Boole (1815–1864) was born of humble parents in Lincoln, England. His father was a cobbler with an active mind who was interested in mathematics and optical instru?ments. He provided an early influence on his son by teaching him mathematics, and teaching him how to make optical instruments. George Boole (Fig. 2.1) inherited his father’s interest in knowledge, and was self-taught in mathematics. He was taught Latin by a tutor, but was self-taught in Greek. He taught in various schools near Lincoln, and developed his mathematical knowledge by working his way through Newton’s Principia, as well as applying himself to the work of mathematicians such as Laplace and Lagrange. He published regular papers from his early twenties onwards, and these included contributions to probability theory, differential equations, and finite differences. He is especially remembered for his major contribution to Boolean algebra which is the foundation for modern computing. Boole is therefore considered (along with Babbage) to be one of the fathers of computing. Boole, never actually built any computer, as he lived well before the computer age. However, the mathematical foundation provided by Boole was exactly what was required for telephone switch?ing and computing, and it was Claude Shannon who saw the potential of Boole’s work, and who brought it to practical fruition. Boole’s work remains important today Boole was interested in formulating a calculus of reasoning, and in 1847 he pub?lished a pamphlet titled “Mathematical Analysis of Logic” [Boo:48]. This article developed novel ideas on a logical method, and he argued that logic should be con?sidered as a separate branch of mathematics, rather than being considered a part of philosophy. Boole argued that there are mathematical laws to express the operation of reasoning in the human mind, and he showed how Aristotle’s syllogistic logic could be rendered as algebraic equations. This publication was well received, and the British mathematician and logician De-Morgan1 spoke highly of it. There was regular correspondence between Boole and De-Morgan on logic. Boole was inter?ested in obtaining a university position, but due to his lack of a formal university education it was therefore difficult for him to achieve this goal. However, the value of his publications were recognized in Britain, and in view of the excellence of his work,2 and especially the pamphlet on Logic, he was awarded the position as the first professor of mathematics at the newly founded Queens University of Cork,3 Ireland in 1849. Boole’s influential paper on a calculus of logic introduced two quantities 0 and 1. He used the quantity 1 to represent the universe of thinkable objects (i.e., the universal set), with the quantity 0 representing the absence of any objects (i.e., the empty set). He then employed symbols such as x, y, z, etc., to represent collections or classes of objects given by the meaning attached to adjectives and nouns. Next, he introduced three operators (+, ?, and ?) that combined classes of objects. For example, the expression xy (i.e., x multiplied by y) combines the two classes x, y to form the new class xy (i.e., the class whose objects satisfy the two meanings represented by the classes x and y). Similarly, the expression x+y combines the two classes x, y to form the new class x +y (that satisfies either the meaning represented by class x or class y). The expression x ? y combines the two classes x, y to form the new class x ? y. This represents the class (that satisfies the meaning represented by class x but not class y). The expression (1 ? x) represents objects that do not have the attribute that represents class x. Thus, if x = black and y = sheep, then xy represents the class of black sheep. Similarly, (1 ? x) would represents the class obtained by the operation of selecting all things in the world except black things; x(1? y) represents the class of all things that areblack but not sheep; and (1 ? x) · (1 ? y) would give us all things that are neither sheep nor black. Similarly, if z = goats then y + z represents the class that are either sheep or goats. He showed that these symbols obeyed a rich collection of algebraic laws and could be added, multiplied, etc., in a manner that is similar to real numbers. Boole showed how these symbols could be used to reduce propositions to equations, and algebraic rules could be used to solve the equations. The algebraic rules satisfied by his system included: 1. x + 0 = x (Additive Identity) 2. x + (y + z) = (x + y) + z (Associativity) 3. x + y = y + x (Commutativity) 4. x + (1–x) = 1 (Not operator) 5. x1 = x (Multiplicative Identity) 6. x0 = 0 7. x + 1 = 1 8. xy = yx (Commutativity) 9. x(yz) = (xy)z (Associativity) 10. x(y + z) = xy + xz (Distributive) 11. x(y–z) = xy–xz (Distributive) 12. x 2 = x (Idempotent) 13. x n = x These operations are similar to the modern laws of set theory with the set union operation represented by “+”, and this is the “or” operation in modern Boolean algebra. The set intersection operation is represented by multiplication, and this is the “and” operation in Boolean Algebra. The universal set is represented by “1” is denoted by U in modern set theory, and the empty set “0” is denoted by ?. The laws of associativity and distribution hold. Finally, the set complement operation is given by (1 ? x), and this is the “not” operation of Boolean Algebra. Boole applied the symbols to encode propositions of Aristotle’s Syllogistic Logic, and he showed how the propositions could be reduced to the form of equa?tions. These equations allowed conclusions to be derived from premises by elimi?nating the middle term in the syllogism. Syllogistic logic had been in use for over 2000 years, and was the main form of logic employed in the nineteenth century. It remains of historical interest today as it has largely been replaced by predicate logic. Boole refined his ideas further in his book “An Investigation of the Laws of Thought” [Boo:58] published in 1854. This book and his earlier paper on logic contain the concepts which have come to be known collectively as Boolean alge?bra. His book aimed to identify the fundamental laws underlying reasoning in the human mind, and to give expression to these laws in the symbolic language of a calculus. Boole considered the equation x 2 = x to be one of the fundamental laws of though. It allows the principle of contradiction (i.e., that it is impossible for a being to possess an attribute and at the same time not to possess it) to be derived from this fundamental law of thought. For example, if x represents the class of horses then (1 ? x) represents the class of “not-horses”. The product of two classes represents a class whose members are common to both classes. Hence, x(1 ? x) represents the class whose members are at once both horses and “not-horses”, and the equation x(1 ? x) = 0 expresses that fact that there is no such class whose members are both horses and “not-horses”. That is, there are no members of such a class and it is the empty set. Boole made contributions to other areas apart from logic. He published papers on differential equations and on finite differences.4 He also published a book on differential equations which was used as a text-book at Cambridge University in England. He also contributed to the development of probability theory. Boole married Mary Everest in 1855 and they lived in Lichfield Cottage in Ballintemple, Cork. She was a niece of the surveyor of India, Sir George Everest, after whom the world’s highest mountain5 in the Himalayas is named. The Booles had five daughters, and one of their daughters was the author of the “The Gadfly”.6 Boole died prematurely (at the early age of 49) from pneumonia in 1864. He got soaked while walking two miles in the rain from his home in Ballintemple to teach his class at Queens University, Cork. He is buried in the graveyard of St. Michaels Church in Blackrock, Cork. Queens University Cork honoured his memory by installing a stained glass win?dow in the Aula Maxima of the college. This shows Boole writing at a table with Aristotle and Plato in the background. University College Cork is the modern name for Queens University Cork, and it named the new library after Boole in 1983. The Mathematics department at University College Cork holds an annual competition that is open to students of mathematics or engineering, and the winner is awarded the annual Boole prize. The prize recognises the recipient as having potential in mathematics. Further information on George Boole including his work and working relationship with Queens University Cork is in [Bar:69, McH:85]. Boole left a major legacy to the world as the design of all modern binary digital computers is dependent on Boolean algebra. The Boolean logical operations are implemented by electronic AND, OR and NOT gates, and from these fundamental building blocks more complex circuits may be designed for operations such as arith?metic. In view of this legacy, Boole is rightly considered to be one of the founding fathers of computing.