Sequent Calculi for Intuitionistic Gödel–Löb Logic

Abstract

This paper provides a study of sequent calculi for intuitionistic Gödel-Löb logic (iGL), which is the intuitionistic version of the classical modal logic GL, known as Gödel-Löb logic. We present two different sequent calculi, one of which we prove to be the terminating version of the other. We study those systems from a proof-theoretic point of view. One of our main results is a syntactic proof for the cut-admissibility result for those systems. Finally, we apply this to prove Craig interpolation for intuitionistic Gödel-Löb logic.

The G4i Analogue of a G3i Sequent Calculus

“…Therefore, to apply the method in this paper to such logics, an order different from the Dyckhoff order has to be found with respect to which G4iX is terminating. Gödel-Löb Logic and Strong Löb Logic are examples for which that can be done, as shown in [8,9] using an ingenious order introduced by Bílková [1]. For reasons that are out of the scope of this paper we conjecture that for G4iK4 and G4iS4 such an order does not exist but leave it for future work to settle that conjecture.…”

“…Acknowledgements. I thank Iris van der Giessen for the pleasant collaboration that lead to two papers [8,9] that were the inspiration for the material presented in this paper. I thank an anonymous referee for the useful remarks on an earlier draft of this paper.…”

“…The general result presented in this paper grew out of the research carried out in [8,9,15], in which G4ip based calculi are developed for four concrete logics, namely the intuitionistic versions of the logics K, KD, Gödel-Löb Logic GL and Strong Löb Logic SL. The results for the first two logics follow from the results in this note, while the results for the other two logics require a slight adjustment of the method presented in this note, as explained in Section 4.…”

On Provability Logic of HA

“…• iGL := iK4L. (A sequent calculus for iGL is provided in [van der Giessen and Iemhoff, 2021]) Note that in this setting, iGL and iK4 are closed under necessitation:…”