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

TL;DR

This paper explores a multi-dimensional type theory called Nabla to model natural language syntax and semantics, allowing for multiple layers such as pragmatics, with a focus on rules, categories, and combinators.

Contribution

It introduces a multi-dimensional type framework that integrates syntax and semantics using categorial grammar and combinators, extending traditional Montague grammar.

Findings

Nabla provides a concise interpretation and sequent calculus.

The framework supports multiple dimensions, including syntax and semantics.

It offers a basis for implementing natural language processing systems.

Abstract

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.