A Terminating Sequent Calculus for Intuitionistic Strong Löb Logic with the Subformula Property

Abstract

AbstractIntuitionistic Strong Löb logic $$\textsf{iSL}$$ iSL is an intuitionistic modal logic with a provability interpretation. We introduce $$\textsf{GbuSL}_{\Box } $$ GbuSL □ , a terminating sequent calculus for $$\textsf{iSL}$$ iSL with the subformula property. $$\textsf{GbuSL}_{\Box } $$ GbuSL □ modifies the sequent calculus $$\textsf{G3iSL}_{\Box } $$ G 3 iSL □ for $$\textsf{iSL}$$ iSL based on $$\textsf{G3i} $$ G 3 i , by annotating the sequents to distinguish rule applications into an unblocked phase, where any rule can be backward applied, and a blocked phase where only right rules can be used. We prove that, if proof search for a sequent $$\sigma $$ σ in $$\textsf{GbuSL}_{\Box } $$ GbuSL □ fails, then a Kripke countermodel for $$\sigma $$ σ can be constructed.