Pregroup Grammars, their Syntax and Semantics

TL;DR

This paper explores the use of pregroup grammars for syntax and semantics, addressing the ambiguity problem in set-theoretic semantics and proposing finite dimensional vector spaces as a solution.

Contribution

It investigates the application of finite dimensional vector spaces to pregroups, proposing a new approach to composition in semantics beyond traditional set-theoretic methods.

Findings

Finite dimensional vector spaces can model pregroups' semantics.

Tensor product may be suitable for composition in vector space semantics.

Addresses ambiguity issues in set-theoretic semantics of pregroups.

Abstract

Pregroup grammars were developed in 1999 and stayed Lambek's preferred algebraic model of grammar. The set-theoretic semantics of pregroups, however, faces an ambiguity problem. In his latest book, Lambek suggests that this problem might be overcome using finite dimensional vector spaces rather than sets. What is the right notion of composition in this setting, direct sum or tensor product of spaces?