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.

Non-contradiction fails: ¬(φ ∧ ¬φ) not sub-true for borderline φ. Some point makes φ true, another makes ¬φ true. So φ ∧ ¬φ is sub-true somewhere.

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 ↔ non-contradiction failure. One frame, two consequence relations.

Symbols.

Symbol	Unicode	Name	Meaning
S, ◇	U+25C7	Sub-true	True on some admissible precisification
□	U+25A1	Super-true	Dual (supervaluation)
Δ	U+0394	Glut	Contingent across precisifications
W	—	Frame	Admissible precisifications
R	—	Admissibility	Accessibility relation
⊨ˢ	—	Sub-validity	Sub-truth consequence

Metatheory. Sub-truth is the ◇-dual of super-truth on the same frame, inheriting its frame conditions (KT/S4/S5) dually. Consequence is paraconsistent: non-contradiction fails as excluded middle fails for supervaluationism, and explosion is invalid. Non-truth-functional. Higher-order vagueness again weakens the frame below S5.

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

Limitations. Less standard than supervaluationism. Requires a paraconsistent consequence relation. Non-truth-functional. The frame of admissible precisifications must be fixed. Sacrifices classical meta-rules dually.