly such points of intersection are the exception, not the rule. This was alreadyobvious by the end of the last chapter, when we considered various cases in which thetwo approaches failed to inter-sect. Yet there is a tendency to overlook this divergence,to assume that Tarski‟s analysis of the logical properties is correct because it

guaran¬tees, say, that logical truths will be true in all possible worlds, that they will benecessarily or analytically true. This might be a defensible position if Tarski had indeedgiven us a“reduction of possible worlds to models,” if his analysis required logical

truths to turn up true in all the models appearing in an adequate representationalsemantics. But here Tarski‟s analysis is clear and unequivocal; it is the model-theoretictranslation that engenders the confusion.

In Chapter 2,1 remarked on the obvious interest we may have in an account of “x is true

in W” for a fixed language L. The importance of such a theory comes not from a general

account of logical truth or logical consequence but from the illumination it may shed onthe semantic rules of the language. At first glance, there is considerably less interest



attaching to an account of the complementary “x is true in L” for a fixed world W. One

simple reason for this is that with a sufficiently broad range of languages to choosefrom, all sentences are precisely on a par. That is, for any true sentence of (say) English,we can devise some languages in which it is false; similarly, any false sentence canalways find a home in which it happens to be true. This in spite of any logical orsemantic properties the sentence may originally have had. Sentences, at least in thesense in which these are things that can wander from language to language, do not carrywith them the semantic characteristics necessary to ensure any truth value.

Now, Tarski‟s account changes a superficially uninteresting study into a potentially

important investigation. If Tarski‟s analysis is cor -rect, then we have a standardtechnique for narrowing in on a limited range of languages against which the relation “x

is true in L” gains considerable significance. But here again we must not get the signifi-cance turned around. Our goal in applying Tarski‟s account is not simply to specify, by

whatever means available, some range of lan-guages whose shared truths happen to bethe logical truths of the original language. Of course, there will always be such acollection of languages: at worst, we could treat all expressions as variable and take alllogical truths as meaning postulates. But this is simply to give up Tarski‟s generalaccount of the logical properties and, so it would seem, to undermine any interest thatmay originally have motivated an account of “x is true in L.” This is the sacrifice we

risk when we resort to cross-term restrictions.

6

Modality and ConsequenceSo far, what we have by way of extensional evidence for Tarski‟s analysis is a rather

mixed and confusing bag. It is clear that with the extremely simple languages ofChapters 2 and 3, the definitions pro-duce an intuitively plausible yield of logical truthswhen we hold fixed the right expressions, specifically, when we hold fixed „or‟ and

„not,‟ the two terms traditionally considered “logical constants.” But when we make

other choices for when we treat other expressions as logical constants, the account produces strikingly counterintuitive results. Tarski himself was the first to note this fact.

The situation is considerably more perplexing with quantified lan-guages, such as thelanguage of Chapter 5. Here, no single selection of fixed terms produces a uniformly plausible distribution of the logical properties. With these languages, the only way toget a reasonable extension is by using cross-term restrictions. Yet these restrictionsseem inconsistent with the analysis itself.

To complicate matters even further, it is clear that no matter what language we mayconsider, any given valid argument will be declared valid on some selection of fixedterms. For at the very least, we can include in % every atomic expression appearing inthe particular argu¬ment. Likewise any given invalid argument will be declared such on

some choice of fixed terms; excluding all expressions from $ will guarantee this. But wehave no assurance that there will be any onr selection of fixed or “logical” terms that



produces the right assessment for every argument expressible in the language. And that, presumably, is what we are after.

Now, this extensional evidence is all rather hard to assess, and it will only get morecomplicated as we move to increasingly complex lan¬

guages. But one thing is clear: we do not, at this point, have enough such evidence toconclude either that the account is right or that it is wrong. The cases where it clearlyworks — simple truth-functional lan-guages with connectives held fixed — hardly inspireconfidence that the account will work for arbitrary languages. On the other hand, theremay be very good explanations for those cases where it seems to fail. For example, wemay be able to explain the haphazard behavior of the definitions when we vary ourselection of fixed terms — say, by finding some characteristic that makes certainexpressions suitable for inclusion in ft and others not. And in the end, we may evenuncover some insight that shows certain cross-term restrictions to be perfectlyconsistent with the account, and so find these problems to be surmoun-table as well.

In any event, let us set these questions aside for the moment, and consider Tarski‟s own

justification of his account. Tarski does not base his justification on extensionalevidence; as I mentioned, he discusses no specific applications of the definitions.Rather, he argues that the analysis successfully captures “the essentials” of the ordinary

concept of consequence. Such an intuitive or conceptual justification is obvi¬ouslyquite important, since extensional evidence will bear at most on a single language, whilethe account is meant to work with any language for which satisfaction can be defined.

Since we can hardly survey all possible languages to which the definitions may beapplied, we clearly need a different kind of evidence, evidence of a more conceptualsort, to show that the definitions get the right extension in any such lan¬guage. Tarski‟s

argument is meant to provide such evidence.

Necessity

The most important feature of logical consequence, as we ordinarily understand it, is amodal relation that holds between implying sen-tences and sentence implied. The

premises of a logically valid argu-ment cannot be true if the conclusion is false; such

conclusions are said to “follow necessarily” from their premises. That this is the single most prominent feature of the consequence relation, or at any rateof our ordinary understanding of that relation, is clear from even the most cursorysurvey of texts on the subject. We find modal characterizations of logical consequencein the very earliest works on logic:

A syllogism is discourse in which, certain things being stated, something

oilier than what is stated follows of necessity from their being so. I mean

by i IK* Iasi phrase dial diey produce I he consequence, and by this, that no



further term is required from without in order to make the consequence necessary.(Aristotle, 24° 18-22)

in modern textbooks geared for the basic, nontechnical course:

A deductive argument is valid when ... it is absolutely impossible for the premises to betrue unless the conclusion is true also. (Copi, 1972, p. 23)

in more advanced texts aimed at intermediate students:

An argument is sound if and only if it is not possible for its premises to be true and itsconclusion false. (Mates, 1965, p. 3)

and in texts directed toward the most proficient and mathematically inclined:

What makes [the conclusion] a logical consequence of [the premises] is the fact that if

[the premises] are true then [the conclusion] must be true as well. (Bell and Machover,1977, p. 5)

This modal characteristic, however dimly perceived and poorly un-derstood, is clearlycentral to our intuitive understanding of the conse¬quence relation. It is, at the veryleast, a necessary condition for the relation to hold: if it is possible for the members ofK to be true while 5 is false, then 5 cannot be a logical consequence of K. Whether it isalso a sufficient condition, as is suggested in the above quotations, is harder to say.Thus, most logicians would agree that the continuum hypothe¬sis, even if true, is not aconsequence of the pair-set axiom.1 Yet if the former is true, it is (presumably)

necessarily so, and hence it would be impossible for the latter to be true and the formerfalse. Observations of this sort suggest that the modality at issue is really of a moreepistemic sort. But in any event, some such modality, whether alethic or epistemic, isclearly crucial to the relation of logical consequence.

In this section I would like to emphasize two points before con-sidering Tarski‟s

justification of his account. The first point has, I hope, already been made. The point isthat an account of consequence will indeed capture an essential feature of our

pretheoretic notion if it offers some guarantee that arguments declared valid display thedis-tinctively modal feature invariably attributed to such arguments. We need not be tooconcerned about the exact nature of this modality; for present purposes, we can leavesuch issues unresolved. What is impor-tant is just this simple observation: For anargument to be genuinely valid, it does not suffice for it to have a true conclusion or afalse premise, for it simply to “preserve truth.” The truth of the premises must somehow

guarantee the truth of the conclusion. It is this guaran-tee of truth preservation that givesrise to the familiar modal descrip-tions of the consequence relation. The exact source ofthe perceived guarantee, whether it be the meanings of the expressions contained in theargument, brute logical intuition, or something else entirely, need not concern us at themoment.



The second point is that Tarski himself, not surprisingly, recognized this guarantee to bethe central feature of the “ordinary concept” of consequence, the concept his analysiswas meant to capture. To appre¬ciate this, we need only consider the initial remarks inwhich Tarski motivates his analysis. Most of this discussion is aimed at showing that no

purely syntactic or “formal” definition captures our common un¬derstanding ofconsequence, or is even extensionally equivalent to that notion. This discussion is worthrecounting.

Tarski‟s first piece of evidence for the inadequacy of syntactic defi-nitions is theexistence of theories that are “a>-incomplete.” A theory is a>-incomplete if it displaysthe following peculiarity. Employing only the normal rules of inference, we can derivethe following sentences from the axioms of the theory:

Ao. 0 possesses the property P.

Aj. 1 possesses the property P.

And, in general, we can deduce all sentences of the form

An. n possesses the property P,

where V is any symbol that denotes a natural number. However, an a>-incompletetheory does not allow us to derive, according to the standard rules of inference, theuniversal claim

A. Every natural number possesses the property P.

The phenomenon of a>-incompleteness shows that the universal sen-tence A is notdeducible, using the standard syntactic rules, from the sentences A0, A1, . . . , An, . . .After pointing this out, Tarski con¬cludes:

This fact seems to me to speak for itself: it shows that the formalized concept ofconsequence, as it is generally used by mathematical logicians, by no means coincideswith the ordinary concept. For intuitively it seems certain that the universal sentence Afollows in the ordinary sense from I he totality of particular sentences AQ, Ai,, An ... :

provided all these sentences are true, the sentence A must also be true.2

Tarski‟s gloss here of the “ordinary” concept of consequence is the familiar one: A

follows in the ordinary sense from A0, A\> . . . be¬muse, ;is he puts it, provided thelatter are all true, the former must be true as well. According to Tarski, this shows thatstandard syntactic characterizations of consequenc e are not even extensionallyadequate.

For there are arguments that are valid in this “ordinary” sense but whose condusions

cannot be deduced from their premises.

Tarski admits that a formal characterization of consequence could be supplemented withan infinitary rule of inference that would avoid this particular failing, the so-called (o-



rule. But such an addition, due to its infinitary nature, would involve a significantdeparture from stan¬dard systems of deduction. A more reasonable alternative would beto supplement the system with a new rule that allows the derivation of A from the(single) claim that all the An are provable using the remaining rules, a claim that caneasily be encoded into sufficiendy powerful languages. This new rule, though morecomplex than standard rules, can still be considered “purely syntactic” or “structural.”

Furthermore, the resulting set of rules generates new consequences not provable fromthe original set, and so is a genuine step in the right direction. However, suchsupplementation is ultimately of no avail. The futility here, Tarski claims, follows fromGodel‟s incompleteness results:

In every deductive theory (apart from certain theories of a particularly elementarynature), however much we supplement the standard rules of inference by new purelystructural rules, it is possible to construct sen¬tences which follow, in the ordinary

sense, from the theorems of this theory, but which nevertheless cannot be proved in thistheory on the basis of the accepted rules of inference.3

Tarski‟s point here is this. Even if we add additional rules of the above sort to ourformal system of deduction, there will remain theo-ries from which we cannot deduceall the intuitive consequences. Specifically, we will not be able to derive the Gõdelsentence, G, of the theory, even though we can easily see that, as Tarski puts it,

provided all the sentences of the theory are true, the sentence G must be true as well.Again G is, at least in this ordinary sense, a consequence of the sentences contained inour theory. Yet G cannot be derived according to the structurally specified rules that

were meant, by hypothesis, to be extensionally equivalent to our ordinary concept ofconsequence. Here again there are necessary consequences that our syntacticcharac¬terization fails to capture.

Tarski takes these considerations to show that no “purely structural” characterization

can, in principle, even agree in extent with our ordi¬nary concept of consequence. Thus,he concludes:

In order to obtain the proper concept of consequence, one that is close in essentials tothe ordinary concept, we must resort to quite different methods and apply quite different

conceptual apparatus in definm# it.'1

With this, Tarski turns to the stated task of his article: giving a precise definition which,unlike syntactic accounts, captures the essential features of our ordinary concept ofconsequence.

Now, Tarski‟s argument that any syntactic characterization of conse¬quence will be

extensionally inadequate may strike us as problematic in several respects. Perhaps themost obvious is that in neither case is the sentence cited by Tarski (that is, A or G) astandard model-theoretic consequence of the theory from which it allegedly follows.

Indeed, both of Tarski‟s examples involve the consequence relation for first- orderlanguages, where the model-theoretically defined relation co¬incides with the



syntactically defined relation. How can a semantic account be judged extensionallysuperior to the usual syntactic charac¬terization if the two are, in fact, extensionallyequivalent? It would seem that the complaints Tarski has about the extensionaladequacy of the syntactic characterization will ricochet off the completeness theo¬remand strike his own account with equal force.

In fact this is not true, for reasons I have already mentioned. In any of these cases, theintuitive consequence will emerge as a Tarskian consequence if we include a sufficientnumber of expressions in the set of fixed terms. So, for example, the a>-rule comes outvalid, model- theoretically, if we include in ft the expression „every natural number‟ as

well as the collection of numerals „0,‟ „1,‟ „2,‟ and so forth. I assume this is why Tarski

does not consider his account subject to precisely the same criticism he directs atsyntactic definitions.

What is important for our purposes, though, is not the specific examples Tarski employsin his argument but his emphasis on the intuitive consequence relation, the relation hecharacterizes using the familiar modal terms. When one sentence is, in the ordinarysense, a logical consequence of others, then it must be true provided the others are trueas well. That is, the truth of the premises must guarantee the truth of the conclusion.However vague and poorly understood this guarantee may be, it is clearly an essentialfeature, if not the essential feature, of our ordinary concept of consequence.

Tarski's Fallacy

Any intuitively valid argument (K, S) will come out logically valid, ac¬cording to

Tarski‟s account, on some choice of fixed terms. The argu¬ment would not be valid ifall the members of K were true while 5 was false, and hence the argument will at leastsatisfy Tarski‟s definition when all of its component expressions are included in ft. This

observa¬tion gives us the following implication:

If 5 is a consequence (in the ordinary sense) of K, then 5 is a Tarskian consequence of Kon some selection of

Now, if the converse of this implication could be demonstrated, we would have a ratherimpressive result:

(L) 5 is a consequence (in the ordinary sense) of K if and only if Sis a

Tarskian consequence of K on some selection of

Needless to say, if equivalence (L) could somehow be shown, then Tarski‟s definition

of consequence could hardly be faulted. But in order to show that the equivalence holds,we must show that if 5 is a Tarskian consequence of K, then it is a consequence "in theordinary sense.” That is, we must show that if all the members of K are true, 5 must be

true as well.



After proposing his account, Tarski offers the following justification of his definition. Itcontains a simple argument that appears, at first glance, to give us exactly theimplication we need:

It seems to me that everyone who understands the content of [my] definition must admit

that it agrees quite well with ordinary usage. This becomes still clearer from its variousconsequences. In particular, it can be proved, on the basis of this definition, that everyconsequence of true sentences must be true, and also that the consequence relation ... iscompletely indepen¬dent of the sense of the extralogical constants which occur in thesesentences.5

The proof that Tarski is referring to is quite straightforward. Suppose that (K', S') is theargument form corresponding to the argument (K, S), and that (K', S') is satisfaction

preserving on all sequences. As-sume further that all the sentences in K are actually truewhile 5 is actually false. Our goal will be to derive a contradiction from this assumption.

The contradiction is virtually immediate. For we know that there must be at least onesequence that assigns to each variable occurring in (K', S') that member of theappropriate satisfaction domain which corresponds to the expression the variablereplaced. Since we have assumed that the members of K are true but that 5 is false, itfollows that on this assignment (K', S') cannot be satisfaction preserving. But thiscontradicts the hypothesis that the argument form was satisfaction preserving on allsequences, and so we conclude our proof.

We have shown that if 5 is a Tarskian consequence of K, and if all of the members of K

are true, then 5 must be true as well. Furthermore, we can see from our proof that thisholds quite independently of our selection of logical constants. Thus, it would seem, wehave exactly the result we need. If 5 is a Tarskian consequence of K (on any selection offt), then 5 is a consequence of K in the ordinary sense. This gives us biconditional (L).

But (L) is so obviously false that something has clearly gone wrong. We know that anytruth-preserving argument is logically valid, accord¬ing to Tarski‟s definition, on some

selection of ft. Thus, „Lincoln had a beard‟ is a Tarskian consequence of„Washington

was president‟ when all the component expressions are held fixed. For then the

corre¬sponding argument form is just the argument itself, and this argument issatisfaction preserving on all sequences simply because the conclu¬sion is itself a truesentence — hence, satisfied by any sequence. But it is clear that the former sentence isnot a genuine consequence of the latter. We would hardly say that, provided„Washington was president‟ is true, „Lincoln had a beard‟ must be true as well.

It is perfectly clear that with many selections of ft, there are Tarskian consequences thatare not genuine consequences, and hence that (L) is simply false. Yet our proof thatevery Tarskian consequence of true sentences must be true is perfectly correct. The

problem is not with our proof, but with thinking that this proof shows that any modal

relation holds between the premises and conclusion of the argument (K, S). To showthat all Tarskian consequences are consequences in the ordinary sense, we would need



to prove a theorem with an embedded modality. Specifically, we would have to showthat, for any K and 5, if

(1) 5 is a Tarskian consequence of K (for some ft) then the following are jointlyincompatible:

(2) All the members of K are true

(3) 5 is false.

But of course all we can show is that for any K and S, the following three conditions are jointly incompatible:

(1) 5 is a Tarskian consequence of K (for some ft)

(2) All the members of K are true

(3) 5 is false.

Now, it should be clear from a purely abstract point of view that the jointincompatibility of (1), (2), and (3), plus the truth of (1), does not entail the jointincompatibility of (2) and (3). Here we need only note the fallaciousness of anyinference from

Necessarily (if P and Q then not R) to V

If P then necessarily (if Q then not R).

More concretely, we can note that the argument with „Lincoln had a beard‟ as

conclusion and „Washington was president‟ as sole premise could not come out valid on

any selection of ^ if it did not in fact preserve truth — that is, have either a false premiseor true conclusion. But the mere fact that this argument does come out valid on someselection of ^ certainly does not imply that it is a necessarily truth- preserving argument,that it is valid in the ordinary sense.

The fallacy here may be emphasized by sketching the parallel con-sideration forTarski‟s definition of logical truth, for there the problem is even more transparent. With

logical truth there is also an important modal feature of our ordinary concept. A logicaltruth must be true — that is, it is necessarily true. Thus, it would be a strong point infavor of a definition of logical truth if we could show that sentences satisfying thedefinition are necessarily true, that they have the intuitive modal property. And indeedwe can prove that if a sentence satisfies Tarski's definition of logical truth then it must

be true. After all, if it were not true, it would not satisfy the definition. Unfortunately,this does not guarantee that the sentence has any peculiar modal properties, any morethan the trivial observation “if a sentence is true then it must be true” shows every truth

to be a necessary truth.

Obviously, the proof in question does not show that every Tarskian consequence is aconsequence “in the ordinary sense.” It is only through an illicit shift in the position of



the modality that we can imagine ourselves demonstrating of any Tarskian consequencethat it is entailed by the corresponding set of sentences. This fallacy becomes quiteapparent when we consider the arguments that come out valid when we include allexpressions in But it is crucial to recognize that the inference remains fallacious, and forexactly the same reasons, regardless of our choice of fixed terms. The fallacy may beeasier to spot when we include names and predicates in but the inference is no lessfallacious when we only hold fixed (say) the truth functional connectives. The argumentdoes not depend on and it does not get better or worse according to what we suppose themembers of ^ to be.

A parallel justification for Tarski‟s account can be given, but is just as fallacious, when

we replace the alethic reading of “must” with a purely epistemic reading. Although the

most common pretheoretic descrip¬tions of logical consequence involve necessity, wefind many in which the “must” takes on a more epistemic cast. For example, the

following, from Quine‟s introductor y text, is a familiar description:[Among the] relations of statements to statements, one of conspicuous importance is therelation of logical implication: the relation of any statement to any that follows logicallyfrom it. If one statement is to be held as true, each statement implied by it must also beheld as true. (Quine, 1972, p. 4)

If you accept the premises of a valid argument, you must also accept the conclusion (towhich we sometimes add “on pain of irrationality”). This epistemic characteristic issometimes thought to be more impor¬tant than, and perhaps to underlie, our intuitions

about the alethic modality involved in valid arguments. For example, some wouldclaim, not implausibly, that it is only due to the a priori relation between the premisesand conclusion of a valid argument that we judge the latter to follow necessarily fromthe former, and hence that we judge the argument valid. On this view, a necessaryconsequence that could not be recognized as such a priori would never qualify as alogical conse¬quence. And this certainly seems right.

Can we show that this epistemic feature follows from the definition? Again, the best wecan offer is a version of Tarski‟s fallacy. We can note, quite accurately, that it would be

irrational to believe that an argument satisfies Tarski‟s definition (for any ft) but has

true premises and a false conclusion. Or we can point out that if you accept the premisesof an argument, and also accept that it passes Tarski‟s test for validity, then you must

accept the conclusion. But neither of these shows that any peculiar epistemic relationholds between the premises and conclusion of these arguments. These observationsshow only that it is a genuine consequence of Tarski‟s definition that the argument in

question either has a false premise or a true conclusion, that it indeed preserves truth.But they do not show that it would be irrational to accept the premises and deny theconclusion; they show only that if you did, you could no longer hold that the argumentsatisfied the definition.

The last point brings out the real weakness of this justification, regardless of what thesought-after modality may be. Tarski‟s account demands, first and foremost, that any



argument declared valid preserve truth; those that do not do not pass muster. Theaccount shares this characteristic with Bolzano‟s, and it is perhaps easier to see with the

simpler, substitutional definition. With Bolzano‟s account, this feature is incorporated

directly into the definition, by virtue of the fact that any argument is, for any ft, a permissible substitution instance of itself. So Bolzano‟s demand that all substitution

instances of (K, S) preserve truth can be divided into two requirements:

(a) that (K, S) have either a false premise or true conclusion, and

(b) that all members of some (possibly empty) collection of related arguments preserve truth as well.

Tarski‟s move from substitutioirto satisfaction will at most increase the stringency of

clause (b), allowing consideration of arguments drawn from expansions of the originallanguage.

Now, Tarski‟s proof that “every consequence of true sentences must be true” depends

only on clause (a) of his account. This is why the choice of $ is entirely irrelevant; theset appearing in (b) may as well, and often will, be empty. And it is also why a parallelconsideration can be offered as equal justification for any definition of consequence thatincorporates requirement (a).6

The fact that Tarski‟s proof depends only on clause (a) shows how little bearing this

initially impressive consideration really has on the adequacy of the analysis. Indeed, wecould offer precisely the same justification for the following “definition” of logical

consequence: 5 is a logical consequence of Xjust in case either 5 is true or somemember of K is false. Now, this analysis is certainly far off track. However, we mightnote, first of all, that every intuitive consequence will obviously be a “logical

consequence” according to this trivial definition, thus giving us at least one direction of biconditional (L). And for the con¬verse implication we can point out that anyconsequence of true sen¬tences indeed must be true: after all, if a sentence is not true, itwill only turn up a “logical consequence” of sets containing at least one false sentence.But of course this “must” has nothing to do with any modal or epistemic property of the

genuine consequence relation, or with any guarantee of truth preservation. Yet Tarski

can, on this particular score, give us nothing more to commend his own account. Thisnew, obviously incorrect account has as much claim to biconditional (L) as Tarski‟s,

and for precisely the same reasons.

Tarski's Reasoning

Was Tarski guilty of the modal fallacy I have described? Did he really believe that his proof that “every consequence of true sentences must be true” assures us that the right

sort of relation will hold between the premises and conclusion of arguments satisfyingthe definition? Or was he simply making a very weak claim for his definition — namely,that it will not designate as “logically valid” any argument with true premises and a



false conclusion? This latter claim follows trivially from the definition, but hardlyseems much evidence that it “agrees quite well with ordinary usage.”

Needless to say, the crucial sentence is ambiguous. This is not sur-prising, since the keyto the fallacy itself is the ambiguous scope of the modality in question. But this same

ambiguity leaves open the possibil¬ity that Tarski himself was not misled by thefallacy, though perhaps guilty of a misleading turn of phrase.

A careful reading of the article seems to suggest otherwise. It is significant that we findno comparable ambiguity in Tarski‟s initial

observations about the “ordinary concept” of consequence. Recall, for example, how he

characterizes this concept in his discussion of a>-incompleteness:

[The existence of co-incomplete theories] shows that the formalized con¬cept of

consequence ... by no means coincides with the ordinary concept. For intuitively itseems certain that the universal sentence A follows in the ordinary sense from thetotality of particular sentences A0, A\, ...,

An : provided all these sentences are true, the sentence A must also be true.1

There is only one way to construe the modality that Tarski here identifies with theordinary concept of consequence. Obviously, he is not simply noting that the argumentin question happens to have a false premise or a true conclusion; since no specificsentences have been given, such a construal would not even make sense. The observa-tion clearly concerns the modal or epistemic relation between these sentences, the factthat arguments of this form are guaranteed to preserve truth. Here, the scope of themodality is clear and un-equivocal.

Now consider again Tarski‟s justification of his account, this time paying particular

attention to his exact choice of terms:

It seems to me that everyone who understands the content of the above definition mustadmit that it agrees quite well with ordinary usage. This becomes still clearer from itsvarious consequences. In particular, it can be proved, on the basis of this definition, thatevery consequence of true sentences must be true.8

Set next to his earlier remarks, it is hard not to see the fallacy at work in this justification. It is hard to overlook Tarski‟s use of precisely the same expressions to

describe, in the first passage, the modality central to consequence “in the ordinary

sense” and, in the second, the alleged agreement of his definition “with ordinary usage.”

But as we have seen, thinking that any such modality is a consequence of thedefini¬tion is a simple confusion.

Tarski clearly saw the importance of the modal features of our ordinary concept ofconsequence. Indeed, his article is peppered with modal and quasi-modal descriptions of

this relation. Some of these display the same scope ambiguity as his justification, while





of logically valid arguments that even those most skeptical of modal notions recognizeas essential. Now, if we equate logical validity with mere truth preser¬vation, assuggested in the last section, we obviously miss this essential characteristic of validity.For in general, it will be impossible to know both that an argument is “valid” (in this

sense) and that its premises are true, without antecedently knowing that the conclusionis true. This is why such arguments as

(B) Washington was president So, Lincoln had a beard

are incapable of justifying their conclusions. For although this argu-ment preservestruth, there is no guarantee of this fact independent of the specific truth values of itsconstituent sentences. Consequently, any doubts we may have about the truth of theconclusion translate directly into doubts about the argument‟s “validity.”

Tarski‟s account equates validity with the joint truth preservation of a collection of

arguments. In the extreme case, the collection will contain only the argument itself, andthen the account reduces to the trivial one above. In other cases, though, the collectionwill contain other arguments. But whichever is the case, Tarski‟s equation still misses

the essential feature of validity. For in general, it will be impossi¬ble to know whetheran argument is a member of such a collection of truth-preserving arguments, hencewhether it is “valid,” without ante¬cedently knowing the specific truth values of its

constituent sentences. If we know that the premises of an argument are true, then anydoubts about the truth of its conclusion will translate directly into doubts about whetherthis argument, and any others in the associated collec¬tion, are “valid.” Simply moving

from a collection of one to a collection of many does not change this in any significantway. We still have no assurance that arguments satisfying the definition will be capableof justifying their conclusions, and hence no assurance that they will be genuinely valid.Tarski‟s fallacy obscures this omission, by noting that arguments declared valid are

indeed guaranteed to preserve truth. But this is not the required guarantee: it is backedup only by the definition of validity, not by any characteristic of the argument itself,whether modal, epistemic, or semantic. Consequently, it leaves such argumentsimpotent as a meanp of justifying their conclusions.

Tarski‟s brief justification oKhis account, when properly under -stood, adds very little to

the rather ambiguous, extensional evidence surveyed at the beginning of this chapter.What we can say with certainty is simply this. First of all, the definition will never saythat an argument with true premises and a false conclusion is logically valid. Second, ifan argument is declared valid by the definition, then so too will be any other argumentthat results by replacing expressions that are not members of $. In other words, theassessments made by the account are, as Tarski puts it, “independent of the sense” of

those expressions not held fixed. We have no assurance, however, that argu¬mentsdeclared valid carry with them any independent guarantee of truth preservation, whethermodal or epistemic or semantic, nor that validity is an enduring characteristic of these

arguments. To think otherwise is to succumb to one form or another of Tarski‟s fallacy.7



The Reduction Principle

So far we have seen two reasons, both bad, for accepting Tarski‟s account of the logical

properties. The first is the conflation of the Tarskian or model-theoretic definitions withrepresentational seman-tics. It is perfectly obvious why an adequate representational

semantics can yield necessary truths, and hence logical or analytic truths, insofar asthese are a species of those. But interpretational semantics is not representationalsemantics; what we get trivially from the latter should not be considered a deep orsignificant upshot of the former. The second reason is the argument I have calledTarski‟s fallacy, an argu¬ment that seems to have played a role in Tarski‟s own

adoption of the analysis.

Still, pointing out bad reasons for accepting an account is a far cry from giving goodreasons for rejecting it. In Chapter 1,1 claimed that Tarski‟s definitions are, in a sense,

obviously mistaken. It is time I explained what I consider the obvious mistake: animplausible prin-ciple on which both Tarski and Bolzano base their accounts. Since it issimpler to discuss this principle when treating sentences rather than arguments, let usonce again direct our attention to the definition of logical truth. This is just a matter ofconvenience, though; the points can be made, with the obvious changes, about theanalysis of logical consequence.

Quantificational Accounts

Both Bolzano and Tarski propose quantificational accounts of logical truth: both equatethe logical reuth of a sentence within a given lan-guage with the ordinary truth of a

universally quantified sentence appearing in a (perhaps) expanded version of thelanguage. The difference between the two accounts comes down to the nature of theuniversal quantifiers in the associated sentence: for Bolzano, these quantifiers aresubstitutional; for Tarski, objectual.

Suppose S' is a sentential function obtained by uniformly replacing the variable terms in5, those expressions not contained in the set ft of fixed terms, with variables ofappropriate type. Recall that according to Bolzano, 5 will be logically true (with respectto ft) if all the permissi¬ble substitution instances of S' are true. Notice that these are

simply the truth conditions for the universal “substitutional” closure of S'— that is, forthe sentence obtained from S' by appending an initial string lit»! .. . Uvn ofsubstitutionally interpreted universal quantifiers bind¬ing each free variable in S'.1Thus, Bolzano equates the logical truth of 5 with the ordinary truth of the universalgeneralization

Uvi . . . Ilvn[ S' ].

Now, the intent of Tarski‟s move from substitution to satisfaction is not to alter the

“quantificational” nature of the account, but to insist that the associated sentence be the

universal objectual closure of the sentential function S'. Thus, recall that according toTarski, sentence 5 is logically true (with respect to the same choice of ft) if the



sentential function S' is satisfied by all sequences. But these are, once again, simply thetruth conditions for the following universal closure of S':

Vvi... Vvn[ S' ].

But here, the universal quantifiers are of the standard, “objectual” sort, the variablesranging over objects within the appropriate satisfac¬tion domains.2

It will often be the case that neither of these universal closures is a sentence of ouroriginal language ££, simply because ££ may not contain the needed quantifiers. But ifwe can apply either Bolzano‟s or Tarski‟s account of logical truth to ££, then we can

define truth for an expanded language ££+ containing the appropriate universal closureamong its sentences. In the case of Bolzano‟s account, this simply follows from the

assumption that each variable term is associated with a well-defined substitution classand that we already have a truth predicate for the original language ££. With Tarski‟s

account, it follows from the fact that if the satisfaction relation has been defined for a particular senten¬tial function S'y then the standard clause for „V‟ will guarantee that it

is defined for the sentential function ViaS'.

The logical truth of 5 comes down to the ordinary truth, in ££+, of the respectiveuniversal closure of S' — at least according to our two quantificational accounts oflogical truth. This is easiest to see in the trivial case in which $ contains all the atomicexpressions occurring in the sentence 5. Here, the sentential function of interest is just 5itself. And since S, being a sentence, contains no free variables, it happens to be its ownuniversal closure (either substitutional or objectual). Conse¬quently, according to both

Bolzano and Tarski, the logical truth of 5 will here correspond to its simple truth. Thus,if $ contains both „Abe Lincoln‟ and „was president,‟ then according to either

quantificational account, the sentence „Abe Lincoln was president‟ will be logically true

just in case the following universal closure is true:

[Abe Lincoln was president].

Needless to say, there is no need here to resort to any expanded language ££+ in orderto express the above “universal generalization” of our original sentence.3

Suppose now that we are testing for logical truth with respect to an $ containing allexpressions except names — that is, with names as the only variable terms. Here, as wehave seen, the logical truth of „Abe Lincoln was president‟ is, for Bolzano, determined

by the truth values of the permissible substitution instances of the sentential function „x

was president‟: the original sentence is logically true just in case all the substitution

instances are true. Or, what comes to the same thing, this sentence will be logically trueif the following generalization is true:

Ihc[x was president].

For Tarski, on the other hand, the sentence is logically true on this choice of $ just incase lx was president‟ is satisfied by all sequences, or equivalently, in case it is satisfied



by all individuals in the name domain of the satisfaction relation. But this is just to saythat the logical truth of the original sentence is tagged to the ordinary truth of theobjectually quantified sentence

Vx[x was president].

Notice that neither of these closures actually occurs in the language of Chapter 3. But itis easy to see that in either case there is a simple language ££+ which contains theassociated generalization and in which the logical truth (with respect to the same $) of„Abe Lincoln was president‟ will be equivalent to the ordinary truth of that universal

closure. As we have seen, the substitutionally quantified sentence will be true, theobjectually quantified sentence false.

To take one final example, if $ contains neither of the lexical com-ponents appearing inthe sentence „Abe Lincoln was president,‟ then according to Bolzano‟s quantifiteational

account, this sentence will be logit ally true just in case the following closure is true:

EbcIIPt xP].

According to Tarski‟s account, the same sentence will be logically true in case thecorresponding objectual generalization is true:

VxVP[ xP].

Clearly, neither of these sentences is true. Consequently, neither Bolzano nor Tarskideclares „Abe Lincoln was president‟ to be logically true on this selection of ft.

Three Principles

Emphasizing the quantificational nature of these two accounts opens up a promisingavenue for assessing them. We can now consider in abstraction the various principlesthat govern the relation between universally quantified sentences and their instances.Clearly, there are at least two principles concerning this relation that we can endorsewithout hesitation. First is a simple principle of logic, the principle of universalinstantiation:

(i) If a universally quantified sentence is true, then all of its in-stances are true aswell.

Second is a principle, so to speak, about logic:

(ii) If a universally quantified sentence is logically true, then all of its instances arelogically true as well.

This second principle follows from the fact that the set of logical truths is itself closedunder logical consequence and, as the first principle states, an instance is indeed aconsequence of its generalization. I will call (i) the instantiation principle and (ii) the

closure principle. Neither (i) nor (ii) is the least bit controversial.



It should by now be apparent that a quantificational account of logical truth is based ona third principle, quite different from either of these, and at first glance considerablymore surprising. The principle is this:

(Hi) If a universally quantified sentence is true, then all of its in¬

stances are logically true.

I will call (Hi) the reduction principle. Of course, both of our quantifica¬tional accountstake logical truth to be relativized to a choice of fixed terms, of “logical constants,” and

so the underlying principle will be a somewhat modified form of (Hi). There are acouple of ways this modification might go, and I will consider both in due course. Fornow,

though, it is important to see that Bolzano and Tarski both base their accounts on this

rather unlikely principle, in some form or other. Indeed, the substantial technical andmathematical attraction of Tarski‟s account derives directly from principle (in). For,

assuming his analysis is right, it is this principle that allows the direct application ofwell-known techniques for defining truth to the task of defining logical truth.

This is an important selling point for Tarski‟s account. Our ordinary concepts of logical

truth and logical consequence involve various no¬tions that are notoriously difficult to pin down, notions like necessity, a prioricity, analyticity, and so forth. But if thequantificational account is correct, what it achieves is a truly remarkable reduction ofobscure notions to mathematically tractable ones. If it is right, the analysis shows that

we can in fact sidestep all of these difficult concepts, that we can give a mathematically precise definition of the logical truths of a language if we can just define the notion oftruth for a slightly ex¬panded language — or, what comes to the same thing, if we candefine the notion of truth relative to an arbitrary interpretation or d-sequence.

This is a tremendous advantage, one we should not undervalue. And it is an advantagenot shared by representational semantics. When we are doing representationalsemantics, we appeal to modal notions from the very outset, in assessing the adequacyof our class of models and our definition of truth in a model. In contrast, Tarski‟s

account equates the logical truth of a sentence with the ordinary truth of another

sentence, one that makes a nonmodal, nonepistemic, nonse- mantic claim about theworld, about the world as it actually happens to be. The source of this advantage is, ofcourse, the reduction principle (Hi). Unfortunately, we cannot construe this strikingtechnical advantage as support for Tarski‟s analysis itself: we can hardly argue that the

analy¬sis is correct because it would simplify our lives if it were correct. Still, it isimportant to acknowledge this benefit and to locate its source.

Now consider for a moment principle (in). I will not spend much time discussing theabstract acceptability of this principle. Unadorned and unmodified, its implausibilitycould hardly be more apparent. Our natural inclination is to reject the principle out ofhand, to reject it for a very simple reason: universal generalizations have no particular



claim to logical truth; they, like any sentences, can be true by mere happen¬stance. Andwhen such a sentence just happens to be true, there is no guarantee that its instances will

be logically true. Some might, but then again some might not.

The problem with the redufciion principle is that the mere truth of a universal

generalization can, in general, guarantee nothing more than

the truth of its instances. It cannot guarantee that its instances have any otherdistinguishing characteristics. In particular, it cannot guar¬antee that the instances willhave any of the distinctive features, whether modal or epistemic or semantic, ordinarilythought to set logical truths apart from common, run-of-the-mill truths. Of course, if thegeneralization itself is logically true, then the instances will be logically true as well.This is guaranteed by the closure principle (ii). But if the generalization is not logicallytrue — if it is, say, a historical truth, or an arithmetical truth, or a truth of physics — thenthe in-stances will presumably be just as historical or arithmetical or physical.

Modifying the Principle (Part One)

Unmodified, the reduction principle is simply false. But as I have stated it, this principlemakes no mention of the set ft of fixed terms. So before we count it too heavily againstTarski‟s analysis, we should decide how the selection of ft figures into the principle. As I said earlier, there are two ways this might work. Exactly which way we go will dependon whether we construe Tarski‟s account as a completed analysis of a fundamentally

relational notion, one that varies with an arbitrary choice of ft, or as an incompleteanalysis of a more or less fixed notion of logical truth. If the latter, then the analysis

must be supple¬mented with some account of how we go about making the “proper”selection of fixed terms.

It is clear that Tarski was not, himself, entirely sure which way to view the account. Hisexamples of a>-incomplete theories and the Gõdel incompleteness results pull in theformer direction. For in order to declare the o>-rule logically valid, or to get a Gõdelsentence to come out a consequence of its corresponding theory, we have to presupposegreat leeway in our selection of fixed terms. On the other hand, when we allow suchleeway, we often get extremely counterintuitive results. This fact pushes in the latter

direction, toward thinking that there is something — as yet unaccounted for — that makessome selections of ft definitely “wrong” and others definitely “right.” Tarski‟s

ambivalence on this question comes out most clearly in his concluding remarks, wherehe describes this problem as the most important open question left by his account (1956,

pp. 418-419).

I will consider the two possibilities in turn. The first is clearly the less plausibleconstrual, and so should be easier to set aside. Still, there are some importantobservations to be made, even here. I will turn to the second, more plausible construalin the following chapter.



According to the first view, Tarski has given us a completed analysis of an irreduciblyrelational notion, the notion of logical truth with respect to an arbitrary selection $ offixed terms. If this is how we construe the account, then the required modification of

principle (Hi) is straight¬forward. It runs as follows:

(Hi') If a universally quantified sentence is true, then all of its in-stances are logicallytrue with respect to those expressions not bound by the initial universal quantifiers.

The way we arrive at this modification should be clear. The selection of fixed termsdetermines which expressions do not get replaced by variables in our move from 5 to 5',and hence which expressions do not get bound by the quantifiers in the associateduniversal closure, Vt»i . . . Vvn[S']. If the closure is true, then Tarski‟s account declares

the original sentence logically true with respect to that selection of $ — indeed, withrespect to any selection that includes all the unbound expressions in the closure. If theclosure is not true, then the sentence does not come out logically true on that sameselection.

Now, is this modified principle any more plausible than the original reduction principle?It is certainly harder to assess. And the reason for that is pretty apparent: it simply is notclear what intuitive notion this is meant to be an analysis of Do we really have a conceptof logical truth that is keyed to an otherwise arbitrary selection of expressions? Can asentence be logically true with respect to some expressions, but not logically true withrespect to others? If our answer is no, then principle (Hi') must be rejected out of hand;it simply makes no sense as a description of any ordinary concept of logical truth.

There are, though, some likely possibilities, some ways of viewing our ordinary notionof logical truth as relativized in the manner sug-gested by principle (Hi'). I will considerwhat seems the most natural; similar remarks can be made about any reasonablealternative.

When we describe a simple logical truth, for example,

Either Lincoln was president or Lincoln was not president

we often say that the sentence is true solely by virtue of the meanings of a certain subset

of its component expressions. In this case, the expres¬sions in question are „or‟ and„not.‟ Our intuition here is twofold. First, it seems that the truth of this sentence does not

depend on how the world happens to be, on whether Lincoln was, in fact, president.This is the import of our judgment that the sentence owes its truth solely to themeanings of its constituent expressions. But there is a clear sense in which the meaningsof some of the expressions play a secondary role in this judgment. Indeed, the fact thatthis sentence is logically true does not depend on the specific meanings of either thename „Lincoln‟ or the predicate „was president.‟ So long as the name is a genuine name

and the predicate a genuine predicate — assuming, for example, that neither behaves like

„Nix‟— the sentence will still be true no matter how the world happens to be. Thus, oursecond intuition is that the logical truth of this sentence is dependent on the meanings of



the terms „or‟ and „not‟ in a way that it is not dependent on the meanings of„Lincoln‟

and „was president.‟ This difference, the fact that the meanings of the latter two

expressions are pretty much irrelevant, makes it natural to describe this sentence as true by virtue of the meanings of„or‟ and „not‟ alone. Just so, it would not be unnatural tosay that the sentence is logically true with respect to these two expressions alone.

Now contrast this with the sentence

Either everyone is happy or someone is not happy.

This second sentence also strikes us as true solely by virtue of meaning: to recognize itstruth, there is no need to check who is and is not happy. But in this case our judgmentdoes not rely only on the meanings of„or‟ and „not.‟ To be sure, the specific meaning of

„is happy‟ does not matter— no expression of the same semantic category, no genuine predicate, would alter this judgment. But the meanings of „everyone‟ and „someone‟

most certainly do matter. For example, if„everyone‟ had meant something else— say, ifit had meant no one or every dog — then this sentence could easily have been false.Clearly, the logical truth of this sentence depends on the specific meanings of„everyone‟ and „someone‟ in a way in which it does not depend on the specific meaning

of„is happy.‟ Thus, we might say that this sentence, unlike the previous one, is notlogically true with respect to „or‟ and „not‟ alone. Instead, it is logically true with

respect to the four expressions, „or,‟ „not,‟ „everyone,‟ and „someone.‟

Take one more example. Consider the sentence

Either Leslie is a man or Leslie is not a bachelor.

Once again, this sentence strikes us as true simply by virtue of the meanings of certainof its component expressions. But here, the rele-vant expressions include the predicates„is a man‟ and „is a bachelor.‟ Again, the specific meaning of „Leslie‟ makes little

difference; this sentence is true solely by virtue of meaning, and remains true so long as„Leslie‟ functions as a genuine name. Thus, it would be natural to describe the sentence

as true by virtue of the meanings of„or,‟ „not,‟ „is a man,‟ and „is a bachelor.‟ And by

analogy with the earlier cases, we might say this sentence is logically true with respectto these same four expressions.

What we have here is a perfectly understandable notion that could well pass for arelativized version of logical truth, for the notion of

logical truth with respect to an arbitrary selection of fixed terms. Of course, it seemsmost natural to apply the term logical truth when this concept is relativized toexpressions of traditional interest to logicians, expres¬sions like „or‟ and „everyone.‟

When the collection includes such ex¬pressions as „is a man‟ and „is a bachelor,‟ we

might want to revert to a term like analytic; “logical truth” seems a bit of a stretch. But

what is important is that the basic idea, the idea of a sentence that owes its truth to

nothing more than the meanings of some (perhaps proper) subset of its expressions, iseasily extended to the general case. The concept that results is something like this.



Certain sentences are true merely by virtue of the meanings of their expressions. Withsome of these sentences, this fact will depend on the meanings of all of the sentence‟s

constituent expressions equally. But with others, including all the examples we havelooked at, the fact depends only on the specific meanings of certain of thoseexpressions, plus very general assumptions about the semantic categories of theremaining expres¬sions. This is what gives rise to an intuitive relativization of ourconcept of logical truth, what makes sense of the notion of a sentence being logicallytrue with respect to an arbitrary selection of expressions. The selected expressions arethose whose specific meanings, as opposed to general semantic category, are relevant tothe sentence‟s analytic truth.

When the selected expressions include only those of a traditional “logical” sort—

connectives, quantifiers, and so forth — this relative notion coincides with one standardconception of logical truth. The basic idea of this conception is often described as

follows: a sentence is logically true if it is true solely by virtue of the meaning of thelogical vocabulary it contains. To the credit of the revised principle {Hi'), this notionadmits of a natural extension to completely arbitrary selections of the logicalvocabulary. In the extreme, if all the expressions in a sentence are considered “logical,”

the sentence will be logically true just in case it is analytic.

Now, does Tarski‟s analysis capture this notion of logical truth? The answer is clearlyno. We can see this very easily from principle (in'). According to this view of logicaltruth, principle (Hi') says that if a universally quantified sentence is true, then all of itsinstances are true solely by virtue of the meanings of the expressions not bound by the

initial quantifiers. But that claim is patently false: the mere truth of the univer¬sallyquantified sentence gives us no guarantee of this. This is easy to see, whether inabstraction or by looking at the multitude of simple counterexamples. A sentence of theform Vt»i .. . Vt»n[5'] can be true f or any number of reasons. Most obviously, indeed

paradigmatically, it might be true simply because themembers of the appropriatesatisfac¬tion domains, /«r whatever diverse reasons, all happen to satisfy S'. Thus, everyindividual satisfies „if x was president then x was a man‟— some because they weremen, some because they were never president, some for both reasons at once. But thisgives us no license to conclude that the instances of this sentential function are true

solely by virtue of the meanings of their constituent expressions. Indeed, they are not.The fact that all individuals satisfy this particular sentential function is not guaranteed by the meanings of„if. . . then,‟ „was president,‟ and „was a man‟; it is simply a matter of

historical fact. And the fact that its instances are true is obviously no more a matter ofmeaning alone.

For any sentence 5 and set ft of expressions, it makes sense to ask whether 5 is truesolely by virtue of the meanings of the members of ft. If ft contains every expression in5, then this simply comes down to the question of whether 5 is analytic, true by virtue ofthe semantic rules of the language. But if ft does not contain certain expressions

appearing in 5, — say, e\y ... , en — then the question is somewhat more compli¬cated.Presumably, what we want to know is whether 5 is true by virtue of the particular



meanings of the members of ft, given our background assumptions about the semanticcategories of e\ through en. Now, if this is our general conception of logical truth, thenit is clear that Tarski‟s account does not capture it. For all that Tarski‟s account requires

is that the following closure be true:

(G) Vi/, . . . Vi/n[ S(e/v) ]

But this requirement is, as we have seen, too weak to guarantee that 5 is true solely dueto the meanings of the expressions in ft. For this closure could well be true for allmanner of reasons, reasons quite apart from the purely semantic characteristics of its

parts. It might be a mere historical truth, an obscure arithmetical or set-theoretic truth,even a purely coincidental truth. In none of these cases will its instances be logically oranalytically true.

Now, it is important to see that (G) is by no means irrelevant to the logical or analytic

truth of 5. On the one hand, the mere truth of (G) clearly provides insufficient groundsfor concluding that 5 is true solely by virtue of meaning. But it should be equally clearthat if S owes its truth to nothing more than meaning, and if this fact depends only onthe semantic categories of e\ through en, then (G) will indeed be true. Think of it thisway. Suppose that (G) is false, but that 5 is in fact logically true. Then it is obvious thatthe logical truth of 5 must depend on the specific meanings of one or more of theexpressions e\ through en. For the falsity of (G) can arise only if there is at least one d-sequence that falsifies 5 simply by assigning different interpretations to e\. .. en. But ifwe have constructed the satisfaction domains properly, these interpre¬tations do not

change the semantic categories of e.\ through e. So either 5 was not logically true in thefirst place, or its logical truth depended on the particular meanings of e\ through en. Ineither case, 5 would not be logically true with respect to the members of ^ alone.

A couple of examples here might help. First, consider again the sentence

Either Leslie is a man or Leslie is not a bachelor.

This sentence seemed logically true, in our extended sense, with re-spect to the fourexpressions „or,‟ „not,‟ „is a man,‟ and „is a bachelor.‟ Since our judgment here

presupposes only that „Leslie‟ behaves se-mantically as a name, it is clear that the

following closure must be true:

Vx[x is a man or x is not a bachelor].

If we found a counterexample to this generalization — say, a female bachelor — thiswould show either that the original sentence is not analytically true or that its analyticitysomehow depends on the specific meaning of the name „Leslie.‟ Of course, the fact thatwe do not find a counterexample does not alone allow us to conclude that the originalsentence is true solely by virtue of meaning. To see that, we need only apply the sametest to the sentence

Either Leslie was a man or Leslie was not president.



Thus, the simple truth of the above generalization, though clearly necessary, is notsufficient to show that its instances are logically true on this selection of expressions.

The situation is similar with our other examples. Take, for instance, the sentence

Either Lincoln was president or Lincoln was not president.The logical truth of this sentence does not depend on the specific meanings of „Lincoln‟

or „was president.‟ Consequently, it is clear that the corresponding universal closure

must be true:

VxVP[x P or not x P].

Any counterexample to this generalization would show either that the original sentencewas not logically true, or that its logical truth de-pended on the specific meanings of thename „Lincoln‟ or the predicate „was president.‟ Thus, the fact that the original sentence

is true solely by virtue of the meanings of „or‟ and „not‟ does guarantee the truth of I liecorresponding closure. But the mere truth of the closure does not assure us that itsinstances are Xrue by virtue of meaning alone, any more than it did in the previous case.

In both of these examples if we generalize further —say, if we replacc “is a bachelor‟ or

“not‟ with appropriate variables— the result¬

ing closures turn out to be false. This simply shows that the logical or analytic truth ofthe original sentences did indeed depend on the specific meanings of these expressions,not just on their general se-mantic category. But notice that even if the closures had in

fact been true, this would not alone have guaranteed that the original sentences weretrue solely by virtue of the meanings of the remaining expres-sions. It would mean theywere either true by virtue of meaning alone, or true by virtue of facts about the world.But of course that is true of any sentence whatsoever.

We do not usually think of logical truth, the ordinary concept, as relativized to acompletely arbitrary selection of expressions. This in itself might be sufficient reasonnot to construe Tarski as offering a completed analysis of an irreducibly relationalnotion. Still, it is clear that there are various ways we can view logical truth as sorelativized, without doing injustice to the intuitive notion. In the above discussion, Ihave considered the most natural: take logical truth to be a form of analytic truth andrelativize to the semantic function of the selected expressions. But we could havechosen to emphasize some other dis-tinctive feature of the ordinary concept — say, thefact that logical truths are necessarily true or can be known a priori. Thus, the aboveexamples all express necessary truths, and the fact that they manage to do so is

peculiarly dependent on certain of the expressions they con-tain. In which case wemight take the notion of logically true with respect to the members ofi§ to meansomething like “expresses a necessary truth by virtue of the expressions in ft.” But none

of these relativized ver¬sions of our ordinary concept is captured by Tarski‟s analysis.

This, of course, is quite obvious, once we give it a moment‟s thought: „Abe Lincoln was president‟ is not a necessary truth, or an a priori truth, or an analytic truth— even when



we take into consideration all of its component expressions. Yet it comes out logicallytrue, according to Tarski‟s account, if we include all of these expressions in ft.

What is important to note, though, is that if we construe Tarski‟s account in this way—

or misconstrue it, as the case may be — it becomes perfectly clear exactly how the

analysis has gone astray. The account takes a merely necessary condition for logicaltruth to be a sufficient condition. For if a sentence 5 is logically true with respect to a setft of expressions, where this notion is taken in any of the ways suggested above, then itfollows that the corresponding universal closure Vt»i . . . Vt»n[ S' ] will indeed be true.But the converse, namely principle (Hi'), simply does not hold. Recognizing this will

become important in Chap¬ter 11, when we discuss the significance of thecompleteness theorem for first-order logic. For the moment, though, let us put it on the

back burner.

8

Substantive Generalizations

According to the reduction principle, any instance of a true universal generalization islogically true. Unmodified, this principle is clearly wrong. And it is equally clear thatthe basic problem with the principle is not solved — indeed, is not really evenaddressed — by the modifica-tion incorporated into (in'). Before looking at thealternative modifica¬tion, we should first consider just what we are up against. What,exactly, is the defect that our modified principle must avoid?

The key problem is this. When we equate the logical truth of a sentence with theordinary truth of a universal generalization of which it is an instance, we risk an accountwhose output is influenced by f acts of an entirely “extralogical” sort. Clearly, the

question of whether the sentence

(1) If Leslie was president then Leslie was a man

is a logical truth does not depend on the sorts of historical facts that determine the truthor falsity of the generalization

(2) Vx[if x was president then x was a man].

As it happens, (2) is true, and so any account that equates the logical truth of (1) withthe simple truth of (2) will mistakenly declare the former logically true. But of coursethe basic problem with the account would remain even if (2) happened to be false. Inthat case the account would issue the right assessment of (1), but certainly not because itcoincides with our ordinary understanding of logical truth, or even of fers a reliable testfor that property. The analysis would be just as liiulty — it would still entrust thelogicalstatus of (1) to the political contingencies described by (2) — though in that, case thedefect would not show up in the actual assessment of (1). But only thanks to the way

those contingencies happened to fall out, not thanks to the definition itself.



This point is a simple one, but still easy to overlook. Let me change the exampleslightly to emphasize it. Suppose we were presented with a definition that ties thelogical truth of

(3) If Leslie is a member of the Senate then Leslie is a man to the ordinary truth of

(4) Vx[if x is a member of the Senate then x is a man].

Clearly, we would consider this analysis unacceptable, whether or not there were anywoman senators. During those congressional terms in which there are none, the flaw inthe account would be highlighted by the incorrect claim that (3) is logically true. Butduring terms in which there are woman senators — as is now the case — the very samedefini-tion would not suddenly be judged an adequate account of logical truth. Thiseven though it would not — and at present does not — issue a faulty assessment of (3). Allwe need note is that (3) is not logically true, and neither would it have been logically

true had all senators been men, that is, had (4) come out true.

This simple observation shows two things. Most obviously, it shows that the suggestedaccount does not capture, or even come close to capturing, the ordinary concept oflogical truth. For the extension of that property, as we ordinarily understand it, iscompletely indepen-dent of the makeup of the Senate. Indeed, none of the keycharacteris-tics that we attribute to logical truths depend on the substantive, extra-logical facts that determine the truth or falsity of (4). Thus, (3) is not a necessary truth,or an a priori truth, or an analytic truth, and neither would it have been any of these had(4) come out true.

More important, though, our observation shows that the suggested account is not areliable test for logical truth, one whose extension is sure to be right. Indeed, byequating the logical status of sentences like

(3) with the simple truth or falsity of substantive, nonlogical claims like

(4) , we clearly forfeit any hope of an internal guarantee of extensionaladequacy. For the extension of the account is determined by facts — here, facts of a

political or historical sort — that are entirely indepen¬dent of either the analysis itself or

of the property of logical truth. Obviously, there is no way to tell whether ourassessment of (3) comes out right, and hence whether the definition is extensionallycorrect, without checking the makeup of the Senate. The analysis itself, on its own,simply cannot guarantee this.

It is because of these failings that accounts based on principle (in), at least in itsunmodified form, seem intuitively unacceptable. The key to their failure is thedependence of their assessments on extralogical features of the world. If our assessmentof the logical status of a sentence rests on substantive facts about Abe Lincoln, or aboutthe presidency, or about anything of the sort, then either the assessment will as a matter

of fact be wrong, or it would have been wrong had the facts in question been otherwise.



Either possibility is equally damaging to our claim to have captured the ordinaryconcept of logical truth or to have devised a reliable test for that property.

Modifying the Principle (Part Two)

It is clear that with certain selections of ft, the assessments made by Tarski‟s account aresubject to extralogical influence, depending, as they do, on the truth values ofsubstantive generalizations like (2) and

(4) . This is most obvious when we include names or predicates among the fixedterms: if „Abe Lincoln‟ or „was president‟ are included in ft, then it is hardly surprising

when we find that the output of the account depends on nonlogical facts about Lincolnor the presidency. But it seems equally apparent that if we exclude these expressionsfrom ft, and if they are not definable in terms of the remaining expressions, then ourassessments will not depend on facts about that particular individual or that particular

property. What this suggests is obvious: maybe we can avoid an account that is subjectto extralogical influence by imposing restrictions of some sort on the selection of ft.Specifically, perhaps we can sidestep the basic problem with principle (Hi) byin¬cluding in ft only expressions of a distinctively “logical” sort. The plausibility of thesecond construal of Tarski‟s account rests on exactly this assumption.

We need to explore this possibility in some detail. The idea is that we should not equatethe logical truth of a sentence with the truth of just any generalization of which it is aninstance. Rather, the logical status of a sentence should be tied only to generalizationsof a very special sort: those that contain in their matrix1 nothing but variables bound to

the initial universal quantifiers and constant expressions of a distinc¬tively logical sort.Thus, the logical truth of „Abe Lincoln was presi¬dent‟ will not be determined by the

truth values of:

[Abe Lincoln was president]

or:

Vxfx was president]

or:

VPfAbe Lincoln P] but rather by the truth value of:

VxVP[x P].

This simply because the first three closures contain expressions of an intuitivelynonlogical sort, and so the outcome of our test might de-pend on facts of a similarlynonlogical sort.

If we construe Tarski‟s account in this second way, the assumed modification of

principle (in) is fairly clear. The modified principle would run roughly like this:



(Hi") If a universally quantified sentence is true, and the constant expressions appearingin its matrix are of a distinctively logical sort, then all of its instances are logically true.

Now, the notion of a “distinctively logical” expression is rather vague, but we will not

worry about that just yet. Our present concern is whether this construal of the account,

however it is ultimately spelled out, could render the analysis immune to the kind ofextralogical influence that plagues the general, quantificational approach. Will theoutput of the definition still depend, as we would expect from prin¬ciple (Hi), onintuitively irrelevant features of the world — say, historical or physical or mathematicalfacts — or has that dependence been se¬vered by the more careful choice of fixed termsreflected in (Hi")?

However reasonable (Hi") may initially seem, it in fact suffers from exactly the samedefect as the original principle (in). In other words, even when we hold fixed onlydistinctively logical expressions, the output of Tarski‟s account remains dependent on

completely non-logical facts. This is true no matter how narrowly we construe thenotion of a “logical” expression, and whether or not we impose the cross-termrestrictions employed in the standard, first-order semantics.

The reason this has been overlooked is that with very weak lan-guages, such as thosewe have considered so far, we can so arrange it that the definitions are extensionallycorrect in spite of this faulty dependence. But that speaks no more in favor of theanalysis than the observation that we can safely tie the logical truth of

(3) If Leslie is a member of the Senate then Leslie is a man to the simple truth of

(4) Vx[if x is a member of the Senate then x is a man] so long as the voting publiccooperates.

The Size of the World

Viewed in this second way, Tarski‟s account rests on a quite straight-forwardassumption. The assumption is that facts of a nonlogical sort can influence the outcomeof his test only if expressions of a nonlogical sort are included in the set of fixed terms.This assumption is neither obviously right nor obviously wrong. What is clear is that the

converse of the assumption is true: if ft contains expressions like names and predicates,then the account will certainly be subject to all manner of extralogical influence.Further, it is clear, or at any rate relatively clear, that certain kinds of extralogicalinfluence can be excluded by banning names and predicates from ft. Certainly, it is hardto see how facts involving specific individuals and particular properties could affect theoutcome of the definition if there were no way to refer to those individuals or those

properties.

Still, not all facts of a nonlogical sort involve specific individuals or properties. Let us begin by considering the most obvious examples: facts concerning the size of the

universe, the number of individuals that exist. Take, for instance, the followingsentences, touched upon briefly in Chapter 5:



cr2: 3X33>(X =É 31)

0-3: 3x33>3z(x i^y/\y + z/\x + z)

For each n, the sentence crn says that there are at least n objects. According to the

standard conception, none of these should come out logically true: the size of theuniverse is surely not a matter of logic.2 But if we consider the existential quantifier, theidentity predicate, and the truth functional connectives to be distinctively logicalexpressions, then Tarski‟s account equates the logical truth of these sentences with thesimple truth of the following (trivial) closures:

[<r2]: [3x3ji(x *ji)]

[0-3]: [3x33>3z(x ^â^^zax^z)]

Clearly, some of these sentences are true. Exactly how many de-pends, of course, on the

size of the universe — that is, on how many objects there happen, in fact, to be. If theuniverse is infinite then all of the sentences are true, and so will be mistakenly judgedlogically true. If it is finite, then only a finite number of them will be judged logicallytrue. But the important point is not how many of these sentences the definition getswrong, but rather th^fact that the assessments here are clearly dependent on a nonlogicalstate of affairs. This is exactly the del cel that seemed so apparent when we firstconsidered the un¬modified principle (Hi). As we noted then, whether a sentence islogi¬cally true should not depend on substantive, extralogical facts, whether historicalor physical or mathematical. This is the problem with the original principle (Hi), and it

still infects (Hi"), at least if this new principle ties the logical truth of crn to the simpletruth of [cr]. Though perhaps true, [crn] is not a logical truth, and neither of course isits “instance,” crn.

On their own, these examples are not tremendously persuasive. For there are two quitedifferent responses that immediately suggest them-selves. First, none of the abovesentences comes out logically true in the standard interpretational semantics for first-order languages, and this fact is indeed independent of how many individuals theuniverse as a whole happens to have. Of course, since the standard semantics treats thequantifiers as variable and then imposes the cross-term restrictions that this move

ultimately necessitates, this response presupposes some resolution of the problemsdiscussed in Chapter 5. Still, we should look for a solution to the present difficulty inthis direction. Second, even if cross-term restrictions cannot be made consistc.it withTarski‟s analysis, we could always question whether the identity predicate is in fact a

“distinctively logical” expression, and so whether it should be included in Here, too, we

might find a solution to this problem. Let us explore these two suggestions in turn, tosee whether the assessments made by the resulting accounts are independent ofquestions about the size of the universe.

The first suggestion would have us vary the domain of quantifi-cation — that is, theinterpretation of 3. Once we do this, the logical status of the sentences 0-2, cr?, . . . no



longer depends in any way on the size of the universe. The reason for this is simple: nomatter how small or large we assume the universe to be, whether finite or infinite, wecan always interpret 3 in such a way that it quantifies over a very small subcollection ofthe actually existing objects, including a subcollection of one. Thus, if 3 is interpreted tomean, say, some sixteenth president of the United States, rather than the unrestrictedsomething, then all of the <jn will clearly be false. For in fact there were not twodistinct sixteenth presidents, or three, or four, and so forth.

Instead of talking about reinterpreting 3, consider how this same tactic converts to thequantificational framework. To do this, we need an existential quantifier variable, E,whose satisfaction domain consists of various subcollections of the universe, variousquantifier restriction sets.3 According to the present suggestion, the logical truth ofcr«>, 0-3, ... is dependent on the truth values of the following closures:

V£[o-2(3/£) ]: VE[ExEy(x * y)]

V£[cr,s(3/£) ]: VE[ExEyEz(x ±y/\yizAx±z)]

These closures say, in effect, that every subcollection of the universe — that is, everyquantifier restriction set — contains at least two (three, four, . . .) members. Butsequences that assign a singleton to E satisfy none of the closures‟ constituent sentential

functions. Conse¬quently, all of these closures come out false, no matter how large theuniverse actually happens to be. This is how Tarski‟s account avoids declaring any of

our original sentences logically true when we take the existential quantifier as a variableterm. This is the expedient built into the standard interpretational semantics for first-

order languages.4

Now, does this tactic solve the problem with Tarski‟s account, or does it simply treat

one of the symptoms? Admittedly, since the truth values of the above closures areindependent of the size of the uni-verse, the assessments of the crn no longer depend onthis extralogical fact. And this is as it should be. But this hardly guarantees that theresulting account escapes such influence elsewhere. Indeed, we do not have to look farto find sentences whose assessment still depends on precisely the same fact. Consider,for example, the negations of our original sentences:

— icr2: — i3x3;y(x ± y)

— icr3: — i3x3;y3z(x ̂ aj^za^z)

For each n, —icr says that there are fewer than n objects in the universe. Once again,

none of these should come out logically true, no matter how large or small the universehappens to be. But consider how the standard account assesses these sentences. Treatingthe existential quantifier as variable, the definition tags the logical status of thesesentences to the ordinary truth values of the following closures:

V£[-ict-2(3/£)]: VE[ÊxEy(x ± y)]



V£[ — ict3(3/£)]: V£[ — \ExEyEz(x ± y Ay ± z a x =£ z)]

Notice what these closures say. For each n, the sentence

(5) V£[ — icrn(3/£)]

claims that every subcollection of the universe contains fewer than n objects. And thiswill be true just in case the largest subcollection of the universe — namely, the universeitself — contains fewer than n objects.5 Thus, if the universe is finite, the present accountwill mistakenly pronounce an infinite number of the sentences — icr2, — 103,.. .logically true; in that case, the account will be extensionally incorrect. Of course, if theuniverse is infinite, Wne of these sentences will be declin ed logically true. But is that

because the account has captured our ordinary notion of logical truth? Alter all, thesesentences are not in [art logically true*, but nc*itlu*i would thc*y be logically true* ifthe

universe were finite. Yet according to the standard account, the sen-tence — icrn is notlogically true only because sentence (5) is false — that is, only because there are morethan n objects in the universe.

Sentence (5) makes a perfectly ordinary claim about the world, one that has little, ifanything, to do with logic. If the world has fewer than n objects, then (5) is true; if morethan n, then it is false. When we trust the logical status of — icrn to the truth or falsity ofthe substantive claim made by (5), we put ourselves in the same position as we werewith (1) and (2) or (3) and (4). If there are fewer than n objects, our situation is parallel

to (1) and (2) — or (3) and (4), during terms in which there are no woman senators.Then, our account will be extensionally incorrect, thanks to the truth of (5). If there aremore than n objects, our situation is more like (3) and (4) — or (1) and (2), had a woman

been elected president. In that case, the defect in the analysis remains, though it isdisguised by the fact that (5) is false. But whichever is the appropriate parallel, it is clearthat with the current selection of fixed terms — that is, the selection employed by thestandard, first-order semantics — the assessments made by our test are still influenced bythis extralogical state of affairs. The account still suffers from the basic defect of

principle (Hi).

How does the standard semantics deal with this problem? After all, the sentences — icr2, — icr3, ... do not come out logically true according to the usual, model-theoreticaccount, at least as it is ordinarily pre-sented; if they did, the analysis would have few, ifany, defenders. Exactly what feature of the standard semantics assures us that none ofthese is declared a logical truth? Is there some subtlety about the account that we havesimply overlooked?

The answer is that nothing about the standard semantics assures us of this, nothingwhatsoever. We get our “assurance” from an assump-tion made quite independently ofour account of the logical properties — an assumption, needless to say, about the size ofthe uni-verse. When we present the standard first-order semantics, we gener-ally do so



within some set-theoretic framework or other. We build our models out of objectscobbled together from the set-theoretic universe, and in doing so we naturally assumevarious facts about that universe. Generally, the specific framework we presuppose isthat of Zermelo- Fraenkel set theory, but of course nothing about the analysis dictatesthis particular choice, or even that our background theory should be a set theory ratherthan a class theory or category theory or property theory.

Now, in the standard presentation, the only thing that assures us that none of the abovesentences comes out logically true is the axiom of infinity assumed in the underlying settheory. It is this axiom thai guarantees the existence of infinite models (that is, infiniterestriction

sets for interpreting our quantifiers), and so that guarantees interpre¬tations of 3 inwhich all of these sentences come out false. Or, to put it in quantificational terms, it isthis axiom that entails, for any n, the falsity of (5), and so assures us that the accountdoes not wrongly declare — icrn a logical truth. If the very same semantics weredeveloped within a theory that does not presuppose that axiom — say, Kripke- Platek settheory — then our assurance would instantly evaporate. The analysis itself provides noguarantee; the outside assumptions are what do the trick.

It is important to see exactly what is going on here. The basic problem to be addressedis this. Our assessments of the logical status of — icr2, — 10-3,. . . , like our earlierassessments of (1) and (3), are depen¬dent on a nonlogical feature of the world, the sizeof the universe. So long as this dependence remains, our definition clearly has not

cap¬tured the ordinary concept of logical truth, nor do we have any inter¬nal guaranteeof its extensional adequacy. For none of these sentences is a logical truth, and neitherwould any of them be logical truths if the universe were finite. Whether or not thesesentences are logically true simply does not depend on the truth of substantive claimsabout the size of the universe. But when we give the standard, first-order seman¬tics forthis language, do we solve this problem? On the contrary, far from severing the faultydependence, we simply annex to our seman¬tics a sufficiently powerful assumptionabout “the world”— namely, the axiom of infinity — which then provides us with theright as¬sessments of these particular sentences. This is equivalent to “solving” our

incorrect assessment of (1) by electing a woman president: the tactic may get the

assessment right, but it hardly corrects the underly¬ing defect.

We might dramatize the point in the following way. Suppose the standard,interpretational semantics really did capture our ordinary understanding of the logical

properties. If this were the case, then it would be inconsistent — not just wrong, butinconsistent — for a finitist to hold that none of the sentences — 1 cr2, ~10-3, ... islogically true. But of course there is nothing whatsoever about the finitist‟s basic

assumption that makes this an incoherent position; quite to the contrary, it is the onlyreasonable stand to take. The finitist is perfectly within his rights to claim that the

universe could have been larger than it happens to be: although there might in fact beexactly n objects (both physical and mathematical), there might have been n+1 objects,



or n+2, and so forth. From this, the finitist should obviously be allowed to conclude thatnone of these sentences is logically true. But the standard account of logical truth wouldrule out this\sondusion: if there happen to be no more than n objects, then —icr+|, —

icrn+2, . . . are all declared logical truths. This in spite of the fact that they are not, evenfrom the finitist‟s perspective, either necessarily true, or analytically true, or knowable a

priori. This in spite of the fact that they have none of the distinctive features ordinarilyattributed to logical truths.

Note that this point does not depend on our agreeing with the finitist‟s position. Indeed,

when it comes to the “ontology” of mathe-matics, I personally tend toward a rathernaive platonism. But if we even acknowledge that the finitist‟s position is a coherent

one, then it follows that the standard account has gotten things wrong. For clearly, fromthe finitist‟s perspective, no claim about the specific size of the universe is logically

true, even though some such claim might, purely as a matter of fact, be true. Appealing

to the finitist‟s position is just a handy way to emphasize the defect in the account, toemphasize that generalizations like (5) do indeed make substantive, extralogical claims.This defect remains even if the finitist is in fact wrong.6

Indeed, we can put this point even more strongly. The problem these sentences bringout remains even if we consider the finitist to be necessarily wrong — that is, even if wetake the axiom of infinity to be a necessary truth. All we need recognize is that theaxiom of infinity, and its various consequences, are not logical truths. This is all that isre¬quired to see that the output of Tarski‟s account is still influenced by extralogical

facts — in this case, by the set-theoretic fact expressed by the axiom of infinity. It isexactly such potential influence that makes the original reduction principle (in) seem soimplausible, and it is clear that the influence survives the modification built into (Hi").

Let us pause for a moment and take stock. When we consider the existential quantifier,the identity predicate, and the truth functional connectives to be “distinctively logical”

expressions, the output of Tar¬ski‟s account clearly depends on the size of the universe:

whether crn comes out logically true is determined by whether there are more than nobjects in the universe as a whole. To block this dependence, and the faulty assessmentsit would yield, the standard, model-theoretic ac¬count varies the domain of the

quantifier. But the output of the resulting account is no less dependent on the size of theuniverse: whether — icrn comes out logically true is still determined by the number ofobjects in the universe as a whole. Here, though, the standard account offers no remedy:it simply appeals to an external assumption about the size of the universe, and leaves thefaulty dependence intact.

Now, if we still want to claim that the standard semantics avoids the intuitive defect in principle (Hi), there seems only one recourse avail¬able. We must claim that the axiomof infinity does not express an “extralogical” claim, and so that our account is not, at

least on this score, subject to extralogical influence. But this response is implausible inthe extreme. For if it is a logical truth that there are infinitely nuiny objects, then it must



equally be a logical truth that there are at least twenty-seven. So to execute this defenseconsistently we would have to argue that, contrary to our initial impressions, all of thecrn really should be judged logically true. We might put it this way. The claim that thereare at least twenty-seven objects (cr27) is not a logical truth, by anyone‟s lights. Neither

is the claim that there are fewer than twenty-seven ( — 10-27). Yet if Tarski‟s account of

logical truth is right, — icr27 is not logically true only because V£[ — io-27(3/£)] isfalse — that is, only because there are at least twenty-seven objects. Clearly, the outcomeof Tarski‟s account here depends, by anyone‟s lights, on matters of a nonlogical sort.

The assessments made by the standard, first-order semantics are influenced by at leastone kind of nonlogical fact: the size of the universe. In a moment we will see that theydepend on other such facts as well. At this point, though, let us briefly look at thesecond sug-gestion for dealing with the crn: treating the identity predicate as anonlogical expression. I will return to the standard account in a moment.

If we exclude the identity predicate from Tarski‟s account equates the logical truth of

cr2, the claim that there are at least two objects, with the ordinary truth of the closure

(6) Vi?[3x3;y(^ci?;y)].

Roughly speaking, (6) says that no relation relates everything with everything. Ofcourse, the exact claim made by this sentence will depend on how we specify thesatisfaction domain for the relation variable R — in particular, whether we take thatdomain to consist of relations themselves, or instead extensions of relations, that is, setsof ordered pairs. But intuitively, the claim seems false: certainly, the relation of

coexistence, or that of being either identical or not identical with, relates absolutelyeverything in the universe with absolutely every¬thing in the universe. Andconsequently, the suggested account would not mistakenly judge cr2 logically true, nomatter how large the uni¬verse happens to be. This holds as well for the other crn.7

Once again, our assessments of the crn have been divorced from questions about thesize of the universe. And this is as it should be. But once again, the same problemsimply recurs elsewhere. To see this, consider the following three sentences.

a: VxV;yVz(x is taller than y a y is taller than z

y: V)»3x(x is tiillcr than y)

The first two sentences say that taller than is transitive and irreflexive, which of courseit is; the third sentence claims (falsely, let us suppose) that there is no tallest thing. Nowconjoin these sentences and negate the result:

(7) -i(a a p a y)









If the only expressions we consider “distinctively logical” are the truth functional

connectives, then Tarski‟s account equates the logical status of this sentence with the

ordinary truth value of the closure

(9) VxV)/VP[x P-+yP].

If we take the satisfaction domain for the predicate variable P to consist of arbitrary sets,then (9) will be true if and only if the universe as a whole contains just one object. (Ifthe satisfaction domain consists of properties, (9) will be true just in case the universecontains only one type of object — that is, if all objects share the same properties.) Itturns out that for any n, we can construct a sentence rn which, like T\, contains onlytruth functional operators, and whose corresponding closure is true just in case there are,as a matter of fact, no more than n objects (or n types of objects). But none of thesesentences is logically true, and neither would any of them be logically true if theuniverse were finite (or if it had finitely many types).

Now, my point here is to emphasize that the problem with Tarski‟s account has nothing

to do with the question of whether we are mistak-enly including in ft expressions of anonlogical sort. Surely, if any terms deserve the status of “distinctively logical”

expressions, the truth functional connectives do. The problem does not come from anincor¬rect choice of ft, but rather from the assumption that we can exclude

the influence of nonlogical facts by excluding nonlogical expressions from the set offixed terms. This assumption is simply mistaken. Prin¬ciple (Hi") suffers from exactlythe same defect as the original princi¬ple (Hi).

Other “Extralogical” Influence

So far, I have emphasized how facts about the size of the universe influence theassessments made by Tarski‟s account, and how this influence remains no matter how

tightly we construe the notion of a “logical” expression. It would be a mistake, though,

to think that this is the only type of extralogical fact determining the output of thedefini¬tions. Indeed, it is not. It just so happens that this influence is relatively striking,since it forces the standard account to rely on the axiom of infinity, an obvious appeal toa nonlogical assumption.

Actually, it should by now be clear that many other assumptions, besides the axiom ofinfinity, influence the outcome of the account. We can emphasize this by taking a closerlook at any of our examples involving predicate or relation variables. Consider, forinstance, the sentence rj, and its corresponding closure (9). Notice that simply assumingthat the universe has more than one object, or even infinitely many objects, does notalone guarantee that (9) is false. What we need to know in addition is that two or moreof the existing objects are distinguishable by members of the predicate domain of thesatisfaction relation. As I mentioned earlier, if the predicate domain consists of

properties, then (9) would be true in any completely homogeneous world, regardless ofits size.



In the standard, interpretational semantics, what facts guarantee that closure (9), and itscounterparts for r2, T3, ... , all come out false? First is the assumption of the axiom ofinfinity; this is what guarantees that there are two (three, four,...) objects to bedistinguished. But this alone does not guarantee that the objects are “distinguishable.”

When we take the predicate domain to consist of arbitrary sets, what does this job is the pair-set axiom. This axiom assures us that every object is (in the now-relevant sense)distinguishable from every other, each being the only member of its own singletonset.10 Given this assumption, the truth value of (9) depends only on the size of theuniverse; without this assumption, the closure raises additional issues as well.

Now, nothing about Tarski‟s analysis leads us to construe the vari-able P as rangingover sets rather than properties, and this decision makes for a considerable difference inthe claims made by generaliza-tions like (9). When we interpret such claims in the latterway, their appeal to extralogical facts seems quite blatant: clearly, questions about how

many different types of objects exist is not a matter for logic to decide. But even whenwe settle on the set-theoretic construal of these variables, as is standardly done, theclaims that result are, not surprisingly, of a set-theoretic, not a logical, sort. When wegive the standard semantics, we do nothing to prevent this dependence, but simplyimport all of our background assumptions about the universe of sets in order to carry outour assessments. The assessments end up depending on all of these assumptions, fromthe most powerful to the most mundane. The only difference is that, as the underlyingassump-tions become more powerful, the faulty dependence becomes increas-ingly hardto overlook.

Let us briefly look at another example of the same problem, one that has frequently been discussed but repeatedly misdiagnosed. It is often said that when we move to asecond-order language, one that allows quantification of predicate variables, logicaltruth becomes a relative notion, one that depends on the underlying set-theory

presupposed. The reason people say this is that when we apply the standard account oflogical truth to a full second-order language, there are sentences that come out logicallytrue if we assume (say) the continuum hypothe¬sis, but that do not come out logicallytrue if we assume its negation.11 This is often taken to be an objectionable feature ofsecond-order logic: after all, why should the logical truth of a sentence depend on such

highly abstract set-theoretic claims, claims that are not, intuitively, a matter of logic atall?

By now, what is going on here should not surprise us in the least. What we have is asentence of our second-order language, call it C, whose logical status is being tied to theordinary truth or falsity of a certain generalization, say,

(10) Vi/, ... Vi/n[C'].

It turns out, though, that the facts described by (10) are of an entirely extralogical sort:whether (10) is true depends on nothing more nor less than the continuum hypothesis —

and clearly, neither it nor its negation is a logical truth. But this is simply the faulty



principle (Hi) at work. The fact that our assessment of C depends on the substantiveclaim made by (10) is no different from the fact that our assessment of

(7) depends on the size of the universe (the substantive claim made by

(8) ), or that our assessment of (3) depends on the makeup of the Senate (thesubstantive claim made by (4)).

Of course there is a difference between (10) and (8), but not one of much import. Thetruth value of (8) is guaranteed by the axiom of infinity, which, though certainly not amatter of logic, is nonetheless a far more comfortable assumption to make than eitherthe continuum

hypothesis or its negation. It is because of this that we find it easier to overlook this particular appeal to a nonlogical state of affairs when giving our first-order semantics.

In contrast, when we move to a second-order language and come up against thesomewhat daunting visage of the continuum hypothesis, the appeal is rather hard not tonotice. But the difference here does not show that there is anything peculiar about thelogic of second-order languages, or that, as it is sometimes put, second-order logic isreally “set-theory in disguise.” The problem lies with our faulty account of the logical

properties, which mistakenly equates the logical status of C with the ordinary truth orfalsity of (10). But this is no more or less mistaken than equating the logical status of(7) with the truth value of (8), or the logical status of (3) with the truth value of (4).Sentences like C are not logically true, and it is only our allegiance to a faulty accountof logical truth that makes us think that they are (or would be, if the continuum

hypothesis were true).

Unmodified, principle (in) is obviously false. The mere truth of a universalgeneralization can only guarantee that its instances are true; it cannot guarantee thatthey are logically true. Of course, if the generalization itself is logically true, then theinstances will be logically true as well: that was the substance of the uncontroversialclosure principle (ii). But if the generalization is not logically true, if it makes, say, asubstantive historical or physical or set-theoretic claim, then neither will its instances belogical truths. The second construal of Tarski‟s account relies on the assumption that no

such substantive claims can be made by generalizations of a particular sort — namely,those that contain in their matrix only distinctively logical expressions. But thatassumption is simply false, as can be seen from the claims made by such generalizationsas (5), (8), (9), and (10); even if true, none of these claims is logically true. And neither,of course, are their instances.

9

The Myth of the Logical Constant

I have not claimed that when we apply the standard account of the logical properties to

certain simple, first-order languages we get an incorrect extension. Indeed, as I explainin Chapter 11, the sentences that come out true in all models of the standard first-order



semantics do in fact owe their truth to nothing more than the meanings of theconnectives, the quantifiers, and the identity predicate (assuming that we employ theusual cross-term restrictions and that all the usual axioms of set theory are true). Andthere is a perfectly understandable reason for this. But the reason is not that we have areliable account of logical truth and logical consequence, one whose extension is sure to

be correct. Rather it is due to a combination of the weakness of our first-order languageand the strength of our underlying set-theoretic assumptions. The world, in effect,simply compensates for our faulty analysis.

Let me give an analogy. Suppose we applied Tarski‟s account to a language containing

names, truth-functional connectives, and the fol-lowing three predicates: „is a man,‟ „is

a bachelor,‟ and „is a senator.‟ Suppose further that we included in $ all expressions

except names. Thus, for example, we would equate the logical truth of

(11) Leslie is a senator — > Leslie is a man with the ordinary truth or falsity of

(12) Vx[x is a senator —► x is a man].

Similarly, the logical truth of

(13) Leslie is a bachelor -^Leslie is a man would be tied to the ordinary truth of

12 6 The Myth of the Logical Constant

(14) Vx[x is a bachelor —► x is a man].

Notice that this application of Tarski‟s account would have a quite reasonableextension. Indeed, the only sentences that would be de-clared logically true are those —

like (13) — that are true solely by virtue of meaning. All others, for example (11), wouldnot come out logically true, thanks to the falsity of their corresponding generalizations.This is because, given the way the world happens to be — that is, given the fact that thereis at least one woman senator, and at least one married man, and at least one unmarriedsenator, and so forth — the only true generalizations that we will encounter are those,like (14), that are true by virtue of meaning alone. The world, or rather that very limited

part of the world describable in our language, is sufficiently varied that all of thesubstantive generalizations, such as (12), happen to be false. Of course, there is nonecessity to this: there might not have been women senators or married men; it justhappens that there are.

Now, no one would suggest that the above account captures either our ordinary notionof logical truth or our notion of analytic truth (if these are different). After all, it is clearthat the proper extension here is not due either to a proper analysis or to a properselection of fixed terms. On the contrary, the world happens to be compensating for anobviously incorrect account of logical truth. We might emphasize this in a couple ofways. First, we might note that had the relevant facts been otherwise — say, had all male

senators been married, or had all senators been women — the account would have produced a large number of faulty assessments. What stands in the way of an incorrect



assessment of, for example, (11) is not an adequate analysis but a woman senator.Second, we might observe that if the expressive power of our language were slightlyincreased — say, if we added the pre-dicate „is president‟ or „is from New Jersey‟— thenonce again the extension of the account would be thrown completely off. Then, truegeneralizations like (2) would bring out the error in our analysis:

(2) Vx[if x was president then x was a man].

This is precisely what is happening with the standard, interpreta-tional semantics forfirst-order languages. The problem is not that the account gets the wrong extensionwhen applied to such languages. Indeed, assuming all the standard axioms of Zermelo-Fraenkel set theory, the only true generalizations that we encounter are in fact logicallytrue — and so too are their instances, by the closure principle (ii). But here again theworld — that is, the set-theoretic universe — is simply compensating for an incorrectanalysis of the logical properties. What stands in the way of a large number of faultyassessments is simply the variety afforded us by the set-theoretic universe — or ratherthe limited portion of that universe describable in our language.

We can emphasize this in the same two ways. First, we can note that with different background assumptions — say, if we assume a finite rather than an infinite universe —

the extension of the account is clearly wrong. For then, sentences like (7) are declaredlogically true. Second, we can observe that by simply increasing the expressive powerof our language — say, by adding second-order quantifiers — the ex-tension of theaccount is, once again, thrown completely off. Then, generalizations like (10) are what

bring out the failure of the analysis. Now, it is important to see that in this case, as in the previous one, we are notguaranteed a correct extension either by Tarski‟s general analy¬sis or by our particular

selection of fixed terms. What makes for the correct extension are such things as theexistence of an infinite number of objects, the assumed distinguishability of thoseobjects, the existence of transitive, irreflexive relations with and without minimalelements, and so forth. The reason it is crucial to recognize this is that our attention is soeasily drawn away from the real defect in the account — namely, the reduction principle,however modified — and toward the supposed issue of how to go about making the

“proper” selection of The assumption is that the reason Tarski‟s account works in thiscase must be some special characteristic of the expressions we have held fixed, and thatthe reason it fails in other cases is that we have held fixed expressions without that

peculiar characteristic. This is the source of the so-called problem of the logicalconstants.

It is understandable how this issue comes to seem the key point. On the one hand, it is perfectly clear that Tarski‟s general account— that is, when we allow arbitrary selectionsof $ — does not capture our ordinary notion of logical truth. This is apparent both fromthe faulty assessments it yields, and from simple reflection on the unmodified principle

(in). On the other hand, it also seems that when we include in $ only first-orderquantifiers, truth functional connectives, and the identity predicate, the definition



produces a plausible extension — modulo our background assumptions about theuniverse, and the special treatment given the quantifiers. From this it is all too easy toconclude that the analysis is basically correct, but in need of a sup-plement. What ismissing, we assume, is an account of the distinctive characteristic that makes it “right”

to hold fixed such expressions as truth functional connectives, but “wrong” to hold

fixed names, pre-dicates, and (according to some) second-order quantifiers.

Now, of course, for any finite number of expressions, it will always be possible to find properties shared by all and only those expressions. At the very least' we can simply listthe expressions and take the property of being a member of that list. But when we seeour goal as that of supplementing Tarski‟s analysis, it seems clear that not just any

property will do. The reason for this is simple. Since Tarski‟s general account captures

none of the modal, epistemic, or semantic character¬istics of logical truth and logicalconsequence, it seems that these characteristics must somehow emerge from the sought-

after sup¬plement, from our account of what makes certain expressions “genu¬inelylogical” and others not. It would hardly do, for example, to add the injunction to holdfixed only words spelled with fewer than four letters, even if the injunction seemed towork. Such a supplement would not make up for what is missing from the generalaccount. It would hardly explain why logical truths are, or are commonly thought to be,necessary or a priori or true solely by virtue of meaning. It would hardly persuade usthat the account can be relied on to make the right assessments.

This is why the task of characterizing the logical constants comes to seem at once soimportant and yet so difficult. Indeed, most of the burden of Tarski‟s analysis seems to

shift to exactly this issue. But by now it should be clear that the issue is based on aconfusion — namely, the assumption that when the account works, it works due to some

peculiar property of the expressions included in ft. But this assump-tion is false: there isno property of expressions that guarantees the right extension in these cases, nonewhatsoever. After all, any property that distinguishes, say, the truth functionalconnectives from names and predicates would still distinguish these expressions if theuniverse were finite. But in that eventuality, Tarski‟s account would be exten¬sionally

incorrect. This observation alone is enough to show that it is not any property of theexpressions we hold fixed, the so-called logical constants, that accounts for the

occasional success of Tarski‟s defini¬tions. Here our earlier analogy will help drive the point home. Imagine that our goal is toexplain why Tarski‟s account produces a plausible extension when we hold fixed „is a

senator,‟ „is a man,‟ and „is a bache¬lor,‟ but not when we also hold fixed „is president.‟

It would clearly be misguided to look for our explanation in some characteristic ofex¬pressions that distinguishes the former terms from the latter. Cer¬tainly, it would beeasy to find a variety of properties that distinguish these expressions; at worst, we couldappeal to a list. But we will not find any property guaranteeing our success when wehold fixed the first expressions while explaining our failure when we hold fixed the

second. For what accounts for that difference is not a property of expressions at all, but





Now, in some cases we may be fortunate: if the relevant portions of the world aresufficiently varied, if none of the substantive generaliza-tions come out true, then theaccount will not issue any faulty declara-tions of logical truth. For the only sentences

that will be judged logi-cally true will then be instances of generalizations that arelogically true, and these instances will, by the closure principle (ii), be logically true aswell. In these cases, but only in these cases, Tarski‟s definition will yield a reasonable

assortment of logical truths. But we succeed here not because principle (Hi) is correct,however modified, or because we have chosen the right “logical constants.” Our success

is due to principle (ii) and simple good fortune.

Extensional Adequacy

When we apply Tarski‟s account to an arbitrary language, there is no way to guarantee

that it will be extensionally correct. No matter how tightly we constrain the selection offt, the output of the definition will still depend on the truth values of varioussubstantive, nonlogical claims. If any of these turn out to be true, then their instanceswill mistakenly be declared logically true. And that, of course, is beyond our control:we can decide which expressions to include in ft, but we can hardly just “decide” that

the resulting, substantive generalizations must all be false. That decision is not up to us, but up to the world, to the historical or physical or mathematical matters described bythe generalizations.

We cannot guarantee, antecedently, so to speak, that a given appli-cation of Tarski‟s

definition will not overgenerate, that it will not declare sentences logically true andarguments logically valid that in fact are not. To be sure, in the paradigmatic, first-ordercase, I have already alleged that the account does not, as a matter of fact, over-generate.But it is important to see that this is by no means obvious, and certainly does not followfrom anything we have said so far. In Chapter 8, we surveyed a few of the morestraightforward generaliza-tions on which the extension of this application depends.Those partic-ular claims happened to be false, given the set-theoretic assumptions westandardly make. But of course the extension of the account de-pends on infinitelymany such substantive generalizations, most far more complex than those we bothered

to look at. Nothing we have seen so far precludes one of these coming out true, perhapsthanks to some complex but universal characteristic of sets, or due to a subtle andsurprising algebraic fact akin to Wedderburn‟s theorem. And if any of these

generalizations happen to be true, then even this paradig-matic application of the model-theoretic account will have the wrong extension. Clearly, if we think its extensionaladequacy is somehow obvious, even assuming the usual axioms of set theory, we aresimply fooling ourselves.

The claim that the standard semantics for first-order languages does not overgeneraterequires an external justification. It does not follow from any characteristic of Tarski‟s

definition, or of the expressions held fixed, or of the language itself. So far, the onlyevidence for this claim that we can point to is, to use Hilbert‟s term, experimental: we



have not yet run into any examples of sentences declared logically true or argumentsdeclared logically valid that in fact are not. Better evidence would be a proof showingthat, in this particular case, the only true closures are in fact logically true. How we canget such a proof is a topic we will take up later. At the moment, though, our concern ismore general than the first-order case.

What can we say, in a general vein, about the extension of Tarski‟s account? First of all,

we can say exactly why an application of the account overgenerates, when it does. As Ihave pointed out, the ac-count can make this mistake only when it associates asubstantive, nonlogical generalization with some sentence, and when that general-ization turns out to be true. Clearly, it will not make the faulty as-sessment if thegeneralization is false, for then the sentence will not be judged logically true. And ofcourse if the associated generalization does not make an extralogical claim, if it is itselfa logical truth, then its instance will be as well. But the fact that a generalization

contains only expressions of a traditionally “logical” sort in no way precludes itsmaking a substantive, nonlogical claim: this is, demonstrably, a myth, one that failsalready with the truth functional connectives.

The second thing we can say is that certain applications of the account are guaranteed toovergenerate, and so guaranteed to have the wrong extension. With certain languages,and certain selections of $, we will find among the associated generalizations both asubstantive claim and its negation (or some other claim equivalent to its negation). Thisis what happens, in the first-order case, when we hold fixed the interpretation of boththe identity predicate and the quantifiers. For then we find the following sentences

among the relevant closures:

[3x3y(x 4 y)]

[ — i3x3;y(x 4 >)]

Since both of these make substantive claims, the account will over-generate if eithercomes out true. But since one or the other of them must be true, the account is sure tomake a faulty assessment. When we vary the interpretation of 3, we replace theseclosures with the follow¬ing two:

VE[ExEy(x 4 y)]

VE[ÊxEy(x 4 ?)]

These still make substantive claims, but since they are not straightfor¬ward negations ofeach other, the account is no longer sure to fail. If both turn out to be false, as I trustthey do, then neither of their instances will be wrongly accused of logical truth.

As we move to increasingly powerful languages, this problem be-comes harder toavoid. For example, the situation crops up with full, second-order languages whether or

not we vary the domain of quanti-fication. Thus, it turns out that here we find amongthe associated generalizations not only sentences equivalent to the continuum hy-



pothesis, but also sentences equivalent to its negation.1 Consequently, whichever waythe hypothesis goes, this application of Tarski‟s defini¬tion will overgenerate, declaring

some sentences logically true because of their true, but not logically true, closures. Thisis hardly surprising, since as we increase the expressive capacity of the members of ft,we increase the likelihood that some substantive claim and its negation (or a claimequivalent to its negation) will be among the generalizations determining the extensionof the account.

Both of these remarks have to do with the problem of overgene-ration. What can we sayabout the complementary problem of un-dergeneration? Is it possible f or Tarski‟s

definition to judge valid arguments invalid, or to judge sentences not logically truewhen in fact they are? The answer, of course, is yes. But this can happen only when thelogical truth of the sentences, or the validity of the arguments, depends essentially onthe meanings of one or more expressions not included in ft. To repeat a trivial example,

if the interpretation of„or‟ is not held fixed, then the logical truth Lincoln was president or Lincoln was not president

will not be so declared.

Though this example is trivial, the general problem can hardly be shrugged off. If ourgoal is to study the logical properties of a given language, the only way to ensure thatTarski‟s definition will not undergenerate is to include every expression in ft. But assoon as we do this we are sure to encounter the opposite problem, that of overgeu-eration. Suppose, for example, that our target language is (or includes) the language of

elementary arithmetic. When we apply the standard, interpretational semantics to thislanguage, our specification of the logical properties falls short of their genuineextension — as Tarski himself, indirectly, pointed out. For example, instances of the o>-rule (even perfectly effective versions of it) are wrongly judged invalid. Of course, it iseasy to see why these instances are not declared valid — their validity depends on themeanings of various expressions not held fixed — but this does not give us a solution toour problem, namely, specifying the genuine extent of the consequence relation for thegiven language. If this is our goal, Tarski‟s account does not, in gen¬eral, allow us to

steer a course between the complementary hazards of over- and undergeneration.

This is not to say that an application of the model-theoretic account can never get theextension exactly right. This can happen if, first of all, no valid arguments expressible inthe language depend for their validity on any expressions not included in $, and if, inaddition, all the associated generalizations that determine the extension of the accountare either false or, if true, logically true. These two circumstances will sometimes cometogether. They come together, for instance, when we apply the account to the simplelanguage of Chapter 3, holding fixed only the truth functional connectives. It happensthat no valid argu¬ments expressible in this language depend on the specific meaningsof „Lincoln,‟ „Washington,‟ „was president,‟ and „had a beard.‟ This might not have

been the case had we included other predicates as well — perhaps „was an electedofficial‟ or „sported unshorn facial hair‟— or had we included expressions of other



semantic categories, such as adjectives and adverbs. But we did not. It also happens thathere the only true generalizations are indeed logically true. Of course, many of therelevant closures make substantive claims, for example (9), but these turn out to befalse.

(9) VxVyVP[x P y P]

These two circumstances do not come together, however, either in the language ofelementary arithmetic (where we run into the first prob-lem) or in languages withhigher-order quantifiers (where we run into the second).

What is important to recognize is that there is no reason to expect, for an arbitrarylanguage, that there will be any single selection of $ that gets the extension exactlyright, that neither overgenerates nor undergenerates. The reason for this should, by now,

be clear. To avoid undergeneration we cannot exclude from $ any expression whose

meaning plays an essential role in any valid arguments expressible in the language.(This will depend\not only on the expression itself, but on the entire vocabulary of thelanguage.) But this general characteris¬tic of expression ̂— figuring essentially intovalid arguments — has nothing to do with the question of whether substantive claims can

be made employing just those expressions plus an initial string of quanti¬fiers. And ithas even less to do with whether those substantive claims turn out to be true — in otherwords, with whether the account will, on that selection of ft, overgenerate.

Clearly there can, and often will, be an irreconcilable tension be-tween our two goalshere, the one inclining us toward a more inclusive ft, the other pushing in the opposite

direction. Indeed, it is a trivial exercise to construct simple languages in which noselection of “logical constants” can possibly yield the right extension. For example,

sup¬pose we supplement the language of Chapter 3 with a binary connec¬tive, O, withthe following semantics:

p O q is true iff Lincoln had a beard and either p is true or q is

true.

When we do this, we end up with a language in which every sentence is logically

equivalent to some sentence of our original language. Yet it is easy to show (and notsurprising in the least) that no selection of f t yields a set of “logical truths” containing

exactly the right sentences — that is, those equivalent to logical truths of thé originallanguage. If we include O in ft, the account overgenerates; if we exclude it, the accountundergenerates.

Of course, there is no reason to resort to such fabricated examples to make this point.Indeed, it is precisely this tension that underlies the need for cross-term restrictions inthe case of languages with quantifi¬ers. As we saw in Chapter 5, there is no way toinclude in ft all of the expressions that figure essentially into the logically valid

arguments expressible in standard first-order languages, without also declaring manysentences logically true (or many arguments logically valid) that in fact are not. To



combat this we are forced to exclude certain expres¬sions from ft even though they dofigure essentially into valid argu¬ments, but then we must so restrict theirinterpretations that these valid arguments are still so declared. The same tension arises,

but is not so easily solved, in the case of higher-order logics, or in any first-orderlanguage with a rich stock of logically related predicate or function expressions.

Clearly, when we do not avail ourselves of cross-term restrictions, Tarski‟s definition

fails more often than not: only with the very sim-plest languages will there be aselection of ft that neither overgenerates nor undergenerates. Not surprisingly, when weallow arbitrary cross- term restrictions, our chances of getting an extensionally adequatespecification of the logical properties improves considerably. But re¬gardless ofwhether such restrictions are imposed, and regardless of

how the restrictions are chosen, extensional adequacy is by no means automatic. For theassessments made by the account will still depend on the outcome of a wide range ofsubstantive, extralogical claims. Until these are all shown to be false, there is no way to

be sure that the account‟s positive assessments are not in error. The question of whether

the account‟s negative assessments are in error, of whether it undergenerates, can be

even less straightforward. Roughly speaking, this will depend on whether all of thelogically relevant relations among expressions are captured, either because theexpressions are included in $, and so their interpretations never varied, or because ofcross-term restrictions that constrain the permissible variations. I re¬turn to thisquestion in Chapter 11.

10Logic from the Metatheory

Various characteristics distinguish logical truths from common, run- of-the-mill truths,and logically valid arguments from those that hap-pen to have a false premise or a trueconclusion. But Tar ski‟s analysis does not capture any of these characteristics,

regardless of how tightly we constrain the selection of ft. Furthermore, we are not evenguaran¬teed that the definition will be extensionally correct when applied to a givenlanguage, not even in the paradigmatic, first-order case. What, then, makes Tarski‟s

account seem so persuasive? Why has it received such widespread acceptance? No doubt to some extent, this acceptance is due to the conflation already noted betweenrepresentational and inter pretational seman¬tics. And perhaps it is partly due to Tarski‟s

fallacy, in its various versions. But there is a more subtle reason the account seems so persuasive, one that I suspect has been by far the most influential. The reason is this. Inits standard application to simple first-order sen¬tences, Tarski‟s account is capable of

entirely persuading us both that a sentence which passes the test is indeed logically true,and that one which does not pass the test is not logically true. In other words, in this

particular case the account seems capable of convincing us of the genuine logical status

of individual sentences to which it is applied. Faced with this fact, it is hard not toassume that, one way or another, the account must surely be getting at some essential



feature of our ordinary notion of logical truth. How else would we be convinced of thecorrectness of its individual assessments?

To understand what is going on here, we need to review two impor¬tant points. The

first is that Tarski‟s account does provide a necessary (but not sufficient) condition forthe relativized notion of logical truth. That is, if a sentence S is true solely by virtue ofthe meanings of the members of $ (or expresses a necessary truth thanks to the membersof $, and so on), then the universal generalization the account associ¬ates with S mustindeed be true. And this holds no matter what expres¬sions are kept fixed. Thus, thesimple truth of the closure

[Abe Lincoln was president]

is indeed a necessary, but not sufficient, condition for the logical (necessary, a priori,

analytic) truth of

Abe Lincoln was president.

This is just to say that this sentence cannot be logically true unless it is true. Lesstrivially, the simple truth of the closure

Vx[If x is a bachelor then x is a man]

is again a necessary, but not sufficient, condition for the logical truth (with respect to„if.. . then,‟ „is a bachelor,‟ „is a man‟) of the sentence

If Leslie is a bachelor then Leslie is a man.

Here, if there is a counterexample to our generalization, then the latter sentence either isnot logically true or, if it is, this fact must somehow depend on the specific meaning ofthe name „Leslie.‟ To see that it is not also a sufficient condition, we need only consider

such generalizations as

Vx[If x is a man then x is a bachelor].

This generalization would be true if all men were bachelors. But of course its instances

would not, even then, be logically true. Finally, the simple truth of the closure

VxVP[x P or not x P]

is a necessary, but not sufficient, condition for the logical truth (with respect to „or‟ and

„not‟) of the sentence

Leslie was president or Leslie was not president.

Again, a counterexample to the above generalization would show that our sentence isnot true simply by virtue of the meanings of „or‟ and „not.‟ But again, the condition is

not sufficient, as witness such general-izations as



VxV^VPtx P or not y P].

This generalization would be l;rue if the universe contained only one (type of) object.But of coursers instances would not, even then, be logically true. ^

The significance of this first point should be apparent. When we show that a sentencefails Tarski‟s test (for any selection of ft), then we have genuinely shown that the

sentence is not logically true with respect to the expressions held fixed. Thus, in thestandard semantics, when we produce an interpretation of the names and predicates ofour first-order language that falsifies a given sentence, we can rest assured that thesentence is not true solely by virtue of the meanings of the traditional “logical”

expressions, those we kept fixed in the process. The problem with Tarski‟s account is

that the mere absence of such an interpretation, or, alternatively, the mere truth of theassociated gen¬eralization, cannot guarantee that our sentence is logically true.Simi¬larly, if we can find an interpretation in which the premises of a given argumentare true but the conclusion false, then we have genuinely shown that the conclusiondoes not follow solely by virtue of the expressions held fixed. But the absence of suchan interpretation does not guarantee that it does so follow.

Given this, it is perfectly understandable how the account persuades us that a sentence isnot logically true or that an argument is not logically valid, at least with respect to theexpressions held fixed. Indeed, here we are simply relying on a traditional techniqueem¬ployed in independence proofs, familiar from the axiomatic method. What needsexplanation, then, is how Tarski‟s account can possibly persuade us that a sentence is

logically true. And it frequently seems to do just that.Here is where the second point comes in. As we have seen, it some¬times happens, witha given language and a given selection of fixed terms, that the only generalizationswhich come out true are those that are, themselves, logically true. In such situations theaccount does not mistakenly dub any sentence logically true. But this is simply becausethe variety of the world compensates for our faulty analysis by falsify-ing the othergeneralizations. It is not because what the analysis actu¬ally requires, the mere truth ofthe associated generalizations, in any way guarantees the logical truth of their instances.

This is all well and good. But when we apply Tarski‟s account to an individual, first-order sentence, say,

(15) Lincoln was president or Lincoln was not president

it strikes us that, somehow or other, the logical truth of this sentence has been quiteclearly and unequivocally demonstrated. The question is why we would have thisimpression if Tarski‟s account cannot pro¬vide such a guarantee. Where would the

assurance come from? There is a perfectly straightforward answer, and it has nothing todo with the adequacy of Tarski‟s account. Rather, it has to do with the adequac y, albeittrivial adequacy, of any account that replaces the reduction prin¬ciple (Hi) with theclosure principle (ii):



(ii) If a universally quantified sentence is logically true, then all of its instances arelogically true as well.

As we will see, principle (ii) is where we derive the sensed guarantee. Carnap'sObservation

Let us start with a simple observation. It seems clear that if the truth of a sentencefollows logically from the recursive definition of truth for the language in which itoccurs, then that sentence must be logically true. For if the sentence expressed ahistorical or physical or set- theoretic claim, we would need some historical or physicalor set- theoretic premises in addition to the bare characterization of truth in order toestablish that the claim is in fact true. But if its truth simply follows from the semantic

properties of such expressions as „or,‟ „not,‟ and „all,‟ the properties we characterize

when we specify the truth conditions of our sentences, then the sentence must surely bea logical truth.

This is a common observation, one appealed to repeatedly, for example, in various ofCarnap‟s later works. Indeed, we might call it Carnap‟s observation, just to give it a

name.

Carnap's observation: If the truth of a sentence is a logical consequence of the definitionof truth for the language in which it occurs, then that sentence is logically true.

Of course, since Carnap‟s observation presupposes the notion of logi¬cal consequence,

it will never yield a general account of the logical properties of the sort Tarski hoped to

achieve. But the observation seems right nonetheless — perhaps not tremendouslysignificant, but in accord with our ordinary understanding of logical truth.

Now consider how Carnap‟s observation, along with principle (ii), can provide theassurance that principle (Hi) cannot. Clearly, if it is possible to show, without appeal toany “extralogical” premises, that the truth of a sentence of the form

Vi/, ... Vvn[ S' ]

is a purely logical consequence of the semantic clauses in our definition of truth, thenwe can rest assured that this generalization is logically true. This simply by virtue ofCarnap‟s observation, not due to any significant account of logical trutK And naturally,

by principle (ii) we are then assured that its instances are logically true as well. Thisintu¬

itive assurance does not arise from any general account of logical truth, certainly notTarski‟s, but just from our plausible observation plus the unexceptional principle (ii).

When we apply Tarski‟s account to a sentence like (15), what con-vinces us that thissentence really is a logical truth? The key lies in the way we show that all sequencessatisfy the sentential function „xP or not x P.‟ Our reasoning here takes the following

line. First we note that



For any f either/satisfies „x P‟ or/does not satisfy „x P.‟

This is just an elementary logical truth of the metatheory. But it follows from thislogical truth, in tandem with our clause for „not‟ in the definition of satisfaction, that

For any f either/satisfies „x P‟ or/satisfies „not x P.‟ Finally, given our recursive clause for „or,‟ we have as an immediate consequence that

For any/,/satisfies „x P or not x P.‟

This, of course, is what had to be shown in order for Tarski‟s account to issue adeclaration of logical truth (with respect to „or‟ and „not‟) for sentence (15). That is, it is

precisely the observation we need in order to demonstrate the truth of the associatedclosure

(16) VxVP[x P or not x P]

The important point is that in the process of showing that all se-quences satisfy „x P or

not x P‟ we did not have to appeal to any intuitively “empirical” facts (say, Lincoln‟s

presidency or the size of the universe), or even to any set-theoretic claims (say, the pair-set axiom or the continuum hypothesis). Indeed, we do not even have to know what asequence is in order to carry out our demonstration: all that is required are the semanticclauses and the logic of the metatheory. Of course, this would not always be the case,and certainly is not a require¬ment of Tarski‟s analysis. Thus, if the universe had fewer

than n objects, all sequences would also satisfy the sentential function

— \<jn(3!E)

but to show this we would have to go out and count the existing objects; logic, plus ourdefinition of satisfaction, would no longer suffice. Simi-lar appeals to “extralogical”

assumptions would be required in all the other examples we have discussed.

Now, the fact that the above demonstration requires no appeal external to the semanticsof the language and the logic of the meta-theory provides us with a genuine assurance,quite independent of

Tarski‟s account, of the logical truth of the associated universal closure

(16) . This is just Carnap‟s observation. And this independent assur -ance, linked with principle (ii), is where we find our guarantee of the logical truth of (15). The sensedguarantee could not flow from the faulty principle (in), and conversely should not bethought to lend it, or Tarski‟s analysis, any plausibility. For we are not assured that thissentence is logically true because the associated generalization is true — that is, by virtueof the sentence‟s satisfaction of Tarski‟s defini¬tion. This is clear from the fact that our

assurance would instantly evaporate if, in establishing the truth of (16), there were an

essential appeal to some extralogical fact — say, the fact that all past presidents have been men, or the fact that the universe has more than twenty- seven objects. Rather, we



are guaranteed that the original sentence is logically true thanks to our assurance that thecorresponding closure is itself logically true.

In the first-order case, as in any application of Tarski‟s account, the output of the

definition depends on a large number of substantive, nonlogical facts. But this particular

application is one of those “fortu-nate” cases: the only true generalizations turn out to bethose that are, themselves, logically true. This does not mean that we do not appeal tosubstantive facts when applying the definition to first-order sentences and arguments.What it does mean, though, is that the appeals do not arise while showing that asentence is logically true or that an argument is logically valid. These demonstrationsrequire only the logic of the metatheory and the semantics of our language, and this iswhy we find them persuasive.

Where the appeals come into play is in demonstrating the con¬verse — for instancewhen we show, say, that the sentence

— 1y: — iV;y3x(x is taller than 3;)

is not a consequence of the sentences

a: VxV;yVz(x is taller than y a y is taller than z

—► x is taller than z)

/3: Vx — i(x is taller than x).

Here we rely on (among other things) the axiom of infinity in con-structing aninterpretation that satisfies the latter without satisfying the former. But in suchdemonstrations our reliance on extralogical facts does not diminish our persuasion, anymore than it does when we point to a woman senator as proof^that (11) is not a logicaltruth.

(11) Leslie is a senator — > Leslie is a man.

In these cases, the existence of the counterexample, however obtained, is sufficient forthe purpose at hand. But once again, the mere absence of such counterexamples does

not suffice to show the converse.The most persuasive feature of Tarski‟s account, or, equivalently, of interpretationalsemantics, is its capacity to convince us of the logical status of simple, first-ordersentences to which it is applied. And half of that capacity is genuine: since the accountemploys a necessary condi-tion for the relativized notion of logical truth, the fact that asentence fails Tarski‟s test really does show that it is not logically true— relative, at anyrate, to the particular expressions held fixed. But the other half is just illusion. For whatconvinces us that a sentence is logically true is not the fact that its associated closurehappens to be true, or that there happen to be no interpretations in which the sentence

comes out false. What convinces us is the fact that the absence of such interpretationscan be shown purely on the basis of the semantic rules of the first-order language, those



embodied in our definitions of satisfaction and truth. This is what assures us that thesentence in question is logically true. But the credit for this assurance does not go toTarski‟s account of logical truth. It belongs instead to principle (ii) and the logic of themetatheory.

Strengthening the Account

Tarski‟s definition of logical truth is based on a faulty principle, the principle that the

instances of a true universal generalization are not simply true, but logically true.Unmodified, this principle is obviously wrong: any true sentence is, if only trivially, aninstance of a true universal generalization, namely itself, and so the requirement builtinto the principle cannot possibly distinguish logical truths from ordi-nary, run-of-the-mill truths. But what is important to see is that the principle does not gain any

plausibility when the generalization is less trivial, or if we require that the generalizationcontain only the tradi-tional “logical” expressions. We still have no reason to expect itsinstances to be logically true.

The definition does occasionally get things right, though. Specifi-cally, it gets thingsright in precisely those cases where all the substan-tive generalizations associated withsentences happen to be false. Then, the only positive assessments made by Tarski‟s

account will be directed toward instances of logically true generalizations, and theseassessments will of course be correct. This is not to say that it will be obvious when wehave such a fortuitous application: as I have empha-sized, we have not yet seen any firmevidence that the account does not overgenerate even in the paradigmatic, first-order

case. Still, in these cases our positive assessments can generate conviction; we may feelassured that individual sentences that pass Tarski‟s test really are logi-cally true. Andthis is what misleads us. For the felt conviction issues not from the fact that thesesentences pass the test, but from the way they pass it: in these cases, the truth of therelevant generalization follows logically from the semantic rules of the language,without appeal to intuitively extralogical facts. From this we can conclude that thegener-alization, and so its instances, really are logically true.

Let me conclude by noting two lessons to be learned from the fact that Tarski‟s account

succeeds and fails exactly where it does: a lesson about what would be required to

correct the account, and a related lesson about the futility of trying. The only reliableway to avoid the problem of overgeneration would be to incorporate some generalguarantee that the truth of substantive, extralogical claims — say, those of a historical,

physical, or mathematical sort — could not influence its output. If we could do this, theaccount would no longer be based on the faulty principle (11V), but instead on some

principle along these lines:

If a universal generalization is true, but does not make a substan¬tive claim, then all ofits instances are logically true.

This new principle seems basically right. Indeed, it seems right be¬cause it is nothingmore than a vague restatement of principle (ii):



(ii) If a universal generalization is logically true, then all of its instances arelogically true as well.

This, of course, brings us to the second lesson. The only hope for coming up with animproved version of Tarski‟s account, a version guaranteed to produce correct results, is

in effect to replace principle

(iii) with principle (ii). But once we recognize this, the futility of the project becomesapparent. For in order to use principle (ii), we first need an account of what it means forthe generalization mentioned in the antecedent to be logically true, for its truth not to bea historical or physical or mathematical matter. But if we already had such a charac-terization of logical truth, the remainder of our new, “improved” account— that is, the

part left over from Tarski‟s original definition— would be completely unnecessary. Socorrecting the “de¬fect” in the account turns out to be precisely equivalent to solvingthe original problem de novo. An adequate analysis of logical truth will not be found bymodifying Tarski‟s reduction principle.

11

Completeness and Soundness

The reduction principle is the linchpin of the model-theoretic account of consequence. Ifsome version of it were correct, the account would certainly deserve the esteem inwhich it is held. But we have seen that this principle is irremediably flawed, and that, asa result, we have no guarantee that an application of the account to any particular

lan¬guage will be extensionally correct. The definition can both over¬generate andundergenerate, declaring arguments logically valid that in fact are not, and declaringthem not when they actually are.

In the case of first-order languages, the completeness and soundness theorems have been taken as providing a proof that a particular deductive calculus correctlycharacterizes the logical conse¬quence relation for these languages. The soundnesstheorem is tradi¬tionally viewed as showing that the calculus does not overgenerate,that whenever a sentence 5 is derivable from a set K of assumptions, 5 does indeedfollow logically from K. Conversely, the completeness theorem is thought to show that

the calculus does not undergenerate, that if 5 follows logically fromX, then there is a proof of this fact within the calculus. The goal of this chapter is to reconcile the moralsof the preceding chapters with the intuitions at work here.

It is clear that the traditional construal of completeness and soundness can no longer bemaintained. If the model-theoretic analy¬sis can overgenerate, a soundness theorem byitself does not guarantee the soundness of the deductive calculus in question. Just so, ifthe model theory can undergenerate, a completeness theorem does not, by itself,guarantee the completeness of the deductive system. The usual interpretation of thesetheorems clearly presupposes the exten¬sional adequacy of the model-theoretic accountof consequence, a presumption that is simply unjustified.



We have seen that the model-theoretic account will get things right in one set ofcircumstances. If, first of all, none of the substantive generalizations on which theoutput of the account depends turns out to be true, then the definition will notovergenerate. Second, if none of the valid arguments expressible in the language dependfor their validity on expressions whose interpretations we vary, then the defini¬tion willnot undergenerate, either. However, a second‟s thought shows that the relationship

between these principles and the soundness and completeness theorems is far fromstraightforward.

What work are the soundness and completeness theorems doing? Do they in factguarantee anything at all about the intuitive notions of logical truth and logicalconsequence? To answer these questions I will make a slight detour. I am not the first

person to raise questions about the significance of the completeness theorem for first-order logic. In a well-known article entitled “Informal Rigour and Completeness

Proofs,” Kreisel distinguishes between what he calls “intuitive” validity and the model-theoretic notion of truth in all set-theoretic structures. This distinction leads Kreisel toan alternative view of the significance of the completeness theorem. Although Kreisel‟s

starting point is in-correct, for reasons that will become clear, his strategy is one we willfind useful in our own reconciliation.

Kreisel‟s Observation

The main thrust of Kreisel‟s article is to emphasize that we can prove rigorous resultsabout informal notions, a contention with which I wholeheartedly agree. As a case

study, he considers the intuitive no¬tions of logical validity (what I have been calling“logical truth”) and logical consequence. Kreisel‟s aim is to show that, in the case offirst- order logic, we can rigorously establish that the intuitive notion of validity, whichhe abbreviates as Val, is extensionally equivalent to the set-theoretic definitionstandardly given.

The definition that Kreisel has in mind (which he denotes by V) is that a sentence has property V just in case it is true in all models (or structures, as Kreisel prefers to callthem), where the domain of quantifi-cation is a set in the cumulative hierarchy.KreisePs worry is that this does not correspond exactly to the notion Val. As he

expresses the problem:

The intuitive meaning of Val differs from that of Vin one particular: V(a) (merely)asserts that a is true in all structures in the cumulative hierar¬chy, . . . while Val(a)asserts that a is true in all structures. (1969, p. 90)

To drive home the difference between these two notions, Kreisel considers a sentence ain the language of set theory. Intuitively, it seems that if Val(a) then a must be true as astatement about the cumulative hierarchy — that is, where the domain of quantification isthe collection of all sets. But V(a) assures us only that it must be true in all set-theoretic

structures. But the cumulative hierarchy itself is “too big” to be among these structures.



Because of this, Kreisel rightly points out, it is not at all clear that sentences true in allset-theoretic structures will be true in all structures.

Kreisel‟s main point is that, insofar as V takes into consideration fewer structures than

Val, it cannot be a trivial matter to go from V(a) to Val(a). His use of a set-theoretic

example introduces an additional point, though. For in this case, it turns out that theintended interpreta-tion of the language is among the structures canvassed by Val, butnot among those canvassed by V. Of course, as I have emphasized several times, one ofTarski‟s key requirements is that the intended interpreta-tion of an expression be in thesatisfaction domain associated with that expression, otherwise we risk declaringsentences logically true that in fact are false. This requirement is violated when ourlanguage is the language of set theory and our model-theory uses only structures fromwithin the cumulative hierarchy.

Kreisel claims that in spite of these seeming difficulties, the com-pleteness theoremallows us to establish that V has the same extension as the intuitive notion of logicalvalidity. His reasoning is as follows. He first notes that, since the standard deductiverules of first-order logic are intuitively sound, we know that if a first-order sentence a isde-rivable, D(a), then it is logically valid.1 That is,

(1) Va(D(a) Val(a)).

Second, since it is obvious that all set-theoretic structures are struc¬tures, truth in allstructures (Val) implies truth in all set-theoretic structures (V). That is,

(2) Va(Va/(a) V(a)).

However, the completeness theorem for first-order logic tells us that any sentence truein all set-theoretic structures is derivable. That is,

(3) Va(V(a) -* D(a)).

Putting these three together, we see that Val, V, and D are, for first- order languages,extensionally equivalent:

Va(Val(a) «-> V(a) «-> D(a)).

For our purposes, there is a serious flaw in Kreisel‟s argument. The problem has to do

with the interpretation of Val. If Val simply means truth in all structures, then theargument is correct, though its moral is not exactly what Kreisel implies. But if Valreally is the intuitive notion of logical validity (or logical truth), then step (2) is quitedubious. The problem is that Kreisel simply identifies, without argument, the intu-itivenotion with the model-theoretic notion of truth in all structures. Needless to say, this is

precisely the identification against which I have been arguing.

Let us reexamine the two steps of Kreisel‟s argument that involve Val. But to avoid the

above conflation, I will reserve Val for the notion of truth in all structures, and



introduce LTr for the intuitive notion of logical truth or validity. With thisdisambiguation, step (1) splits into two possible claims, namely:

(1) Va(D(a) Val(a))

(1') Va(D(a)-+LTr(a)).It turns out that both of these are legitimate, though they require slightly different

justifications. (1') holds simply because the deductive system is intuitively sound — thatis, it allows us to derive only logically true sentences. To recognize the truth of (1),however, we need to observe that the validity of the rules of our deductive system holdsin all of the interpretations canvassed by Val. In the case of a standard first-order systemof deduction, both of these follow by a routine examination of the rules on a case-by-case basis. But notice that (1) is more sensitive than (1') to the details of the deductivesystem in ques¬tion. For example, if our deductive system included the o>-rule, or a

rule allowing us to conclude „Man(x)‟ from „Bachelor(x),‟ then (1) would certainly fail,even though (1') might not.

How about step (2)? Here we have the following split:

(2) Va(Val(a) V(a))

(2') Va(LTr(a)-+V(a)).

Clearly, (2) follows trivially from the fact that every model in the cumulative hierarchyis a model, the same reason we gave before. But (2') is quite another matter: it is simply

the bald assertion that logical truths are true in every model in the cumulative hierarchy.But in fact we do not know that the logical truths of any given first-order language will

be a subset of either V or Val. To suppose that they are is just to suppose that themodel-theoretic account (whether V or Val) does not undergenerate. If there is anargument for this contention, it must be something quite specific to the first-orderlanguage in question, since we have seen that it does not hold in general.

Kreisel‟s argument goes through for the notion Val. What it shows is that, in the first-order case, truth in all structures is equivalent to truth in a restricted collection ofstructures. To drive this point home, let us generalize the argument a bit. Let M, be anyclass of models and write Valjn for truth in all structures in Ü. Thus V is the specialcase of Valji where M, consists of all structures whose domain is a set in the cumu-lative hierarchy.

What do we need in order for the argument given above to general-ize? We need toknow that the class M is rich enough to serve as a basis for the proof of thecompleteness theorem. More fully, let us call a class rich if it satisfies the condition thatany first-order sentence 5 which is true in all models in M is derivable. For example, thecollection of countable structures is rich. We can clearly replace V by Valji throughout

Kreisel‟s proof, and the argument goes through so long as Jii is rich, since then we haveVal(a) — > VaU(a) — > D(a) —> Val(a). Thus, one moral of Kreisel‟s argument is that



when we apply the model- theoretic account to first-order languages, it does not matterwhether we use all structures, all set-theoretic structures, or all countable struc-tures.These will all have precisely the same extension.

What Kreisel‟s argument does not show, however, is that this exten-sion coincides with

the set of logical truths of any given first-order language — say, the logical truths of thelanguage of elementary arith-metic. For, in spite of the argument, it is far from clear thatVaU (or Val itself) does not undergenerate. To see this, we need only recall that richclasses are in general forced to contain unintended models for our first-order language.Thus, for example, if our language is the lan¬guage of first-order arithmetic, then a richclass will perforce contain nonstandard models of arithmetic. But what guarantee do wehave that an intuitive logical truth in the language of arithmetic will be true in thesenonstandard models, or that an intuitively valid argument will preserve truth in them?For example, if some version of the o>-rule is logically valid, as Tarski argued, then

there will indeed be logical truths which are not true in all models in M, let alone true inall models.

The Problem of Overgeneration

Recall that the goal of this chapter is to determine the significance of completeness andsoundness theorems for the intuitive notions of logical truth and logical consequence. In

particular, what bearing do they have on the question of whether a given application ofthe model- theoretic account either overgenerates or undergenerates?

If Kreisel‟s argument were correct when construed as an argument about LTr, then it

would settle both of these questions; as it is, though, it does not directly address either.All it shows us is that the three notions Val, V, and D, are coextensive. This is asignificant observation, to be sure, but not the one we are after. It tells us nothing abouthow the intuitive notions of logical truth and logical consequence relate to their model-theoretic (or proof-theoretic) counterparts.

Still, it does suggest a partial solution. Indeed, as the reader may already have noticed,Kreisel‟s argument can be combined with (1') to settle the overgeneration question, at

least in the first-order case. Recall that (1') is the observation that the deductive system

used in the proof of completeness is intuitively sound, that only genuine logical truthsare derivable in the system.

(1') Va(D(a) LTr(a)).

This observation holds for any first-order language, whether the lan-guage ofelementary arithmetic, the language of set theory, or the simple language of Chapter 5.But Kreisel has shown, using complete¬ness, that any first-order sentence that is true inall models is derivable:

Va(Va/(a) D(a)).





Of course, we know the strategy cannot always succeed, because the model-theoreticaccount does sometimes overgenerate. For example, I argued in Chapter 9 that there aresecond-order sentences which are not logical truths but which are declared such by themodel-theoretic account. If so, then it follows that there is no sound deductive system(effective or not!) that is complete with respect to the standard, second- order modeltheory. This is partially substantiated by the well-known result that no such effectivesystem exists, a consequence of Godel‟s incompleteness results.

In Chapter 9, we saw that there is no “internal” guarantee that an application of the

model-theoretic account will not overgenerate. Even when it does not, there is no way torecognize this fact from the analysis itself, from characteristics of the language, or fromthe expressions held fixed. What we can now see, though, is that an external guaranteecan sometimes be found, a guarantee derived from the presumed soundness of ourdeductive calculus, in tandem with a completeness theorem showing that the semantic

account reaches no further than the syntactic.The Problem of Under generation

The reason the completeness theorem is so called is that it purports to establish that agiven deductive calculus does not undergenerate, that it is “complete.” We have used

the theorem, in contrast, to show that an application of the model-theoretic account ofconsequence does not overgenerate, in effect to show that our semantic account is, inthe first- order case, sound. Is there any way to prove the converse, to show that ourfirst-order model theory (or an extensionally equivalent deductive system) does not

undergenerate?The most straightforward answer, unfortunately, is no. If our aim is to characterize theset of logical truths (or the logical consequence relation) for an antecedently given first-order language, then there is no general way, short of fixing all of the expressions in thelanguage, to guarantee that the model theory captures them all. Indeed, once we focuson any interesting first-order language, such as the language of elementary arithmetic, itseems clear that the standard model theory does undergenerate. It is only our uncriticaladoption of the model- theoretic analysis that has obscured this simple point.

Still, it is possible to extract from the model-theoretic account some substantiveobservations about the intuitive notions of logical truth and logical consequence. Thetrick is to shift attention from the logical properties of any particular language to thelogical properties com¬mon to a range of languages.

In Chapter 7, we noted that Tarski‟s unmodified definition of logical truth (that is, prior

to the use of cross-term restrictions) provides a necessary condition for what we therecalled the relativized concept of logical truth. That is, if a sentence 5 is logically true,and if this fact depends only on the meanings of some subset ft of its constituentexpressions, then 5 will remain true however we reinterpret the other expressions (so

long as our reinterpretations do not change the seman¬tic categories of thoseexpressions). We took this as showing that Tarski‟s original definition would never



undergenerate with respect to the notion of logical truth relativized to the set ft of fixedterms.

We can reconstrue this as an observation about the logical truths common to acollection of languages, those languages canvassed by the model theory. In the case of

Tarski‟s original account, these are the languages that arise when our models provide(semantically well- behaved) interpretations of the expressions not in ft. Construed thisway, the observation is that the set of sentences that are logically true in every suchlanguage will be a subset of the set of sentences declared logically true by the modeltheory.

It turns out that when we cast our observation in this form, it becomes completelyindependent of any details of Tarski‟s account. Indeed, suppose we have any collection

È = {LmWíi of languages that share the same set of sentencesVbut differ in how thesesentences are interpreted. Note first of all that for any language LM in this collection,

the logical truths of LM must clearly be a subset of the truths of LM. Modifying ourearlier notation in an obvious way, we can express this as follows:

(6) LTr(LM) C Tr(LM).

It follows from this simple fact that the logical truths common to the languages in ££must be a subset of the common truths of the languages. That is:

(7) HltXI*) C QMTr(LM).

Or, equivalently:

QMLTr(LM) C ValM.

Now, the model-theoretic account takes the set appearing on the right-hand side of (7)to be the set of logical truths of each and every language in ££. While this simpleidentification is mistaken, what (7) shows us is that, at least as an account of thecommon logical truths of the canvassed languages, the account will not undergenerate.And unlike our earlier observation, this observation holds even if our semanticsemploys cross-term restrictions. Interestingly, it is entirely indepen¬dent of how the

model theory specifies the collection ££ of languages: it does not even matter ifexpressions retain the same semantic catego¬ries as we move from interpretation tointerpretation.

This puts us in a position to give a Kreisel-like argument showing that, in the first-ordercase, we can characterize exactly the set of logical truths common to all languages ofthe form LM, where M ranges over any rich collection M of models. Let us writeCLTrjâ) to indicate that a is one of these common logical truths — that is,

CLTrM(a) <-► a e L7V(LM).

Our argument will use the following three steps.



(1") Va(D(a)-+ CLTrM(a))

(2") Va(CLTrM(a)-+ValM(a))

(3") Va(ValM{a)-+D(a)).

All of these observations have, in fact, been made earlier in the chap¬ter. Step (1") issimply the observation that our deductive system is sound, independent of which first-order language LM is under consid¬eration. Thus, if a is derivable in the system, it must

be a common

logical truth of these languages. Step (2"), on the other hand, is a restatement of (7),which holds for absolutely any collection of lan-guages. Finally, step (3") is a statementof the completeness theorem, and follows from our assumption that M is a richcollection of models. Combining these gives us a result analogous to Kreisel‟s:

(8) Va(CLTrM(a) ++ ValM(oc) ** D(a)).

It is not entirely clear how significant this result really is, for all its elegance. If ourconcern is to explicate the logical properties of a specific first-order language, then (8)is of limited interest. Indeed, it seems likely that the most significant logical truths andlogically valid arguments of a given language will be filtered out by shifting attention tothat portion of its logic common to a rich collection of languages. From this perspective,we have done little more than redefine the notions under investigation, and in such away that the resulting task has been stripped of many of the intuitions that motivated the

pio¬neers of modern logic, intuitions clearly at work in Tarski‟s original attempt to

characterize the consequence relation.

On the other hand, there is a different project in the context of which (8) is ofconsiderable interest. It would be misleading to think of model theory as motivatedsolely by the goal of analyzing logical properties and relations. A large part of itsmotivation can be under¬stood only in relation to modern algebra. Indeed, a centralconcern of the discipline from Tarski and Robinson on has been the systematicunderstanding of notions and techniques of abstract algebra.

One of the most striking features of modern algebra is the technique of simultaneouslystudying a wide collection of mathematical struc-tures, as when we investigate the

properties of abelian groups. A key insight was that one and the same proof can often beinterpreted as applying to all structures in the specified collection. By isolating thecommon truths on which such a proof depends, we can obtain results of strikinggenerality. As a result, the practice in algebra is to group structures together by meansof a set of core truths called “axioms,” and to construct proofs that rely solely on the

core truths together with the logical properties common to any interpretation of thesetruths.

From this perspective, a key concern is exactly the logical properties common to acollection of interpreted languages, and so (8) acquires added significance. It assures us



that so long as our collection of algebraic structures can be characterized by first-orderaxioms,3 the consequence relation simultaneously captured by the model theory and

proof theory coincides with the specialized notion of consequence used by the algebraistwherrreasoning about a range of structures. This positive Result is in striking contrast tothe case where the collec¬tion of structures in question cannot be characterized usingfirst-order axioms, as in the case of torsion groups, archimedean fields, or finite divisionrings. In these cases, the notion of consequence used by the algebraist clearly outstripsthat captured by the notions related in (8).

Recapitulation

In previous chapters, we saw that the model-theoretic account of, logical truth andlogical consequence will regularly and predictably go astray: some applicationsovergenerate, others undergenerate, and in some cases it fails both ways at once. In thischapter, I have tried to reconcile these general observations with the intuition that, atleast in the first-order case, the analysis gets something right, and that the completenessand soundness theorems play an important role in dem¬onstrating that fact.

By modifying an argument of Kreisel‟s, we saw that for first-order languages themodel-theoretic account does not overgenerate: no argument declared valid by themodel theory will be invalid. The crucial observation is that the completeness theoremallows us to trans¬fer the intuitive soundness of the deductive system over to the modeltheory. The theorem assures us that any model-theoretically valid argument is provablein the deductive system, and so is genuinely valid if this system is sound. Note that,

somewhat ironically, the real guaran¬tee of validity is carried by the presumedsoundness of the deductive calculus, and not by the declarations of the model theoryitself.

Reassuring as this is, it is also a bit unsatisfying. After all, one thing we might hope forin a semantic account of consequence is an expla¬nation of why valid arguments arevalid, an explanation not given to us by syntactic characterizations of this notion. Butwe now see that even in cases where we can demonstrate that the model theory does notovergenerate, our proof hinges on the presumed soundness of the syntacticcharacterization.

Still, a proof is a proof, and we are better off in this case than we are in the absence of acompleteness theorem. In those cases where the model theory outstrips the deductivecalculus, we have no general way of determining whether it is because the model theoryovergenerates or the deductive calculus undergenerates. Indeed, with second-orderlogic, we seem to have both. The model theory declares the continuum hypothesis (or

perhaps its negation) to be a logical consequence of the pair-set axiom — hardly a plausible assessment. On the other hand, any effective deductive calculus for thelanguage will, if sound, fall short of the intuitive consequence relation for thelanguage.4 Here, the genu¬ine consequence relation must fall somewhere in between

the deduc¬tive and model-theoretic accounts.



When we turn from the problem of overgeneration to the problem of undergeneration,the situation is even less satisfactory. If we main¬tain our original interest in the notionof consequence for a fixed language, the model-theoretic account does undergeneratefor all but the most trivial languages, and so of course there is no way to show that itdoes not. The only way to get around this is, in effect, to define away the problem, byshifting attention to the notion of the logical truths and logically valid argumentscommon to the range of languages can¬vassed by the model theory. Then, although themodel theory can still overgenerate, it is guaranteed not to undergenerate for verysimple and straightforward reasons.

In cases where we have a completeness theorem, this trick allows us to view our modeltheory (and our deductive calculus) as both sound and complete, relative to thisalternative notion of “common logical validity.” The thing to remember here is that the

role of the complete-ness theorem is to show the soundness of the model theory relative

to this new notion. The “completeness” of the model theory is simply built into thedefinition of the alternative notion. Still, this gives us a construal of the first-ordercompleteness theorem that sheds some light on the notion of consequence that is ofinterest to practitioners of modern algebra.

12

Conclusion

In the early part of this century, it was not uncommon for philoso-phers and logicians toconflate the notion of logical consequence with that of derivability in a deductive

calculus. For example, Carnap often promoted the view that languages, both natural andartificial, came equipped with three sorts of rules. Two of these fell under the generalheading of syntax: the rules of grammatical syntax determined which strings of symbolswere grammatically correct sentences, and the rules of logical syntax determined whichsequences of sentences were logi¬cally valid arguments. The third set of rulesgoverned, among other things, the semantics of the language, but at the time Carnap hadlittle to say about these additional rules.

According to Carnap‟s picture, a deductive system for a language, its logical syntax,

was essentially independent of the language‟s seman¬tics. The question of whether onesentence followed logically from another came down to the question of whether aderivation of the one from the other could be constructed by means of theconventionally adopted logical rules, just as the question of whether a given string ofwords made up a sentence came down to whether it could be formed using theconventionally adopted grammatical rules. Of course, nei¬ther the logical syntax nor thegrammatical syntax could be entirely divorced from the semantics. Presumably, thesemantics would not declare a string of symbols to be meaningful if the grammaticalsyntax declared it ill-formed. Similarly, if the logical syntax declared modus ponens avalid rule, then the semantics could hardly assign the mean¬ing or to the symbol „if .. .

then.‟ But the view was that the syntactic rules fixed the logic, and thereby placedconstraints on the semantics, not the other way around.



Nowadays, we see this identification as a confusion. We recognize that the logicalconsequence relation is determined not by an indepen¬dent deductive regime but by thesemantic rules of the language. After all, if the term „and‟ expresses the usual truth

function, then nothing mor e is needed to guarantee the validity of an inference from „A

and B‟ to „A.‟ Independent rules of “logical syntax” could at best reiterate that fact, and

at worst contradict it. If the former, the rules would be idle; if the latter, flatly wrong.

The real harm in identifying consequence with derivability is that it distracts attentionfrom genuine issues in logic toward artificial ones that arise from the conflation. Forexample, according to Carnap‟s early picture it makes no sense to ask whether a

deductive system, a language‟s logical syntax, is sound and complete. Since the

deductive system is what gives rise to the consequence relation for the language, itautomatically gets that relation exactly right. From this perspective, a more pressingissue might be the question of which deductive system, among sundry equivalent

systems, corresponds to the language‟s “gen¬uine” logical syntax: Does that syntax takethe form of an axiomatic system, or is it instead a system of natural deduction? Is a

particular rule primitive to the system, or is it a derived rule instead?

Clearly, our understanding of deductive techniques has changed considerably in theintervening decades. This is not to say that such techniques have been, or ever should

be, abandoned. But we now see them as serving a rather different purpose. A deductivesystem pro-vides a way to study a language‟s consequence relation, to prove results

about it, perhaps even mechanize it. But it does not determine or give rise to thatrelation. This is why the question of whether a particular deductive system for a

particular language is sound and complete is always a sensible, and indeed important,one to ask.

Identifying logical consequence with model-theoretic consequence is as mistaken asidentifying it with derivability. The question of whether one sentence follows logicallyfrom another does not come down to whether there are interpretations that make thelatter true and the former false; logically valid arguments can fail this test, while invalidarguments can slip by it. Though the model-theoretic account may sometimes get theextension exactly right, as may deductive char¬acterizations, this is not because eitherof them captures, or comes close to capturing, the genuine concept.

Tarski‟s conflation spawns as many confusions, as many distracting issues, as Carnap‟s.

Take, for example, the so-called problem of the logical constants. We saw how thisalleged problem immediately evap¬orates once we recognize exactly why the model-theoretic account is sometimes right and sometimes wrong. The reason has nothing todo with any shared characteristic of the expressions held fixed, but rather with factsabout the world. The effort spent trying to find such a characteristic, trying to maintainthe analysis while making sense of its haphazard behavior, would be more profitablyspent on genuine issues surrounding logical consequence.

Another example, and perhaps a more important one, is the much- debated question ofwhether second-order logic is really “logic.” What motivates this odd question is the



fact that claims like the continuum hypothesis are declared logically true by the standardmodel theory, and yet such claims seem clearly beyond the scope of logic. But once werecognize this as a case where Tarski‟s account overgenerates — and more generally,once we recognize overgeneration as a natural and predictable hazard of the mod el-theoretic technique — the issue takes on an entirely different light. Every genuinelanguage has its conse¬quence relation, its sentences that follow logically from others.This is as obviously true of higher-order languages as it is of languages where model-theoretic techniques yield more plausible results. And whether or not we have sure-fireways to characterize this relation, it seems clear that the relation is a legitimate concernof logic. To claim otherwise, to say that the logic of some languages is not logic, is justto abdicate the discipline‟s natural charter.

Similar remarks can be made in cases of undergeneration. It is a mistake to think thatthe logic of, for example, the language of elemen¬tary number theory is confined to that

characterized by the usual model theory, or that the consequence relation that arisesfrom the meanings of predicate or function terms is any less significant than the logic ofconnectives and quantifiers. Once again, it is only the con¬flation of logicalconsequence with model-theoretic consequence that inclines us to think otherwise. Onceagain, there is more logic to be studied than we might otherwise have thought.

It is always important to ask whether our model theory overshoots or undershoots thelogic of a particular language. And the answer to this question will frequently be yes.But as with deductive techniques, this does not mean that model theory should simply

be abandoned. For as we have seen, model-theoretic techniques, when properly un-

derstood, can yield genuine insight into a language‟s consequence relation. Forexample, combined with an intuitively sound deductive system and a proof ofcompleteness, the model-theoretic account al-lows us to precisely specify significant

portions of that relation, the portions common to the range of languages surveyed by themodel theory.

Properly understood, both deductive and model-theoretic tech-niques can be put to gooduse. Both provide tools that can profitably be used in studying the consequence relation.But it is in no one‟s interest to identify the consequence relation with either the model-theoretic or the proof-theoretic notion. To do so buys only an illusion: the illusion that

the relevant technique is incapable of going astray. In the end, this illusion can haveonly one of two results: either we uncritically accept the technique‟s faulty declarations

or we confine the scope of logic to domains where the technique happens to work. Ineither event, we shortchange the vision that motivated the founding fathers of modernlogic.

Notes

1. Introduction

1. Though the model-theoretic definitions have come to be standard, theterminology still varies considerably. I have adopted the terminology used by Chang



and Keisler (1973). Models are also sometimes called structures, valuations,assignments, interpretations, or model structures. Terms for the relation of truth in amodel vary accordingly, with holds in and is satisfied by sometimes replacing is true in.

2. Tarski (1936); all page references in this book are to the English transla¬tion in

Tarski (1956). Some writers attribute the model-theoretic defini¬tions to Tarski‟smonograph on truth (1933), but this is simply an error.

3. See Hilbert (1929), p. 8. The remark was made in order to motivate thecompleteness problem for first-order logic, the problem solved by Gõdel that same year.

4. For a more detailed discussion of the historical relationship between Tarski‟s

analysis and the model-theoretic definitions, see Etchemendy (1988).

5. That is, N is the intersection of every set (hence, the “smallest”) that has the

following two properties: (1) it contains 0; and (2), if it contains a number x, it alsocontains j(x) = x + 1. In symbols:

N = fl{ A I 0 E A a Vx(x E A — > s(x) E A)}

2. Representational Semantics

1. Toward the end of his article, Davidson hedges this claim, remarking that “

„absolute‟ truth goes relative when applied to natural language” (1973, p. 85). The

hedge is needed because of indexical sentences: Davidson allows that these are trueonly “relative to” a speaker, time, and place of utterance. I think the properSnove here is

not to reladvize truth to an occasion of us^, but rather to l „cognize that the ordinarynotion of truth applies not to sentences but to statements (the actual uses of sentences)or to propositions (the claims made by such uses). Of course, if the property of truthapplies primarily to statements or propositions, and only deriv¬atively to sentences, thesame must presumably be said of logical truth and logical consequence. I will set asidesuch issues in this book, though, since they are irrelevant to my objections to Tarski‟s

analysis.

2. It may be unfair to say that Davidson implies that relational truth has no bearingon absolute truth, since at one point he says there is a “perfectly clear” sense in which

absolute truth is a special case of relative truth (1973, p. 79). Davidson does not,however, explain what sense he has in mind, and later, when discussing the conceptsilluminated by theories of relational truth, truth itself is conspicuously absent (cf. p. 79and p. 83).

3. Of course, even a Davidsonian theory of “absolute” truth does not alone give us

the “absolute” values of our sentences. Before we can know whether the monadic truth

predicate is applicable to a given object lan¬guage sentence, we must know more than just the appropriate T-sentence, or the theory from which it “falls out.” But naturally the

goal of a semantic theory is not to tell us such things.



4. I assume throughout this book that the sample languages, though frag¬ments ofEnglish, are syntactically as well-behaved as artificial languages. In particular I assumeunique readability, though I try to avoid parentheses.

5. Naturally our theory of “truth in a row” could easily have used a standard

“maximal” reference column, so long as the vocabulary of atomic sen¬tences remainedfinite. We can think of the construction of a reference column for a given targetsentence as providing the class of models for the smallest truth-functional fragmentcontaining that sentence. Seen in this way, truth tables embody a general technique for

building the minimal model-theoretic semantics capable of handling a particularsentence. Of course, they are still incapable of illuminating any semantic properties ofinfinite sets of sentences.

6. Kaplan (1975), p. 216. Few mathematical logicians view Tarski‟s analysis this

way, since they generally come to it via an entirely different tradition, that of abstractalgebra. It would seem quite anomalous to view a structure satisfying, say, the groupaxioms as having much of anything to do with possible worlds.

7. I am setting aside here certain important issues about the notions of analyticityand necessary truth —in particular, issues that arise with sen¬tences, like „I am here

now,‟ that express contingent propositions but cannot, by virtue of their meaning, beuttered falsely (see Kaplan, 1978). With languages containing such sentences, thesituation is more compli¬cated, and requires a finer taxonomy than these traditionalnotions provide.

3. Tarski on Logical Truth

1. Bolzano (1837); page references are to the translation (1973).

2. Bolzano actually considers logical truth (Allgemeingultigkeit) to be a relation between propositions (Satze ansich) and component ideas (Vontellungen an

sich), not between sentences and expressions. To facilitate comparison between Tarskiand Bolzano, I gloss over this difference. As I note later (Chapter 3, note 5), this isactually unfair to Bolzano‟s account, but my purpose is to illuminate Tarski‟s, not

Bolzano‟s, analysis.

3. Bolzano (1973), pp. 187ff.

4. Notice, though, that this assumption places a heavy burden on our gram¬mar.For example, when we replace the expression „snow‟ with the expres¬sion „grass is

pink and snow‟ we get the false sentence „Grass is pink and snow is white or grass is

pink and snow is not white.‟ So for the assumption to be correct, the grammar cannot

judge „snow‟ and „grass is pink and snow‟ to be of the same grammatical type. Further,

such judgments must be motivated by reasons other than the fact that the substitution of

the latter for the former is capable of rendering a logically true sentence false, sinceotherwise we risk a disguised circularity in our definition of logical truth. In the present



case it seems plausible that such a motivation can be found; for one thing, substitutionof the latter for the former often renders a grammatically proper sentenceungrammatical (e.g., „I hate snow‟ becomes „I hate grass is pink and snow‟). But

occasionally the grammatical motiva¬tion is not nearly so clear, as when Quine, asupporter of a Bolzano-style, “substitional” definition of logical truth, classifies

expressions like „or‟ and „and‟ as syncategorematic, in effect placing each of them in a

category of one (see Quine, 1970, especially pp. 27ff and 49ff). I will not discuss this problem at length, but will assume that if the exchange of two expressions always, or byand large, preserves grammaticality, then the expressions are of similar grammaticaltypes. In fact, though, proponents of the substi¬tional definition need a much strongerassumption than this when dealing with natural languages; see Chapter 3, note 13.

5. Here, my characterization of Bolzano‟s theory in terms of sentences and

expressions, rather than the original propositions and ideas, does the account some

injustice. Bolzano‟s original definition makes logical truth dependent not on theexpressive resources of any particular language, but rather on what might be called theconceptual resources of the realm of ideas. I will, however, continue attributing toBolzano the simplified, linguistic version of the theory.

6. “Sentential function” is the term Tarski uses, by analogy with Russell‟s notion of

a propositional function. The expression now sounds a bit odd, “open formula” or “open

sentence” having become standar d.

7. If a language contains variable binding operators, such as quantifiers, and

explicitly displays bound variables, then sentences must be taken to be sententialfunctions with no unbound or free variables.

8. Using schematic machinery analogous to that introduced for (1), Tarski‟s T-schema would come out as:

(T) \ . .‟ is true (in L) if and only if ...

Tarski generally used “X” wheresl have “ „...‟ ” and “/>” where I have “...”, requiring

that ‟jX” be replaced with a name of the sentence replacing "p.” Thus, Tarski‟s actual

statement of the schema ran as follows: (T') X is true (in L) iff p.

Tarski‟s (T') is, by itself, less perspicuous than (T), but (T) has the draw¬back that its

initial symbol (i.e., “ „. ..‟ ”) might be mistaken for a quotation name of a succession ofthree dots (though only by a philosopher). In fact, I have used “ ..‟ ” as an independent

schematic device (like “X”) whose intended relation to the other schematic placeholder

(the three dots on the right) is set forth in the instantiation conditions. My policy is tomake the actual expression of schemas as perspicuous as possible, though this mayinvolve employing certain symbols (e.g., the single quotation marks in (T) and (1)) asmere orthographic components of a larger schematic device (“ „...‟ ” and “ „... n . ..‟ ”

respectively).



9. See Chapter 3, note 8, for a statement of the T-schema. It is what provides theconnection between the right hand side of (1) and the right hand side of (2), so long asthe replacement for “.. . n ..is a sentence of L.

10. This is not a standard account of sequences. (Finite sequences are gener¬ally

taken to be ordered n-tuples, and infinite sequences to be functions whose domain is theset of natural numbers.) Tarski adopts the standard notion of a sequence, assuming anappropriate ordering of the variables. Allowing sequences to be functions directly fromvariables simplifies things considerably, in particular when we turn to sententialfunctions with vari-ables of different types.

11. It should be emphasized that the expressions „/(V)” and „/C/)” are com¬plex

names (akin to “Tom‟s father” and “Sam‟s father”), and that they do not contain

variables (as do „Vs father” and “/s father”). The expression “V ” occurring in „/(V)”

names an object (the variable “x”), just as the expression “Tom” occurring in “Tom‟s

father” names an object (the per¬son Tom).

12. Linguistic entities can, of course, stand in the relation of satisfaction to certainsentential functions —for example, „was president‟ does indeed satisfy „x is a predicate.‟

But the relation that would emerge from (3.2) would preclude the right sorts oflinguistic entities even here. Thus, on analogy with (3.2) we would have to say that “

„was president‟ ” satisfies „x is a predicate‟; “ „was president‟ ” is, however, the name ofa predicate, not itself a predicate.

13. This has been considered an important advantage of the substitutional account,

but in fact the advantage is illusory. In natural languages, exam¬ples abound in whichexpressions that seem to be of the same grammatical category differ semantically. Thus,

perhaps the clearest grammatical cate¬gory in English is that of noun phrases, but thisincludes such expressions as „George Washington,‟ „July 4, 1776,‟ „every president,‟

and „no presi¬dent.‟ The radical diversity here shows that even the genuine “substitu-tionalists” (Bolzano and Quine) must assume some implicit semantic crite¬rion of

substitutability. If the noun phrase „every president‟ were taken to be substitutable for

the noun phrase „George Washington,‟ it would wreak havoc on the substitional test:

„Every president had a beard or every president did not have a beard‟ would then be a

false substitution instance of„George Washington had a beard or George Washingtondid not have a beard.‟

14. This remark is intended as a simple observation, not a rejection of a particular philosophical tradition. We can certainly devise languages in which all expressionsname objects of various types and in which the concatenation of any two of theseexpresses, say, function application (or perhaps, taking a cue from Wittgenstein, depictssome concrete relation). But it is clear that ordinary English — and hence fragments ofordinary English like the object language we are considering — do not in fact operate inthis way. For one thing, if they did so function we would have no need for the various

complex techniques of nominalizing verb phrases in order to place them in subject position.



15. It also, we should note, commits us to the view that properties are a type ofobject, and hence capable of being named by expressions of the meta¬language.

16. There are also alternatives available with names. We could, for exam¬ple, takenames to denote collections of properties, following Richard Montague. I will not

explore these possibilities.

17. For example, if a language contains only the basic expressions „Snow is white,‟

„Grass is green,‟ and „and,‟ then it will in fact have no logical truths whatsoever. There

will, however, be logically valid arguments — for exam¬ple, the inference from „Snow is

white and grass is green‟ to „Grass is green.‟

18. Thus, if we added „if. .. then‟ to the language of note 17, we would not encounter

the first problem. However, the valid argument mentioned there does not depend in anyway on the expression „if... then,‟ and so we might expect it to come out valid even if

this expression were excluded from ft. But the conditional would not then be logicallytrue.

19. There are actually standard counterexamples to this. Thus, some would claimthat all sequences satisfy „It is not the case that Ralph believes x is a spy,‟ though

„Ralph believes the shortest spy is a spy‟ is true. I will not consider these so-called“opaque” contexts, except to note that the notion of satisfaction implicit in such

judgments is immediately ruled inadequate by schema (1). Whether this should be heldagainst the judgments or the schema I leave to the reader to decide; my inclinationwould be to hold it against the latter.

4. Interpretational Semantics

1. I should mention that Tarski uses the term “model” in his article, though not in

the same way I have used it here. Stated in my terminology, Tarski‟s use is the

following: a d-sequence is a model of a set K of sentences just in case it d-satisfiesevery member of K. This corresponds, as Tarski points out, to a standard use of“model” in mathematics; if a d-sequence provides an interpretation of a set of axiòms onwhich they all come out true, then it is commonly saieKto be a model of those axioms.

5. Interpreting Quantifiers

1. Henceforth, I will take predicates to be interpreted by sets rather than properties,as is usually done.

2. We commonly define truth in a model by employing the auxiliary notion of thesatisfaction of a formula by a sequence in a model. Here a slight confusion might arise,for on the original Tarskian conception, models are them¬selves sequences (d-sequences) and truth in a model is itself satisfaction (d-satisfaction). John Kemeny wasthe first to employ such auxiliary se¬quences and the notion of satisfaction-in-a-model

(note: not “by a model”). See Kemeny (1956). His technique has since become standard.



3. In a language that explicitly displays bound variables, the satisfaction clausewould run as follows (with *£‟ an existential quantifier variable and V a name variable):Sequence / satisfies ExM iff for some E/x- variant f off, /' satisfies M. We take/' to be an£/x-variant of/just in case/'(V) is a member of/(*£‟) and f(v) = f (v) for all v 4 V.

4. The set-theoretic paradoxes force a certain idealization here. On the in¬tendedinterpretation, „something‟ quantifies over a class too large to be a genuine set: the class

containing everything. Thus, in any traditional set theory the satisfaction domain wehave described must actually omit the intended interpretation of our expression. For

purposes of giving a se¬mantics for natural languages, these facts may be viewed asquirks of the mathematical theory of sets, however important they are to that theory.The designers of natural language were, after all, unaware of the set- theoretic

paradoxes.

5. When our semantics is applied to standard symbolic languages, the twoalternatives emerge in the following way. On the one hand we can take the specificationof a universe set as providing the appropriate interpretation for the variable term 3 as itappears in sentences of the form 3xM. Alterna¬tively, we can think of 3 as fixed andtake all sentences of this sort as abbreviations for their relativization to an implicit

predicate U, where the relativization (3xAi)* of 3xM is 3x(Ux a M*). According to thisview, we are providing an interpretation of the variable expression U, while theinter¬pretation of 3 is held fixed.

6. Of course, we could avoid that by treating identity as variable, but this in turn

gives counterintuitive results, declaring invalid such arguments as: Fa, a = b, so Fb.6. Modality and Consequence

1. The continuum hypothesis is the claim that any infinite set whose cardi¬nality isless than that of the reals has the same cardinality as the natural numbers. The pair-setaxiom says that for any objects x and y, there is a set whose only members are x and y.Well-known results of Gõdel and Cohen show that there are models of the standardaxioms of set theory in which the first-order statement of the continuum hypothesis istrue and others in which it is false. This shows that the continuum hypothesis is not a

model- theoretic consequence of those axioms. The question of whether it also showsthat it is not a logical consequence of the axioms depends, of course, on the relation between model-theoretic consequence and logical conse¬quence. But quite apart fromthese results, even the most Platonistic set-theoretician would not claim that thecontinuum hypothesis (or its negation) is a logical consequence of the extremely weak,

pair-set axiom.

2. Tarski (1956), p. 411, my translation. For the German text of this passage, seeChapter 6, note 7.

3. Ibid., pp. 412-413, my translation.

4. Ibid., p. 413, my translation.





S(v/e) is true for each expression e in the appro¬priate substitution class. The definition presupposes that truth is well- defined for sentences not containing the substitutionalquantifier. For a detailed discussion of substitutional quantification, see Kripke (1976).

2. For a discussion of objectual quantifiers that bind nonstandard variables (e.g.,

predicate variables) see Boolos (1975). Boolos gives a treatment of satisfaction similarto the one I develop in Chapter 3.

3. For simplicity, I will use the term “universal generalization” to apply to any

sentence that begins with a string of universal quantifiers, even though that string mayhave length zero. For purely heuristic purposes, I indicate the universal closure of asentence (that is, a sentential function with no free variables) by enclosing it in brackets.The reader is free to imagine a vacuous universal quantifier standing in front of these

brackets.

8. Substantive Generalizations

1. The “matrix” of the closure Vui ... Vun[S'] is the sentential function S'.

2. Of course, until we clarify what sorts of individuals we count as part of theuniverse, it is hard to say what kind of fact the size of the universe is. The size of the

physical universe — say, the number of elementary particles — is presumably acontingent, physical fact. The size of the set-theoretic uni¬verse is presumably anoncontingent, set-theoretic fact. But neither of these are issues to be settled by logicalone; both are substantive, extralogical facts. The point I will make does not depend

on which way we go here.

3. For a definition of satisfaction for sentential functions containing quanti¬fiervariables, see Chapter 5, note 3.

4. Note that we can here ignore the cross-term restrictions used in the stan¬dardsemantics, since our sentences contain only the identity predicate, whose interpretationwe are holding fixed.

5. For simplicity, I am assuming that the range of the variable E consists ofarbitrary subcollections of the universe. If the range consists of sets, then the relevantquestion is about the size of these, not the size of the universe as a whole. Similar pointscan be made, though, whichever way we go.

6. If we expand the language to include other cardinality quantifiers — for example,“there exist uncountably many” or “there exist inaccessibly many”— all of the same

points can be made. But then the outcome will depend on whether there are sets withuncountable (or inaccessible) car¬dinalities, rather than just some infinite cardinality.Many mathematicians who accept the existence of infinite sets still question thesestronger as¬sumptions.

7. In fact, the truth value of (6) is not as clear as it might seem. Indeed, if I lie*



satisfaction domain for the relation variable consists of all sets of ordered pairs, and thesatisfaction domain for the individual variables consists of all objects (including sets),then (6) is actually true according to standard set theories. (This is due to the set/classdistinction imposed on us by the set-theoretic paradoxes.) In which case, the presentaccount would still mistakenly declare 02 logically true (and the rest of the <rn> aswell). This is a bit ironic, since the usual set-theoretic assumptions are what we earlierrelied on to get a proper assessment of — icrn; here, they would result in an improperassessment of <rn. To get the right assessment while keeping the set-theoretic construalof (6), we would again have to vary the interpreta¬tion of 3.

8. If something is at least as tall as everything else, then we say it is a minimalelement of the taller than relation. A relation can have more than one minimal element;for example, if everything were precisely the same height, each individual would be aminimal element of both the taller than and the shorter than relation.

9. Once again, I should emphasize that my appeal to the finitist‟s position is simplymeant to dramatize the problem with Tarski‟s account. The prob¬lem does not depend

on any endorsement of the position, or even on the assumption that the axiom ofinfinity, and the existence of noncommu- tative division rings, are contingent truths.Even if our views about mathe¬matical objects lead us to conclude that these arenecessary truths, which I happen to believe, they are surely not logical truths. (If theywere, then so too would be (r2, 03, ...) Our judgment of the logical status of suchsentences as (7) is surely not dependent on our belief in the axiom of infinity, a fact

brought out nicely by the finitist‟s position.

10. The pair-set axiom says that for any x and y, there is a set whose only membersare x and y. Since 'x' and „y‟ can be instantiated to a single object a, this axiomguarantees the existence of the singleton set {a}.

11. There are many ways of arriving at such sentences. For example, let N(X) andR(X) be second-order formulas that are satisfied by a set iff it is isomorphic to thenatural numbers or the real numbers, respectively. Since the relation Card(X) < Card(Y)is also definable in the second-order lan¬guage, the closure

vxvr VZ[JV(X) a R(Y) a Card(X) < Card(Z)Card(Y) < Card(Z)]

is equivalent to the continuum hypothesis, and any instance of it will be declaredlogically true if and only if the continuum hypothesis is true. (Once again, as with (8)and (8'), it does not matter here if we impose the standard cross-term restrictions.)

9. The Myth of the Logical Constant

1. For example, using N, R, an"ú^Card as in Chapter 8, note 11, one of the

following generalizations will be true, depending on which way the contin¬uumhypothesis goes:



VDV£ = 9(D) [ — i£X, Y,Z(N(X) a R(Z) a Card(X) <Card(Y) < Card(Z))]

VDV£ = 9(D) IEZ(R(Z)) -> £X, K, Z(N(X) a R(Z) a Card(X) < Card(K) < Card(Z))]

Here, D ranges over interpretations (domains) for the first-order quanti¬fiers and E

ranges over interpretations (domains) for the second-order quantifiers. The usual cross-term restriction is imposed by requiring that the latter be the powerset of the former.When we move to so-called generalized structures this restriction is loosened, and bothof the resulting generalizations, though still substantive claims, come out false.

11. Completeness and Soundness

1. Note that the appeal here to “intuitive soundness” is not a reference to the

soundness theorem, Va(D(a) — > V(a)), which would not give Kreisel what he needs.Rather, it is an observation about the intuitive correctness of the first-order deductive

calculus in question.2. I say “roughly” because the following argument ignores complications arising

from cross-term restrictions. The argument can be extended to cover that case as well, but only at the cost of obscuring the main point.

3. More precisely, what we need is a set/i of axioms such that (1) each of ouralgebraic structures satisfies K, and (2) every structure M from our rich, collection Msatisfying K is one of the algebraic structures in question.

4. The argument here is basically the same as that given by Tarski for the first-

order language of arithemetic, using Godel‟s incompleteness results.

Bibliography

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Extensional adequacy, 3-4, 6, 8-9, 11, 80-81, 83, 85, 108, 130-135, 144-155

Fixed terms, 28, 30, 32. See also Logical constants

Gõdel, Kurt, 5, 7, 161n3, 166-167nl,

171. See also Completeness theorem; Incompleteness theorems

Hilbert, David, 6, 7, 171

Identity predicate, treating as nonlogical constant, 117-118 Incompleteness theorems,84, 100, 150, 170n4

Infinity, axiom of, 114-117 Instantiation principle, 98 Intended interpretation, 56, 146,166n4 Interpretational semantics, 51-56, 65-68; vs. representational semantics, 51, 57-64, 66, 77-79. See also Cross-term restrictions

Kaplan, David, 23, 162n7, 171 Keisler, H. J., 161nl, 171 Kemeny, John, 166n2, 172Kreisel, Georg, 11, 145-155, 172 Kripke, Saul, 168nl, 172 Kripke-Platek set theory, 115

Logical constants, 28, 80, 100, 109-110, 125-130, 157-158 Logical truth vs. logicalconsequence, 11, 47

Machover, M 82, 171 Mates, Benson, 82, 171 Meaning postulates, 71-72, 77. See alsoCross-term restrictions Model-theoretic account of logical truth/ consequence, I, 55Model-theoretic semantics, 51. See also Interpretational semantics; Representa¬tionalsemantics Montague, Richard, 165nl6

Natural numbers, inductive definition of, 9

Necessity, 24-26, 78, 81-94, 106, 108 Nix, 40-41

w-incompleteness, 83-84, 100 o)-rule, 84, 100, 133 Opaque contexts, 165nl9Overgeneration, 8, 130-135, 144-145, 148-150, 154-155, 158

Padoa, Allessandro, 7, 172 Pair-set axiom, 82, 122 Persistence, 30-32, 36-38, 48-50;and cross-term restrictions, 69-70 Possible models, 119-120 Possible worlds, 12, 23, 25,

78 Principle (i). See Instantiation principle Principle (»). See Closure principle Principle(in). See Reduction principle Principle (in'). See Reduction principle, first modificationof Principle (iif). See Reduction principle, second modification of Propositions vs.sentences, 161-162nl, j^l62n7, 162-163n2, 163n5

Quantification: and interpretational se-mantics, 65-79; and logical truth, 95- 124;substitutional, 96, 168nl; objectual, 96, 168n2 Quine, W. V. O., 88, 163n4, 164-165nl3, 167n6, 172

Recursion theory, 5

Reduction principle, 98-99; first modifi-cation of, 101-106; second modification of,110-124 Representational semantics, 10, 20-26; and logical properties, 25; vs. interpre-



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