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;
-
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<s," and since this is not a valid argumcxnt-form, we could then \\it11 equal plausibility (or unplausihility) conclude that " = " is a logical constant. This tliffictilty cannot be avoided either t)y stipulating that the formalization should be as abstract or generic as possible. For such a stipulation is presumably equivalent t o the demand that the range of the variables used for formalization should correspond to the logical type of the 1-slues in cluestion. But what is meant by saying that class C is the logical type to \vhich entity s belongs, if not that "s is a member of C" is true, pro~idcdit is siyn
to say, unlike arithmclzcal t ~ r m s geometrical , terms are descriptive of observable features of the 11 orld. I t is, moreover, somen-hat naive to say flatly .'arithmetic is reducible to logic," or "geometry is not reducible to 10gic."~_$part from the consideration that the confines of the language of logic are largely determined hy the intuitive acceptal~ilityof certain definitions-somebody might, for example, reject Russell's contextual definition of descriptive phrases in terms of logical primitives as intuitively inadequate, and on that ground hold that statements about mathematical functions are not really reducible to logic-arguments by i\hich such reducibility is commonly proved involve 1 = 2" a subtle element of circularity. To say, for example, that "1 is really a truth of logic, is to say that it can be derived from the primitive propositions of Princzpza with the help of the logistic definitions of the specifically arithmetical terms "1 ," "2," ." Rut on what grounds are those definitions accepted as adequate? Either the grounds are intuitive, in which case one is left nithout a logical argument by ~vhichthe reducibility of arithmetic to logic could be proved in the face of objections from the intuitive inadeqliacy of, say, the definition of the number one, or else one will have to argue: the definitions are adequate in the sense that they enable the demonstration of the theorems of arithmetic from logical premises alone. Considering, then, the vagueness of the expression "logical term," I do not find the claim that arithmetic (or any other science which contains, prior to reduction to primitive notation, nonlogical terms) is logic any more convincing than the contrary claim. The contrary claim might be supported by two reasons: (a) the I\ hole reduction is circular if thc underlying definitions can he defended only by showing that they lead to the desired result, (b) the definitions themselves express s!jnthctzc propositions, for a biconditional joining n proposition in purely logical notation with a proposition in partly arithmetical notation cannot hc analytic unless arithmetic~has already been reduced to logic; but if this reduction could be effected independently of such definitions, there would be no need of setting the latter up as instruments of reduction. By way of clarifying this type of argument (~vhich,it should be kept in mind, is not so much intended as a proof of the esiztence of synthetic a prlori proposition than as a proof that, in default of a clear definition of
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What I mean, here, 111 $1retiuction of a, system of geometry to logic is a derivation of the postulates of such :t sl-stern from postulates conta~nlugonly logical terms I t is frrquently said that piire geometrl, whose propositions ale formal imphcations in nhich specifically grometrical terms occur vacuously, IS part of logic. This is, however, a rather t r i v ~ n loi)servat~onslnce the word "geometrv" here has itself a v:tcuous occulrence in the sanlr sense pure mechanics, pure deductive sociolog\-, etc , form part of l o g ~ c
510
PHILOSOPIIY AND PHESOMESOLOGIC~AL REYE-~RCH
'clogical constant," it is to a certain measure arbitrary whether a given a priori proposition be called "analytic" or "synthetic"), let us suppose that the confines of a system of logic are drawn, in the manner familiar to formal logicians, with the help of recursive, not explicit, definitions. We arbitrarily mark out some terms, say "not" and "or," as logical constants, and then specify that any term exclusively definable with the help of "not" and "or" is also a logical constant. This kind of definition leaves us, of course, in the dark as to \\-hat "logical constant" means, yet it provides an effective procedure for deciding questions of the form "is x a logical constant in the system L?" If contextual definitions are permitted, then such terms as "all," "and," '*the," "there is," etc., ivould all be logical constants in this system.' Yet, hoiv ~vouldone decide whether a proposed definition of a derived constant in terms of the selected primitive3 mas adequate? I t seems to me that the proof of adequacy mould rest on the intuitively accepted validity of certain forms of argument. Hot17 could we prove, for example, that "class X has exactly one member" is logically equivalent to "there is an u such that x is a member of A and for any y, if y is a member of A then y is identical with x?" &I11we can say in the end is that the two statements evidently entail each other. Since what is in question is just a definition of "one" with the help of which the eutcnsion of the term '*logical constant," and therewith of the term "logical truth," could be increased, it would be circular to attempt to show that this equivalence is analytic. But what guarantee, now, do we have that any new logical constant which we might discover in the process of analyzing arguments would be definable in terms of our logical primitives? What if what appears to belong to the logical skeleton of our language should defy definitional reduction to the selected primitives? If such a situation should arise, two alternatives would be open to us. We might say that the terms in question are logical constants since they evidently have an essential ocscurrencein some valid arguments, and that therefore statements containing these constants along \\ ith already accepted citizens of the vocabulary of logic are logical truths. Or, we might say that such terms are not logical constants, just because they are not definable in terms of our primitive logical constants, ant1 that therefore statements which contain them essentially are synthetic. Take, for example, the intuitively valid argument: s and y are distinct points in i~ plane; therefore there exists just one straight lint, (in that plane) \\-tlich contains both x and y (where "point" and "straight line" and "plane" are riot predicate cariables but have the customary georneti.ic.al meanings). 'To he sure, I might, using "point" as an undefined term, define a straight In order to retluce "the" to those primitiyc C O I I S ~ : L I I ~0118 S ueeds, though, the symhol of identity n-liich can be redured to the prinlitive c.orist:~nts only if qu:r~itifiratiorl over predicate-v:tri:tt,lm is permitteti.
line (in a plane) as a class of points which is uniquely determined by any two members of itself, and relatively to this definition, of course, the argument ~vouldbe formally valid. But such a definition would obviously be question-begging in the present context of discussion, just as a definition of "cube" as "regular solid with txelve edges" would be question-begging if it were used as a refutation of Langford7s point. If we assume that "point" and "straight line" and "plane" belong to the undefined vocabulary (which assumption is justified in view of the fact that the meanings of these terms are commonly understood only by virtue of ostensiuc definition), we see a t once that this argument is not formally valid: if "point7' and "straight line7' and "plane" and "x contains y" are replaced by variwe say, then, that ables, the resulting argument-form is i n ~ a l i d Should .~ this is a case of synthetic, or material, entailment since the implication is not derivable from what has been defined as "logic," but that the essential occurrence of the geometrical terms calls for an estension of our definition of "logical constant?" Or should we say that these geometrical terms could never be called "log~calconstants" a t all, just because they are not definable in terms of the accepted primitive vocabulary of logic? Most logicians would seize the latter alternative, in line with the view that geometrical axioms, unlike the postulates of arithmetic, cannot be reduced to pure logic. But then they might also have refused to admit, say, "all" as a logical constant into a tiuth-functional logic, since it can be defined in terms of "and" only if infinity (expressed by " . . . ") is admitted as a logical concept-and would it not be arbitrary to do so? And arguments involving "all" essentially would, relatively to such a narrow definition of "logic," have to be counted as synthetic entailments. I t might be objected that an explicit definition of "plane" could be constructed which ~ ~ o u bring l d the analytic character of this axiom of Euclidean plane geometry into evidence. Suppose we wanted t o verify whether a given surface was plane or curved, mould we not endenvor to determine whether one and only one straight line could be drawn through any couple of points in i t ? And does this not suggest the definition of a plane surface as a surface such t h a t any couple of points in it uniquely determines a straight line? The trouble with this attempt a t ezplicit definition is t h a t it lands us in a vicious circle. For one could similarly ask how one 15-ould distinguish a straight line from any other type of line? And with the same plausibility one might sny t h a t i t is that t)-pe of line which, i~aa plane, is uniquely determined by n, couple of points. I t is, indeed, sometimes said t h a t the geometrical primitives "reciprocally" define each other, but I have never bee11 able to understand how such "reciprocal" defirlition differs from viciously circu1:tr definition. We are then left, and only two alternatives: either the primitives are as far as I can see, with t ~ - o predicate variables, in which case the axioms are.no propositions a t all; or else they have an empirical reference through ostensive definition. I t should be noted, however, t h a t the contingent character of such empiricnlly interpreted axioms cannot be inferred from the fact tlint they refer to qualities of sense experience, no more than i t folloxvs that "2 2 = 4" is contingent from the fact that i t is applicable to classes of empirical objects
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The same point may ivell be illustrated further in terms of Langford's second candidate to the honorable (or decadent?) title of "synthetic a priori truth," viz. the proposition "whatever is red is colored." Langford argues that "x is colored" cannot be formally deduced from "x is red," since a man who is able to understand the meanings of the extra-logical terms in the premises of a formal argument should also be able to understand the meanings of the extra-logical terms in its concllision (at least if the argument is written out in primitive notation, ant1 does not have some such trivial form as "p, therefore p or q"); and he argues (convincingly, I think) that a man might n-ell understand the meaning of "red" xvithout understanding the meaning of L'colored." I shall offer an independent, though perhaps similar, argument for the view that this proposition is synthetic. I t is natural to suppose that the conditional "if s is red, then x is colored" has the form "if p, then p or q," since to say "s is colored" is to say "x is blue or green or retl o r . . ." But could such a disjunctive definition, as n-e may call it, ever be written out? Suppose \ye wrote it out, eliminating the convenient dots, by putting in as a disjunct each and every color that happens to have received a name. And suppose that thereafter me observed for the first time an object that had a color which, unknown as it mas, failed to have a name; ~vouldwe not want to say that this object is colored? We certainly ~vould,yet if "colored" meant what we defined it to mean, we could not say that. I conclude that "colored," like "red," must be regarded as a term whose meaning is grasped only through ostensive definition and which therefore belongs to the primitive vocabulary. But in that case the argument "x is red, therefore x is colored" xvould be formally valid only if all arguments of the form "x is P, therefore x is Q" were formally valid. Have I shown that the true statement "\\-hatever is red is colored" (and all similar species-genus statements, like "whatever is round has shape," "n~hateveris hot, has temperature") is synthetic. (though presumably necessary)? Only if it can he assumed that "red" and "colored" are not logical constants. This assumption seems innocent moligh: if these are not descriptive terms, what terms are? Yet, in the first place, the customary manner of distinguishing logical terms from nonlogical terms as purely syntactic elements of language from denotative elements of language is, as pointed out already, not quite safe: numbers are, up to a certain point, observable properties of collections, still they are regarded as logical concepts; secondly, if all we can say by way of defining "logical constant" is that logical constants are those terms which ocac.uressentially in some valid arguments,1° then "red" \vould have to be admitted as a logical constant. I n One might think thnt much embarrassment coultl Iic spared by replacing "valid," here, with "forrn:tlly valid." But such a rrplnre~ncnt\vould make the defini-
To be sure, this word occurs inessentially in many valid arguments, but the same is true, as was shown, of such full-fledged members of the x-ocahulary of logic as the identity sign. Perhaps the proper conclusion to be drawn from these observations is that it does not make sense to represent the distinction between logical and nonlogical expressions as absolute. Perhaps all that can be significantly said in answer to the cluestion of what a logical espression is, is that an espression functio?zs logically in the context of an argument in which it occurs essentially (in the already explained sense of Quine's phrase "essential occurrence"). But in that case the whole problem of whether any necessary propositions c*ould fail to be analytic., i.e., certifiable hy reference to logzcal principlrs alone, is ill-defined. If, on the other hand, the meaning of "logical principle" should be, more or less arbitrarily, rnade precise by laying do~vna number of postulates and rules of derivation and defining "logical prinriple" as any proposition derivable in this system (including the postulates which are derivable from themselves!), then I think Langford's proof is irrefutable, since any number of necessary propositions (necessary, that is, in terms of ordinary, pre-analytic usage of the term) could easily be produced which, like "all cubes have twelve edges," cannot be demonstrated except ~ r i t hthe help of extra-logical postulates." Of course, what inevitably happens is that such a logical system which, judged in the light of the small number of postulates and primitive terms, appears rather meager, expands enormously with the help of definitions: just think of the way in which logic, in Russell and Whitehead's magrLum opus tion grossly circular, since formal validity is defined in terms of a set of formal rules of deduction which cannot I)e forml~lateduntil after a choice of logical constants has been rnade for the given language. " Whether Langford is right or wrong may well depend on the precise extension assigned t o the term "definition" in the characterization of analytic propositions as propositions demonstrable with the sole help of definitions. Specifically, if postulates (or, rather, sets of postulates) should be described as i m p l i c i t d e j n i t i o n s of the primitives they contain, then n.h:tt Langford calls a. "synthetic a priori" proposition is for those \\rho use the term "definition" more liberally simply a species of :tr~:tlytic proposition. Once, however, one operotes ivith the concept of implicit definition, the extension of the concept of logical truth is in danger of berorning paradoxically large, especially if implicit definitions are claimetl, as I)y Schlick i r ~his rillge,neine E r k e n r ~ t nislehre, t o prev:til not only in formal mtrthcm:~ticallangu:tgest)ut also in the 1angu:tge of science. Thus o ~ might ~ e hold t h a t , like the primitives of a system of geometry, the primitives of :t system of mechnnics ("particle," "mass," "force." etc..), a1.e "hIet 11implicit,ly defined 1)y the postul:ttes of the science (sce e.g., Hcrlri A'fargen:~~, odology of JIoderrl Physics," P h i l o s o p h y of Science, Val. 11); in fact, 1'oirlc:~rP has impressively shown ho\l- attempt,^ a t explicit definition ent:r,ngle one in logical circles. I3ut the eml)arrassing consequence of this position is that the laws which ;tppe:tr ;is thcorenls in the postul:~t,ion;tldevelopment of the science would then h:~vet o Iw r i x g:trtlctl either :is proposition:~lfunctions or as 1ogic:rl truths !
sir-allows up the huge system of classical mathematics! But I see no escape from the conclusion, well worth repeating, that those definitions which syntactically function as rules of translation from one universe of discourse to another and thus enable incorporation of more and more material into logic, express (in a sense of "express" sufficiently clear to go without analysis in this discussion) necessary or a priori propositions; and since the necessity of those propositions is the ground which makes those definitions cognitively acceptable, it would be circular to prove that they are analytic by reference to the very definitions which they are to support.
ARTHUR PAP.
Con frecuencia se ha negado la posibilidad de proposiciones que a la vex Sean sintkticas y necesarias (a priori), fundindose en que el Bnico sentido no psicol6gico en que una proposicidn pueda llamarse necesaria es en el sentido de "verdadera por definici6n." I'ero esas definiciones a que se alude se entienden como "adecuadas." CAdecuadas para yue? Para deducir formalmente proposiciones que se sabe son necesarias independientemente de la deducci6n formal. De un modo parecido se llega a reconocer que hay proposiciones necesarias yue no son formalmente demostrables, bajo pena de incurrir en circulo vicioso; o sea mediarlte el anhlisis del concepto de "verdad 16gica." Esta se define usualmente en tkrminos del concepto de una "constante 16gica." Pero las constantes 16gicas se introducen en la argumentaci6n deductiva examinando si la cridente validez intuitiva del reciocinio se mantiene con la substituci6n de ciertos tkrminos por variables. Puesto que, a1 afirmar la validez de un raciocinio, se afirma como necesaria una cierta implicaci6n dada, la definicidn de "necesario" en tkrminos de "verdad 16gica" resultaria de igual mod0 implicitamente circular. Xparte del problema del conocimiento sintktico a priori, la mismadistinci6n entre proposiciones analiticas y sintkticas no es en modo alguno tajante, cuando se aplica a lenguajes naturales. Depende de la viabilidad de una definici6n de "constante 16gica' (y por ello de la "verdad 16gican); y existen razones de peso para dudar de que pueda ofrecerse tal definicidn, a menos clue ella tome la forma de una peticidn de principio, y rcsulte una enumeracitin de tkrminos.
