Boole’s Algebra Isn’t Boolean Algebra

Abstract

To Boole and his mid-nineteenth century contemporaries, the title of this article would have been very puzzling. For Boole’s first work in logic, The Mathematical Analysis of Logic, appeared in 1847 and, although the beginnings of modern abstract algebra can be traced back to the early part of the nineteenth century, the subject had not fully emerged until towards the end of the century. Only then could one clearly distinguish and compare algebras. (We use the term algebra here as standing for a formal system, not a structure which realizes, or is a model for, it—for instance, the algebra of integral domains as codified by a set of axioms versus a particular structure, e.g., the integers, which satisfies these axioms.) Granted, however, that this later full degree of understanding has been attained, and that one can conceptually distinguish algebras, is it not true that Boole’s “algebra of logic” is Boolean algebra?