Algebraic Logic

Paul R. Halmos

• Publisher: Dover Publications
• Year: 2016
• Print ISBN: 9780486801452
• eBook ISBN: 9780486810416
• Edition: 1
• Topic: Mathematics
• Subtopic: Algebra
• Index: Mathematics

Beginning with an introduction to the concepts of algebraic logic, this concise volume features ten articles by a prominent mathematician that originally appeared in journals from 1954 to 1959. Covering monadic and polyadic algebras, these articles are essentially self-contained and accessible to a general mathematical audience, requiring no specialized knowledge of algebra or logic.
Part One addresses monadic algebras, with articles on general theory, representation, and freedom. Part Two explores polyadic algebras, progressing from general theory and terms to equality. Part Three offers three items on polyadic Boolean algebras, including a survey of predicates, terms, operations, and equality. The book concludes with an additional bibliography and index.

V

GENERAL THEORY

Algebraic logic (II)

Homogeneous locally finite polyadic Boolean algebras of infinite degree

Table of contents

Introduction,

§1.Monadic algebras,

§2.Functional polyadic algebras,

§3.Substitutions,

§4.Polyadic algebras,

§5.Locally finite algebras,

§6.Supports and independence,

§7.Quasi-polyadic algebras,

§8.Algebraic theory,

§9.Logics,

§10.Functional representation,

§11.Dilations,

§12.Constants,

§13.Factorization and commutativity,

§14.Constants from endomorpbisms,

§15.Examples of constants,

§16.Rich algebras,

§17.Representation,

Appendix,

Introduction

The purpose of this paper is to do for the lower functional calculus what the first paper of the series did for the monadic special case. To a very great extent, however, the two papers are independent of each other. The situation is similar to that in the theory of differential equations. Once the basic but relatively elementary concept of differentiation is known, it can be used to give a mathematical formulation of the problems of classical Newtonian mechanics, usually in the form of ordinary differential equations. It can also be used to give a mathematical formulation to some of the more sophisticated problems of modern physics, usually in the form of partial differential equations. Similarly, once the algebraic version of the logical operation of quantification is defined, it can be applied to the algebraic study of either classical or modern logic. The analogy is quite close. Traditional Aristotelean logic can be viewed as the theory of propositional functions of a single variable; the lower functional calculus, on the other hand, treats propositional functions of any number of variables.

The analogy between physics and logic can be used to illuminate not only the role of this paper in the series, but, for that matter, the purpose of algebraic logic as a whole. It is fashionable nowadays to give a mathematical exposition of quantum mechanics in terms of certain algebraic systems. The concepts occurring in such an exposition, the terms used to describe them, and the problems selected for special emphasis are all suggested by the facts of physics. The algebra does not pretend to solve physical problems or to give new information about the state of the universe; its primary aim is to study a mathematical subject by mathematical methods. The mathematic...

Table of contents

• Cover
• Title page
• Copyright
• Contents
• Preface: Being a Letter from Sir William Betford, of Bampton St. Mary, in Dorsetshire, to George Trevelyan; of Lincoln’s Inn Fields, London
• I. Introduction
• II. General Theory
• III. Representation
• IV. Freedom
• V. General Theory
• VI. Terms
• VII. Equality
• VIII. General Theory
• IX. Terms and Equality
• X. Brief Summary
• Additional Bibliography
• Index