Tableaux and hypersequents for justification logics

Abstract

Justification logic is a new generation of epistemic logics which along with the traditional modal knowledge/belief operators also consider justification assertions ‘$t$ is a justification for $F$.’ In this paper, we introduce a prefixed tableau system for one of the major logics of this kind S4LPN, which combines the Logic of Proofs (LP) and epistemic logic S4 with an explicit negative introspection principle $\neg t:F\to \square \neg t:F$. We show that the prefixed tableau system for S4LPN is sound and complete with respect to Fitting-style semantics. We also introduce a hypersequent calculus HS4LPN and show that HS4LPN is complete. We establish this fact by using a translation from the prefixed tableau system to the hypersequent calculus. This completeness result gives us a semantic proof of cut-admissibility for S4LPN. We conclude the paper by discussing a subsystems of S4LPN, namely S4LP.