Categorical Vector Space Semantics for Lambek Calculus with a Relevant Modality

Abstract

We develop a categorical compositional distributional semantics for Lambek Calculus with a Relevant Modality !L∗, a modality that allows for the use of limited editions of contraction and permutation in the logic. Lambek Calculus has been introduced to analyse syntax of natural language and the linguistic motivation behind this modality is to extend the domain of the applicability of the calculus to fragments which witness the discontinuity phenomena. The categorical part of the semantics is a monoidal biclosed category with a !-functor, very similar to the structure of a Differential Category. We instantiate this category to finite dimensional vector spaces and linear maps via "quantisation" functors and work with three concrete interpretations of the !-functor. We apply the model to construct categorical and concrete semantic interpretations for the motivating example of !L∗: the derivation of a phrase with a parasitic gap. The efficacy of the concrete interpretations are evaluated via a disambiguation task, on an extension of a sentence disambiguation dataset to parasitic gap phrase one, using BERT, Word2Vec, and FastText vectors and relational tensors.