Results concerning same modal systems that contain S2

Abstract

It is well known that if a postulate LLCpp or a rule of procedure ├α→├Lα (from a thesis α of a considered system to infer a thesis Lα of that system) is added to Lewis's modal system S3, we get his system S4 (see [7], p. 148). It is also known that the addition of this rule to S2 and SI has the effect of converting these systems into the Gödel-Feys-von Wright system T (M). My purpose in this paper is first to draw attention to some other ways of converting S3 and S2 into S4 and T respectively, as well as to extend a theorem of Halldén on S3, S4 and S7 to S2, T and S6. The results reached will also apply to the systems S3.5, S5 and S7.5, where S3.5 and S7.5 are obtained, respectively, from S3 and S7 by the addition of the postulate CNLpLNLp; an answer will be given to the question of irreducible modalities in S3.5. Moreover, two results will be proved that bear on the problem whether the systems S2 and T can be axiomatized by means of a finite number of axiom schemata and material detachment as the only primitive rule of inference.