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    A Modern Perspective on Type Theory: From Its Origins Until Today (Applied Logic) (Hardback) By (author) Fairouz D. Kamareddine, By (author) Twan Laan, By (author) Rob Nederpelt

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    DescriptionThis 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.


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    Title
    A Modern Perspective on Type Theory
    Subtitle
    From Its Origins Until Today
    Authors and contributors
    By (author) Fairouz D. Kamareddine, By (author) Twan Laan, By (author) Rob Nederpelt
    Physical properties
    Format: Hardback
    Number of pages: 374
    Width: 155 mm
    Height: 235 mm
    Thickness: 22 mm
    Weight: 708 g
    Language
    English
    ISBN
    ISBN 13: 9781402023347
    ISBN 10: 1402023340
    Classifications

    BIC E4L: MAT
    Nielsen BookScan Product Class 3: S7.8
    B&T Book Type: NF
    LC subject heading:
    B&T Merchandise Category: TXT
    BIC subject category V2: PBCD
    LC subject heading:
    B&T General Subject: 710
    B&T Modifier: Region of Publication: 04
    B&T Modifier: Academic Level: 02
    Ingram Subject Code: MA
    LC subject heading:
    DC22: 511.3
    LC subject heading:
    LC classification: QA76.9.M35
    BISAC V2.8: COM014000, PHI011000, MAT018000
    Warengruppen-Systematik des deutschen Buchhandels: 16230
    BISAC V2.8: COM018000
    LC subject heading:
    Libri: MATH3575, LOGI1500, PHIL8054
    LC subject heading:
    LC classification: QA9 .K225 2004, BC1-199, QA1-939, QA8.9-10.3
    Thema V1.0: QDTL, UY, UNC, PBCD
    Edition
    4000
    Edition statement
    2004 ed.
    Illustrations note
    biography
    Publisher
    Springer-Verlag New York Inc.
    Imprint name
    Springer-Verlag New York Inc.
    Publication date
    09 June 2004
    Publication City/Country
    New York, NY
    Review quote
    "This book has evolved from a number of projects by the authors ... into aspects of the evolution of calculus and type theory in Logic, Mathematics and Computation. ... The crown of this collaboration is this very rich and well written book ... . This book not only gives a nice and concise overview ... but also a mathematically precise and faithful reconstruction of the different type theories ... . I expect the book will be a major work of reference ... ." H.C.M. De Swart, Journal of Logic, Language and Information, Vol. 15, 2006
    Table of contents
    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.