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) (assertion). Note the conjunctive form (A ∧ (A → B)) → B — "pseudo modus ponens" — is not a theorem of R; adding it trivializes naive set theory over R (Meyer, Routley, Dunn 1979).
• Distribution of ∧ over ∨, which is what separates R from linear logic

E (Entailment): Weaker than R, not stronger: E ⊊ R. E lacks permutation, (A → (B → C)) → (B → (A → C)), which R has. A → B means "A necessarily entails B"; the modal reading is what motivates dropping permutation.

T (Ticket entailment): Weaker still: T ⊊ E ⊊ R. T drops permutation as well, keeping only prefixing and suffixing as the transitivity principles. Contraction is retained — W is a theorem of T. It is RW, not T, that 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.

Metatheory. R is undecidable — Urquhart, "The undecidability of entailment and relevant implication" (JSL 1984), by reduction through von Neumann's coordinatization of modular lattices; the same argument covers E and T. Craig interpolation also fails for all three (Urquhart 1993, via the non-embeddability of a non-Arguesian projective plane in three-dimensional projective space). 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.

© 2026 Lingenic LLC