Chapter 15 - Classical Type Theory

Abstract

Type theory, otherwise known as higher-order logic, is an extension of first-order logic which has significant advantages over first-order logic for formalizing certain domains, such as parts of mathematics and specifications for hardware and software. A number of versions of type theory have been developed. The chapter provides an introduction to classical type theory, and discusses methods for automatically proving theorems of classical type theory. One can also construct computer-checked proofs in a mode which is primarily interactive. Since for certain applications it is very important to obtain computer-checked proofs, and not currently practical to obtain them automatically, this is a very important activity. Interactive theorem proving can be partially automated through the use of tactics. Type theory is a language for formalizing mathematics which was invented by Bertrand Russell. Type theory includes first-order logic, so type theory is also known as higher-order logic.