On the number of complete extensions of the Lewis systems of sentential calculus

We say that a system S of sentential calculus is an extension of a system R , if R and S have the same class of (meaningful) sentences, and every provable sentence of R is also provable in S . If the two classes of provable sentences do not coincide, we call S a proper extension of R . By a complete system of sentential calculus is meant one which is itself consistent, but has no consistent proper extensions. Thus one cannot add a new independent primitive sentence to a complete system without obtaining an inconsistent system. The usual two-valued sentential calculus is complete in the sense defined. It has been shown by Lindenbaum that every incomplete system of sentential calculus possesses at least one complete extension. In this paper we shall examine how many complete extensions there are of some of the Lewis systems of sentential calculus. We shall show that there is only one complete extension of S4 (and hence also of S5, which is an extension of S4), and that there are infinitely many complete extensions of S2 (and hence also of S1, since S2 is an extension of S1). We leave open the question how many complete extensions there are of S3.