Quantifier Scope in Categorical Compositional Distributional Semantics

Categorical Compositional Distributional semantics (CCDS) adds compositionality to distributional semantics via a functorial passage from the syntax to the semantics of natural language [4]. Both the syntax and the semantics are represented by compact closed categories. The claim is that regardless of how complex the structure of a sentence can be and what bizarre forms the words therein can take, as long as the sentence is represented in the syntax, CCDS will prescribe a vector to represent its semantics. In practice, however, one needs more than a syntax and a functorial passage. Semantic postulates for meanings of words are a requirement. As long as these are of good quality, CCDS will provide a good theoreticalglue. Various distributional semantics can be trusted in their ability to produce vectors for atomic words; functional words, on the other hand, need a special treatments. In our path to provide special treatments for functional words of natural language, in [10] we dealt with negation, in [8, 11] we showed how meanings of verbs and relative pronouns can be represented using Frobenius algebras. Recently, we used bialgebras to represent meanings of quantifiers [6]. Therein, we also developed a relational instantiation for the model and proved that it is sound and complete with regard to the generalised quantifier theory of natural language by Barwise and Cooper [2]. A natural question that arises in every quantifier theory is that of scope. The scope and branching of generalised quantifiers have been discussed in a wealth of literature, e.g. see the collection of papers by Gärdenfors [5], the paper by Westerstahl in that collection [12], and of course the original work of Barwise himself [1]. In this short account, we show how one may deal with scope and branching in our bialgebraic CCDS model of quantifiers.