Neighborhood semantics

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

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

But this is not the only possible choice for equivalence, N can also be defined only with reference to R (and W):

${\displaystyle N'(w)=\{X\subseteq W\mid \{u\in W\mid wRu\}\subseteq X\}.}$

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

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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