A simplification of the Whitehead-Huntington set of postulates for boolean algebras

OP the various sets of postulates that have been given for Boolean logic the most elegant and natural is the set of Huntington's based on Whitehead's "formal laws."* This set may be simplified by reducing the number of its postulates without injuring, the writer feels, the elegance or the naturalness of the original. This reduction is effected by substituting for Huntington's Postulates IIa, II&, and V the following single postulate: POSTULATE X. For any element b in the class there exists an element b juch that, whatever a is, a © (b 0 b) = a and a o (6 e b) = a. Evidently, Huntington's Postulates IIa, II&, and V follow from Postulate X, with the help of Ia and I&. Evidently, also, Postulate X can be derived from IIa, II&, and V, with the help of Ia, lb, IHa, and III&. It is of course seen that by adopting Postulate X in place of IIa, II&, and V, not only is the number of Huntington's postulates reduced from ten to eight, but also the number of postulated special elements is reduced from three ("zero," the "whole," and the "negative") to one (the "negative"). In establishing the independence of the modified set of postulates Huntington's systems for Ia, I&, IVa, IV&, VI can serve for the same numbered postulates in the new set. For Postulate X we can take Huntington's system for V. For IIIa and III&, however, a class of more than two elements is, in each case, necessary. Proof-systems for these two postulates are, respectively, the following: