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 {\bf G4IP} that we extend with operator definitions providing an embedding in intuitionistic propositional logic ({\bf IPC}). With help of a candidate definition formula generator, we discover epistemic operators for which axioms and theorems of Artemov and Protopopescu's {\em Intuitionistic Epistemic Logic} ({\bf IEL}) hold and formulas expected to be non-theorems fail. We compare the embedding of {\bf IEL} in {\bf 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 {\em necessitation rule} for an otherwise successful {\bf 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. Keywords: epistemic intuitionistic logic, propositional intuitionistic logic, Prolog-based theorem provers, automatic synthesis of logic systems, definition formula generation algorithms, embedding of modal logics into intuitionistic logic.