Paraconsistent Logic

Origin. Stanisław Jaśkowski (1948) and Newton da Costa (1963) independently developed logics tolerating contradiction. Motivated by: inconsistent but useful theories (early calculus, naive set theory), real-world databases with conflicts, dialethism (some contradictions are true). Graham Priest developed dialetheism and Logic of Paradox (LP).

Models. Contradictions that don't explode. In classical logic, A ∧ ¬A ⊢ B (explosion/ex falso quodlibet): contradiction implies everything. Paraconsistent logics reject explosion: local contradictions don't pollute the whole system. Can reason about inconsistent information without triviality.

Formalism.

What's rejected: The inference A, ¬A ⊢ B (explosion).

Approaches:

Logic of Paradox (LP, Priest): Three values: t (true only), b (both true and false), f (false only) Designated values: t and b (both count as "true")

¬: t↦f, b↦b, f↦t
∧, ∨: computed on ordering f < b < t
A ∧ ¬A = b when A = b (contradiction without explosion)
Valid: anything true in all valuations where premises are designated

N4 / Nelson's logic: Strong negation ∼ distinct from intuitionistic negation Allows A and ∼A without explosion Constructive variant of paraconsistency

Relevant logic: Rejects not explosion directly, but irrelevant implications A → B requires relevance between A and B No explosion because A ∧ ¬A is not relevant to arbitrary B

da Costa's Cn hierarchy: Weaken consistency principle at different levels C1: contradiction doesn't explode, but can mark "well-behaved" formulas Nested hierarchy of progressively weaker systems

Preserving useful inferences: Paraconsistent logics aim to reject explosion while keeping modus ponens, conjunction introduction, etc.

Symbols.

Symbol	Unicode	Name	Meaning
¬	U+00AC	Not sign	Negation, which may behave non-classically
∼	U+223C	Tilde operator	Strong negation (Nelson's logic)
∧	U+2227	Logical and	Conjunction; adjunction may fail
∨	U+2228	Logical or	Disjunction; addition is usually retained
→	U+2192	Rightwards arrow	Implication, material or relevant depending on the system
∘	U+2218	Ring operator	Consistency operator; ∘A marks A as well behaved
⊢	U+22A2	Right tack	Paraconsistent derivability
⊬	U+22AC	Does not prove	Failure of derivability, as in A, ¬A ⊬ B
⊥	U+22A5	Up tack	Absurdity, which need not entail everything
b	—	Both	True and false simultaneously
LP	—	Logic of Paradox	Priest's three-valued system
FDE	—	First-Degree Entailment	Belnap–Dunn four-valued system
D2	—	Discussive logic	Jaśkowski's non-adjunctive system
Cₙ	—	da Costa's hierarchy	Consistency-operator systems
LFI	—	Logic of Formal Inconsistency	Family taking ∘ as primitive

Metatheory. Paraconsistency is a property of the consequence relation — that A, ¬A ⊬ B for some A and B — and not of truth values, so it is independent of dialetheism, which is a claim about what is true. What constrains every system in the family is C. I. Lewis's argument: from A and ¬A, addition gives A ∨ B, and disjunctive syllogism with ¬A gives B, so blocking explosion means giving up addition, disjunctive syllogism, or transitivity of consequence. The families are classified by which they surrender. LP, FDE and the relevant logics reject disjunctive syllogism; Jaśkowski's D2 rejects adjunction, so premises cannot be combined; da Costa's Cₙ restrict explosion to formulas marked consistent. Decidability varies with the choice: LP and FDE are finite-matrix logics and decidable, while Urquhart (1984) proved R and E undecidable. Blocking explosion is not the same as blocking triviality — Curry's paradox trivializes a naive theory using only contraction and modus ponens — but Brady (1989) showed naive comprehension non-trivial in a suitably weakened relevant logic.

Applies to. Inconsistent databases (query without explosion). Belief revision with contradictory inputs. Legal reasoning (conflicting statutes). Self-referential reasoning (Liar paradox). Fault-tolerant systems. Philosophy (dialethism: some contradictions are true).

Limitations. Paraconsistency is defined negatively and has no positive characterization, so the class contains systems agreeing on nothing but the failure of explosion. Lewis's argument makes the cost unavoidable: something in addition, disjunctive syllogism or transitivity must go, and each loss removes standard mathematical argument, disjunctive syllogism being the ordinary way of reasoning from a disjunction and adjunction the ordinary way of combining premises. Paraconsistent logics are not conservatively extendable by a Boolean negation or an unrestricted consistency operator, since either restores explosion, so they cannot simply be added to a classical system. Non-explosiveness does not secure non-triviality: Curry's paradox needs no negation at all, so contraction must be surrendered as well before a naive truth or membership predicate is safe. And most such logics are strictly weaker than classical logic, so the recapture principles that make ordinary mathematics available have to be supplied from outside.