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Published online by Cambridge University Press: 22 April 2025
Quine and Goodman have described in a recent paper a procedure whereby sets of extra-logical postulates are to be so reformulated that each postulate of a set will become a provable theorem. In its simplest form, the procedure in question may be outlined as follows. Let K be an extra-logical constant, and let P(K) be the conjunction of a set of postulates constructed on K. Let C be a logical constant such that P(C) is a theorem of logic.
1 See Elimination of extra-logical postulates, this JOURNAL, vol. 5 (1940), pp. 104-109, and a review by Barkley Rosser, ibid., vol. 6 (1941), p. 37. The notation here used is that of Rosser.
In the course of formulating the present remarks; I have been indebted to Alonzo Church for important comments and criticisms, which for the most part cannot be here indicated because the text to which they referred has been abandoned. In particular, however, Church has pointed out to me the bearing of Russell's discussion of descriptions and denoting on the procedure proposed by Quine and Goodman; indeed, what is said below in regard to this is a paraphrase of remarks made by Church.
2 Quine and Goodman must of course rely upon the fact, not the hypothesis, that P(K) is true, because it is this hypothesis that is to be eliminated by the procedure under consideration. On the other hand, passing from P(K) to P(K*) does not require that P(C) be provable; it is simply that unless this is so, P(K*) will not be provable. The question of the validity of the procedure could therefore just as well be discussed without this assumption.
It is to be noted, too, that although P(K*) is not a postulate, it replaces P(K) and serves postulational purposes. Having proved P(K*) by reference to P(C), we can then proceed without further recourse to P(C) and prove theorems just as if K* were a primitive constant. Let us call these the K*-theorems, and let us call the theorems which follow from P(K) the K-theorems. There will be a one-to-one correspondence between these two classes of theorems; and it is held that corresponding theorems will be equivalent in their extraformal meanings in such a way that they can do duty for each other in an extra-logical theory. There is, however, also a one-to-one correspondence between K*-theorems and theorems provable from P(K) v P(C), which we may call X-C-theorems: T(K*) will be provable from P(K*) if and only if T(K) v T(C) · P(K) ⊃ T(K) is provable from P(K) v P(C), and this will hold within a formalism in which P(C) is not provable. In such a formalism, each K*-theorem follows from P(K) v P(C) and each K-C-theorem follows from P(K*). If, therefore, P(K*) can replace P(K) for postulational purposes, so can P(K) v P(C).