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 $$\langle W,R\rangle$$ 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


 * $$ M,w\models\square \varphi \Longleftrightarrow (\varphi)^M \in N(w), $$

where


 * $$(\varphi)^M = \{u\in W \mid M,u\models \varphi \}$$

is the truth set of $$\varphi$$.

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


 * $$ N(w) = \{(\varphi)^M \mid M,w\models\Box \varphi\}. $$

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.