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On the Possibility of a Privileged Class of Logical Terms

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On the Possibility of a Privileged Class of Logical Terms 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 consequence, 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. 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 304 GREGRAY (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 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 305 (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
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