In this paper we shall prove theorems about some systems of sentential calculus, by making use of results we have established elsewhere regarding closure algebras and Brouwerian albegras. We shall be concerned mostly with the Lewis system and the Heyting system. Some of the results here are new (in particular, Theorems 2.4, 3.1, 3.9, 3.10, 4.5, and 4.6); others have been stated without proof in the literature (in particular, Theorems 2.1, 2.2, 4.4, 5.2, and 5.3).

The proofs to be given here will be found to be mostly very simple; generally speaking, they amount to drawing conclusions from the theorems established in McKinsey and Tarski [10] and [11]. We have thought it might be worth while, however, to publish these rather elementary consequences of our previous work—so as to make them readily available to those whose main interest lies in sentential calculus rather than in topology or algebra.

The purpose of this paper is to show that some of the difficulties confronting logicians are caused by too narrow an interpretation of the function of mathematical logic. I hope to show that very fruitful results can be obtained by considering a much wider range of problems than usually found in the literature.

I believe that the function of mathematical logic is to study all logical systems. The problems now considered usually concern some particular system or set of systems. Of course we would have to agree first on what a logical system is. I shall suggest a tentative definition. The rest of my paper will proceed on the assumption that we have agreed on this question. However, it is only necessary for this paper that some definition should be accepted, it is not necessary that it should be the one I suasest.

There are logicians who maintain that modal logic violates Leibniz's principle that if x and y are identical, then y has every property of x. The alleged difficulty is illustrated in the following example due to Quine.1

A. It is logically necessary that 9 is less than 10.

B. 9 = the number of the planets.

C. Therefore, it is logically necessary that the number of the planets is less than 10.

The two following papers are in need of certain corrections: Modal functions in two-valued logic and Note on modal functions. I became aware of the need for these corrections while comparing these earlier papers of mine with J. C. C. McKinsey's paper, On the syntactical construction of systems of modal logic.

The first of the two papers should be corrected in such a way that the system of two-valued logic there employed becomes “complete” in the sense that either p or ∼ p is in Τ for every Κ-symbol Τ. This could be achieved by use of an additional axiom that postulates the required completeness. (I mistakenly supposed that the original set of axioms guaranteed this completeness.)