• • ,;;..'. , "l t f ! 1 , SONDERDRUCK aus LANGUAGE AND ONTOLOGY PROCEEDINGS OF mE 6th INTERNATIONAL WfTTGENSTEIN SYMPOSIUM 23 rd TO 30th AUGUST 1981, KIRCHBERG/WECHSEl (AUSTRIA) SPRACHE UND ONTOLOGIE AKTEN DES 6. INTERNATIONAlEN WITIGENSTEIN SYMPOSIUMS 23. BIS 30. AUGUST 1981, KIRCHBERG/WECHSEl (OSTERREICHI , ,1: ~ -l:- .. , ' ,:,./ ' - . .!~ WIEN 1982 HOLDER*PICHLER* TEMPSKY SOME FORMAL MOMENTS OF TRUTH" B. Smith Correspondence theories of truth have. since the time of the Traclafu5 and of The Philo sophy 0/ Logical Atomism. fallen out of favour amongst logicall y minded philosophers. Theorists of truth have indeed normally avoided the issue of correspondence by averting their attent ions from the basic truth-relation be tween individual sentence and world, and concentrating instead upon logically structured sets of sen tences and artificially constructed set-theoretical models. In the present note I wish to give a brief account of some more or less obvious formal characteristics of this almost forgotten basic truth-relat ion. I shall then attem pt to show how this account may be extended to provide elements of a theory of truth wh ich is in keeping with the spi rit of the Tractatus. One principal reason why the correspondence th eory has lost so much of it s former att raction rests on the fact that philosophers have, in the last few decades , extended their interests beyond the narrow corpus of sentences for which the question of correspondence is most obviously appropriate, i.e. simple declarative sentences about the spatio-temporal world . They have sought instead to construct theories of truth which take account of the characteristic properties of sentences of other types, for example of mathematical or fictional sentences, of counterfactuals, or of sentences involving intentional or modal operators. Such experiments are of considerable interest, and they demonstrate the power and scope of set-theoretical semantics. Here however, in keeping with our more modest task, we shall restrict ourselves entirely to descriptive sentences like' John has a headache' or 'atom a (at some specific instant J strikes atom b', which are used to make assertions about objects or object-configurations in the real material world. Simons has argued that most, if not all such sentences are made true by entities which, following Husser!, he calls 'moments'. 1 Wittgenstein and Russell on the other hand saw the role of truth maker as being filled by specific kinds of complexes-states of affairs, or fa cts-complexes which may involve moments, but which manifest an essentially different ontological structure. The ideas sketched in the present note are in fact consistent with either approach and with a range of possible variants. We shall need to assume only that the relations which hold amongst truth-makers (be they moments, states of affairs, or objects of other kinds) are ontological rather than logical; that, in other words, such relations are radically distinct from the kinds of relations which hold amongst sentences, propositions, or other candidate bearers of truth. This assumption reflects a principle of the heterogeneity of logic and ontology, which has been formulated in order to forestall any too ready imputation of logical structure to the objects of the material world . 1 The ontological relation which is most important for our present purposes is the relation of (proper or improper) part to whole. Thus it seems clear that if a makes p true, then every b which includes a as a part will also make p true. In symbols; (1) aFp-+'Vb(ãb-+bFp).J The relation ~ obtains only between objects (names for which occur to the left of the Fconnective). It has no analogue amongst sentences or propositionsin complete accordance with the principle of the heterogeneity of logic and ontology. We can now go on to affirm that (2) a F p . ..... p. 186 But the (3) p ..... 3a . a 1= P. also true? Consider. say, the sentence: 'Jack and Jill went up the hill ILe. not n .. c',.",,,, together)', or, alternatively: 'There were three eclipses of the moon in Erna's lifetime'. It would surely be wrong to assume that there are any single composite objects, events, or states of affairs-a Jack's running up the hill fused. mereologically. with a Jill's running up the hill, or a three-fold eclipse-fusion-which would make these sentences true. Rather we should accept, quite naively. that the given sentences are made true by a relevant manifold or plurality of truth-makers. Such a manifold is not a new, conjunctive entity. (It is not, for example, a set.) There are no conjunctive entities, any more than there are disjunctive, negative or implicative entities. A manifold is, rather, nothing more than the objects it comprehends; a manifold comprehending a single object is therefore simply that object itself. 4 Manifold truth-makers may be represented by means of non-empty lists, 'a, b, C, .•. ,k', of names of individual truth-makers. 'r', 'A', etc., will be used to stand in for lists of this kind. 'a E r' will signify that r comprehends or includes a or, in other words, that a name for a appears as an item in the list' r'. We define the relation £' of mereologica1 inclusion between manifolds as follows: DI. r ~ A: = Va E r . 3 bE A . a or;; b. As possible axioms for the 1= -relation we might now propose: AI. r 1= p. -+ p. A2. p 3 r . r 1= p. A3. r 1= p. -+ V A (r ~ A -+ A 1= p), (a generalisation of (1) above). A3. implies in particular a rule of thinning: (4) r 1= p. V A . r, I:. 1= p. A4. (r t= p./\ I:. 1= q) -+ r, I:. =p/\ q. AS. (r 1= p./\p-+ q)-+ 31:. (Ãr./\ A 1= q). By A3. and AS. we have: (5) (r 1= p. /\ p"" q)"" r 1= q, whence, in particular: (6) rl= p.'" r 1= pVq, from which we can infer: (7) r 1= p. V r 1= q. -+ r 1= p V q, 187 the converse of which we affirm as an axiom: A6. r Fe p q. ~ r p. r Fe q. By (5) we have: (8) r t= p t\ q. -+ r t= p. t\ r t= q. and by A4. we have also: (9) r t= p. t\ r t= q. r 1= p t\ q. We can affirm as axioms: A7. rI=3a.p:-3a.rl=p; A8. r 1= 'Va. p : - 'Va. r 1= p. Logically compound sentences involving negation raise more serious problems (as may be expected within the framework of a theory constructed on the belief that there are no entities to which negated sentences or sentence-parts may correspond). S Wittgenstein solves this most fundamental of all problems facing a correspondence theory of truth by embracing two distinct types of correspondence. We have first of all the relation of direct depiction between Elementarsl1tze (all of which are positive) and states of affairs; and secondly a higher-level truth relation between sentences and facts. The classes of sentences and Elementarsl1tze, like the classes of facts and states of affairs, are mutually exclusive. Sentences are obtained from Elementarsl1tze by successive applications of the logical functions it is true that and it is false that. Facts are obtained from states of affairs by successive applications of the ontological functions the existence of and the non-existence of A fact is, that is to say, the existence or non-existence of states of affairs (d. TLP 2, 2.06, 2.062, 2.11). {; It would take us too far afield to provide a detailed account of Wittgenstein's theory here. We shall content ourselves instead with an approximation to the theory which can be constructed within the framework developed above. We define, first of all, a counterpart of Wittgenstein's relation of direct depiction, which we shall signify by 1=. D2. a 1= p : = (0 a t= p. t\ (ii) 'V q (a t= q . ..... . p .... q) . t\ (iii) 'V r (r t= p . ..... a ~ r). Clause (0 asserts that direct depiction (being made true elementarily) is merely one special case of being made true. Clause (ti) is intended to capture the immediacy of the relation 1= between the sentence p and object-configuration a. It teils us that a proposition made true elementarily involves no logical redundancy or roundaboutness (thus no disjunctive sentence, for example, is elementarily made true). Clause (iii) signifies that the atomic object or state of affairs a which elementarily makes p true is a unique common factor of all truth-makers for p.7 Note that 1= is defined only in relation to atomic or individual truthmakers: no sentence is elementarily made true by a manifold. From D2. and the axioms above we may infer: (10) a t= p. t\ a t= q . ... p q. 188 JUS (as may be that there are no negative md).5 Wittgenstein solves lfY of truth by embracing :ation of direct depiction affairs; and secondly a sentences and Elementarexclusive. Sentences are 19ical functions it is true by successive applications 'of A fact is, that is to 'i, 2,062, 2.11).6 ,Vittgenstein's theory here. theory which can be conof all, a counterpart of rily) is merely one special immediacy of the relation that a proposition made .ness (thUS no disjunctive fies that the atomic object ,ue common factor of all 'tomie or individual truth- (We (11) a p ;\ q. P .... q, Consider the case where a is the mereological sum of two discrete individuals band c, such that b '" p and c '" q. ) (12) a'F p. ;\ b ~ p. Q = b. (13) ãp;\ bFp. -+ ã b. The following now suggests itself as a possible definition of the concept elementary proposition: D3. elem (P) : = p -+ 3 a . Q t= p, whence by clause (iii) in D2., (15) pt\elem(p).-31a.ãp, --every true elementary proposition has a unique (minimal) truth-maker. In a fully developed system we should wish to recognise also theorems like: (16) ã p. - "" E (a) --elementary truth-makers exist only contingently. (17) Q t= p. - "" p, a theorem whieh has no analogue for 1=. (Consider, for example, the sentence 'It is raining or it is not raining'. It is not possible that this sentence be false, yet it is made true-though not, of course, elementarily-by the relevant contingently existing condition of the weather.) And: (18) p t\ ""p. 3a. 3q (a 1= q. t\ p'" q). But there are more radical extensions of the basic theory which suggest themselves. We might, for example, wish to consider the formal moments of the relations: is possibly true in virtue oj, is necessarily true in virtue oj, is made empirically probable by. and so on. And we might wish to consider the interconnections between formal correspondence theory and the characteristic issues of epistemology. These questions must however be postponed until another place. University of Manchester Department of Philosophy Manchester M13 9PL England 189 NOTES .... ho contributed for n'Ul>IJ'CIl"dL'.e I Simons, as Makers" (in this volume), 2 I have defended this principle elsewhere: see B, Smith, "An Essay in Formal Ontology", GraZl'r Philo, ,~or,hische Studil'n, Vol, 6 (1978), pp, 39--62 and B, Smith, "Logic, Form and Matter", Procel'dings of rislotl','ian Society, Supplementary Vol, 55 (1981), pp, 47--62, J -, here, is at least as strong as strict implication, Our theory of part*whole relations is a "''''''''"'','U of the mereology of Lesniewski; see Smith and Mulligan, "Pieces of a Theory", in: B, ParIs and Moments: Studies in and Formal Ontology (Munich (982) and Smith and "Framework for Formal Issue on Lesniewski, forthcoming,) 4 See p, M, Simons, "Number B, Smith (1982), 5 A sentence like There is no God' (if it is perhaps made true byeverrthing, of the to which no fact is a state of affairs, is defended R,*A, Dietrich, lind In Witrgen5teins Tracta/us (Tubingen 1974). 7 Clause (iii) perhaps too mong, for certain kinds of sentences we may wish to allow for a of partially coincident elementary truth makers. Consider, for example, the sentence 'This tone C', which may be seen as made true elementarily not only by the tone as a whole, but also by the specific pitch from out of with a specific individual timbre and loudness) the tone is lUted: See Smith and of a Theory". and also Smith, Mulligan and Simons. "Truth* Makers". Philosophy and Research (forthcoming), ... ... ...