Supervaluationism

Origin. Van Fraassen (1966) introduced supervaluations for truth-value gaps, but the semantics is modal in structure. The space of admissible precisifications is a Kripke frame, and the "definitely" operator D is a normal necessity □ quantifying over accessible precisifications. Classical theorems survive the gaps because every point of the frame is itself a classical valuation.

Models. Points are admissible precisifications: complete classical valuations agreeing on the clear cases. An accessibility relation R relates a point to the precisifications it admits. Dφ (super-truth) holds at a point iff φ holds at every accessible point. Universal R yields S5; restricting R models higher-order vagueness.

Formalism.

Precisification frame: W = {v₁, v₂, ...} admissible precisifications. Each v: complete classical valuation. R ⊆ W × W admissibility relation.

Definiteness modality: (W, R, v) ⊨ Dφ iff ∀w (vRw ⇒ w ⊨ φ). D = □; ▽φ ≡ ¬D¬φ = ◇φ. Super-true = Dφ; super-false = D¬φ.

Classical base: Each point is a classical valuation. So p ∨ ¬p holds at every point. Hence D(p ∨ ¬p) is super-true though p is gappy.

Indeterminacy is contingency: Iφ ≡ ¬Dφ ∧ ¬D¬φ = ◇φ ∧ ◇¬φ. Exactly the contingency operator Δ. Borderline = contingent across precisifications.

Frame conditions: Universal R: S5, higher iterations of D collapse. Drop transitivity or euclideanness: sub-S5, DDφ ≠ Dφ. Higher-order vagueness = a genuine accessibility relation.

Global consequence: Γ ⊨ᵍ φ iff φ super-true whenever all of Γ super-true. D-free fragment: coincides with classical consequence. With D present: contraposition, conditional proof, argument by cases, and reductio fail.

Symbols.

Metatheory. Each point is classical, so D is a normal modality—K minimally, KT/S4/S5 according to the frame conditions. Super-truth is validity in the frame. Bivalence fails for atoms at a point, yet excluded middle is super-true: the modal signature of the view. Global consequence is non-truth-functional and, once D enters the language, diverges from classical logic at the level of meta-inference. Higher-order vagueness corresponds to weakening S5.

Applies to. Vagueness and borderline cases. Semantic indeterminacy. Legal and conversational interpretation. Truth-value gaps without many-valuedness.

Limitations. Higher-order vagueness forces a sub-S5 choice of frame. Non-truth-functional. The admissible precisifications must be fixed in advance. Global consequence sacrifices classical meta-rules. No standard proof theory.

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