Neighborhood models are generalizations of the standard relational models for modal logic invented independently by Dana Scott and Richard Montague in 1970. The underlying idea for neighborhood models comes from point-set topology: In a topology, a neighborhood
of a point \(x\) is any set \(A\) containing \(x\) such that you can "wiggle" \(x\) without leaving \(A\). What exactly it means to "wiggle" \(x\) depends on the underlying topology. Using this idea, a general definition for truth of a modal formula \(\Box\varphi\) runs as follows: \(\Box\varphi\) is true at a state \(w\) just in case the truth set
of \(\varphi\) (i.e., the set of states where \(\varphi\) is true) is a neighborhood of \(w\). Different semantics for modal logic can be classified in terms of what it means to be a "neighborhood" of a state. For example, in the topological semantics of McKinsey and Tarski, a neighborhood is defined with respect to some given topology. In the standard relational semantics, a neighborhood of \(w\) is any set containing the states that are immediately connected to \(w\) via some relation \(R\) on the set of states \(W\). The most general approach are neighborhood models
in which a neighborhood of \(w\) is any element of a distinguished collection of sets (the may not necessarily contain the state \(w\)).
A general criticism of neighborhood models is that they are not well-motivated: It is ad hoc
to simply assert that certain subsets and not others are in the neighborhood of a given state. They do provide a semantics for weak systems of modal logic, but do they do so in a principled way? There is certainly some truth to this criticism. Nonetheless, recent work has demonstrated the usefulness and interest of neighborhood semantics.
- Neighborhood models naturally show up when studying cooperative and non-cooperative game theory.
- Neighborhood models can be given an epistemic interpretation in which the neighborhoods describe the evidence accepted by a rational (or not so rational) agent.
- Neighborhood models offer an interesting new interpretation of the Barcan and Converse Barcan formulas in first-order modal logic.
Finally, one can learn a great deal about normal systems of modal logic by looking at how these systems behave in a more general semantics. Consult the following articles for highlights of this research project:
Johan van Benthem, David Fernández-Duque, and Eric Pacuit (2014). Evidence and plausibility in neighborhood structures, Annals of Pure and Applied Logic, 165:1, pp. 106 - 133. (pdf, bibtex)
Johan van Benthem and Eric Pacuit (2011). Dynamic logics of evidence-based beliefs, Studia Logica, 99:1-3, pp. 61 - 92. (pdf, bibtex)
Helle Hvid Hansen, Clemens Kupke, and Eric Pacuit (2009). Neighbourhood structures: Bisimilarity and basic model theory, Logical Methods in Computer Science, 5:2. (pdf, bibtex)
Horacio Arló-Costa and Eric Pacuit (2006). First-order classical modal logic, Studia Logica, 84:2, pp. 171 - 210. (pdf, bibtex)