Simple example of weak modal logic based on intuitionistic core

TL;DR

This paper introduces a basic weak modal logic built on an intuitionistic core, exploring its semantics with neighborhood and pre-order structures, and characterizing the axiom 4 in this context.

Contribution

It presents a simple, weak modal logic based on intuitionistic logic with a complete semantics and characterizes the axiom 4 within this framework.

Findings

The logic is very weak, lacking regularity and monotonicity rules.

Complete semantics involve possible worlds with neighborhoods and pre-order relations.

A characterization of the axiom 4 in this weak modal setting.

Abstract

In this paper we present simple example of propositional logic which has one modal operator and is based on intuitionistic core. This system is very weak in modal sense - e.g. rules of regularity or monotonicity do not hold. It has complete semantics composed of possible worlds equipped with neighborhoods and pre-order relation. We discuss certain restrictions imposed on those structures. Also, we present characterization of axiom 4 known from logic S4.