Logic as Algebra (Dolciani Mathematical Expositions) by Paul Halmos (PDF)

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.

User’s Reviews

Editorial Reviews: Book Description An introduction to logic from the perspective of algebra. Book Description An introduction to logic that treats 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. Written in the engaging and provocative style that is the hallmark of Paul Halmos.

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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).

⭐This is a nice little book, but it does require some knowledge in abstract algebra to understand. One of the key ideas is the use of ideals to model logical consequence. I did not understand this part until I learned some algebraic geometry, which uses ideals.An ideal is a subset (of a ring) closed under addition. In boolean algebra, the logical implication (ie, the arrow A->B) can be expressed as (not A or B), and logical AND and OR can be expressed as multiplication and addition in an algebra, and logical NOT is expressed as -A. So logical implication is essentially addition or multiplication. To find the logical consequence of a set of axioms is to find the algebraic terms closed under conjunctions or disjunctions (ie addition or multiplication) of the premises. Thus, an ideal in boolean algebra models logical consequence.The ending chapter of the book is a gentle introduction to predicate logic in algebraic form.

⭐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.

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