## An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Probabilities.

London, and Cambridge: Walton and Maberly; Macmillan & Co., 1854.

First edition, the rare first issue, and a fine copy in original state, of Boole’s principal work, in which he gave the first detailed presentation of Boolean algebra. “Bertrand Russell (1872-1970) described *The Laws of Thought* as: the work in which pure mathematics was discovered,’ while the mathematician Garrett Birkhoff (1911-96) stated that: ‘The ‘Boolean Algebra’ of classes, largely originated in this classic book, has had an ever-increasing influence on all branches of mathematics’” (georgeboole.com). “Boole invented the first practical system of logic in algebraic form, which enabled more advances in logic to be made in the decades of the nineteenth century than in the twenty-two centuries preceding. Boole’s work led to the creation of set theory and probability theory in mathematics, to the philosophical work of Peirce Russell, Whitehead, and Wittgenstein, and to computer technology via the master’s thesis of Claude Shannon, who recognized that the true/false values in Boole’s two-valued logic were analogous to the open and closed states of electric circuits” (Hook & Norman, *Origins of Cyberspace*, 224). In this work, Boole “rendered logics into a precise and mathematical form. He set down postulates or axioms for logics for the first time – just as Euclid had done for geometry. Further, he gave to the whole subject an algebraic treatment … The monumental work of Boole was to remain a curiosity for many years, and it was not until Whitehead and Russell wrote their great *Principia Mathematica* in 1910-13 that serious mathematicians took up formal logics. Since then the field has flowered into the stupendous achievements of Gödel and Cohen in our era. In any case 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).

Boole’s first presentation of Boolean algebra appeared in 1847 in his rare pamphlet *The Mathematical Analysis of Logic*. “Boole had often regretted the fact that his first book, *The Mathematical Analysis of Logic*, had been so hurriedly written. Further reflection and the reactions of other mathematicians, sometimes adverse, to his earlier book had convinced him of the possibility of extending his algebra of logic and, moreover, he began to suspect that probability might be analysed in a similar fashion … Boole worked feverishly all during the year 1851 on his book, all the time deepening and perfecting his algebra of logic, while at the same time giving an analogous development of the theory of probability. He kept in close and continual touch with de Morgan throughout this period and there is no doubt that he was strongly influenced by de Morgan’s *Essay on Probabilities* of 1838 and the chapter on probability in de Morgan’s *Formal Logic* of 1847. By July 1852, Boole was able to report that he had five hundred manuscript pages ready for final revision … By 8 December 1852, Boole was able to write to de Morgan: ‘I have agreed with Gill to print my book and to get a good deal of the MS to press before the end of the year’ …

“The preface, dated 30 November 1853, begins as follows: ‘The following work is not a republication of a former treatise by the author, entitled *The Mathematical Analysis of Logic*. Its earliest portion is indeed devoted to the same subject, and it begins by establishing the same set of fundamental laws, but its methods are more general and its range of applications far wider. It exhibits the results, matured by some years of study and reflection, of a principle of investigation relating to the intellectual operations, the previous exposition of which was written within a few weeks after its idea had been conceived’” (MacHale, pp. 145-148). “A major difference from [*The Mathematical Analysis of Logic*] was the status of syllogistic logic. In the earlier book it had provided many of the examples analysed; here it débuts only on p. 226, in the last of the chapters on logic. Boole had realized in the meantime that he had advanced his algebra of logic far beyond the Aristotelian tradition” (Grattan-Guinness, p. 474).

“In Boole’s own words, the design of *The Laws of Thought* is to investigate the fundamental laws of those operations of the mind by which reasoning is performed; to give expression to them in the symbolical language of a calculus and, upon this foundation, to establish the science of logic and construct its method; to make that method itself the basis of a general method for the application of the mathematical doctrine of probabilities; and finally, to collect from the various elements of truth brought to view in the course of these inquiries some probable intimations concerning the nature and constitution of the human mind … In *The Laws of Thought*, he progressed a further step along the road to abstraction by performing algebraic operations on symbols representing entities not hitherto considered as mathematical objects at all. Thus if the symbol *x* represents the class of all ‘white objects’ and the symbol *y*, the class of all ‘round objects’, Boole used the compound symbol *xy* to represent the class of all objects that are simultaneously white and round. He saw that since the class of all objects that are white and round is precisely the same as the class of all objects that are round and white, his symbols obeyed the commutative law *xy = yx* under this particular law of composition. Many of the other laws of ordinary multiplication of numbers are also obeyed.

“In the case where every member of the class *x* is also a member of the class *y*, the law of combination of classes gives *xy = y*, and in the special case *x = y*, an allowable possibility, this equation becomes *xx= x*. By analogy with the multiplication of numbers, this is written *x*^{2} = *x*, an equation which is not in general satisfied by numbers, and this equations marks the point where Boole’s algebra of classes parts company with traditional numerical algebra.

“Again, if *x* and *y* are mutually exclusive classes (that is to say, there is no object which is simultaneously a member of *x* and *y*), Boole defines *x + y* to represent the class of all objects which belong to class *x* or to class *y*. Thus, if *x *is the class of all men and *y* is the class of all women, *x + y* is the class of all people. One immediately sees that *x + y* = *y + x* and that other laws resembling those of addition of numbers are valid. Furthermore, the two operations on classes thus defined are connected by a law which exactly mirrors the distributive law connecting addition and multiplication of numbers, namely *z*(*x + y*) = *zx* + *zy*. For example, if *z* represents the class of all Europeans, we see that the class of all European men and women is exactly the same as the class of European men and European women.

“Boole remarked that the laws of thought which he had stated were incorporated as laws of ordinary language, which he regarded as the product and instrument of thought … Boole made a further important discovery that strengthened this link between language and mathematics – namely that there is a perfect analogy between the calculus of classes and the calculus of simple propositions. Thus, simple propositions such as ‘snow is black’ or ‘London is a city’ could be manipulated and analyzed in combination in precisely the same way as classes could.

“However, it was in the purely symbolic manipulation of classes that Boole’s book made its greatest impact. Consider, for example, the classical implication ‘all As are B, all Bs are C; therefore, all As are C’. In Boole’s notation, the hypothesis would be written *a = ab*, *b = bc*. By substitution, *a = ab = a*(*bc*) = (*ab*) *c = ac*, which gives the desired conclusion … By analogy with numerical algebra, he assigned the symbol 0 to the empty class and the symbol 1 to the universal class. Further, if the class *y* contains the class *x*, he found it natural to represent the class of all objects in *y* but not in *x* by the notation *y – x*. Thus 1 – *x* is the class of all objects not in *x*. The fundamental relationship on which Boole’s results rested, *x*^{2} = *x*, is now easily seen to be algebraically equivalent to *x*(1 – *x*) = 0 …

“*The Laws of Thought* contained a great deal more than the important innovations just referred to. Boole introduced the concept of the development of any function *f*(*x*), where *x* is a logical symbol, writing it as *f*(*x*) = *f*(1)*x* + *f*(0)(1 – *x*); this concept was clearly motivated by his earlier interest in how Taylor’s Theorem applied to entities other than the usual symbols representing numbers. He also gave rules for the development of expressions such as *f*(*x*,*y*) and *f*(*x*,*y*,*z*), which forced him to introduce the symbols 0/0 and 1/0. Other topics treated include the solution of logical equations, methods of abbreviation and reduction, and the symbolical representation of secondary propositions.

“The second half of *The Laws of Thought* is devoted to the theory of probabilities, a subject in which Boole had become intensely interested after the publication of his book of 1847” (MacHale, pp. 149-151).

“The major difference between [*The**Mathematical Analysis of Logic*] and [*The Laws of Thought*] was the appearance in the latter of probability theory; it took up Chs. 16–21, at over 150 pages. Around 1849 he had realized that compound events could be handled in his logic as con- and/or disjunctions of simple ones, and so their consequences determined by his laws and expansions theorems and any attendant probabilities calculated accordingly. By these means he hoped to bring a new level of generality to the theory (p. 265) with probability logic.

“Boole’s construal of probability was epistemic: ‘the word *probability*, in its mathematical acceptation, has reference to the state of our knowledge under which an event may happen or fail. [. . . ] Probability is expectation founded upon partial knowledge’ (p. 244). However, he did not always distinguish the probability of a conditional proposition from conditional probability.

“In a remarkable chapter ‘on statistical conditions’ Boole considered situations in which the values of some or all probabilities may be known only approximately, or to within some upper and/or lower bounds. In his first case he showed that the probability of a dis- junction of events was less than the sum of the probabilities of each event (pp. 297–299), an inequality now named after him as part of his modest influence on the subject. Some of the more elaborate later cases led him to aspects of linear programming, which was not to develop as a mathematical topic for nearly a century although his was not the first anticipation” (Grattan-Guinness, p. 476).

“Boole (1815-64) was given his first lessons in mathematics by his father, a tradesman, who also taught him to make optical instruments. Aside from his father’s help and a few years at local schools, however, Boole was self-taught in mathematics. When his father’s business declined, George had to work to support the family. From the age of 16 he taught in village schools in the West Riding of Yorkshire, and he opened his own school in Lincoln when he was 20. During scant leisure time he read mathematics journals in the Lincoln’s Mechanics Institute. There he also read Isaac Newton’s *Principia*, Pierre-Simon Laplace’s *Traité de mécanique céleste*, and Joseph-Louis Lagrange’s *Mécanique analytique**,* and began to solve advanced problems in algebra.

“Boole submitted a stream of original papers to the new *Cambridge Mathematical Journal*, beginning in 1839 with his ‘Researches on the Theory of Analytical Transformations.’ These papers were on differential equations and the algebraic problem of linear transformation, emphasizing the concept of invariance. In 1844, in an important paper in the *Philosophical Transactions of the Royal Society* for which he was awarded the Royal Society’s first gold medal for mathematics, he discussed how methods of algebra and calculus might be combined. Boole soon saw that his algebra could also be applied in logic” (Britannica).

“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” (Grattan-Guinness, p. 471).

“Developing novel ideas on logical method and confident in the symbolic reasoning he had derived from his mathematical investigations, he published in 1847 a pamphlet, *Mathematical Analysis of Logic*, in which he argued persuasively that logic should be allied with mathematics, not philosophy. He won the admiration of the English logician Augustus de Morgan, who published *Formal Logic* the same year. On the basis of his publications, Boole in 1849 was appointed professor of mathematics at Queen’s College, County Cork, even though he had no university degree. In 1854 he published *An Investigation into the Laws of Thought, on Which Are Founded the Mathematical Theories of Logic and Probabilities*, which he regarded as a mature statement of his ideas. The next year he married Mary Everest, niece of Sir George Everest, for whom the mountain is named. The Booles had five daughters … In 1857 Boole was elected a fellow of the Royal Society. The influential *Treatise on Differential Equations* appeared in 1859 and was followed the next year by its sequel, *Treatise on the Calculus of Finite Differences*. Used as textbooks for many years, these works embody an elaboration of Boole’s more important discoveries” (Britannica).

Our copy of *The Laws of Thought* is the rare first issue. “The probable first issue of Boole’s *Laws of Thought*, of which the *Origins of Cyberspace* copy is an example, has the errata leaf bound in the back, and a binding of black zigzag cloth with blind-stamped border, panel, lozenge, and corner- and side-ornaments. The probable second issue has the errata leaf following the last numbered leaf of preliminaries, an additional printed ‘Note’ leaf following page 424 concerning a complex error, an eight-page Walton and Maberly publisher’s catalogue, and a binding of black blind-panelled zigzag cloth without the central lozenge. Both issues have an integral title-leaf with imprint reading ‘London: Walton and Maberly, Upper Gower-Street, and Ivy Lane, Paternoster Row. Cambridge: Macmillan and Co.’ A later issue has been noted in a green pebble-cloth binding, with a cancel title-leaf and imprint reading ‘London: Macmillan and Co.’”(Hook & Norman).

OOC 224 (1st issue, rebacked); Erwin Tomash B198 (2nd issue); Haskell Norman 266 (3rd issue). Goldstine, *The Computer from Pascal to von Neumann*, 1972. Grattan-Guinness, ‘George Boole, An Investigation of the Laws of Thought on which are Founded the Mathematical Theory of Logic and Probabilities,’ Ch. 36 in *Landmark Writings in Western Mathematics 1640-1940* (Grattan-Guinness, ed.), 2005. MacHale, *The Life and Work of George Boole. A Prelude to the Digital Age*, 2014.

8vo (225 x 143 mm), pp. [10], 424, [2, errata], uncut, text fresh and clean. Original publishers black blind-panelled cloth with gilt spine lettering, bound by Edmonds & Remnants, London (top of hinges with a very small tear, covers with some very light discoloration), entirely unrestored. Rare in such fine condition.

Item #3906

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Price:
$28,500.00
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