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  and . 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.