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.

Symbol	Unicode	Name	Meaning
→	U+2192	Arrow	Relevant implication
⊢	U+22A2	Turnstile	Entailment
R	—	Accessibility	Ternary relation
∘	U+2218	Fusion	Intensional conjunction

Metatheory. Propositional R is undecidable (Urquhart 1984), and so are E and T; the four-variable fragment of R is already undecidable (Urquhart 2007). Craig interpolation fails as well (Urquhart 1993). Variable sharing holds. No disjunctive syllogism with the relevant conditional. Algebraic: De Morgan monoids.

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

Limitations. Weaker than classical logic. Some classical inferences blocked. Multiple systems disagree. Philosophical debates on "relevance." Not mainstream in computing.