Ticket Entailment

Origin. 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.

Models. The antecedent must be used, as in every relevant logic; what T drops relative to E and R is permutation, so the order in which tickets are presented matters. Ternary Routley-Meyer frames.

Formalism.

Ticket intuition: A → B: "Given a ticket for A, can get B." The ticket must be used — T is relevant, so variable sharing holds. Contrast: linear (must use exactly once, no distribution), R (must use, and may permute).

System T axioms: A → A (identity) (A → B) → ((B → C) → (A → C)) (suffixing) (A → (A → B)) → (A → B) (contraction) (A → B) → ((C → A) → (C → B)) (prefixing) A ∧ B → A, A ∧ B → B (simplification) (A → B) ∧ (A → C) → (A → B ∧ C)

Key differences: From relevance logic R: T lacks: A → ((A → B) → B) (assertion) Has: (A → B) → (¬B → ¬A) (contraposition)

Semantics: Ternary relation R(a,b,c). Modified accessibility conditions. T ⊊ E ⊊ R.

Fragment: Implication-only fragment T→. Combinator basis B, B′, I, W — not BCI. C is permutation, which T rejects; B′ is the converse compositor that survives without it, and W is contraction, which T keeps.

Decidability: Full propositional T is undecidable (Urquhart 1984), along with E and R. The pure implicational fragment T→ is decidable — a problem open since the 1950s, settled independently by Bimbó and Dunn (2012) and by Padovani (2013). No PSPACE bound is claimed for either.

Symbols.

Symbol	Unicode	Name	Meaning
→	U+2192	Ticket entailment	Conditional
T	—	System T	Ticket logic
R	—	Relevance	For comparison
¬	U+00AC	Negation	Standard

Metatheory. Undecidable in full, decidable in the implicational fragment — the gap between those two facts is the interesting thing about T. The weakest of R, E, T. Algebraic: T-algebras. Combinator correspondence via B, B′, I, W.

Applies to. Philosophy of entailment. Substructural hierarchy. Conditional logic. Fine-grained implication.

Limitations. Less studied than R. Philosophical niche. Limited applications. Between systems.