Ordered logic

Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties (a family of cyclic orders which may be viewed as a species of structure), and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.

By extension, the term noncommutative logic is also used by a number of authors to refer to a family of substructural logics in which the exchange rule is inadmissible. The remainder of this article is devoted to a presentation of this acceptance of the term.

The oldest noncommutative logic is the Lambek calculus, which gave rise to the class of logics known as categorial grammars. Since the publication of Jean-Yves Girard's linear logic there have been several new noncommutative logics proposed, namely the cyclic linear logic of David Yetter, the pomset logic of Christian Retoré, and the noncommutative logics BV and NEL studied in the calculus of structures.

Noncommutative logic is sometimes called ordered logic, since it is possible with most proposed noncommutative logics to impose a total or partial order on the formulae in sequents. However this is not fully general since some noncommutative logics do not support such an order, such as Yetter's cyclic linear logic. Note also that while most noncommutative logics do not allow weakening or contraction together with noncommutativity, this restriction is not necessary.

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