Product Description
New corrected printing of a well-established text on logic at the introductory level. ... Read more

Customer Reviews (2)

not for newcomers
I used parts of the first two chapters in a class. As with everything in logic, it may take some time to be understood, but it will rewire your brain once you get the point. The book is aimed at mathematicians, so be careful.

Best title ever
One of the most finest books in logics. For academics or just for fun, like me. ... Read more

Product Description

Mathematical logic is a branch of mathematics that takes axiom systems and mathematical proofs as its objects of study. This book shows how it can also provide a foundation for the development of information science and technology. The first five chapters systematically present the core topics of classical mathematical logic, including the syntax and models of first-order languages, formal inference systems, computability and representability, and Gödel’s theorems. The last five chapters present extensions and developments of classical mathematical logic, particularly the concepts of version sequences of formal theories and their limits, the system of revision calculus, proschemes (formal descriptions of proof methods and strategies) and their properties, and the theory of inductive inference. All of these themes contribute to a formal theory of axiomatization and its application to the process of developing information technology and scientific theories. The book also describes the paradigm of three kinds of language environments for theories and it presents the basic properties required of a meta-language environment. Finally, the book brings these themes together by describing a workflow for scientific research in the information era in which formal methods, interactive software and human invention are all used to their advantage.

This book represents a valuable reference for graduate and undergraduate students and researchers in mathematics, information science and technology, and other relevant areas of natural sciences. Its first five chapters serve as an undergraduate text in mathematical logic and the last five chapters are addressed to graduate students in relevant disciplines.

Product Description
This book provides a tour through the main branches of the foundations of mathematics. It contains chapters covering elementary logic, basic set theory, recursion theory, Gödel’s (and others’) incompleteness theorems, model theory, independence results in set theory, nonstandard analysis, and constructive mathematics. In addition, this monograph discusses several topics not normally found in books of this type, such as fuzzy logic, nonmonotonic logic, and complexity theory.

The word "tour" in the title deserves some explanation. This word is meant to emphasize that this is not a textbook in the strict sense. To be sure, it has many of the features of a textbook, including exercises. But it is less structured, more free-flowing, than a standard text. It also lacks many of the details and proofs that one normally expects in a mathematics text. However, in almost all such cases there are references to more detailed treatments and the omitted proofs. Therefore, this book is actually quite suitable for use as a text at the university level (undergraduate or graduate), provided that the instructor is willing to provide supplementary material from time to time.

The most obvious advantage of this omission of detail is that this monograph is able to cover a lot more material than if it were a standard textbook of the same size. This de-emphasis on detail is also intended to help the reader concentrate on the big picture, the essential ideas of the subject, without getting bogged down in minutiae. This book could have been titled "A Survey of Mathematical Logic," but the author’s choice of the word "tour" was deliberate. A survey sounds like a rather dry activity, carried out by technicians with instruments. Tours, on the other hand, are what people take on their vacations. They are intended to be fun. The goal of this book is similar: to provide an introduction to the foundations of mathematics that is substantial and stimulating, and at the same time a pleasure to read. It is designed so that any interested reader with some post-calculus experience in mathematics should be able to read it, enjoy it, and learn from it. ... Read more

Customer Reviews (1)

An in-depth and in-breadth tour of the foundations of mathematics
This tour is both broad and deep; it covers most of what would ordinarily be considered upper level material in the foundations of mathematics. While it contains theorems, proofs and exercises, it does not have the structure of a textbook. It is more on the order of an advanced survey that includes history. There are occasional short sections reserved for biographies of mathematicians of note in the field. For example, there are biographies of Bertrand Russell, Euclid, Georg Cantor, John von Neumann and Julia Robinson.
The chapter headings are:

*) Predicate logic
*) Axiomatic set theory
*) Recursion theory and computability
*) Godel's incompleteness theorems
*) Model theory
*) Contemporary set theory
*) Nonstandard analysis
*) Constructive mathematics

Given the branching away from logic into set theory and analysis, in my opinion, this book is best considered a survey of the foundations of mathematics. Considered in that way, it is a sound primer for people with mathematical maturity. If you have an interest in advanced material in these areas, you will find this book worthy of study.
... Read more

Product Description
A comprehensive and user-friendly guide to the use of logic in mathematical reasoning

Mathematical Logic presents a comprehensive introduction to formal methods of logic and their use as a reliable tool for deductive reasoning. With its user-friendly approach, this book successfully equips readers with the key concepts and methods for formulating valid mathematical arguments that can be used to uncover truths across diverse areas of study such as mathematics, computer science, and philosophy.

The book develops the logical tools for writing proofs by guiding readers through both the established "Hilbert" style of proof writing, as well as the "equational" style that is emerging in computer science and engineering applications. Chapters have been organized into the two topical areas of Boolean logic and predicate logic. Techniques situated outside formal logic are applied to illustrate and demonstrate significant facts regarding the power and limitations of logic, such as:

• Logic can certify truths and only truths.
• Logic can certify all absolute truths (completeness theorems of Post and Gödel).
• Logic cannot certify all "conditional" truths, such as those that are specific to the Peano arithmetic. Therefore, logic has some serious limitations, as shown through Gödel's incompleteness theorem.

Numerous examples and problem sets are provided throughout the text, further facilitating readers' understanding of the capabilities of logic to discover mathematical truths. In addition, an extensive appendix introduces Tarski semantics and proceeds with detailed proofs of completeness and first incompleteness theorems, while also providing a self-contained introduction to the theory of computability.

With its thorough scope of coverage and accessible style, Mathematical Logic is an ideal book for courses in mathematics, computer science, and philosophy at the upper-undergraduate and graduate levels. It is also a valuable reference for researchers and practitioners who wish to learn how to use logic in their everyday work. ... Read more

Customer Reviews (1)

rigorous introduction
For the math major, and perhaps for the serious computer science major, Tourlakis presents a rigorous discourse on logic. Step by step he gives definitions and proofs. Some readers will appreciate the elegance of the presentation.

There are many exercise problems in each chapter; of sufficient difficulty to get the undivided attention of many readers.

The book goes beyond the elementary descriptions of Boolean logic typical of introduction books on logic or computing. It gets up to Godel's theorem and the understanding of what it means for a problem to be computable.

In terms of computing applications, especially SQL, the book describes the normal form. However, it does not go into the various theorems of SQL, involving the different normal forms. ... Read more

Product Description
This lively introduction to mathematical logic, easily accessible to non-mathematicians, offers an historical survey, coverage of predicate calculus, model theory, Godel’s theorems, computability and recursivefunctions, consistency and independence in axiomatic set theory, and much more. Suggestions for Further Reading. Diagrams.
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Customer Reviews (3)

No-nonsense survey of logic
This is an introduction to the main ideas and results of mathematical logic. It is primarily a text for non-logicians but it is still very serious. Practically everything is proved, and the proofs are carefully crafted and not too technical. For a reader with a bit of mathematical background this is far more valuable than the more typical logic-for-casual-readers books, such as for instance "Gödel's Proof" by Nagel & Newman, which are too chatty and trivial and don't really prove anything. By contrast, a high point of this book is a very accessible treatment of the proof of Gödel's incompleteness theorem in a matter of a few pages. On the other hand, this book is perhaps not chatty enough: the clear proofs and discussions of the main results are nicely done, but the discussions of historical background, motivation and context are very sketchy.

Six Rigorous Lectures - Not for the Faint-Hearted
Although this book - What is Mathematical Logic? - is written in an informal and entertaining style, it is unlikely to appeal to a reader not familiar with predicate calculus, recursive functions, and set theory. Despite its innocuous title, this little book is surprisingly rigorous.

The six chapters are derived from a series of lectures given by the five authors - J. N. Crossley, C. J. Ash, C. J. Brickhill, J. C. Stillwell, and N. H. Williams - at Monash University and University of Melbourne in 1971.The lectures were substantially revised for publication.

Only the first chapter, a detailed historical survey of mathematical logic, can be readily appreciated by the non-mathematician. The remaining five chapters examine advanced topics in mathematical logic including the Godel-Henkin Completeness Theorem, Model Theory, Turing machines and recursive functions, Godel's Incompleteness Theorem, and advanced set theory.

Chapter 2 introduces the Godel-Henkin Completeness Theorem, a proof that predicate calculus is complete. Chapter 2 is not easy, but it is essential to acquire a reasonable familiarity with predicate calculus before moving forward.

Chapter 3 offers a detailed look at model theory, the study of relations between formal languages and the interpretation of formal languages. Topics include Predicate Calculus with Identity, the Compactness Theorem, and the Lowenheim-Skolem Theorems. I had substantial difficulty with the details, but I did gain a general understanding and appreciation for model theory.

Chapter 4 addressed in considerable detail a more familiar topic, Turing machines and recursive functions. The discussion concludes with a key proof: there is no algorithm which will enable us to decide, given any particular formula of predicate calculus, whether or not this particular formula is deducible from the axioms of predicate calculus.

Chapter 5 was a detailed examination of Godel's Incompleteness Theorem for formal systems that include arithmetic of the natural numbers. I had less difficulty with this topic as I had previously read Godel's Proof by E. Nagel and J. R. Newman. This chapter would very likely be tough going for a reader entirely new to Godel's exceeding complex and abstruse proof.

Chapter 6, titled Set Theory, might be better named Advanced Set Theory. I was entirely new to the Axiom of Choice and the Generalized Continuum Hypothesis.

I highly recommend this intriguing and lively look at mathematical logic to readers with some familiarity with this rather formidable subject. For readers new to mathematical logic, I suggest that the following books might be better starting points.

Foundations and Fundamental Concepts of Mathematics by Howard Eves is outstanding. The chapter titled Logic and Philosophy is an excellent introduction to mathematical logic.

The Advent of the Algorithm by David Berlinski is an eclectic, rather bizarre introduction to a complex mathematical topic. Although many reader reviewers aggressively criticize this book, I enjoyed puzzling my way through Berlinski's discursive discussions.

Godel's Proof by Ernest Nagel and James R. Newman offers a fascinating look at a mind boggling, incredibly complex, inventive mathematical proof.

Dense but readable
After a 10-page historical survey of logic from the 1850s through the 1960s, similarly brief chapters on Completeness, Model Theory, Recursion Theory, the Incompleteness Theorems, and Set Theory give an idea of what might be covered in an undergraduate course and the first several graduate courses in mathematical logic. (The last 5 pages of the book are an introduction to forcing arguments and a fairly detailed sketch of the consistency of not-GCH.)

Results are clearly and carefully stated; and while sketches of proofs have a hard time staying nontechnical and still meaningful, most such attempts are admirable.

A marvel of brevity while not watering anything down. ... Read more

Product Description
Assuming no previous study in logic, this informal yet rigorous text covers the material of a standard undergraduate first course in mathematical logic, using natural deduction and leading up to the completeness theorem for first-order logic.At each stage of the text, the reader is given an intuition based on standard mathematical practice, which is subsequently developed with clean formal mathematics.Alongside the practical examples, readers learn what can and can't be calculated; for example the correctness of a derivation proving a given sequent can be tested mechanically, but there is no general mechanical test for the existence of a derivation proving the given sequent. The undecidability results are proved rigorously in an optional final chapter, assuming Matiyasevich's theorem characterising the computably enumerable relations. Rigorous proofs of the adequacy and completeness proofs of the relevant logics are provided, with careful attention to the languages involved.Optional sections discuss the classification of mathematical structures by first-order theories; the required theory of cardinality is developed from scratch.Throughout the book there are notes on historical aspects of the material, and connections with linguistics and computer science, and the discussion of syntax and semantics is influenced by modern linguistic approaches.Two basic themes in recent cognitive science studies of actual human reasoning are also introduced. Including extensive exercises and selected solutions, this text is ideal for students in logic, mathematics, philosophy, and computer science. ... Read more

Customer Reviews (3)

A SUPERLATIVE TEXTBOOK OF LOGIC - one that is without headaches !
Such an excellent text. I congratulate logician Wilfred Hodges whose works I have had the honour of studying. This is an excellent text in following sense.

A deep complex work in mathematical logic would be at least 600+ pages of pure mathematical reasoning. I did a Masters course with the brilliant Moshe Machover (A course in mathematical logic, which he wrote with John Bell).

But a rigorous account convering Quantificational Logic, Model Theory, Recursive Functions and large proofs (such as that of Putnam, Davis, Matyasevich's theorem), Formal Set Theory, Non-standard Analysis, would at lease be as long as Machover's book -- in fact, a long-hand explanation of what he did in the book, solutions of exercises would make the book at least 800 pages.

So then, if you don't have a career in logic and logic-related sciences in mind, you can get a great introduction here to logic by one of recent logic's icons. His Model Theory book is something of a bible.

Once you appetite is whetted for Logic by this book, you might next go to Category Theory/Topos approach to logic in the Lawvere's expose: conceptual mathematics (2nd Edition!).

From then on sky is the limit: You can tend to that limit via the work of Saunders Maclane in SHEAVES in Geometry and Logic. But for the non-mathematician, this book is ENTIRELY inaccessible.

A WORD or two of HOPE ! Mathematics is like a large tapestry and one mustn't be too fussy about the EXACT coverage of "everything" you want in a book. The point is that as you read an good book - -like Hodge's -- you will have learnt a portion of that tapestry. The next book will conver more of it, you can skip the parts you have read from Hodges or Lawvere's or find them more easily workable.

The most difficult part of doing logic/mathematics is ballancing the pace of working and the mind-set for studying it. I read a partial account of how in musical training in 18th century getting the mind to be in the right attitude was the central focus - which was part of the environment of creativity.

Poorly written
This is probably the worst math textbook I've read as an undergraduate.There are several typos, and the author is extremely unclear and not precise or rigorous at all in his explanations.Some examples flat out don't make sense at all.This seems like the product of a mathematician who desired to create a textbook without knowing what he was getting himself into but finished it for the sake of finishing it.

Logic: Symbolic & Conceptual
The book is fundamental in the first chapter but quickly moves on to logic diagrams and parsing trees.Chapter 3 is hard for the beginning student and advanced student because of the multiple clauses.The History of Logic is well thought out in introducing several key figures in the history like Charles Pierce, and David Hilbert.The exercises are difficult so I would have to say have another 2-4 logic references with you, one being Quine's Methods of Logic.Does not touch on Wittgenstein's Logic from the Tractatus Logico Philosophicus.There are numerous gaps in the book so this cannot be a comprehensive reference either. ... Read more

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Customer Reviews (1)

Good to teach or learn logic from
I have been using quite intensively this book as part of a Logic for Philosophy Majors Class I am teaching this semester in Bogotá. The approach in the book is excellent - from the beginning it emphasizes various logics (Classical, Constructive, Fuzzy, Comparative, Relevance among others) with many examples and classical motivations (Aristotle on relevance and comparison of truth, etc.).

I particularly like the treatment of the semantics in the book - the fact it does two-valued, three-valued, integer valued, set-valued and topological-valued semantics for various logics. The treatment of the semantics is clear enough - it may be taught for second-semester students at my University.

I like a bit less the treatment of syntax - my impression is that from chapter 12 on, the book seems to provide a picture of syntax less clear, at least for the class I teach this with. That part of the book is very good for self-study and for examples, but my impression is that the treatment of syntactic aspectics is not at the level of the treatment of semantic aspects (superb in this book).

All in all, my impression is that Schechter's Classical and Nonclassical Logics (...) is excellent either as a textbook (though I prefer it in the semantics "half"), as a self-study book or as a basic clear reference of many different logics.

Warning: the book - as the complete title says - is centered on Propositional Calculus - there is essentially no Predicate Calculus. At first that seemed strange to me, but I now understand a bit better the possible reasons for the author's decision. That does not make per se the book worse or better - there is already a lot of material covered - if the author had tried to include Predicate Calculus as well, the book would have probably doubled in volume and it is not clear that it could accomplish what it does: presenting on equal footing many logics, giving the reader many tools and examples to see the differences and the motivations of the various logics. ... Read more

Product Description
A comprehensive one-year graduate (or advanced undergraduate) course in mathematical logic and foundations of mathematics. No previous knowledge of logic is required; the book is suitable for self-study. Many exercises (with hints) are included.

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Customer Reviews (3)

Covers a lot, but not all that well
Bell & Machover is meant for a one-year graduate course and is comparable to the more well-known text by Shoenfield. B&M is larger than Shoenfield, having additional chapters on Boolean algebra, intuitionist logic, and nonstandard analysis. There are a few newer things in B&M and they use the tableau method, which Shoenfield doesn't. Otherwise, they have about the same coverage, trading off as to which has more detail on this or that item. The main difference is in the writing style. The delivery in B&M is less articulate, proofs are terse and schematic, and problems have little setup. I find it harder to follow the train of thought in B&M than in Shoenfield and would not want to try learning anything for the first time from it.

This is the book you need as a logic primer
When I was in my third year of graduate school and was deciding to specialize in set theory, I realized that it was time to get some formal training in first-order logic and model theory. At our school, there were no courses in foundations at all, so I had to find the right book and map out a course of study for myself. I sat in on a seminar for more advanced students that was going through Chang and Keisler (model theory); although I now use this as a basic reference, at the time I needed a more systematic treatment of first-order logic before getting into the details of model theory.

One day I discovered this book by Bell and Machover. It was exactly what I needed. The first three chapters are just what you need to get a solid grounding in first-order logic, covering the soundness, completeness and compactness theorems.

As I was working through the book, I noticed that the incompleteness theorems weren't treated till the 7th chapter -- this, along with the fact that the exercises were so much fun kept me pushing forward one chapter at a time.

Chapter 4 deals with Boolean algebras. I'd alreday had quite a bit of general topology before starting this chapter, so when the duality between Boolean algebras and Stone spaces is explored in some detail in the exercises, it was really a blast. A lot of material is covered in this chapter; the authors never tell you that they chose just the material you will need if you go on to study the Boolean algebra approach to forcing. (They actually give you a taste of Boolean-valued models in Chapter 5, but that's it for forcing in this book; however, the first author, Bell, wrote another book called Boolean-Valued Models and Independence Proofs in Set Theory that provides the most complete treatment of the subject available in book form.)

Chapter 5 gives a careful treatment of the basics of model theory (which was what I needed). Chapter 6 then gives a very fun treatment of recursion theory. The main material covers enough to prove the equivalence of recursive and computable functions (and computable functions are treated using a special kind of register machine rather than the usual Turing machines).

Finally with all the groundwork completed, the authors give a marvelous treatment of the incompleteness theorems, including the usual results about undecidability of Peano arithmetic and the undefinability of truth. The authors go on to develop recursion theory a little further in light of the theorems of Chapter 7 -- enough to solve Post's problem by building two incomparable r.e. degrees.

The rest of the book consists of special topics -- intuitionistic logic, set theory, and nonstandard analysis. All this was very good, though I suppose I prefer other treatments of set theory (Kunen or Jech).

The strength of this book is that it doesn't gloss over any details -- and this is very important when you first get into mathematical logic. This should be every graduate student's first course on logic!

Probably the most comprehensive Course in Mathematical Logic
This is probably one of the most oustanding textbooks on advanced mathematical logic written this side of the century.

But, it is surprising to me how difficult it is to find it in any libarary in USA.Even more surprising, however, is that, after all these years, it is notavailable at an affordable price. (It would be hasty to suggest that somepublishers are motivated by greed than a desire to inform and educate, I amsure there are better reasons -- with the reservations that go with sayingthis, of course.).

On the cheerful side, I had the good fortune to siton most of this course given by Moshe Machover in London. He is anoutstanding logician and teacher. As a human being he is profound and just.He used to say, with a humorous matter-of-factness, something to the effectthat he hoped we were not filled with "malaise" his favouriteword.

The coverage is thorough and deep. The book is sufficientlyadvanced for it to be used as a textbook for a Master Level Introduction inLogic. It wasbeing used in this way at London University (in1979).

Thus it takes you further than most logic books that seek to teachthe same set of topics. After covering early theorems in, say, modeltheory, it goes on to prove advanced theorems well beyond the standardtexts on logic. The book, as I mentioned prepares you, relative to theBritish system, for an MA degree -- and so perhaps, a pre-amble to MPhiland, presuemeably,PhD levels.

You might need to augment this book withsome other books in Topology, to follow some of the topological theorems).

Machover's other logic book, intended for Philosophers is set theory,logic and their limitatins, which locally resembles this book but is on amuch more modest scale and certainly does not cover constructive logic. Hisearliest book was Nonstandard Analysis without tears.

In the class, hewould give handouts for proof of some of the exercises. But, if you cannotdo them on your own you should consult some of the bibliographic sources.For instance on Non-standard models you might consult Robinson'sNon-standard Analysis.

His coverage of Set Theory, Limitative Results(Goedel results etc.) and Intuitionistic logic is fantastic.

Buy it ifyou can afford it. ... Read more

Product Description
One of the pioneers of mathematical logic in the twentieth century was Alonzo Church. He introduced such concepts as the lambda calculus, now an essential tool of computer science, and was the founder of the Journal of Symbolic Logic. In Introduction to Mathematical Logic, Church presents a masterful overview of the subject--one which should be read by every researcher and student of logic. The previous edition of this book was in the Princeton Mathematical Series. ... Read more

Customer Reviews (2)

a classic, but mostly useful as a historical reference
I give this book 5 stars out of respect for its enormous contribution to mathematical logic; for no doubt many of the authors of the more modern math-logic texts were greatly influenced by this book. But with that said, all of the material here is a proper subset of other current books which present the material much more clearly and using better notation. Examples include Burris' "Logic for Mathematics and Computer Science", Ebbinhaus' "Intro. To Math Logic", and Gallier's "Logic for Computer Scientists".

One of the classics
This book, which first appeared in print as an issue in Annals of Mathematics in 1944, is now a classic in mathematical logic, and is still worth perusing in spite of the out-dated notation. The author outlines comprehensively the propositional calculus and predicate calculus. Although the book is mostly formal in its style, the author does introduce the reader to some elementary notions in logic, and some brief commentary on what would now be classified as philosophical logic. He defines logic as the analysis of propositions and their proof according to their form and not their content. He notes also that inductive logic and the theory of partial confirmation should also be included as part of mathematical logic. There are exercises throughout the book, and so it could conceivably be used as a textbook, in spite of its publication date. The book could better be used as a historical supplement to a course in mathematical logic or one in the philosophy of logic.

In the introduction to the book the author defines the terms and concepts he will use in the book, with a discussion of proper names, constants and variables, functions, and sentences. He adopts the Fregian point of view that sentences are names of a particular kind. His discussion of this is rather vague however, for he does not give enough clarification of the difference between an "assertive" use of a sentence and its "non-assertive" use. Readers will have to do further reading on Frege in order to understand this distinction more clearly, but essentially what Church is saying here is that sentences are names with truth values. The existential and universal quantifiers are introduced as well. And here the author also introduces the concepts of object language and metalanguage, along with a discussion of the axiomatic method. The author distinguishes between informal and formal axiomatic methods. The modern notions of syntax and semantics are given a nice treatment here, and the di

scussion is more in-depth than one might get in more modern texts on mathematical logic.

Chapter 1 is a detailed overview of propositional logic, being the usual formal system with three symbols, one constant, an infinite number of variables, rules on how to form well-formed formulas, and the rules of inference. The deduction theorem is proved in detail along with a discussion of the decision problem for propositional logic, with the famous truth tables due to W. Quine introduced here. The notions of consistency and completeness are briefly discussed.

The discussion of the propositional calculus is continued in the next chapter where a new system of propositional calculus is obtained by dropping the constants from the first one and adding another symbol (negation). The two systems are shown to be equivalent to each other using a particular well-formed formula in the second one to replace the constant in the first. Other systems of propositional calculus are also introduced here, using the idea of primitive connectives such as disjunction, along with various rules of inference. Church also outlines an interesting propositional calculus due to J.G.P.Nicod, which assumes only one primitive connective, one axiom, and only one rule of inference (besides substitution). The author also introduces partial systems of propositional calculus, with the goal of showing just what must be added to these systems to obtain the full propositional calculus. He discusses the highly interesting and thought-provoking intuitionistic propositional calculus, due to A. Heyting, which is a formalization of the famous mathematical intuitionism of L.E.J. Brouwer. The system he discusses is a variant of Heyting's and he gives references to the positive solution of the decision problem for this system. The author ends the chapter with a brief discussion of how to construct a propositional calculus by employing axiom schemata.

The author then moves on to what he has termed functional calculi of first order beginning in the next chapter. Called predicate calculi in today's parlance, the author first defines the pure functional calculus of first order, and shows that the theorems of the propositional calculus also follow when considered as part of this system. Free and bound variables are defined, and Church proves explicitly the consistency of this system, and the deduction theorem. The important construction of a prenex normal form of a well-formed formula is discussed, and the author shows that every well-formed formula of the functional calculus is equivalent to some well-formed formula in prenex normal form.

In chapter 4, the author gives an alternative formulation of pure functional calculus of first order, wherein rules of substitution are used and axiom schemata are replaced by instances, making the number of axioms finite. The Skolem normal form of a well-formed formula is defined, which sets up a discussion of satisfiability and validity. The author then proves the Godel completeness theorem, which states that every valid well-formed formula is a theorem. This is followed by a very well written discussion of the Skolem-Lowenheim theorem, and an overview of the decision problem in functional (predicate) calculus.

In the last chapter of the book the author considers functional (predicate) calculi of second order, which is distinguished from the first order case by allowing the variables to range over what its predicates and subjects represent. In second-order functional calculus, propositional and predicate variables can have bound occurrences. The author discusses the elimination problem and consistency for second-order predicate calculus, and gives a proof of the (Henkin) completeness theorem. A fairly detailed discussion of a logical system for elementary number theory is given, but the treatment involves notation that is somewhat clumsy and the discussion is difficult to follow. ... Read more

Product Description

Handbook of Mathematical Induction: Theory and Applications shows how to find and write proofs via mathematical induction. This comprehensive book covers the theory, the structure of the written proof, all standard exercises, and hundreds of application examples from nearly every area of mathematics.

In the first part of the book, the author discusses different inductive techniques, including well-ordered sets, basic mathematical induction, strong induction, double induction, infinite descent, downward induction, and several variants. He then introduces ordinals and cardinals, transfinite induction, the axiom of choice, Zorn’s lemma, empirical induction, and fallacies and induction. He also explains how to write inductive proofs.

The next part contains more than 750 exercises that highlight the levels of difficulty of an inductive proof, the variety of inductive techniques available, and the scope of results provable by mathematical induction. Each self-contained chapter in this section includes the necessary definitions, theory, and notation and covers a range of theorems and problems, from fundamental to very specialized.

The final part presents either solutions or hints to the exercises. Slightly longer than what is found in most texts, these solutions provide complete details for every step of the problem-solving process.

Customer Reviews (3)

A "must have" for everyone whose passion lies in the mathematics!
This book quickly became one of my favorites for it's richness of the examples from all fields of mathematics, ranging from number theory to probability and statistics.
Not only does it provide you with the huge armory for being able to tackle very tricky mathematical statements, it also has a good introduction to each field with needed theorems and proofs.
Also, author's great sense of humor make the book even more enjoyable to read! It is an absolutely "must have" for anyone, who call himself a mathgeek.

Fantastic resource!
This book covers a phenomenal amount of material on the many uses of mathematical induction: from identities, inequalities and sequences, to number theory, set theory, topology, logic, graph theory, recursion, algebra, geometry, probability and many more.

I think this book would make a great resource for any student of mathematics, not only for the wealth of problems (and solutions), but also for the chapters on how to write a proof by mathematical induction and as a guide to common proof techniques.The first part also contains fascinating chapters on the history and theory of induction that provide a detailed exposition of the set theory underlying the principle of mathematical induction.

Working mathematicians and instructors will also find this book amazingly useful, as both a sourcebook of problems and background when teaching mathematical induction and as a reference for the proofs of a huge number of useful theorems in each of the subjects that are covered.

Not a Handbook
Not a Handbook.
Highly disorganized collection of problems and solutions.
Problems scattered through the book.
Solutions in remote pages. ... Read more

Product Description
This classic in the field is a compact introduction to some of the basic topics of mathematical logic. Major changes in this edition include a new section on semantic trees; an expanded chapter on Axiomatic Set Theory; and full coverage of effective computability, where Turing computability is now the central notion and diagrams (flow-charts) are used to construct Turing machines. Recursion theory is covered in more detail, including the s-m-n theorem, the recursion theorem and Rice's Theorem. New sections on register machines and random access machines will be of special interest to computer science students. The proofs of the incompleteness theorems are now based on the Diagonalization Lemma and the text also covers Lob's Theorem and its connections with Godel's Second Theorem. This edition contains many new examples and the notation has been updated throughout. This book should be of interest to introductory courses for students of mathematics, philosophy, computer science and electrical engineering. ... Read more

Customer Reviews (11)

A big mistake
Late in August, the text originally selected for my mathematical logic class became unavailable. On the basis of reviews only, I chose Mendelson's Introduction to Mathematical Logic as the replacement. A disasterous choice. There may be a page without a typo, but I don't expect to find it. The presentation is inconsistent in notation and focus. Concepts are confused and more difficult than they should be. Definitions are not wisely selected. This book reads like something that has been patched for four decades (since 1964). On the positive side it contains interesting supporting material and will be a valuable private source of ideas to the lecturer. Be sure to read sections from chapter 2 and 3 before selecting this as a text.

A must have....
This is a very useful and must have book for every graduate student in logic.Theory covers many fields(logic and computability) and has a lot of exercises (and also solutions to the tough ones)!!!

twisted pants unleashed on men
This is one of the more popular introductory textbooks on mathematical logic, with Enderton's being its biggest competitor. I prefer Mendelson's for its breadth of material and the choice of proofs he uses, which are generally the most intuitive (e.g. Kalmar's for the completeness of the propositional calculus). This is not to say that they are always constructive, as they many of them are in the older texts (e.g. Kleene, Introduction to Metamathemaitcs).

The exercises are thoughtfully chosen. There's a good range of difficulty and a good portion of the answers can be found in the back. Difficult questions are indicated to the reader.

Out of all the mathematical logic texts I have (which are quite a few in number), this is the most oft-referred-to.

Wonderful at the second glance.
Mendelson's Introduction to Mathematical Logic was the textbook for a logic-course I took a couple of years ago. At the time I did not like the book at all. It seemed too difficult and so typographically ugly that I thought I would never use it. Things have changed though. Now, I keep it close at hand on my desk and use it almost every day. Technical questions that used to require a trip to the library and several different books to answer, can usually be resolved by a look in Mendelson's book. It's wonderfully rich and clear! I still don't find everything easy but that's because the material isn't easy and so not something Mendelson can be blamed for. I do find the typography ugly and at times annoying, but that's a small price to pay for a presentation as rigorous and detailed as Mendelson's.
So in summary: it's not the ideal book for the complete newcomer, but once you get past the initial hurdle it's a must read.

Best reference in first step math logic
Mendelson reaches an optimal point between the concision of the expert reference, and the wideness requested to a introductory text. Not in vain it has been the text forced in the universities during forty years.
Nevertheless, I believe to have found an error in the demonstration that does of the theorem of the completeness of the Predicate calculus, in the part in which it tries to demonstrate that all logical truth is
a theorem of the system.
[...] ... Read more

Product Description
Thisintroduction to rigorous mathematical logic is simple enough in both presentation and context for students of a wide range of ages and abilities. Starting with symbolizing sentences and sentential connectives, it proceeds to the rules of logical inference and sentential derivation, examines the concepts of truth and validity, and presents a series of truth tables. Subsequent topics include terms, predicates, and universal quantifiers; universal specification and laws of identity; axioms for addition; and universal generalization. Throughout the book, the authors emphasize the pervasive and important problem of translating English sentences into logical or mathematical symbolism. 1964 edition. Index.
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Customer Reviews (2)

The best book for self teaching logic
I own Symbolic Logic by Virginia Klenk, The Laws of Thought by George Boole, Logic Sets and Recursion by Causey and several discrete math texts including Rosens. Each one of them offers you some insight into logic; however, none of them offer the facility to learn logic like this book does. It is even superior as a first text to Suppes other Introduction to Logic which I also own. I highly recommend it to everyone. I wish I would have found this text 20 years ago it really changed the way I do logic.

Good coverage, but lack of solutions weakens it.
Logic is one of the foundations of mathematics, making quality textbooks essential. This is one of the better ones, with descriptions followed by a large number of exercises. The basic strategy is to present the material in small sections, most of which are two pages or less in length. One unusual aspect of the book is that the authors chose to wait until chapter four to present truth tables. Traditionally the first topic in logic textbooks, leaving truth tables until later forces the reader to learn the operational meanings of the connectives. I approve of this pedagogical technique, undue reliance on truth tables can lead to the masking of some of the concepts of logic. Propositions and predicates are covered, with more ink spent on propositions that predicates and a simple set of axioms for addition is also presented
One negative point is that no solutions to the exercises are included. In my opinion, any book without solutions to some of the exercises is of reduced value as a textbook. Students work problems on their own and it is very important for them to get immediate feedback. This also reduces its' value as a text for self study.
I can recommend this book, the quality of the explanations and the number and detail of the exercises make that easy. However, the lack of solutions means that it can only receive my second highest recommendation.

Published in Journal of Recreational Mathematics, reprinted with permission. ... Read more

Product Description
Designed to provide a firm foundation in mathematical logic, this book provides an elementary yet rigorous textbook for both graduate study and for applications of logic, such as logic programming and formal specification and verification. The text supplies the mathematics often treated sketchily in introductory computer science books, while using the simplest techniques rather than the most general used in mathematical books. ... Read more

Product Description
This user-friendly introduction to the key concepts of mathematical logic focuses on concepts that are used by mathematicians in every branch of the subject. Using an assessible, conversational style, it approaches the subject mathematically (with precise statements of theorems and correct proofs), exposing readers to the strength and power of mathematics, as well as its limitations, as they work through challenging and technical results. KEY TOPICS: Structures and Languages. Deductions. Comnpleteness and Compactness. Incompleteness--Groundwork. The Incompleteness Theorems. Set Theory. : For readers in mathematics or related fields who want to learn about the key concepts and main results of mathematical logic that are central to the understanding of mathematics as a whole. ... Read more

Customer Reviews (2)

Why is this out of print?
The impossible goal of this text is to start from scratch and then cover both incompleteness theorems in a single semester, and under his presentation it would almost be manageable.This is by far the best written text on predicate calculus I have read.Kaye and Goldrei can't really compare, as they contain less material and what they do cover isn't done quite as well.Enderton on the other hand covers more than Leary, but is much more dense and would not serve as well as an introduction.
The main drawback of the book is how much effort the author put into making it fit into a single semester.There is a lot of fascinating material that could have been covered in greater depth than is done.It is worth noting that he almost completely skips over propositional calculus, so if you find yourself struggling at the beginning of the book you may want to read up on that subject in another text (the first half of Goldrei would do nicely).Also the section on the second incompleteness theorem is extremely rushed; some of the properties of peano arithmetic used for the proof are not proven.
Still, it's better than the other options I've seen.You would think with all the mediocre mathematics texts Dover picks up they would have found this gem.

Most Accessible Undergraduate Text Covering Incompleteness
I have used this text in both graduate and undergraduate courses as well as tutorials and independent studies.It is the best text for a one semester course that introduces formal logic and has as its goal the Incompleteness Theorems of Godel.Students have reported it to be very readable and the array of exercises is excellent.Moreover, the author is a really nice fellow. ... Read more

Product Description
Well-written undergraduate-level introduction begins with symbolic logic and set theory, followed by presentation of statement calculus and predicate calculus. First-order theories are discussed in some detail, with special emphasis on number theory. After a discussion of truth and models, the completeness theorem is proved. "...an excellent text."—Mathematical Reviews. Exercises. Bibliography.
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Customer Reviews (2)

I sat at the master's feet
I took the Mathematical Logic class from Dr. Margaris at Rhodes College in the mid-1970s.He used the first edition of this text to teach the class, and it was an outstanding class.While others were perplexed, I found his presentation of the material to be very clear.He also participated in our senior seminar in computability that utilized the algebraic formulation of logic he presents in this text to get at Godel's theorem, etc.

First mathematical logic book I truly enjoyed!
I studied the 1st edition (Blaisdell) of this book from cover to cover in 1985. It was great. The updated Dover edition is just as enjoyable. If you have any passion for the subject at all, get a copy of this book. You can'tbeat it for an introduction. ... Read more

Product Description
Mathematical logic grew out of philosophical questions regarding the foundations of mathematics, but logic has now outgrown its philosophical roots, and has become an integral part of mathematics in general.This book is designed for students who plan to specialize in logic, as wellas for those who are interested in the applications of logic to other areas of mathematics. Used as a text, it could form the basis of a beginninggraduate-level course.There are three main chapters:Set Theory, Model Theory, and Recursion Theory.The Set Theory chapter describes the set-theoretic foundations of all of mathematics, based on the ZFC axioms.It also covers technical results about the Axiom of Choice, well-orderings, and the theory of uncountable cardinals.The Model Theory chapter discusses predicate logic and formal proofs, and covers the Completeness, Compactness, and Löwenheim-Skolem Theorems,elementary submodels, model completeness, and applications to algebra.This chapter also continues the foundational issues begun in the set theory chapter.Mathematics can now be viewed as formal proofs from ZFC.Also, model theory leads to models of set theory.This includes a discussion of absoluteness, and an analysis of models such as H(κ) and R(γ).The Recursion Theory chapter develops some basic facts about computable functions, and uses them to prove a number of results of foundational importance; in particular, Church's theorem on the undecidability of logical consequence, the incompleteness theorems of Gödel, and Tarski's theorem on the non-definability of truth. ... Read more

Customer Reviews (1)

Kunen tales on indecidability of structures
This is the second book written by Kunen I have read. In his book Set Theory An Introduction To Independence Proofs (Studies in Logic and the Foundations of Mathematics), he gives a brilliant exposition of the basic techniques to proof statements to be consistent with Zermelo-Fraenkel Set Theory.

The reason I bought this book is the same reason I bought the first one: I know nothing about the subject and the book looks like a promising way to learn the basic stuff. I am a topology Ph.D. and have found that it is very common to have many Set Theoretic and Model Theoretic tools in the area I want to specialize in. Further, there is no one who will help me learn this at college, so I decided I had to study at least the basic stuff by myself in order to learn what I needed. I think both of Kunen's books are great to learn by oneself. However, many undergraduates have told me that Kunen's first book is hard to read so you must take caution at my words.

One of the most amusing features of Kunen's book is that the way he explains things is really entertaining. This feature is also present in this book in various places, for example: when he explains the notion of Cardinality (p. 17) by talking about ducks and pigs, when he compares the philosophy of mathematics with religion (p. 190) or my favourite quote from the part where he explains the philosophy of the Church-Turing thesis (p. 200):

"Likewise, the ultimate nature of human intelligence and insight is not understood, so it is conceivable that a human, via some insight, perhaps in contact with God or the spirit world, could reliably decide membership in some non-$\Delta_1$-set."

I must of course make clear that this book is serious and even though this quote seems taken from a pseudoscience book, it has philosophycal roots in the reductionist philosophy he then explains.

Now let me give my opinion about each of the chapters.

Set Theory: In this chapter, he develops the basic facts about ZFC set theory in a very detailed fashion, he practically gives all the missing details from his first book. He also talks about some simple models like $R(\gamma)$ and $H(\kappa)$ which will play an important role in chapter 2. One of the most interesting exercises in the book is I.14.14 where he gives you a hint on how to construct a computable (not yet defined) enumeration of hereditarily finite sets. An excelent summary on the subject.

Proof Theory and Model Theory: This chapter is almost 100 pages long but is worth reading. Even though the details of proof theory seem annoyingly tedious, this is the first book in which I venture to read them all. I was really worth it and I finally feel myself "complete" by knowing what the completeness theorem says and how to prove it. Of course, the most interesting parts (in my opinion) are the ones that come next, that are model theoretic. Elementary submodels were practically the reason I wanted to read the book and I think they were neatly explained. After this, he starts talking about absoluteness in models of set theory, something which is in more detail in his previous book. He also manages to introduce the notion of $\Delta_0$ (which for the first time I understood) and $\Delta_1$, which will be used in chapter 4. A really complete survey.

Philosophy: There's not much to say here because the content is purely philosophycal (and really interesting) but non-mathematical, you should read it by yourself.

Recursion Theory: This chapter was kind of boring for me. Although I finally came to understanding the Church-Turing thesis: "A subset of HF is computable if and only if it is $\Delta_1$ in HF", the arguments start becoming hard to follow. Many details are missing which made me sometimes despair on what the arguments really meant. By this part of the book, you have learned that many things you take for granted in the general setting are important details in logic, so it feels bad when Kunen starts skipping details. Of course, the author says himself he will skip more and more details as he advances, so it may be my inexperience in this kind of arguments. Important theorems that I have managed somehow to understand (at an informal level) are Godel's incompleteness theorems and Tarski's undefinability of truth. One thing I can say I learned is that the absoluteness of hereditarily finite sets is everything that matters for this theorems (or so I understood). A good excuse for the apparent lack of "Kunenness" of this chapter is that he refers to the book Incompleteness in the Land of Sets (Studies in Logic) for a more extensive reading.

The bibliography is extense and not only mathematical. There are links to web pages where you can find ancient texts (like one from Ockham) and computer programs that simulate proof theory. The bibliography is definitely worth reading.

One last detail is that I noticed that Kunen does not talk about Category Theory. I have read elsewhere that there are parts of logic that are studied in the categorical point of view. Plus, Kunen says that Set Theory is the "theory of everything". However, there are mathematicians that have done research on trying to axiomatize mathematics using categories. I even remmember there was some kind of (heated) discussion between Mac Lane and Mathias on whether category theory should replace set theory. Of course I am of the idea that set theory is more fun. However, I think Kunen lacks a discussion about this matters. He does mention categories once "in the language of category theory" in some part of the book to make a notion easy to understand (however, I cannot find it anymore) but that is the only time when he talks about this. Thus, I think what this books lacks is a section on categories.

Conclusion: you will find this book interesting if you are interested in a "fast" reading on foundations of mathematics, the references should guide you to more advanced topics and specialization of the ideas presented, read only if you are mathematically mature (perhaps for grad school) ... Read more

Product Description
This textbook is intended for a one term course whose goal is to ease the transition from lower division calculus courses, to upper level courses in algebra, analysis, number theory and so on. Without such a "bridge course", most instructors in advanced courses feel the need to start their courses with a review of the rudiments of logic, set theory, equivalence relations, and other basic mathematics before getting to the subject at hand. Students need experience in working with abstract ideas at a nontrivial level if they are to achieve what we call "mathematical maturity", in other words, to develop an ability to understand and create mathematical proofs. Part of this transition involves learning to use the language of mathematics. This text spends a good deal of time exploring the judicious use of notation and terminology, and in guiding students to write up their solutions in clear and efficient language. Because this is an introductory text, the author makes every effort to give students a broad view of the subject, including a wide range of examples and imagery to motivate the material and to enhance the underlying intuitions. The exercise sets range from routine exercises, to more thoughtful and challenges ones." ... Read more

Customer Reviews (1)

Good intro for beginning
Before this book, I was still wondering through a maze of nonsensical calculus textbooks and couldn't get myself out of that trap. I think this book is a good start for going further in math. This book is clearly written and makes me think clearly, relative to other too concise books. Now I know what's going on when proofs are present in books. It's worth a buy. Also, selected exericise solutions are just enough to get the hint on how to approach a problem and also are motivating enough to solve a problem on my own. This book would probably better as a textbook in a course taught by a good instructor, but self-study is probably possible. You get to peak inside for table of contents and few pages, so why not have a look? ... Read more

Product Description
While many books have been written about Bertrand Russell's philosophy and some on his logic, I. Grattan-Guinness has written the first comprehensive history of the mathematical background, content, and impact of the mathematical logic and philosophy of mathematics that Russell developed with A. N. Whitehead in their Principia mathematica (1910-1913).

This definitive history of a critical period in mathematics includes detailed accounts of the two principal influences upon Russell around 1900: the set theory of Cantor and the mathematical logic of Peano and his followers. Substantial surveys are provided of many related topics and figures of the late nineteenth century: the foundations of mathematical analysis under Weierstrass; the creation of algebraic logic by De Morgan, Boole, Peirce, Schröder, and Jevons; the contributions of Dedekind and Frege; the phenomenology of Husserl; and the proof theory of Hilbert. The many-sided story of the reception is recorded up to 1940, including the rise of logic in Poland and the impact on Vienna Circle philosophers Carnap and Gödel. A strong American theme runs though the story, beginning with the mathematician E. H. Moore and the philosopher Josiah Royce, and stretching through the emergence of Church and Quine, and the 1930s immigration of Carnap and GödeI.

Grattan-Guinness draws on around fifty manuscript collections, including the Russell Archives, as well as many original reviews. The bibliography comprises around 1,900 items, bringing to light a wealth of primary materials.

Written for mathematicians, logicians, historians, and philosophers--especially those interested in the historical interaction between these disciplines--this authoritative account tells an important story from its most neglected point of view. Whitehead and Russell hoped to show that (much of) mathematics was expressible within their logic; they failed in various ways, but no definitive alternative position emerged then or since. ... Read more

Customer Reviews (3)

He hasn't yet given up on the 'paradoxes'
You'll notice that although GG still lists Cantor's "paradox" in his index, in the text he doesn't quite bring himself to say that there is such a thing.Why not?Because he has read Garciadiego's BERTRAND RUSSELL AND THE ORIGINS OF THE SET-THEORETIC 'PARADOXES,' which shows quite clearly that there is no such thing as Cantor's paradox, or the Burali-Forti paradox, or Russell's for that matter.The so-called "set-theoretic paradoxes" were for the most part inventions of Russell, and not a single one from the period, comes out as anything but a meaningless formulation.

The problem this creates for GG is that so-called "set theory" is nonsense, and not much worth wasting time on.Apart from Cantor's own pathetic inability ever to define what a set is, the history is a farce of the blind leading the blind--trying to "avoid" formulations which are not paradoxes or anything else.This is worth writing about?Worth listing 1900 items in a bibliography, about?It's sad, but a good study in how wastes of time and resources occur.

So GG goes ahead and talks about these "paradoxes" as if they really were such, and about people's "responses" to them as if there was anything to respond to.GG still hasn't quite weaned himself from the "paradoxes," although he cites Garciadiego and should have known better.The gist of the book is that the "paradoxes" which led to Godel's argument (and those of the Intuitionists, the Logicists and Formalists as well as their successors), are not paradoxes at all--they are meaningless formulations. This undermines most, if not all, of twentieth-century mathematics, and in particular destroys Godel's very sloppy argument.

Garciadiego cites Richard's own formulation of this "contradiction" (Richard's term) in a letter to Poincare. He also cites Richard reducing the argument to meaninglessness. What does this have to do with Godel? It's simple. For Godel, Richard's "paradox" means that truth in number theory cannot be defined in number theory. On this basis, he distinguishes truth from provability. He combines his idea of Richard's "paradox" with the idea that provability in number theory can be defined in number theory. He arrives at the conclusion that if all the provable formulae are true, there must be some true but unprovable formulae. However, since Richard's "paradox" is without meaning, since it has no logical content whatsoever and is simply letters pulled out of a bag, there is no basis in Godel's argument for distinguishing truth from provability. It turns out that there is no logical content in the idea that if all the provable formulae are true, there must be some true but unprovable formulae.

People are having a hard time getting over the notion that Godel didn't do his homework, and has nothing to say, but really you have to grow up.Get over it.The problem is that Godel was a terrible scholar, and did not apply himself sufficiently to the details of the development of set theory.

Garciadiego's book has implications for all twentieth-century mathematics. Here are just a few examples of horrendous errors which explain a lot about why mathematics today is regarded as the province of clowns.For example, Brouwer based the idea of an infinite ordinal number on the idea that Cantor had proved well-ordering of the ordinal numbers. But not only did Cantor never prove this, but also, he never said he had done so, and never used the term infinite ordinal number.Turing never examines the "paradoxes" in order to determine whether they are simply meaningless formulations. Thus, in an attempt to "prove that there is no general method for determining about a formula whether it is an ordinal formula, we use an argument akin to
that leading the Burali-Forti paradox, but the emphasis and the conclusion are different." As Garciadiego reveals, there is no Burali-Forti paradox. In the context of an attempt to prove the Trichotomy Law, Burali-Forti tried "to prove by reductio ad absurdum that the hypothesis [involved in his own argument] was false and this method required supposing the hypothesis true and arriving at a contradiction. The employment of the hypothesis, as an initial premise, generated the inconsistency. But once the hypothesis is seen to imply a contradiction it is thereby proved to be false." Turing purported to distinguish completeness from decidability, not realizing that the absence of a contradiction made the distinction insupportable. Turing claimed justification for his definition of a computable number in a "direct appeal to intuition." This is not a cavalier reference to intuitionism. In fact, it provides the basis for Turing's use of binary numbers. This base2 system is a metaphor which traces itself back through Turing's own bifurcation of the mathematical process to Brouwer's own bifurcation of the operation of the human mind ("the connected and the separate, the continuous and the discrete")-all in an attempt to "avoid" the "paradoxes." Brouwer's complaint is that the "paradoxes" deprive us of distinctions. Turing's entire apparatus of calculability is designed to "restore" "distinctions." The binary number, and Turing's later restoration of a modified form of completeness in the form of decidability, are assertions by way of distinction.

However, there is no problem against which to assert it. Operating on these numbers with "finite means" (Turing's definition of a computable number) merely takes us back to Richard's response to his own contradiction and no recourse to intuition can rescue us from the consequences of that response: the computable number only has meaning if finite means are defined in totality, and this can only be done with infinitely
many words.

It turns out that Richard's discussion of his own "contradiction" serves as a useful template for evaluating, and then discarding, putative "paradoxes." There is much work to be done in that field. But as for twentieth-century mathematics, to the extent it is based on already-discredited "paradoxes," it loses any logical content. This, unfortunately, is certainly true of Godel's argument. It is even more glaringly true in the case of now-trivial figures such as Carnap and Tarski. After Garciadiego, these names go from the headlines to the footnotes. In general, Garciadiego's book is an indictment of twentieth-century math academics.

The real story is the insidious advance of intutionist-style mathematics through the other disciplines during the twentieth-century.This idea that mathematics is a "natural" part of human experience--an idea nowhere tested or even rigourously used as a hypothesis--provides a crutch for a lot of investigators who were unfamiliar with contemporary mathematics but needed a mathematical expression for their ideas.Thus Sraffa, the economist, whose work came to be expressed in intuitionist math, and Kimura, the biologist, who got his intuitionist math from Malecot, a protege of Boel.Einstein also fell victim to it.Note this passage for his book RELATIVITY:

Up to now our considerations have been referred to a particular body of reference, which we have styled a 'railway embankment.' We suppose a very long train travelling along the rails with the constant velocity v and in the direction indicated....People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system); they regard all events in reference to the train. Then every event which takes place along the line also takes place at a particular point of the train. Also the definition of simultaneity can be given relative to the train in exactly the same way as with respect to the embankment. As a natural consequence, however, the following question arises: Are two events (e.g. the two strokes of lightning A and B) which are simultaneous with reference to the railway embankment also simultaneous relatively to the train? We shall show directly that the answer must be in the negative. When we say that the lightning strokes A and B are simultaneous with respect to the embankment, we mean: the rays of light emitted at the places A and B, where the lightning occurs, meet each other at the mid-point M of the length A -> B of the embankment. But the events A and B also correspond to positions A and B on the train. Let M' be the mid-point of the distance A -> B on the travelling train. Just when the flashes (as judged from the embankment) of lightning occur, this point M' naturally coincides with the point M, but it moves towards the right in the diagram with the velocity v of the train.

This translation is accurate (the French and Italian are not). Einstein really does say "fallt zwar...zusammen." That is, he says that one point "naturally" coincides with another. The "naturally" reveals the intuitionist expression of the concept, for it reflects the belief that the formulations of geometry do not express facts.

Obviously, the logical problem with it is that, regardless of what Einstein may "feel" about mathematical expressions, nowhere in Einstein's writings--either in the 1905 papers or after--is any meaning assigned to 'naturally.' The failure to do so, destroys the idea, and it is easy to see why. If we retain the concept without meaning there is no logical basis on which to proceed beyond it. If we eliminate it, we wind up with a contradiction: the two assumed coordinate systems collapse into one. What is more, when we place this train experiment next to the various other thought experiments, we see that they are simply translations of the same problem into other terms, just as the false 'paradoxes' turn out to be subject to the same problem Richard indicated (reference to an infinite domain which destroys the meaning). In special relativity, natural coincidence can only be defined by infinitely many words. So the distinction collapses.

Thus, GG's account is merely the beginning of an attempt to get out from under bogus "paradoxes" and the ridiculous intuitionist-style mathematics designed to "avoid" them.

Not what you might expect
I hoped that reading this book would give me a better understanding, in an historical context, of the issues involved in the controversies about the foundations of mathematics a century ago.I found this book fairly interesting, and it was a quick read, but it seems to be written for those who already have an essentially complete understanding of those issues, since the ideas themselves were addressed only tangentially.The focus of the book is much more on:who published what paper when, to what journal did he send it, who was the editor of the journal, who refereed the paper, to whom were offprints sent, in what archives can the manuscript be found, who read whose paper when, who met whom at what conference, who used what notation in writing which paper.This is very much a documentary history, and historians of mathematics will probably love it, but I am probably not the only mathematician who will not find this book completely satisfying.

The best book of its kind in existence
I have owned a copy for over a year, and not a week goes by in which I do not consult it. The 50pp bibliography alone is worth the price.

Modern foundational mathematics emerged around 1840, with the work of Boole, De Morgan, and Bolzano. In the 1870s, Cantor, Peirce, Frege made their appearance. In the 1890s, Peano, Hilbert, Russell and Whitehead came on line. The author is an authority on Cantor, Peano, the rise of set theory, on Russell, and Principia Mathematica, and these are covered in great detail. The era closes in the 1930s, with the negative metatheorems of Goedel and Church, and the rise of Quine. All this makes for an exciting human adventure, and this book is the best narrative we have of that adventure.

The book is a gold mine of details little known to most philosophers and to nearly all mathematicians. Here I learned that Husserl was trained as a mathematician, and that much of foundational mathematics can be seen as a meditation on bits of Kant. I should grant that IGG is not fair to everyone: Skolem, for instance, is slighted. Also, this book is far from definitive about Polish logic, which deserves a book of its own. ... Read more

Product Description
Fuzzy logic has become an important tool for a number of different applications ranging from the control of engineering systems to artificial intelligence. In this concise introduction, the author presents a succinct guide to the basic ideas of fuzzy logic, fuzzy sets, fuzzy relations, and fuzzy reasoning, and shows how they may be applied. The book culminates in a chapter which describes fuzzy logic control: the design of intelligent control systems using fuzzy if-then rules which make use of human knowledge and experience to behave in a manner similar to a human controller. Throughout, the level of mathematical knowledge required is kept basic and the concepts are illustrated with numerous diagrams to aid in comprehension. As a result, all those curious to know more about fuzzy concepts and their real-world application will find this a good place to start. ... Read more

Customer Reviews (8)

Excellent book!!!
This book is an EXCELLENT book for teaching this subject.
Students can finally understand what is fuzzy logic and its notation.
It is short and sweet, with great examples, great pictures, great explanations.
Not difficult to read at all.
Actually, quite enjoyable to read for students.

It just has several typos, but thet are easy to detect as typos right away.

A perfect intorduction to fuzzy logic
After spending some time trying to grasp the concepts of fuzzy logic and fuzzy sets, I found this book. This is THE book to start if you want to get a quick introduction to what fuzzy logic is, and how to use fuzzy sets as a tool. I highly recommend this book if you are having problems following other books in fuzzy logic. There is an example for every concept that is introduced, making it really easy to follow and understand

Good introduction, with some errors
This 136 page book provides a brief introduction to fuzzy logic and applications.However, I have to disagree with the comment that symbols are always defined when used, as many are not.

Also, I think the book may have suffered in translation, as there are quite a few errors, especially in the translation of formulas.For example, on page 27 is an incontrovertable mangling of De Morgan's laws.In other places, symbols are left out, subscripts and superscripts are inexplicably moved around, and shading for graphs and tables is mentioned many places in the text but mysteriously not present in the graphs and tables referred to.

Fortunately, Tanaka goes over the same topic from multiple prespectives, in most cases allowing the reader to figure out what is going on.As an introduction, this book would definitly have benefited from a table of symbols.However, overall, a good introduction to (or review of) the topic.

Fast entry to notation
I purchased this book to gain enough information to read a technical paper.Fuzzy logic is new to my industry (petroleum) and my 1970's education did not provide any background.The book took less than an evening to absorb and provided more than enough understanding of the notation and basic operations that I was able to read my paper and start building an interest in deploying Fuzzy logic in my daily work.Other books are needed to fully apply the methodology.However, sufficient demonstration of basic fuzzy arithmetic was provided to know that 2 times 3 divided by 2 is not necessarily 3.

I shared the book with a mathematically oriented associate and she had similar experiances.

Overall, a great introduction with just enough information for a cursory review and enough detail to help determine need or interest for a more detailed presentation.

Breezy Intro to Fuzzy Logic
I was impressed by the fact that most symobology used in the book is defined during its first use.There are a few symbols like 'sup' on page 38 that I'm still unclear about.

There are five basic chapters in the book:
1 Introduction
2 Fuzzy Set Theory
3 Fuzzy Relations
4 Fuzzy Reasoning
5 Fuzzy Logic Control

Chapter 1 is a brief two page intro to the concept.The chapter on Fuzzy Set Theory gets into the basics. I've read the descriptions of Fuzzy Logic in MathLab's Fuzzy Logic module and so was prepared for most what is in this chapter.As such, I'm still somewhat unclear as how Cartesion Products and Extension Principles are applicable to the whole concept of Fuzzy Logic.

In the chapter on Fuzzy Relations, further use of extensions is used along with the properties of composition.Simple matrix math is used in some cases to arrive at results in some of the examples.

The fourth chapter, which is about Fuzzy Reasoning, includes reasoning based upon Mamdani's Direct Method, Takagi & Sugeno's Fuzzy Modelling, and the Simplified Method.It is in this chapter where the earlier mechanisms of composition are utilizied.Defuzzification, which is final step of any fuzzy logic process, is lightly described with a brief reference to the standard centroid calculation.

The final chapter is light on formulas, and offers up a high level description of the superiority of fuzzy logic over PID controllers, and how the former can help the latter obtain better control in some situations.

After having taken a first read of this book, I'll have to go through it again to see if I can better relate Fuzzy Relations to the remainder of the book.In addition, now that I've got a better grasp on fuzzy symbology, I believe I'm ready to move on to the more heavy duty books of the subject area. ... Read more

Product Description
This text aims to unify mathematical logic and symbolic logic, and outlines how mathematical logic emerged from symbolic logic. Derivations are extended to encompass mathematical principles. Godel's theorems are covered, including philosophical and historical issues. ... Read more