BOOLE, GEORGE English logician and math ematician, was born in Lincoln on Nov. 2, 1815, the son of a tradesman of limited means. When about 16 years of age, Boole became assistant-master in a private school at Doncaster. Later he established a successful school at Lincoln, and in 1849 was made professor of mathematics in Queen's college, Cork. Boole's earliest published paper, on the "Theory of Analytical Transformations," printed in the Cambridge Mathematical Journal for 1839, led to a friendship with D. F. Gregory, the editor. Only two systematic treatises on mathematical subjects were completed by him during his lifetime. The well-known Treatise on Differential Equations appeared in 1859 (supplementary posthumous volume, 1865), and was followed, the next year, by a, Treatise on the Calculus of Finite Differences, designed to serve as a sequel to the former work. In the i6th and t7th chapters of the Differential Equations there is a lucid account of the general symbolic method, the bold and skilful employment of which led to Boole's chief discoveries, and of a general method in analysis, originally described in his famous memoir printed in the Philosophical Transactions for 1844. Boole was one of the most eminent of those who perceived that the symbols of operation could be separated from those of quantity and treated as distinct objects of calculation.
With the exception of Augustus de Morgan, Boole was prob ably the first English mathematician since John Wallis who had also written upon logic. Speculations concerning a calculus of rea soning had occupied Boole's thoughts, but it was not till the spring of 1847 that he put his ideas into the pamphlet called Mathemat ical Analysis of Logic. Boole afterwards regarded this as a hasty and imperfect exposition of his logical system, and he desired that his much larger work, An Investigation of the Laws of Thought, on which are founded the Mathematical Theories of Logic and Proba bilities (1854), should alone be considered as containing a mature statement of his views. He did not regard logic as a branch of mathematics, as the title of his earlier pamphlet might be taken to imply, but he pointed out such a deep analogy between the symbols of algebra and those which can be made, in his opinion, to repre sent logical forms and syllogisms that we can hardly help saying that logic is mathematics restricted to the two quantities, o and 1. By unity Boole denoted the universe of thinkable objects; literal symbols, such as x, y, z, v, u, etc., were used with the elective meaning attaching to common adjectives and substantives. Thus, if x= horned and y = sheep, then the successive acts of election represented by x and y, if performed on unity, give the whole of the class horned sheep. Boole showed that elective symbols of this kind obey the same primary laws of combination as algebraical symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, i —x would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 —x) (I —y) would give us all things neither horned nor sheep. By the use of such symbols propositions could be reduced to the form of equations, and the syllogistic conclusion from two premises was obtained by eliminating the middle term according to ordinary algebraic rules.
Still more original and remarkable, however, was that part of his system, fully stated in his Laws of Thought, which formed a general symbolic method of logical inference. Given any propo sitions involving any number of terms, Boole showed how, by the purely symbolic treatment of the premises, to draw any conclusion logically contained in those premises. The second part of the Laws of Thought contained a corresponding attempt to discover a general method in probabilities, which should enable us from the given probabilities of any system of events to determine the consequent probability of any other event logically connected with the given events. Boole died on Dec. 8, 1864.
For Boole's memoirs and detached papers see Catalogue of Scientific Memoirs, published by the Royal Society, and the supplementary volume on Differential Equations, edited by Isaac Todhunter. In the Cambridge Mathematical Journal and its successor, the Cambridge and Dublin Mathematical Journal, and in the Philosophical Magazine there are other papers. The Royal Society printed six important memoirs in the Philosophical Transactions, and a few other memoirs are to be found in the Transactions of the Royal Society of Edinburgh and of the Royal Irish Academy, in the Bulletin de l'Academie de St. Peters bourg for 1862 (under the name G. Boldt, vol. iv. pp. 198-215), and in Crelle's Journal. See R. Harley's article in the British Quarterly Review, July, 1866, No. 87. (W. S. J.)