Higher-Order Logic and Type Theory

This Element is an exposition of second- and higher-order logic and type theory.

It begins with a presentation of the syntax and semantics of classical second-order logic, pointing up the contrasts with first-order logic.

This leads to a discussion of higher-order logic based on the concept of a type.

The second Section contains an account of the origins and nature of type theory, and its relationship to set theory.

Section 3 introduces Local Set Theory (also known as higher-order intuitionistic logic), an important form of type theory based on intuitionistic logic.

In Section 4 number of contemporary forms of type theory are described, all of which are based on the so-called 'doctrine of propositions as types'.

We conclude with an Appendix in which the semantics for Local Set Theory - based on category theory - is outlined.