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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].

##### MSC:
 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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