11 - Types

Abstract

This chapter discusses various notions of type and function that are employed in the semantic enterprise. It also focuses various formulations of the typed lambda calculus and its logical extensions. From Frege and Russell, formal semantics inherited two crucial notions. Frege introduced the idea that certain natural language expressions should be semantically analyzed as mathematical functions. The second important notion that underpins modern semantics is that of type. In Montague semantics, English sentences are directly interpreted as expressions of his Intensional Logic. This is a system of higher order modal logic whose underlying notion of function is supplied by the typed lambda calculus. There are three different versions of the typed lambda calculus, which differ from each other according to the rigidity of the attachment between terms and types. In all three theories, the notion of type is the same: Types are generated from a basic type by forming the type of functions from one type to a second. The language of type expressions thus takes the following form: I (type of individuals) is a type expression; and If T and S are type expressions then so is T → S (the type of functions from T to S). However, the way in which terms get assigned types in the three theories is different. In the first system the types are hardwired into the syntax of terms whereas, the last two are less syntactically constrained in that the type information is not built into the syntax but supplied by rules of type assignment.