University of Alberta Dictionary of Cognitive Science: Boolean Algebra

Boolean algebra is a formal system of logic developed by George Boole (1854). In its modern conception, the various elements of Boolean algebra are functions that are defined over pairs of input variables. The truth or falsehood of a Boolean function depend upon the nature of the function, as well as the truth or falsehood of each a variable. For example, XOR is false is both input variables are true or if both are false; XOR is true if one input variable is true and the other is false.

Boolean algebra is fundamentally important because the logic operators that are at the core of any modern digital computer are Boolean in nature (Hillis, 1998).

While we ordinarily view Boolean algebra as being defined over pairs of variables, Boole introduced his formalism as an algebra of sets. He used addition to define the conjunction of sets, subtraction to define the disjunction of sets, and multiplication to define the intersection of sets. It was this latter move that led Boole to associate his algebra with the binary values of zero and one. A fundamental law of his algebra was that X² = X. There were only two numbers that led to this law being mathematically true: 0 and 1! He then proceeded to use this mathematical result to argue that 0 must mean the null set, and 1 must represent the universal set.

References:

Boole, G. (1854). An investigation of the laws of thought [on which are founded the mathematical theories of logic and probabilities]. London: Walton and Maberley.
Hillis, W. D. (1998). The Pattern on the Stone. New York: Basic Books.

(Added April 2010)