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.

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

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

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.

Using that a subset ${\displaystyle 2^{W}}$ is equivalent to its characteristic function ${\displaystyle W\to 2}$, a neighborhood function ${\displaystyle N}$ can also be understood as a predicate transformer:

${\displaystyle (W\to 2^{2^{W}})\cong (W\to 2^{W}\to 2)\cong (2^{W}\to W\to 2)\cong (2^{W}\to 2^{W})}$

• 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.

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