Book Description
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 (11)

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.

Lots of typesets, and for what purpose?
I do not recommend it as an introduction to mathematical logic.
I found the material to be insufficiently motivated. Unfortunately, the authors take some of their variables from an old english alphabet, which ruins the aesthetics.I am sure that some elegant and glorious principles are expounded here.But it was not written clearly enough for me to see them. ... Read more

Customer Reviews (2)

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

Book 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

Product Description
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 (1)

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

Book Description

Logic is sometimes called the foundation of mathematics: the logician studies the kinds of reasoning used in the individual steps of a proof. Alonzo Church was a pioneer in the field of mathematical logic, whose contributions to number theory and the theories of algorithms and computability laid the theoretical foundations of computer science. His first Princeton book, The Calculi of Lambda-Conversion (1941), established an invaluable tool that computer scientists still use today.

Even beyond the accomplishment of that book, however, his second Princeton book, Introduction to Mathematical Logic, defined its subject for a generation. Originally published in Princeton's Annals of Mathematics Studies series, this book was revised in 1956 and reprinted a third time, in 1996, in the Princeton Landmarks in Mathematics series. Although new results in mathematical logic have been developed and other textbooks have been published, it remains, sixty years later, a basic source for understanding formal logic.

Church was one of the principal founders of the Association for Symbolic Logic; he founded the Journal of Symbolic Logic in 1936 and remained an editor until 1979 At his death in 1995, Church was still regarded as the greatest mathematical logician in the world.

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

Book Description
Epistemic logic has grown from its philosophical beginnings to find diverse applications in computer science as a means of reasoning about the knowledge and belief of agents. This book, based on courses taught at universities and summer schools, provides a broad introduction to the subject; many exercises are included together with their solutions. The authors begin by presenting the necessary apparatus from mathematics and logic, including Kripke semantics and the well-known modal logics K, T, S4 and S5. Then they turn to applications in the contexts of distributed systems and artificial intelligence: topics that are addressed include the notions of common knowledge, distributed knowledge, explicit and implicit belief, the interplays between knowledge and time, and knowledge and action, as well as a graded (or numerical) variant of the epistemic operators. The problem of logical omniscience is also discussed extensively. Halpern and Moses' theory of honest formulae is covered, and a digression is made into the realm of non-monotonic reasoning and preferential entailment. Moore's autoepistemic logic is discussed, together with Levesque's related logic of 'all I know'. Furthermore, it is shown how one can base default and counterfactual reasoning on epistemic logic. ... Read more

Book Description

This introduction to mathematical logic starts with propositional calculus and first-order logic. Topics covered include syntax, semantics, soundness, completeness, independence, normal forms, vertical paths through negation normal formulas, compactness, Smullyan's Unifying Principle, natural deduction, cut-elimination, semantic tableaux, Skolemization, Herbrand's Theorem, unification, duality, interpolation, and definability.

Customer Reviews (1)

used early draft as grad text
I took a great graduate course from Prof. Andrews, way back in the 1970's, where his class lecture notes were titled "To Truth Through Proof", so I assume that was a very very early draft of this book.

If so, this must be a very good book, because his notes were wonderful even back then. ... Read more

Book Description
Here is an introduction to modern logic that differs from others by treating logic from an algebraic perspective. What this means is that notions and results from logic become much easier to understand when seen from a familiar standpoint of algebra. The presentation, written in the engaging and provocative style that is the hallmark of Paul Halmos, from whose course the book is taken, is aimed at a broad audience, students, teachers and amateurs in mathematics, philosophy, computer science, linguistics and engineering; they all have to get to grips with logic at some stage. All that is needed to understand the book is some basic acquaintance with algebra. ... Read more

Customer Reviews (3)

A superb introduction to the glories of Boolean algebra
This book is an undergrad introduction to Boolean algebraic logic and Halmos, who worked hard in the area during the 1950s, is the person to write it. The book includes Halmos's monadic algebra, but remains at the undergrad level because he stops short of his full-blown polyadic algebra (on which, see Halmos's "Algebraic Logic," which AMS keeps in print and is a fine read).

While Halmos does not cover all of first order logic, he does an excellent job of introducing the reader to the great power and depth of Boolean algebra, revealed by Marshall Stone and Tarski in the 1930s, and other Poles in the 1950s. By this I mean Boolean algebra coupled with the notions of filters, ideals, generators, and quotient algebras. The metatheory of the propositional calculus has a very elegant Boolean representation.

Lattice theory is an extremely powerful generalization of Boolean algebra that has not attracted the attention it deserves. If Halmos had written a text on lattice theory, that situation would in all likelihood have ended. Halmos and Givant include an all-too-brief tantalizing chapter on lattices.

If this book has a drawback, it is the relative unsophistication of its first 40 odd pages, an introduction to logic. This is especially disappointing given that Givant is a logician, and an excellent one at that, being a student of Tarski's.

The books main asset is Halmos's lively prose style, unparalleled in modern mathematics. Math PhD students should study this book closely as a superb example of how to exposit mathematics.

Interesting view on logic
In his "automathography" Halmos described his views on logic which he had in the 1960's.He felt that logic, as usually stated, was very un-profound, unrigorous, combinatorial amusement.He felt additionally that logic could be put on a firm algebraic footing through the theory of Boolean rings.At that time he interpreted many things in logic in terms of Boolean rings.This book is, in some sense, the child of these labors.Halmos created this book in his usual easy to read style, and when he said that few prerequisites were assumed, he meant it.I found the (short) book very interesting, but I also found the introductory pages seemed to drag.Perhaps this is because I already know something about logic, but the rest of the book was interesting and self contained.This book was lighter than most logic books I've seen.These books were mainly written by philosophers in some capacity or other, and they never stopped their thick prose.

A Builder of a Solid Foundation in Mathematics
It can be strongly argued that logic is the most ancient of all the mathematical sub-disciplines. When mathematics as we know it was being created so many years ago, it was necessary for the concepts of rigidanalytical reasoning to be developed. Of the three earliest areas, geometrywas born out of the necessity of accurately measuring land plots and largebuildings and number theory was required for sophisticated countingtechniques.Logic, the third area, had no "practical" godfather, otherthan being the foundation for rigorous reasoning in the other two. In theintervening years, so many additional areas of mathematics have beendeveloped, with logic and logical reasoning continuing to be thefundamental building block of them all. Therefore, every mathematicianshould have some exposure to logic, with the simple history lessonautomatically being included. This short, but excellent book fills thatniche.
The title accurately sets the theme for the entire book.Algebra is nothing more than a precise notation in combination with arigorous set of rules of behavior.When logic is approached in that way,it becomes much easier to understand and apply. This is especiallynecessary in the modern world where computing is so ubiquitous.Many areasof mathematics are incorporated into the computer science major, but noneis more widely used than logic. Written at a level that can be comprehendedby anyone in either a computer science or mathematics major, it can be usedas a textbook in any course targeted at these audiences.
The topicscovered are standard although the algebraic approach makes it unique. Onesimple chapter subheading, `Language As An Algebra', succinctlydescribesthe theme. Propositional calculus, Boolean algebra, lattices and predicatecalculus are the main areas examined. While the treatment is short, it isthorough, providing all necessary details for a sound foundation in thesubject. While the word "readable" is sometimes overused in describingbooks, it can be used here without hesitation.
Sometimes neglected asan area of study in their curricula, logic is an essential part of allmathematics and computer training, whether formal or informal. The authorsuse a relatively small number of pages to present an extensive amount ofknowledge in an easily understandable way. I strongly recommend thisbook.

Published in Smarandache Notions Journal reprinted withpermission. ... Read more

Book Description

In Classical Mathematical Logic, Richard L. Epstein relates the systems of mathematical logic to their original motivations to formalize reasoning in mathematics. The book also shows how mathematical logic can be used to formalize particular systems of mathematics. It sets out the formalization not only of arithmetic, but also of group theory, field theory, and linear orderings. These lead to the formalization of the real numbers and Euclidean plane geometry. The scope and limitations of modern logic are made clear in these formalizations.

The book provides detailed explanations of all proofs and the insights behind the proofs, as well as detailed and nontrivial examples and problems. The book has more than 550 exercises. It can be used in advanced undergraduate or graduate courses and for self-study and reference.

Classical Mathematical Logic presents a unified treatment of material that until now has been available only by consulting many different books and research articles, written with various notation systems and axiomatizations.

Amazon.com
This book has been a classic in the former Soviet Union since it was first published in 1956, and it remains just as entertaining today. A master at making math fun for his high school students, Boris Kordemsky loaded this clever collection with a wide variety of math and logic related games and puzzles dealing with magic squares, tricky weights and measures, properties of numbers, mathematical tricks, and more.Number and math game fans are bound to find several new amusements here. Even many of the well-known classics from generations past take on new life with the fresh twists Kordemsky provides.Book Description

Customer Reviews (10)

Funny, challenging, and well written!
I bought this book while working as a gifted children teacher. I liked it so much that I used to keep it in my car and solve riddles whenever I had to wait for someone. It is a great resource for all teachers; children are suddenly made quiet when you present a puzzle to them.

I especially like the stick puzzles, where you can distribute a number of matches to students (by the way, it works with kids, teenagers and adults alike) and give them a puzzle. The advantage of this kind of puzzle is that you can give additional tasks to those fast-solvers; you do not have a story behind it.

The organization of the book is excellent; it is divided by difficulty levels as well as by type of puzzle. For example, you have different levels of geometry problems and of sticks problems.

Great book!

very good
despite of difficulty, I love it because there are various good problems
Thanks you

An absolutely must have for puzzle lovers.

Excellent collection of math puzzles not requiring advanced math - A book for anyone and everyone

With and outstanding collection of 359 mathematical recreations and being lavishly illustrated with more than 400 diagrams and sketches, this book will certainly become a treasure in the personal library of anyone that enjoys solving puzzles.

It's a mammoth puzzle collection, compare with most math teasers and puzzles book available. But what is important is not the quantity, but the quality and charm of the problems presented.

The book is divided in fifteen chapters, as shown:
- Amusing problems.
- Difficult problems.
- Geometry with matches.
- Measure seven times before you cut.
- Skill will find its application everywhere.
- Dominoes and dice.
- Properties of nine.
- With algebra and without it.
- Mathematics with almost no calculations.
- Mathematical games and tricks.
-Divisibility.
- Cross sums and magic squares.
- Numbers curious and serious.
- Numbers ancient but eternally young.
- Solutions.

Everyone will find the type of problems the like most. Often the puzzles are presented in the form of charming stories that provides valuable insights into contemporary Russian life and customs.

Wonderful, charming puzzles teach problem solving.
I was thrilled and surprised to see that this book is still in print. I loved puzzles as a child and spent many hours for fun working the problems in this book (which may have paved the way for my PhD in Computer Science). I fished out my old copy recently to show my 11 year old daughter how I spent my spare hours as a child - *not* playing computer games of dubious educational value. I am sure that working the problems in this book helped increase my problem solving skills -in a different and more general way than I was learning in school.

Of all the puzzle books and puzzles I ever owned, this is the only one I saved.The book has a wide variety of types of puzzles (not all involve numbers). While some are easy, most were challenging.The descriptions were charming, with Russian names of children and towns and quaint puzzle descriptions involving wells, or steam engines or household objects. All in all, a delightful, very educational puzzle collection.

Comprehensive set of math puzzles for various levels
Nice collection of problems which demand some creativity as well as varying degrees of mathematical prowess.Also populated with interesting anedotes regarding mathematicians throughout history.

Although no mathematics beyond the high school level is required, the challenge lies in the ingenious application of even the most rudimentary math and logic necessary to successfully tackle these exercises.The problems range from rather simpleto difficult.Some amount to raw logic riddles requiring little or no math while others offer the opportunity to fine tune one's skills in geometry and algebra. In addition to offering a richvariety of problems which will satisfy the needs of puzzlists at many levels, the editors have made a good point of dividing the problems into categories emphasizing different sets of skills including geometry, algebra, arithmetic operations, spatial visualization and logic.Such a delineation makes it easy evaluate strengths and weaknesses so you can focus on areas of improvement.

Given the long history of this publication, several problems will be familiar to some seasoned puzzle enthusiasts but most will still provide a fresh challenge. ... Read more

Book Description

This book covers all the traditional topicsof discrete mathematics—logic, sets, relations,functions, and graphs—and reflects recent trends incomputer science.Shows how to use discretemathematics and logic for specifying new computerapplications, and how to reason about programs in asystematic way. Describes Prolog, a programminglanguage based on logic, and a section on Miranda,language bad on functions. Features numerous exampleswhich relate the mathematical concepts to problems incomputer science.

Customer Reviews (2)

University Level material
I have used this textbook when delivering a second year University class on this topic and found that the thorough coverage of the topics was appreciated by my students. Even more important to them was the large number of examples that are presented, in detail, throughout the text.

This is a University Level textbook, not a Study Guide, and respects the reader's intellectual maturity by preparing them for subsequent classes. The perception of "density" implies that it is best taken with a liberal dose of classroom instruction - not many students seem to intuitively grasp discrete mathematics and learn the material wholly on their own. I know that I certainly did not when I was an undergrad!

For students who feel that the material is difficult, I always suggest using the library for another point of view. I also recommend the Schaum's Outline for Discrete Mathematics as a companion if the student is having significant difficulty with the concepts.

Obviously, I like the book, so why not a 5? Unfortunately, I don't know of any books that I would grant a 5 - the authors can always do something better :-)

Not for first-time students..
I'm sure this book covers all the stuff that it's meant to and I'm sure that if I was a post-grad in maths this would be a good book to use. However for anyone else this book is way too heavy reading. The authors have made no attempt to keep the material easy to study and understand. The whole book is just a continuous stream of information with the density of a neutron star and where every 5th or so word is a mathematical formula. Then again maybe I'm just biased because I hate the subject matter - I think it's unnecesarily obscure and difficult for a general computer science course.
And look at the price - that's nearly $150 for us Aussies, (although our uni co-op sells it for about A$90) and that doesn't include shipping fees. Don't you hate it the way they jack up the price on these text books because they know that you have to buy it to have any chance of passing the course. ... Read more

Book Description

Customer Reviews (7)

A great introduction to the limits of math
Most causual users of math consider it to be the most unassailable of endeavors.

After, 2 + 2 always has to equal 4 doesn't it?

It turns out that at the periphery of math there are certain inconsistencies that can arise either owing to the use of faulty methods in arriving at a conclusion (what Bunch calls "fallacies") or inconsistencies owing to the limits of math itself (what Bunch calls "paradoxes").

Though one would need recourse to the book itself in order to completely understand what Bunch means by each category, what follows are a couple of examples to help illustrate the kinds of issues this book will treat.

In relation to fallacies, an early example used by Bunch is Aristotle's paradox wherein Aristotle tried to use a deceptively simple experiment to measure the perimeter of two circles.For ease of convenience, let's say he used two coins of different denominations...say a dime and a half dollar.

Obviously, the coins by their size have to have different measures of distance around their perimeters.And yet, according to Aristotle's experiment, they turn out equally.They turn out equally because Aristotle simply placed one on top of the other and rolled them to see which would make a complete turn the earliest.As you may have gleaned they both turned at the same time owing to the particular mathematics of circles.

Bunch's point is that by applying incorrect reasoning Aristotle's "paradoxical" result was simply a fallacy.

In terms of true paradoxes, Bunch discussed Kurt Godel's incompleteness theorem which says that any consistent system will produce so called "formally undecideable propositions."In other words, to the extent that a consistent system produces self referential statements, those statements can defy formal proof.

An oft used English language example is "This sentence is false."Obviously, the sentence is neither be bracketed with all true statements or all false statements owing to its category defying nature.

In turns out that Kurt Godel was able to stand over two millenia of math philosophy on its head by showing that math had its logically limits of proof.

As can be seen from the previous examples, this book is thought provoking even for casual readers who admittedly will have to struggle cracking the hard nutshell of some its more dense arguments.However, those who do so will be richly rewarded for the heightened understanding of the limits of math they have thereby gained in the process.

Informal and engaging
This is a great informal treatment of some of the more notable paradoxes and fallacies of mathematics and mathematical reasoning, old and new. Bunch's prose style is clear and unencumbered and his presentation of each topic - from his easily resolved fallacies and paradoxes of basic algebra and geometry to the deeper and unresolved paradoxes of logic and analysis - is always clean, well-illustrated and engaging.

At a glance, he treats:
The Liar paradox and Godel's Incompleteness theorems
Zeno's and the Sorites paradoxes and the conceptual difficulties associated with the continuum
The existence of irrational magnitudes and some basic philosophical issues associated with existence proofs
The Petersburg paradox
The paradoxes of Infinity and the Formalist and Intuitionist responses to them
The set theoretic paradoxes of Cantor, Russell, and Burali-Forti
The paradoxes of the axiom of choice including the Cantor diagonilisation, Skolem, Hausdorff and Tarski-Banach parodoxes

and a range of thought experiments which highlight the difficulties that may be asociated with applying abstract reasoning to the real world - notably those of the Thompson lamp experiment and Tarski-Banach golden sphere manufacturing plant.

If you want a good popular treatment of the subject matter with a detailed and informal emphasis on the key themes mathematical logic, then this is the book for you. The informal description Godel's first Incompleteness theorem is excellent, as is the discussion of the paradoxes of self reference as they appear in set theory and logic. As such, I would recommend it as excellent recreational reading for anyone with a budding interest in mathematical logic, whether they be math graduates or high-school students.

This is a Great Book for Math Fans
This is a great book for people who love mathematics, including: recreational math enthusiasts, math teachers, professors and other university level math instructors, curious and self-motivated students, etc. This book provides numerous examples of how seemingly logical steps can lead to mathematically fallacious results. The level of math ranges from advanced high school to college level math, but the level is not really important ... what is important is the insights one can get from looking at common mathematical mistakes.

This book may also be of interest to neuroscientists, cognitive scientists, and psychologists who are interested in how human beings learn and apply mathematics. On a somewhat related note, I have noticed that (for some strange reason) this book has attracted a set of rather bizarre reviewers (see below). Please ignore them and buy this inexpensive and insightful book on math.

Zeno and set theory
It is the paradoxes that keep us honest in mathematics. Tarski with Banach found a basic flaw in the axiom of choice in set theory. Zeno has puzzled children for two thousand years... Time travel paradoxes are the modern "new" problem of tacyonic loops and the Hawking conjecture. Without examples of critical thinking doctrine rules and men become fools!

Interesting
I would recommend this book to anyone interested in Mathematics.The fallacies are interesting, including the author's.For example, on page 94 regarding Oscar Wilde's epigram : "The only way to get rid oftemptation is to yield to it".Mr. Bunch suggests this to be afallacy due to the key word "only", and offers an example such assuicide to show "only" to be invalid.But would not suicide be atemptation as well?Or for that manner, anything one would try? ... Read more

Customer Reviews (8)

Extremely satisfying brain-meltdown
If you're a fan of logic & puzzles, this is definitely worth picking up.It starts simple and works its way up to nearly impossible, so there should be something on everyone's level.The book is kind of a narrative, but its main purpose is to display and explain more and more advanced logic.

I first found this book when I was 15 or so among my Uncle's things when I was visiting.I stayed up half the night working through the puzzles, but after about halfway through the book, I was completely lost. Years later, after studying symbolic logic in college, I go back and the book gives me new pleasure, though I still can't get through 'all' of it - not quite ;PIts somewhat of a Logic Quest, really.

If you enjoy a challenge, definitely pick this one up.

Fun
Great book of puzzles. Perfect for the whole family. It begins with simple yet confusing conundrums and progresses onto complex and more confusing ones! Great way to exercise those little gray cells.

Intellectually stimulating
This is a fine book with independent chapters that can be dipped into when you want some good intellectual company. The first few puzzles got me hooked. Smullyan is very good at giving a new twist to old puzzles.

This is a workout for the brain!
The author starts out this book of logic puzzles with fairly easy stuff and moves the reader along so gradually that you are able to figure out solutions you never thought you could. But it gets REALLY complex toward the end and in fact lost me part of the time. So don't get too frustrated if you can't figure them all out. If you like logic puzzles, you don't want to miss this one.

An Eternal Testament to the Wonders of Logic and Mathematics
Dr. Raymond Smullyan once again spins a yarn, in which professional logic and tantalizing puzzles blend seemlessly into his tasteful story-line, taking us from King X and his logical/macabre sense of humour, to theinsane assylums of Doctor Tarr and Professor Feather, then onward toTransylvanyia and the Mysterious Islands, finally to stand along-side Craigas he faces his most challenging puzzle yet in the Mystery of the MonteCarlo Lock (which I am still solving).A must-read. ... Read more

Book Description
Elementary set theory accustoms the students to mathematical abstraction, includes the standard constructions of relations, functions, and orderings, and leads to a discussion of the various orders of infinity. The material on logic covers not only the standard statement logic and first-order predicate logic but includes an introduction to formal systems, axiomatization, and model theory. The section on algebra is presented with an emphasis on lattices as well as Boolean and Heyting algebras. Background for recent research in natural language semantics includes sections on lambda-abstraction and generalized quantifiers. Chapters on automata theory and formal languages contain a discussion of languages between context-free and context-sensitive and form the background for much current work in syntactic theory and computational linguistics. The many exercises not only reinforce basic skills but offer an entry to linguistic applications of mathematical concepts. For upper-level undergraduate students and graduate students in theoretical linguistics, computer-science students with interests in computational linguistics, logic programming and artificial intelligence, mathematicians and logicians with interests in linguistics and the semantics of natural language. ... Read more

Customer Reviews (2)

Comprehensive
This book is well written and detailed.I found it particularly useful for my semantics course.It covers the necessary logic one need to do semantics.It also discuses type theory and the lambda calculus. This book is a great complement to any semantics text.

Not Just for Linguists
Mathematical Methods in Linquistics is far more about mathematical methods than about linguistics, although in many places linquistics is used as a source of examples.

Instead it covers such mathematical topics as sets (including infinite sets), relations, a good deal of mathematical logic,
automata (up to turing machines), the lambda calculus, lattices and more.

This would be an excellent book for an advanced undergraduate or graduate student in either mathematics or computer science to use
either as a review text, or as a study guide for further investigation. ... Read more

Book Description
Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course.It is a major element in theoretical computer science and has undergone a huge revival with the every- growing importance of computer science.This text is based on a course to undergraduates and provides a clear and accessible introduction to mathematical logic.The concept of model provides the underlying theme, giving the text a theoretical coherence whilst still covering a wide area of logic.The foundations having been laid in Part I, this book starts with recursion theory, a topic essential for the complete scientist.Then follows Godel's incompleteness theorems and axiomatic set theory.Chapter 8 provides an introduction to model theory.There are examples throughout each section, and varied selection of exercises at the end.Answers to the exercises are given in the appendix. ... Read more

Book Description

Customer Reviews (3)

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

Book Description
Logic forms the basis of mathematics, and is hence a fundamental part of any mathematics course. In particular, it is a major element in theoretical computer science and has undergone a huge revival with the explosion of interest in computers and computer science.This book provides students with a clear and accessible introduction to this important subject. The concept of model underlies the whole book, giving the text a theoretical coherence whilst still covering a wide area of logic. ... Read more

Customer Reviews (1)

OK but Hard
You'll find this very hard unless you are a competent math major at one of the better universities. Similar to Elliot Mendelson's text, but not quite as good. Good chapter on Boolean algebra as a
piece of pure math; Halmos and Givant is gentler, though.

Interesting topic covered: the resolution so dear to the AI crowd. Unlike most mathematicians, Cori and Lascar have time for
the way computer scientists think. At the same time, this book does not cover tableau methods (see Smullyan), natural deduction, Genzen's ideas, and so on. For pure logic at the advanced undergrad level, you're better off with Bostock.

Haven't seen Part II, so cannot comment on the treatment of set theory. This is something Mendelson and Machover already do well. ... Read more

Book Description

Book Description

Customer Reviews (1)

Still an interesting read....
Those interested in mathematical logic will appreciate this book written by one of the main contributors to the field in the twentieth century. The technique of "currying" in higher order logic is named after the author, wherein unary functions can be used to emulate functions with many parameters. The book was first published in 1963, reprinted in 1977, and so is not a up-to-date treatment of mathematical logic, but it could still be used as an historical supplement to a course in this subject. The reader should be aware though the terminology employed by the author is very idiosyncratic and therefore it may not reflect what is currently used in the literature.

The first chapter of the book could be considered an introduction to the philosophy of logic and mathematics. The author though views "philosophical logic" as the study of the principles of valid reasoning, and this is to be distinguished from "mathematical logic", wherein mathematical systems are constructed to study (formally) the principles of valid reasoning. One can also according to the author view logic as a theory in itself, and many "models" of it can be studied, in much the same way as many different models of geometry can be considered. The author also discusses very succinctly the logical paradoxes, and the different schools of thought in mathematics, such as Platonism, intuitionism, and formalism. The author clearly advocates the formalist school of thought in this book.

In chapter 2, the author gets more into the details of formal reasoning, the field of semiotics is outlined, and the author first begins defining the grammar and symbols for the upcoming discussion. A theory is defined as a class of statements, and consistency and decidability of theories is defined. The idea of a deductive theory is also defined, and the author defines the notion of such a theory being complete. The notions of consistency, decidability, and completeness are the familiar ones now entrenched in current textbooks on mathematical logic. A formal system, according to the author, is a theory in which the parameters of the statements of the theory are introduced as unspecified objects, and the statements of the theory make assertions on the properties of the parameters and their relations. The author considers syntactical systems, wherein the formal objects are taken from some object language, and what he calls Ob systems, which are essentially the systems considered in modern mathematical logic.The author employs the familiar Godel numbering scheme to numerically represent formal objects. The notion of algorithm is brought in here as an effective procedure to manipulate the formal objects of a system.

The next chapter is basically an introduction to the analysis of what would now be called the metalanguage of a formal system. This analysis is done in terms of what the author calls epistatements and epitheorems. Examples of these epitheorems include the Godel incompleteness theorem and the Skolem-Lowenheim theorem. The author introduces and classifies variables, and defines free and bound variables. A brief introduction to the lambda calculus and combinatory logic is given.

Then in chapter 4, the author discusses logical systems which are relational but with no bound variables. These are called logical algebras by the author, and the reader will encounter the famous truth tables and lattices in this chapter. A discussion of the Heyting algebra is given in the notes to the chapter. The reader interested in the more exotic types of algebraic logic, such as quantum logic, could benefit greatly from the reading of this chapter.

The logic of propositional calculus in terms of algebraic logic is discussed in chapter 5. Called propositional algebras by the author, the author proves the deduction theorem for such systems in this chapter. Interestingly, the L systems introduced by Gentzen are also discussed in this chapter. Although there are much better overviews of Gentzen's work in the current literature, a reader may still profit from a perusing of this chapter. L-systems where negation is added is then the subject of the next chapter.

Quantification in formal systems is taken up in chapter 7, considered both in the usual predicate calculus and in L systems. Prenex normal forms, the Herbrand-Gentzen theorem, and the completeness theorem are discussed in fairly good detail, albeit with old-fashioned notation.

The last chapter covers the interesting concept of modal logic. First considered by Aristotle, the author discusses it in the context of L systems, with the presentation being the shortest in the book. ... Read more