# 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.

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