Epistemic extensions of combined classical and intuitionistic propositional logic

Logic L was introduced by Lewitzka [8] 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 embedding φ 7→ φ. 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. Our systems involve, in adapted form, some of the epistemic principles of Intuitionistic Epistemic Logic presented by Artemov and Protopopescu [1]. In particular, the implications “intuitionistic truth ⇒ knowledge ⇒ classical truth” are given as theorems φ → Kφ and Kφ → φ of our logic EL3. Intuitionistic and classical (epistemic) principles are combined within a single system. Finally, we show that a slight modification of our semantics yields algebraic models for the systems of Intuitionistic Epistemic Logic studied in [1].