Logic of Paradox

Origin. Priest (1979). Three-valued paraconsistent logic underwriting dialetheism—the view that some contradictions are true. Same tables as Strong Kleene/K3, but with the glut value designated.

Models. Three values: true only (T), false only (F), both (B). Gluts are designated ("true enough"), so contradictions are tolerated without explosion. Classical reasoning is recaptured when no gluts occur.

Formalism.

Truth values and designation: {T, F, B}; designated D = {T, B}. B is designated: a glut still counts as true.

Connectives (strong Kleene tables): ¬T = F, ¬F = T, ¬B = B. ∧ = min, ∨ = max on the order F < B < T. v(A → B) = v(¬A ∨ B).

Relation to K3: Same tables as K3; K3 designates only T (gaps), LP designates T and B (gluts). The two are duals: paracomplete vs paraconsistent.

Paraconsistency: A, ¬A ⊭ B. Contradictions do not explode. B ∧ ¬B is designated yet entails nothing arbitrary.

Classical recapture: On glut-free premises and conclusion, ⊨_LP coincides with classical consequence. Gluts are the exceptional case.

Material conditional and its repair: A → B := ¬A ∨ B is not detachable: modus ponens fails when A takes value B. Priest supplements LP with a relevant conditional to restore detachment.

N4 and FDE: Adding a gap value to the glutty three yields the four-valued N4 = First-Degree Entailment—paracomplete and paraconsistent together.

Dialetheism and the catuṣkoṭi: Liar sentences are read as gluts. Priest reads the third koṭi of Nāgārjuna's catuṣkoṭi as a true contradiction (see First-Degree Entailment).

Symbols.

Symbol	Unicode	Name	Meaning
B	—	Both	Glut (true and false)
D	—	Designated	{T, B}
N4	—	Four-valued	FDE extension
⊨_LP	—	LP consequence	Designated-preservation

Metatheory. Three-valued, decidable. Paraconsistent: explosion invalid. Classical recapture on glut-free arguments. The material conditional is non-detachable, motivating a relevant conditional. Algebraically a Kleene algebra with {T, B} designated; the dual designation gives K3.

Applies to. Semantic paradoxes (Liar). Dialetheism. Paraconsistent mathematics. Inconsistent but non-trivial theories. Cross-cultural readings of the catuṣkoṭi.

Limitations. Weak material conditional; modus ponens needs a relevant supplement. Some classical validities fail. Dialetheism is philosophically contested.