Local Intuitionistic Modal Logics and Their Calculi

We investigate intuitionistic modal logics with locally interpreted $\mathrm{\square }$ and $\mathrm{◊}$. The basic logic LIK is stronger than constructive modal logic WK and incomparable with intuitionistic modal logic IK. We propose an axiomatization of LIK and some of its extensions. We propose bi-nested calculi for LIK and these extensions, thus providing both a decision procedure and a procedure of finite countermodel extraction.