Modal logic

Neighborhood semantics

Neighborhood semantics, also known as Scott–Montague semantics, is a formal semantics for modal logics. It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic. Whereas a relational frame consists of a set W of worlds (or states) and an accessibility relation R intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame still has a set W of worlds, but has instead of an accessibility relation a neighborhood function that assigns to each element of W a set of subsets of W. Intuitively, each family of subsets assigned to a world are the propositions necessary at that world, where 'proposition' is defined as a subset of W (i.e. the set of worlds at which the proposition is true). Specifically, if M is a model on the frame, then where is the truth set of A. Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic K. (Wikipedia).

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Related pages

Modal logic | Accessibility relation | General frame | Kripke semantics | Classical modal logic | Normal modal logic