A modern perspective on type theory : from its origins until today
著者
書誌事項
A modern perspective on type theory : from its origins until today
（Applied logic series, v. 29）
Kluwer Academic, c2004
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注記
Includes bibliographical references (p. 337-348) and indexes
内容説明・目次
内容説明
This book provides an overview of type theory. The first part of the book is historical, yet at the same time, places historical systems 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. The third part proposes new systems that bring more advantages together.
目次
Preface. Preliminaries. Overview of this book. Acknowledgements.
Introduction.
I: The Evolution of Type Theory until the 1940s.
1. Prehistory. 1.a. Paradox threats. 1.b. Paradox threats in formal systems.
2. Type theory in Pricipia Mathematica. 2.a. Principia's propositional functions. 2.b. The Ramified Theory of Types RTT. 2.c. Properties of RTT. 2.d. Legal propositional functions. Conclusions.
3. Deramification. 3.a. History of the deramification. 3.b. The Simple Theory of Types STT. 3.c. Are orders to be blamed? Conclusions.
II: Propositions as Types, Pure Type Systems, AUTOMATH.
4. Propositions as Types and Pure Type Systems. 4.a. Propositions as types and proofs as terms (PAT). 4.b. Lambda calculus. 4.c. Pure type systems.
5. The pre-PAT and STT in PAT-style. 5.a. RTT in PAT-style. 5.b. STT in PAT-style.
6. A correspondence between RTT and the system Nuprl. 6.a. On the role of orders. 6.b. The Nuprl type system. 6.c. RTT in Nuprl. Conclusions.
7. Automath. 7.a. Description of AUTOMATH. 7.b. From AUT-68 towards a PTS. 7.c. lambda68. 7.d. More suitable pure type systems for AUTOMATH. Conclusions.
III: Extensions of Pure Type Systems.
8. Pure type systems with definitions. 8.a. Definitions in contexts. 8.b. Definitions in the terms and the contexts.
9. The Barendregt cube with parameters. 9.a.On parameters in the Barendregt cube. 9.b. The Barendregt cube refined with parameters.
10. Pure Type Systems with parameters and definitions. 10.a. Parametric constraints and definitions. 10.b. Properties of terms. 10.c. Properties of legal terms. 10.d. Restrictive use of parameters. 10.e. Systems in the redefined Barendregt cube. 10.f. First-order predicate logic. Conclusions: yet another extension of PTSs? Practical motivation. The heart of type theory. Future work.
A: Type Systems in this Book. A.a. Pure Type Systems. A.b. The Barendregt cube. A.c. The Ramified Theory of Types. A.d. The Simple Theory of Types. A.e. Church's simply typed lambda-calculus lambda-->Church. A.f. A fragment of Nuprl in PTS-style. A.g. AUTOMATH. A.h. Pure Type Systems with definitions. A.i. Pure Type Systems with parametric constants. A.j. A CD-PTS and its subsystems.
Bibliography. Subject Index. Name Index. List of Figures.
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