Boole, George See BOOLEAN ALGEBRA, LOGICAL. FOR. Boolean algebra, (1) an ordered triple (B,†,3), where B is a set containing at least two elements and † and 3 are unary and binary operations in B such that (i) a 3 b % b 3 a, (ii) a 3 (b 3 c) % (a 3 b) 3 c, (iii) a 3 † a % b 3 † b, and (iv) a 3 b = a if and only if a 3 † b % a 3 † a; (2) the theory of such algebras. Such structures are modern descendants of algebras published by the mathematician G. Boole in 1847 and representing the first successful algebraic treatment of logic. (Interpreting † and 3 as negation and conjunction, respectively, makes Boolean algebra a calculus of propositions. Likewise, if B % {T,F} and † and 3 are the truth-functions for negation and conjunction, then (B,†,3) – the truth table for those two connectives – forms a two-element Boolean algebra.) Picturing a Boolean algebra is simple. (B,†,3) is a full subset algebra if B is the set of all subsets of a given set and † and 3 are set complementation and intersection, respectively. Then every finite Boolean algebra is isomorphic to a full subset algebra, while every infinite Boolean algebra is isomorphic to a subalgebra of such an algebra. It is for this reason that Boolean algebra is often characterized as the calculus of classes. See also SET THEORY, TRUTH TABL. G.F.S.