Modality and possibility in some intuitionistic modal logics
In the modal logic based on classical logic, the modal operators L (necessity) and M (possibility) are considered to be dual each other and mutually definable as \(M\leftrightarrow \neg L\neg\) and \(L\leftrightarrow \neg M\neg\). However in the modal logic based on intuitionistic logic, the duality disappears because of having weak negation, and both equivalences cannot remain valid. In this paper, some intuitionistic modal logics are studied, that is, L is adapted as a primitive symbol with implicative S4-type axioms and M is defined as \(\neg L\neg\). (L\(\leftrightarrow \neg M\neg\) does not hold generally.) The purpose of this paper is to analyze the behaviour of M, and to show the use of algebraic models to obtain logical properties of the modal systems. Concerning the behavior of M, there are two points. One is the study of all different modalities, and the other is the possible definition of intuitionistic modal logics analogous to S5-type. The concerning algebraic models are the topological pseudo-Boolean algebras in the author's earlier paper in Models and Sets, Proc. Logic Colloq., Aachen 1983, Part I, Lect. Notes Math. 1103, 169-192 (1984; Zbl 0562.03034).
scientific article; zbMATH DE number 4039855
scientific article