DisCoCat for Donkey Sentences

Montague semantics is a compositional method to translate the semantics of written language into first order logic. As a simple example one can understand the meaning of the sentence "(all) dogs eat snacks" as x, y.dogs(x) snacks(y) eats(x, y). However, when translating the meaning of the sentence "Every farmer who owns a donkey beats it", the variable representing the donkey cannot be bound by the existential quantifier coming from the determiner'a'. This issue was studied by Geach [4], using it as a counterexample to the scope of Montague semantics. Many have created systems that form semantic representations of donkey sentences, to name a few we have dynamic predicate logic [7], where the binding rules of quantifiers in first order logic are relaxed, discourse representation theory [11] where an collection of'discourse referents' keep track of individuals' mentions and are identified to keep track of references, as well as an approach using dependent type theory [18], exploiting dependent sums to differentiate between ambiguous readings of donkey sentences. However, none of the models mentioned above are type-logical grammars which poses the question whether it is possible to parse donkey sentences and form usable representations of them using type logical grammars? We propose to model donkey sentences using (an extension of) Lambek calculus, L. In the following section, we explain how a type-logical analysis of natural language works, and in sections 1.3,1.4,1.5 how to extend it to model more exotic linguistic phenomena, culminating in a parse of a donkey sentence. Then we introduce relational semantics and vector space semantics of the extended Lambek calculus in sections 3.1 and 3.3 respectively, demonstrating how donkey sentence is interpreted as a relation or as a linear map.