Quine, W. v. Carnap and Logical Truth (2)

Carnap and Logical Truth Author(s): W. V. Quine Source: Synthese, Vol. 12, No. 4 (Dec., 1960), pp. 350-374 Published by: Springer Stable URL: http://www.jstor.org/stable/20114356 Accessed: 10/06/2010 08:18 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=springer. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission. JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Springer is collaborating with JSTOR to digitize, preserve and extend access to Synthese. http://www.jstor.org W. V. QUINE CARNAP AND LOGICAL TRUTH1) I Kant's 'How are synthetic judgments a priori possible?' pre question the Critique of Pure Reason. Question and answer notwith cipitated standing, Mill and others persisted in doubting that such judgments were possible at all. At length some of Kant's own clearest purported instances, drawn from arithmetic, were sweepingly disqualified (or so it seemed; but see ? II) by Frege's reduction of arithmetic to logic. Attention was thus and indeed logically prior question, upon the less tendentious 'How is logical certainty possible?' It was largely this latter question that the form of empiricism which we associate with between-war precipitated which began with Wittgenstein's Vienna - a movement Tractatus and reached its maturity in the work of Carnap. forced Mill's position on the second question had been that logic and math ematics were based on empirical generalizations, despite their superficial to the contrary. This doctrine may well have been felt to do appearance sciences less than justice to the palpable surface differences between the deductive on the one hand, and the empirical of logic and mathematics, sciences ordinarily so-called on the other. Worse, the doctrine derogated from the certainty of logic and mathematics Mill may not have been ;but one to be excessively disturbed by such a consequence. Perhaps classical mathematics infinitistic to experience then than now; at any rate the of set theory, which are so fraught with speculation and so remote from any possible experience, were unexplored in his day. did lie closer reaches it is against just these latter-day mathematical extravagances that And !) This paper was written early in 1954 at the request of Professor Schilpp, for inclusion in a volume in Italian appeared vol. 48 (1957), pp. have appeared also Books, New York, on Carnap which he had been planning. The paper has since as 'Carnap e la verit? translation Rivista di Filosof?a, l?gica', 3-29. to somewhat Selected less than half, portions, running in American at Work Philosophers (Sidney Hook, ed.), Criterion 1956. 350 CARNAP AND LOGICAL TRUTH empiricists outside the Vienna Circle have since been known to inveigh,1) in much the spirit in which the empiricists of Vienna and elsewhere have inveighed against metaphysics. What now of the empiricist who would grant certainty to logic, and to the and yet would make a clean sweep of other non whole of mathematics, theories under the name of metaphysics? The Viennese solution empirical of this nice problem was misuse meaningless through tautologous use of language. As an answer to the question on predicated of language; was language. Metaphysics was certain through logic 'How is logical certainty possible?' this doctrine of logical truth has its attractions. For there can be no linguistic doubt that sheer verbal usage is in general a major determinant of truth. Even so factual a sentence as 'Brutus killed Caesar' owes its truth not only to the killing but equally to our using the component words as we do. Why then should a logically true sentence on the same topic, e.g. 'Brutus killed Caesar or did not kill Caesar', not be said to owe its truth purely to the fact that we use our words (in this case 'or' and 'not') as we do? for it depends not at all for its truth upon the killing. The suggestion is not, of course, that the logically true sentence is a contingent truth about verbal usage ;but rather that it is a sentence which, 'Brutus killed becomes true, whereas given the language, automatically on the alleged Caeser', given the language, becomes true only contingently killing. Further accrues to the linguistic doctrine of logical truth plausibility reflect on the question of alternative logics. Suppose someone forward and uses a consistent the principles of which are logic when we puts contrary to our own. We are then clearly free to say that he is merely in other than the 'and', 'all', or whatever, using the familiar particles familiar senses, and hence that no real contrariety is present after all. There may of course still be an important failure of intertranslatability, of certain of our logical particles is incapable of being in his system or vice versa. If the translation duplicated by paraphrases in this sense is possible, from his system into ours, then we are pretty sure to protest that he was wantonly using the familiar particles 'and' and in that the behavior A) An Scripta example is P. W. vol. Bridgman, 2, 1933-4, 4A physicist's pp. 101-117, second 224-234. reaction to Mengenlehre,' Mathematica, 351 W. V. QUINE 'all' (say) where me might unmisleadingly have used such and such other familiar phrasing. This reflection goes to support the view that the truths of logic have no content over and above the meanings they confer on the logical vocabulary. Much the same point can be brought out by a caricature of a doctrine of according to which there are pre-logical peoples who accept Levy-Bruhl, as true. Over-simplifying, no doubt, certain simple self-contradictions it claimed that these natives accept as true a certain let us suppose sentence of the form 'p and not p'. Or - not to over-simplify too much that they accept as true a certain heathen sentence of the form 'q ka bu q' the English translation of which has the form 'p and not p'. But now is this, and what may the lexicographer's just how good a translation method have been? adoption natives' acceptance are left with the meaninglessness of 'and' and If any evidence can count against a lexicographer's 'not' as translations of 'ka' and 'bu', certainly the of 'q ka bu q' as true counts overwhelmingly. We of the doctrine of there being pre is a trait injected by bad translators. This logical peoples; prelogicality is one more illustration of the inseparability of the truths of logic from the meanings of the logical vocabulary. We thus see that there is something to be said for the naturalness of the linguistic doctrine of logical truth. But before we can get much further we shall have to become more explicit concerning our subject matter. II Without either the linguistic doctrine, thought of any epistemological or another, we may mark out the intended scope of the term 'logical truth', within that of the broader term 'truth', in the following if not otherwise, what way. First we suppose indicated, by enumeration doctrine are to be called 'and', 'all', 'every', words 'then', logical words; 'only', 'some'. typical The ones logical are 'or', 'not', truths, then, 'if, are those true sentences which this means logical did not kill Caesar'), 1 ) Substantially 'Bolzano's essentially. What is that any other words, though they may also occur in a truth (as witness 'Brutus', 'kill', and 'Caesar' in 'Brutus killed or involve only logical words can be varied at will without is traced of back a century propositions,' engendering and a quarter Methodos, falsity.1) by Yehoshua vol. 2, 1950, this formulation definition Bar-Hillel, analytic 352 CARNAP AND LOGICAL TRUTH to language, the above clarification truths owe their truth to language. we have thus far is only a delimitation What of the class, per accidens if the linguistic doctrine of logical truth, which is an you please. Afterward formulated with reference Though does not of itself hint that logical doctrine, goes on to say that logical truths are true by epistemological or intended usage, of the logical virtue purely of the intended meanings, words. Obviously if logical truths are true by virtue purely of language, the logical words are the only part of the language that can be concerned in the matter; for these are the only ones that occur essentially. systematized nowadays, Elementary logic, as commonly comprises truth-function theory, and identity theory. The theory, quantification for this part, as commonly rendered for technical logical vocabulary to 'or', 'and', purposes, consists of truth-function signs (corresponding 'not', etc.), quantifiers and their variables, and *='. The further part of logic is set theory, which requires there to be classes The one sign needed among the values of its variables of quantification. to elementary in set theory, beyond those appropriate logic, is the Additional 'e' of membership. connective signs, though commonly used can be eliminated in well-known for convenience, ways. or logical syntax, out of account. I leave metatheory, In this dichotomy For, either it treats of special objects of an extralogical kind, viz. notation or else, if these are made to give way to numbers by al expressions, it is reducible via number theory to set theory. arithmetization, I will not here review the important contrasts between elementary logic and set theory, except for the following one. Every truth of elementary logic is obvious (whatever this really means), or can be made so by some state in its present series of individually obvious steps. Set theory, I am not alluding here to G?del's anyway, is otherwise. incompleteness principle, but to something right on the surface. Set theory was straining But note that the formulation vol. 16, 1950, pp. 91-117). fails pp. 32-55 (= Theoria, to provide of its purpose is understood 'can be varied at will,' above, unless the phrase not only singly but also two or more at at time. E.g., the sentence for varying the words can be turned are angels' some animals are angels into a falsehood 'If some men by simultaneous substitution for 'angels' alone, nor For this observation some indebtedness 1955; thus one year but not by any for substitution for 'men' and 'angels', the non-existence of angels). (granted 'men', nor for 'animals' I am indebted who expresses to John R. Myhill, and illustration - I of this footnote added most in turn to Benson Mates. in May, after the rest of the essay left my hands. 353 W. V. QUINE at the leash of intuition and with the added ever since Cantor snapped. so far as is known, impetus of the paradoxes set theory has now Comparative no consistent the higher infinites; of set theory the leash was long been the trend; for, set theory is both adequate to the discovered purposes envisaged for set theory and capable of substantiation by steps true principles. What we do is of obvious from obviously reasoning develop one or another set theory by obvious reasoning, or elementary first principles which are set down, whether for logic, from unobvious good or for the time being, by something very like convention. the contrasts between elementary logic and set theory are Altogether, so fundamental that one might well limit the word 'logic' to the former (though I shall not), and speak of set theory as mathematics exclusive of logic. To adopt this course is merely to deprive status in a sense 'e' of the of arithmetic would of a logical word. Frege's derivation then cease to count as a derivation from logic; for he used set theory. At any rate we should be prepared to find that the linguistic doctrine of logic and fails for set theory, or logical truth holds for elementary versa. Kant's readiness to see logic as analytic and arithmetic as is not superseded by Frege's work (as Frege thetic, in particular, if 'logic' be taken as elementary logic. And for Kant posed x)) certainly did not include set theory. vice syn sup logic Ill Where happens sentences, someone that we which disagrees with can convince he does us as to the truth of a sentence, it often him by getting the sentence from other accept, by a series of steps each of which he which cannot be thus resolved I shall call deductively accepts. Disagreement if we try to warp the linguistic doctrine of logical truth irresoluble. Now a first like an experimental around into something thesis, perhaps irresoluble disagreement as to a will run thus: Deductively approximation logical truth is evidence of deviation in usage (or meanings) of words. This since one term of the affirmed rela is not yet experimentally phrased, is in dire need of an independent tionship, viz. 'usage' (or 'meanings'), *) See ?? 87f., 109 of Gottlob Frege, Foundations of Arithmetic (New York: Philosoph ical Library, (Breslau, and Oxford: 1884) with 1950), a reprint Blackwell, translation by J. L. Austin. of Grundlagen der Arithmetik 354 CARNAP AND LOGICAL TRUTH seem to be fair enough would the formulation However, its limits; so let us go ahead with it, not seeking more sublety until need arises. or potential of elementary obviousness the obviousness logic Already criterion. within can be seen to present experimental meaning irresoluble truth. Deductively would count as evidence simply because get. insuperable to the linguistic dissent of deviation an obstacle doctrine from an elementary logical truth over meanings if anything can, but truism is as extreme as dissent can to our assigning any of elementary logical dissent from a logical The philosopher, like the beginner in algebra, works in danger of finding reduces to '0 = 0'. Such is the threat to the that his solution-in-progress linguistic theory of elementary logical truth. For, that theory now seems to imply nothing that is not already implied by the fact that elementary logic is obvious or can be resolved into obvious steps. which were adduced in ? I, to show the naturalness The considerations of the linguistic doctrine, are likewise seen to be empty when scrutinized that in the present spirit. One was the circumstance are inseparable practically from mere change in usage Another was that illogical cultures are indistinguishable are adequately ones. But both of these circumstances mere of logics of logical words. from ill-translated alternative accounted for by without of a linguistic obviousness help logical principles, doctrine of logical truth. For, there can be no stronger evidence of a change in usage than the repudiation of what had been obvious, and no earnest of bad translation than that it translates stronger evidence into obvious falsehoods. affirmations Another point in ? I was that true sentences generally depend for their to the traits of their truth on the traits of their language in addition and that logical truths then fit neatly in as the limiting subject matter; case where the dependence on traits of the subject matter is nil Consider, or '(*) (x = x)'. the logical truth 'Everything is self-identical', however, on traits of the language (specif We can say that it depends for its truth '= ically on the usage of '), and not on traits of its subject matter ;but we can also say, alternatively, that it depends on an obvious trait, viz. self identity, present I have been using its subject matter, viz. everything. reflections is that there is no difference. of the vaguely psychological word The tendency of our 'obvious' non-technic 355 W. V. QUINE ally, assigning it no explanatory value. My suggestion is merely that the linguistic doctrine of elementary logical truth likewise leaves explanation I do not suggest that the linguistic doctrine is false and some unbegun. doctrine reality Turning doctrine of ultimate and inexplicable traits of insight into the obvious is true, but only that there is no real difference between these two pseudo-doctrines. away now from elementary logic, let us see how the linguistic of logical truth fares in application to set theory. As noted in ? II, we may think of 'e' as the one sign for set theory in addition to those of elementary the version of the linguistic doctrine logic. Accordingly which was italicized at the beginning of the present section becomes, in to set theory, this: Among persons already in agreement on as to a truth of set irresoluble disagreement elementary logic, deductively is evidence of deviation in usage (or meaning) of's'. theory This thesis is not tiivial in quite the way in which the parallel thesis for elementary logic was seen to be. It is not indeed experimentally significant application as it stands, simply because of criterion for usage or meaning. But the lack, noted earlier, of a separate it does seem reasonable, by the Any reasoning. evidence of usage or meaning of words must reside acceptable in the observable circumstances under which the words are surely either uttered (in the case of concrete terms referring to observable individuals) or in the affirmation and denial of sentences in which the words occur. following is relevant to 'e'. Therefore any evidence of Only the second alternative on of 'e' must reside in disagreement deviation in usage or meaning sentences containing 'e'. This is not, of course, to say of every sentence over it establishes deviation 'e' that disagreement in usage or containing of 'e\ We have to assume in the first place that the speaker meaning under investigation agrees with us on the meanings of words other than 'e' in the sentences in question. And itmight well be that, even from among the sentences containing only V and words on whose meanings he agrees with that he us, there is only a select species S which is so fundamental cannot dissent from them without in his usage deviation betraying or meaning of 'e'. But S may be expected surely to include some (if not all) of the sentences which contain nothing but 'e' and the elementary logical particles; for it is these sentences, insofar as true, that constitute (pure, or unapplied) set theory. But it is difficult to conceive of how to be 356 CARNAP AND LOGICAL TRUTH toward the truths of set theory. In exposition we other than democratic may select some of these truths as so-called postulates and deduce others from them, but this is subjective discrimination, variable at will, ex and not set-theoretic. We do not change our meaning of 'e' pository the page where we show that one particular truth is deducible by elementary logic from another and the page where we show the con verse. Given this democratic outlook, finally, the law of sufficient reason between leads us to look upon S as including all the sentences which contain only 's' and the elementary logical particles. It then follows that anyone in on set on elementary agreement logic and in irresoluble disagreement theory is in deviation was the thesis. The with respect to the usage or meaning of's' ;and this effect of our effort logical nothing applied tion of the otherwise to inject content into the linguistic doctrine of truth has been, up to now, to suggest that the doctrine says worth saying about elementary logical truth, but that when to set-theoretic truth itmakes for a reasonable partial condensa vaporous notion of meaning as applied to 'e'. IV The linguistic doctrine of logical truth is sometimes expressed by saying if this be so, that such truths are true by linguistic convention. Now certainly the conventions are not in general explicit. Relatively few persons, that engendered the time of Carnap, had ever seen any convention truths of elementary logic. Nor can this circumstance be ascribed merely to the slipshod ways of our predecessors. For it is impossible in principle, in an ideal state, to get even the most elementary part of logic of conventions stated in advance. by the explicit application before even exclusively The difficulty is the vicious regress, familiar from Lewis Carroll,1) which I have elaborated elsewhere.2) Briefly the point is that the logical truths, being infinite in number, must be given by general conventions rather than singly; and logic is needed then to begin with, in the meta to individual cases. theory, in order to apply the general conventions !) What the tortoise said to Achilles,' Mind, vol. 4,1895, pp. 278ff. 2) 'Truth by convention,' in O. H. Lee (ed.), Philosophical Essays for A. N. Whitehead (New York, 1936), pp. 90-124. Reprinted inH. Feigl andW. Sellars (eds.), Readings in Philosophical Analysis (New York: Appleton, 1949). 357 W. V. QUINE from the and explicitness the attributes of deliberateness 'In dropping in the afore I went on to complain notion of linguistic convention,' force mentioned paper, 'we risk depriving the latter of any explanatory seem that to call elementary and reducing it to an idle label.' It would is to add nothing but a metaphor logic true by convention doctrine of logical truth which, as applied to elementary come to seem rather an empty figure (cf. ? III). The to the linguistic logic, has itself is different on both counts. For set case of set theory, however, theory the linguistic doctrine has seemed less empty (cf. ? III); in set in quite the ordinary sense seems to be convention theory, moreover, has a serious claim pretty much what goes on (cf. ? II). Conventionalism if only because of set of mathematics, in the philosophy to attention was encouraged in the though, conventionalism theory. Historically, rather by the non-Euclidean of mathematics geometries and philosophy abstract algebras, sequent purposes is deferred to ? V. about to sub little good reason. We can contribute this situation. Further talk of set theory by surveying with of truths In the beginning there was Euclidean geometry, a compendium form and void; and its truths were not based on convention might, begging the present question, apply (except as a conventionalist Its truths were in practice presented to everything mathematical). this tag I shall not (including axioms; postulates out of and the selection of truths for this role of postulate, of of truths of Euclidean geometry, was indeed a matter from so-called by deduction distinguish); the totality The truths were there, But this is not truth by convention. convention. and what was conventional was merely the separation of them into those at hand) to be taken as starting point (for purposes of the exposition and those to be deduced from them. The non-Euclidean without geometries came of artificial deviations from Euclid's thought (to begin with) of true interpretation. These postulates, were a for Euclid's postulates departures were doubly conventional; selection from among the truths of geometry, and then the conventional devised in turn. But still were arbitrarily or conventionally departures because there was no truth. there was no truth by convention, a non-Euclidean geometry, one might conveniently make Playing within and true; but even such believe that his theorems were interpreted conventional make-believe is not truth by convention. For it is not really 358 CARNAP AND LOGICAL TRUTH truth at all; and what are true by non-convention. is conventionally pretended is that the theorems Non-Euclidean geometries have, in the fullness of time, received serious interpretations. This means that ways have been found of so construing terms as to identify the at first conventionally the hitherto unconstrued with some genuine truths, and truths pre chosen set of non-sentences sumably not by convention. The status of an interpreted non-Euclidean geometry differs in no basic way from the original status of Euclidean geometry, noted above. systems became quite the fashion after the advent of non Uninterpreted Euclidean geometries. This fashion helped to cause, and was in turn to mathematics. formal approach by, an increasingly encouraged Methods formal to make up for the unavailability, served systems, of intuition. Conversely, disinterpretation as a crude but useful device (until Frege's syntactical approach came to be appreciated) for achieving formal rigor uncorrupted by intuition. in uninterpreted geometries as true by con tendency to look upon non-Euclidean vention applied to uninterpreted systems generally, and then carried over from these to mathematical oped to look systems generally. A tendency indeed devel all mathematical upon systems as, qua mathematical, This tendency can be accounted for by the increase of as a heuristic aid the use of disinterpretation in an effort to make some sense of mathematics recourse was had with had to become more The uninterpreted. formality, together with to formalization. Finally, thus drained of of all to the shocking the elementary quibble logic merely to uninterpreted which leads from uninterpreted theorems.1) postulates is shocking about this is that it puts arithmetic qua interpreted What theory of number, and analysis qua interpreted theory of functions, and alto geometry qua interpreted theory of space, outside mathematics interpretation, identifying mathematics gether. The substantive and Russell elementary reduction of mathematics to logic by Frege, Whitehead, thing. It is a reduction not to of genuine quite another to set theory; and it is a reduction from arithmetic onward. interpreted mathematics, logic but is of course x) Bertrand Russell, Principles ofMathematics (Cambridge, 1903), pp. 429f.; Heinrich *Sind die mathematischen Behmann, vol. 4, 1934, pp. 8ff.; and others. Urteile Analytisch oder synthetisch?' Erkenntnis, 359 W. V. QUINE V and get back to set theory. Set Let us then put aside these confusions as interpreted mathematics, like arithmetic is pursued and theory it is to set theory that those further branches are reducible. analysis ; indeed, about certain immaterial In set theory we discourse entities, real or alleged, viz. sets, or classes. And it is in the effort to make up truth and falsity of sentences about these about genuine that we find ourselves engaged in something very like convention objects sense of the word. We find ourselves in an ordinary non-metaphorical erroneously our minds making deliberate choices attempt at justification and their logical called postulates, These adoptions, are true until further notice. elementary logic), So here is a case where and setting them forth unaccompanied by any other than in terms of elegance and convenience. consequences (via can plausibly be looked on as con postulation stituting truth by convention. But in ? IV we have seen how the philosophy can be corrupted by supposing that postulates of mathematics always that role. Insofar as we would epistemologize and not just mathe as follows. Uninterpreted we might divide postulation matize, postulates may be put aside, as no longer concerning us; and on the interpreted side we may distinguish between legislative and discursive postulation. play Legislative illustrated institutes truth by convention, and seems plausibly postulation set theory. On the other hand discursive in contemporary from a preexisting is mere selection, postulation body of truths, of certain ones for use as a basis from which to derive others, initially known or unknown. What discursive postulation fixes is not truth, but only some particular ordering of the truths, for purposes perhaps of pedagogy or perhaps of inquiry into logical relationships ('logical' in the sense of All postulation is of course conventional, but only elementary logic). postulation properly hints of truth by convention. to recognize, if only for its distinctness, yet a further way in can enter; viz., in the adoption of new notations which convention for old ones, without, as one tends to say, change of theory. Truths contain legislative It is well are conventional of sentences true ing the new notation transcriptions in question. They depend for their truth partly apart from the convention on language, but then so did 'Brutus killed Caesar' (cf. ? I). They come into being through a conventional of a new sign, and they adoption 360 CARNAP AND LOGICAL TRUTH true through conventional definition of that sign together with the corresponding sentences in the old notation true. in a properly narrow sense of the word, is convention in a Definition, narrow sense of the word. But the phrase 'true by definition' properly must be taken cautiously; in its strictest usage it refers to a transcription, become whatever made such a sentence by the definition, of a truth of elementary logic. Whether on whether is true by convention the logical truths themselves depends be reckoned as true by convention. Even an outright equation or bicon the definiens and the definiendum ditional connection is a definitional = x' or = transcription of a prior logical truth of the form 'x 'p p\ so-called is not thus narrowly conceived, and must Definition commonly for present purposes be divided, as postulation was divided, into legislative a notation definition introduces hitherto Legislative or used only at variance with the practice proposed, or used also unused, so that a convention at variance, is wanted to settle the ambiguity. and discursive. Discursive of definition, interchangeability familiar usage. A frequent purpose chosen part of language can be made on the other hand, or coextensiveness sets forth a preexisting relation between notations in already of this activity is to show how some to serve the purposes of a wider purpose is language instruction. part. Another frequent It is only legislative definition, and not discursive definition nor discursive a conventional that makes to the truth of contribution postulation, sentences. Legislative postulation, finally, affords truth by convention unalloyed. the word 'definition' connotes the formulas of definition Increasingly which appear in connection with formal systems, signalled by some extra systematic sign such as '=d/. Such definitions are best looked upon as correlating economical two systems, lexicon and two notations, the other for is prized for its its brevity or familiarity of ex can be either legislative or discursive in one of which so used pression.1) Definitions their inception. But this distinction wisely; and not for it is a distinction germane translation. is in practice left unindicated, and acts of definition, only between particular to the definition as an enduring channel of inter the legislative of View The distinction between a Logical and the discursive refers thus to *) See my From Point (Cambridge, Mass.: Harvard, 1953), pp. 26f. 361 W. V. QUINE in the case of postulation the act, and not to its enduring consequence, as in the case of definition. This is because we are taking the notion of for lack of an truth by convention fairly literally and simple-mindedly, a passing trait, is So conceived, conventionality intelligible alternative. the front of science but useless in classifying significant at the moving sentences behind the lines. It is a trait of events and not of sentences. Might we not still project a derivative trait upon the sentences themselves, if its first thus speaking of a sentence as forever true by convention adoption as true was a convention? No; this, if done seriously, involves us in the most contributes unrewarding historical conjecture. Legislative postulation truths which become integral to the corpus of truths; the of their origin does not linger as a localized quality, but If a subsequent expositor singles out those once this signifies nothing; truths again as postulates, legislatively postulated he is engaged only in discursive postulation. He could as well choose his from elsewhere postulates his expository ends. in the corpus, and will if he thinks this serves artificiality suffuses the corpus. VI Set theory, currently so caught up in legislative postulation, may some - even a strain of - and lose all obviousness, day gain a norm perhaps trace of the conventions in its history. A day could likewise have been when our elementary logic was itself instituted as a deliberately from something ventional deviation earlier, instead of evolving, shifts of form and emphasis coupled con as with it did, mainly by unplanned casual novelties of notation. Today indeed deviations there are dissident from propounding deviations, level, logicians even at the elementary the law of the excluded middle. These for serious use and not just as uninterpreted are as clear cases of legislative postulation as the ones in set systems, For here we have again, quite as in set theory, the propounding theory. of a deliberate choice unanccompanied (conceivably) by any attempt at justification other than in terms of convenience. This example from elementary logic controverts no conclusion we have to ?? II and III, the departure from the law of the reached. According excluded middle would count as evidence in ?HI, of revised usage though disqualified of 'or' and 'not'. (This judgment was upheld as evidence insofar as meant 362 CARNAP AND LOGICAL TRUTH for the linguistic doctrine of logical truth.) For the deviating logician or defamiliarized; and his 'or' and 'not' are unfamiliar, the words decisions regarding truth values for their proposed contexts can then be just as genuinely a matter of deliberate convention as the decisions of creative set theorist regarding contexts of's'. The two cases are indeed much alike. Not only is departure from classical logic of 'or' and 'not' evidence of revised usage of 'or' between 'not'; likewise, as argued at length in ? III, divergences the the and set theorists may reasonably be reckoned to revised usage of 'e'. Any such a matter and can be is conspicuously of convention, revised usage declared by legislative postulation. We to at a loss to give substance or to of elementary particularly logical truth, familiar truths of logic are true by convention. sense in the notion of truth by convention, but have been the linguistic the doctrine We have doctrine, that the some found only as attaching to a and not as a significant of adoption, viz. legislative postulation, process sentence. Surveying current lingering trait of the legislatively postulated we note legislative postulation in set theory and, at a more events, elementary level, in connection with the law of the excluded middle. And do we not find the same continually in the theoretical hypotheses of in set theory science itself? What seemed to smack of convention at any rate, was 'deliberate choice, set forth unaccompanied (? V), by other than in terms of elegance and con any attempt at justification natural of natural science might venience' ; and to what theoretical hypothesis not this same character be attributed? For surely the justification of any theoretical hypothesis consist in no more can, at the time of hypothesis, the elegance or convenience which the hypothesis brings to the of laws and data. How then are we to delimit the category containing body than of legislative The postulation, short of including under it every new act of scientifichypothesis? in natural situation may seem to be saved, for ordinary hypotheses there being some indirect but eventual confrontation with science, by can be remote; and* con this confrontation empirical data. However, versely, some such remote confrontation with experience may be claimed even for pure mathematics and elementary logic. The semblance of a in this respect is largely due to over-emphasis of departmental For a self-contained boundaries. theory which we can check with difference 363 W. V. QUINE experience hypotheses mathematics drawn includes, in point of so-called natural as it makes of fact, not only its various theoretical science but also such portions of logic and use of. Hence I do not see how a line is to be truth by convention and which confer hypotheses to the former hypotheses which do not, short of reckoning all hypotheses save perhaps those actually derivable or refutable by elementary category logic from what Carnap used to call protocol sentences. But this version, between to an unwelcome degree on the debatable notion of depending too inclusive to suit anyone. sentences, is far protocol our troubles are waxing. We had been trying to make sense Evidently in a priori knowledge. Now the very distinction of the role of convention besides between a priori and empirical begins to waver and dissolve, at least as a distinction sentences. could of course still hold as a between (It between factors in one's everywhere.) adoption of a sentence, but both factors might be operative distinction VII Whatever ceded from our difficulties over the relevant distinctions, it must be con different from do seem qualitatively that logic and mathematics the rest of science. Logic and mathematics hold conspicuously aloof to observation and experiment. any express appeal, certainly, thus nothing external to look to, logicians and mathematicians Having : to expressions, look closely to notation and explicit notational operations terms, substitution, cancellation, transposition, clearing of fractions, and the like. This concern of logicians and mathematicians with syntax times it has become but in modern it) is perennial, increasingly searching and explicit, and has even prompted, as we see, a of logical and mathematical truth. linguistic philosophy On the other hand an effect of these same formal developments inmodern (as Carnap calls (other not considerations logic) from any peculiarly notational to natural science. By this Imean that mathematics can as it can be handled be handled at all) by axiomatization, (insofar elsewhere in science; and outwardly quite like any system of hypotheses can then be left to extract the theorems. elementary logic logic, curiously, than elementary equally relevant The consequent affinity between mathematics and systematized natural has been to show how to divorce mathematics 364 CARNAP AND LOGICAL TRUTH his P-rules by Carnap when he propounded recognized Yet he did not look upon his L-rules or meaning alongside postulates. the P-rules as engendering analytic sentences, sentences true purely by to sustain this distinction has been very much our language. How science problem agement. was in these pages, appreciated a difference and one on which we have found little encour Carnap this problem, in Logical Syntax, as a problem of in kind between the P-rules (or the truths thereby finding specified) and the L-rules (or the L-truths, analytic sentences, thereby an ingenious solution.1) In effect he he proposed specified). Moreover as the the logical (including mathematical) characterized vocabulary such that (1) there are sentences which contain only largest vocabulary as true or that vocabulary and (2) all such sentences are determinable false by a purely syntactical condition i.e., by a condition which speaks of concatenation of marks. Then he limited the L-truths in effect to only Truths just the logical vocabulary essentially.2) given by P-rules were supposedly excluded from the category of logical truth under this criterion, because, though the rules specifying them are formally stated, the vocabulary involved can also be recombined to give sentences whose truth values are not determinate under any set of those involving rules formally formulable in advance. one can object (pending a further expedient of Carnap's, At this point which I shall next explain) that the criterion based on (1) and (2) fails of its purpose. For, consider to begin with the totality of those sentences which are expressed purely within what Carnap (or anyone) would want to count as logical (and mathematical) vocabulary. Suppose, in conform (2), that the division of this totality into the true and the false is in purely syntactical terms. Now reproducible surely the adding of one general term of an extra-logical kind, say 'heavier than', is not going to alter the situation. The truths which are expressible in terms of just 'heavier than', together with the logical vocabulary, will be truths of only heavier than x)\ z D. x is heavier and \x)iy)iz)ix is heavier than y y is heavier than than z)'. The division of the truths from the falsehoods ity with themost general kind, such as '(3 x) (3 y) ix isheavier thany)\ \x) ~ (jcis x) Carnap, Logical Syntax of Language, ? 50. for certain 2) Cf. ? I above. Also, see ? IX on 'essential predication.' reservations conveniently postponed at the moment, 365 W. V. QUINE in syntactical domain can probably be reproduced in this supplementary terms if the division of the original totality could. But then, under the criterion based on (1) and (2), 'heavier than' qualifies for the logical vocabulary. And it is hard to see what whole collection of general terms of natural The further of Cartesian particular science might expedient, not qualify likewise. by which Carnap met this difficulty, was his use this procedure, each spatio-temporal a class K of quadruples of real the class of those quadruples which are the coordinates numbers, viz., of c. Further let us write K[t] for the class of of component point-events is that class of triples which with t appended belong to K; thus K[t] state is associated with the momentary triples of real numbers which of object c at time t. Then, in order to say e.g. that c\ is heavier than c