Product Description
Undergraduate students with no prior classroom instruction in mathematical logic will benefit from this evenhanded multipart text by one of the centuries greatest authorities on the subject. Part I offers an elementary but thorough overview of mathematical logic of first order. The treatment does not stop with a single method of formulating logic; students receive instruction in a variety of techniques, first learning model theory (truth tables), then Hilbert-type proof theory, and proof theory handled through derived rules. Part II supplements the material covered in Part I and introduces some of the newer ideas and the more profound results of logical research in the twentieth century. Subsequent chapters introduce the study of formal number theory, with surveys of the famous incompleteness and undecidability results of Gödel, Church, Turing, and others. The emphasis in the final chapter reverts to logic, with examinations of Gödel's completeness theorem, Gentzen's theorem, Skolem's paradox and nonstandard models of arithmetic, and other theorems. Unabridged republication of the edition published by John Wiley & Sons, Inc. New York, 1967. Preface. Bibliography. Theorem and Lemma Numbers: Pages. List of Postulates. Symbols and Notations. Index.
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Customer Reviews (4)

Still very usefull
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.

Excellent Excellent Book
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.

Excellent introduction
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.

Not for the autodidact
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. ... Read more

Product Description
Noted expert selects 70 "short" puzzles. The Returning Explorer, The Mutilated Chessboard, Scrambled Box Tops, and 67 more. Solutions included.
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Customer Reviews (13)

Fun book to bring on vacation
This is a must bring when you go on vacation and just need some time to chill at the beach or at the end of a busy day, in between day and evening plans. If you happen to be interviewing in the technical fields you may run across interviewers that ask brain teasers. You'll find many of those problemsin this book. For technical interviewing this makes a good study guide. I have collected quite a few puzzle books and was never fully satisfied with any of them until I heard about the authoer and decided to pick up one his books. The puzzles are all differnt from each other, ranging from word problems to geometry problems.The solutions given are very complete and actually take up the largest emphasis.

Very good puzzles book
I didn't expect much from a 5-dollar book, but I was pleasantly surprised. Other reviewers have already covered the quality of the book, so I won't go into those details, but I have the following suggestion for you.

Store this item in your wishlist and use it when you need a filler item to qualify for free shipping.

Highly recommend the book.

Don't waste your money
While the puzzles are intriguing and thought-provoking, the solutions don't explain how the answers were reached.I was highly disappointed.

Great bathroom reading!
Very nice.I like the old-fashioned approach (I think the author has been writing books like this since the 1960s) and the problems are interesting and varied; most of them you can do in your head (hence an ideal "bathroom book") but some do make you break out the pencil and paper just to double-check.Highly recommended for interested people who studied Maths to around age 18 or beyond.

Good for warming up your brain.
Nice collection of puzzles with varying difficulties, which do not require any special knowledge of mathematics. ... Read more

Product Description
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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Customer Reviews (13)

Best Intro. Logic Book Ever!
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.

NOT A GOOD CHOICE FOR THE BUKKAKE LOVER
The title says it all. If you are one of the chosen few, you are better off buying a different logic book.

Fascinating material, poor proofs
Maybe it's because I'm only in undergrad, but I found a lot of the proofs in this book to be incomplete and hard to penetrate.Sometimes he would simply write "induction" and be through with it.That being said, this book covers a lot more material than other logic books, and the majority of it is extremely interesting.Much of it is, again, hard to penetrate (section 2.7 almost made me want to give up), but I found it to be a very worthwhile read.It covers things other authors simply hand-wave away such as the proof for the recursion theorem and the unique-readability theorem.I would recommend this to anyone with suitable mathematical maturity, but don't expect an easy read.For someone at the undergrad level there are better places to start.

From the point of a CS student
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.

John Wilson
Keen students may find if they study and parse both editions of Enderton's

Logic they may find much of interest. Getting to the root of a problem

can be of use in many situations. So best of luck. ... Read more

Product Description

This is a short, distinctive, modern, and motivated introduction to mathematical logic for senior undergraduate and beginning graduate students in mathematics and computer science. Any mathematician who is interested in knowing what logic is concerned with and who would like to learn Gödel’s incompleteness theorems should find this book particularly convenient. The treatment is thoroughly mathematical, and the entire subject has been approached like a branch of mathematics. Serious efforts have been made to make the book suitable for the classroom as well as for self-reading. The book does not strive to be a comprehensive encyclopedia of logic. Still, it gives essentially all the basic concepts and results in mathematical logic. The book prepares students to branch out in several areas of mathematics related to foundations and computability such as logic, axiomatic set theory, model theory, recursion theory, and computability. The main prerequisite for this book is the willingness to work at a reasonable level of mathematical rigor and generality.

Product Description

Customer Reviews (4)

An Important Work in Logic History
This book has a lot of interesting remarks. I, however, feel that it is a bit too wordy.

Perhaps the best written written elementary book of logic
I bought the book just because my teacher of elementary philosophy in the university respected Tarski as a master of formal logic. It took me 26 years to get this book in my hands. What makes Tarski unique is, that he was a great logician and a great teacher, too.

I belive that there still are no better guide for a student who wants to understand logic, not just try to remember basic rules of it. The beauty of logic has never been exposed in a better way.

The fifth star was spared to a new, annotated edition of this classic among the field of logic. I hope I can find one some day.

TIMELESS CORE HOLDING IN ANY LOGIC LIBRARY
This timeless classic by one of the five greatest logicians of all time should be owned by anyone who cares about logic - especially at this illogically low price.The Greek philosopher Aristotle (384-322 BCE), the English mathematician George Boole (1815-1864), the German mathematician Gottlob Frege (1848-1925), the Austrian-American mathematician Kurt Gödel and the Polish mathematician Alfred Tarski (1901-1983) are considered to be the five greatest logicians of history.Today it is difficult to appreciate the astounding permanence of what is accomplished in the works of Aristotle, Boole, and Frege without seeing their ideas surviving in the work of a modern master.Of the two modern master logicians Tarski is by far the most suitable for this purpose since he was by far the one most interested in the articulation of the conceptual basis of logic, he was by far the one most interested in history and philosophy of logic, and he was the only one to write an introductory book attempting to explain his perspective in accessible terms. This book, together with Aristotle's Prior Analytics and Boole's Laws of Thought, should form the core of any logic library. All three are still in print and available in inexpensive paperback editions.Hackett publishes an excellent up-to-date translation of Prior Analytics by Robin Smith and Prometheus recently reprinted Laws of Thought with an introduction by John Corcoran.- Frango Nabrasa.

I will always keep it as a reference
This is one of the classic introductory mathematics books. When I was learning logic, I relied on it heavily, although it was not the text for the course. Over my years as a teacher, I have consulted it often and when I was working on a recent book on logic, there were very few days when I did not open it in search of an idea or clarification.
All of the basics of logic are covered in one of the most readable texts I have ever opened. Exercises are given at the end of each chapter, although no solutions are included. This is one of those books that will always be on my key shelves of reference works and it will no doubt receive a great deal of use. ... Read more

Product Description

Retaining all the key features of the previous editions, Introduction to Mathematical Logic, Fifth Edition explores the principal topics of mathematical logic. It covers propositional logic, first-order logic, first-order number theory, axiomatic set theory, and the theory of computability. The text also discusses the major results of Gödel, Church, Kleene, Rosser, and Turing.

New to the Fifth Edition

• A new section covering basic ideas and results about nonstandard models of number theory
• A second appendix that introduces modal propositional logic
• An expanded bibliography
• Additional exercises and selected answers

This long-established text continues to expose students to natural proofs and set-theoretic methods. Only requiring some experience in abstract mathematical thinking, it offers enough material for either a one- or two-semester course on mathematical logic.

Product Description
Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. ... Read more

Customer Reviews (11)

Largely trivial
Lakatos' motives for writing this book seem to have been:
(a) "Under the present dominance of formalism, ... the history of mathematics, lacking the guidance of philosophy, has become blind, while the philosophy of mathematics, turning its back on the most intriguing phenomena in the history of mathematics, has become empty." (p. 2)
(b) "present mathematical and scientific education is a hotbed of authoritarianism and is the worst enemy of independent and critical thought" (pp. 142-143)
I passionately agree, but still found the actual book quite bland. It consists in a fairly amusing, semi-historical dialogue on Euler's formula V-E+F=2, intended to illustrate the very trivial thesis that creative mathematics is based on informal reasoning, heuristics, conjectures, counterexamples, etc., while also noting some general patterns of thought within this framework. Illustrations of similar patters in the history of the foundations of the calculus are also pointed out briefly; e.g., "the exception-barring method," is exemplified by Abel's reaction to his discovery of counterexamples such as sin(x)+sin(2x)/2+sin(3x)/3+... to Cauchy's theorem that the limit function of a convergent series of continuous functions is always continuous: "His response to these counterexamples is to start guessing: 'What is the safe domain of Cauchy's theorem?' ... All the known exceptions to this basic continuity principle were trigonometrical series so he proposed to withdraw analysis to within the safe boundaries of power series, thus leaving behind Fourier's cherished trigonometrical series as an uncontrollable jungle." (p. 133).

the heuristic of mathematical discovery
In a footnote to chapter 2 (much of the content of "Proofs and Refutations" is in the footnotes) Lakatos writes: "Until the seventeenth century, Euclidians approved the Platonic method of analysis as the method of heuristic; later they replaced it by the stroke of luck and/or genius." That stroke of luck and/or genius is a slight of hand that hides much of the story of the unfolding of mathematical research.

In "Proofs and Refutations," Lakatos illustrates how a single mathematical theorem developed from a naive conjecture to its present (far more sophisticated) form through a gruelling process of criticism by counterexamples and subsequent improvements. Lakatos manages to seemlessly narrate over a century of mathematical work by adopting a quasi-Platonic dialogue form (inspired by Galileo's "Dialogues"?), which he thoroughly backs up with hard historical evidence in the voluminous footnotes. The story he tells explores the clumsy and halting heuristic processes by which mathematical knowledge is created: the very process so carfully hidden from view in most mathematics textbooks!

The participants of Lakatos' dialogue argue over questions like "when is something proved?", "what is a trivial vs. severe counterexample?", "must you state all your assumptions or can some be thought of as implicit?", "in the end, what has been proved?",etc.. The answers to these questions change as the theorem under consideration is successively seen in a new light. Throughout, Lakatos is at pains to point out that the different perspectives adopted by his characters are representative of viewpoints that were once taken by the heroes of mathematics.

nice reading for the general public
Very nice book if you are in high school or in college and would like to see how mathematics evolves. It makes a very pleasant reading although the mathematical ideas behind are not trivial.

It discusses polyhedra in 3 (or more) dimensions and Euler's formula that describes their numbers of vertices, edges, faces, e.t.c. The challenge is to determine what specific kinds of polyhedra satisfy the formula and conversely, how one could generalize the formula so as to describe more (if not all) polyhedra. Lots of historical references illustrate the fact that the discussion is not naive and that reflects the actual history of the subject.

One can realize through this book that math people are not Gods and do not produce theories out of nowhere, but they experiment with their objects like any other scientist, and then try to summarize in an elegant/rigorous way.

a study in mathematical thought
I want to add a few words to the brief comment by the reader in Monroe (who gave this book one star). I tend to agree that "Proofs and Refutations" isn't a primer in mathematical proof-writing; it's certainly not a textbook for beginning mathematicians wanting to know how to practice their craft.

However, for those readers (including beginning mathematicians) who are interested in the broader picture, who are interested in the nature of mathematical proof, then Lakatos is essential reading. The examples chosen are vivid, and there is a rich sense of historical context. The dramatised setting (with Teacher and students Alpha, Beta, Gamma, etc) is handled skilfully. Now and then, a foolish-seeming comment from one of the students has a footnote tagged to it; more often than not, that student is standing in for Euler, Cauchy, Poincare or some other great mathematician from a past era, closely paraphrasing actual remarks made by them. That in some ways is the most important lesson I learned from this book; "obvious" now doesn't mean obvious then, even to the greatest intellects of the time.

Although "Proofs and Refuatations" is an easy book to begin reading, it is not an easy book per se. I have returned to it repeatedly over the last ten years, and I always learn something new. The text matures with the reader.

Excellent Critical Reasoning Framework
As a lay reader of mathematics, I am prone to read for more for analogy and thought methods instead of, for example, the real implications of variations on Eulers Formula: for any convex polyhedron, the number of vertices and faces together is exactly two more than the number of edges.

Displaying solid content with artful execution, this book interested me in both the math of the thing and the acompanying thought processes.

Content:This book has near-poetic density and elegance in arguing a non-linear approach to mathematical development and, for me, to just plain thinking.Our tendency (as born worshippers of linearity and causality) is to discover a brick for the building then immediately look for the next to stack on top.Lakatos contends that PERHAPS you have discovered a brick worthy of the building, now let's see what truly objective tests we will put to this brick and before giving it a final stamp of approval.It seems obvious to say "always question", but the exercise in this book will take you through the process and show you what you may take for granted in this simple concept.For example, do you observe HOW you question? See his discussion throughout on global vs. local counterexamples, just as a start.

Execution of the text:This is the beautiful part.Mr. Lakatos has written this book as theater: characters with definite identities, plot, drama. The narrative flows in the voices of students and a professor who proves to be a sound moderator, intervening at timely points, i.e. those where questions may be crystallized or thoughts prodded to that point.This is where learning takes place, in a heated, moderated debate over Euler's formula.What was most interesting to me about this method was that it lent itself easily to isolating a particular thread of discussion. I literally chose certain characters to research from beginning to end in order to follow the evolution or confirmation of their thinking.

You emerge with a good framework that makes this book excellent reference material for problem-solving.

One last, but important note.This book will have you praising the lowly footnote.I would buy it for that alone.You will read along with the discussion, then get off and examine a footnote, and then pick the dialogue back up not having lost a step.On the contrary, Mr. Lakatos deepens your context with on-point explanations and math history. ... Read more

Product Description
This introductory graduate text covers modern mathematical logic from propositional, first-order, higher-order and infinitary logic and Gödel’s Incompleteness Theorems to extensive introductions to set theory, model theory and recursion (computability) theory.

Based on the author’s more than 35 years of teaching experience, the book develops students’ intuition by presenting complex ideas in the simplest context for which they make sense. He also provides extensive introductions to set theory, model theory and recursion (computability) theory, which allows this book to be used as a classroom text, for self-study, and as a reference on the state ofmodern logic. ... Read more

Customer Reviews (2)

Where's the proof?
Quoting the author Hinman (page xi):"A notable lacuna is Proof Theory,
which fails to appear largely due to the incompetence of the author in this area".
And he is correct: there is no proof theory in this book, no Hilbert axioms,
no Gentzen natural deduction nor sequent calculus, nothing (except for a cursory
13 page section out of 878).
So, on the one hand, we have the largest logic book I have ever seen
-- and yet, ironically, the most incomplete.
I'm sure there's a lot of good stuff in this book and it is written well but it's
missing half the story, i.e., it is missing an exposition of how one manipulates
symbols formally to prove theorems. Even the semantic, model-theoretic side is incomplete:
there's no semantic tableaux, no resolution.Also, at 878 pages, plausibly
a considerable portion is at an advanced level.
So, this is not a good introduction to logic.
By far the best introduction to logic I've found is "Mathematical Logic for
Computer Science" by Mordechai Ben-Ari.Serious/pure mathematicians of course will
want to continue with the likes of "An Introduction To Mathematical Logic"
by Elliott Mendelson.

A Comprehensive Graduate Text
If I were a young graduate student in mathematics looking for that one "perfect" graduate text on mathematical logic to purchase with my (very) limited income, I would buy a copy of Professor Hinman's book.In just under 900 pages, Hinman provides an extremely well written and informedintroduction to propositional logic, first order mathematical logic, axiomatic set theory, model theory, and recursion theory.Indeed, the book is written so well that a motivated student with the requisite background can easily profit from independent study---a statement that simply cannot be made about many of the other "classic" references in this difficult field.One great virtue of having a single reference that introduces these diverse but interconnected areas is the uniformity of notation and definitions;the reader need not pull his hair out cross-referencing between texts that use wildly different notation and, occasionally, different definitions.

I studied mathematical logic at the University of Colorado--Boulder in the late 1970s.In those days, the logic students all depended on a standard list of references to prepare for the PhD qualifying examinations, and it is significant that all or nearly all of those works are still in print.At the introductory level we read the magnificent books on mathematical logic and set theory by Herbert Enderton.At the graduate level, we read Shoenfield, Monk, Mendelson, and Manin for mathematical logic, Chang and Keisler for Model Theory, Jech (and to a lesser extent, Kunen) for set theory, and Hartley Rogers for recursive function theory.In the course of plodding through these references, I discovered a wonderful comprehensive text by John Bell and Moshe Machover and quickly elevated it to primary status on my reading list.Bell and Machover remains my favorite among the older references today, nearly thirty years later, both in terms of comprehensive coverage and clarity of prose;when I reach for a reference to clarify an issue onfoundations, Bell and Machover is the first book I turn to.

The new book by Hinman achieves the same comprehensive goals of Bell and Machover, providing a rigorous and coordinated introduction to logic, set theory, recursion theory and model theory.However, Hinman incorporates some research topics that have emerged in the years since the 1977 publication of Bell and Machover, and it includes some more traditional topics that were difficult to find in the earlier texts.To give one example, Hinman provides a brief introduction tothe axiom of determinacy. This topic was made available to non-specialists in two papers published in the AMS Notices of June and July, 2001, where Hugh Woodin of Berkeley discussed the axiom of projective determinacy and other hypotheses within the context of possible enlargements of ZFC that would resolve Cantor's famous continuum hypothesis.A second example is Hinman's very lucid treatment of forcing;this writer has always had difficulty understanding the very few presentations of Paul Cohen's forcing technique that have been available in the older texts, but I found Hinman's treatment exceptionally clear and easy to follow.

Professor Hinman states that this book resulted from his nearly 40 years of experience teaching mathematical logic to graduates and undergraduates.The truth of this claim is reflected in the exceptional clarity of the prose and the coherence as one skims across different chapters.It is apparent that serious thought, consideration for the reader, and years of experience in the classroom shaped the final form of this text.Given the paucity of new texts in mathematical logic and foundations, the publication of this book is truly a cause for celebration.If you can only afford one text on the subject, purchase this one;if you are burdened with an abundance of spare change, I recommend buying Bell and Machover as a second reference to supplement Hinman. ... Read more

Product Description
A brief introduction to switching patterns in mathematical logic. ... Read more

Customer Reviews (2)

Logic for children
In an absence of any detailed reviews, I just got this on faith and it's wonderful.Starting by introducing the concept that many things have only two options (the light is either on or off, the drawbridge is either up or down), it then moves on to more complicated patterns in logic, dealing with more than one option.The majority of the book tells the story of a castle with drawbridges.The illustrations and cute story takes the reader through understanding when it's possible for different groups of people to reach the castle depending on what part of the bridge is up or down.The story introduces AND patterns and OR patterns.It also gives several more examples of where to find them and how to use them.This, along with Anno's Hat Tricks, are some of the only books I know of that present logic directly for young children.This is a book that kindergarteners could get something out of, but would also help older elementary children learn about logic.

Why this book is great
I found this book to be very good because it made this paticular part of math understandable and interesting. ... Read more

Product Description
Logic concepts are more mainstream than you may realize. There’s logic every place you look and in almost everything you do, from deciding which shirt to buy to asking your boss for a raise, and even to watching television, where themes of such shows as CSI and Numbers incorporate a variety of logistical studies. Logic For Dummies explains a vast array of logical concepts and processes in easy-to-understand language that make everything clear to you, whether you’re a college student of a student of life. You’ll find out about:

• Formal Logic
• Syllogisms
• Constructing proofs and refutations
• Propositional and predicate logic
• Modal and fuzzy logic
• Symbolic logic
• Deductive and inductive reasoning

Logic For Dummies tracks an introductory logic course at the college level. Concrete, real-world examples help you understand each concept you encounter, while fully worked out proofs and fun logic problems encourage you students to apply what you’ve learned. ... Read more

Customer Reviews (8)

The most readable introductory-level book on logic I've read
If you're a certain breed of nerd/geek (e.g., me), there's always a longing to nail down "logic" so as to better evaluate statements and arguments that one comes across in books, on blogs and webboards, etc.

In an attempt to fulfill this longing, I have purchased several introductory texts on logic in the past, which were somewhat helpful. However, I noticed gaps in the material presented in these books that made it difficult to make it all the way through. It seemed the material wasn't dumbed down enough.

Enter Logic for Dummies.

This is the most readable introductory-level book on logic I've read to date. In typical Dummies fashion, few paragraphs in the book exceed more than three or four sentences and the author is always indicating how the current material builds on what has already come and how it leads to what will eventually follow.

The focus in Logic for Dummies is sentential logic and quantifier logic - formal/symbolic logic. So if you're looking for a book on informal logic, this book is not it. You'll have to go elsewhere for that.

Despite my generally positive review, it should be pointed out that if you don't have a mathematical-oriented mind and a strong enthusiasm for the subject matter, chances are you won't make it through this book. Formal/symbolic logic is not for the faint of heart. But even so, this book may be useful as a handy reference for such folks.

Easy to understand logic!
This book makes it easy to understant logic.It is a very nice way to learn.Starts simple, but in a few chapters you will find yourself thinking in a different language!

Very good book.

Simple Logic
Though I have not yet finished this book, I already know that it will be a useful and interesting challenge.I am reading it only for my own interest and am not in any class or group setting.The explanations are easy to understand and are straight forward.This is a new area for me but this book will make a valuable contribution to my knowledge.
KSR

Good for leisure reading
This is a nice book for newcomers to logic. It reads easily. Unfortunately there are a dozen or so mistakes that may make the first time logic student confused and frustrated.

Beware of typos and errors
While I thought this book was laid-out well and served as a good introduction, I was disappointed to see such a large number of errors in the book.It made me wonder if anyone had bothered to proof-read it at all.These mistakes can potentially be severely misleading to the reader unless you are alert and recognize what Zegarelli meant to say.

For example, he clearly lays out in his truth table that value for the biconditional, F <--> T is F.Yet two pages later, there is an error that states that F <--> T is T.In another spot, a parentheses is omitted which completely alters the value of the statement.In another, the statement he is evaluating contains a biconditional <--> but the sentence below the statement refers to an &-operator that isn't even in the statement!And then there's this one: he states, "But when P is true and Q is false, the statement is false", when in actuality he has completely mixed up the truth values of P and Q according to his own truth table.Needless to say, unless you already have a background in logic or are adept enough to pick up on these errors, the reader can easily be confused by these apparent contradictions.

Potential buyers should also be aware that there are a large number of references to pop-culture such as Kelly Clarkson, Jennifer Lopez, and Hell's Kitchen.I'm sure Zegarelli uses these as a poor attempt at humor and to try to make readers relate to the subject-matter, but his assumptions about the knowledge and interests of his audience may not necessarily be applicable to everyone.

Despite these misgivings, the book does give a good basic refresher to the material at hand. ... Read more

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Oldie But Goodie
I first read this book in 1972 for a college course in the Philosophy of Mathematics. I was a philosophy major, mainly because I thought I was no good at math. By the end of the first chapter, I understood geometry for the first time. By the end of the second chapter, I was hooked. Mathematics has been a favorite hobby and occupation ever since.

Unfortunately, my prized copy of this text was lost in a flood in 1982. So I am delighted that this reprint is available. If you enjoy math puzzles, if you are interested in Gödel or non-Euclidian geometry, or if you simply want to stretch your mind a bit, I heartily recommend this book. ... Read more

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Traditional logic as a part of philosophy is one of the oldest scientific disciplines and can be traced back to the Stoics and to Aristotle. Mathematical logic, however, is a relatively young discipline and arose from the endeavors of Peano, Frege, and others to create a logistic foundation for mathematics. It steadily developed during the twentieth century into a broad discipline with several sub-areas and numerous applications in mathematics, informatics, linguistics and philosophy.

This book treats the most important material in a concise and streamlined fashion. The third edition is a thorough and expanded revision of the former. Although the book is intended for use as a graduate text, the first three chapters can easily be read by undergraduates interested in mathematical logic. These initial chapters cover the material for an introductory course on mathematical logic, combined with applications of formalization techniques to set theory. Chapter 3 is partly of descriptive nature, providing a view towards algorithmic decision problems, automated theorem proving, non-standard models including non-standard analysis, and related topics.

The remaining chapters contain basic material on logic programming for logicians and computer scientists, model theory, recursion theory, Gödel’s Incompleteness Theorems, and applications of mathematical logic. Philosophical and foundational problems of mathematics are discussed throughout the text. Each section of the seven chapters ends with exercises some of which of importance for the text itself. There are hints to most of the exercises in a separate file Solution Hints to the Exercises which is not part of the book but is available from the author’s website.

Customer Reviews (2)

Complete and demanding
For a motivated student, the best way to learn logic - in my opinion - is to study this book of Prof. Rautenberg. Thanks to a clever path through the subject of logic, the maximum result is obtained with the minimum effort. Every theorem is stated in the most general form and each proof appears to be the best.
This is neither a quick introduction nor an easy book, rather a dense and complete introduction to logic, demanding time and good will for a rewarding result.

The best intro to logic to-date
This is the best introductory text in logic available. It only lacks a coverage of set theory, and I advise the author to include it as chapter 4 of the third edition. ... Read more

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Starting with the concept that mathematical logic is not a collection of vaguely related results, but a method of attacking some of the most interesting problems which face the mathematician, the author sets the tone for this classic introduction. The basic concepts are presented in an unusually clear and accessible fashion, keeping in mind the original purpose of mathematical logic to build the foundations of this vast edifice of knowledge in a way that helps and intrigues the working mathematician as much as the philosophically minded student of logic. This book has served as a rite of passage to many mature and accomplished researchers. ... Read more

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Standard by default
Almost four decades after being written, this is still the standard graduate survey text. A large part of the reason is that there is little competition, but it is also a good book on its own merits. The author writes with a clarity and concision you rarely see in a math (or any) textbook. Proofs are straighforward, not tricky or convoluted. There are many excercises and with detailed setup. The exercises are often quite hard, requiring significant extension from the text.

Although the writing is good, that doesn't mean it is easy. He progresses deliberately through the details, rarely giving an overview. I think he is just expecting that you already have a good sense of context from the undergrad logic course you took (didn't you?). Sometimes he seems to belabor a point. There is also a dearth of examples, just five in the whole book, three of them in the appendix. There are no references at all. The age of the book makes it, not wrong, but inadequate in some areas. Still, I have looked at alternatives and haven't found something better for a graduate survey text in English.

Rock-solid introduction to Mathematical Logic
Since my first contact with mathematical logic, I've always seen it as a kind of brainwashing, forcing one's mind to work based on several little pieces of thought. Nevertheless, it can be described as "a necessary evil", because the mindless use of mathematical logic throughout mathematics is very treacherous, as it can be seen in the problems regarding the axiom of choice, the Banach-Tarski paradox inmeasure theory, the issues about the undecidability of certain assumptions in set theory, and the very limitations of mathematical logic.

Usually, of course, most work in mathematics doesn't require a deep knowledge of rigorous mathematical logic, but it's always a good thing to a serious mathematician to have some acquaintance with it, even if it's just to avoid boobytraps. Then, it's hard to find a better choice than Shoenfield's book. After a long absence from the book market, A K Peters made the wise decision of reprint this masterpiece. Although most of its contents are fairly standard for a book on mathematical logic (unlike the equally marvellous out-of-print book of Yu.I. Manin, which has a more philosophical slant and concerns itself with issues such as quantum logic, literature, etc.), it provides proofs for many propositions that in most of the literature are only stated. It has, of course, some extras not generally found in other books, as for example issues concerning constructibility of sets.

But the most important characteristic of this book is its clarity and precision. Itdoesn't waste time in unnecessary stuff, and shows why we need mathamaticallogic at all. Although it lacks some topics (for example, it doesn't discuss otheraxiomatic set theories besides Zermelo-Fraenkel. This is not so nice, because itlacks the distinction between classes and sets, one of the tenets of the Goedel- -Bernays-von Neumann set theory, although it is conceptually easier than thislast one. But maybe it's a pedagogical choice, because the set theory we allintuitively know is more or less based in Zermelo-Fraenkel), its main concern ispedagogy, so this limitation has a sound reason: this book exposes mainly the logic present in the math most mathematicians and alike scientists (mathematicalphysicists, etc.) use. Its solidity and razor-sharp precision is great to instruct thesepeople to be more careful with the math they use.

Besides that, some of the missing topics can be complemented by Mendelsson's "Introduction to Mathematical Logic", which is a bit more "merciful" book, which, by the other side, welcomes the thoroughness of Shoenfield. ... Read more

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AGood Mathematical Logic Text
This book is suitable foradvanced undergradutes and graduate students for learning mathematical logic.It contains a wide selection of exercises. In the back of the book, the author gave answers to selected exercises. But I think the notational conventions is a little of old, not stylish. I don't like this kind of notational convensions in the book.

I've worked throught the first two chapters
There are 5 chapters in total, of which I've worked through the first two on my own. These take you clearly through the metatheory of propositional and predicate calculus. I'm not a mathematician, so I was looking for something clear. I got this from the library along with several others including Enderton and Mendelsohn, and this was the one I ended up working with because it struck me as the clearest. Robbin does a good job of getting to the essentials without getting bogged down in a mass of theorem proving, but at the same time doesn't skip anything that you really need to know. This is a good, fast intro to the subject. ... Read more

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Lucid, non-intimidating presentation of propositional logic, propositional calculus and predicate logic by Russian scholar. Topics of concern in a variety of fields, including computer science, systems analysis, linguistics, etc. Accessible to high school students; valuable review of fundamentals for professionals. Exercises (no solutions). Preface. Three appendices. Indices. Bibliogaphy. 14 figures.
... Read more

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Awesome
the package came just as better than promised. This being an old book can be said o be a new one.

Not for beginners
Logic has an elegant simplicity to it that this author tries valiantly to capture, but fails. If one is already acquainted with propositional and predicate calculus, this is a joy to read for its elegance (something often attributed to Mates, but I think erroneously). This is a second-course, not a first encounter, text. But once the foundations of logical theory are laid, this is worthy of beholding the same concepts from a purely mathematical perspective. ... Read more

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( 20 ) OF EXPRESSION AND INTERPRETATION. A Proposition iB a sentence which either affirms or denies, as, All men are mortal, No creature is independent. A Proposition has necessarily two terms, as men, mortal; the former of which, or the one spoken of, h culled the subject; the latter, or tha! which is affirmed or denied of the subject, the predicate. These are connected together by the copula w, or m not, or by some other modification of the substantive verb. The substantive verb is the only verb recognised in Logic; all others are resolvable by means of the verb to be and a participle or adjective, e. g. " The Romans conquered"; the word conquered U both copula and predicate, being equivalent to "were (copula) victorious" (predicate). A Proposition must either be affirmative or negative, and must be also either universal or particular. Thus we reckon in ail, four kinds of pure categorical Propositions. 1st. Universal-affirmative, usually represented by A, Ex. All Xs

About the Publisher

Forgotten Books is a publisher of historical writings, such as: Philosophy, Classics, Science, Religion, History, Folklore and Mythology.

Forgotten Books' Classic Reprint Series utilizes the latest technology to regenerate facsimiles of historically important writings. Careful attention has been made to accurately preserve the original format of each page whilst digitally enhancing the difficult to read text. Read books online for free at www.forgottenbooks.org ... Read more

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What sould I do?
I'm 16 years old and I'm a boy and I'm from Switzerland, that's it ... Read more

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Good book
I have been reading this book off and on for years.It is beautiful.However, I am not well read in mathematical logic, and the comments of a mathematical logician as to whether the proofs are correct and what should be read next would be helpful to readers interested in mathematical logic.I read the book to understand Godel.There are better books for that.However, once I starting reading this book, I appreciated the eloquence of Prof. Quine and the beauty of the axioms, definitions and proofs in the book.

Very good but be aware of omissions
This book is indeed much shorter than Principia, mainly because it is derived for lecture notes for a 1 semester PhD course. It is also a lot clearer than PM. But the notation is largely the same, which makes for hard reading if your are under 50. Quine's proof format doesn't take up much space, but has always eluded me. This book contains the best treatment of truth functional and quantificational logic prior to natural deduction and truth trees.

I like the set theory of this book, but I warn you that it is very nonstandard. Even ardent lovers of Quine's NF theory hate
the ML theory of this book.

The weakness of this book is its treatment of metatheory:
consistency, completeness, decidability, categoricity. The treatment of Godel's incompleteness is detailed and highly original (altho' it owes more to Tarski than to Godel). But it is very difficult, and Smullyan (1991) is much better.
Quine also had no clue re model theory or recursion.

I respect the historical remarks a lot. Just one big omission: Quine, like nearly everyone of his generation, missed that
math logic as we know and love it does not descend from Frege, but from an 1885 article by C S Peirce.

In Depth Look at Logic
Try this book when you know a bit about the basics of logic.The descriptions are much more lucid than those in Principia, even if the ideas are less earthshattering for there time.Quine, as he always does, gives amasterful, detailed look at logic.If you are a fan of logic and thefoundations of math, this book is not to be missed. ... Read more

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The book starts with an elementary introduction to formal languages appealing to the intuition of working mathematicians and unencumbered by philosophical or normative prejudices such as those of constructivism or intuitionism. It proceeds to the Proof Theory and presents several highlights of Mathematical Logic of 20th century: Gödel's and Tarski's Theorems, Cohen's Theorem on the independence of Continuum Hypothesis. Unusual for books on logic is a section dedicated to quantum logic.

Then the exposition moves to the Computability Theory, based on the notion of recursive functions and stressing number{theoretic connections. A complete proof of Davis{Putnam{Robinson{Matiyasevich theorem is given, as well as a proof of Higman's theorem on recursive groups. Kolmogorov complexity is treated.

The third Part of the book establishes essential equivalence of proof theory and computation theory and gives applications such as Gödel's theorem on the length of proofs. The new Chapter IX, written for the second edition, treats, among other things, categorical approach to the theory of computation, quantum computation, and P/NP problem. The new Chapter X, written for the second edition by Boris Zilber, contains basic results of Model Theory and its applications to mainstream mathematics. This theory found deep applications in algebraic and Diophantine geometry.

Yuri Ivanovich Manin is Professor Emeritus at Max-Planck-Institute for Mathematics in Bonn, Germany, Board of Trustees Professor at the Northwestern University, Evanston, USA, and Principal Researcher at the Steklov Institute of Mathematics, Moscow, Russia. Boris Zilber, Professor of Mathematics at the University of Oxford, has been added to the second edition.

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David Hilbert was particularly interested in the foundations of mathematics. Among many other things, he is famous for his attempt to axiomatize mathematics. This now classic text is his treatment of symbolic logic. It lays the groundwork for his later work with Bernays.This translation is based on the second German edition, and has been modified according to the criticisms of Church and Quine. In particular, the authors' original formulation of Gödel's completeness proof for the predicate calculus has been updated.In the first half of the twentieth century, an important debate on the foundations of mathematics took place. Principles of Mathematical Logic represents one of Hilbert's important contributions to that debate. Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic. ... Read more

Customer Reviews (2)

Good old book
I learned logic from this book, and so I am very fond of it.As a presentation of what was known at that time, it cannot be beaten.The only problem with it is that a lot more has been discovered, so a modern treatment of the subject is better for the beginner who wants to be properly informed.For this I suggest the books by Copi or even Kleene.But if you don't care about modernity, or have an interest in the way things used to be done, I strongly recommend this book.Also, I might mention that Hilbert's "Geometry and the Imagination" is good even for the modern mathematician.

classic
Brief though it is, _Priniciples_ manages to cover not only the usual topics (sentential calculus, first-order predicate calculus, completeness, decidability), but also:the monadic predicate calculus in relation toAristotelian logic; second-order logic; set theory and the Fregean conceptof number; and the theory of types (logics of higher order).You might saythat Hilbert covers the same ground in 160 pages that Russell and Whiteheadlabor over for 3 volumes.The bottom line:a treat for anyone interestedin logic, especially in the period from Frege to Godel. ... Read more

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This junior/senior level text is devoted to a study of first-order logic and its role in the foundations of mathematics: What is a proof? How can a proof be justified? To what extent can a proof be made a purely mechanical procedure? How much faith can we have in a proof that is so complex that no one can follow it through in a lifetime? The first substantial answers to these questions have only been obtained in this century. The most striking results are contained in Goedel's work: First, it is possible to give a simple set of rules that suffice to carry out all mathematical proofs; but, second, these rules are necessarily incomplete - it is impossible, for example, to prove all true statements of arithmetic. The book begins with an introduction to first-order logic, Goedel's theorem, and model theory. A second part covers extensions of first-order logic and limitations of the formal methods. The book covers several advanced topics, not commonly treated in introductory texts, such as Trachtenbrot's undecidability theorem. Fraissé's elementary equivalence, and Lindstroem's theorem on the maximality of first-order logic. ... Read more

Customer Reviews (12)

Will not suggest to anyone.
Not a very good text for beginners. There are other good books like the ones by Mendelson, A. Margaris and tons of others.

Excellent Choice for Teaching Mathematical Logic
This is a truly excellent book -- one I've used (along with other other books) to teach mathematical logic for 20 years.(The new edition provided welcome coverage of logic programming.)Traditionally, logic pedagogy has tended to revolve around which colleges or universities are involved.You will need to have sharp students to take full advantage of this textbook.In addition, some proof construction environment/proof checker is a good thing to have accompany the textbook; the same would hold of model finders.For grad students in my lab, I require familiarity with the book, sooner or later.

The steepest on-ramp to the fast lane of logic
Learning mathematical logic from this textbook is a little like learning to rock-climb by going straight to the half-dome.Most likely, you'll fall to your death.But if you're strong enough and lucky enough to endure the climb, you'll look back on how far you've come and have an "OH MY GOD I ACTUALLY DID THAT???" moment of clarity like nothing else you've ever experienced :-)

Should be the standard undergrad introduction
Intended for a one-semester course, it ignores some of the usual topics in a survey course so it can give a deeper treatment of the nature and adequacy of mathematical proofs. It slights number theory, second-order logic, nonstandard analysis, and set theory. There is only enough on recursion and computability to support the main topic, but it goes deeper than usual on limitative results.

What it does cover it does very well. Motivation is rich and exercises follow well from the text. Proofs are very clear. Overall, there is much greater coherence in the development of ideas than you usually see in a survey text.

While the writing is very good, there is a shortage of definitions, examples, and exercises. Notation is not always clearly introduced and they adopt so many abbreviations it's hard to keep track of what things mean. I also thought that it was not as clear in the second half, maybe due to the multiple authors. Still, I would choose it over Enderton unless you need lots of exercises for class use.

Reads like Mathematical Poetry
As others have pointed out, this book is not for beginners, but is very well suited for those with some confidence in formal logic and axiomatized set theory. The book is just great if you want to deepen your understanding of the subject beyond what can be had from undergrad level courses on the topic. It should be required reading for any student of computational logic.

The question this book addresses is not "why logic?", or "what is a formal logic?", but more specifically, "why is first-order predicate calculus with equality such a good foundation for mathematics?"

The formal mathematics is organized and presented so clearly and precisely that I felt I was admiring a fine crystal structure.
The notation used may seem excessive to some, but it actually is the least amount of notation that could be gotten away with without resorting to glossing over fine distinctions.For example, many logic books assume a fixed countably infinite number of function and predicate symbols, which leads to some confusion when comparing different axiomatizations of the natural numbers, or of groups.This book on the other hand is crystal clear on how such different axiomatizations are related to each other.Another subtle point I never noticed before about first-order predicate logic but that is pointed out in the footnote on page 73 is that one might think it possible that just because a formula can be proven with one choice of predicate and function symbols, it might not be provable with a different choice of symbols.It turns out that this cannot happen as a simple consequence of the completeness theorem! (p. 85)

The book explores second-order predicate logic and makes explicit some of the difficulties, such as incompleteness and even the problem of how closely the truth of a formula in second order logic depends on what we take as true in set theory: different axiomatizations of set theory lead to different semantics for second-order predicate logic!

There is a great chapter on the incompleteness theorems, and in addition to Goedel's theorems, there is a section on Register Machines (a version of Turing Machines) and a proof of the undecidability of arithmetic using the halting problem, as well as a more general theorem about the undecidability of any theory that can encode the workings of a Register Machine.

The next section is a reasonable presentation of the mathematical underpinnings of logic programming.

The book concludes with an algebraic characterization of elementary equivalence followed by two deep theorems by Lindstrom that demonstrate the uniqueness of first order predicate calculus among formal languages with set theoretic semantics. ... Read more