Epistemic extensions of combined classical and intuitionistic propositional logic

TL;DR

This paper introduces epistemic extensions of a combined classical and intuitionistic logic, providing completeness results and connecting classical knowledge with intuitionistic truth through modal principles.

Contribution

It defines epistemic extensions of Lewitzka's combined logic, proves their completeness, and relates classical knowledge to intuitionistic truth within a modal framework.

Findings

Established completeness w.r.t. algebraic semantics.

Connected classical knowledge with intuitionistic truth via modal principles.

Provided algebraic models for systems of Intuitionistic Epistemic Logic.

Abstract

Logic $L$ was introduced by Lewitzka [7] as a modal system that combines intuitionistic and classical logic: $L$ is a conservative extension of CPC and it contains a copy of IPC via the embedding $\varphi\mapsto\square\varphi$. In this article, we consider $L3$, i.e. $L$ augmented with S3 modal axioms, define basic epistemic extensions and prove completeness w.r.t. algebraic semantics. The resulting logics combine classical knowledge and belief with intuitionistic truth. Some epistemic laws of Intuitionistic Epistemic Logic studied by Artemov and Protopopescu [1] are reflected by classical modal principles. In particular, the implications "intuitionistic truth $\Rightarrow$ knowledge $\Rightarrow$ classical truth" are represented by the theorems $\square\varphi\rightarrow K\varphi$ and $K\varphi\rightarrow\varphi$ of our logic $EL3$, where we are dealing with classical instead of intuitionistic knowledge. Finally, we show that a modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic introduced in [1].