Subvaluationism

Origin. Hyde (1997), Varzi: the dual of supervaluationism over the same precisification frame. Sub-truth is the possibility modality ◇—truth on at least one admissible precisification—so that borderline cases become truth-value gluts and the resulting consequence is paraconsistent.

Models. The same Kripke frame of admissible precisifications, read through ◇ rather than □. Sub-truth = ◇φ: φ holds at some accessible point. Borderline φ is both sub-true and sub-false (◇φ and ◇¬φ), a glut. Where supervaluationism reads the frame through necessity and gets gaps, subvaluationism reads it through possibility and gets gluts.

Formalism.

Shared precisification frame: W admissible precisifications, each classical. R ⊆ W × W admissibility relation.

Sub-truth modality: Sφ ≡ ◇φ: ∃w (vRw ∧ w ⊨ φ). Sub-false ≡ ◇¬φ. Dual to super-truth: Sφ = ¬D¬φ.

Gluts at the border: Borderline φ: ◇φ ∧ ◇¬φ. Both sub-true and sub-false. = contingency Δφ, read as glut not gap.

Adjunction fails: Every precisification is classical, so ¬(φ ∧ ¬φ) is true at each one and remains sub-true; φ ∧ ¬φ is sub-true nowhere. What fails is aggregation: one point makes φ true, another makes ¬φ true, and no point makes both true. So ◇φ ∧ ◇¬φ without ◇(φ ∧ ¬φ) — φ, ¬φ ⊭ˢ φ ∧ ¬φ.

Paraconsistent consequence: Γ ⊨ˢ φ iff φ sub-true whenever all of Γ sub-true. Explosion fails: gluts do not trivialize. Dual to the supervaluationist gap logic.

Duality: Super-truth □ ↔ sub-truth ◇. Excluded-middle failure (super) ↔ adjunction failure (sub). One frame, two consequence relations.

Symbols.

Metatheory. Sub-truth is the ◇-dual of super-truth over the same frame, so it inherits the frame conditions dually: whatever D validates as a necessity, S validates as the corresponding possibility, and the S5 case is again the one in which all iterations collapse. Consequence is paraconsistent in the non-adjunctive manner of Jaśkowski's D2, and for the same structural reason — φ and ¬φ may each be sub-true while φ ∧ ¬φ is sub-true at no point, so explosion never obtains a contradiction to work from. Varzi (2000) made the duality exact: subvaluational consequence coincides with classical consequence for single-premise arguments and diverges only where premises must be aggregated, mirroring supervaluationism, which is classical for single-conclusion arguments and diverges only where conclusions must be taken disjunctively. Neither relation is truth-functional, and higher-order vagueness weakens the frame below S5 on both readings.

Applies to. Vagueness treated by gluts. Paraconsistent handling of borderline cases. Dialetheic semantics. Semantic indeterminacy.

Limitations. The paraconsistency is non-adjunctive, which is the weakest variety: the logic never has a live contradiction to reason from, since φ ∧ ¬φ is sub-true nowhere, so it supplies no account of inference within a glutty region — from φ and ¬φ one may draw the consequences of each separately and never those of their conjunction. That makes the treatment of borderline cases a bookkeeping device rather than a theory of how inconsistent information is used, and it means the gluts do no work the gaps were not already doing. The duality is exact, so every structural complaint against super-truth has a mirror image here, and nothing in the frame settles which reading to take: one frame supports both consequence relations equally. The set of admissible precisifications remains an unexplained primitive, and no proof theory for ⊨ˢ is standard.

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