An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Applied Logic Series) [Hardcover]

Peter B. Andrews (Author)

Book Description

Publication Date: July 31, 2002 | ISBN-10: 1402007639 | ISBN-13: 978-1402007637 | Edition: 2nd

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability. The last three chapters of the book provide an introduction to type theory (higher-order logic). It is shown how various mathematical concepts can be formalized in this very expressive formal language. This expressive notation facilitates proofs of the classical incompleteness and undecidability theorems which are very elegant and easy to understand. The discussion of semantics makes clear the important distinction between standard and nonstandard models which is so important in understanding puzzling phenomena such as the incompleteness theorems and Skolem's Paradox about countable models of set theory. Some of the numerous exercises require giving formal proofs. A computer program called ETPS which is available from the web facilitates doing and checking such exercises. Audience: This volume will be of interest to mathematicians, computer scientists, and philosophers in universities, as well as to computer scientists in industry who wish to use higher-order logic for hardware and software specification and verification.

Product Details

• Hardcover: 408 pages
• Publisher: Springer; 2nd edition (July 31, 2002)
• Language: English
• ISBN-10: 1402007639
• ISBN-13: 978-1402007637
• Product Dimensions: 9.3 x 6.6 x 1.1 inches
• Shipping Weight: 1.6 pounds (View shipping rates and policies)
• Average Customer Review: 3.7 out of 5 stars  See all reviews (3 customer reviews)
• Amazon Best Sellers Rank: #1,336,180 in Books (See Top 100 in Books)

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Customer Reviews

Most Helpful Customer Reviews

1 of 2 people found the following review helpful:

5.0 out of 5 stars great book on logic, August 16, 2011

By 

symbolic (NJ USA) - See all my reviews

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This review is from: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Applied Logic Series) (Hardcover)

I love this book. I have quite a few logic books and I have studied it for years.

The question I wanted to investigate when I got this book was a very special

purpose one: I wanted to read about what Kleene calls replacement theorems.

This Andrews book has the best explained, best proved, most general replacement

theorems I have seen anywhere. I was so thrilled I went back and read a good

portion of the earlier part of the book. Andrews writes clearly and concisely.

His presentation of logic is nicely ordered, economical and to the point.

It is packed with a good and wide assortment of important theorems and concepts.

The exercises do play an important role in extending the results of the text

and are ordered in such a way that they often build naturally on their predecessors.

However, I will say that the book is probably pitched at too high a mathematical

level for a first or beginning course in logic and it gets increasingly so

as the page numbers increase. For good beginner introductions to logic,

I recommend The Logic Book by Bergmann, Moor, and Nelson (which is so repetitive

and wordy that beginners will have a hard time missing the point; and that is

not really a criticism -- that book is not in its 5th Edition for nothing)

and Ben-Ari's Mathematical Logic For Computer Science 2nd Ed. (see my review).

In any event, I find it useful to read multiple books on what is nominally

the same topic,especially for a topic that is as rich, subtle, and deep as logic

-- and where every author has a different take on it,

and so I am very pleased to have found and be reading this Andrews book as it

very nicely extends, complements and reinforces my previous understanding of logic.

2 of 4 people found the following review helpful:

2.0 out of 5 stars meh., November 19, 2010

By 

David Kim - See all my reviews

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This review is from: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Applied Logic Series) (Hardcover)

The book itself for the most part is ok.

Would not recommend it for beginner though since the way logic is presented through the book is very rigorous and thus unnecessarily complex is its wording.

There are absolutely no solutions to any of the problems which means this book has no point of having been published since only the writer's students may buy it and actually use it effectively.

Since the author assumes many things and not all of the material presented within the book is conventional some example of usage would have cleared up the proofs and theorems but again this book is lacking this.

In brief:

- No solutions to a single problem and hardly any examples (if any).

- Very wordy, horrible worded statements lead to misunderstandings.

4 of 13 people found the following review helpful:

4.0 out of 5 stars used early draft as grad text, December 6, 2005

By 

Christopher K. Koenigsberg (Norman, OK USA) - See all my reviews
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This review is from: An Introduction to Mathematical Logic and Type Theory: To Truth Through Proof (Applied Logic Series) (Hardcover)

I took a great graduate course from Prof. Andrews, way back in the 1970's, where his class lecture notes were titled "To Truth Through Proof", so I assume that was a very very early draft of this book.

If so, this must be a very good book, because his notes were wonderful even back then.