The System S9

Abstract

The Lewis systems of modal logic S1–5 were originally constructed as N-K-M systems, i.e. with the help of primitive operators for negation (N), conjunction (K) and possibility (M). As a result they are normally considered as strengthenings of classical two-valued logic (PC), since the axioms and rules of inference of PC are easily derivable in even the weakest of them. If, however, S1–5 are reformulated as C-N-K systems in which the primitive operator C of strict implication replaces M (the latter then being definable via the definition Mα = NCαNα), the Lewis systems appear in quite a different light. Since every thesis which holds of strict implication holds also of material implication (but not vice versa), S1–5 emerge as progressively stricter fragments of PC rather than as containing it. Furthermore, they are then properly speaking systems of propositional logic rather than systems of modal logic, though of course the modal operators L and M are definable in them, and their characteristic modal theses derivable. Although Lewis (with an assist from Hugh MacColl) is the founder of modern modal logic, there is good evidence that he himself preferred to regard the Lewis systems as formalizations of the notion of implication rather than of possibility and necessity: furthermore, the notion of implication Lewis had in mind was arrived at by restricting and so to speak cutting some of the fat off material implication. Hence Lewis would approve of the treatment of his systems in this paper.

The authors would like to thank A. T. Tymoczko of Harvard University and the members of philosophy 292, University of Pittsburgh, 1967 (especially Mr. Karatay, Mr. Pottinger, Miss Roper and Miss Smith) for help with the axiomatization of matrix 1.