Minimal modal logics, constructive modal logics and their relations

Abstract: We present a family of minimal modal logics (namely, modal logics based on minimal propositional logic) corresponding each to a different classical modal logic. The minimal modal logics are defined based on their classical counterparts in two distinct ways: (1) via embedding into fusions of classical modal logics through a natural extension of the G\"odel-Johansson translation of minimal logic into modal logic S4; (2) via extension to modal logics of the multi- vs. single-succedent correspondence of sequent calculi for classical and minimal logic. We show that, despite being mutually independent, the two methods turn out to be equivalent for a wide class of modal systems. Moreover, we compare the resulting minimal version of K with the constructive modal logic CK studied in the literature, displaying tight relations among the two systems. Based on these relations, we also define a constructive correspondent for each minimal system, thus obtaining a family of constructive modal logics which includes CK as well as other constructive modal logics studied in the literature.