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 $⟨W,R⟩$ 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 $⟨W,N⟩$ still has a set W of worlds, but has instead of an accessibility relation a neighborhood function

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

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

$M,w\models ◻\phi ⟺(\phi {)}^M\in N(w),$

where

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

is the truth set of $\phi$.

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

$N(w)=\{(\phi {)}^M\mid M,w\models ◻\phi \}.$

But this is not the unique possible choice: there also corresponds an equivalent one with neighborhood function defined only with reference to R (and W):

$N^{\prime }(w)=\{\{w^{\prime }\in W\mid wR{w}^{\prime }\}\mid w\in W\}.$

For any w, N'(w) contains N(w) but may be strictly bigger, since some element of it may not be the truth set in M of any formula.

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:

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

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.