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Topological semantics of justification logic. (English) Zbl 1138.03015
Hirsch, Edward A. (ed.) et al., Computer science – theory and applications. Third international computer science symposium in Russia, CSR 2008 Moscow, Russia, June 7–12, 2008. Proceedings. Berlin: Springer (ISBN 978-3-540-79708-1/pbk). Lecture Notes in Computer Science 5010, 30-39 (2008).
Summary: The Justification Logic is a family of logical systems obtained from epistemic logics by adding a new type of formulas \(t:F\) which reads “\(t\) is a justification for \(F\)”. The major epistemic modal logic S4 has a well-known Tarski topological interpretation which interprets \(\square F\) 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 \(\square F\) 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.
For the entire collection see [Zbl 1136.68005].

03B42 Logics of knowledge and belief (including belief change)
03B45 Modal logic (including the logic of norms)
03F45 Provability logics and related algebras (e.g., diagonalizable algebras)
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