It is a generalization, developed independently by Dana Scott and Richard Montague, of the more widely known relational semantics for modal logic.
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
Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic K. 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 The fact that the converse fails gives a precise sense to the remark that neighborhood models are a generalization of relational ones.