Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce

Introduction

Figure 1.  Left to right: Pioneers of Boolean algebra George Boole, John Venn, and Charles Sanders Peirce (Source: MacTutor History of Mathematics Archive)

On virtually the same day in 1847, two major new works on logic were published by prominent British mathematicians: Formal Logic [3] by Augustus De Morgan (1806–1871) and The Mathematical Analysis of Logic [1] by George Boole (1815–1864). Both authors sought to stretch the boundaries of traditional logic by developing a general method for representing and manipulating logically valid inferences or, as De Morgan explained in an 1847 letter to Boole, to develop "mechanical modes of making transitions, with a notation which represents our head work" [7, p. 25]. Unlike the method proposed by De Morgan, Boole’s approach took the significant step of explicitly adopting algebraic methods for this purpose. As De Morgan himself later proclaimed, “Mr. Boole’s generalization of the forms of logic is by far the boldest and most original . . . ” (as quoted in [4, p. 174]).

Although explicitly algebraic in nature, Boole’s bold and original approach led to a very strange new system of algebra. For example, among the laws which hold in the system one finds both the standard distributivity law of multiplication over addition, \(x(y+z)=xy+xz\) and the unusual looking dual distributivity law of addition over multiplication, \(x+yz=(x+y)(x+z).\) In his mature work on logic, An Investigation of the Laws of Thought [2] published in 1854, Boole further explored the ways in which the laws of this algebraic system both resemble and differ from those of standard algebra, as well as the reasons why it satisfies these various laws. Somewhat ironically, Laws of Thought was not initially well received; Boole and a friend who bore the expense of its initial printing probably did not recover their costs. The abstract structure of a boolean algebra which eventually did evolve from Boole’s work has, however, become not only an important field of study in mathematics, but also a powerful tool in the design and study of electronic circuits and computer architecture.

From the writings of the numerous individuals who contributed to the tale of boolean algebra’s birth and development, we have created three primary source project modules appropriate for students in introductory or intermediate discrete mathematics courses:

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

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 Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce 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.

Figure 2.  The lower one-third of a stained glass window dedicated to George Boole in Lincoln Cathedral in Lincoln, England, Boole's birthplace (Source: Wikimedia Commons)

Notes to the Instructor

The project “Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce” is designed for an introductory or intermediate course in discrete or finite mathematics that includes a study of elementary set theory. Without explicitly introducing modern notation for operations on sets, the 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. Although the project focuses on what would now be called ‘introductory set theory,’ it also lays the groundwork for a more abstract discussion of boolean algebra as a discrete axiomatized structure. Accordingly, this project may also be used as an introduction to one or both of the companion projects described below in any course which considers boolean algebra from either a mathematical or computer science perspective. Beyond a certain level of mathematical maturity, commensurate with a typical Calculus I background, there are no pre-requisites for this project. Strong students at a pre-calculus level could also complete the earlier sections of this project.

Figure 3. John Venn at a somewhat more advanced age than in Figure 1 (Source: Convergence Portrait Gallery)

Beginning with Boole’s writings on the use of symbolic algebra to represent logical classes in his An Investigation of the Laws of Thought [2] (Section 2), this project introduces the operations of logical addition (i.e., set union), logical multiplication (i.e., set intersection) and logical difference (i.e., set difference) and examines certain restrictions placed on their use by Boole which have since been lifted. The basic laws governing these operations are also introduced, as they were developed and justified by Boole; these justifications relied in part on his definitions of the operations and in part on the analogy of his symbols with those of ‘standard algebra.’ The project then follows refinements to Boole’s system made by John Venn in his Symbolic Logic [8] (Section 3) and Charles Sanders Peirce in his “On an Improvement in Boole’s Calculus of Logic” [5] (Section 4), with the level of abstraction steadily increasing through these sections. The project concludes (Section 5) with a summary of how Boole’s ‘Algebra of Logic’ relates to elementary set theory as it is typically presented today, and discusses how elementary set theory (when viewed as an algebraic structure) serves as a concrete example of a boolean algebra. Standard (undergraduate) notation and properties for set theory operations in use today are included and compared to standard (undergraduate) notation and axioms for a boolean algebra in that section. Issues related to language use and set operations which pose difficulties for many students, but which are ignored or unrecognized by contemporary textbook authors, are also explicitly considered in the writings of Boole and Venn (Sections 3 and 4), and further explored through the project questions in those sections.

Figure 4. Charles Sanders Peirce at somewhat more advanced age than in Figure 1 (Source: Convergence Portrait Gallery)

By following the refinements made to Boole’s symbolic algebra in the hands of Venn and Peirce, this project provides students with an opportunity to witness first-hand how the process of developing and refining a mathematical system plays out, the ways in which mathematicians make and explain their choices along the way, and how standards of rigor in these regards have changed over time. Thus, in addition to developing the properties of set theory as a specific concrete example of a boolean algebra, the project is able to explore a variety of mathematical themes, including the notion of an inverse operation, the concept of duality, issues related to mathematical notation, and standards of rigor and proof. By following one or more of these themes through the project, instructors have considerable leeway in tailoring the project to their goals for a particular group of students. 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. For example, Section 4 could be completely omitted, if an instructor elected not to examine proofs of elementary set properties from the more abstract perspective that Peirce adopted. To complete the project in its entirety requires approximately two weeks.

Two companion projects on boolean algebra are also available as follow-up to this introductory project, either or both of which could also be used independently of the Boole-Venn-Peirce project. Additionally, either of the two companion projects could be used as a preliminary to or as a follow-up to the other companion project.

The companion project Boolean Algebra as an Abstract Structure: Edward V. Huntington and Axiomatization explores the early axiomatization of boolean algebra as an abstract structure through readings from Huntington’s 1904 paper “Sets of Independent Postulates for the Algebra of Logic.” In addition to introducing the now standard axioms for the boolean algebra structure, the project illustrates how to use these postulates to prove some basic properties of boolean algebras. Specific project questions also provide students with practice in using symbolic notation, and encourage them to analyze the logical structure of quantified statements. The project also examines 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 — as a model to establish the independence and consistency of one of his postulate sets. The final section of the project discusses modern (undergraduate) notation and axioms for boolean algebras, and provides several practice exercises to reinforce the ideas developed in the earlier sections.

The companion project Applying Boolean Algebra to Circuit Design: Claude Shannon, based on Shannon’s ground-breaking paper “A Symbolic Analysis of Relay and Switching Circuits” [6], 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 properties 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 of 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.

Download the project Origins of Boolean Algebra in the Logic of Classes: George Boole, John Venn and C. S. Peirce 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]  De Morgan, A., Formal Logic: or, The Calculus of Inference, Necessary and Probable, Taylor and Walton, London, 1847.

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

[5]  Peirce, C. S., On an Improvement in Boole’s Calculus of Logic, Proceedings of the American Academy of Arts and Sciences, 7 (1867), 250-261. Reprinted in Collected Papers of Charles Sanders Peirce, Volume III: Exact Logic, C. Hartshorne and P. Weiss (editors), Oxford University Press, London, 1967, 3-15.

[6]  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.

[7]  Smith, G. C., The Boole-DeMorgan Correspondence, 1842-1864, Clarendon Press, Oxford, 1982.

[8]  Venn, J., Symbolic Logic, MacMillan, London, 1894. Reprint Chelsea, Bronx, 1971.

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.