`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
