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 □(φ ↔ ψ) → (□φ ↔ □ψ)
M: adds □⊤
C: adds □φ ∧ □ψ → □(φ ∧ ψ)
N: adds □(φ → ψ) → (□φ → □ψ)
K = E + M + C + N (normal modal logic)

Supplemented vs. augmented:

Supplemented: closed under supersets
Augmented: supplemented + contains W
Augmented + closed under intersections = Kripke

Monotonic modalities: □(φ → ψ) → (□φ → □ψ) fails in general. Non-monotonic logics capture "most," "typically," etc.

Symbols.

Symbol	Unicode	Name	Meaning
□	U+25A1	Box	Necessity (via neighborhood)
◇	U+25C7	Diamond	Possibility
N(w)	—	Neighborhood	Sets necessary at w
⟦φ⟧	—	Extension	Worlds where φ true
⊆	U+2286	Subset	Set inclusion
∈	U+2208	Member	In neighborhood
P(W)	—	Powerset	All subsets

Metatheory. Neighborhood semantics is more general: every Kripke frame is a neighborhood frame. Complete for various non-normal logics. Decidability typically preserved. Finite model property depends on logic. Classical modal logic E is weaker than K. Adding conditions on N recovers normal logics.

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. More complex than Kripke semantics. Intuitions from Kripke don't transfer. Non-normal logics are less well-developed. Axiomatizations can be complex. The neighborhood is abstract — not always clear what it "means." Some properties of Kripke models (bisimulation) require modification.