Contributions of the Logicians

Contributions of The Logicians. Part I. From Richard Whately to William Stanley Jevons Stanley N Burris Draft: March 3, 2001 Author address: Department of Pure Mathematics, University of Waterloo, Waterloo, Ontario, Canada, N2L 3G1 E-mail address: [email protected] URL: www.thoralf.uwaterloo.ca Contents Preface v Prelude: The Creation of an Algebra of Logic 1 Chapter 1. Richard Whately, Archbishop of Dublin (1787{1863) 9 1. Elements of Logic (1826) 9 Chapter 2. George Peacock (1791{1858) 21 1. Treatise on Algebra: Vol. 1 1842, Vol. 2 1845 21 Chapter 3. Augustus De Morgan (1806{1871) 25 1. Formal Logic (1847) 27 2. The A±rmatory Syllogisms of Formal Logic 43 3. On the Syllogism 45 4. Concluding Remarks 52 Chapter 4. George Boole (1815{1864) 55 1. The Mathematical Analysis of Logic (1847) 56 2. The Laws of Thought (1854) 68 Chapter 5. William Stanley Jevons (1835{1882) 77 1. Pure Logic (1864) 78 2. The Substitution of Similars (1869) 85 3. On the Mechanical Performance of Logical Inference (1869) 90 4. The Principles of Science (1874) 91 5. Studies in Deductive Logic (1880) 93 Appendix 1: The Proof System 95 BR Appendix 2: The Proof System 101 BA Appendix 3: The Proof System 105 1. The system AB 107 2. The Rule of AB0 and 1 109 3. Boole's Main Theorems 112 4. Boole's Method 114 5. A More Sophisticated Approach 114 6. A Remark on Hailperin's System 115 7. Tying the Proof System to the Real World 116 8. The Connection with Boolean Rings 118 9. The Connection with Boolean Algebra 119 10. Boole's Use of Division 120 Appendix 4: Existential Import 123 iii iv CONTENTS Bibliography 125 Preface As an undergraduate I read portions of Boole's 1854 classic, The Laws of Thought, and came to the conclusion that Boole was rather inept at doing Boolean algebra. Much later, when studying the history of logic in the 19th century, a conversation with Alasdair Urquhart led me to the 1976 book of Theodore Hailperin that shows why Boole's methods work. This was when I ¯nally realized that trying to superimpose Boolean algebra, or Boolean rings, on Boole's work was a real mistake. Boole uses the ordinary number system, and not a two-element algebra. With the exception of Hailperin's book I do not know of a single history of logic or mathematics that properly explains Boole's work. Furthermore, even though the books and papers devoted to the development of mathematical logic in the mid 1800s are readily available today, the essential facts have not been written up in a concise yet comprehensive form for a modern audience. These notes intend to round out the picture with a brief but substantial account of the transition from Aristotelian logic to mathematical logic, starting with Whately's revival of Aristotelian logic The Elements of Logic in 1826, continuing through the work of De Morgan, Boole and Jevons, and concluding with modern versions of equational proof systems for Boolean algebra, Boolean rings, and a discussion of Hailperin's expos¶e of the work of Boole. Many of the sections are devoted to discussing a single book. Within such sections plain page number references (without a source cited) apply to the book being discussed in the section. Stanley Burris Waterloo, Ontario March, 2001 v vi PREFACE Prelude: The Creation of an Algebra of Logic The successful introduction of mathematical techniques to handle traditional Aristotelian logic was based entirely on the development of an equational logic for classes. We will trace this development through the works of De Morgan, Boole and Jevons.1 By studying equations involving the operations of union, intersection, and complement, they created an algebraic approach that led to an extension of traditional Aristotelian logic. The development of logic for mathematics was initiated by e®orts to ¯nd the axioms and rules of inference needed for an equational logic of classes. Some of the most fundamental perspectives of modern logic were not available to these pioneers. Above all they did not have a clear separation of logic into its syntatic and semantic components. For example, in 1874 Jevons says that A = B and B = A are really just \one equation accidentally written in two di®erent manners." The modern viewpoint says that the two equations are di®erent (as strings of symbols), but they have the same truth value under any given interpretation. De Morgan was fascinated by the study of equality insofar as it led to new forms of categorical propositions, but he did very little in the way of an equational logic formulation. The famous De Morgan laws he stated as ([4], p. 118): The contrary of PQ is p,q; that of P,Q is pq. Although he had many ideas regarding how one could make use of symbolism to give a fresh approach to syllogistic logic, it is curious that he did not use a symbol for equality in his work and continue to develop an equational logic of classes. Instead this was done, in a rather ba²ing manner, by Boole. A transparent approach would have to wait till the early 1860s, with the work of De Morgan's student Jevons. De Morgan was an inspiration for his younger friend, the school teacher Boole, as well as for Jevons. De Morgan's feud with the Scottish philosopher Sir William Hamilton of Edinburgh2 caused Boole to put aside his work in traditional mathematics (di®erential equations, the calculus of variations, etc.) in 1847 in order to apply algebra to logic. Revising this initial work on logic would be Boole's main interest till 1854. Then he returned to traditional mathematics. Boole views classical logic as reasoning about classes: given a collection 1, : : : , n of propositions about classes one wants to ¯nd propositions about (some of) these classesP thatP follow from , : : : , . Boole's new approach to logic is builtPon two beliefs: P1 Pn 1. the meaning of such propositions can be expressed by equations, and 2. one can apply the symbolic methods of an augmented ordinary algebra to the equations of the premises to ¯nd the equations of the conclusions. The ¯rst item is handled convincingly by Boole, at least for universal propositions, but the second item is not. In the 1847 book he simply describes his algebraic methods and employs them. Seven years later, in his 1854 book, he is caught up in trying to explain why it is permissible to use terms in his equations that have no interpretation as classes. His explanations were, and still 1The names Boole and De Morgan are well known in logic circles, but not that of Jevons. Jevons is much better remembered for his contributions to economics|he is considered one of the fathers of mathematical economics, introducing the methods of calculus to the subject. In this arena one of his least successful and most memorable contributions was to tie economic cycles to sunspot cycles. 2Not to be confused with the mathematician of quaternion fame, Sir William Hamilton of Dublin. 1 2 PRELUDE: THE CREATION OF AN ALGEBRA OF LOGIC are, most unconvincing. In essence his defence was just that he was using the `symbolic method', and the requests for clari¯cation were mainly met with a number of new examples showing how to use his system. Boole discovered that ordinary algebra leads to a powerful algebraic system for analyzing logical arguments, but he really did not know why it worked. Indeed, it seems that the fact that it did work in many examples was quite enough to convince Boole that it would work in general|a masterful application of inductive reasoning to provide an equational approach to classical deductive reasoning. The fact that his methods provide an equational logic that can indeed correctly handle arguments about classes was not established until Theodore Hailperin's book appeared in 1976.3 Hailperin uses signed multisets to extend the collection of classes, after identifying classes with idempotent multisets. In this setting Boole's operations become meaningful, namely class + class may not be a class, but it is always a signed multiset. The justi¯cation of Boole's system likely requires more sophistication in modern algebra than was available in the 19th century. This would lead Jevons in the 1860s to create an alternative algebra of logic, essentially what is now called Boolean algebra.4 Boole's work is a clever interplay between traditional logic and the algebra of numbers. With Jevons, logic abandons the ties to numbers and joins its new companion, `Boolean' algebra, for a mature and enduring relationship. To have a good idea of where Boole and De Morgan were starting from we begin these notes with an overview of the inﬂuential 1826 logic text of Richard Whately. The notion of a class was well established in the traditional literature on Aristotelian logic, as well as the use of a single letter to denote a class, for example, using M to denote the class of men. However, using symbols to describe combinations of classes, for example, using A + B for the union of A and B, was not part of the traditional literature of either logic or mathematics. Variables and Constants In modern equational logic5 we use two kinds of symbols as simple names for the objects in the universe of discussion, namely variables, usually letters x; y; z; : : : from the end of the alphabet, and constants, usually letters a; b; c; : : : from the beginning of the alphabet, as well as the numerals 0; 1; : : : . The reason for the two kinds of symbols is that we treat the variables in an equation as universally quanti¯ed, but the constants are not quanti¯ed.

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