Customer Reviews

Mathematical Logic (Dover Books on Mathematics)

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This book was written by one of the great American mathematical minds of this century. I've read it cover to cover and it happens to be my favorite logic book for its scope, depth, and clarity. Kleene uses a combined model-theoretic and proof-theoretic approach, and derives many interesting results relating the two (he also gives mention to special axioms for Intuitionistic logic). Although his focus in the first part of the book is on a more or less mathematical treatment of standard first-order predicate logic (augmented later by functions and equality), he also spends considerable time discussing the ways in which formal logic can and should be used to analyze "ordinary language" statements and arguments. After setting the groundwork, he moves onto subjects such as set theory, formal axiomatic theories, turing machines and recursiveness, Godel's incompleteness theorem, Godel's completeness theorem, and just about every interesting subject relating to logic in the first half of the twentieth century.

For the mathematically inclined self-teacher, Kleene's exposition should not be difficult at all, in fact I found it remarkably clear compared to other mathematical treatments of the subject (which are necessary if one wants to understand the deeper results). I suppose less mathematically inclined readers could try Irving Copi's "Symbolic Logic" as a start, although even that requires some mathematical proficiency, and since it doesn't cover many of the things you will want to know about, you'll end up coming back to a book like Kleene's anyway. So to summarize, if you want to learn the hard stuff (from the first half of the twentieth century--which includes just about everything the layman/philosopher wants to know), there is no better or easier way.

Ten years ago, I took an undergraduate course in symbolic logic. Wishing recently to refresh my (extremely rusty) memories of the propositional calculus and the first-order predicate calculus, I picked up this meaty text and was extremely dismayed to find myself soundly defeated within the first few pages. Kleene does not even make a pretense of holding the reader's hand: either you get it or you don't. There is nothing even remotely "user-friendly" about this book's presentation of its material.

If one were to read this book under the guidance of a teacher, I think it might be worthwhile. It may not be fair for me to blame the author for my inability to understand his writing. If you're smarter than I am, you might breeze right through it.

I cannot recommend this book, though, good though it may be, for anyone who wishes to teach him/herself logic, nor for anyone who wishes to brush up on the subject. There are exercises for the reader to test his/her understanding of the material, but no answer key is provided. This is heavy-duty stuff, and not well-suited to the self-teacher.

There are many books on logic, but this, written by one of the chief logicians of the 20th century, deserves a place on your bookshelf. The informations and the exposition style are solid, clear and still new to many. The author also explains the historical motivations for each new concept he talks about, and this alone should make up your mind about buying this book.
But I have seen more modern approaches, and maybe better ones. It's possible to learn so much from this book, and so much that you'll need no other book, least you'd prefer a contemporary way to talk about those things. I would not adopt it for my classes, but I strongly recommend it to my fellows.

My problem with Kleene's Mathematical Logic is simply that its layout is terrible. There is hardly any white space in the text, so the entire book reads like a giant run-on sentence. This makes it difficult to find information that you want when you want to refer to a previous topic. There is also no glossary (defined terms are italicized in otherwise normal sentences, and hardly stand out at all. I generally highlighted them, since that was my only hope for referring to them later, short of typing my own glossary). There is also no answer key, which in my opinion is an extremely important element of any textbook.

That being said, all of the information that you could want is there. Simple concepts are built up appropriately before more abstract ones are given. Proofs of all important theorems are provided. Examples and exercises are abundant. In order to motivate you, the author simply insults your intelligence every once and a while, and you are thus determined to prove him wrong (I actually did enjoy this aspect of Kleene's style).

In summary - If you are determined, you can teach yourself mathematical logic from this book. However, you can make this undertaking much easier on yourself by getting a more readable textbook.

I bought this book based on the short altough extremely positive review in the bibliography section of Hofstadter's GEB. And it has proven to be one of the my best acquisitions. The books is wonderfully written, very detailed indeed, so it makes it easy to follow the proofs. It comes along with exercises at the end of every paragraph (chapters are divided into paragraphs) so you can process and reinforce what has been learned.
I have worked with plenty of logic books (among them: Suppes', Gamut's, Machover's, etc) but I am loyal to Kleene's.
What it attracks me the most is that it not only contains detailed and rigorous proofs of the most important theorems of logic, but that it also comes with philosophical considerations of one of the best logician's of the 20th century. There are other parts too where Kleene includes a piece of history, for instance, about the Löwenheim-Skolem theorem: delightful.
Buy it 'cause you won't regret it.

I have to agree with the more recent reviewer and disagree with the first one. I don't even have college-level maths; in fact, I failed abysmally in my school-leaving maths exams (I think I got an F). I wanted to read this because, now in my mid-30s, I had got very interested in various mathematical topics (game theory, number theory, logic) and was sick of just reading popular scientific books about them that assumed that you didn't know how to read the symbols. I ordered this to get me started on logic.

Kleene does an excellent job of introducing a novice like me to the first principles; it's true that he doesn't hang about, and he has a way of bullying his readers into making the effort to understand by dropping sarcastic little remarks like 'Anyone who cannot follow this is clearly mentally sluggish', years of teaching logic in Madison, WI clearly finding payback right there. Some readers may find that kind of thing overbearing, but I found it bracing. I admit that I'm only on page 14, but already I can find the scope of a propositional connective, and when I woke up this morning I had never heard of such a thing.

I thoroughly recommend this book; a brisk, clear, ruthlessly no-nonsense introduction to the subject. Maybe it's not 'Mathematical Logic for Dummies', but Kleene would probably crack that dummies shouldn't be attempting the subject in the first place.