Neighborhood-Sheaf Semantics for First-Order Modal Logic

This paper extends neighborhood semantics for propositional modal logic to the first-order case, by unifying topological-sheaf semantics (in [Awodey, S. and K. Kishida, Topology and modality: the topological interpretation of first-order modal logic, Review of Symbolic Logic 1 (2008), pp. 146–66]) for first-order S4 and Kripke-sheaf semantics (see [Goldblatt, R., “Topoi: The Categorial Analysis of Logic”, North-Holland, Amsterdam and New York, 1979] and [Gabbay, D. M., V. Shehtman and D. Skvortsov, “Quantification in Nonclassical Logic”, Volume 1, Elsevier, Oxford, 2009], just for instance) for quantified K. It will be shown how to take a sheaf-like structure over a neighborhood frame, and the resulting semantics properly generalizes the two preceding sheaf semantics; it has a weaker modal logic (in which the rule N fails) sound and complete, while accommodating classical, full first-order logic with equality and function symbols.