Semantic Interpretation by CC Functors for LFG Glue Semantics

The ‘glue semantics’ that has been developed for Lexical-Functional Grammar (LFG) is based on the idea of using linear logic proofs to effect the semantic assembly of the meaning of a sentence from the meanings of its parts. The production of the meanings or model-theoretic interpretations themselves is standardly achieved by labeling the nodes of the deduction with terms in a combination of linear and non-linear lambda-terms, raising questions of exactly what is the relationship between these two systems, and suggesting that there might be a somewhat arbitrary symbolic intermediary sitting between the glue proofs and the actual meanings. Here I will show how, if the meanings themselves are inhabitants of a closed cartesian category (CCC), as is often (but not always) the case, we can make the relationship between the glue proof and the assembly of the actual meanings more principled, by implementing it with a closed cartesian (CC) functor from a CCC based on the glue proof to the CCC that the model-theoretic interpretations reside in, which, furthermore, can go via a functor generated by the semantic types and the lexicon. Furthermore, if desired, this functor can be factored through some intermediate levels which, from a mathematical point of view, exist independently, and possibly have empirically relevant properties. I will first give some background for LFG, to provide an account of the context in which the ideas are meant to function. A real explanation of the category theory on the other hand, developed in Lambek and Scott (1986:4160), henceforth LS, is probably not a realistic prospect for inclusion in a