Semantical Analysis of Intuitionistic Modal Logics between CK and IK

TL;DR

This paper explores the differences in semantics and axioms of intuitionistic modal logics between CK and IK, extending Kripke semantics and formalizing results in Coq to clarify their conservativity and axiomatisation.

Contribution

It extends Kripke semantics for CK with frame conditions for IK axioms and formalizes the findings in Coq, addressing open questions on conservativity.

Findings

Extended Kripke semantics for CK with IK axioms

Identified which logics are conservative over intuitionistic modal logic

Formal proof development in Coq

Abstract

The intuitionistic modal logics considered between Constructive K (CK) and Intuitionistic K (IK) differ in their treatment of the possibility (diamond) connective. It was recently rediscovered that some logics between CK and IK also disagree on their diamond-free fragments, with only some remaining conservative over the standard axiomatisation of intuitionistic modal logic with necessity (box) alone. We show that relational Kripke semantics for CK can be extended with frame conditions for all axioms in the standard axiomatisation of IK, as well as other axioms previously studied. This allows us to answer open questions about the (non-)conservativity of such logics over intuitionistic modal logic without diamond. Our results are formalised using the Coq Proof Assistant.