Proof Theory for Intuitionistic Strong Löb Logic

Abstract

This paper introduces two sequent calculi for intuitionistic strong Löb logic iSL✷: a terminating sequent calculus G4iSL✷ based on the terminating sequent calculus G4ip for intuitionistic propositional logic IPC and an extension G3iSL✷ of the standard cutfree sequent calculus G3ip without structural rules for IPC. One of the main results is a syntactic proof of the cut-elimination theorem for G3iSL✷. In addition, equivalences between the sequent calculi and Hilbert systems for iSL✷ are established. It is known from the literature that iSL✷ is complete with respect to the class of intuitionistic modal Kripke models in which the modal relation is transitive, conversely well-founded and a subset of the intuitionistic relation. Here a constructive proof of this fact is obtained by using a countermodel construction based on a variant of G4iSL✷. The paper thus contains two proofs of cut-elimination, a semantic and a syntactic proof.

Terminating Calculi and Countermodels for Constructive Modal Logics

“…As an example, an extraction of relational countermodels from failed proofs in a G4-calculus for Intuitionistic Strong Löb Logic with only is presented in[9].…”