## The mathematical analysis of logic, being an essay towards a calculus of deductive reasoning.

Cambridge: Macmillan, Barclay & Macmillan, 1847.

First edition, very rare in commerce, of Boole’s first book, the birth of modern symbolic logic and the first presentation of ‘Boolean algebra’. “Boole’s work also contains what Bertrand Russell called the greatest discovery of the nineteenth century: the nature of pure mathematics” (OOC). “Self-taught mathematician George Boole (1815–1864) published a pamphlet in 1847 – *The Mathematical Analysis of Logic* – that launched him into history as one of the nineteenth century’s most original thinkers” (Introduction to the CUP reprint). “*The Mathematical Analysis of Logic *marks the beginning of symbolic logic in the modern sense. Boole showed that classical logic was actually a branch of mathematics which gave rise to a hitherto unconsidered type of algebra. Boole’s book however went considerably further. It threw a great deal of light on the nature of pure mathematics; it opened up possibilities of an extension of the subject into totally new and unexpected areas – classical mathematics had concentrated on the notions of shape and number and even when symbols were employed, they were generally interpreted in terms of number. Boole had now introduced the notion of interpreting symbols as classes or sets of objects, a concept breathtaking in its scope because it meant that the study of all well-defined sets of objects now came under the realm of mathematics … By enlarging the horizons of mathematics so enormously, Boole unwittingly (but perhaps subconsciously, wittingly) highlighted a topic that has come to influence virtually every aspect of present-day life – the storage and processing of information, which in turn has led to the development of computer science. Not only is Boole’s algebra the ‘correct’ and most economical tool for handling information, but the electronic machines which now do the work actually operate according to principles determined by that self-same algebra. Boole has been called the ‘Father of Symbolic Logic’ and the ‘Founder of Pure Mathematics’, but he is just as deserving of the title, ‘Father of Computer Science’” (MacHale, p. 82). ABPC/RBH list only two copies since Honeyman: the OOC copy, Christie’s 2005, $10,800, and Bonham’s 2013, £25,000 = $38,000. Both of these copies were in modern bindings (the OOC copy with the original front wrapper bound in).

“Boole’s contribution to logics made possible the works of subsequent logicians including Turing and Von Neumann … Even Babbage depended a great deal on Boole’s ideas for his understanding of what mathematical operations really are … Since Boole showed that logics can be reduced to very simple algebraic systems – known today as Boolean Algebras – it was possible for Babbage and his successors to design organs for a computer that could perform the necessary logical tasks. Thus our debt to this simple, quiet man, George Boole, is extraordinarily great … His remark about a ‘special law to which the symbols of quantity are not subject’ is very important: this law in effect is that *x ^{2} =x* for every

*x*in his system. Now in numerical terms this equation or law has as its only solution 0 and 1. This is why the binary system plays so vital a role in modern computers: their logical parts in effect carrying out binary operations. In Boole’s system 1 denotes the entire realm of discourse, the set of all objects being discussed, and 0 the empty set. There are two operations in this system which we may call + and ×; or we may say

*or*and

*and*. It is most fortunate for us that all logics can be comprehended in so simple a system, since otherwise the automation of computation would probably not have occurred – or at least not when it did” (Goldstine, pp. 37-38).

“Early in the spring of 1847, Boole's long-dormant interest in the connections between mathematics and logic was dramatically reawakened. At this time, a furious controversy was raging between the supporters of de Morgan and those of Sir William Hamilton, the Scottish philosopher and metaphysician (not to be confused with Sir William Rowan Hamilton). Hamilton was a logician who distrusted mathematics, but he was an innovator in logic and had, about this time, introduced the notion of 'quantification of the predicate' which was to lead to a widening of the scope of logic. Classical logic had concentrated on the 'four forms' of statement — all A are B, no A are B, some A are B, some A are not B. In Hamilton's approach, the predicate, or second term B, is quantified by considering statements of the type: all A are all B, any A is not some B, and so on. De Morgan too was at this time working on a more mathematical theory of logic which included a notion equivalent to quantification of the predicate. Hamilton at once accused de Morgan of plagiarism, despite the fact that the notion in question was not original to either of them nor, as it transpired, of any great significance *per *se in the development of logic. Hamilton's charges were unjust, even absurd, but controversy raged for many years and attracted a great deal of attention …

“From his neutral position, Boole was able to judge the merits and defects of the approaches of both Hamilton and de Morgan. Though Hamilton disliked mathematics and even poured scorn on the subject, yet his approach seemed to suggest that logic should concentrate on 'equations' connecting 'collections of objects or classes'. De Morgan's approach, on the other hand, seemed to concentrate on a purely symbolic representation of logical processes, yet his notation was cumbersome and unwieldy. Why not, thought Boole, synthesise the two approaches by representing each class of objects by a single symbol and allow relations between classes to be expressed by algebraic equations between the symbols? This devastatingly simple but ingenious notion intrigued and excited him and he set to work at once on a book expounding a new mathematical theory of logic” (MacHale, p. 79-80).

“The priority dispute triggered Boole to write his first book; but its content was much influenced by his researches on differential operators. Partly drawing upon his work of his friend the Cambridge mathematician Duncan Gregory, he produced a long paper on these methods which he submitted to the Royal Society in 1844. After wondering about rejecting the manuscript they published it [‘On a general method in analysis’, *Philosophical Transactions*, Vol. 134, pp. 225–282], and then awarded him a Gold Medal for his achievement!

“This theory was one of the early algebras in which the ‘objects’ were neither numbers nor geometrical magnitudes; and it had met controversy in its algebraic laws, such as identifying powers with orders (that is, D^{2} for D on D, not D times D). Aware of the mystery, Boole (and before him Gregory) tried to bring light by highlighting the principal desirable properties … and using them in solving various differential equations” (*Landmark Writings*, p. 472).

“Boole wrote his book at a furious pace and it was ready in a matter of months. He entitled it *The Mathematical Analysis of Logic, being an Essay Towards a Calculus of Deductive Reasoning *[MAL] … The book was then published by Macmillan, Barclay and Macmillan of Cambridge, with a preface dated 29 October 1847” (MacHale, p. 81).

“The Introduction chapter starts with Boole reviewing the symbolical method. The second chapter, First Principles, lets the symbol 1 represent the universe which “comprehends every conceivable class of objects, whether existing or not.” Capital letters *X*, *Y*, *Z*, … denoted classes. Then, no doubt heavily influenced by his very successful work using algebraic techniques on differential operators, and consistent with De Morgan's 1839 assertion that algebraists preferred interpreting symbols as operators, Boole introduced the elective symbol *x* corresponding to the class *X*, the elective symbol *y* corresponding to *Y*, etc. The *elective symbols* denoted election operators—for example the election operator red when applied to a class would elect (select) the red items in the class …

“Then Boole introduced the first operation, the *multiplication* *xy* of elective symbols. The standard notation *xy* for multiplication also had a standard meaning for operators (for example, differential operators), namely one applied *y* to an object and then *x* is applied to the result. (In modern terminology, this is the *composition* of the two operators) …

“The first law in *MAL* was the *distributive law* *x*(*u*+*v*) = *xu* + *xv*, where Boole said that *u*+*v* corresponded to dividing a class into two parts. This was the first mention of addition. On p. 17 Boole added the *commutative law* *xy* = *yx* and the *idempotent law* *x*^{2} = *x* (which Boole called the *index law*). Once these two laws were secured, Boole believed he was entitled to fully employ the ordinary algebra of his time, and indeed one sees Taylor series and Lagrange multipliers in *MAL*. The law of idempotent class symbols, *x*^{2} = *x*, was different from the two fundamental laws of symbolical algebra—it only applied to the individual elective symbols, not in general to compound terms that one could build from these symbols. For example, one does not in general have (*x*+*y*)^{2} = *x*+*y* in Boole's system since, by ordinary algebra with idempotent class symbols, this would imply 2*xy* = 0, and then *xy* = 0, which would force *x* and *y* to represent disjoint classes. But it is not the case that every pair of classes is disjoint.

“Boole focused on Aristotelian logic in *MAL*, with its 4 types of categorical propositions and an open-ended collection of hypothetical propositions. In the chapter Of Expression and Interpretation, Boole said that necessarily the class not-*X* is expressed by 1−*x*. This is the first appearance of *subtraction*. Then he gave equations to express the categorical propositions. The first to be expressed was All *X* is *Y*, for which he used *xy* = *x*, which he then converted into *x*(1−*y*) = 0. This was the first appearance of 0 in *MAL*—it was not introduced as the symbol for the empty class. Indeed the empty class did not appear in *MAL* …

“Beginning with the chapter Properties of Elective Functions, Boole developed general theorems for working with equations in his algebra of logic—the Expansion Theorem and the properties of constituents are discussed in this chapter. Up to this point his sole focus was to show that Aristotelian logic could be handled by simple algebraic methods, mainly through the use of an elimination theorem borrowed from ordinary algebra.

“It was natural for Boole to want to solve equations in his algebra of logic since this had been a main goal of ordinary algebra, and had led to many difficult questions (e.g., how to solve a 5th degree equation). Fortunately for Boole, the situation in his algebra of logic was much simpler—he could always solve an equation, and finding the solution was important to applications of his system, to derive conclusions in logic. An equation was solved in part by using expansion after performing division. This method of solution was the result of which he was the most proud—it described how to solve an elective equation for one of its symbols in terms of the others, and it is this that Boole claimed (in the Introduction chapter of *MAL*) would offer “the means of a perfect analysis of any conceivable set of propositions”” (*Stanford Encyclopedia of Philosophy*).

Seven years after publishing the present work, Boole gave a more elaborate treatment of Boolean algebra in *An Investigation of the Laws of Thought*. His aim was partly to address criticisms of the earlier work by de Morgan and others; he also considered the application of his theory to probability, a topic not treated in the earlier work.

“The range and depth of the achievements of George Boole are especially remarkable when one notes not only the shortness of his life but also the disadvantageous circumstances of his background. He was born to an intelligent tradesman who however was so poor that George had to become the main breadwinner in his 20th year when he opened a school. Nevertheless, he found time to teach himself advanced mathematics, and also Greek, Latin, French and German, especially in order to read important works. His research papers began to appear in the early 1840s, and his principal interest soon turned to an English specialty: the ‘calculus of operations’, now called ‘differential operators’, where differentiation was represented by the letter ‘D’, higher-order differentiation by ‘D^{2}, D^{3}, . . .’, integration by ‘D^{-1}’, and so on. This tradition had developed under the influence of the algebraised calculus propounded by J. L. Lagrange, initially by some French mathematicians; but from the 1810s this algebra and related topics were prosecuted in England by Charles Babbage and John Herschel as part of the revival of research mathematics there. Boole was to become a major figure in this movement in the next generation; as we have seen, it was to affect his work on logic” (*Landmark Writings*, pp. 470-471).

Goldstine, *The Computer from Pascal to von Neumann*, 1972. MacHale, *The Life and Work of George Boole. A Prelude to the Digital Age*, 2014. *Landmark Writings in Western Mathematics 1640-1940*, 2005 (Chapter 36). Smith, ‘Boole’s Annotations on ‘The Mathematical Analysis of Logic’,’ *History and Philosophy of Logic* 4 (1983), 27-39.

8vo (217 x 133 mm), pp. [i:blank] [ii:title], [1-3], 4-82, errata slip tipped in after title. 20th century cloth with gilt spine lettering. Clean and fresh throughout.

Item #4365

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Price:
$18,000.00
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