© 1980

Logic and Structure


Part of the Universitext book series (UTX)

Table of contents

  1. Front Matter
    Pages I-IX
  2. Dirk van Dalen
    Pages 1-3
  3. Dirk van Dalen
    Pages 4-57
  4. Dirk van Dalen
    Pages 58-105
  5. Dirk van Dalen
    Pages 106-153
  6. Dirk van Dalen
    Pages 154-164
  7. Dirk van Dalen
    Pages 165-167
  8. Dirk van Dalen
    Pages 168-168
  9. Dirk van Dalen
    Pages 169-169
  10. Back Matter
    Pages 170-174

About this book


A book which efficiently presents the basics of propositional and predicate logic, van Dalen’s popular textbook contains a complete treatment of elementary classical logic, using Gentzen’s Natural Deduction. Propositional and predicate logic are treated in separate chapters in a leisured but precise way. Chapter Three presents the basic facts of model theory, e.g. compactness, Skolem-Löwenheim, elementary equivalence, non-standard models, quantifier elimination, and Skolem functions.

The discussion of classical logic is rounded off with a concise exposition of second-order logic.

In view of the growing recognition of constructive methods and principles, one chapter is devoted to intuitionistic logic. Completeness is established for Kripke semantics. A number of specific constructive features, such as apartness and equality, the Gödel translation, the disjunction and existence property have been incorporated.

The power and elegance of natural deduction is demonstrated best in the part of proof theory called `cut-elimination' or `normalization'. Chapter 6 is devoted to this topic; it contains the basic facts on the structure of derivations, both classically and intuitionistically.

Finally, this edition contains a new chapter on Gödel's first incompleteness theorem. The chapter is self-contained, it provides a systematic exposition of primitive recursion and partial recursive functions, recursive by enumerable sets, and recursive separability. The arithmetization of Peano's arithmetic is based on the natural deduction system.


Goedel's theorem basic model theory computability theory first-order logic with natural deduction intuitionistic logic and semantics model theory normalisation of first-order logic predicate logic proof proof theory recursive functions second order logic set theory

Authors and affiliations

  1. 1.Mathematisch InstituutRijksuniversiteit UtrechtUtrechtThe Netherlands

About the authors

Dirk van Dalen studied at the University of Amsterdam, where he obtained his PhD. He has taught since 1960 at Utrecht University, where he is full professor. He also taught at MIT and Oxford. His technical work is mostly in the area of intuitionistic mathematics and logic. He uses to call attention to the benefits and challenges of constructive methods. His current project is a biography of L.E.J. Brouwer and the editing of Brouwer's correspondence.

Bibliographic information

  • Book Title Logic and Structure
  • Authors Dirk van Dalen
  • Series Title Universitext
  • DOI
  • Copyright Information Springer-Verlag Berlin Heidelberg 1980
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Softcover ISBN 978-3-540-09893-5
  • eBook ISBN 978-3-662-08402-1
  • Series ISSN 0172-5939
  • Series E-ISSN 2191-6675
  • Edition Number 1
  • Number of Pages IX, 174
  • Number of Illustrations 1 b/w illustrations, 0 illustrations in colour
  • Additional Information 4th ed., corrected 2nd printing.
  • Topics Mathematical Logic and Foundations
  • Buy this book on publisher's site
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From the reviews of the fourth edition:

"This book teaches logic to mathematicians in just the way I would have wished. … Beginning with the propositional calculus by means of truth-tables, i.e. the semantics, it proceeds to the syntactics in the form of Gentzen’s natural deduction. … this fourth edition has a long final chapter added, on Gödel’s incompleteness theorem. … The chapter follows traditional lines but preserves the excellent quality of the earlier chapters. This is a delightful textbook, with plenty of examples for the reader." (C.W. Kilmister, The Mathematical Gazette, Vol. 89(515), 2005)

"This is the fourth edition of van Dalen’s classic textbook on logic in the popular ‘Universitext’ -series. … this book explains clearly all aspects of logic which a novice in this matter should learn by heart. … Reading this book was a real delight. A lot of the fun was in the exercises … heartily recommend this excellent textbook; … Current students may have little interest in formal mathematics … the problem will solve itself when they all have a copy on their personal bookshelf." (Pieter Audenaert, Bulletin of the Belgian Mathematical Society, Vol. 12 (3), 2005)