Polarized Montagovian Semantics for the Lambek-Grishin calculus

TL;DR

This paper develops a polarized Montagovian semantics for the Lambek-Grishin calculus by adapting Girard's double negation embedding, providing a linguistically motivated analysis of extraction and comparing it to continuation semantics.

Contribution

It introduces a novel polarization-based semantic framework for LG, integrating Girard's embedding and addressing linguistic extraction within a display calculus.

Findings

Linguistic motivation for linear distributivity of tensor over par

Comparison with continuation semantics of Bernardi & Moortgat

Application of polarity-sensitive embedding to LG

Abstract

Grishin proposed enriching the Lambek calculus with multiplicative disjunction (par) and coresiduals. Applications to linguistics were discussed by Moortgat, who spoke of the Lambek-Grishin calculus (LG). In this paper, we adapt Girard's polarity-sensitive double negation embedding for classical logic to extract a compositional Montagovian semantics from a display calculus for focused proof search in LG. We seize the opportunity to illustrate our approach alongside an analysis of extraction, providing linguistic motivation for linear distributivity of tensor over par, thus answering a question of Kurtonina&Moortgat. We conclude by comparing our proposal to the continuation semantics of Bernardi&Moortgat, corresponding to call-by- name and call-by-value evaluation strategies.