A Modern Perspective on Type Theory: From its Origins until Today

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Springer Science & Business Media, Mar 10, 2006 - Mathematics - 360 pages
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`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
 

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Contents

Introduction Avoiding the paradox in type theory The approach Avoiding the paradox in set theory I The Evolution of Type Theory until the 1940s
1
Type theory in Principia Mathematica 2a Principias propositional functions 2a1 2a2 2a3 Definition Principias propositional functions as Principias pf...
19
3
69
Propositions as Types Pure Type Systems AUTOMATH
102
The prePAT RTT and STT in PATstyle 5a RTT in PAT style 5a1 5a2 5a3 5a4 5a5 5a6 An introduction to The system Metaproperties of Interpreting ...
125
rameter types The different treatment of constants and variables The definition system and the translation using 7c 7c1 7c2 7c3 7c4 7c5 Definition a...
153
182
182
7d1 with parameters and
224
Pure Type Systems with definitions
233
A Type systems in this book
311
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About the author (2006)

Rob Nederpelt was Lecturer in Logic for Computer Science until his retirement. Currently he is a guest researcher in the Faculty of Mathematics and Computer Science at Eindhoven University of Technology, The Netherlands.

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