NONTRANSITIVE
Logics individuated by restricting or dropping cut—the transitivity of the consequence relation. The move is the youngest in the division and the most counterintuitive: transitivity looks less like a structural convenience than like what consequence is for, and a logic without it seems to lose the right to chain arguments at all.
The motivation is the semantic paradoxes. Every other response to the Liar gives up something in the object language—a truth value, a connective's classical behaviour, the disquotation schema. The nontransitive response gives up nothing there and drops cut instead: all classical inferences remain valid, the T-schema holds in full, and what fails is the licence to compose two valid inferences into a third. Curry's paradox gets the parallel treatment from the noncontractive side, which drops contraction rather than cut.
Entries here are classical in every local respect and non-classical in exactly one global one, which is why they are placed by the structural rule rather than by the algebra: their matrices are ordinary and their metainferences are not.