23 May 2012, A|C seminar, Kohei Kishida

This talk extends Tarski's classical topological semantics for propositional modal logic to first-order modal logic, with respect to the following two aspects: (i) It takes a sheaf over a topological space, and shows that such structures (or the category of them) model first-order modal logic by equipping points of the space with domains of individuals. (ii) It is also shown how topological semantics extends to the more general case of neighborhood semantics, at the level of sheaf semantics. These extensions provide semantics for the simple unions of first-order logic with S4 modal logic and with more general modal logics. Corresponding to the point-set and algebraic formulations of Tarski's topological semantics, the semantics of this paper will be presented in both point-set and topos-theoretic formulations.

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