Church's Type Theory

Church’s type theory, aka simple type theory, is a formal logical language which includes classical first-order and propositional logic, but is more expressive in a practical sense. It is used, with some modifications and enhancements, in most modern applications of type theory. It is particularly well suited to the formalization of mathematics and other disciplines and to specifying and verifying hardware and software. It also plays an important role in the study of the formal semantics of natural language. When utilizing it as a meta-logic to semantically embed expressive (quantified) non-classical logics further topical applications are enabled in artificial intelligence and philosophy.

A great wealth of technical knowledge can be expressed very naturally in it. With possible enhancements, Church’s type theory constitutes an excellent formal language for representing the knowledge in automated information systems, sophisticated automated reasoning systems, systems for verifying the correctness of mathematical proofs, and a range of projects involving logic and artificial intelligence. Some examples and further references are given in Sections 1.2.2 and 5 below.

Type theories are also called higher-order logics, since they allow quantification not only over individual variables (as in first-order logic), but also over function, predicate, and even higher order variables. Type theories characteristically assign types to entities, distinguishing, for example, between numbers, sets of numbers, functions from numbers to sets of numbers, and sets of such functions. As illustrated in Section 1.2.2 below, these distinctions allow one to discuss the conceptually rich world of sets and functions without encountering the paradoxes of naive set theory.

Church’s type theory is a formulation of type theory that was introduced by Alonzo Church in Church 1940. In certain respects, it is simpler and more general than the type theory introduced by Bertrand Russell in Russell 1908 and Whitehead & Russell 1927a. Since properties and relations can be regarded as functions from entities to truth values, the concept of a function is taken as primitive in Church’s type theory, and the λ-notation which Church introduced in Church 1932 and Church 1941 is incorporated into the formal language. Moreover, quantifiers and description operators are introduced in a way so that additional binding mechanisms can be avoided, λ-notation is reused instead. λ-notation is thus the only binding mechanism employed in Church’s type theory.