# Logic and Structure

• Dirk van Dalen
Book

Part of the Universitext book series (UTX)

1. Front Matter
Pages I-X
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 165-199
8. Dirk van Dalen
Pages 200-202
9. Dirk van Dalen
Pages 203-203
10. Dirk van Dalen
Pages 204-204
11. Back Matter
Pages 205-210

### Introduction

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.

### Keywords

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

• Dirk van Dalen
• 1
1. 1.Mathematisch instituutRijksuniversiteit UtrechtTA UtrechtThe Netherlands

### Bibliographic information

• DOI https://doi.org/10.1007/978-3-662-02382-2
• Copyright Information Springer-Verlag Berlin Heidelberg 1983
• Publisher Name Springer, Berlin, Heidelberg
• eBook Packages
• Print ISBN 978-3-540-12831-1
• Online ISBN 978-3-662-02382-2
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675