# 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.

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