Underspecified Intepretation in a Curry-Typed Representation Language

In previous work we have developed Property Theory with Curry Typing (PTCT), an intensional first-order logic for natural language semantics. PTCT permits fine-grained specifications of meaning. It also supports polymorphic types and separation types.(1) We develop an intensional number theory within PTCT order to represent proportional generalized quantifiers like 'most', and we suggest a dynamic type-theoretic approach to anaphora and ellipsis resolution. Here we extend the type system to include product types, and use these to define a permutation function that generates underspecified scope representations within PTCT. We indicate how filters can be added to encode constraints on possible scope readings. Our account offers several important advantages over other current theories of underspecification.