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Deduction Systems

  • Rolf Socher-Ambrosius
  • Patricia Johann

Part of the Graduate Texts in Computer Science book series (TCS)

Table of contents

  1. Front Matter
    Pages i-xii
  2. Rolf Socher-Ambrosius, Patricia Johann
    Pages 1-7
  3. Rolf Socher-Ambrosius, Patricia Johann
    Pages 8-23
  4. Rolf Socher-Ambrosius, Patricia Johann
    Pages 24-35
  5. Rolf Socher-Ambrosius, Patricia Johann
    Pages 36-45
  6. Rolf Socher-Ambrosius, Patricia Johann
    Pages 46-64
  7. Rolf Socher-Ambrosius, Patricia Johann
    Pages 65-90
  8. Rolf Socher-Ambrosius, Patricia Johann
    Pages 91-131
  9. Rolf Socher-Ambrosius, Patricia Johann
    Pages 132-164
  10. Rolf Socher-Ambrosius, Patricia Johann
    Pages 165-198
  11. Back Matter
    Pages 199-206

About this book

Introduction

The idea of mechanizing deductive reasoning can be traced all the way back to Leibniz, who proposed the development of a rational calculus for this purpose. But it was not until the appearance of Frege's 1879 Begriffsschrift-"not only the direct ancestor of contemporary systems of mathematical logic, but also the ancestor of all formal languages, including computer programming languages" ([Dav83])-that the fundamental concepts of modern mathematical logic were developed. Whitehead and Russell showed in their Principia Mathematica that the entirety of classical mathematics can be developed within the framework of a formal calculus, and in 1930, Skolem, Herbrand, and Godel demonstrated that the first-order predicate calculus (which is such a calculus) is complete, i. e. , that every valid formula in the language of the predicate calculus is derivable from its axioms. Skolem, Herbrand, and GOdel further proved that in order to mechanize reasoning within the predicate calculus, it suffices to Herbrand consider only interpretations of formulae over their associated universes. We will see that the upshot of this discovery is that the validity of a formula in the predicate calculus can be deduced from the structure of its constituents, so that a machine might perform the logical inferences required to determine its validity. With the advent of computers in the 1950s there developed an interest in automatic theorem proving.

Keywords

Syntax automated deduction calculus complexity logic proving semantics

Authors and affiliations

  • Rolf Socher-Ambrosius
    • 1
  • Patricia Johann
    • 2
  1. 1.FB Elektrotechnik und InformatikFachhochschule OstfrieslandEmdenGermany
  2. 2.Pacific Software Research CenterOregon Graduate InstituteBeavertonUSA

Bibliographic information

  • DOI https://doi.org/10.1007/978-1-4612-2266-8
  • Copyright Information Springer-Verlag New York 1997
  • Publisher Name Springer, New York, NY
  • eBook Packages Springer Book Archive
  • Print ISBN 978-1-4612-7479-7
  • Online ISBN 978-1-4612-2266-8
  • Series Print ISSN 1868-0941
  • Series Online ISSN 1868-095X
  • Buy this book on publisher's site