The purpose of this article is to introduce the notion of “recursive realizability.”

Let P be some property of natural numbers. Consider the existential statement, “There exists a number n having the property P.” To explain the meaning which this has for a constructivist or intuitionist, it has been described as a partial judgement, or incomplete communication of a more specific statement which says that a certain given number n, or the number n obtainable by a certain given method, has the property P. The meaning of the existential statement thus resides in a reference to certain information, which it implies could be stated in detail, though the trouble is not taken to do so. Perhaps the detail is suppressed in order to convey a general view of some fact.

The information to which reference is made should be thought of as possibly comprising other items besides the value of n or method for obtaining it, namely such items as may be necessary to complete the communication that that n has the property P.

In an m-valued propositional calculus, or a formalization of such a calculus, truth-value functions are allowed to take any truth-value t where 1 ≦ t ≦ m and m ≧ 2. In working with such calculi, or formalizations thereof, it has been decided to distinguish those truth-values which it is desirable for provable formulas to have from those which it is not desirable for provable formulas to have. The first class of truth-values is called designated and the second undesignated. This specification of certain of the m truth-values as designated and the remainder as undesignated is one of the distinguishing characteristics of m-valued propositional calculi, and it should be observed at the outset that two m-valued propositional calculi will be considered to differ even if they differ only in respect to the number of truth-values which are taken as designated.

This paper is intended to serve a twofold purpose. Its ultimate aim is the presentation of a syntactical definition of the degree of confirmation of a hypothesis on the basis of given evidence, a notion which is known to be of outstanding significance for the logic of inductive reasoning in the empirical sciences. The theory of confirmation to be developed here has as its foundation the theory of probability, and in order to make that foundation sufficiently secure, it was found expedient to present the theory of probability in axiomatic form and to introduce a syntactical interpretation of probability suitable for the intended application in the theory of confirmation. In this sense then to clarify the logical foundations of probability theory is this paper's first aim.

The notation that I shall use here for the logic of quantification is of the familiar kind wherein the letters ‘p’, ‘q’, etc. stand in place of unspecified statements and the letters ‘ƒ’, ‘g’, etc. stand in place of unspecified predicates. The present section will deal with the significance of this notation; the purpose and scope of the paper as a whole can better be indicated afterward, in §2.

By “predicates” I mean, not properties (or classes) and relations, but merely certain notational expressions. As a first approach they may be thought of as expressions like ‘walks,’ ‘is red,’ ‘touches,’ ‘gives to.’ Where ‘ƒ’, ‘g’, ‘h’, and ‘k’ represent these four predicates, we may read ‘ƒx’ ‘gy’, ‘hxy’, and ‘κxyz’ as ‘x walks,’ ‘y is red,’ ‘x touches y,’ ‘x gives y to z.’

The present note is conceived as a sequel to Mostowki's paper, Über die Unabhängigkeit des Wohlordnungssatzes vom Ordnungsprinzip (which will be quoted as “M.”).

In that paper Mostowski considers a system of axioms , very close to the system of axiomatic set theory of Bernays, but where the axiom of choice is not supposed to hold. Mostowski proves that the well-ordering theorem (Wohlordnungssatz) cannot be derived from the system plus the principle of simple ordering (Ordnungsprinzip).

Consider an axiomatic set theory in which there is a distinction between “sets” and “classes,” only sets being allowable as elements. How can one define a finite sequence of classes? This problem was proposed to me by A. Tarski, and a solution is given in this note. We shall assume the axiom system Σ used by Godei in his study of the continuum hypothesis, and shall use the same notation.1

When C. I. Lewis developed his theory of strict implication, he left open the question which of his various systems should be regarded as being closest to our intuitions—though he was inclined to favor the system S2. There are to be found in the literature numerous discussions of this question; most of these have condemned S2 as being too strong, and have proposed ways of weakening it.

In the present paper I shall attempt to throw some light on this question by setting up a syntactical definition of “possibility.” I shall show that every system of modal logic constructed on the basis of this definition is at least as strong as the Lewis system S4.

As the intuitive basis for the syntactical definition of possibility, I take the position that to say a sentence is possible means that there exists a true sentence of the same form. Thus, for example, it would be said that the sentence, “Lions are indigenous to Alaska,” is possible, because of the fact that the sentence, “Lions are indigenous to Africa,” has the same form and is true.

Wiener, in 1914, reduced the theory of relations to that of classes by construing relations as classes of ordered pairs and defining the ordered pair in turn on the basis of class theory alone.1 The definition, as improved by Kuratowski,2 identifies the ordered pair x;y with uxyi(ix U iy).

In terms of Russell's theory of types, x;y in the above sense is two types higher than x and y. Even when we abandon Russell's theory of fixed types of objects in favor of a theory of stratified formulae,3 there is still significance in saying that ‘x;y’ is of type 2 relative to ‘x’ and ‘y’—meaning that a test of the stratification of any context involves assigning a higher number by 2 to ‘x;y’ than to ‘x’ and ‘y’.