Semantical Analysis of Modal Logic Ii. Non-normal Modal Propositional Calculi

This paper continues the investigations of Kripke [63]. Not only is Kripke [63] presupposed; the reader is advised to have it on hand for ready reference. The notations and terminology of that paper are used freely; in particular, P, Q, R, ... are atomic formulae, and A, B, 0, ... are arbitrary formulae built from them using the connectives A, ~, O. All propositional calculi in this paper have the same formation rules as those of Kripke [63]. However, they in general will lack the rule of necessitation of Kripke [63] and thus will not be normal. Hence (in our opinion), they are intuitively somewhat unnatural; but nevertheless they have an elegant model theory. Among these systems aTe, notably, Lewis's S2 and S3; this paper extends the results of Kripke [63] to these and other systems. The results of this paper were announced in Kripke [63, abstract]. All systems considered here will be formulated with axiom schemata; substitution is a derived rule.