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A Mathematical Introduction to Logic [Hardcover]

Herbert Enderton , Herbert B. Enderton
1.0 out of 5 stars  See all reviews (1 customer review)
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Book Description

23 Jan 2001 0122384520 978-0122384523 2
A Mathematical Introduction to Logic, Second Edition, offers increased flexibility with topic coverage, allowing for choice in how to utilize the textbook in a course. The author has made this edition more accessible to better meet the needs of today's undergraduate mathematics and philosophy students. It is intended for the reader who has not studied logic previously, but who has some experience in mathematical reasoning. Material is presented on computer science issues such as computational complexity and database queries, with additional coverage of introductory material such as sets.

* Increased flexibility of the text, allowing instructors more choice in how they use the textbook in courses.
* Reduced mathematical rigour to fit the needs of undergraduate students

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Product details

  • Hardcover: 317 pages
  • Publisher: Academic Press; 2 edition (23 Jan 2001)
  • Language: English
  • ISBN-10: 0122384520
  • ISBN-13: 978-0122384523
  • Product Dimensions: 1.9 x 16.3 x 23.1 cm
  • Average Customer Review: 1.0 out of 5 stars  See all reviews (1 customer review)
  • Amazon Bestsellers Rank: 523,151 in Books (See Top 100 in Books)
  • See Complete Table of Contents

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"This is easily the BEST intro logic book ever written.It's the only MATH book on logic out there that is easy to understand and yet formal enough to be considered 'mathematical.'", May 26, 2013 Reasons for This Book's Success "Rigor, integrity and coherence of overall purpose, introducing students to the practice of logic ..." --Douglas Cannon, University of Washington "The book is clearly and carefully written. I adopted this text because of its detailed and rigorous treatment of the predicate calculus, detailed and optimal treatment of the incompleteness phenomena, standard notation as developed by the Berkeley school." --Karel Prikry, University of Minnesota "It is mathematically rigorous [and] it has more examples than other books ... I definitely would use a new edition of this book." --Sun-Joo Chin, University of Notre Dame

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Most Helpful Customer Reviews
2 of 4 people found the following review helpful
... contrary to what I read among the wealth of dithyrambic adjectives concerning that book !!!

FIRST : As I reached half the book it was already giving signs of a strong "desire" to fall apart, with the front pages almost ripped off and the next pages soon to follow... Academic Press/Elsevier should try to get a training in the UK on how to provide a decent structure for a book in that price range.

SECOND : impractical numbering of sections, theorems, subsections + no mention of sections at the top of the page, making the search difficult + a very dull layout ...

THIRD : A very peculiar way of proving theorems : quite a personal interpretation of induction and recursion (a way for Enderton to free himself from the burden of really getting at the bottom of things...). It seems like Enderton had enrolled in a marathonian effort to give tortuous proofs, often incomplete and based on fistulous definitions, which turn the reading into a continual second-guessing exercise, with its load of annotations...
Added to the annoying game of transferring part of the theory to a bunch of exercises.

FOURTH : Wiith a horrific set of notations, chapter 3 (on undecidability) is simply unreadable and I wish good luck to those who want to understand Gödel's theorems via such confused and confusing text...

FIFTH : I am perusing chapter 4 with the faint hope that it isn't a second-order magma...

I really wish that Peter Smith (his excellent "Introduction to Gödel's theorems", see my review) decide, one day, to write a book on mathemetical first- and second-order logic !!!
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Most Helpful Customer Reviews on (beta) 3.4 out of 5 stars  17 reviews
58 of 62 people found the following review helpful
4.0 out of 5 stars Still the best. 22 Sep 2003
By Jason T - Published on
I review the classic FIRST EDITION. If you buy only one book on mathematical logic, get this one. It's by far the best logic book (see my other reviews) that is both 1)introductory and 2)sufficiently broad in scope and complete. The exposition is very clear and succinct- its suitable for beginners without getting wordy. Enderton always clearly explains what he's doing and why, keeping the reader focused on the big picture while going through the details. He helps to place topics in perspective, and has organized the book so readers can skip some of the more involved proofs and sections on the first reading.

Besides being easy to learn from, it's also the most rigorous introductory book I've seen- a rare combination. The proofs are detailed and complete, instead of the usual hand-waving or leaving everything as an exercise for the reader. There are some weak points in it, but overall you're not going to find a better book. It requires a little more 'mathematical sophistication' than most intro books- but if you've had some logic in a computer science course, or a little combinatorics or abstract algebra you'll be more than ready. Familiarity with automata/computability theory will help you in a few of the sections. Although Enderton is very good, it always helps to get several books on a subject- I'd recommend you pick up cheap copies of Boolos & Jeffrey's _Computability and Logic_ and Smullyan's _First-order logic_ as supplements.

Here is the complete table of contents for the first edition, c1972:

Chapter Zero - USEFUL FACTS ABOUT SETS . . . .1
Chapter One - SENTENTIAL LOGIC/ Informal Remarks on Formal Languages 14 /The Language of Sentential Logic 17/ Induction and Recursion 22/ Truth Assignments 30/ Unique Readability 39/ Sentential Connectives 44/ Switching Circuits 53/ Compactness and Effectiveness 58

Chapter Two - FIRST-ORDER LOGIC/ Preliminary Remarks 65/ First-Order Languages 67/ Truth and Models 79/ Unique Readability 97/ A Deductive Calculus 101/ Soundness and Completeness Theorems 124/ Models of Theories 140/ Interpretations between Theories 154/ Nonstandard Analysis 164

Chapter Three - UNDECIDABILITY/ Number Theory 174/ Natural Numbers with Successor 178/ Other Reducts of Number Theory 184/ A Subtheory of Number Theory 193/ Arithmetization of Syntax 217/ Incompleteness and Undecidability 227/ Applications to Set Theory 239/ Representing Exponentiation 245/ Recursive Functions 251

Chapter Four - SECOND-ORDER LOGIC/ Second-Order Languages 268/ Skolem Functions 274/ Many-Sorted Logic 277/ General Structures 281
Index 291
12 of 12 people found the following review helpful
3.0 out of 5 stars Quite readable, but misses the forest in favor of trees 8 May 2012
By Eric "Siggy" Scott - Published on
Format:Hardcover|Verified Purchase
There are two types of mathematical texts: source code (definition-theorem-proof-remark-definition-...), and books intended to educate via explanations of where we came from, where we're going, and why we should care. Enderton's (2nd edition) text is an actual *book,* albeit not a superb one (compare to Simpson's free text on Mathematical Logic at [...], which fits my definition of "source code"). For this he automatically earns 2 stars -- though in any field except mathematics, this would earn him nothing.

The prose itself is easy to follow, and makes suitable use of cross-references -- you will not find yourself stumped for 30 minutes trying to substantiate a casual statement made half-way through the book, as with some mathematical authors. High-minded ideas such as effectiveness and decidability appear (briefly) at the end of chapter one, so you don't have to read 180 pages before any "cool" things are presented, and there are occasional (but too few) sentences explaining what the goal of a formalism is before it is developed. Chapter 1, which covers sentential (propositional) logic, also has a short section on applications to circuit design, providing some much-welcome motivation for the material. Model theory is also integrated with the discussion of first-order logic in chapter 2, which is preferable to having it relegated to a later section as in some texts. The book also gives heavy emphasis to computational topics, and even gets into second-order logic in the final chapter -- a very complete coverage for such a small introductory text. These virtues combine to earn it a third star.

My primary complaint is the manner in which rigor is emphasized in the text to the neglect (rather than supplement) of a coherent big picture -- losing two full stars.

For instance, in chapter 1, 10 pages are spent very early on induction and recursion theorems, to put intuitive ideas like "closure" on firm ground. And yet the words "deduction" and "completeness" -- arguably the whole reason we want to study logic in the first place -- do not appear until after the entirety of the rigorous discussion of propositional logic, and even then only as an exercise. Most readers will reach page 109 before realizing that logicians care about deduction or soundness at all.

41 pages from chapter 2 are given over to defining models/structures, truth, definability, homomorphisms and parsing in first-order logic. These complex and highly detailed definitions remove ambiguity from mathematical discourse, and are essential -- but are best viewed as fungible reference material. After all, many alternative renditions of the formalism exist. This is not the essence of mathematical logic -- but to Enderton, they appear to be the field's first-class content.

I found it difficult to see the forest for the trees in this book. I would have much preferred to see examples of deduction proofs -- with exercises in making use of axioms of natural deduction, discharged assumptions, etc -- and a brief discussion of completeness up front. *Then* I would have enjoyed being told "okay, now that we've seen how FOL works in practice, it's important to note that we have not yet set it on a rigorous footing. The next three sections will set to that task via many small steps. We'll see how it all comes together in the end." It is amazing what a difference just a few sentences like that can make in a book on mathematics -- guiding your reader is vital.

I would also have loved to see some more high-level discussion on the history of FOL and justification for it's prominence, the decline of syllogistic logic, the origins of Boolean algebra, etc. But perhaps that is too much to ask, since mathematics educators are (uniquely in academia) not accustomed to contextualizing their material as part of a wider intellectual enterprise.
19 of 22 people found the following review helpful
5.0 out of 5 stars Excellent Textbook with lots of examples 31 July 2002
By M. Vishnu - Published on
I used this book for self study of Mathematical Logic with the aim of understanding Godel's incompleteness theorem. I also referred to other introductory Mathematical Logic books. In my opinion, this book is by far the best among them. Very readable and contains lots of carefully selected examples.
9 of 9 people found the following review helpful
5.0 out of 5 stars Best Intro. Logic Book Ever! 11 Sep 2010
By A Customer 2000 - Published on
This is easily the BEST intro. logic book every written. (Yes, I sound horribly biased.) This books covers everything from Sentential Logic to 1st Order to Recursion to a bit of 2nd Order Logic. It's the only MATH book on logic out there that is easy to understand and yet formal enough to be considered "mathematical." Even the treatment of Sentential Calc. brings interesting tidbits (ternary connectives, completeness, compactness, etc). Truth and models (the heart of it) are treated incredibly clearly. Extra topics such as interpretations between theories and nonstandard analysis keep things exciting (for a math book). His treatment of undecidability is well-written and lucid. The second order stuff is fun.

I loved this book. As far as math teachers go, Enderton is top notch. Even someone as unacquainted with math as I was when I studied the book (and as I still am now, I guess) understood what was going on. To be honest though, I did have one advantage, I was a student of the master, Enderton, himself. I learned so much about logic (and math in general) from this great book. I was fortunate enough to study some more with Enderton throughout my years as a student. Of course, I went through his "Elements of Set Theory" which is also fantastic. Too bad he never wrote a book on model theory...But, you never know; maybe someday he will.
12 of 15 people found the following review helpful
3.0 out of 5 stars From the point of a CS student 6 Sep 2008
By Daniel Wong - Published on
It's very hard to review a book like this without letting personal interest in the subject bias you... but I'll try ;).

I used this book in my fourth year at Berkeley. Being a CS major, I found the chapter on sentential (aka boolean) logic very pedantic. I feel that most people are going to be able to easily navigate that part by sheer intuition.

On the other hand, first-order logic (the real meat of the course) comes with little motivation from Enderton. He simply dives into the syntax, as if the semmantics will be just as obvious as in sentential logic.

One of the main points of this class that I didn't understand until late in the semester, was that mathematical logic is merely an attempt to model (using symbols) the logic most mathematician use proofs, which are written in words. In turn, this gives us a framework to reason about mathematical logic itself, creating a whole new branch of mathematics in its own right (perhaps you can see why it took me a while to understand all this). The only attempt that Enderton makes to explain this is a poorly drawn diagram of "meta-theorems" on top, which are the results of mathematical logic, and theorems, which are the subjects of mathematical logic, on the bottom.

The oddest thing about this book was its treatment of algorithms, which is one of the most interesting aspects of this subject. Any (meta)theorems about those were marked with a star, because a precise definition of an algorithm is never given. I'm guessing most reviewers who praise the rigor of this book tend to overlook this weakness, because they come from math departments and not CS departments. If you take a course in computability and complexity theory, you'll see the two subjects are intimately intertwined.

This may be the best book on the subject, but I did not feel it guide me very much through the course, esp the later half about first-order logic.
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