The theory of relations was discussed, along with the theory of classes, by Peirce and Schröder. While the calculus of classes has subsequently been presented (by Couturat, and by Huntington) as an abstract mathematical system, no similar formulation has been published of the calculus of relations. Professor Alfred Tarski has proposed to me, the problem of developing a similar formulation for the theory of relations. In this paper I present a set of postulates for the calculus of relations.

The postulates have been chosen so as to enable us to prove Theorem A, below, which asserts a kind of completeness of the system. A set of postulates is called semi-categorical, if every two realizations which involve the same number of elements are isomorphic. From Theorem A one can easily prove that my postulates are semi-categorical, if one assumes the following theorem from Mengenlehre: if α and β are cardinal numbers such that , then α=β. This theorem has not been proved, however, except on the assumption of the axiom of choice and the generalized continuum hypothesis, and is not assumed in the present paper.

It should be pointed out, that the sort of completeness I have chosen is not the only sort, or even necessarily the most interesting sort, of completeness which a set of postulates for the calculus of relations could possess. For example, the question arises, whether it would be possible to find a set of postulates from which all true equations (involving free variables and the operations used by Schröder) could be derived, and which was the weakest possible such set. The present work should therefore be regarded only as a first step in the investigation of the axiomatization of the calculus of relations.

Under certain circumstances a postulate can be eliminated in favor of a mere definition, or convention of notational abbreviation. Suppose e.g. that a given system presupposes the machinery of ordinary logic and contains in addition a single extra-logical primitive: the relation Pt, “is a spatial part of.” Suppose this relation is governed by a single extra-logical postulate, to the effect that Pt is transitive:

Now instead of Pt, the relation O of spatial overlapping might be taken as primitive; Pt could then be defined in terms of O as follows:

The transitivity of Pt then follows by purely logical principles from the definition. The statement (1) with its ‘Pt’ clauses expanded according to (2) is a purely logical theorem, demonstrable independently of any stipulations or assumptions concerning the properties of O. Thus, through a change involving neither increase nor decrease of primitive ideas, the need of adopting (1) as a postulate is removed.

Or again, consider a system comprising ordinary logic and the sole extra-logical primitive S, the relation of simultaneity, governed by a sole postulate to the effect that S is symmetrical:

Here instead of S the relation N, “is no later than” might be taken as primitive; S could then be defined in terms of N as follows:

Postulation of (3) then becomes unnecessary, for (3) is an abbreviation, according to (4), of a theorem which is demonstrable within pure logic.