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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. ==Correspondence between relational and neighborhood models== To every relational model M = (W,R,V) there corresponds an equivalent (in the sense of having point-wise equivalent modal theories) neighborhood model M' = (W,N,V) defined by : The fact that the converse fails gives a precise sense to the remark that neighborhood models are a generalization of relational ones. Another (perhaps more natural) generalization of relational structures are general relational structures. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「neighborhood semantics」の詳細全文を読む スポンサード リンク
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