Models for stronger normal intuitionistic modal logics

This paper, a sequel to "Models for normal intuitionistie modal logics" by M. Bo~i6 and the author, which dealt with intuitionistic analogues of the modal system K, deals similarly with intuitionistic analogues of systems stronger than K, and, in particular, analogues of $4 and $5. For these propositional logics Kripke-style models with two accessibility relations, one intuitionistic and the other modal, are given, and soundness and completeness are proved with respect to these models. It is shown how the holding of formulae characteristic for particular logics is equivalent to conditions for the relations of the models. Modalities in these logics are also investigated. w 1. Introduct ion. As in [3], f irst we shall deal with systems with [] primitive, and next with systems with 0 primitive. Unlike what we have ~n modal logic with a classical basis, these two kinds of systems are not reducible to each other. This means t h a t with the usual semantical definitions of the holding of [:]A and OA we cannot use the same models for both kinds of systems. Some of the systems we shall consider are already known in the literature. For example, systems equivalent to our H S 4 [ ] (see w were considered in [11] and [8] (see [3] for a survey of works on intui t ionist ic modal logic), t towever, we do not presuppose for this paper an acquaintance with these earlier works. W h a t we presuppose for it is a certain acquaintance with Kr ipke models for intui t ionist ic propositional logic and Kr ipke models for normal modal logics based on classical logic (for the former the reader m a y consult [7] or [1], Ch. 9, and for the la t ter [6]). We also presuppose an acquaintance with [3]. Nevertheless, in order to make this paper more self-contained, we shall review briefly in the next section some of the terminology and results of [3]. (A reader reading this paper in conjunct ion with [3] m a y skip this section.) w 2. Review of terminology and results of [3]. The language L [] is the language of propositional moda4 logic with denumcrably m a n y propositional variables, for which we use the schemata 29, q, r , p l , . . . , and the connectives -% ^, v, -q and [] ( ~ is defined as usual in terms of and ^, and in formulae ^ and v bind more strongly than -+ and ~ ) . * This paper presents results of an investigation of intuitionistic modal logio conducted in collaboration with Dr Milan Bo~i6.