Polarized Montagovian semantics for the Lambek-Grishin calculus

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 and Moortgat. We conclude by comparing our proposal to that of Bernardi and Moortgat, where alternative semantic interpretations of LG are considered on the basis of call-by-name and call-by-value evaluation strategies.