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 \langle W,R\rangle 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 \langle W,N\rangle still has a set W of worlds, but has instead of an accessibility relation a neighborhood function

 N : W \to 2^{2^W}

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

 M,w\models\square A \Longleftrightarrow (A)^M \in N(w),

where

(A)^M = \{u\in W \mid M,u\models A \}

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[edit]

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

 N(w) = \{(A)^M: M,w\models\Box A\}.

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.

References[edit]

  • 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.