A Modern Perspective on Type Theory

From its Origins until Today

  • Fairouz Kamareddine
  • Twan Laan
  • Rob Nederpelt

Part of the Applied Logic Series book series (APLS, volume 29)

Table of contents

  1. Front Matter
    Pages i-xiv
  2. Introduction

    1. Pages 1-6
  3. The Evolution of Type Theory until the 1940s

    1. Front Matter
      Pages 7-7
    2. Pages 3-18
    3. Pages 69-101
  4. Propositions as Types, Pure Type Systems, AUTOMATH

  5. Extensions of Pure Type Systems

  6. Back Matter
    Pages 311-359

About this book

Introduction

`Towards the end of the nineteenth century, Frege gave us the abstraction principles and the general notion of functions. Self-application of functions was at the heart of Russell's paradox. This led Russell to introduce type theory in order to avoid the paradox. Since, the twentieth century has seen an amazing number of theories concerned with types and functions and many applications. Progress in computer science also meant more and more emphasis on the use of logic, types and functions to study the syntax, semantics, design and implementation of programming languages and theorem provers, and the correctness of proofs and programs. The authors of this book have themselves been leading the way by providing various extensions of type theory which have been shown to bring many advantages. This book gathers much of their influential work and is highly recommended for anyone interested in type theory. The main emphasis is on:

- Types: from Russell to Ramsey, to Church, to the modern Pure Type Systems and some of their extensions.

- Functions: from Frege, to Russell to Church, to Automath and the use of functions in mathematics, programming languages and theorem provers.

- The role of types in logic: Kripke's notion of truth, the evolution and role of the propositions as types concept and its use in logical frameworks.

- The role of types in computation: extensions of type theories which can better model proof checkers and programming languages are given.

The first part of the book is historical, yet at the same time, places historical systems (like Russell's RTT) in the modern setting. The second part deals with modern type theory as it developed since the 1940s, and with the role of propositions as types (or proofs as terms), but at the same time, places another historical system (the proof checker Automath) in the modern setting. The third part uses this bridging in the first two parts between historical and modern systems to propose new systems that bring more advantages together. This book has much to offer to mathematicians, logicians and to computer scientists in general. It will have considerable influence for many years to come.' - Henk Barendregt

Keywords

Computer calculus computer science logic predicate logic programming programming language proof type theory

Authors and affiliations

  • Fairouz Kamareddine
    • 1
  • Twan Laan
    • 2
  • Rob Nederpelt
    • 3
  1. 1.Heriot Watt UniversityEdinburghScotland
  2. 2.EindhovenThe Netherlands
  3. 3.Eindhoven University of TechnologyThe Netherlands

Bibliographic information

  • DOI https://doi.org/10.1007/1-4020-2335-9
  • Copyright Information Springer Science + Business Media, Inc. 2005
  • Publisher Name Springer, Dordrecht
  • eBook Packages Mathematics and Statistics
  • Print ISBN 978-1-4020-2334-7
  • Online ISBN 978-1-4020-2335-4
  • Series Print ISSN 1386-2790
  • About this book