Neighborhood semantics explained

\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 functionN : 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.

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.