Higher-Order Logic and Type Theory

The Element provides a wide-ranging, but unified account of higher-order logic and contemporary type theory.

John L. Bell (Author)

9781108986908, Cambridge University Press

Paperback / softback, published 31 March 2022

75 pages
22.8 x 15.2 x 0.5 cm, 0.14 kg

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.

1. Second- Order Logic and Higher-Order Logic
2. Type Theory and its Origins
3. Local set theory
4. Newer Forms of Type Theory Based on the Doctrine of 'Propositions as Types'
Appendix
The Semantics of Local Set Theory/Intuitionistic Higher-Order Logic.

Subject Areas: Set theory [PBCH], Philosophy: logic [HPL], Philosophy [HP]