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. Because every point of the frame is a complete classical valuation, D is a normal modality: the K axiom and the rule of necessitation hold outright, and T, 4 and 5 hold according to the conditions imposed on R, with universal R giving S5 and collapsing all iterations of D. Super-truth is validity in the frame, and Fine (1975) is the canonical development, adding penumbral connections and the hierarchy of higher-order vagueness. Bivalence fails for atoms at a point while excluded middle is super-true — D(p ∨ ¬p) holds though neither Dp nor D¬p does — and that combination is the modal signature of the view. In the D-free language global consequence coincides with classical consequence for single-conclusion arguments; the divergence is multiple-conclusion, since p ∨ ¬p is super-true while the sequent ⊨ p, ¬p fails. Once D is in the language, contraposition, conditional proof, argument by cases and reductio all fail.

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

Limitations. The claim to preserve classical logic holds for theorems and not for inference: with D in the language, contraposition, conditional proof, argument by cases and reductio all fail, so the classical meta-rules are lost exactly where the theory does its work. Higher-order vagueness forces the frame below S5, and if "definitely" is itself vague the hierarchy of D-iterations has no principled stopping point; Williamson has argued that the natural closure assumptions collapse it rather than stabilizing it. The set of admissible precisifications is a primitive of the model and not derived by the semantics, so a sharp boundary is reintroduced one level up — the boundary between admissible and inadmissible ways of making a vague term precise. Super-truth is non-truth-functional, and no proof theory for global consequence is standard, since the meta-rule failures block the obvious sequent formulations.

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