Multi-dimensional Type Theory: Rules, Categories, and Combinators for Syntax and Semantics

We investigate the possibility of modelling the syntax and semantics of natural language by constraints, or rules, imposed by the multi-dimensional type theory Nabla. The only multiplicity we explicitly consider is two, namely one dimension for the syntax and one dimension for the semantics, but the general perspective is important. For example, issues of pragmatics could be handled as additional dimensions. One of the main problems addressed is the rather complicated repertoire of operations that exists besides the notion of categories in traditional Montague grammar. For the syntax we use a categorial grammar along the lines of Lambek. For the semantics we use so-called lexical and logical combinators inspired by work in natural logic. Nabla provides a concise interpretation and a sequent calculus as the basis for implementations. . . . Lambek originally presented his type logic as a calculus of syntactic types. Semantic interpretation of categorial deductions along the lines of the Curry-Howard correspondence was put on the categorial agenda in J. van Benthem (1983) The semantics of variety in categorial grammar, Report 83-29*, Simon Fraser University, Canada. This contribution made it clear how the categorial type logics realize Montagues Universal Grammar program — in fact, how they improve on Montagues own execution of that program in offering an integrated account of the composition of linguistic meaning and form. Montagues adoption of a categorial syntax does not go far beyond notation: he was not interested in offering a principled theory of allowable ‘syntactic operations’ going with the category formalism. * Revised version in [2] M. Moortgat (1997) Categorial Type Logics, in J. van Benthem & A. ter Meulen (eds.) Handbook of Logic and Language, Elsevier. Full paper of short presentation at the International Workshop on Constraint Solving and Language Processing (CSLP 2004). This research was partly sponsored by the IT University of Copenhagen and the CONTROL project: CONstraint based Tools for RObust Language processing http://control.ruc.dk