Adapting Type Theory with Records for Natural Language Semantics

Abstract

In this paper we will go through the version of type theory TTR (Type Theory with Records) that we have proposed in a number of publications (including, Cooper, Res Lang Comput, 3:333–362, 2005a, J Log Comput, 15(2):99–112, 2005b, Handbook of the philosophy of science, volume 14: Philosophy of linguistics, 2012) and discuss the motivation for some choices that we have made which make it differ from other more standard modern type theories. We will relate TTR to the kind of type theory used in traditional formal semantics, characterizing more modern type theories as rich type theories with a greater variety of types. TTR, unlike many rich type theories, allows objects to be of several types and introduces both a kind of intensionality and modality. While TTR uses the idea that propositions should be modelled by types, it does not complete follow the Curry-Howard Correspondence introducing intersection and union types for a more classical treatment of conjunction and disjunction. It uses record types in place of \(\varSigma \)-types and uses dependent types for several aspects of linguistic analysis.