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 a merely false antecedent.

Lewis systems: S1: minimal strict implication. S2: adds ◇(A ∧ B) ⥽ ◇A. 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). Reflexive already in S1: ⊢ A ⥽ A.

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

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

Symbols.

Symbol	Unicode	Name	Meaning
⥽	U+297D	Right fish tail	Strict conditional: necessarily, if A then B
□	U+25A1	White square / Box	Necessity
◇	U+25C7	White diamond	Possibility
→	U+2192	Rightwards arrow	Material conditional
¬	U+00AC	Not sign	Negation
∧	U+2227	Logical and	Conjunction
:=	—	Definitional equality	A ⥽ B is defined as □(A → B)
⊂	U+2282	Subset of	Proper containment of one Lewis system in the next
A, B, C	—	Schematic letters	Arbitrary formulas
S1–S5	—	Lewis systems	The 1932 hierarchy; S1–S3 non-normal, S4 and S5 normal

Metatheory. The Lewis systems form a strictly increasing chain S1 ⊂ S2 ⊂ S3 ⊂ S4 ⊂ S5, each properly contained in the next, and all five are decidable. Only S4 and S5 are normal: S1 through S3 fail the rule of necessitation or the K axiom, and Kripke's (1965) completeness proofs for S2 and S3 require frames with non-normal worlds at which every boxed formula is false, since no class of ordinary Kripke frames characterizes them. Within S4 and S5 the strict conditional is transitive, contraposable, and monotone in the antecedent — (A ⥽ B) → ((A ∧ C) ⥽ B) is a theorem — and these are exactly the three principles that fail for the variably strict conditional of Stalnaker and Lewis, which is why ⥽ cannot serve as a counterfactual. The paradoxes are relocated rather than removed: □B → (A ⥽ B) and ¬◇A → (A ⥽ B) are theorems of every system in the chain.

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

Limitations. The construction trades the paradoxes of material implication for their exact analogues one level up: an impossible antecedent strictly implies everything and a necessary consequent is strictly implied by everything, so a contradiction still entails an unrelated proposition. Anderson and Belnap's variable-sharing requirement is unattainable by any conditional defined as □(A → B), since □ is closed under classical consequence and cannot track subject matter — this is what motivated relevance logic as a separate enterprise rather than a repair. Being monotone in the antecedent, ⥽ cannot represent counterfactuals, which fail strengthening. And the choice among S1 to S5 does not change ⥽ itself: all five define it identically and differ only over iterated necessity, so the hierarchy answers a question about □ rather than about implication. S1 to S3 are non-normal, placing them outside frame correspondence and the Sahlqvist machinery entirely.