George Boole
George Boole [ ˌ bu ː dʒ dʒɔ ː l] ( born November 2, 1815 in Lincoln, England; † December 8, 1864 in Ballintemple, County Cork, Ireland ) was an English mathematician ( self-taught ), logician and philosopher.
Life
Boole was originally a teacher. Due to his scientific work, he was in 1848 professor of mathematics at Queens College in Cork (Ireland). From the Royal Society, he was in 1844 awarded the Royal Medal. In 1857 he was elected a member ( "Fellow" ) of the Royal Society.
George Boole is the father of Ethel Lilian Voynich writer (1864-1960) and of Alicia Boole Stott (1860-1940), who as an amateur mathematician succeeded without formal education to classify the regular polyhedra in four dimensions.
Main work
Boole created in his book The Mathematical Analysis of Logic of 1847 the first algebraic calculus of logic, arguing that modern mathematical logic that stands out from the traditional philosophical logic by a consistent formalization. He formalized the classical logic and propositional logic and developed a decision procedure for the true formulas of a disjunctive normal form. Boole took it - because of the decidability of classical logic completeness and consistency follows - already a good 70 years before Hilbert's program for a central logic area to solve the problems posed by David Hilbert in advance. As a generalization of Boole's calculus of logic called Boolean algebra and Boolean ring were named after him later.
Boole's original calculus
Boole used for his logical calculus the then known algebra, which is now specified as a power series ring over the field of real numbers. In this algebra it embedded a classical logic by the conjunction "x and y" as a multiplication xy and the negation of " not X" 1-x formalized. It is a real embedding, in which not all terms have a logical sense; for the logically meaningful terms he demanded the idempotency xx = x, which does not apply generally in algebra, for example, not for the usual addition x y, which is why Boole called uninterpretable. He thus used partial operations and spoke at the logical significant terms and operators of elective symbols, elective functions, elective equations.
Boole designed his calculus class primarily as a logic in which one the universe ( the universal class ) and the unknowns x, y, represent, for .. classes. Within this class calculus, he then presented the traditional syllogistic represents the two basic syllogistic predicates he represented by equations, ie " All x are y" with x = xy and "No x are y" by xy = 0 These equations served as rules by which he derived the Aristotelian- scholastic syllogisms in the form of systems of equations on metalogischer level.
Secondary used Boole 's calculus. Well as propositional logic, in which the unknowns x, y, represent such statements .. Enclosing the disjunction "x or y" he formalized by the expression x y -xy, and the exclusive disjunction " either x or y " with x y 2xy. They arise via the following algebraic derivations:
With equations he recognized the truth and falsity of statements, namely "x is true" by x = 1 and " x is false " by x = 0 He used here ie 0 and 1 as truth values . His logical decision process via a normal form, he added with an equivalent semantic decision procedure with truth - value substitutions in Boolean functions that map each occupied logic term a truth value. This method corresponds to the decision-making process with truth tables, which is used to identify tautologies.
Modifications of Boole's calculus
Boole's original calculus can be modified so that there is no logically meaningless terms longer occur, namely a power series ring over the idempotent body from the bits { 0,1} with Boole Idempotenzgesetz xx = x. The result is a so-called Boolean ring, Ivan Ivanovich Schegalkin 1927 introduced and the Marshall Harvey Stone in 1936 gave the name. In it you can do without the minus sign, since it is self-inverse and x = x and x x = 0 apply; This makes the addition of the exclusive disjunction synonymous " either x or y ". Boolean rings are mathematically elegant, because here the school known calculation rules apply. The necessary for the decidability of a normal form formula arises here simply by multiplying distributive and brushing twice factors and summands with xx = x and x x = 0
Under the Boolean algebra is today not Boole's original algebra understood, but the so-called Boolean association, which is equivalent to the Boolean ring, but was earlier developed by Boole 's successors, especially by Ernst Schröder in 1877 and Giuseppe Peano, 1888. Boolean association is in of propositional logic and set theory widely used and works with the conjunction, negation and inclusive disjunction.
Writings
- George Boole: The mathematical analysis of logic: being an essay towards a calculus of deductive reasoning, 1847 Transferred from the English, annotated and with an afterword by Tilman Recover: The mathematical analysis of logic. Hall shear Verlag, 2001. 195 pp. ISBN 3-929887-29-0
- Reduced and transferred from the English printed in Karel Berka, Lothar Kreiser: logic texts. Commenting selection on the history of modern logic, Berlin: Akademie 4th edition, 1986, page 25-28
- George Boole: An Investigation of The Laws of Thought, New York: Dover 1958 ISBN 0-486-60028-9.