Neighborhood semantics
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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.
[edit] References
- Scott, D. "Advice in modal logic", in Philosophical Problems in Logic, ed. Karel Lambert. Reidel, 1970.
- Montague, R. "Universal Grammar", Theoria 36, 373-98, 1970.
- Chellas, B.F. Modal Logic. Cambridge University Press, 1980.
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