Logic and Structure [Paperback]

Dirk Van Dalen (Author)

Book Description

ISBN-10: 0387578390 | ISBN-13: 978-0387578392 | Publication Date: June 27, 1997 | Edition: 3rd/Augmnt

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.

Editorial Reviews

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) --This text refers to an alternate Paperback edition.

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. --This text refers to an alternate Paperback edition.

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Product Details

• Paperback: 215 pages
• Publisher: Springer-Verlag; 3rd/Augmnt edition (June 27, 1997)
• Language: English
• ISBN-10: 0387578390
• ISBN-13: 978-0387578392
• Product Dimensions: 9.2 x 6 x 0.5 inches
• Shipping Weight: 12 ounces
• Average Customer Review: 4.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #8,990,514 in Books (See Top 100 in Books)

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3 of 3 people found the following review helpful:

5.0 out of 5 stars On the Selection of Topics, April 7, 2011

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Customer (Baltimore) - See all my reviews

This review is from: Logic and Structure (Paperback)

There are some things not explicit in the posted summary that buyers of this book should know. Researchers will also find these comments valuable

This book is an introductory treatment of mathematical logic, written entirely from the perspective of GENTZEN natural deduction. Standard texts are written from the viewpoint of Hilbert axiomatics or from (an alternative from Gentzen) natural deduction. Thus, for one wanting any discussion of the Gentzen calculus ND this book is the only full, start from scratch, treatment that I know of. Note, however, that the Gentzen sequent calculus is NOT discussed.

Another (great) feature of the text is the chapter on ND in intuitionistic systems. Again, other than texts on structural proof theory (e.g. Negri and Von Plato.), such discussions are hard to find in an introductory setting. It also beats digging through journals or symposium proceedings.

Also, another (great) component, found only in the 4th edition, is the treatment of Godel's first famous result, but entirely handled via the aforementioned Gentzen calculus.

Finally, to give a general comment, there are are reasons for wanting to treat Hilbert systems rather than the Gentzen systems; these are most easily extended to study theories of arithmetic. But from the strictly logical, proof theoretic aspects, Hilbert systems prove to be less appealing. Note however that Gentzen systems are being pushed further all the time to handle arithmetic theories, though these still might be less elegant than their counterparts. Anyway, whether you choose Gentzen or the more standard treatment of Hilbert systems, one will unavoidably be making some concessions; but if Gentzen is what you want, there really isn't another alternative than this. Anecdote: Sometimes I hear, Gentzen is the more pedagogically effective route, but I'm not sure about this.

2 of 2 people found the following review helpful:

4.0 out of 5 stars not for newcomers, July 2, 2010

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CaRaPr - See all my reviews

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This review is from: Logic and Structure (Paperback)

I used parts of the first two chapters in a class. As with everything in logic, it may take some time to be understood, but it will rewire your brain once you get the point. The book is aimed at mathematicians, so be careful.

2 of 8 people found the following review helpful:

5.0 out of 5 stars Best title ever, August 25, 2009

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Guilherme B. G. Lima "Shop Addicted" (São Paulo, SP Brazil) - See all my reviews
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This review is from: Logic and Structure (Paperback)

One of the most finest books in logics. For academics or just for fun, like me.