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Logic as Algebra [Paperback]

Paul, and Givant, Steven Halmos (Author)
4.0 out of 5 stars  See all reviews (1 customer review)

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Front Cover | Copyright | Table of Contents | Excerpt | Index | Back Cover
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4.0 out of 5 stars A superb introduction to the glories of Boolean algebra, May 15 2003
This review is from: Logic as Algebra (Paperback)
This book reviews some ideas Halmos worked on in the 1950s: the algebraization of predicate logic. The result was polyadic algebra, which has been unfairly neglected since. Tarski, Henkin, and their Berkeley students worked on a rival research program that culminated in the better known cylindric algebras. The treatment remains at the undergrad level, because Halmos stops short short of polyadic predicates. Halmos's "Algebraic Logic," which AMS keeps in print and is a fine read, contains all of Halmos's professional writings on polyadic algebra.

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.
For that matter, the completeness of first order logic has a nice polyadic algebra translation.

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.

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Amazon.com: 4.0 out of 5 stars (3 customer reviews)

10 of 10 people found the following review helpful
4.0 out of 5 stars A superb introduction to the glories of Boolean algebra, May 15 2003
By galloamericanus "galloamericanus" - Published on Amazon.com
This review is from: Logic as Algebra (Paperback)
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.

13 of 15 people found the following review helpful
4.0 out of 5 stars A Builder of a Solid Foundation in Mathematics, Feb 8 2000
By Charles Ashbacher - Published on Amazon.com
This review is from: Logic as Algebra (Paperback)
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 rigid analytical reasoning to be developed. Of the three earliest areas, geometry was born out of the necessity of accurately measuring land plots and large buildings and number theory was required for sophisticated counting techniques. Logic, the third area, had no "practical" godfather, other than being the foundation for rigorous reasoning in the other two. In the intervening years, so many additional areas of mathematics have been developed, with logic and logical reasoning continuing to be the fundamental building block of them all. Therefore, every mathematician should have some exposure to logic, with the simple history lesson automatically being included. This short, but excellent book fills that niche.
The title accurately sets the theme for the entire book. Algebra is nothing more than a precise notation in combination with a rigorous set of rules of behavior. When logic is approached in that way, it becomes much easier to understand and apply. This is especially necessary in the modern world where computing is so ubiquitous. Many areas of mathematics are incorporated into the computer science major, but none is more widely used than logic. Written at a level that can be comprehended by anyone in either a computer science or mathematics major, it can be used as a textbook in any course targeted at these audiences.
The topics covered are standard although the algebraic approach makes it unique. One simple chapter subheading, `Language As An Algebra', succinctly describes the theme. Propositional calculus, Boolean algebra, lattices and predicate calculus are the main areas examined. While the treatment is short, it is thorough, providing all necessary details for a sound foundation in the subject. While the word "readable" is sometimes overused in describing books, it can be used here without hesitation.
Sometimes neglected as an area of study in their curricula, logic is an essential part of all mathematics and computer training, whether formal or informal. The authors use a relatively small number of pages to present an extensive amount of knowledge in an easily understandable way. I strongly recommend this book.

Published in Smarandache Notions Journal reprinted with permission.


2 of 2 people found the following review helpful
4.0 out of 5 stars A Solid Introduction, April 12 2011
By Bryan Goodrich - Published on Amazon.com
Amazon Verified Purchase(What's this?)
This review is from: Logic as Algebra (Paperback)
I found the first two or three chapters of the book to be a great introduction to logic and algebraic reasoning. From there, the reader should probably have some familiarity with modern algebra to fully appreciate the ideas being introduced (e.g., kernals, ideals, morphisms, lattices). The first few chapters are easy to read, and unlike other introductions to logic, the tedium of proofs doesn't drown out the concepts. Due to space, the presentations are brief, but none of the concepts are difficult: sit down and iron out the proofs in your head or sit down with a pencil and paper. Nevertheless, they give quick work of the main properties of the propositional calculus, building it up from 6 symbols, 4 axioms, and one rule of inference. Rigor isn't always practiced, but mathematicians should be comfortable with that!

The book is definitely pleasing to a mathematician that wants to refine their understanding and perception of logic, and it is good for the logician that could benefit from a mathematical mindset. The first chapters develop a propositional calculus genetically, but then branch off into approaching it structurally from a representative algebra. This book lays the path to take that algebraic approach to monadic (single variable) predicate calculus, and prepares the student to look at Halmos' continuing work presented in "Algebraic Logic", a collection of papers on polyadic (more than one variable) predicate calculus from the (Boolean) algebraic perspective.

I give a rating of 4 out of 5 stars because this book lacks the "wow" factor that makes it a 5 star book. There's little to complain about in the book that is of any seriousness. Minor flaws are to individual tastes. Nevertheless, this isn't a comprehensive analysis with great insights or major contributions. It's a nice introduction and approach to logic from a mathematical perspective. I never used this book for a class. I'm a student of both logic and mathematics. Halmos is a great mathematical logician and well known for his excellent writing (e.g., see his "Naive Set Theory"). The book was just a great addition to my library and I still return to it now and again to refine the basics of my logical reasoning, because the main thrust from the beginning chapters is that logic can be greatly benefited when approached structurally (in algebraic terms) than the usual genetic perspective (that really comprises the first part of the book).
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