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HO W NO T TO DEFINE TRUTH-FUNCTIONA L ITY James A . MARTIN
V irt u a lly all modern logic books, and most philosophers, have as p a rt o f th e ir conceptual apparatus the notion o f t r u t h -tional compoundedness, i.e., o f a sentence's being truth-funcf u n c - compound. ( M y use o f 'sentence re f le ct s n o philosotionally phical vie w, o r prejudice; t h e reader is in vit e d t o substitute 'statement', 'proposition', 'judgement', o r whatever, throughout — w i t h a f e w cle a r exceptions.) L o g ic te xts, f o r example, typ ica lly ma ke a d istin ctio n between simp le a n d compound sentences; and among those that are compound a distinction is made between th o se w h i c h a re tru th -fu n ctio n a l a n d th o se which a re not. No r a re examples o f philosophical disputes in which the notion o f truth-functionality is a focal p o in t hard to find. A n y d isp u te co n ce rn in g t h e va rio u s p u t a t ive co u n te r examples t o t h e thesis o f exte nsionality (i.e., t h e v i e w t h a t every meaningful compound either is, or can be "analyzed into", a truth-functional compound) w i l l do. Such controversies in clude, f o r example, th o se fo cu sin g o n sentences in v o lv in g modalities, nomological and general sentences, and sentences about propositional attitudes (e.g., 's believes p', and 's is ce rtain that p l. Unfortunately, discussions o f the notion o f tru th functionality are often marred b y a la ck of cla rity and, indeed, of accuracy, wh ic h sca rce ly b e fits i t s apparent importance. What I should like to do in this paper is to indicate where the lack of cla rity (and of accuracy) lies, and to make some progress toward getting things right. I begin wit h some b rie f remarks o n compoundedness. Mo st logic books te ll us that a sentence is compound if and o n ly if it contains with in it another, different sentence. 'I t is false that p' is compound because it contains 'p', wh ich is a different sentence fro m 'I t is false that p'. 'p and cr, 'S believes that p', and 'I f (I) I hav e been helped a great deal in my t hink ing o n t he topic s o f this paper b y discussions w i t h Da v id Sanford.
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p had happened, then q wo u ld have happened' are other examples wh ich are compound f o r simila r reasons. So far, so good. But consider something o f the f o rm 'p o r p', e.g., 'Eith e r Susie has freckles o r Susie has freckles'. He re is a case that shows that i t makes a difference whether we couch o u r re ma rks o n logical th e o ry in terms o f sentences, statements, propositions, or whatever. Fo r cle a rly 'Susie has freckles' is a different sentence fro m the longer one of wh ich it is a disjunct. But it is b y no means obvious that it is (o r makes, o r expresses) a different statement or proposition. On the other hand, the sentence 'Susie and Anne were both there' makes o r expresses wh a t seems an obviously co mp o u n d statement o r proposition, b u t sin ce i t contains n o (g ra mma tica l) sentence d iffe re n t f ro m th e wh o le sentence it does n o t satisfy the requirement (as stated above) for being a compound sentence. Cla rity in the concept of compoundedness is p a rtly a function of the outcome of disputes over the relative merits of the above mentioned " lo g ica l units", as we ll as o f the decisions o r discoveries we make concerning such matters as the crite ria o f in dividuation, identity, and so on, t o be employed in connection with the various competing notions (sentence, proposition, etc.). We cannot solve the lit t le puzzle I 've raised u n t il we have a solution to some b ig ones. There are other problems with compoundedness. Consider the notion o f " co n ta in me n t" e mp lo ye d i n i t s d e fin itio n . O n a straightforward understanding of this word, the sentence 'I will play yo u t o mo rro w afternoon o n the f ie ld behind the lib ra ry at t wo o 'clo ck' contains (n o t co u n tin g itse lf) a t le a st seven different sentences: ' I w i l l , ' I w i l l play', ' I w i l l p la y you', and so on. I t does not, however, seem to be a compound. No r can we avail ourselves of an easy wa y out by simp ly defining 'compound' in te rms o f the truth-functional connectives, and 'simple' a s it s complement; th is wo u ld ma ke i t too transparently analytic that a ll compounds are truth-functional, wh e n in fact, this is a philosophically controversial doctrine. A p la u sib le suggestion, f o r wh ich I a m indebted t o Da vid Sanford, is that a sentence is compound if and only if it contains component sentences different f ro m i t wh ich can be replaced
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by a n y in te llig ib le sentence wit h o u t yie ld in g nonsense. Thus, 'I will' is not a component, in the required sense, o f 'I will p la y you to mo rro w' because replacing it wit h T w o plus t wo equals four' yields 'Two plus two equals four play you to mo rro w' a b it of nonsense. The in tu itive p la u sib ility o f this suggestion is enhanced b y an interesting parallel wit h truth-functionality. Just as i n a tru th -fu n ctio n a l compound, a component ca n b e re placed by any other sentence with the same truth value without altering the tru th value o f the compound, so in any compound a component ca n b e re p la ce d b y a n y in t e llig ib le sentence without decreasing the in t e llig ib ilit y of the compound. A truthfunctional compound is a mo re re stricte d kin d o f compound, and so here we e mp lo y a test in wh ich not o n ly in t e llig ib ilit y but t ru t h va lu e mu st be preserved. Some d ifficu ltie s re ma in : again, h o w sh a ll we handle cases lik e 'Susie and An n e we re both there' ? But such questions seem to spring from the deeper problems th e sentence-statement-proposition issue, a n d so d o not, as far as I can see, constitute a legitimate complaint against the suggestion mentioned. Assuming th a t we have an adequate grasp o f the notion o f compoundedness, wh a t sh a ll w e sa y about b e in g tru th -fu n ctio n a lly compound ? Again, ma n y lo g ic books sa y something like this: A sentence is truth-functionally compound ju st in case its t ru t h o r fa lsity depends on (is a function of, is determined by) th e tru th o r fa lsity o f the component sentences ( definition is defective, however, i n th a t i t can be taken in a t 2 ) . S t wo least u cways, h onea o f wh ich is quite wrong. First, i t ma y be taken i n such a wa y as t o re q u ire th a t the t ru t h va lu e o f a component o f a truth-functional sentence be a determinant o f the truth value of the compound. This, as I shall show, is a common b u t wro n g interpretation. Second, th e d e fin itio n ma y be read as asserting that o n ly the tru th value of a component of a truth-functional compound can be relevant in determining the truth value o f the compound. Th is wa y is right. The f irst ve rsio n is exceedingly co mmo n and is o fte n e m( bolic 2 Logic , 2nd edition, p. 10, o r BARKER'S The Elements o f Logic , p. 102. ) S e e , f o r e x
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ployed in discussions in vo lvin g sentences about propositional attitudes, e.g., b e lie f sentences. Wh e n we focus o n sentences like 'S believes p', we may be tempted to suggest that the truth value of the component must play some role in determining the truth value of the compound; such sentences seem to be something more or less than truth-functional because the truth value of p can be altered without altering the truth value of the whole sentence. Fo r example, Urmson, in h is Philosophical An a lysis says that 'if p is an element in the complex proposition then the truth or fa lsity of the complex proposition will be in part determined b y the truth o r fa lsity of p'. The problem wit h sentences like 'S believes p', he goes on to say in the same place, is that the tru th o f the b e lie f sentence i n no wa y depends on the truth of p' (p. 72). The same point is urged in both of the above cited Copi books (in the vicin it y of the definitions of 'truth-functional') and in numerous other places ( this 3 ve rsio n b y wh ich it s e mp lo yme n t ca n b e recognized, i s something ). A slike t a'beliefs, n d aafter r dall, can be true or false'. The point of s thlis pronouncement o g a n seems to be th a t since the tru th va lu e of sentence is independent o f the tru th value o f the o the belief f sentence believed, b e lie f sentences a re n o t truth-functional. This wa y o f construing the definition is wrong. I f belief sentences were truth-functional this wo u ld not entail that the truth values o f such sentences a re even p a rt ly determined b y th e truth values o f the believed sentences. Fo r ma n y tru th -fu n ctional sentences a re independent i n t h is w a y o f t h e ir co mponents. I do not mean just that in some circumstances the value of the component is irre le va n t to the value o f the compound. It is true that if q is false, the value of p is irre le va n t in determining the value of (p and 11), i.e., one ma y alter the truth value of p with o u t altering the value o f the compound. But there is a line in the tru th table for (p and q) where the value of p makes (3) Some prominent examples: J ohn PASSMORE'S A Hundred Years of Britis h Philosophy, p. 235; Ma x BLACK'S A Companion t o Wittgens tein's Trac tatus , p. 298; A lb e r t E. BLUMBERG'S a rt ic le ent it led «Modern Logic » i n v o l. 5 o f The Enc y c lopedia o f Philos ophy , P a u l Edwards , (ed.), p . 15; a n d G eorge PITCHER'S The Philos ophy of Wittgens tein, p. 149.
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all the difference. Ob vio u sly the interpretation taken b y Copi, Pitcher, Urmson, and others, ought to be understood in this way. Let us suppose that in saying that a sentence is not truth-functional i f the va lu e o f a component is irre le va n t t o th e va lu e of the compound, wh a t is meant is th a t the compound is n o t truth-functional if on no assignment of truth values to the components does the altering of the value of some component alter the value of the compound. But this is st ill wrong. Consider (p o r not-p). On no assignment of values to the components o f th is sentence does th e a lte rin g o f th e va lu e o f p alter th e va lu e o f the compound. I f we ma ke o u r crite rio n a matter of the alteration of the value of the compound b y altering the value o f the component, the consequences w i l l include the absurd result that there are no truth-functional tautologies (o r self-contradictions). There does seem to be a sense in which the value o f p determines the value o f (p o r not-p), i n wh ich we want to say that the truth of this compound is somehow p a rtly constituted by, o r a function of, the value o f p. I wo n 't bother here t o explicate th a t sense; t h e need can be skirte d wit h a different example. Consider the compound ((p o r not-p) o r q). Here, the value of q is completely irre le va n t to the value of the compound. A lt e rin g it makes no difference; n o r does the tru th of the wh o le depend, as fa r as I can see, in a n y wa y upon it : it is not "constructed" fro m it. It is possible, o f course, th a t one mig h t be able to extricate the problematic interpretation f ro m th is predicament b y constructing mo re complicated definitions o f 'truth-functional i n which tautologies and self-contradictions are treated differently from o th e r so rts o f sentences. Sin ce th e sp o ilin g examples I have thus far employed are all lo g ica lly necessary propositions, it might seem that a definition in which such propositions me rit a special clause could succeed at least in the min imu m task o f putting th e d istin ctio n in question we re we wa n t it . B u t th is appearance is misleading; such a n approach wo u ld be ra th e r more complicated than m y examples thus f a r wo u ld suggest, for i t is e a sy t o f in d examples o f truth-functional, lo g ic a lly contingent propositions which contain " irre le va n t" components, e.g., ((p o r not-p) and q), and ((p and not-p) o r q). Ne ith e r corn-
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pound is lo g ica lly true, o r false; ye t in neither does th e tru th value o f p p la y a role. Thus, th e relevance o f a component's truth value to the tru th value o f the compound is n o t a necessary condition of truth-functionality. It is wo rth noting in vie w of the common definitions of truthfunctionality, that it is n o t a sufficient condition either. To see this, consider such sentences as (A) ' S kn o ws th a t p' (B) 'S is under the mistaken impression that r'. These d iffe r f ro m the standard b e lie f examples i n p re cise ly the a lle g e d ly cru cia l respect: ( i f t h e y a re t ru e ) w e ca n n o t change the tru th values o f p and r with o u t altering the values of (A) and (B). I f we substitute any false sentence f o r (true) p, (A) is thereby falsified; the same remarks hold mutatis mutandis for (B). Nevertheless neither sentence is truth-functional. Again, i t is possible th a t some v e ry complicated p rin cip le will wo rk. But a fa r easier wa y o f dealing wit h the problem is available. W e mu st abandon th e f irst in te rp re ta tio n and re ly on the second, viz., that if a component does p la y a ro le in determining the tru th value of a compound, it is o n ly in virtu e of its tru th value, i.e., i t is in virtu e of o n ly its tru th value. Some standard remarks about truth-functional components reflect just this p o in t. Some examples: I n such compounds a component sentence can be replaced b y any other sentence wit h the same truth value with o u t altering the value of the compound, o r: We can t h in k o f the components a s t ru t h values. Th e imp o rta n t point wh ich is captured in such remarks is that nothing other than the t ru t h va lu e o f a component is re le va n t to th e t ru t h or f a lsit y o f a truth-functional compound. Let us p u t it in terms o f sentences like 'S believes p'. I t isn't the irrelevance of p's truth value to the value of the compound that makes th e b e lie f sentence non-truth-functional, i t is th e relevance of something besides truth value. I t is not that beliefs can be false as we ll as true, but rather that we cannot substitute for the believed sentence another sentence wit h identical tru th value, and be certain that the truth value of the compound will
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remain as it was. S imila rly, the cru cia l point about (A) and (B), the fa ct in virt u e o f wh ich neither is truth-functional, is th a t the tru th value o f neither compound is necessarily preserved through substitution of like tru th values. I f 'S knows p' is true, p is true; and although, as noted earlier, we can falsify the compound b y substituting f o r p a n y fa lse sentence, w e ca n n o t substitute f o r p ju st a n y tru e sentence, n o r f o r r in the (true) sentence 'S is under the mistaken impression that r' ju st a n y false one. A person wh o kn o ws one th in g doesn't necessarily know everything, and the ma n wit h one mistaken belief need not have them all. The reason wh y (A) and (B), and more o rd in a ry belief sentences, are non-truth-functional is that something about the components o th e r than t h e ir t ru t h va lu e is re le va n t t o th e t ru t h value o f the compound; wh a t th e y say is relevant. (Indeed, in some cases one might wa n t to say that h o w wh a t is said gains expression ca n be important. I f ' w but l S doesn't kn o w this, i t wo u ld be misleading, o r even false, to ' say a nthat d S knows ' w that x is w it is wo, and when he i2 w h e n w h wo a u ld t deny, o r refuse to assent to, ' x is WI h' a e r e w s To y osunm uup: o nIlt isy dcommmo o n ly u thought that wh a t I have called the ss ,'relevance' a yo f a component's tru th value to the value o f a compound i sis both a necessary and a sufficient condition o f the compound's being On this vie w, a component t h a truth-functional. t occurs i n a compound tru th -fu n ctio n a lly ju st in case it s tru th value so me h o w p a rt ly constitutes, o r p a rt ly determines t h e truth value o f the compound. I have shown that th is provides neither a necessary n o r a su fficie n t co n d itio n o f tru th -fu n ctionality. Wh a t i s necessary i s t h a t n o th in g o th e r th a n t h e truth value of a component can be relevant to the tru th o r falsity o f the compound. Perhaps the best wa y to p u t it is this: I f a compound contains a component tru th -fu n ctio n a lly then, i f anything about that component is relevant to the truth value of the compound, it is its tru th value. Dartmouth College
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