Logic and Structure (Universitext) - Softcover

Van Dalen, Dirk

Softcover

ISBN 10: 0387578390 ISBN 13: 9780387578392

Publisher: Springer-Verlag, 1994

Synopsis

A book which efficiently presents the basics of propositional and predicate logic, van Dalen's popular textbook contains a complete treatment classical logic on the basis of Gentzen's Natural Deduction and the traditional two-valued semantics, culminating in the completeness theorems. The first chapter, containing a leisured treatment of propostional logic, is followed by an equally elaborate chapter on predicate logic. On the basis of the material of the first of two chapters the completeness theorem is established and an excursion is made into model theory. The main facts of model theory, e.g. compactness, Skolem-Loewenheim, elementary equivalence, non-standard models, quantified elimination and Skolem functions are covered in chapter Three. The exposition 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. This chapter contains a completeness proof for Kripke's semantics and a number of specific constructive features have been incorporated, e.g. a study of equality and apartness the disjunction and existence property, the Goedel translation. A new chapter has been added at the end of this edition, with the basics of the proof theory of natural deduction; derivations are studued for their own sake and weak normalisation is proved. A choise of exercises is added ranging from simple applications of the definitions to more sophisticated problems.

About this title

Review:

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)

From the Back Cover:

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.

About this edition

• Publisher: Springer-Verlag
• Publication date: 1994
• ISBN 10: 0387578390
• ISBN 13: 9780387578392
• Binding: Paperback
• Edition number: 3