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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 [[Kripke semantics|relational semantics]] for modal logic. Whereas a [[Kripke frame|relational frame]] <math>\langle W,R\rangle</math> 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''' <math>\langle W,N\rangle</math> still has a set ''W'' of worlds, but has instead of an accessibility relation a ''neighborhood function''
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| : <math> N : W \to 2^{2^W} </math> | |
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| 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
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| : <math> M,w\models\square A \Longleftrightarrow (A)^M \in N(w), </math> | |
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| where
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| : <math>(A)^M = \{u\in W \mid M,u\models A \}</math>
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| is the ''truth set'' of ''A''.
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| Neighborhood semantics is used for the classical modal logics that are strictly weaker than the [[normal modal logic]] '''K'''.
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| ==Correspondence between relational and neighborhood models==
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| 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
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| : <math> N(w) = \{(A)^M: M,w\models\Box A\}. </math>
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| 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 frame|general relational structures]].
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| ==References==
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| * Scott, D. "Advice in modal logic", in ''Philosophical Problems in Logic'', ed. Karel Lambert. Reidel, 1970.
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| * Montague, R. "Universal Grammar", ''Theoria'' 36, 373-98, 1970.
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| * Chellas, B.F. ''Modal Logic''. Cambridge University Press, 1980.
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| [[Category:Modal logic]]
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| {{logic-stub}}
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