Proto-Boolean rings

We introduce the family of R-based proto-Boolean rings associated with an arbitrary commutative ring R. They generalize the protoBoolean algebra devised by Brown [4] as a tool for expressing in modern language Boole’s research in The Laws of Thought. In fact the algebraic results from [4] are recaptured within the framework of proto-Boolean rings, along with other theorems. The free Boolean algebras with a finite or countable set of free generators, and the ring of pseudo-Boolean functions, used in operations research for problems of 0–1 optimization, are also particular cases of proto-Boolean rings. The deductive system in Boole’s Laws of Thought [3] involves both an algebraic calculus and a “general method in Logic” making use of this calculus. Of course, Boole’s treatises do not conform to contemporary standards of rigour; for instance, the modern concept of Boolean algebra was in fact introduced by Whitehead [12] in 1898; see e.g. [9]. Several authors have addressed the problem of presenting Boole’s creation in modern terms. Thus Beth [1], Section 25 summarizes the approach of HoffHansen and Skolem, who describe the algebra devised by Boole as the quotient of an algebra of polynomial functions by the ideal generated by x − x, y − y, z − z, . . . . Undoubtedly the most comprehensive analysis of the Laws of Thought is that offered by Hailperin [5] in terms of multisets. Quite recently, Brown [4], taking for granted the existence of formal entities described as polynomials with integer coefficients, subject to the usual computation rules except that the indeterminates are idempotent (xi = xi), analyses Chapters V-X of [3] within this elementary framework, which he calls proto-Boolean algebra.