On Proof Realization on Modal Logic

Artemov’s Logic of Proof, LP, is an explicit proof counterpart of S4. Their formal connection is built through the realization theorem, that every S4 theorem can be converted to an LP theorem by substituting proof terms for provability modals. Instead of the realization of theorems, what is concerned in this paper is the realization of proofs. We will show that only a subclass of S4 proofs, called non-circular proofs, can be realized as LP proofs in this way. Furthermore, we introduce a numerical version of LP, called S4∆, to constructively prove that every S4 theorem has a non-circular proof. These results provide a new algorithmic proof of the realization theorem.