Categorial grammar and lexical-functional grammar

This paper introduces $\lambda$-grammar, a form of categorial grammar that has much in common with LFG. Like other forms of categorial grammar, $\lambda$-grammars are multi-dimensional and their components are combined in a strictly parallel fashion. Grammatical representations are combined with the help of {\em linear combinators}, closed pure $\lambda$-terms in which each abstractor binds exactly one variable. Mathematically this is equivalent to employing linear logic, in use in LFG for semantic composition, but the method seems more practicable. While $\lambda$-grammars could be used to formalize many approaches to grammatical theory, they are certainly natural as a basis for the formalization of LFG. This leads to a theory I would like to call $\lambda$-LFG. In this paper it will be shown how the standard components of LFG can be set up in the framework. We will have descriptions of c-structure, descriptions of f-structure, and semantics. The difference between defining and constraining information will be explained in terms of entailment, and requirements on long-distance paths in f-structure will be explained in terms of entailment in the presence of a simple set of axioms.