Topological Semantics of Justification Logic

Abstract

The Justification Logic is a family of logical systems obtained from epistemic logics by adding new type of formulas t:F which reads as t is a justification for F. The major epistemic modal logic S4 has a well-known Tarski topological interpretation which interprets □F as the interior of F (a topological equivalent of the 'knowable part of F'). In this paper we extend the Tarski topological interpretation from epistemic modal logics to justification logics which have both: knowledge assertions □F and justification assertions t:F. This topological semantics interprets modality as the interior, terms t represent tests, and a justification assertion t:F represents a part of F which is accessible for test t. We establish a number of soundness and completeness results with respect to Kripke topology and the real line topology for S4-based systems of Justification Logic.