Relevance Logic

Origin. Ackermann's "rigorous implication" (1956). Developed extensively by Alan Anderson and Nuel Belnap (Entailment, 1975, 1992). Motivated by "paradoxes of material implication": in classical logic, A → B holds whenever A is false or B is true, regardless of connection. Relevance logic requires the antecedent be relevant to the consequent.

Models. Implication with genuine connection. Classical logic validates: A → (B → A), (A ∧ ¬A) → B, A → (B → B). These are "paradoxes" because the antecedent is irrelevant to the consequent. Relevance logic rejects them. A → B requires that the truth of A genuinely bears on the truth of B.

Formalism.

Rejected principles:

A → (B → A) — why should A imply that B implies A?
(A ∧ ¬A) → B — explosion (ex falso quodlibet)
A → (B ∨ ¬B) — A doesn't determine excluded middle

Relevance criterion (variable sharing): A → B is a theorem only if A and B share a propositional variable. This syntactic criterion tracks semantic relevance.

Main systems:

R (Relevance logic):

Axioms for ∧, ∨ as usual
A → A (identity)
(A → B) → ((B → C) → (A → C)) (transitivity)
(A → (A → B)) → (A → B) (contraction)
A ∧ (A → B) → B (modus ponens as axiom)
Distribution, but carefully

E (Entailment): Stronger than R; adds necessity-like features. A → B means "A necessarily entails B."

T (Ticket entailment): Weaker; drops contraction.

Ternary relation semantics (Routley-Meyer): Models M = (W, R, *, V) where:

W: worlds/situations
R ⊆ W³: ternary accessibility
*: involution on W (for negation)
M, a ⊨ A → B iff for all b, c: Rabc and M, b ⊨ A implies M, c ⊨ B

The ternary relation captures "combining" information from two situations.

Algebraic semantics: De Morgan monoids, residuated lattices. The fusion operation (∘) models relevant combination.

Symbols.

Symbol	Unicode	Name	Meaning
→	U+2192	Relevant implication	Entailment with relevance
∘	U+2218	Fusion	Relevant combination
¬	U+00AC	Negation	De Morgan negation
∧	U+2227	Conjunction	Extensional and
∨	U+2228	Disjunction	Extensional or
t	—	Truth constant	Ackermann constant
*	—	Star	Routley star (negation)
R	—	Ternary relation	Accessibility for →

Metatheory. R is decidable (EXPSPACE). Variable sharing is necessary but not sufficient for theoremhood. R has no "fallacies of relevance" — can't derive B from A unless A is used. Disjunctive syllogism (A ∨ B, ¬A ⊢ B) fails in R (related to paraconsistency). R and E have well-developed proof theory (natural deduction, sequent calculus).

Applies to. Philosophy of logic (what should implication mean?). Information flow (where does conclusion depend on premises?). Paraconsistent reasoning (R is paraconsistent). Legal reasoning (relevant precedent). Relevance in natural language.

Limitations. More complex than classical logic. Multiple competing systems (R, E, T, etc.). Ternary semantics is less intuitive than Kripke semantics. Some classically valid reasoning becomes unavailable. Practical application is limited; classical logic remains dominant. The notion of "relevance" is formalized syntactically, which may not capture all intuitions.