Proof figures and structural operators for categorial grammar

Abstract

Use of Lambek's (1958) categorial grammar for linguistic work has generally been rather limited. There appear to be two main reasons for this: the notations most commonly used can sometimes obscure the structure of proofs and fail to clearly convey linguistic structure, and the calculus as it stands is apparently not powerful enough to describe many phenomena encountered in natural language.In this paper we suggest ways of dealing with both these deficiencies. Firstly, we reformulate Lambek's system using proof figures based on the 'natural deduction' notation commonly used for derivations in logic, and discuss some of the related proof-theory. Natural deduction is generally regarded as the most economical and comprehensible system for working on proofs by hand, and we suggest that the same advantages hold for a similar presentation of categorial derivations. Secondly, we introduce devices called structural modalities, based on the structural rules found in logic, for the characterization of commutation, iteration and optionality. This permits the description of linguistic phenomena which Lambek's system does not capture with the desired sensitivity and generality.

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