Epistemic extensions of combined classical and intuitionistic propositional logic

Logic |$L$| was introduced by Lewitzka (2017, J. Logic and Comput., 27, 201–212) as a modal system that combines intuitionistic propositional logic IPC and classical propositional logic CPC: |$L$| is a conservative extension of CPC, and for any propositional formula |$\varphi$|⁠, |$\varphi$| is a theorem of IPC iff |$\square\varphi$| is a theorem of |$L$|⁠. 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 (2016, Rev. Symbol. Logic, 9, 266–298) 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 (Artemov and Protopopescu, 2016, Rev. Symbol. Logic, 9, 266–298).