Higher-Order DisCoCat (Peirce-Lambek-Montague semantics)

TL;DR

This paper introduces a novel higher-order DisCoCat framework where word meanings are diagram-valued functions, enabling a diagrammatic approach to complex natural language semantics including quantifiers and negation.

Contribution

It presents a new higher-order categorical model for natural language semantics that extends DisCoCat with diagram-valued functions and connects to Montague semantics via lambda calculus.

Findings

Provides a formal definition of higher-order DisCoCat models

Demonstrates translation from Lambek calculus to Peirce's system beta

Includes a proof-of-concept implementation in DisCoPy

Abstract

We propose a new definition of higher-order DisCoCat (categorical compositional distributional) models where the meaning of a word is not a diagram, but a diagram-valued higher-order function. Our models can be seen as a variant of Montague semantics based on a lambda calculus where the primitives act on string diagrams rather than logical formulae. As a special case, we show how to translate from the Lambek calculus into Peirce's system beta for first-order logic. This allows us to give a purely diagrammatic treatment of higher-order and non-linear processes in natural language semantics: adverbs, prepositions, negation and quantifiers. The definition presented in this article comes with a proof-of-concept implementation in DisCoPy, the Python library for string diagrams.