# Intuitionistic Non-Normal Modal Logics: A general framework

## TL;DR

This paper introduces a broad family of intuitionistic non-normal modal logics, providing axiomatizations, sequent calculi, and semantic models, extending existing systems like Constructive K and Dynamic Logic.

## Contribution

It develops a general framework for intuitionistic non-normal modal logics, including monomodal and bimodal systems with new interaction semantics, and proves their decidability.

## Key findings

- Defined a family of intuitionistic non-normal modal logics
- Provided Hilbert axioms and cut-free sequent calculi
- Established semantic neighborhood models for these logics

## Abstract

We define a family of intuitionistic non-normal modal logics; they can bee seen as intuitionistic counterparts of classical ones. We first consider monomodal logics, which contain only one between Necessity and Possibility. We then consider the more important case of bimodal logics, which contain both modal operators. In this case we define several interactions between Necessity and Possibility of increasing strength, although weaker than duality. For all logics we provide both a Hilbert axiomatisation and a cut-free sequent calculus, on its basis we also prove their decidability. We then give a semantic characterisation of our logics in terms of neighbourhood models. Our semantic framework captures modularly not only our systems but also already known intuitionistic non-normal modal logics such as Constructive K (CK) and the propositional fragment of Wijesekera's Constructive Concurrent Dynamic Logic.