Ordered Logic

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**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.

Joachim Lambek proposed the first noncommutative logic in his 1958 paper *Mathematics of Sentence Structure* to model the combinatory possibilities of the syntax of natural languages.^{[1]} His calculus has thus become one of the fundamental formalisms of computational linguistics.

David N. Yetter proposed a weaker structural rule in place of the exchange rule of linear logic, yielding cyclic linear logic.^{[2]} Sequents of cyclic linear logic form a ring, and so are invariant under rotation, where multipremise rules glue their rings together at the formulae 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 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.^{[3]} 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.

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.

Structads are an approach to the semantics of logic that are based upon generalising the notion of sequent along the lines of Joyal's combinatorial species, allowing the treatment of more drastically nonstandard logics than those described above, where, for example, the ',' of the sequent calculus is not associative.

**^**Lambek, Joachim (1958). "The Mathematics of Sentence Structure".*The American Mathematical Monthly*.**65**(3): 154-170. CiteSeerX 10.1.1.538.885 . ISSN 0002-9890. JSTOR 2310058. doi:10.2307/2310058.**^**Yetter, David N. (1990). "Quantales and (Noncommutative) Linear Logic".*The Journal of Symbolic Logic*.**55**(1): 41-64. ISSN 0022-4812. JSTOR 2274953. doi:10.2307/2274953.**^**Retoré, Christian (1997-04-02). "Pomset logic: A non-commutative extension of classical linear logic". In Philippe de Groote, J. Roger Hindley (eds.).*Typed Lambda Calculi and Applications*. Lecture Notes in Computer Science. Springer Berlin Heidelberg. pp. 300-318. CiteSeerX 10.1.1.47.2354 . ISBN 978-3-540-62688-6. doi:10.1007/3-540-62688-3_43.

- Non-commutative logic I: the multiplicative fragment by V. Michele Abrusci and Paul Ruet, Annals of Pure and Applied Logic 101(1), 2000.
- Logical aspects of computational linguistics (PS) by Patrick Blackburn, Marc Dymetman, Alain Lecomte, Aarne Ranta, Christian Retoré and Eric Villemonte de la Clergerie.
- Papers on Commutative/Non-commutative Linear Logic in the calculus of structures: a research homepage from which the papers proposing BV and NEL are available.

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