Logic and the Synthetic A Priori

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Logic and the Synthetic A Priori Arthur Pap Philosophy and Phenomenological Research, Vol. 10, No. 4. (Jun., 1950), pp. 500-514. Stable URL: http://links.jstor.org/sici?sici=0031-8205%28195006%2910%3A4%3C500%3ALATSAP%3E2.0.CO%3B2-D Philosophy and Phenomenological Research is currently published by International Phenomenological Society.

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LOGIC * I S D THE SYXTHETIC I: PIZIORI The distinguished .\merican logician, C. H. Langford, recently published a paper (Journal o j Philosophy, January 6, 1949), as brief as alarming in what the title, "*IProof that Synthetic d Priori Propositions Exist," claims for it. Although this publication has, to my knowledge, had no noticeable repercussions in the literature of analytic philosophy, it deserves credit for reopening (for open minds, that is) an issue which according to the logical positivists has been decided once and for all. One of the merits of logical positivism which I ~vouldbe the last one to deny is to have revealed a typical character of philosophical disagreements, vix., the fact that many (or most, or all?) philosophical controversies are rooted in differences of verbal usage. I am fairly sure that Langford's paper constitutes, indeed, further confirmation of this positivistic thesis, for a positivist is not likely to deny the cogency of Langford's proof of the existence of synthetic a priori propositions in Langford's sense of "synthetic a priori." He would rather criticize Langford for having suggested hy his terminology an accomplishment which he cannot really claim. I hope, therefore, to shed some light on this issue by scrutinizing the Iiantian concepts involved in terms of modern logic. Indeed, it seems to me just as futile to discuss the nature of logic without a clear understanding of the distinctions which Kant strove (though rather unsuccessfully) to clarify as to discuss those distinctions without regard (be it ignorance or oblivion) to modern logic. The line of attack against the Icantian theory of geometry most popular with modern analytic philosophers has been to call attention to the distinction between pure and physical geometry, and to shon- that synthetic a priori propositions disappear from geometry once this distinction is ohserved. The "axioms" of pure geometry (more aptly called "postulates") are propositional functions, so are the derived theorems, and the concepts synthetic-analytic, empirical-a priori significantly apply only to propositions. The propositiol~sof pure geometry really belong to logic (and are hence analytic), since they have the form "if the axioms are true, for a given interpretation of the predicate variables (the so-called prirnitive terms of the axiom set), then the theorems are true, for that interpretation." On the other hand, once the axioms are interpreted, one obtains either analytic propositions or empirical propositions; if specifically an empirical interpretation is given, the interpreted deductive system refers to physical space (physical geometry) or to some other empirical subject matter. Kow, Langford admits, in the cited paper, that if the postulates ~vhich

have to he added to :ln "adequate" definition of "cube" in or(1c.r to tlcrive the theorem "all cuhcs have tnelve edges" are propositional function\, then it cannot he bupposed that thih geometrical theorem expresser L: proposition a t all, and that if the postulates are interpreted in terms of physical space, the theorem is not (or, a t least, may not be) a priori. Yet, he claims it to he a priori if an interpretation of the postulates in terms of uisual space is assumed. Rut thus he must hold that, however mistaken Kant's vie~vsabout physical space may have been (specifically the view, suggested by the apparent finality of Kewtonian physics, that physical space must necessarily conform to the axioms and theorems of Euclidean geometry), Kant n-as right in holding that there is such a thing as "pure intuition" which makes a priori kno~vledgeof synthetic geometrical propositions possible. Langford emphasizes, indeed, that his proof '.does not require that all theorems of Euclidean geometry should become true a piiori n i t h an appropriate interpretation." But it seems to me evident that his proof, if valid at all, establishes that all theorems of geometry ivhidl require for their demonstration postulates (containing specifically geometrical terms), in addition to esplicit definitions, are synthetic a priori propositions, provided only that we suppose them to refer to visual ( = idealised?) space. Kot that 1,angford is committed to the Iiantian view that \\ e are graced with a power of specifically spatial intuition which puts geometrical knowledge into a category hy itself. Consider, for example, a proposition of phenomenological acoustics, like "if x is higher in pitch than y , and y is higher in pitch than z, then x is higher in pitch than z." This proposition, which we are all inclined to regard as necessary on intuitive grounds (the contradictory is inconceivahle) is certainly not derivahle from logical principles with the help of definitions: the relational predicate involved admits only of ostensive definition, it defies analysis. *Ind if so, then this proposition \rould he analytic only if the predicate "higher in pitch than" occurred inessentially in it, i.e., if the proposition "for every It, s, y, z, if ully ant1 yKz, then sRz" I~elongedto logic-~rhich, of course, it clocs not. Some \fill no doubt sag: "Granting, for the sake of the argument, that Langford has established the synthetic character of the proposition 'all cuhes have tnelve edges' ii.t., that it is not dtmonstrahle with the help of explicit definitions, which do not beg the question, alone); has he given any argument a t all for the claim that it is a priori (neceswy)?" Indeed, Langford just takes it for granted that the proposition is not empirical. And I can hear those n ho hold all necessary propositions to he definitional truths argue: "Khat could one mean hy saying 'p is necessary' if one at thc same time admits that p is not demonstrable nith the help of logical princ,iples alone? Surcly, the c20nc.eptof necessity one has in mind must

then be purely psychological, something like the inconceivability of the falsehood of p." Kow, for the sake of those ~vhothinli, for some such reasonb as I just outlined. that Langford's proof stands and falls (or, rather fall51 uith his hy and large discredited Kantian assumption of a faculty of pure intuition of visual space, I want to she\\ as tersely as I can that it mu.t be possible to kno~vsome propositions to be necessary before any can he knomn to be analytic;' and that if the concept of synthetic entailment? he held to be psychological, the concept of analytlc truth (as applied to natural languages) ~1~111 be no better off Firit, what sort of a statement does a philosopher intend to malie when llc says "all necessary propositions are analytic (and, of course, con~.ersely)?"An empirical statement, like "all dogs have the power to bark>" 0111-iouslynot. He intends this statement as an explzcatio~z(to use Carnap's term) or analysis (to use Moore's term) of the concept of logical necessity. I t therefore rather resembles such statements as "all cubes are regular solltl. hounded hy square surfaces," "all fathers are male parents." I shall non attempt to show (a) that a definition of "logically necessaryn3 in terms of "analytic" is untenable since it suffers from implicit circularity, (h) that if :I semantic system of logic is taken to include its meta-language (metaloglri, it must he held to contain synthetic propoiitions (which, of course, are not empirical). Suppose we define an analytic statement as one which is demonstrable nith the help of adequate definitions and without the use of extra-logical premises (Langford's "postulates7'). The use of the word "demonstrable" in this definition should make it clear that the concept herc defined is a selnantic one (corresponding to Carnap's L-truth), since demonstration, as commonly understood, involves the assertion of the premises from which a deduction has heen made as true. apparently equil d e n t definition is the one preferred by Quine: a true statement is analytic if either it contains only logical constant$, or, provided it is wr~tten s l yit, i.e., the out in p r i m i t i ~ enotation, descriptive terms occur u a c ~ ~ o ~ iin statement II ould remain true if the descriptive terms \\ere arbitrarily replaced hy otheri that are semantically admissible in the context. I n a n t -I her :tlso t h e author's article " Are :ill Seccssar \ Propos~tion\A n : ~ lt\ ~ c ' " Phtlos o p h z ~ ( ~Reczeui, 1 J u l y , 1949. ?JT7hat1s here called "synthehc e r l t ~ ~ l n l e n has t " n o t h ~ n gt o d o w ~ t ht h e rlot~onof s , c:tus,tl cau\,~lcr~tailmentn hich is allegedly involved 1x1 sul.)jurlct~vcc o n d ~ t ~ o n a lfor t~c staternrnts are a t a n y rate emp111c,t1,n h ~ l ea ~ t n t e r n c r ~ t\ j ) r ~ > s:I~~n ~g n t h eentallmerit 1%ould be necessary usagr f o ~tlic eupresslon "log~call\ nccesThere e x ~ s t st,o be sure, at1 esti~l.)l~shetl sarx" .recording t o vhlcll ~t 1s synon\ mous n ~ t h"anal\ t ~ c "or " l o g ~ c n l l \t~~ u e " , i usage here 13ut I+(' ol)v~ouslyneed a y u a l ~ f- ~ and I am obviousl\~d e p a r t ~ n gf ~ o n thls Ing n d j e c t ~ v e1x1 order t o d ~ s t ~ n g u i tshhe sense of "neccss:~r\" u r l d e ~discussion frorn other, irrelevant, senses 11he " f , i c t n ~ l n e c r s s ~ t ," \ " I I I t c t I ( * L I n r c e s s ~ t ,\ " :tntl so fo1 tll

to show that if in these definitions "necessary" were substituted for "analytic," the definitions would become circular on two accounts. I t is important to realize, in the first place, that unless the classification of a statement as belonging to logic or an empirical science respectively is to be wholly arbitrary and devoid of philosophical interest, it will not do to define "analytic" as a predicate which accrues to statements relatizvly to arbitrary deji~zitionsof their constituent terms. We obviously want to say, for example, that uhile one could arbitrarily define "man" in such a way that "all men are mortal" would become a logical truth, the statement as commonly understood simply is empirical. Elit in that case the definitions from which analytic truth derives will have to be characterized as in some sense adequate. What, then, are the criteria of adequacy of definitions? Extensional equi\ralence is obviously an insufficient criterion, other\vise it would be adequate to define "equilateral triangle" as meaning "ccluiangular triangle," and, worse still, any proposition of physics that has the form of an equivalence (an "if and only if" proposition, in other nords) could be made out as analytic and thus belonging to logic. This consideration suggests a further negative criterion of adequacy of definitions, to be added to extensional equivalence of dejiniendum and dejiniens: the definition should enable the logical demonstration only of such propositions as are not empirical. But to call a proposition nonempirical is the sxme as calling it necessary, hence the concept of adequate definition which was lised to define analytic truth leads us back to the concept of a necessary proposition, and if "necessary," here, \\ere synonymous with "anal.vtic," the definition ~vouldbe \riciously circular."^ illustrate: in constructing a definition of propositional truth, one will be guided by the criterion "to say of a proposition that it is true is equivalent t o asserting that proposition" (if and only if "p" is true, then p), i.e., no definition of truth will be accepted as adequate unless it entails the mentioned proposition. Why not choose as a criterion of adequacy the proposition "if p is true, then, if p is asserted, p is believed by the speaker?" The obvious reason is that this proposition is not necessary, i.e., it is conceivable that people should assert true propositions n~hichthey fail to believe (say, because they are liars who, contrary to their knowledge, happen to disbelie~retrue propositions). Let us, non-, illustrate the same point with regard to the definition of a

' I t may be noted in p:~ssingthat C. I. Lewis's definitioil of anitl~.tictruth, supplemented by his identification of analytic and a priori truth, suffers precisely from Ch. IT.) Analytic statethis circularity. (cf., A I Ldr~alt,sis of Rnowleclgear~dT7al~iatior~, ments are defined as stz~tementsderivable from principles of logic with the hclp of definitio~lsn.hicli are not ;~rtjitraryterminological convrntio~lshut "esplicativc, s t : ~ t e mrnts." Explicative st:ttrments are said to be staterncrlts t,o the effect th:tt the in"1'" and "Q," are identical. But then we are told t h : ~ t"P" trnsions of t ~ terms, o 2nd "d)" have the samr i~itcnsionif t h r y arc inter-de111icit)lei.c, if the fornlnl eclrlivaencc " ( s ) [ P s = Qx]" is atlalytic!

logical constant, say, "nut." \Thy is the definition ernbodied in the conventional truth-table (if p is true, then not-p is false, and if p is falsc, then not-p is true) considered adequate? What does it rnean to say, in answer t o this question, that it "conforms to ordinary usage?" Suppose we wanted to decide which of the following two definitions of the function "I /;now that p" conforms to ordinary usage: (a) I believe that p, and p ; (b) I belielye that p, and p is highly prot~ableon the available evidence. We ~ o u l d , or shoulti, :irgue somewhat as follox~s:the proposition "if I lino~vthat p, then p is true" is clearly necessary, in other words, it would be self-contradictory to claim knowledge of false propositions; but :~ccordingt o definition (b) it nould be logically possible that falsc propositions should be linou-n, since a false pl-oposition may be highly prok)al)le on the available evidence; hence definition (a), not (b) is correct In order to caonform to the ordinary usage of the defined term, a definition, then, must enable the (lemonstration of such sentences, and only wch sentences, involving the defined term as are ordinarily held to express necessary propositions. Aiccaortlingly, the test of atlequacy of the definition of ',notn is that it enables the logical demonstration of certain fundamental trecessary proposit~onsinvolving the defined constant, such as the lan- of the excluded middle and the law of noncontradiction. And if n e argued that what makes these principles necessary is the fact that they are demonstrable with the help of adequate definitions of the logical constants involved, our argument ~ o u l d evidently be circular. Since the definitions of such logical constants do not form part of the logical system as such but belong to the meta-language, it may he reasonable to demand that the criteria of adequacy themselves he formulated in the meta-language. Thus, one would properly distinguish the tautology of the propositional calculus "for every p, p or not-p" from the meta-linguistic statement "every proposition has either the truth-value 'true' or the truth-value 'false' " (here "true" and "false" are meta-linguistic terms and hence the "either-or" of this statement is different from the "either-or" which is used, but not mentioned, in the calculus). But now we face the folloning situation: unless this meta-linguistic statement is acrepted as necessary, no instruments, as it were, are provided for proving that the l a ~ vof excluded middle of the object-language is analytic;

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Incidentall>, i t scems t o me semanticall\ inaccur:rt~t o d~stiiiyuish,its phllosopliers frequeritlj do, t n o klnds of hnowledpe cerlazn knou ledge and probable kriou 1edge What could be mesrit by saying "I krion n i t h high 1)rob:tt)ility t h a t the sun wlll rise"' I t i$ould not mean that on the orie hand I know, but on the other hand it is (or I am) not certain, for that surely sounds s e l f - c o r ~ t r a d ~ c t o Ir . ~think what the intended dlstinctiori comes to IS merely this. sometimes tlic propositiori "I k n o ~th a t or not) is certa~rl p" (which is always empirical regardless of whether p is en~pirici~l and sometimes it is only protxrhle on the evidence In the latter case one might approj1rl:rte1y 5 s ) "I : ~ m not r e r t . ~ that ~ r ~ I 1,non p, but ~t is p~ol):tt>le t h ~ It do "

and if the meta-language is not formalized in telms of a meta-meta-language, the meta-linguistic L.E.31. cannot be analytic; and since me cannot go on building meta-meta . . . meta-languages forever, some meta-language will have t o contain an analogue of the object-linguistic L.E.X. which is a t once synthetic and necessary. I anticipate the objection that my argument is completely worthless since it proceeds on the assumption that "analytic" is an absolutr cor~cept.The opposition might, indeed, demonstrate the absurdity of my argument by comparing it to the follo~ving:a body can be said to move only relatively to a reference-body; but unless we Itnew that the referencebody is absolutely a t rest, we could not be sure that the first body reall!/ moves; hence, if we \\ant to say that some body really moves, we shall have t o assume that some other body is absolutely a t rest, i.e., at rest regardless of what happens to any other body. The point of the analogy \vould presumably be that it is just as meaningless to call a statement of a given language synthetic relatively to no meta-language a t all, as it is t o speak, in old Seutonian fashion, of absolute rest (and the same analogy \vould, of course, hold for "analytic" and "in absolute motion.") To which 1 reply: If "analytic" is to be short for "an:~lytic in I,," ~I-here1, is formalized in terms of some meta-language 1vhic.h includes, among other rules, definitions of the defined terms of I,, then "analytic" cannot be regarded as an esplzcatrim of the common notion of logical truth, since by a suitable choice of definitions any statement could then be made out as a logical truth. Thus, to escape from this "conventionalism," as Lewis calls it, one nil1 have to make reference to a priuzleyed meta-language (just as the earth is the privileged reference-frame tacitly referred to in commonsense statements of the forrn "s moves" or "s does not move"); and in order to mark out this privileged meta-language one n-ill have to introduce precisely that notion of adequate definitions which leads up to necessary propositions defying formal demonstration. KOIT, to my second argument for the proposition that a definition of logical necessity in terms of analyticity would be circular. 1,angford defines "analytic," as most logicians ~vould,in terms of "logical principle"; and "logical principle" is defined, in the paper already referred to, as "principle involved in the extended function calculus." But how are we to decide whether a given proposition is involved in the extended function calculus? Either n e enumerate all such propositions, so that "logical principle" becomes an abbreviation for a finite disjunction of propositions, or else we shall have t o state a common and distinctive property of such propositions by virtue of vhich they :ire classifiable as "logical principles." The former method of definition is impossible since (a) the number of propositions belonging t o a system of iogic that can he fabricated is unlimited, owing t o

repeated applicability of the rule of substit~~tion, (b) given such an enumeration of propositions which defines "logical principle," it would be selfcontradictory to suppose that one day a new logical principle should he discovered (or manufactured), since this would mean that an element both belonged and did not belong to the same collection. We are, therefore, faced with the necessity of supplying an explicit definition of "logical principle."" I t seems that such a definition will have to make use of the notion of logical constant, e.g., "a logical principle is a true proposition containing only logical constants." But what do we mean by "logical constant?" While I am unable to give a satisfactory explicit definition, I am sure that such a definition ~vouldinvolve the concept of validity (as predicated of deductive arguments), for the follo~t-ingreason: The rules of formal logic contain no descriptive terms (this is the reason why they are called formal ~ules),hence before they can he applied to the test of the validity of specific arguments, the latter must be formalized; which means that specific descriptive terms in the argument are replaced with variables until a more or less abstract schema is left over. S o t all evpressions can he replaced by variables, hourever, since otherwise it could not be said that the argument has one logical form rather than another: in order to have a specific form of argument, we need some constants. How, then, are we to tell ~vhich terms may be replaced by variables (of appropriate type) and which may not? Consider, for example, the syllogism "if Socrates is a man, then Socrates is mortal; Socrates is a man; therefore Socrates is mortal." We see that if in this argument the proper name "Socratrs" is replaced by any other name or description of an individual, the resl~ltingargument TI-ould still be valid. Hence, instead of considering this spec~ificargument, n-e consider any argument of the form "if x is a man, then s is mortal; x is a man; therefore x is mortal." But then we also notice that the validity of the argument would not be destroyed if any other predicates of the first type were substituted for "man" and "mortal"; \vhicsh leads us to consider any Some may thirlli that this corlclusior~car1 be avoided since the general concept, of t:tutologp admits of ~ e c u r s i c edefinition, thus: one first cnurnerates a set of prinlitive propositiorls n-hich one calls "tautologies," arltl thrrl cstends the term "tautology" to any proposition which is derivable from thcsc primitive propositions with the help of specified rules of derivation. I think, howevclr, it is perfeotlj- evident l tautologies ant1 not t o an exp1ic:ttiorl of the that this amounts to a s t n t e n ~ e t ~ahout meaning of "tautology." The Tery choice of prinlitivc propositions as well as of rules of derivation must 1)e guided by :L prior underst:a~ltlir~g of \vhat a tautolog)- is. We might want to saJr,for example, that not enough rulcs of tic~rivation,or not enough primitive propositions iintl I ~ c r n]:rid down, since tlicrc :IT(, t;~utologicswhich coultl riot I)[: derived in the constructen sjrstern; but this ~voultl1)c :I self-contradictory statc~mrntif "t:tutology" ncenict "~)ropositionderiv:~l)lt:i r ~111econstriicted system."

argument of the form "if l's, then Qx; P x ; therefore Qs." And one further degree of abstraction is seen to be possible: it does not matter what the forms of the constituent propositions are; hence we may introduce propositional variables and consider any argument of the form "if p, then q ; p ; therefore (1." Hut n hat prevents us from pushing this process of formalization one step further by introducing a variable whose values are binary connectives (like "if-then," "and," "or"), and studying the schema: pCq (where "C" is a binary-connecti1.e-variable); p ; therefore q ? The anSIT er is obvious: TVC see a t once that not all the argument-forms resulting from substitution of a binary connective for "C" are valid (as, e.g., p or q ; p; therefore q). I t is no doubt such observations that led logicians to stress the distinction between logical constants, as terms on whose specific meanings the validity of an argument depends, and descriptive constants which occur incssentially (to borrow Quine's term) with respect to the validity of the argument- -although they may ocvur essentially with respect to the factual truth of propositions entering intc the argument. But t o say that an argument is valid is to say that its conclusion necessarily follows irom its premises; uhich is to sag that the implication from premises to conclusion is a necessary proposition. Which completes the vicious circle I have been endeavoring to demonstrate. If the distinction between analytic and synthetic truth rests, as has been suggested, on the distinction 1,etween logical and descriptive constants, and if there should esist no sharp criterion by nhich the two types of expressions could be distinguished, then the distinction analytic-gynthetic is less clear-cut than is commonly supposed. Superhcially it looks as though the ahove descriprion of the f ~ ~ n c t i oofn logical constants in formal logic suggested a perfectly simple explicit definition of "logical ?onstant" : a logical conhtant is a term uhic.11 cannot be replaced by a variable in the process of formally testing the validity of arguments which contain it. This definition, however, breaks do\^-n under the weight of three objections: (a) an expression which occurs essentially in one argument, may occur inessent ially in another aixgument.Talie, for example, the identity sign. I n the argument '(x = y ; tl~creforenot-(x # y)" it occurs inessentially, since any argument of the form .'xRy; therefore not-not-xRy" is valid (indeed, the word "not" is thc only expression in this argument which occurs essentially!). On the other hand, in the context '(x = y ; P s ; therefore Py" the identity sign has an essential occurrence, since "xRy; P x ; therefore Py" is not generally valid. ( b ) On the proposed definition, it will depend on the kind of \-ariables that are available for formalizing arguments, whether a n espression belongs to the vocabulary of logic or not. Suppose that we introduced symmetric-rc1atio11-vziriables, i.e., \.at-iubles taking the names of srmmctric relation> :,. \:tli~c-. S, I-;'. S f ' , etca. Iit th:rt case the argument

"u = y ; therefore y = a" might be regarded as a substitution-instance of the argument "\Sy; theiel'ore ySu," and since the latter is generally valid, " = " ~vouldbe classified as a descriptive (inessential) constant. Hut if the variables a t our tlisposal are less variegated, and \re can use unly generic relation-variables It, I t f , etc., then the above argument will have to be considered as an instance of "xRy; therefore yl