README⤓ .txt 2026-10-22T165042.936 000000000000768 Logics that drop weakening in order to enforce relevance: the antecedent of a valid implication must actually be used in deriving the consequent, blocking the paradoxes of material and strict implication. The characteristic semantics is the Routley-Meyer ternary relation, with the variable-sharing property as a hallmark.
Ackermann Logic⤓ .md 2026-08-20T055942.000 000000000044760 Wilhelm Ackermann, "Begründung einer strengen Implikation" (Journal of Symbolic Logic, 1956), introduced the connective ⥽ of rigorous implication — strenge Implikation — together with the axiomatic systems Π and Π′. The aim was an implication free of the paradoxes of material and strict implication, in which the antecedent must bear on the consequent. Anderson and Belnap took Π′ as their point of departure: their system E of entailment (Entailment, 1975) is Π′ with Ackermann's rule γ removed, and R was then obtained by adding permutation. Ackermann's paper is where the relevant tradition begins as a formal enterprise rather than a complaint.
Contraction-Free Logic⤓ .md 2026-10-22T090631.000 000000000041824 Grišin (1982) showed naive comprehension is consistent without contraction; Girard's linear logic (1987) made the rule's absence systematic; Restall, "How to be really contraction free" (1993); Rogerson and Restall (2004) on the residual paradoxes. No contraction rule. Resource sensitivity. Curry paradox avoidance. Foundation of substructural semantics.
Mingle Logic⤓ .md 2026-10-22T090714.000 000000000036176 The matrices are Sugihara's (1955); Anderson and Belnap named and studied RM = R + mingle; Dunn (1970) proved completeness for the Sugihara algebras. Intermediate system: R ⊊ RM ⊊ classical.
Relevance Logic⤓ .md 2026-10-22T085314.000 000000000047256 Ackermann's "rigorous implication" (1956). Developed extensively by Alan Anderson and Nuel Belnap (Entailment, 1975, 1992). Motivated by "paradoxes of material implication": in classical logic, A → B holds whenever A is false or B is true, regardless of connection. Relevance logic requires the antecedent be relevant to the consequent.
Relevant Implication⤓ .md 2026-10-22T085328.000 000000000036184 Moh Shaw-Kwei (1950), Anderson and Belnap (1975). Antecedent must be "used" in deriving consequent. Rejects A → (B → A). Variable sharing property. Foundation for relevance logic family.
RM3⤓ .md 2026-08-20T065209.000 000000000045592 The three-element Sugihara matrix, from Sugihara's work on mingle (1955); the logic R-mingle (RM) is Anderson and Belnap's (Entailment I, 1975, §29), and RM3 is its three-valued characteristic matrix, due to Dunn (1970), who proved RM's completeness for the Sugihara algebras and RM3's role among them.
Substructural Logics⤓ .md 2026-10-22T090936.000 000000000048416 Identified through Gentzen's sequent calculus (1934) by analyzing structural rules. Lambek calculus for linguistics (1958). Relevance logic (Anderson & Belnap, 1960s-70s) rejected irrelevant implications. Linear logic (Girard, 1987) rejected contraction and weakening. Unified perspective emerged recognizing the family structure.
Ticket Entailment⤓ .md 2026-10-22T092439.000 000000000038600 Anderson and Belnap (1975). System T, the weakest of the main Anderson–Belnap systems: T ⊊ E ⊊ R. The name is from Ryle's "inference tickets" — a conditional is a licence to infer, not a further premise, and T is built so that only genuine ticket-uses license the arrow.
CRITERIA⤓ .txt 2026-10-22T165042.936 000000000000784 Not sufficient: Variable sharing imposed as a content condition without a substructural calculus (Hyperintensional). Dropping contraction alone (Affine) or exchange alone (Ordered). A linear calculus keyed to exact resource use rather than relevance (Linear).