Neighborhood Semantics

Origin. Dana Scott and Richard Montague independently developed neighborhood semantics (1970). Generalizes Kripke semantics to non-normal modal logics. Each world has a neighborhood: sets of propositions that are "necessary" at that world. Captures modalities that don't satisfy all normal modal axioms. Foundation for various non-standard modalities.

Models. Necessity without closure properties. Kripke semantics: □φ iff φ holds at all accessible worlds. This forces □(φ ∧ ψ) ↔ (□φ ∧ □ψ) and □(φ → ψ) → (□φ → □ψ). Neighborhood semantics: □φ iff the set of φ-worlds is in the neighborhood. Neighborhoods need not satisfy these closure properties.

Formalism.

Neighborhood frame: F = (W, N) where:

• W: set of worlds
• N: W → P(P(W)) assigns each world a neighborhood (set of proposition-sets)

Neighborhood model: M = (W, N, V) with valuation V.

Truth:

• M, w ⊨ □φ iff ⟦φ⟧ ∈ N(w) (The set of worlds where φ holds is in w's neighborhood)
• M, w ⊨ ◇φ iff W \ ⟦φ⟧ ∉ N(w) (for classical ◇) Or: ◇ gets its own neighborhood for non-classical

Compared to Kripke: Kripke: N(w) = {X ⊆ W : R[w] ⊆ X} where R[w] = {v : wRv} So Kripke frames are special neighborhood frames (closed under supersets).

Non-normal logics:

• E (classical modal logic): just the congruence rule — from φ ↔ ψ infer □φ ↔ □ψ
• M: adds □(φ ∧ ψ) → (□φ ∧ □ψ) (monotonicity)
• C: adds □φ ∧ □ψ → □(φ ∧ ψ)
• N: adds □⊤
• K = E + M + C + N (normal modal logic)

Supplemented vs. augmented:

• Supplemented: closed under supersets (validates M)
• Augmented: supplemented + ⋂N(w) ∈ N(w)
• Augmented frames are exactly the Kripke frames

Monotonic modalities: Monotonicity — from ⊨ φ → ψ infer ⊨ □φ → □ψ — fails without supplementation. Non-monotonic logics capture "most," "typically," etc.

Symbols.

Metatheory. The minimal classical logic E, closed only under the congruence rule from φ ↔ ψ infer □φ ↔ □ψ, is sound and complete with respect to the class of all neighbourhood frames, proved by a canonical neighbourhood model in which N(w) is read off from the maximal consistent set at w; Segerberg (1971) and Chellas (1980) extended this to every combination of M, C and N, each corresponding to a first-order closure condition on N. A neighbourhood frame is augmented — supplemented and containing its own core ⋂N(w) — exactly when it is the neighbourhood form of a Kripke frame, which is why K is complete for augmented frames and why Kripke semantics is the special case rather than the general one. These logics have the finite model property by filtration and are decidable. Dropping the K axiom lowers complexity rather than raising it: Vardi (1989) showed that satisfiability for the non-normal classical and monotone systems is NP-complete, below the PSPACE-completeness of K. Neighbourhood frames are coalgebras for the double contravariant power-set functor, and Hansen, Kupke and Pacuit (2009) identified the matching notion of behavioural equivalence and proved a Hennessy–Milner theorem for monotone models.

Applies to. Non-normal modalities (evidence, belief without closure). Conditionals (Stalnaker's theory). Deontic logic (permissions without closure). Game logic (non-normal game operations). Majority/generic quantification. Topology (neighborhoods in topological sense).

Limitations. The generality comes at the loss of content: every logic closed under the congruence rule is complete for some class of neighbourhood frames, so completeness is nearly automatic and tells one almost nothing about the logic's structure. There is no correspondence theory of Sahlqvist's strength here — the truth condition quantifies over sets rather than worlds, so the standard translation lands in monadic second-order logic instead of a decidable first-order fragment, and most neighbourhood conditions have no elementary frame correspondent. Kripke bisimulation is neither sound nor complete for neighbourhood models, so invariance arguments and the usual filtration and unravelling constructions must be rebuilt. Without the monotonicity axiom M one cannot infer □φ → □ψ from the validity of φ → ψ, which blocks nearly all derived reasoning and forces most applications to reinstate M. Most importantly, neighbourhood semantics does not remove logical omniscience: the congruence rule is already validated, so necessarily equivalent propositions remain interchangeable inside □, and genuine hyperintensionality requires impossible worlds or structured propositions instead.

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