# 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.**

| Symbol | Unicode | Name | Meaning |
|--------|---------|------|---------|
| → | U+2192 | Conditional | Connexive implication |
| ¬ | U+00AC | Negation | Negation |
| ∧ | U+2227 | Conjunction | And |
| ∨ | U+2228 | Disjunction | Or |
| AT | — | Aristotle | ¬(A → ¬A) |
| BT | — | Boethius | (A → B) → ¬(A → ¬B) |

**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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