A Mathematical Introduction to Logic

Wonderful introduction

This is a 336-page book that ships with about 190 pages. It claims to have four chapters. Chapters 3 and 4 have one page each with a 100-page numbering skip. On the parts that do exist, there are inexcusable formatting errors that render basic and essential formulas unreadable

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.

Explains well

This book was my introduction to Logic. I can't say much more than has already been said on it. It's one of probably the top 2 or 3 intro to Logic books you'll hear recommended, and for good reason. I found its development of concepts to be very good, and the exercises to be a healthy mix of approachable and demanding.The book was in fairly poor physical condition, but as it was used this was not totally unexpected.

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.

The copy I get is a defeat product. This book is supposed to be 317pages book, but it only has 182 pages. Skipped chapter 3 and chapter 4.

Enderton's writing is the best I've seen in any introductory math textbook; he is lucid, well organised, comfortably paced but free of expository flab. The exercises (judging from chapters 2 and 3) are not terribly difficult, but quite useful in building one's intuition and connecting logic to other mathematics. I had the book for my Logic class as a first-semester sophomore with very little experience with proofs and no abstract algebra, and found it quite accessible. I guess the book starts off with an advantage, being about a subject as interesting as logic, but that does not seriously detract from its merit.