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, E and T are complete with respect to the Routley–Meyer ternary relational semantics, and R is algebraically complete with respect to the variety of De Morgan monoids, which are commutative residuated lattices carrying a De Morgan involution. R is undecidable — Urquhart, "The undecidability of entailment and relevant implication" (Journal of Symbolic Logic, 1984), by a reduction through von Neumann's coordinatization of modular lattices — and the same argument covers E and T; Craig interpolation fails for all three (Urquhart 1993, using the non-embeddability of a non-Arguesian projective plane in three-dimensional projective space). The implicational fragment T→ is the exception: Bimbó and Dunn settled its long-open decision problem affirmatively in 2013. Variable sharing holds for all three — a theorem A → B has a propositional variable common to A and B — but it is necessary and not sufficient for theoremhood. Disjunctive syllogism is not valid in R, which is what makes R paraconsistent, yet the rule γ is admissible: Meyer and Dunn (1969) showed that adding it as a rule on theorems yields nothing new.

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. Undecidability rules out any general decision procedure for R, E or T, and the failure of Craig interpolation removes the standard route to definability results. Variable sharing is only necessary, so it does not characterize relevance, and no known syntactic criterion is both necessary and sufficient. Disjunctive syllogism is invalid, so eliminating a disjunct through negation is unavailable inside the conditional, and γ recovers it only for theorems, never under hypotheses. The deduction theorem holds only in the relevant form, requiring that the discharged hypothesis actually be used. Adding pseudo modus ponens (A ∧ (A → B)) → B trivializes naive set theory over R (Meyer, Routley and Dunn 1979). Relevant arithmetic R# does not contain PA (Friedman and Meyer 1992), so the relevant base does not recover classical number theory. And the Routley star has no accepted independent interpretation; it is a device that makes the negation clause work.

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