Polarized Montagovian Semantics for the Lambek-Grishin Calculus

Grishin ([9]) proposed enriching the Lambek calculus with multiplicative disjunction (par) and coresiduals. Applications to linguistics were discussed by Moortgat ([14]), who spoke of the Lambek-Grishin calculus (LG). In this paper, we adapt Girard’s polarity-sensitive double negation embedding for classical logic ([7]) to extract a compositional Montagovian semantics from a display calculus for focused proof search ([1]) 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 ([3]), thus answering a question of [10]. We conclude by comparing our proposal to that of [2], where alternative semantic interpretations of LG are considered on the basis of call-byname and call-by-value evaluation strategies. Inspired by Lambek’s syntactic calculus, Categorial type logics ([13]) aim at a proof-theoretic explanation of natural language syntax: syntactic categories and grammaticality are identified with formulas and provability. Typically, they show an intuitionistic bias towards asymmetric consequence, relating a structured configuration of hypotheses (a constituent) to a single conclusion (its category). The Lambek-Grishin calculus (LG, [14]) breaks with this tradition by restoring symmetry, rendering available (possibly) multiple conclusions. §1 briefly recapitulates material on LG from [14] and [15]. In this article, we couple LG with a Montagovian semantics. Presented in §2, its main ingredients are focused proof search [1] and a double negation translation along the lines of [7] and [19], employing polarities to keep the number of negations low. In §3, we illustrate our semantics alongside an analysis of extraction inspired by linear distributivity principles ([3]). Finally, §4 compares our approach to the competing proposal of [2]. 1 The Lambek-Grishin calculus Lambek’s (non-associative) syntactic calculus ((N)L, [11], [12]) combines linguistic inquiry with the mathematical rigour of proof theory, identifying syntactic categories and derivations by formulas and proofs respectively. On the logical side, (N)L has been identified as (non-associative, )non-commutative multiplicative intuitionistic linear logic, its formulas generated as follows: 1 Understanding Montagovian semantics in a broad sense, we take as its keywords model-theoretic and compositional. Our emphasis in this article lies on the latter.