This paper is a sequel to , and the terminology of  is largely presupposed here. In , the algebraic methods of McKinsey-Tarski were employed and extended to yield semantic results of a Kripke kind for a class of relatively weak modal logics, the strongest of which was the Feysvon Wright system T. Deontic versions of both T and E2, called T(D) and D2, and even weaker systems, were handled. The main aim of the present paper is to extend these results to stronger systems of modal logic. Thus the Lewis systems S2–S5, the Brouwersche system B of Kripke , the systems E3–E5 of , and Łukasiewicz's modal logic, as well as certain new systems, are considered.
Certain modifications of the method of  have proved convenient. Thus in Section I, some further results concerning model structures are proved in order that the relationship between S2 and E2, S3 and E3, can be properly stated; in particular, the notions of a refined and connected model structure play a pervasive role throughout.