Relevant Implication

Origin. Moh Shaw-Kwei (1950), Anderson and Belnap (1975). Antecedent must be "used" in deriving consequent. Rejects A → (B → A). Variable sharing property. Foundation for relevance logic family.

Models. Relevant derivation: every premise contributes. A → B requires shared variables. Ternary accessibility relation R in Routley-Meyer semantics. Rejects weakening for implication.

Formalism.

Relevance constraint: ⊢ A → B only if A and B share at least one propositional variable. Not "every variable in A appears in B" — that is far too strong and is not the variable-sharing property.

Invalid in relevant logic: A → (B → A) (positive paradox) (A ∧ ¬A) → B (explosion) A → (B → B) (vacuous implication)

System R (core relevant logic): Axioms: A → A (A → B) → ((B → C) → (A → C)) A → ((A → B) → B) (A → (A → B)) → (A → B)

Rules: modus ponens, adjunction.

Variable sharing: If ⊢ A → B, then var(A) ∩ var(B) ≠ ∅.

Routley-Meyer semantics: R(a,b,c): combining info at a,b gives info at c. a ⊨ A → B iff ∀b,c(Rabc ∧ b ⊨ A ⇒ c ⊨ B)

Relevant deduction theorem: Γ, A ⊢ B implies Γ ⊢ A → B only when A is actually used.

Symbols.

Metatheory. Belnap's variable-sharing theorem (1960) establishes the criterion by a matrix argument: a finite De Morgan matrix validates every theorem of E while refuting any A → B whose antecedent and consequent have no variable in common. The property therefore holds for E and for R, but it is necessary and not sufficient, so it constrains theoremhood without characterizing it. R's implication is the residual of fusion — A ∘ B → C is a theorem exactly when A → (B → C) is — which makes commutative residuated lattices with a De Morgan involution the algebraic semantics and yields algebraic completeness; the Routley–Meyer ternary models supply the relational completeness. Only the relevant form of the deduction theorem holds: Γ, A ⊢ B licenses Γ ⊢ A → B just when A is actually used. Propositional R is undecidable (Urquhart 1984), as are E and T, and four propositional variables already suffice (Urquhart 2007); Craig interpolation fails for all three (Urquhart 1993). The implicational fragment T→ is the known decidable case (Bimbó and Dunn 2013).

Applies to. Philosophy of logic. Paradox avoidance. Relevant conditionals. Avoiding explosion. Information flow. Connection-based reasoning.

Limitations. Variable sharing is only a necessary condition: formulas satisfying it need not be theorems, so the criterion constrains relevance without defining it, and no known syntactic condition is both necessary and sufficient. Weakening fails, so A → (B → A) is unavailable and an irrelevant premise cannot be adjoined; correspondingly the deduction theorem applies only when the discharged hypothesis is genuinely used. Disjunctive syllogism is invalid for the relevant conditional, so eliminating a disjunct by its negation cannot be carried out inside →. R retains contraction, and Curry's derivation needs nothing else, so unrestricted comprehension over R is trivial: the consistent naive theories require weaker, contraction-free relevant bases. Undecidability denies any general decision procedure and sets in already with four propositional variables, while the failure of Craig interpolation removes the standard definability and modularity results.

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