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. 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 formulas 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. Although most noncommutative logics do not allow weakening or contraction together with noncommutativity, this restriction is not necessary.
The Lambek calculus Joachim Lambek proposed the first non-commutative logic in his 1958 paper
Mathematics of Sentence Structure to model the combinatory possibilities of the
syntax of
natural languages. In his subsequent 1961 paper
On the calculus of syntactic types, he extended the analysis to cover
non-associativity as well. His calculus has since become one of the fundamental formalisms of
computational linguistics.
Cyclic linear logic David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic. Sequents of cyclic linear logic form a cycle, and so are invariant under rotation, where multipremise rules glue their cycles together at the formulas described in the rules. The calculus supports three structural modalities, a self-dual modality allowing exchange, but still linear, and the usual exponentials (? and !) of linear logic, allowing nonlinear structural rules to be used together with exchange.
Pomset logic Pomset logic was proposed by Christian Retoré in a semantic formalism with two dual sequential operators existing together with the usual tensor product and par operators of linear logic, the first logic proposed to have both commutative and noncommutative operators. A sequent calculus for the logic was given, but it lacked a
cut-elimination theorem; instead the sense of the calculus was established through a denotational semantics.
BV and NEL Alessio Guglielmi proposed a variation of Retoré's calculus, BV, in which the two noncommutative operations are collapsed onto a single, self-dual, operator, and proposed a novel proof calculus, the
calculus of structures to accommodate the calculus. The principal novelty of the calculus of structures was its pervasive use of
deep inference, which it was argued is necessary for calculi combining commutative and noncommutative operators; this explanation concurs with the difficulty of designing sequent systems for pomset logic that have cut-elimination. Lutz Straßburger devised a related system, NEL, also in the calculus of structures in which linear logic with the mix rule appears as a subsystem. ==See also==