The typed lambda calculus and its operation of type restricted application are familiar to anyone who has worked in formal semantics. But there are largely unexamined questions about the nature of types, their relations to formulas of logical form, and the effect of rules of type shifting on logical form. We need to look at these questions in detail.

Questions about types

Let us first turn to questions about types. For one thing, are our types all atoms or are there types that have a structure and that are constructed from “type constructors” together with other types? Another question is, what is the interpretation of our types?

Montague Grammar has an answer to our questions. Montague Grammar starts with two basic types, the type of entities e and the type of truth values t and then closes the collection of types under the recursive rule that if a and b are types, then so is a ⇒ b. The type a ⇒ b is one that, given an argument of type a, produces an object of type b. Montague Grammar converts these extensional types into intensional types as follows: if a is an extensional type, then s ⇒ a is its intensional correlate, where s is the type of worlds or more generally indices of evaluation.