The topology of justification

Justification Logic is a family of epistemic logical systems ob- tained from modal logics of knowledge by adding a new type of formula t:F , which is readt is a justification forF. The principal epistemic modal logic S4 includes Tarski's well-known topological interpretation, according to which the modality 2X is read the Interior ofX in a topological space (the topological equivalent of the 'knowable part ofX'). In this paper, we extend Tarski's topological interpretation from S4 to Justification Logic systems with both modality and justification assertions. The topological semantics interpretst:X as a reachable subset ofX (the topological equivalent of 'testt confirmsX'). We establish a number of soundness and completeness results with respect to Kripke topology and the real topology for S4-based systems of Justification Logic.