Categorical compositional distributional semantics is a model of naturallanguage; it combines the statistical vector space models of words with thecompositional models of grammar. We formalise in this model the generalisedquantifier theory of natural language, due to Barwise and Cooper. Theunderlying setting is a compact closed category with bialgebras. We start froma generative grammar formalisation and develop an abstract categoricalcompositional semantics for it, then instantiate the abstract setting to setsand relations and to finite dimensional vector spaces and linear maps. We provethe equivalence of the relational instantiation to the truth theoreticsemantics of generalised quantifiers. The vector space instantiation formalisesthe statistical usages of words and enables us to, for the first time, reasonabout quantified phrases and sentences compositionally in distributionalsemantics.
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