Belief and Propositions

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Belief and Propositions

Arthur Pap Philosophy of Science, Vol. 24, No. 2. (Apr., 1957), pp. 123-136. Stable URL:

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Belief and Propositions Arthur Pap Philosophy of Science, Vol. 24, No. 2. (Apr., 1957), pp. 123-136. Stable URL: Philosophy of Science is currently published by The University of Chicago Press.

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The repudiation of propositions as "obscure entities" which is prevalent among logicians and philosophers of "nominalistic" persuasion, is frequently justified by pointing out that no agreement seems ever to have been reached about the identity-condition of propositions. And if we cannot specify, so they argue, under what conditions two sentences express the same proposition, then we use the word "proposition" without any clear meaning. Quine, for example, feels far less uneasy about quantification over class-variables than about quantification over attribute-variables and propositional variables, because there is a clear criterion for deciding whether we are dealing with two classes or with only one class referred to by different predicates: two classes are identical if they have the same membership. And such a criterion of identity is alleged to be lacking for intensions. The problem is a serious one and cannot be disposed of by applying a generalized Leibnizian principle of identity of indiscernibles: t,wo propositions are identical if they agree in all properties. For this would mean that two sentences express the same proposition if one can be substituted for the other in any context vithout change of truth-value. But either this criterion is applied to extensional contexts only, or it is also applied to modal and intentional1 contexts. If the former, it leads to the untenable conclusion that all sentences of the same truth-value express the same proposition (('proposition", then, becomes a redundant term; to ask what proposition is expressed by a given sentence would simply be asking whether the sentence is true or false). If the latter, then the criterion seems to be ineffective for several reasons. First, a non-extensional language permits the construction of such non-extensional sentences as "the proposition that p = the proposition that q" (such sentences are non-extensional with respect to "p" and "q" because obviously different propositions may have identical truth-values). And how is one to decide whether "p" is substitutable for "q" on the right hand side of this identity without changing the truth-value of the identity-statement, unless one already knows whether or not "p" and "q" express the same proposition? Secondly, consider a modal context like "it is necessary that (A is a father if and only if A is a male parent)". This modal statement is true if and only if "A is a father" is L-equivalent to "A is a male parent", but although L-equivalence is not, as I shall explain presently, in general a sufficient condition for identity of propositions, I doubt whether the L-equivalence of these two sentences could be established inde-

* Received

September, 1956. By an intentional context of "p" I mean a statement like "A believes that p", "A doubts that p" etc. 1




pendently of the assumption of their synonymy2-which is the question to be decided by the substitution test. I t may be replied that Leibniz' principle leads to similar difficulties when applied to individuals, and that the intensionalist, therefore, is really no worse off than the nominalist who uses the concept of "individual" without philosophical qualms. Indeed, there is something in this objection. If the predicatevariable in " ( P )(Px, = Py)", the Leibnizian definiens for "s = y", is completely unrestricted except as to type, then peculiar consequences follow: in the first place, "s = y" would be a value of "PJ", hence the attempt to decide an identity-statement on the basis of such a definition would be circular. Secondly, if an intentional function "A knows that x = y" is admitted as value of "Px", it would even follow that no identity-statement about individuals can be both inforrnativc and true. For, since undoubtedly everyone knows that r = x, whatever individual x may be, "r = y" could not then be true unless everyone knew that x = y. This is a generalization of the paradox of analysis and of what Carnap has called the "antinomy of the name-relati~n",~ anticipated by Russell's observation that king George wished to know whether Scott was the author of TVaverley, but not whether Scott was Scott. Quine has shown that unrestricted substitutivity of identity also leads to paradox if modal functions are admitted: it used to be thought that the morning star is identical with the evening star, but this was to overlook that the morning star is necessarily identical with itself whereas it is not necessarily identical with the evening star.* Nevertheless, the nominalists are right in feeling that identity of individuals is less problematic than identity of intensions, such as propositions. For, whether or not a consistent calculus of modal functions-to be distinguished from inodal operators, prefixed to names of propositions-be possible, none of the mentioned paradoxes would arise if the Leibnizian definition were restricted to first order functions in the sense of the ramified theory of types. This does not mean that the ramified theory of types, which is widely held to be dead, must be resuscitated in order to rescue the concept of identity of individuals. Expressions like "all the properties (of a given type, but of any order) of x" may be admitted as perfectly meaningful, yet a definition of "x = y" as "a and y share all extensional first-order properties, i.e. extensional properties not defined in terms of a totality of properties" is perfectly adequate. Nobody, for Given the synonymy of "father" and "male parent", the above modal statement is of course derivable from the truth of modal logic "it is necessary t h a t A is a father if and only if A is a father". See (11, $31. * Incidentally, some logicians would solve Quine's paradox of necessary identity by pointing out t h a t it cannot arise in the primitive notation in which the identity sign does not occur between descriptions but between logically proper names: there the identitystatements have either the form '(a = a" or the form "a = b", and hence are L-determinate since different individual constants are dejined as names of different individuals. But as the informative identity-statements of natural language always involve descriptions, I doubt whether this solution would satisfy anyone who doubted on philosophical grounds whether an identity-statement like "the morning star is identical with the evening star" could be true.



example, would seriously doubt the identity of Scott and the author of Waverley just because a king some time ago was doubtful of this identity while being perfectly certain of Scott's self-identity. I t is perfectly satisfactory, then, to define identity of individuals as agreement with respect to all extensional firstorder properties, whereas a similar restriction of substitutivity of identity does not, as we have seen, work in the case of intensions. ihccording to Carnap two designators have the same intension if and only if they are L-equivalent. A special case of this criterion is that two declarative sentences express the same proposition if and only if they are L-equivalent. But this mill not do either. For, an obvious criterion of adequacy which an explication of "proposition" (via the explication of s y n o n y m y of declarative sentences) should satisfy is that "A believes that p, and p = q" should entail "A believes that q". Yet, any two logically necessary statements are L-equivalent, but it could hardly be maintained that, where "p" and "q" are logically necessary, "A believes that p" entails "A believes that q". For example, anybody mith a rudimentary knowledge of the propositional calculus will believe a simple tautology like "((p 3 q).-q) 3 ~ p ' ) yet , there are tautologies with respect to which he could profess neither belief nor disbelief because he does not recognize them as tautologies. I n general, it is surely possible that, being familiar mith the semantic and syntactic rules for the symbols of a logical system, one understands the sentences of the system, i.e. knows what propositions they express, yet does not know whether the propositions expressed are logically necessary. We all understand the sentence "for n greater than 2, there are no yn = x7"', i.e. know what proposition it exsolutions for the equation: xn presses, but according to my information it is not yet known whether the proposition is logically necessary. But according to the L-equivalence criterion of propositional identity, we already believe this proposition if it is logically necessary! In order to bring the L-equivalence criterion into accord with the concept of (propositional) belief, one mould in fact have to stipulate, as a postulate partially defining "belief": ([A believes that p] and p entails q) entails (A believes that g). But this postulate is obviously not satisfied by the concept ordinarily meant by "belief". I t is only " A believes that p" together with "A believes that p entails q" that could be said to entail " A believes that q". The L-equivalence criterion does not seem to be satisfied by contingent propositions either. This can again be shown in terms of the evident requirement that "A believes that p, and p = q" should entail "A believes that q". I t will surely be granted that it is impossible to have a propositional attitude, whether belief or disbelief or doubt or any other, towards a proposition some of whose constituent concepts one does not "have". A man who does not understand the meanings of color predicates, e.g., cannot have a propositional attitude towards a proposition containing a color concept. Now, take any true contingent proposition p; it is L-equivalent to the proposition "if anyone asserted that p, he would make a true assertion." I t seems to me to be logically possible that a man who "has" the descriptive and logical concepts that are constituent of p should fail to have the concept of asserting, and therefore




believe p without having a propositional attitude towards this conditional proposition mhich is L-equivalent to it. I am not, however, convinced by this argument for the thesis that even in the case of contingent propositions Lequivalence is not a sufficient condition for identity. For its validity depends on the notion of "concept constituent of a proposition", mhich requires to be clarified. 1s the concept expressed by "red", for example, a constituent of the proposition expressed by "a is round and red, or round and not red"? Now, it is not according to a plausible definition of "constituent concept": the concept expressed by a constituent expression of a sentence is a constituent of the proposition expressed by the sentence, if and only if the expression occurs essentially in the sentence. Clearly, "red" does not occur essentially in the above disjunction of conjunctions. But it might be argued that "assert" likewise fails to occur essentially in the sentence "if anyone asserted that p, he would make a true assertion". For, the latter can be transformed into "if anyone asserted that p, he would assert a true proposition". Now, a verb occurs inessentially in a sentence if any consistent and yram~naticallyadmissible substitution of a different verb for it leaves the truth-value of the sentence unchanged. 13ut the only verbs that can be inserted for the dots in "soandso . . . that p" without producing nonsense are intentional verbs, i.e. verbs like "assert", "believe", "disbelieve", "presuppose" etc., and it is clear that any such substitution into "if anyone . . . that p, he would . . . a true proposition" is truth-preserving. Nevertheless, I do not think that by reference to inessential occurrence of terms the failure of logically equivalent contingent propositions to be identical can always be shown to be merely apparent. "x is orange" is logically equivalent to "n: is intermediate-in-color between Red and Yellow", but all the descriptive terms occur essentially, hence we can argue that a man having a propositional attitude towards the proposition expressed by the first sentence might fail to have a propositional attitude towards the proposition expressed by the second sentence, since he might, say, have a concept of the color Orange without having a concept of the color lied: for the latter concept is a genuine constituent of the proposition expressed by the second sentence. Let us return to our adequacy criterion: A believes that p, and p = q, entails that A believes that q. Would it be satisfied if we strengthened the identitycondition by requiring, instead of merely L-equivalence, intensional isomorphism of "p" and "q"? I t has been argued recently that even this extremely strong identity-condition-I shall argue presently that it is much too strongdoes not necessarily satisfy it. Thus Benson Mates5 seems to think that, though whoever believes that all Greeks are Greeks believes that all Greeks are Greeks, it is not necessarily the case that whoever believes that all Greeks are Greeks believes that all Greeks are Hellenes; yet, on the assumption that "Greek" and '(Hellene" are L-equivalent, the sentence "whoever believes that all Greeks are Greeks, believes that all Greeks are Greeks" (D)is intensionally isomorphic with the sentence "whoever believes that all Greeks are If we substitute D and D' Greeks, believes that all Greeks are Hellenes" (D').



for "p" and "q", and agree with Mates that it is possible to doubt that D' but not possible to doubt that D, then I suppose me must agree with him that no explication a t all of synonymy which allo~vsdiferent sentences to be synonymous could satisfy our adequacy criterion. However, while it is of course conceivable that a person respond affirmatively to the question "do you believe that everybody believes that all Greeks are Greeks" yet negatively to the question "do you believe that everybody believes that all Greeks are Hellenes", this does not establish Mates' contention. For if the subject were asked to support his doubt whether everybody believes that all Greeks are Hellenes, he could only do so by pointing out that somebody may have an imperfect knomledge of the English language so as to fail to know that "Greek" and "Hellene" are synonyms, and therefore fail to knom that the proposition expressed by "all Greeks are Greeks" is the same as the proposition expressed by "all Greeks are Hellene~".~ But then Mates' argument is simply based on the confusion between "A believes that p" and " A believes that 'p' expresses a true proposition" (as I have argued in (10)). Mates' argument does not prove that no explication of synonymy that is compatible with applicability of "synonymous" to pairs of distinct sentences must fail to satisfy our adequacy criterion (substitutivity in belief sentences). If it proves anything, it proves the inadequacy of a superficial behavioristic analysis of "belief" according to which a subject's response t o questions about his beliefs is conclusive evidence for or against hypotheses about his beliefs.' I t would indeed be strange if an experimental linguist reported that some people do not accept the law of identity, his evidence being that some people who professed belief when they were asked whether all Greeks are Greeks, expressed doubt when they mere asked whether all Greeks are Hellenes although the two sentences express (to him, the interrogating linguist!) the same proposition. Such a response would normally be taken as evidence, rather, that "Greek" and "Hellene" do not mean the same t o the subject. On the other hand, intensional isomorphism in Carnap's senses is too strong an explicatum for "synonymy", because a simple designator cannot be intensionally isomorphic with a compound designator. I believe that "father" is synonymous with "male parent", "ignoramus" with "me do not knom", "x3" with "x.x.z"; further, " p and q" with " q and p H , "p or q" with " q or p'), ('some A are not B" with "some d are non-B". And since none of these pairs are intensionally isomorphic, I conclude that either Carnap's explication is incorrect, or else his explicandum is not a concept of synonymy that is of interest for philosophical analysis. Carnap's reply t o a similar objection by Linskyg was "t is but for the sake of the argument that I assume here t h a t a trivial tautology of the form "all A are A" expresses a proposition a t all. It is interesting t o notice, though, that on Carnap's behavioristic analysis of "belief", viz. "A believes t h a t p = A is disposed to an affirmative response t o some sentence t h a t is intensionally isomorphic t o 'p' ", i t is logically impossible t h a t A should believe t h a t D, yet fail t o believe t h a t Dr. See (I), $$14, 15. O See (8).

that there is a family of stronger and weaker synonymy-concepts and that it is unfair t o criticize an explication of a stronger synonymy-concept on the ground that it does not fit a weaker synonymy-concept. But this reply is insufficient, since the explicandum which analytic philosophers are interested in is a semantic relation of which the pair "father, male parent", e.g., is an instance; and sec.ondly, since in a language in which "x3" is explicitly introduced as abbreviation for "x.r.x", 's33"is surely synonymous with "3.3.3" in precisely the same sense in which, say, " 3 . 3 . 3 " is synonymous with "3 X 3 X 3". The time is ripe for suggestion of a new approach to the problem of propositional identity. To begin with, it is utopian to look for an absolute criterion. Instead, propositional identity should be relativized to specified kinds of substitution-contexts. The weakest concept of propositional identity is propositional identity relative to extensional contexts: relative to an extensional language, "p q" is a sufficient condition for "f(p) =f(q)". Hence there is here no need to distinguish propositions from truth-values: the range of the sentential variables in propositional calculus comprises only two values, the True and the False. The idea that a law of propositional calculus is a truth-functional tautology could then be expressed as i'ollo~vs: if 'tf(p)" is a tautology, then "f(T).f(P)" holds, if "f(p, y)" is a tautology, then "f(T, T).f(F, T).f(T, F ) . f(F, F)" holds, etc. The next stronger concept of propositional identity is propositional identity relative to modal contexts: relative t o modal logic, strict equivalence, not material equivalence, is the sufficient condition for truthpreserving substitutivity of statements. "If p is L-equivalent to q, then f(p) = f(y)" expresses this stronger identity-condition. Accordingly, there are distinct propositions of the same truth-value in rnodal logic. However, relative t o modal logic there is but one necessary proposition and but one impossible proposition, since all necessary statements are strictly equivalent and all impossible statements are strictly equivalent. Now, \z e have seen that i t is the adequacy condition concerning belief which forces us t o distinguish one necessary proposition from another. This is t o say that L-equivalent statenients are not necessarily interchangeable in intentional contexts. The strongest identity-condition for propositions accordingly reads: if "x believes that p"I0 is strictly equivalent to "a believes that q"-which entarls that "p" is strictly equivalent to "q"-, then f(p) f(q). Or alternatively, j ( p ) = f(q) if p is strictly equivalent to q and inoreover B(x, p) B(x, q). Relative to an extensional language, propositional identity rneans material equivalence, relative to a modal language it means strict equivalence, relative t o an intentional language it rneans strict equivalence of the corresponding statements of belief. The objection is likely to be raised that no operational ("effective") criterion of synonymy 112s been provided a t all. For the only possible strict proof of the strict equivalence of "a believos that p" and "z believes that q" mrould rest on the proof of ".p = q", and hcnce the whole procedure would be circular. Now, in the firstplace, the same circularity arises already in connection with the weaker criterion af propositional identity which is in lo

This statement-form will henceforth be al~breviated:B ( x , p).



effect Carnap's criterion: in order to prove, e.g., that "father" is synonymous with "male parent" one ~vouldhave to prove that the biconditional joining them is just as necessary as the trivial biconditional "all and only fathers are fathers", which proof could be conducted only by substitution of "male parent9' for one occurrence of "father" in the trivial biconditional, which substitution could be justified only by the assertion of synonymy which is in question. The lesson to be learnt from this, it seems to me, is that the clarification achieved by analyzing "proposition" (via "same proposition") in terms of modal or intentional concepts, in a sense of "analysis" which requires that the analysans be clearer than the analysandum, is illusory. In the following, I shall propose an alternative method of, as it were, simz~lta~zeoz~s clarification of the category-term "proposition" and the non-extensional operators, both modal and intentional, through axiomatic definition. I n the meantime, it might be pointed out that a t least negative conclusions about synonymy can be arrived a t mithout apparent circularity through the test of substitutivity in intentional contexts. I think the following type of argument is perfectly respectable: "p" and "q" do not express the same proposition, because it is possible to know that p l ~ i t h o u kt nowing that g. I t is by this type of argument that I try to convince students who are accustomed to a loose use of the word "definition", that so-called definitions of color predicates in terms of wave-lengths do not express the ordinary meanings of the color-predicates, and that physicalistic definitions of nlentalistic terms likewise do not express the meanings of the latter: it is possible to Itnow that a thing is blue, or that one is feeling sad, without linowing which is the wave-length of the light the thing is disposed to reflect, and without knoving anything about one's physiological condition or behavior a t a time one 1s feeling sad. And just to forestall an irrelevant debate about the criterion of "knowledge", I hastily add that the argument remains unaffected by a substitution of "belief" for "kno~~ledge": if "p" and "q" are synonymous, then the statement "it is possible to believe that p without believing that q" is self-contradictory. Hence, if the latter statement is not selfcontradictory, the sentences are not synonymous. Nelson Goodmann has decried this negative test of synonymy as a pseudo-test, on the follo~vingground: with respect to any two statements a t all, it is possible that one should know one to be true without lino~vinsthe truth-value of the other, since one might not understand the other. And if the test is applied only to persons who understand both statements, then i t becomes redundant since one could not be said to understand both of two statements unless one knovrs whether or not they are synonymous. N o ~ ~ ~ e vGoodman's er, objection does not apply to my formulation of the test, for the latter involves statements of the form "A linows that p H , which are object-linguistic and non-semantic, not nietalinguislic statements of the form "A knows that ' p ' is true". The first part of Goodman's argument presupposes that "11 knows that 'p' is true" entails " A knows what proposition is expressed by 'p' ". But such semantic linowledge is obviously not presupposed by A's linowlcdge of the proposition that p. Kow, it must be admitted that to say two sentences express the same proposi-



tion if they are interchangeable, not only in extensional contexts, but also in modal contexts, is not illuminating unless the meanings of the modal operators, L< necessary", "possible"12 etc. are understood. But the relevant meaning of "possible", e.g., can, apart from illustrations, be indicated only by giving a set of axioms which are satisfied by the intended meaning: if p, then possibly p; if necessarily p, then possibly p ; if possibly (p and q), then possibly p and possibly q, but not conversely (for "not-p" is substitutable for "q") etc. Such an axiomatic definition13 a t the same time defines "proposition": the propositions to which modal logic is "committed" are the values of the sentential variables in the axioms of a system of modal logic. If, indeed, the variables are given completely unrestricted ranges, then the terms which ~vouldotherwise be used in the meta-language to characterize the subject-matter referred to by the axioms simply appear as additional primitives in the axioms. For example, one could diminish the nurnber of primitives in the axioms of formal Euclidean geometry, by using special variables ranging over points, special variables ranging over straight lines, and special variables ranging over plancs. This would be analogous to using the term "proposition" in the metalanguage to dcfine the ranges of the bound variables of modal propositional logic. But Ive could alternatively put it as a primitive into the axioms, just the way one normally handles "point", "straight line", "plane" as primitives along with the relational predicates that are used to make assertions about these entities; and then it would be obvious that "proposition" is defined simultaneously with the modal operators. Following this approach, let us formulate the stricter requirements of propositional identity that are imposed by belief-contexts by laying down axioms of a logic of belief, and saying that propositions are the values of the sentential variables in those axioms. The latter may be conceived as implicitly defining a t once "belief" and "proposition". 1 proceed to formulate five axioms of such a logic of belief which in terms of the intended meaning of "belief" are selfevident-indeed this is to assert a tautology since the axioms serve the purpose of explicating the meaning of this new primitive.

Eotice that only one primitive modality is required, e.g. "possible". I t is sometimes objected to axiomatic (or "implicit") definitions that they are not unique, since there is more than one model for any consistent (and nontrivial) set of axioms. But neither is there any guarantee t h a t an explicit definition is unique in a sense in which an implicit definition is not. For either the primitives occurring in the definiens are implicitly defined by the axioms of the deductive theory in which the defined term occurs, and then their ambiguity is communicated t o the defined term (e.g., since "straight line" as a primitive of a geometrical system admits of several interpretations, so does "triangle" which is explicitly defined in terms of "straight line"). Or else they are interpreted by means of ostensive definition. But there is no guarantee that an intended meaning is really communicated by means of ostensive definition, since the latter limits it only to a property shared by all the instances pointed to (and absent from all the negative instances pointed to, if such are used). At any rate, ostensive definition is applicable only to descriptive terms, not t o logical constant,^, hence only the first alternative is relevant in the present context. 14 Strictly speaking, "believes" requires a time-variable, so t h a t a triadic, not dyadic, l2 '3


The sense of A1 is that we cannot analytically infer from the fact that a person believes p that he believes q, if p does not entail q-though "p entails q" is compatible with "somebody believes p but does not believe q". Of course, if a person believes p without believing q although the latter proposition is entailed by the proposition he believes, this must be because he is not aware of the entailment. Hence A2 says, not that "p entails q" warrants the inference of " A believes q" from "A believes p", but that "A believes that (p entails q)" does. As is likely to be disputed by those who insist that people frequently believe contradictory propositions; they may say that what is here implicitly defined is rational belief, not actual belief. I would justify the axiom, however, by the consideration that if a man believes incompatible propositions, this is because he is not aware of the incompatibility. As a matter of fact, from A8 together with A:, we can deduce:

which means that it is impossible to believe propositions which one believes to be incompatible. This theorem may seem to be much weaker than -O(B(x, P.--P))


deducible from A4 and Asx5,since substitution of ('--p" for ('q" in T I yields 3 the apparently more cautious entailment: (B(x, p) .B(x, p 3 --p)) --(B(x, ~ p ) )But . the caution is unnecessary since a man who grasps the ordinary meaning of "not" cannot fail to see that the propositions expressed by "p" and by "not-p" are contradictory. I t would be futile to try to prove empirically that a man may (at the same time) believe explicitly contradictory propositions by obtaining affirmative responses from the same person to two sentences which, as interpreted by the interrogator, are contradictory. I should think one would inevitably infer from such responses that the subject has misinterpreted a t least one sentence. If I found a man insisting with great earnestness that the same part of a surface can a t the same time be both green and red, I would conclude that he is either an excellent actor or else does not mean predicate should be used to express the concept of belief. The axioms are of course meant as universally quantified with respect to the (omitted) time-variable.-As is added i n order ta differentiate belief from knowledge, for A]-A4 are satisfied by the relation of knowledge as well as by the relation of belief. I t should be noticed t h a t the possibilities of interpretation of the axioms are severely limited by the meta-linguistic explanation t h a t "x" ranges over persons and "p" and "q" take declarative sentences as substituends. l 6 Indirect proof: Suppose B(A, (p.p)). By A4, this entails B(A, p). B(A, -p). By simplification, B(A, -p). By A3 this entails -(B(A, p)). Since the hypothesis thus entails the contradiction B(A, p). -(B(A, p)), i t is self-contradictory.



by ((green" or "red" what one usually means. However, it is possible to believe propositions which are incompatible if their incompatibility is not self-evident. Therefore the inference from an affirmative response to several sentences which in their intended interpretation are contradictory to the subject's ~nisinterpretation of a t least one sentence, is not always warranted. This is the reason why the second conjunct of the antecedent of AZ is " B ( x , (p 3 q))" instead of "p 3 q". A definition of "understanding 'p' " from which it follows that one does not understand "pH unless one knows all that is entailed by "p", is both arbitrary and ineffective-though it cannot be denied that the test of whether one has understood "p" includes a test for awareness of some consequences of "p". The specified axioms serve as adequacy criteria for behavioristic interpretations of "belief", in much the same way as interpretations of "probability" will usually be accepted as adequate only if they satisfy the axioms of the calculus of probability. By a behavioristic interpretation is meant an interpretation of "belief" as a disposition manifested in responses to specified stimuli. For example, "A believes that p = A is disposed to respond affirmatively to some sentence synonymous with 'p' ", as suggested by Carnap in Meaning and Necessity. When combined with Carnap's L-equivalence criterion of propositional identity, this interpretation leads to the contradictory consequence that a person may believe and also not believe (at the same time) one and the same proposition, since he may respond affirmatively to some sentence synonymous in L with "p" yet fail to respond affirmatively to some sentence synonymous in L with "q" though "p" and '(q" are L-equivalent in L. This would of course be due to his not recognizing the L-equivalence in question, but as already pointed out it would be gratuitous hence to infer that he misinterprets a t least one of the two sentences. This contradiction can be avoided by strengthening the requirements for propositional identity, so as to make "p = q" entail ('A believes that p if and only if A believes that q". For if p = q in the sense that p ,, and "q" are interchangeable even in belief-contexts, then obviously it ( (

must be the case that a person believes that p if and only if he believes that q. And furthermore, on this assumption of strong synonymy of "p" and "q", it is logically necessary that A is disposed to respond affirmatively to some sentence synonymous with "p" if and only if he is so disposed towards some sentence synonymous with "q". But having defined synonymy in terms of belief and belief implicitly in terms of a set of axioms, we face the question whether the behavioristic definition of belief satisfies those axioms. Now, it obviously does not satisfy As, for example, since it is quite possible that A respond affirmatively to some sentence synonymous with ( ' ~ p "and also to some sentence synonymous with "p", because he does not interpret these sentences as contradictory. This indicates that the behavioristic definition is inadequate unless an important "mentalistic" condition is added to the linguistic stimulus: a t best we may infer "A believes that p" from "A was asked to respond affirmatively or negatively to the sentence S , and A interpreted S to mean the proposition p, and A responded affirmatively to 8". I say "at best" because even this



better grounded inference is not necessary: A may assert the proposition that p without believing it; he may be lying, in other words. This consideration shows that an interpretation of "belief" in behavioristic terms can hardly be adequate if it takes the form of an explicit definition in terms of a causal implication, unless the antecedent of the latter contains a "ceteris paribus" clause which covers our ignorance of relevant "intervening variables" (such as correct interpretation). The same objection would apply to introduction of "belief" by reduction-sentences conceived as analytic, like Carnap's "bilateral" reduction sentencesI6. Carnap, therefore, is surely on the right track in advocating, more recently117 that "belief" be treated as a theoretical construct which is implicitly and incompletely defined by the postulates of a psychological theory-though whatever theoretical postulates Carnap may have in mind must be supplemented by reduction-sentences which tie the term to the observation-language in the loose way of indefinite probabilityimplications. The important point in the present context is that "proposition", being defined correlatively with "belief", will inevitably share the latter's openness of meaning.18 Notice that the terms "belief" and "intensionw-a proposition being a special kind of intension-are almost inseparable in reductionsentences connecting them with behavioristic terms. As we saw, no reliable inference can be drawn either from a belief-hypothesis to a verbal response or conversely unless an assumption about an act of interpretation is warranted. But likewise, a hypothesis about a habit of interpretation-e.g., A is in the habit of interpreting "dog" to refer to a quadruped of kind K-is not highly confirmable by bare observation of linguistic behavior, for it is highly relevant to the question what a person means by an expression to know what beliefs motivate him to apply the expression to such and such objects. This interconnection of the constructs "belief" and "intension" might be expressed by the following postulate: if predicate "P" has property P as its intension for A, then, ceteris paribus, A applies "P" to an object x if and only if A believes that x has P. The "ceteris paribus" proviso covers such, not exhaustively known, conditions as suitable stimuli to a verbal utterance, a desire to express the belief and not to mislead, absence of relevant inhibitions etc. The postulates for other kinds of intensions, like propositions conceived as intensions of declarative sentences, would be similar. Such postulates are perhaps less "analytic" than the axioms stated above, in the sense that observations of human responses would be more likely to lead to their revision than to a revision of the more "logical" axioms. But a strict analytic-synthetic distinction must be abandoned if a postulational method of meaning-specification is adopted.lg I t remains to contrast the outlined approach to the definition of "intensional" conceptswhich in its recognition of semantical "constructs" that are not del 8 See

(4), see. 8. See (2). '8 For a detailed discussion of "openness of meaning", see my articles (12) and (13); also (6), section 11. l9 This is lucidly argued by Hempel, in (7). I?



h a b l e in the nominalistic vocabulary is akin to the liberalization of strictly positivistic meaning criteria with respect to physics and psychology-with the approach of the logical-constructionis.ts. According to the theory of Principia Matlzematica, classes are logical constructions in the sense that names of classes as well as class-variables are contextually eliminable. I t is in this sense that according t o old-fashioned phenomenalism material objects are logical constructions out of sense-data: expressions referring to material objects were supposed to be in principle eliminable by translation of material-object-statements into the sense-data language. In just the same sense Ayer, for example, regards propositions as logical constructions out of sentences-a convenient fagon de parler that can in principle by dispensed with. Since I have advanced detailed arguments against this kind of "reductionism" elsewhere,20 I shall confine myself here to one fundamental criticism. The logical-constructionists hold that statements ostensibly referring to such pseudoentities as propositions are shorthand lor metalinguistic statements about classes of synonymous sentences. And this implies that they are translatable into a meta-language containing sentential variables and names of sentences (formed by means of the familiar quotes), but no propositional variables nor names of propositions. But surely the statement "there are propositions which are not expressed" is not so analyzable. Therefore the logical-constructionists could make good their claim only if they could show that, unlike statements about expressed propositions, this statement is meaningless. Now, the latter is incomplete in two respects: it does not specify a time, nor a specific language. Let us take it in its strongest form: there are propositions which are not expressed in any language a t any time. I t is not my purpose to argue for the truth, but only for the rneaningfulness of this statement. For this purpose we may begin with the innocent and undoubtedly true statement: there Bras a time when nobody believed, or even thought of the proposition that the earth is round, and when no language a t all existed. The proposition that the earth is round, therefore, was not expressed a t that time. Therefore there is a proposition which was not expressed a t that time. But that a proposition comes to be expressed in a language, is a contingency. It is therefore possible that this proposition should have remained unexpressed for ever-though we could not have mentioned this possibility had it been actualized. And if "it is possible that p is a t all times unexpressed" is true, then " p is a t all times unexpressed" must be a meaningful statement. That it is a pragmatically self-refuting statement, in the sense that its falsehood follows from its assertion, is irrelevant. In the same sense "I am not asserting anything now" is pragmatically self-refuting, yet it is logically possible that the person denoted by "I" should not be asserting anything a t the time denoted by "now". The enemies of propositions will be quick to point to the fallacy in the foregoing argument: the step from "the proposition that pl is not expressed a t t" to "there is a p such that p is not expressed a t t" begs the question. It presupposes that "that pl" is a genuine name, which it can be only if there are propo20

See (ll),esp. sec. 3.



sitions. I t is like the inference from "roundness is a property shared by all nickels" to "there is a property which is shared by all nickels". But since the apparent name "roundness" is contextually eliminable by translation of the above sentence into "all nickels are round", the apparent basis for existential generalization disappear^.^^ Indeed, if the claim of the contextual eliminability of propositional names and variables can be made good, then the nominalist will be justified in saying that there are no propositions-or in a more tolerant vein, that we need not assume that there are. Since I have expressed serious doubts about the possibility of such contextual elimination, some may label (or libel?) me as one holding a metaphysical belief in propositions as "real entities". I would like to conclude, therefore, by inquiring what it means to believe that "there are propositions". Quine has offered a much debated criterion of a man's "commitment" to a special kind of ontology: Look a t the variables of quantification in his language; if they belong to the primitive notation, then the user of that language is committed to the belief that the entities over which they range exist. But he has not, to my knowledge, discussed the question what it means to say that such entities exist. Now, within a realistic language we encounter only qualified existence assertions, not what might be called categorial existence assertions: "there are attributes (or propositions) satisfying the function f(4) (or f(p))"-e.g., "there are propositions which nobody believes", "there are attributes which are possessed by only one individual",-not "there are attributes (or propositions)". I n Carnap's tern~inology,categorial existence assertions are external to a given language.22We may not like Carnap's claim that the corresponding questions like "are there propositions" are devoid of cognitive meaning (unlike the questions concerning qualified existence which are formulated within a given language), but whosoever claims the contrary should offer an interpretation of such questions. I suggest that we take Quine's criterion of ontological commitment as the very definition of ontological commitment and thereby assign a cognitive meaning to categorial existelice assertions. That is, to say [[there are propositions" is to say that the propositional names and variables which we employ in order to say what we want to say are not contextually eliminable, that they belong to the "ultimate furniture" of our language. Thus nebulous questions about the ultimate furniture of the (extralinguistic) universe are reduced to less nebulous questions about the ultimate furniture of cognitive language. The question whether propositions and other abstract entities exist is not, indeed, decidable empirically, not even in the indirect empirical way in which scientists decide whether atoms and electrons exist, but it is nonetheless a cognitive question: it is decidable the way questions of semantic analysis are decidable, by examining whether proposed translations into a language with a specified primitive vocabulary preserve the meanings of the translated statements. Of course, in what precise sense of "meaning" such philosophical translations into ideal languages are required 2'

See (14).


See (3).



to preserve meanings, is a,notller question-fortunately this paper.

beyond the scope of

REFERENCES 1. R . CARNAP, Meaning and iiecessity, Chicago 1947. 2. "On Belief Sentences", reply t o Alonzo Church, in Philosophy and Analysis, ed. M. Macdonald, Oxford 1954. 3. "Semantics, Empiricism and Ontology", in Linsky (ed.), Semantics and the Philosophy of Language, Urbana (Ill.), 1952. 4. "Testability and Meaning", in Philosophy of Science 1936/1937. 5. N . GOODMAN, "On a Pseudo Test of SynonymyJJ,Philosophical Studies, Dec. 1952. 6. C. G. HEMPEL,Fundamentals of Concept Formation in Empirical Science, I n t . Encyclopedia of Unified Science, vol. 11, no. 7. 7. "A logical appraisal of operationism", Scientific Monthly, Oct. 1954. 8. L. LINSKY,"Some notes on CarnapJs concept of intensional isomorphism and the paradox of analysis", Philosophy of Science 1949. 9. B. MATES,"Synonymity", in Linsky (ed.), loc. cit. (see (3)). 10. A. PAP, "Belief, Synonymity and Analysis", Philosophical Studies, Jan. 1955. 11. "Necessary Propositions and Linguistic Rules", in Semantica (Archivio di Filosofia, Rome 1955). 12. "Reduction Sentences and Disposition Concepts", forthcoming in T h e Philosophy of Rudolf Carnap (Library of Living Philosophers, ed. A. Schilpp). 13. "Reduction Sentences and Open Concepts", Methodos vol. V, no. 17. 14. W. V. QUINE, "Designation and Existence", in Feigl and Sellars, Readings in Philosophical Analysis.