Neighborhood semantics
Encyclopedia
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
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
still has a set W of worlds, but has instead of an accessibility relation a neighborhood function
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
where
is the truth set of A.
Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic
K.
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 relational structures
.
Dana Scott
Dana Stewart Scott is the emeritus Hillman University Professor of Computer Science, Philosophy, and Mathematical Logic at Carnegie Mellon University; he is now retired and lives in Berkeley, California...
and Richard Montague
Richard Montague
Richard Merett Montague was an American mathematician and philosopher.-Career:At the University of California, Berkeley, Montague earned an B.A. in Philosophy in 1950, an M.A. in Mathematics in 1953, and a Ph.D. in Philosophy 1957, the latter under the direction of the mathematician and logician...
, of the more widely known relational semantics
Kripke semantics
Kripke semantics is a formal semantics for non-classical logic systems created in the late 1950s and early 1960s by Saul Kripke. It was first made for modal logics, and later adapted to intuitionistic logic and other non-classical systems...
for modal logic. Whereas a relational frame
consists of a set W of worlds (or states) and an accessibility relationAccessibility relation
In modal logic, an accessibility relation is a binary relation, written as R\,\! between possible worlds.-Description of terms:A statement in logic refers to a sentence that can be true or false...
R intended to indicate which worlds are alternatives to (or, accessible from) others, a neighborhood frame
still has a set W of worlds, but has instead of an accessibility relation a neighborhood function
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
where
is the truth set of A.
Neighborhood semantics is used for the classical modal logics that are strictly weaker than the normal modal logic
Normal modal logic
In logic, a normal modal logic is a set L of modal formulas such that L contains:* All propositional tautologies;* All instances of the Kripke schema: \Box\toand it is closed under:...
K.
Correspondence between relational and neighborhood models
To every relational model M = (W,R,V) there corresponds an equivalent (in the sense of having point-wise equivalent modal theories) neighborhood model M' = (W,N,V) defined by
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 relational structures
General frame
In logic, general frames are Kripke frames with an additional structure, which are used to model modal and intermediate logics...
.