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.
S4 enriched multimodal categorial grammars are context-free
