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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 <math>\langle W,R\rangle</math> consists of a set <math>W</math> of worlds (or states) and an accessibility relation <math>R</math> intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame <math>\langle W,N\rangle</math> still has a set <math>W</math> of worlds, but has instead of an accessibility relation a neighborhood function

<math> N : W \to 2^{2^W} </math>.

So <math>N</math> assigns to each <math>w\in W</math> a set of subsets of <math>W</math>. The idea is that we can give a definition of truth in the modal case via this assignment. Specifically, if <math>M</math> is a model on the frame, then

<math> M,w\models\square A \Longleftrightarrow (A)^M \in N(w), </math>

where <math>(A)^M = \{u\in W \mid M,u\models A \}</math>, the truth set of <math>A</math>.

[edit] References

• Scott, D. "Advice in modal logic", in Philosophical Problems in Logic, ed. Karel Lambert. Reidel, 1970.

• Montague, R. "Universal Grammar", Theoria 36, 373-98, 1970.

• Chellas, B.F. Modal Logic. Cambridge University Press, 1980.