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Family of formalisms in natural language syntax
Categorial grammar is a family of formalisms in natural language syntax that share the central assumption that syntactic constituents combine as functions and arguments. Categorial grammar posits a close relationship between the syntax and semantic composition, since it typically treats syntactic categories as corresponding to semantic types. Categorial grammars were developed in the 1930s by Kazimierz Ajdukiewicz and in the 1950s by Yehoshua Bar Hillel and Joachim Lambek. It saw a surge of interest in the 1970s following the work of Richard Montague, whose Montague grammar assumed a similar view of syntax. It continues to be a major paradigm, particularly within formal semantics.

The basic ideas of categorial grammar date from work by Kazimierz Ajdukiewicz (in 1935) and other scholars from the Polish tradition of mathematical logic including Stanisław Leśniewski, Emil Post and Alfred Tarski. Ajdukiewicz's formal approach to syntax was influenced by Edmund Husserl's pure logical grammar, which was formalized by Rudolph Carnap. It represents a development in the historical idea of universal logical grammar as an underlying structure of all languages. A core concept of the approach is the substitutability of syntactic categories—hence the name categorial grammar. The membership of an element (e. g., word or phrase) in a syntactic category (word class, phrase type) is established by the commutation test, and the formal grammar is constructed through series of such tests. The term categorial grammar was coined by Yehoshua Bar Hillel (in 1953). In 1958, Joachim Lambek introduced a syntactic calculus that formalized the function type constructors along with various rules for the combination of functions. This calculus is a forerunner of linear logic in that it is a substructural logic. Montague grammar uses an ad hoc syntactic system for English that is based on the principles of categorial grammar. Although Montague's work is sometimes regarded as syntactically uninteresting, it helped to bolster interest in categorial grammar by associating it with a highly successful formal treatment of natural language semantics. More recent work in categorial grammar has focused on the improvement of syntactic coverage. One formalism that has received considerable attention in recent years is Steedman and Szabolcsi's combinatory categorial grammar, which builds on combinatory logic invented by Moses Schönfinkel and Haskell Curry. There are a number of related formalisms of this kind in linguistics, such as type logical grammar and abstract categorial grammar.

A variety of changes to categorial grammar have been proposed to improve syntactic coverage. Some of the most common are listed below.

Most systems of categorial grammar subdivide categories. The most common way to do this is by tagging them with features, such as person, gender, number, and tense. Sometimes only atomic categories are tagged in this way. In Montague grammar, it is traditional to subdivide function categories using a multiple slash convention, so A/B and A//B would be two distinct categories of left applying functions, that took the same arguments but could be distinguished between by other functions taking them as arguments.

Rules of function composition are included in many categorial grammars. An example of such a rule would be one that allowed the concatenation of a constituent of type A/B with one of type B/C to produce a new constituent of type A/C. The semantics of such a rule would simply involve the composition of the functions involved. Function composition is important in categorial accounts of conjunction and extraction, especially as they relate to phenomena like right node raising. The introduction of function composition into a categorial grammar leads to many kinds of derivational ambiguity that are vacuous in the sense that they do not correspond to semantic ambiguities.

Many categorial grammars include a typical conjunction rule, of the general form X CONJ X → X, where X is a category. Conjunction can generally be applied to nonstandard constituents resulting from type raising or function composition

The grammar is extended to handle linguistic phenomena such as discontinuous idioms, gapping and extraction.