In This Article Expand or collapse the "in this article" section Categorial Grammar

  • Introduction
  • Foundations: 1935–1960
  • The Dark Ages: 1960–1985

Linguistics Categorial Grammar
Glyn Morrill
  • LAST REVIEWED: 18 August 2022
  • LAST MODIFIED: 28 August 2019
  • DOI: 10.1093/obo/9780199772810-0244


The term “categorial grammar” refers to a variety of approaches to syntax and semantics in which expressions are categorized by recursively defined types and in which grammatical structure is the projection of the properties of the lexical types of words. In the earliest forms of categorical grammar types are functional/implicational and interact by the logical rule of Modus Ponens. In categorial grammar there are two traditions: the logical tradition that grew out of the work of Joachim Lambek, and the combinatory tradition associated with the work of Mark Steedman. The logical approach employs methods from mathematical logic and situates categorial grammars in the context of substructural logic. The combinatory approach emphasizes practical applicability to natural language processing and situates categorial grammars within extended rewriting systems. The logical tradition interprets the history of categorial grammar as comprising evolution and generalization of basic functional/implicational types into a rich categorial logic suited to the characterization of the syntax and semantics of natural language which is at once logical, formal, computational, and mathematical, reaching a level of formal explicitness not achieved in other grammar formalisms. This is the interpretation of the field that is being made in this article. This research has been partially supported by MINICO project TIN2017–89244-R. Thanks to Stepan Kuznetsov, Oriol Valentín and Sylvain Salvati for comments and suggestions. All errors and shortcomings are the author’s own.

Foundations: 1935–1960

The prehistory of categorial grammar lies with G. Frege (functional analysis of language), B. Russell (the theory of types), E. Husserl and S. Lesniewski (semantic categories), and the between-wars Polish school of logic. Casadio 1988 provides an overview of these origins. Undoubtedly, however, the inaugural publication of the field is Ajdukiewicz 1935, the first paper in the first issue of the journal Studia Philosophica. Ajdukiewicz defined nondirectional fractional types and a cancellation rule intended to provide a necessary, although not sufficient, condition for grammaticality. Ajdukiewicz was a philosopher and a logician and cannot have failed to note the resemblance of his cancellation rule to Modus Ponens, although he did not remark upon the relation. Bar-Hillel 1953 in the journal Language defined a directional variant intended to provide a necessary and sufficient condition for grammaticality. Bar-Hillel was perhaps the first computational linguist and was struck by the resemblance of cancellation to arithmetic division and multiplication. But the field was independently refounded by Lambek 1958, in the journal American Mathematics Monthly, which combined directional residual and product types in a system with not only rules of use (such as Modus Ponens) but also rules of proof (such as Conditionalization), thereby fully establishing the logical basis of the categorial enterprise. Lambek’s seminal paper proposed to classify expressions by types constructed with directional residuals and a product that is associative for which he provided appropriate logical rules. The young mathematician and linguist Lambek only found the precedents for his work in a library search after he had made his contribution. Later he considered the title of his paper to have been pretentious, but it is difficult to overstate the importance of the contribution.

  • Ajdukiewicz, Kazimierz. 1935. Die syntaktische Konnexität. Studia Philosophica 1:1–27.

    Ajdukiewicz classified expressions by nondirectional fractional types. Then it is proposed that a necessary condition for an expression to be grammatical is that there exists a permutation of types of its words that cancels to the distinguished type S by the cancellation rule. Translated in McCall, Storrs, ed. 1967. Polish logic: 1920–1939 (pp. 207–231). Oxford: Oxford Univ. Press.

  • Bar-Hillel, Yehoshua. 1953. A quasi-arithmetical notation for syntactic description. Language 29:47–58.

    DOI: 10.2307/410452

    Bar-Hillel represents a refinement of Ajdukiewicz in which expressions are classified by directional fractional types. Then it is proposed that a necessary and sufficient condition for an expression to be grammatical is that an ordered sequence of types of its words cancels to the distinguished type by directional cancellation rules. This system has been called AB in the literature.

  • Casadio, C. 1988. Semantic categories and the development of categorial grammars. In Categorial grammars and natural language structures. Edited by Richard T. Oehrle, Emmon Bach, and Deirdre Wheeler. Dordrecht, The Netherlands: D. Reidel.

    A philosophical and historical survey of the origins and development of categorial grammar.

  • Lambek, Joachim. 1958. The mathematics of sentence structure. American Mathematical Monthly 65:154–170.

    The paper covers the following technical ground: categorical calculus for the divisions and product system; Gentzen sequent calculus for the divisions and product system, retrospectively recognizable as the multiplicative fragment of non-commutative intuitionistic linear logic without empty antecedents (“Lambek’s restriction”); proof of the equivalence of these categorical and Gentzen sequent calculi; constructive proof of “Cut-elimination” for the Gentzen sequent calculus—proof that every theorem has a Cut-free proof (Cut is a rule of generalized transitivity that must be admissible but that is problematic computationally.) Lambek’s proof of Cut-elimination has as a corollary decidability. This syntactic calculus has been called L in the literature.

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