Connexive Logic

Origin. Historical roots in Aristotle's and Boethius's theses. Modern revival by McCall (1960s), Wansing (1990s-present). Non-classical logic where negation and implication interact. ¬(A → ¬A) and (A → B) → ¬(A → ¬B) are valid. Counter to material implication paradoxes.

Models. Connection between antecedent and consequent. Material implication: A → B true when A false. Connexive: rejects this — implication requires connection. Aristotle's thesis: ¬(A → ¬A) (nothing implies its negation). Boethius's thesis: (A → B) → ¬(A → ¬B).

Formalism.

Aristotle's theses: AT: ¬(A → ¬A) — A doesn't imply not-A AT': ¬(¬A → A) — not-A doesn't imply A

Boethius's theses: BT: (A → B) → ¬(A → ¬B) — if A implies B, A doesn't imply not-B BT': (A → ¬B) → ¬(A → B) — if A implies not-B, A doesn't imply B

Failure in classical logic: Classically, (⊥ → ⊥) and (⊥ → ¬⊥) both true. So ¬((⊥ → ⊥) → ¬(⊥ → ¬⊥)) — BT fails.

Connexive semantics: Various approaches:

• Worlds with "incompatibility" relation
• Belief revision style
• Nelson's logic extensions

Wansing's C: Combines features of Nelson's N4 (strong negation) with connexive theses. Four-valued approach.

Conditionals: Connexive logic often uses non-material conditional. A → B: A relevantly implies B.

Symbols.

Metatheory. Multiple connexive systems exist. Some are paraconsistent. Aristotle's and Boethius's theses validated. Standard semantics varies. Cut-elimination: some systems have it. Decidable for propositional fragments.

Applies to. Philosophy of logic. Conditional reasoning. Negation theories. Relevance and connection. Non-monotonic reasoning. Legal conditionals. Counterfactual reasoning.

Limitations. Multiple competing systems. Non-standard — unfamiliar. Some validities controversial. Limited tool support. Small research community. Integration with other logics unclear.

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