George Boole Develops Boolean Algebra

In 1847 English mathematician and philosopher George Boole published a pamphlet entitled The Mathematical Analysis of Logic—his first exposition of Boolean algebra. Seven years later in 1854, Boole published a much longer exposition entitled An Investigation of the Laws of Thought, on Which are Founded the Mathematical Theories of Logic and Probabilities. This work contains the full expression of the first practical system of logic in algebraic form.

"He [Boole] did not regard logic as a branch of mathematics, as the title of his earlier pamphlet [The Mathematical Analysis of Logic (1847)] 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 represent logical forms and syllogisms, that we can hardly help saying that (especially his) formal logic is mathematics restricted to the two quantities, 0 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 algebraic symbols, whence it followed that they could be added, subtracted, multiplied and even divided, almost exactly in the same manner as numbers. Thus, (1 - x) would represent the operation of selecting all things in the world except horned things, that is, all not horned things, and (1 - x) (1 - 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, formed a general symbolic method of logical inference. Given any propositions 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" (Wikipedia article on George Boole, accessed 01-09-2008).

Though the audience for Boole's highly specialized work would have been judged to be small, and the edition size reduced accordingly, the existence of three issues of the first edition, all dated 1854, would suggest that the edition may have required several years to sell. The points of the issues are as follows:

1. Probable first issue: London: Walton and Maberly, Upper Gower-Street, and Ivy Lane, Paternoster-Row. Cambridge: Macmilan and Co., errata leaf bound in the back, and binding of black zigzag cloth with blindstamped border, panel, central lozenge and corner and side ornaments.

2. Probable second issue: London: Walton and Maberly as above, but with the errata after the last numbered leaf of preliminaries, an additional printed "Note" leaf following 2E4 concerning a more complex error, an eight-page Walton and Maberly catalogue of "Educational Works and Works in Science and General Literature" and a binding of black blind-panelled zigzag cloth without the central lozenge.

3. Third issue: London: Macmillan and Co. Errata on recto of last unsigned leaf, and bound in green cloth, gilt-lettered spine. This may be a later, or remainder binding

Hook & Norman, The Haskell F. Norman Library of Science and Medicine (1991) no. 266.