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''

: $N : W \to 2^$

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 = \$

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

: $N(w) = \.$

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.

Relation to predicate transformers

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

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.

Modal logic

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