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

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

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

${\displaystyle M,w\models \square \varphi \Longleftrightarrow (\varphi )^{M}\in N(w),}$

where

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

is the truth set of ${\displaystyle \varphi }$.

Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic K.