A fully labelled proof system for intuitionistic modal logics

Abstract Labelled proof theory has been famously successful for modal logics by mimicking their relational semantics within deductive systems. Simpson in particular designed a framework to study variety of intuitionistic integrating binary relation symbol the syntax. In this paper, we present labelled sequent system such that there is not only one but two symbols appearing sequents: accessibility associated with Kripke normal and pre-order logic. This puts our close correspondence standard birelational logics. As consequence, it can be extended arbitrary Scott–Lemmon axioms. We show soundness completeness, together an internal cut elimination proof, encompassing wider array than any existing system.