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	Negation	(may behave non-classically)
∼	U+223C	Strong negation	(Nelson's logic)
∧ ∨ →	—	Connectives	(as usual)
b	—	Both	True and false simultaneously
⊢	U+22A2	Consequence	Paraconsistent entailment
⊥	U+22A5	Absurdity	May not entail everything
∘A	—	Consistency	da Costa's well-behavedness

Metatheory. Paraconsistent logics are typically weaker than classical logic (fewer theorems). Some reject disjunctive syllogism (A ∨ B, ¬A ⊢ B) or related principles. LP is decidable. Paraconsistency is about the consequence relation, not truth values necessarily. Different paraconsistent logics make different trade-offs about what to preserve.

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. Losing explosion means losing some useful inferences. If disjunctive syllogism is rejected, reasoning becomes harder. Different systems disagree on what to preserve. Paraconsistency is a negative property (not explosive); positive characterization is elusive. Critics argue contradictions should be resolved, not tolerated. Dialethism (true contradictions) is philosophically controversial.