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 valuations assigning no atom the glut value, LP evaluation is classical. LPm (Priest 1991) restricts attention to minimally inconsistent models. Classically consistent premises then yield classical consequence, at the cost of monotonicity.

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.

FDE: Adding a gap value to the glutty three yields four-valued First-Degree Entailment (Belnap–Dunn)—paracomplete and paraconsistent together. Not to be confused with Nelson's N4, which is FDE plus an intuitionistic implication.

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
¬	U+00AC	Not sign	Negation; ¬B = B
∧	U+2227	Logical and	Minimum on the order F < B < T
∨	U+2228	Logical or	Maximum on the order F < B < T
→	U+2192	Rightwards arrow	Material conditional, ¬A ∨ B; not detachable
⊨	U+22A8	Satisfaction	Consequence: preservation of a designated value
⊭	U+22AD	Not true	Failure of consequence, as in A, ¬A ⊭ B
T	—	True only	Designated
F	—	False only	Undesignated
B	—	Both	Glut (true and false); designated
D	—	Designated set	{T, B}
LP	—	Logic of Paradox	The matrix ({T, B, F}, {T, B})
LPm	—	Minimally inconsistent LP	Priest's nonmonotonic strengthening (1991)
FDE	—	Four-valued	Belnap–Dunn: gaps and gluts
⊨_LP	—	LP consequence	Designated-preservation in LP

Metatheory. Priest (1979) built LP on the strong Kleene tables with the glut designated, and the effect is exactly the reverse of K3's. The LP-valid formulas are precisely the classical tautologies: given any LP valuation, read relationally as assigning each atom a non-empty subset of {true, false}, choose a classical valuation selecting one member at each atom; the De Morgan clauses guarantee that the selected value is among the values the LP valuation gives to every formula, so a classical tautology must come out true and hence designated. Dually to K3, which has no valid formulas, LP has no unsatisfiable ones — the valuation sending every atom to B designates everything. Validity is therefore coNP-complete, as in classical propositional logic. What changes is consequence, which is strictly weaker: explosion, disjunctive syllogism, modus ponens for the material conditional and reductio all fail, while contraposition survives because negation is symmetric. Algebraically LP is the three-element Kleene algebra with D = {T, B}; the dual designation on the same algebra gives K3, and the two consequence relations are related by A ⊨_K3 B iff ¬B ⊨_LP ¬A. Priest (1991) recovers classical consequence for classically consistent premises by restricting to minimally inconsistent models, at the cost of monotonicity.

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

Limitations. Having every classical tautology as a theorem is worth less than it looks, because the conditional does not internalize consequence: ⊨ A → B holds without A ⊨ B, since modus ponens fails at a glut, so the deduction theorem runs only one way and a relevant conditional has to be added from outside. Disjunctive syllogism and reductio go with it, which removes standard mathematical argument forms rather than just the pathological ones. LP cannot deny: there is no connective distinguishing "false only" from "both", and adding a Boolean negation or a consistency operator restores explosion and destroys paraconsistency — so the extended Liar, the sentence saying of itself that it is not true, is not disarmed. Recapture requires abandoning monotonicity. And truth-functionality means a glut is a glut to the same degree everywhere, with no account of why particular sentences are ones.