ORDERED LOGIC
Logics that drop the structural rule of exchange, so that the order of hypotheses is significant and cannot be permuted. Implication splits into two directed forms—one seeking its argument on the left, one on the right—and the resulting calculi are non-commutative.
Entries center on the Lambek calculus and its extensions. Categorial grammar in the Ajdukiewicz-Bar-Hillel style supplies the applicative core—directional slashes with cancellation rules but no hypothetical withdrawal—which the Lambek calculus completes with introduction rules; it is cross-listed into Type, where its slashes read as a type assignment for syntactic categories. The Lambek calculus itself gives the residuated, order-sensitive core; the Lambek-Grishin and displacement calculi add symmetry and discontinuity; pregroup grammars recast the calculus with left and right adjoints in an ordered monoid; non-commutative and ordered logics generalize the discipline. The primary motivation is linguistic word-order sensitivity, but the systems are studied as substructural calculi in their own right.