Modality Definition Synthesis for Epistemic Intuitionistic Logic via a Theorem Prover

Abstract

We derive a Prolog theorem prover for an Intuitionistic Epistemic Logic by starting from the sequent calculus G4IP that we extend with operator definitions providing an embedding in intuitionistic propositional logic (IPC). With help of a candidate definition formula generator, we discover epistemic operators for which axioms and theorems of Artemov and Protopopescu's Intuitionistic Epistemic Logic (IEL) hold and formulas expected to be non-theorems fail. We compare the embedding of IEL in IPC with a similarly discovered successful embedding of Dosen's double negation modality, judged inadequate as an epistemic operator. Finally, we discuss the failure of the necessitation rule for an otherwise successful S4 embedding and share our thoughts about the intuitions explaining these differences between epistemic and alethic modalities in the context of the Brouwer-Heyting-Kolmogorov semantics of intuitionistic reasoning and knowledge acquisition.