Ackermann Logic

Origin. Ackermann (1956). Rigorous implication. No paradoxes of implication. Relevance precursor. Foundation of relevant logic.

Models. Implication without paradoxes. Strict relevance. Modal interpretation. Precursor to Anderson-Belnap.

Formalism.

Rigorous implication: A ⥽ B: A rigorously implies B. Rejects: ¬A ⥽ (A ⥽ B). Rejects: B ⥽ (A ⥽ B). No irrelevant implications.

Key rejections: Not: A ⥽ (B ⥽ A). Not: (A ∧ ¬A) ⥽ B. Not: A ⥽ (B ∨ ¬B). Paradoxes of material/strict implication rejected.

Positive part: Transitivity: (A ⥽ B) ⥽ ((B ⥽ C) ⥽ (A ⥽ C)). Modus ponens: A, A ⥽ B ⊢ B. Conjunction: A ⥽ B, A ⥽ C ⊢ A ⥽ (B ∧ C).

Modal interpretation: A ⥽ B ↔ □(A → B) ∧ ◇A. Strict implication + antecedent possibility. Avoids vacuous truth.

Propositional version: Π′: Ackermann's system. Weaker than R. Relevance requirement strong.

Relation to R: Ackermann's work inspired Anderson-Belnap. R systematizes rigorous implication. Ackermann: precursor. Less algebraically clean.

Disjunction: Ackermann careful with disjunctive syllogism. Controlled use. Relevance preserved.

Symbols.

Metatheory. Rigorous implication. Paradox rejection. Modal interpretation. Relevance.

Applies to. Relevance logic. History of logic. Implication theory. Paradox avoidance.

Limitations. Less systematic than R. Historical interest. Incomplete treatment. Modal complications.