Analytic Implication

Origin. Parry (1933), Fine (1986), Ferguson. Proscriptive principle. Containment of content. Anti-paradox of relevance. Strongest content connection.

Models. Conclusion content contained in premises. Proscriptive principle: no new vocabulary. Analytically valid = content preserving. Strictest relevance condition.

Formalism.

Proscriptive principle: A → B valid only if var(B) ⊆ var(A). Conclusion introduces no new variables. Conclusion "contained" in premises. Parry's Analytic Implication.

Variable sharing (relevance): Requires var(A) ∩ var(B) ≠ ∅. Weaker than proscriptive. Proscriptive entails sharing. Sharing doesn't entail proscriptive.

Parry's AI: Validates: A → A. Validates: A ∧ B → A. Invalidates: A → A ∨ B (if B new). Strong containment.

Problems with AI: A → A ∨ A valid (same variables). But A → A ∨ (A ∧ B) invalid. Counter-intuitive distinctions. Led to refinements.

Fine's semantics: Content inclusion relation. A includes B: B's content part of A's. Modal + content constraints. Truthmaker approaches.

Analytic containment: A analytically contains B. Not just entailment. Internal conceptual relationship. Hyperintensional.

Kit Fine's refinement: State-based semantics. Truthmaker containment. More discriminating than Parry. Handles disjunction better.

Symbols.

Symbol	Unicode	Meaning
→ᴬᴵ	—	analytic implication
var(A)	—	variables in A
⊆ᶜ	—	content containment
⊩ₛ	—	state support

Metatheory. Content containment. Proscriptive principle. Hyperintensionality. Analytic connection.

Applies to. Philosophy of logic. Relevance logic. Containment. Truthmaker theory.

Limitations. Strong conditions. Technical complexity. Counter-examples. Limited adoption.