GREG RAY

ON THE POSSIBILITYOF A PRIVILEGEDCLASS OF LOGICALTERMS*

(Received 7 August 1995)

INTRODUCTION

Alfred Tarski's (1936) semantic account of the logical properties (, logical truthand logical consistency) makes essential appeal to a distinction between logical and non-logical terms. John Etchemendy (1990) has recently argued that Tarski's accountis inadequatefor quite a numberof differentreasons. Among them is a brief argumentwhich purports to show thatTarski 's reliance on the distinctionbetween logical and non-logical terms is in prin- ciple mistaken.) According to Etchemendy,there are very simple (even first order) languages for which no such distinction can be made. This is a surprisingresult, and an importantone, if true. Since Tarski's account does indeed depend on such a distinction, Etchemendy'sargument, if correct, would rule out definitively the received view on logical truth(as well as logical consequence and logical consistency). But his argumentis not correct,and it is the job of this paperto show that.

TARSKI'SACCOUNT

Etchemendyargues for his claim by giving an exampleof a first-order languagethat is such that,no matterhow you divide up its termsinto "logical"and "non-logical",the applicationof Tarski'sdefinition of logical truth (or logical consequence) will give the wrong results. Before looking at this argumentin detail, it is importantto review how it is that Tarski'saccount of the logical propertiesdepends on making a distinctionbetween logical and non-logical terms.

Philosophical Studies 81: 303-313, 1996. ? 1996 KluwerAcademic Publishers. Printed in the Netherlands.

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(Dxy & Dyz )- Dxz

s < ...... >

Figure 1. Relation between a formula'p and a sequence.

Forease of exposition,we will focuson just logical truth. Tarski's definitionof logicaltruth is closelyrelated to his definitionof truth. In fact,Tarski's idea was thatdefinitions of all the logicalproperties can be obtainedsimply as an extendedapplication of the methods he usedin definingtruth. According to Tarski,a sentenceis truejust in case it is satisfiedby every sequence.For example,the formula '(Dxy & Dyz) -? Dxz' (read "If x is a descendent of y and y is a descendentof z, then x is a descendentof z") is satisfiedby the sequences justin case a certaintriplet of itemsin thatsequence have the rightsort of descendencyrelations. See Figure1. In particular, lettingf be a functionwhich tells us which variablesto associate with whichpositions in the sequence,we can give the satisfaction conditionof, say,just the subformula'Dxy' as follows. s satisfies 'Dxy' if andonly if thething in theposition f('x') of s is a descendent of the thingin positionf('y') of s. Notice thatthe satisfactioncondition I just gave presupposeda choice of a function,f, which assignsvariables of the languageto aninteger (representing a sequence position). We normally suppress thisparameter, and the reason it is alrightto do thatis thatit doesnot matterwhich such function we choose(so long as we stickwith it). Thereason any old suchfunction will do is thatthe functionhas no independentsignificance, it is purelyan artifact of therepresentation we have chosen, and representsnothing about the conceptbeing treated,in thiscase truth. The situationchanges, however, when we turnto a treatmentof logical truth.Again, Tarskiemploys the satisfactionrelation, but this time it is not the satisfactionbehavior of the targetsentence itselfthat matters. After all, if the targetsentence (call it ',o') is so muchas true,then its satisfactionbehavior will simplyconsist in its being satisfiedby all sequences.But it sharesthis distinctionwith

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(Dab & Dbc) - Dac F~~~~~~~~~

(Xxy & XYZ ) XXZ f <' f >1

Figure 2. Relationbetween a formulaF(p) and a sequence. all true sentences. So, the satisfactionbehavior of p by itself cannot be a distinguishingmark of p's logical truth. Instead, on Tarski's account, ,o'slogical truthis markedby the satisfactionbehavior of a formulawhich can be obtainedfrom o by substitution,namely that formulawhich resultsfrom applyinga special replacementfunction, F, which maps each term of the language either onto itself or onto an extra-linguisticvariable. See Figure 2. Forexample, the sentence '(Dab & Dbc) -? Dac' will be said to be logically truejust in case every sequencesatisfies this new hybridfor- mula '(xxy & Xyz) -- xxz'. Notice, then,that the treatmentof logical truthpresupposes a choice of two functions.We still have the func- tion, f, from variablesto positions in sequences,just as we did in the case of truth.But we also now have the functionF which determines what termsof the originalsentence get replacedby variablesprior to evaluation. While this additionalparameter is likewise suppressed in modem presentationsof the Tarskianor model-theoreticaccount of logical truth,this is not because the choice of function F doesn't matterto the outcome. It does matter,and, moreover,the functionF does representsomething about the notion(s) we are treating.F must divide the logical terms from the non-logical terms. In particular, F must slate all and only non-logical termsfor replacementwith extra-linguisticvariables. Call a replacementfunction which meets this condition, a logical termfunction. Why does Tarski'saccount, which relies crucially on a choice of replacementfunction, F, also requirethat F be a logical term function?Here is one way to moti- vate this requirement.The following argumentseeks to establish

This content downloaded from 128.227.195.219 on Fri, 13 Dec 2013 12:06:07 PM All use subject to JSTOR Terms and Conditions 306 GREG RAY an appropriatelyclose relationbetween Tarskianlogical truthand a certainintuitive characterization of logical truth. Let F (p) be the formulathat results from applyinga replacement function,F, to a sentence p. Now, it can be shown (thoughI will not do so here) that, if the formulaF(o) is satisfiedby every sequence, then the original sentence p is true. Put somewhat differently,the truthof o is determinedby the satisfactionbehavior of the formula F(p). Now, if F is also a logical term function, then it is also the case that the satisfaction behavior of F(o) can in no way depend on the non-logical terms of po,since, of course, these will have all been replaced by variables in F(o). In short, if F is a logical term function, and F(o) is satisfied by every sequence, then the truth of o is determinedby the logical form of o independentlyof the particular importof any of its non-logical terms.2This is one good thing to mean by the time-honoredphrase "truein virtue of logical form",which has been used as a characterizationof logical truth. I have just presented an argument to the effect that Tarski's definitioncaptures one good sense of 'logically true', i.e. if Tarski's definitionapplies to a given sentence, then that sentence is a logical truth.But this argumentmakes crucial appealto a logical termfunc- tion, F. So, there had betterbe such functions for the languages the accountis supposedto cover. Thatis ,therehad betterbe a distinction between logical and non-logical terms for such languages. Tarski's definitionsof the other logical properties,logical consequence and logical consistency, likewise depend on this distinction.3In 1936 when Tarski formulatedhis account of the logical properties,he did not have any account to offer of what made a term logical or non-logical. Of course, one can pointto examples, but this is no sub- stitutefor giving a principlefor distinction.Thus, Tarskiconsidered his account incomplete in 1936.4 Tarski'saccount of the logical propertiespresupposes that there is a logical termfunction F. Consequently,Tarski's account depends on an existence claim, unsubstantiatedin 1936, to the effect that there exist logical term functions (for some broadclass of languages, i). Etchemendy'sargument would show thatthis crucialexistence claim is false. Etchemendy'saim is to show thatthere are very simple languages for which the distinctionbetween logical and non-logical termscan-

This content downloaded from 128.227.195.219 on Fri, 13 Dec 2013 12:06:07 PM All use subject to JSTOR Terms and Conditions LOGICALTERMS 307 notbe madesuccessfully. That is notjust to say thatwe arenot sure how to drawthe distinctionin suchlanguages, but that no division of termswill makeTarskian consequences in the languagecorre- spondto the logical consequences.5Thus, Etchemendy holds that philosophersand logicianswho have workedtoward identifying a privilegedclass of logicalterms have been pursuing a chimera.

THE "BULLET"ARGUMENT

Let us takea closerlook at the argumentEtchemendy offers for his claim.First, we will needtwo closely-relatedlanguages with which to work.Following Etchemendy,

let L be a very simplelanguage with two names('George Washington'and 'Abe Lincoln'),two predicates('was president'and 'has a beard'),three truth-functional oper- ations ('V', 'A' and '-1'), where all these terms have their usualmeaning. LetL* be a languagejust like L, exceptthere is, in addition, a binaryconnective, '*', withthe following truth condition

for any sentencesfo and b of L*, r(( * '0)- is trueiff Lincolnhad a beardand either p is trueor , is true. Etchemendymakes his case,using these languages, in thefollowing way. [L*is] a languagein whichevery sentence is logicallyequivalent to somesentence of [L].Yet it is easy to show(and not surprising in theleast) that no selectionof F yieldsa set of "logicaltruths" containing exactly the right sentences - thatis, those equivalentto logicaltruths of the originallanguage. If we include'.' in F, the accountovergenerates; if we excludeit, theaccount undergenerates. (Etchemendy, 1990,p. 134) I will show thatthis argumentis not sound.For definiteness, let us restrictour attentionto two candidatesfor the logicalterrns of L*. Let F be such that the only F-termsof L* are '-", 'A' and ' v '. Let F* be such that the only F*-terms of L* are 'A','V' , and '.'. Assume,as is reasonable,that the logical termnsof L arejust 'm', 'A' and 'V'. Whatare the logicalterms of L*? Presumably,all the logicaltenns of L arealso logicalterms of L*, sincetheir meanings

This content downloaded from 128.227.195.219 on Fri, 13 Dec 2013 12:06:07 PM All use subject to JSTOR Terms and Conditions 308 GREG RAY do not differ between the two languages. So, the logical terms of L* must be either the F-terms or the F*-terms of L*. It all comes down to whether'.' is to be countedas a logical termor not. I gather that this is the sort of set-up that Etchemendyhas in mind for the argumenthe alludes to. According to Etchemendy,if we take the logical terms of L* to be the F*-terms and so count '.' as logical, some sentences which are not genuinely logical truths will nonetheless be F*-true, and so satisfy Tarski's conditions for logical truth. For example, any sentenceof L* of theform -(.(p( ())1 wouldbe declareda Tarskian logical truth, and this certainly seems wrong, since that sentence entails that Lincoln had a beard. From this we can conclude that '.' may not be treatedas a logical term. This result is not by itself worrisome,however, since, '.' is not a likely candidatefor a logical term anyway. The troublewith Etchemendy'sargument surfaces when we spell out its second half. There the claim is that, if we take the logical terms of L* to be the F-terms, some sentences which are gen- uinely logical truths of L* will not be F-true, and so will not satisfy Tarski'sconditions for logical truth.Etchemendy thinks that there must be such logical truths since there will be sentences of L* which are not F-true, but which are nonetheless logically equivalentto sentences of L which are themselves logical truthsof L. Evidently, then, Etchemendy'sargument relies crucially on the following assumption.

If p* in L* is logically equivalent to b in L and 4 is a logical truthof L, then * is a logical truthof L*. Or, as we might put it, sentences logically equivalentto logical truthsare themselves logical truths. But what notion of "logical equivalence" is being invoked here? ThoughEtchemendy does not spell out his argumentin detail, I will hazarda guess as to how the full reasoningis supposedto go. Let 'B' be short for 'Lincoln had a beard'. Let 9 be any sentence common toLandL*. 1a. We can derive from the semantic stipulationsof L* that r((p* --)- is true(in L*) iff Lincoln had a beardand either p is true (in L*) or o is not true (in L*).

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lb. We can likewise derive from the semantic stipulationsof L that -(BA(o V -y))' is true(in L) iff Lincolnhad a beard and either (S is true (in L*) or f is not true (in L*). 2. It follows easily that r---BV(p* -o) is true (in L*) iff r_-BV(BA(oV - )) is true(in L). So, these are"logically equivalent"sentences. 3. But --,BV(BA(pV -'))5 is certainlya logical truthof L. X. Sentences logically equivalentto logical truthsare them- selves logical truths. 5. From (2), (3), and (X), it follows thatr1-BV((p * -) is a logical truthof L*.

6. However, r--iBV(o * -)' is not an F-truthof L*. Thus, using F, Tarski'scondition fails to recognize at least one logical truthof L*. The troublewith this argumentis the notion of "logical equivalence" thatmust be invokedto make the argumentvalid. Howeverplausible principle(X) may seem, thereis good reason to thinkthat it is false, given the sense of "logically equivalent"being employed. In order for Etchemendyto make good on his argument,he needs something at least as strongas the following to hold.

[EQ] For sentence p in L' and 4 in L", if it follows from the satisfaction(or truth)conditions of L' and L" that p is true (in L') if and only if 4 is true (in L"), then ypand 4 are "logically equivalent".6 The of Etchemendy'sargument depends on some such con- dition being true. The argumentneeds to establish that a "logical- status-preserving"relation obtains between r--,BV(o * -y)- and the logical truth r1--BV(BA(oV -p))5. Etchemendycalls this relation "logical equivalence",but whatever we call it, it is a relation that must be determinedto hold between these two sentences in virtue of assumptionsthat have been made by the argumentup through step (2). (EQ) merely codifies this. (EQ) says, in essence, that two sentences that meet the currentassumptions on our two target sen- tences bear the specifiedrelation to each other,and (X) says thatthe

This content downloaded from 128.227.195.219 on Fri, 13 Dec 2013 12:06:07 PM All use subject to JSTOR Terms and Conditions 310 GREGRAY specified relation is logical-statuspreserving. It should be obvious that both these claims are essential to the argument.However, we can reduce the joint assumptionof (EQ) and (X) to absurdityas follows.7 1. Supposethat the languageL+ has the predicates,'is unmar- ried', and 'is a male', where these have their usual mean- ing. Our language will have one other predicate, 'is a bachelor'. And we will outfitthe language otherwisewith just the classical logical operators. 2. Suppose that 'is a bachelor'has the following satisfaction conditionsin L+. A thing satisfies 'x is a bachelor' iff that thing is both unmarriedand a male. 3. Applying (EQ), it will follow that'all bachelorsare unmar- ried' is "logically equivalent"to 'all unmarriedmales are unmarried'. 4. 'all unmarriedmales are unmarried'is a logical truth(in the targetpre-theoretic sense, as well as the Tarskianand model-theoreticsenses). 5. Applying principle (X), we conclude 'all bachelors are unmarried'is a logical truth. But, of course, 'all bachelors are unmarried'is not a logical truth (in the targetpre-theoretic sense, or in either the Tarskianor model- theoreticsenses), so I take thatfor a reductio. Put somewhat differently, the above argument reveals that Etchemendy'sbullet argument presupposes that necessary (or at least analytic) truths are logical truths.I cannot see how Etchemendy's argumentcould run without employing this false presupposition, so I conclude that Etchemendy's"bullet" argument against Tarski's accountis hopelessly flawed.

CONCLUSION

Thereis yet room for Etchemendyto object to this line of argument. He might claim for true, whathis argumentpresupposes, that neces- sary (or at least analytic) truths,like the one about bachelors, are logical truths.8I don't think that Etchemendycan go very far with

This content downloaded from 128.227.195.219 on Fri, 13 Dec 2013 12:06:07 PM All use subject to JSTOR Terms and Conditions LOGICALTERMS 311 this move, on pain of his completely missing the targetof Tarski's account. Tarski'starget notions are formal-logical notions, i.e. are "completely independentof the sense of the extra-logical terms." Necessary truthand analytic truthare not formal-logicalnotions in this sense. The necessary truthof the bachelorsentence is just not a formal matter. This is not to say that Etchemendy cannot object to Tarski's account on the grounds that it does not treat analytic truths as a (perhapsimproper) subclass of logical truths.However, that objec- tion requires no argumentto register and is not furtheredby the sort of argumentthat Etchemendy gives. Tarski explains early in his 1936 essay that the notions he is interestedin are formal-logical notions. Etchemendycan complain about this characterization,but neitherhis complaint, nor his case against Tarski,can be strength- ened by mounting an argumentagainst Tarski that simply applies this disagreement.In particular,the resultingargument could not be telling us anythingabout the existence of logical term functions,nor about whetherTarski succeeded in capturingthe notions he sought to capture,nor, for thatmatter, whether the model-theoreticaccount succeeds in capturingthose notions.9 Perhapsthe importantpoint for ourpresent discussion is to see that principle(X), understoodas it mustbe for Etchemendy'sargument, is question-beggingin the contextof the debate.Etchemendy wanted to show that it was in principlemistaken to think that logical truthwas somehow based on a distinction between logical and non-logical terms. Yet, we have seen that the principle, (X), which drives his argument,simply assumes a notion of "logical equivalence"which is not formal-logical, and likewise entails that logical truth is not formal-logical,either.

NOTES * This paperwas presentedat the Pacific Division meeting of the AmericanPhilo- sophical Association, San Francisco (March 1995), and also, as part of a longer talk, at the annual meeting of the Society for Exact , Austin (May 1994). My thanks to all who attendedthose sessions, especially my commenta- tors PatriciaBlanchette and Cory Juhl, as well as John Etchemendy,Christopher Menzel, and EdwardZalta. I have also benefittedfrom written comments from and/orhelpful discussions with Gary Curtis,Charles Chihara, Kirk Ludwig, John Peoples, StewartShapiro, , and Jan Wolenski.

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1 This is one of two general argumentsEtchemendy (1990) gives for his con- tentionthat basing logical truthon such a distinctionis in principle mistaken.The second is a sort of modal argumentgiven variouslyin Chapters8 & 9, and which I criticize elsewhere (Ray, forthcoming,?4). I also critically discuss some of the other complaintsbrought by Etchemendy. 2 Naturally,I am simply takingthe "logical form"of p to be somethingthat F(f ) may be said to represent.F(fo) retains the grammaticalstructure of S?,and all of its logical terms within that structure.F(fo) also represents(by the use of typed extra-linguisticvariables) the semantictype of each of the non-logical termsin its place within that grammaticalstructure. One might wonder why, if this F parameteris so important, it is always suppressedin modem presentationsof the Tarskianor model-theoreticcharac- terizationof the logical properties.This is because a particularchoice of function has been settled on (for first-orderlanguages, anyway) namely the one which treatsthe boolean connectives and quantifiersas logical terms. The identity sign has somethingof a double life - sometimes treatedas logical, sometimes as non- logical. 4 It would be 30 years before Tarskiwould returnto this issue, and offer a theory of logical terms (Tarski,1986). 5 It should also be clear thatrestricting K to avoid Etchemendy'scounterexample is not a live option,because the resultingrestricted account would be insufficiently general,being inapplicableeven to some first-orderlanguages. For simplicity, (EQ) is strongerthan strictly necessary.For example, it is only conditioned on the identity of the truthconditions of the target sentences, even thoughI have, in settingup the argument,specified enough to determinethe mean- ings of the sentences. My argumentwould work as well for any such weakening of the condition.In particular,my argumentwould still work if we worked with a variant,suggested by PatriciaBlanchette, which only pertainedto cases where S0 and / were translationsof each other. 7 Thus, my criticism of Etchemendy'sclaim will not rest on attributingto him the detailedargument which I hazardedabove on his behalf. Rather,my criticism will stand provided that Etchemendy'sargument commits him to (X) and (EQ). (Even this is somewhattoo general. Cf. note 6.) 8 Etchemendydoes hold this view. CharlesChihara has pointed out to me that a sentence stating the transitivityof spatial betweeness is said to be a logical truth in (Barwise & Etchemendy,1990). So, presumably,the same is to be said of our earlierdescendency sentence, as well as the bachelorsentence above. The view is also suggested in (Etchemendy,1983). 9 A presuppositionsimilar to that uncovered here for logical truth underlies the argumentfor at least one other importantcritical claim in Etchemendy's book, namely his claim that Tarskicommitted a modalfallacy when arguing that his account captures our pre-theoretic concept of logical consequence, and so obscured an essential weakness of the account. For discussion, see ?5 of (Ray, forthcoming).

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REFERENCES

Barwise, Jon, and Etchemendy, John. 1990. Language of First-Order . Stanford:CSLI. Etchemendy,John. 1983. "Doctrine of Logicas Form".Linguistics and Philoso- phy 6: 319-334. Etchemendy,John. 1988. "Models, and Logical Truth". Linguistics andPhilosophy 11: 91-106. Etchemendy,John. 1988. "Tarski on Truthand Logical Consequence". Journal of SymbolicLogic 53: 51-79. Etchemendy,John. 1990. The Conceptof Logical Consequence.Cambridge: HarvardUniversity Press. Ray,Greg. Forthcoming. "Logical Consequence: A Defenseof Tarski".Journal of PhilosophicalLogic. Tarski,Alfred, 1933. "Onthe Conceptof Truthin FormalizedLanguages" in Logic, Semantics,Metamathematics. 2nd ed. Trans.Woodger, J.H. Indianapolis: HackettPublishing, 1983, pp. 152-278. Tarski,Alfred. 1936. "On the Concept of LogicalConsequence" in Logic,Seman- tics, Metamathematics.2nd ed. Trans.Woodger, J.H. Indianapolis:Hackett Publishing,1983, pp. 409-420. Tarski,Alfred. 1944. "The Semantic Conception of Truthand the Foundations of Semantics".Philosophy and Phenomenological Research 4: 341-375. Tarski,Alfred. 1986. "WhatAre LogicalNotions?". History and Philosophyof Logic7: 145-154.

Departmentof Philosophy Universityof Florida Gainesville,FL 32611 USA

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