Philosophy: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 [math]\displaystyle{ \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]\displaystyle{ \langle W,N\rangle }[/math] still has a set W of worlds, but has instead of an accessibility relation a neighborhood function

[math]\displaystyle{ N : W \to 2^{2^W} }[/math]

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

[math]\displaystyle{ M,w\models\square \varphi \Longleftrightarrow (\varphi)^M \in N(w), }[/math]

where

[math]\displaystyle{ (\varphi)^M = \{u\in W \mid M,u\models \varphi \} }[/math]

is the truth set of [math]\displaystyle{ \varphi }[/math].

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 pointwise-identical modal theories) neighborhood model M' = (W, N, V) defined by

[math]\displaystyle{ N(w) = \{(\varphi)^M \mid M,w\models\Box \varphi\}. }[/math]

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 frames.

References

• Chellas, B.F. Modal Logic. Cambridge University Press, 1980.
• Montague, R. "Universal Grammar", Theoria 36, 373–98, 1970.
• Scott, D. "Advice on modal logic", in Philosophical Problems in Logic, ed. Karel Lambert. Reidel, 1970.