Strict Conditional

Origin. Lewis (1912, 1918). Material conditional critique. Modal implication. S1-S5 systems. Foundation of modal logic.

Models. Necessity of conditional. Possible worlds. Lewis systems. Avoids paradoxes of material implication.

Formalism.

Strict conditional: A ⥽ B := □(A → B). Necessarily if A then B. Stronger than material. Modal definition.

Motivation: Material: A → B true when A false. Paradox: false implies anything. Strict: requires necessity. No vacuous truth from impossible antecedent.

Lewis systems: S1: minimal strict implication. S2: adds ◇(A ∧ B) → (◇A ∧ ◇B). S3: adds □(A → B) → (□A → □B). S4: adds □A → □□A. S5: adds ◇A → □◇A.

S2 characteristic: Consistency of conjunctions. If ◇(A ∧ B) then ◇A and ◇B. Modal distribution.

Strict implication properties: Transitive: (A ⥽ B) ∧ (B ⥽ C) → (A ⥽ C). Contrapositive: (A ⥽ B) → (¬B ⥽ ¬A). Not reflexive for S1.

Remaining paradox: □A → (B ⥽ A). Necessary truths strictly implied by anything. ◇¬A → (A ⥽ B) in some systems. Impossibilities strictly imply anything.

Relevance critique: Anderson-Belnap: strict not enough. Need variable sharing. Relevance logic response.

Symbols.

Symbol	Unicode	Meaning
⥽	U+297D	strict conditional
□	U+25A1	necessity
S1-S5	—	Lewis systems
→	U+2192	material conditional

Metatheory. Modal implication. Lewis hierarchy. Necessity. Paradox reduction.

Applies to. Modal logic. Counterfactuals. Conditionals. History of logic.

Limitations. Paradoxes remain. Relevance issues. Which system? Material paradoxes not fully resolved.