On Some Intuitionistic Modal Logics

Some modal logics based on logics weaker than the classical logic have been studied by Fitch [4], Prior [18], Bull [1], [2], [3], Prawitz [17] etc. In this paper, we treat modal logics based on the intuitionistic prepositional logic, which we call intuitionistic modal logics (abbreviated as IML's). Our main concern is to compare properties of several IML's of S4or S5-type 03^ using some model theoretical methods. The study of modal logics based on weak logics seems to reveal to us various properties of classical modal logics, especially of S5, which will be indistinguishable by dealing them only on the classical logic. We will introduce some IML's in the Hilbert-style formalization in § 2. Then we will define IML's in the form of sequent calculi, all of which are given by restricting or modifying the sequent calculi S4 and S5 of Ohnishi-Matsumoto [15]. We will show the proper inclusion relationship between these IML's by using a kind of algebraic models. In §§3 and 5, we will introduce two kinds of models for IML's. One of them is a natural extension of Kripke models for the intuitionistic Jogic and the other is for modal logics (see [11], [12]). Then we will prove the completeness theorem with respect to these models. In § 4, the finite model property for some IML's will be shown. We would like to thank M. Sato for his valuable suggestions.