Necessary Propositions and Entailment-Statements

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Necessary Propositions and Entailment-Statements P. F. Strawson Mind, New Series, Vol. 57, No. 226. (Apr., 1948), pp. 184-200. Stable URL: http://links.jstor.org/sici?sici=0026-4423%28194804%292%3A57%3A226%3C184%3ANPAE%3E2.0.CO%3B2-T Mind is currently published by Oxford University Press.

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V1.-NECESSARY P R O P O S I T I O N S A N D ENTAILMENT-STATEMENTS.

AT the end of his recent excellent paper on entailment, Mr. Korner reiects the view advanced bv Professor Moore that " the proposition that p entails q is not itself a necessary proposition ; but, if it is true, then the proposition that p 3 q is a necessary proposition ".I To me, on the contrary, it seems that Professor Moore's distinction between necessary propositions and entailment-statements contains the germ of a more satisfactory account of " entailment " and "necessity" than has yet, as far as I know, been explicitly advocated. Taking that distinction as my starting-point, I shall summarise as follows the position I propose to expound and defend. (1) Statements descriptive of the use of expressions are contingent. (" Contingent " is to be used as equivalent to " not(either necessary or self-contradictory) "). (2) Entailment-statements (i.e. statements in which the main verb is " entails ") are " intensional contingent statements ". " Other intensional contingent statements are statements in which the main verb is the verb " t o be " and the complement of that verb is some expression such as " necessary ", " impossible ", 6< consistent ", used in the senses " logically necessary ", " logically impossible ", " logically consistent ". Such words as " entails ", and, when used in this way, " impossible ", " consistent ", " necessary ", etc., will be referred to as " intensional words ".2 Korner, " On Entailment ", Proceedings of the Aristotelian Society, 1946-1947, p. 161. Cf. also G. E. Moore, " External Relations " in Philosophical Studies, p. 302. The word "intensional" is chosen because of its recent use in connexion with the question discussed in the final section of this paper, viz. the alleged antithesis between an " intensional " and an " extensional " logic. " Modal " and " non-modal " have sometimes done duty in the same connection. Perhaps " modal " is the more generally familiar term in application to some of the words here classified as "intensional But the ordinary use of "modal" is both too wide and too narrow for the present context; too wide, because modal expressions have a non-intensional as well as an intensional use ; too narrow because not all the words here classified as "intensional " are normally called " modal 184

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(3) In making contingent intensional statements, we use the main intensional words, and mention the other expressions they contain. These statements are of a higher order than statements in which the expressions mentioned in the contingent intensional statements are used, but not mentioned. They are contingent as any statement about the use of expressions is contingent, but they differ from other statements about the use of expressions in certain ways to be referred to later on. (The fact that an expression is mentioned will be consistently indicated by putting it in quotation-marks. It will be necessary later to distinguish between different kinds of " mention ", but not, I think, to introduce a special notation to mark them.) (4) Some expressions mentioned in some contingent intensional statements are themselves intensional expressions. (Korner's " E2-propositions" are expressed in statements of this kind.l But it is necessary to distinguish, as Korner does not distinguish, two classes of E2-statements. It is necessary to distinguish E2-statements mentioning two expressions in which the intensional components are identical from those mentioning two expressions in which the intensional components are not identical. Of these the latter, but not the former, may be held, in Korner's language, to "determine the logical meaning of entailment ". The failure to distinguish between them is the foundation, or a part of the foundation, of C. I. Lewis's erroneous claims for an " intensional logic " (see below, last section). (5) To every true entailment-statement there corresponds a necessary statemenk2 This correspondence is not logical equivalence. It can be described by saying that every true entailment-statement is logically equivalent to another contingent intensional statement which mentions a necessary statement or its contradictory. Thus every contingent intensional statement of the form " ' p ' entails ' q ' " is logically equivalent to another contingent intensional statement of the form " ' p :, q ' is necessary " ; or to a contingent intensional statement of the form " ' p . not-q ' is impossible ". When an entailment-statement is false, the logically equivalent contingent intensional statement of the form " ' p 3 q ' is necessary " fails to mention a necessary Cf.KKner, op. cit., pp. 145-162. I shall often use the word " statement " to refer to what it would be more correct to call, say, a " statement-form " (an expression, containing free variables, which could be turned into a statement by " binding " the variables or substituting appropriate constants). A statement-form js necessary when every statement formed from it in one of these ways is necessary. a

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statement in spite of the claim to do so. Thus, " ' X is red ' entails ' X is round ' " is false : and " X is red a X is round " is not necessary. A false entailment-statement is not impossible or self-contradictory, but simply false ; just as a true entailmentstatement is not necessary, but simply true. (6) I propose so to use the word "entails " that no necessary statement and no negation of a necessary statement can significantly be said to entail or be entailed by any statement. That is, the function " ' p ' entails ' q ' " cannot take necessary or self-contradictory statements as arguments. The expression " ' p ' entails ' q ' " is to be used to mean " ' p 3 q ' is necessary, and neither ' p ' nor ' q ' is either necessary or self-contradictory " ; or " ' p . not-q ' is impossible and neither ' p ' nor ' q ', nor either of their contradictories, is necessary ". It will be seen that the outstanding features of these suggestions are the two following : (i) the assertion that entailmentstatements are contingent ; (ii) the denial that necessary statements (or their contradictories) entail or are entailed by any statement. These conventions are recommended on the " general ground that they enable us to clear up some confusions in our use of " entailment " and " necessary " without denying any facts about deductive inference. More specifically, their acceptance has the following advantages : (i) it provides a simple solution to the so-called " paradoxes of implication " ; (ii) it points in another way to the truth to which what has been variously called the " linguistic " or the " conventionalist " theory of a prior4 propositions points ; (iii) it indicates the source and nature of the mistaken belief that there is an antithesis between " extensional " and " intensional " logic. I shall now attempt to show that these advantages are :ecured and that no seriously unwelcome consequences are involved.

The " Paradoxes of Implication". Nobody now accepts-if indeed anyone ever made-the identification of the relation symbolised by " a " with the relation which Moore called " entailment ". That is, " p a q " (i.e. " i t is not the case that p and not-q ") is rejected as an analysis of " ' p ' entails ' q ' ", because it involves such paradoxical consequences as that any false proposition entails any proposition and any true proposition is entailed by any proposition. It is a commonplace that C. I. Lewis's amendment had consequences scarcely less paradoxical. For if p is impossible (i.e. self-contradictory), it is impossible that p and not-q ; and if q-is necessary, not-q is impossible and it is impossible that

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" ' p ' entails ' q ' " means " it is impossible that p and not-p ", then any necessary proposition is entailed by any proposition and any self-contradictory proposition entails any proposition. On the other hand, Lewis's definition of entailment (i.e. of the relation which holds from " p " to " q " whenever " q " is deducible from " p ") obviously commends itself in some respects, and I have, indeed, asserted something very like it in (5) above. Now it is clear that the emphasis laid on the " expression-mentioning " character of the intensional contingent statement by writing, as in (5) above, q " strictly implies " q " ' is a law of the system " means merely that " p . p > q " entails " q ", then what on earth is the significance of calling " p . p > q : > . q " a necessary proposition or a law ? This is not in any case an open alternative, since " ' p > q ' is necessary " is to be rendered, in Lewis's terms, as " it is impossible that not-(p > q) ", which is to be read as " ' p . not-q ' is impossible ", which is equivalent to " ' p ' entails ' q ' " or " ' q ' is deducible from ' p ' ". Briefly, for any expression of the form " p > q " which Lewis obtains as a law in his system, he also obtains " i t is impossible that not-(p > q) " as a law. Now not only is it absolutely plain that, from the point of view of information about deducibility, one of these expressions is redundant. It is also plain that Lewis is committed to the possibility of obtaining an infinite series of such laws. And indeed it is only by the adoption of the convention here recommended (viz. that intensional statements are contingent), that such an infinite series is avoidable. For if " ' p ' is necessary " is necessary whenever it is true, then " ' " p " is necessary ' is necessary " is necessary, and so on. There seems to be no virtue in such an indefinite multiplication of orders of necessity. Lewis and Langford, op. cit., pp. 245 246.