Relevance and Paraconsistent Logics

Relevance logics demand a genuine connection between premises and conclusion, while paraconsistent logics deny that a contradiction entails everything.

Definition

A relevance logic invalidates inferences in which premises are irrelevant to the conclusion; a paraconsistent logic is one whose consequence relation is non-explosive, so that not everything follows from a contradiction.

Scope

This topic covers two related departures from classical logic. Relevance (relevant) logic rejects the paradoxes of material and strict implication by requiring that the antecedent be relevant to the consequent. Paraconsistent logic rejects ex falso quodlibet (explosion), so that inconsistent theories need not be trivial. It also covers dialetheism — the radical view that some contradictions are true — and the Routley-Meyer relational semantics underpinning these systems.

Core questions

• What kind of relevance must hold between premises and conclusion for a valid entailment?
• Should logic permit inconsistent but non-trivial theories?
• Are any contradictions actually true, as dialetheism claims?
• What semantics validates relevant and paraconsistent consequence?

Key concepts

• ex falso quodlibet (explosion)
• relevance condition
• entailment vs. material implication
• paraconsistency
• dialetheism
• Routley-Meyer semantics

Key theories

Relevant entailment

Anderson and Belnap develop systems of entailment in which a valid implication requires the antecedent to be used in deriving the consequent, blocking the classical principle that a contradiction or a truth implies anything.

Dialetheism and the logic of paradox

Priest argues that some sentences (notably the Liar) are both true and false, and develops the paraconsistent Logic of Paradox in which such dialetheias do not trivialize a theory.

History

Relevance logic was systematized by Anderson and Belnap from the 1950s, building on Ackermann's work, and given a relational semantics by Routley and Meyer in the 1970s. Paraconsistent logics developed in parallel (da Costa, Priest), and Priest's defence of dialetheism made the live possibility of true contradictions a major topic.

Debates

Can a contradiction ever be true?

Whether dialetheism is coherent — accepting some true contradictions to dissolve the semantic paradoxes — or whether the law of non-contradiction is indispensable and paraconsistency should be used only to quarantine inconsistency, not endorse it.

Key figures

• Alan Ross Anderson
• Nuel Belnap
• Graham Priest
• Richard Routley (Sylvan)
• Robert Meyer

Seminal works

• andersonbelnap1975
• priest1979
• priest2006contradiction

Frequently asked questions

What is explosion, and why reject it?

Explosion (ex falso quodlibet) is the classical principle that from a contradiction, any statement whatsoever follows. Paraconsistent logicians reject it because it makes any inconsistent theory trivial — implying everything — which seems too strong, since we often reason sensibly within theories that contain hidden inconsistencies.