  We develop a categorical compositional distributional semantics for Lambek
Calculus with a Relevant Modality !L*, which has a limited edition of the
contraction and permutation rules. The categorical part of the semantics is a
monoidal biclosed category with a coalgebra modality, 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 coalgebra modality. 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
effectiveness of the concrete interpretations are evaluated via a
disambiguation task, on an extension of a sentence disambiguation dataset to
parasitic gap phrases, using BERT, Word2Vec, and FastText vectors and
Relational tensors.
