Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization

Figure 1. Edward V. Huntington (1874-1952) (Source: MacTutor History of Mathematics Archive)

Introduction

In 1847, British mathematician George Boole (1815–1864) published a work entitled The Mathematical Analysis of Logic [1]; seven years later he further developed his mathematical approach to logic in An Investigation of the Laws of Thought [2]. Boole’s goal in these works was to extend the boundaries of traditional logic by developing a general method for representing and manipulating logically valid inferences. Towards that end, Boole employed letters to represent classes (or sets) and a system of operational symbols (\(\times,+)\) on these classes defined to represent “the operations of the mind by which reasoning is performed” [2, p. 3]. As described by Augustus De Morgan, whose own work on logic was eclipsed by that of Boole, “Mr. Boole’s generalization of the forms of logic is by far the boldest and most original . . . ” (as quoted in [4, p. 174]).

Figure 2. George Boole (1815-1864) (Source: MacTutor Archive)

In the years following publication of Laws of Thought, the symbolical algebra introduced by Boole attracted considerable attention from other mathematicians. Alongside various refinements and ex- tensions made to this system, mathematics itself underwent significant changes as mathematicians endeavored to make sense of various strange new discoveries — geometries in which Euclid’s Parallel Postulate was violated, algebras in which basic laws like commutativity failed or strange new laws such as Boole’s ‘idempotent law’ \(x^2 = x\) held, ‘transfinite’ number systems involving different sizes of infinity. These discoveries in turn encouraged mathematicians to introduce even greater levels of abstraction in their work, as well as greater formality in their approach to proof.

In line with the general trend towards greater abstraction and formality in mathematics came a loosening of the ties between the algebraic system introduced by Boole and logic as a concrete application of that system. In his 1904 paper, “Sets of Independent Postulates for the Algebra of Logic,” for example, the American postulate theorist Edward V. Huntington (1874-1952) declared [3, p. 288]:

The algebra of symbolic logic, as developed by Leibniz, Boole, C.S. Peirce, E. Schröder, and others, is described by Whitehead as the only known member of the non-numerical genus of universal algebra. This algebra, although originally studied merely as a means of handling certain problems in the logic of classes and the logic of propositions, has recently assumed some importance as an independent calculus; it may therefore be not without interest to consider it from a purely mathematical or abstract point of view, and to show how the whole algebra, in its abstract form, may be developed from a selected set of fundamental propositions, or postulates, which shall be independent of each other, and from which all the other propositions of the algebra can be deduced by purely formal processes. In other words, we are to consider the construction of a purely deductive theory, without regard to its possible applications.

Figure 3. Huntington built on the work of George Boole (pictured above) and, left to right, Gottfried Wilhelm Leibniz, Charles Sanders Peirce, and Ernst Schröder, among others. (Source: MacTutor Archive)

Huntington’s 1904 paper fully encapsulated the formal rigor which became a hallmark of twentieth-century mathematics as mathematicians endeavored to make sense of the strange new wonders of their world. Only later in the twentieth century, as individuals interested in problems outside of mathematics proper gained exposure to boolean algebra and its unique properties through works like that of Huntington, did the algebra of symbolic logic once again find employment as an applied tool.

From the writings of the numerous individuals who contributed to the development of boolean algebra as both an abstract structure and an applied tool, we have created three primary source project modules appropriate for students in introductory or intermediate discrete mathematics courses:

• a project which develops elementary set theory through original source excerpts from Boole’s own writings and that of two of the earliest mathematicians to make modifications in Boole’s approach to logic, presented in the Convergence article, Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce;
• a project which explores the application of boolean algebra to the problem of circuit design, presented in the Convergence article, Applications of Boolean Algebra: Claude Shannon and Circuit Design; and
• the present project, “Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization,” which examines the structure of a boolean algebra as an abstract axiomatized structure through excerpts from Huntington’s 1904 paper. 


All three projects are part of a larger collection published in Convergence, and an entire introductory discrete mathematics course can be taught from a selection of projects in this collection. For additional projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science. 


Our project, Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization, is ready for students, and the Latex source is also available for instructors who may wish to modify the project for students. The comprehensive “Notes to the Instructor” presented next are also appended to the project itself.

Notes to the Instructor 


The project “Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization” is designed for an introductory or intermediate course in discrete or finite mathematics that considers boolean algebra from either a mathematical or computer science perspective. Some or all of the project could also be used in a course on abstract algebra or model theory. The project does assume some (minimal) familiarity with the set operations of union and intersection and their representation by Venn diagrams. This pre-requisite material may be gained by completing the companion (Boole) project described below, through reading a standard textbook treatment of elementary set operations, or via a short class discussion/lecture. 


Although primarily based on an early paper in the study of axiomatizations of boolean algebras, the introductory section of this project provides a concise overview of George Boole’s original work on ‘the logic of classes’ in order to provide students with a connection to a concrete example of a boolean algebra on which they can draw. In section 2, the goal of formal axiomatics is introduced through select readings from Huntington’s 1904 paper “Sets of Independent Postulates for the Algebra of Logic” [3]. Section 2 also introduces the now standard axioms for the boolean algebra structure and illustrates how to use these postulates to prove boolean algebra basic properties. Specific project questions also provide students with practice in using symbolic notation, and encourage them to analyze the logical structure of quantified statements. In Section 3, the use of models is illustrated in Huntington’s use of the two-valued Boolean algebra on \(K=\{0,1\}\) — first studied by George Boole in his work on the logic of classes — to establish the independence and consistency of one of his postulate sets. Instructors who choose not to consider these issues could safely omit this section, and still include the epilogue section. In that epilogue, modern (undergraduate) notation and axioms for boolean algebras are introduced, and additional practice exercises are provided to reinforce the ideas developed in the earlier sections. It is advisable for an instructor to work through all exercises in advance in order to determine which, if any, she may wish to omit. To complete the project in its entirety requires approximately one week.

Two other projects on boolean algebra are available as companions to this project, either or both of which could also be used independently of this project. The first companion project Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce, is suitable as a preliminary to the Huntington project. Without explicitly introducing modern notation for operations on sets (until the concluding section), that project develops a modern understanding of these operations and their basic properties within the context of early efforts to develop a symbolic algebra for logic. By steadily increasing the level of abstraction, that project also lays the groundwork for a more abstract discussion of a boolean algebra as a discrete structure. Other project questions prompt students to explore a variety of other mathematical themes, including the notion of an inverse operation, issues related to mathematical notation, and standards of rigor and proof.

The second companion project, Applying Boolean Algebra to Circuit Design: Claude Shannon, is suitable either as a preliminary to or as a follow-up to the Huntington project on axiomatization. Based on Shannon’s groundbreaking paper, “A Symbolic Analysis of Relay and Switching Circuits” [5], that project begins with a concise overview of the two major historical antecedents to Shannon’s work: Boole’s original work in logic and Huntington’s work on axiomatization. The project then develops standard properties of a boolean algebra within the concrete context of circuits, and provides students with practice in using these to simplify boolean expressions. The two-valued boolean algebra on \(K=\{0,1\}\) again plays a central role in this work. The project closes with an exploration the concept of a ‘disjunctive normal form’ for boolean expressions, again within the context of circuits.

Implementation with students of any of these projects may be accomplished through individually assigned work, small group work and/or whole class discussion; a combination of these instructional strategies is recommended in order to take advantage of the variety of questions included in the project. For the Huntington project, the instructor is encouraged to provide students with copies of Huntington’s axioms and the definitions of his various models for consistency and independence on a separate sheet; these are included in Appendices B and C of this project.

Download the project Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization as a pdf file ready for classroom use.

Download the modifiable Latex source file for this project.

For more projects, see Primary Historical Sources in the Classroom: Discrete Mathematics and Computer Science.

Bibliography

[1] Boole, G., Mathematical Analysis of Logic, MacMillan, Barclay & MacMillan, Cambridge, 1847. Reprint Open Court, La Salle, 1952.

[2] Boole, G., An Investigation of the Laws of Thought on Which are Founded the Mathematical Theories of Logic and Probabilities, Walton and Maberly, London, 1854. Reprint Dover Publications, New York, 1958.

[3] Huntington, E. V., Sets of Independent Postulates for the Algebra of Logic, Transactions of the American Mathematical Society, 5:3 (1904), 288-309.

[4] Merrill, D., Augustus De Morgan and the Logic of Relations, Kluwer, Dordrecht, 1990.

[5] Shannon, C. E., A Symbolic Analysis of Relay and Switching Circuits, American Institute of Electrical Engineers Transactions, 57 (1938), 713-723. Reprinted in Claude Elwood Shannon: Collected Papers, N. J. A. Sloane and A. D. Wyner (editors), IEEE Press, New York, 1993, 471-495.

Acknowledgment

The development of curricular materials for discrete mathematics has been partially supported by the National Science Foundation's Course, Curriculum and Laboratory Improvement Program under grants DUE-0717752 and DUE-0715392 for which the authors are most appreciative. Any opinions, findings, and conclusions or recommendations expressed in this material are those of the authors and do not necessarily reflect the views of the National Science Foundation.