MODERN LOGIC: THE BOOLEAN PERIOD: BOOLE

George Boole (1815–1864) was the founder of modern mathematical logic. Nevertheless, few of his ideas are currently accepted in mainstream logic in the forms originally proposed by him. His learned and fertile mind conceived of several important hypotheses, the testing and modification of which changed the face of logic irrevocably. One of his most important hypotheses was that every proposition can be expressed using an algebraic equation suitably reinterpreted: that logic and algebra share a common uninterpreted formal language and thus also that they have similar problem types and similar methods.

The universal affirmative, or A proposition, "Every square is a rectangle" was expressed by x = xy, where x is the class of squares, y the class of rectangles, and xy the "Boolean or logical product" of x with y, the class of common members of x and y. The universal negative, or E proposition, "No rectangle is a circle" was expressed by yz = 0, where z is the class of circles and 0 is the empty class—an idea Boole introduced into logic. The conclusion "No square is a circle," xz = 0, which Aristotle and previous logicians deduced in one "intuitive" step, was derived by Boole using a chain of algebraic manipulations—illustrating another of his hypotheses, namely that on some level reasoning was mechanical or algorithmic.

He used 1 for the universe, or "universe of discourse," a ubiquitous expression in modern logic that Boole coined. He used the minus sign for "logical subtraction": 1 − x is the class of objects in the universe that are not in the class x. Using the above symbols, expression of the particular affirmative, or I proposition, "Some rectangle is a square" and the particular negative, or O proposition, "Some rectangle is not a square" as inequalities would have been easy: yx ≠ 0 and y (1 − x ) ≠ 0. This is a point that Boole never mentioned and probably did not notice—Boole's hypothesis was that algebraic equations were sufficient. Instead, he conceived of a logical operator, now called Boole's vee, or the vee, which was to produce from a class x a resultant class vx supposed by him to be "indefinite in every respect except that it contains some individuals of the class [x in this case] to whose expression it is prefixed." Using the vee, Boole "expressed" the above vy = vx and vy = v (1 − x ). The vee itself as well as the two "translations" have been criticized by later logicians—mainstream logic has not adopted Boole's vee, although its similarity to other more recent nonstandard operators has been noted—for example, the Hilbert epsilon.

Using the algebraic formal language, Boole was able to express several "laws of thought" analogous to laws of algebra; indeed some were expressed by the same equations used for laws of algebra—for example, the commutative law xy = yx. He employed his laws of thought in two unprecedented ways. First, regarding the equations as conditions on "unknowns," he created a wholly new theory of logical equation-solving using the laws of thought the way laws of algebra are used in numerical equation-solving. Second, regarding the most basic of his laws of thought as laws of logic, he created an axiomatization of logic. Boole realized that no "class logic" as such could treat the arguments now dealt with in truth-functional proposition logic. To meet this deficiency he proposed an ingenious reinterpretation of his system that, in his view, transformed it into something akin to propositional logic. In the process, he discovered key ideas now incorporated into modern truth-function logic, establishing himself as the first modern figure in any history of propositional logic. These are but three of Boole's many revolutionary innovations.

See also Aristotle; Boole, George; Propositions.

John Corcoran (2005)