We now proceed to define certain terms which we shall apply to sets of axioms and derive a few properties. A great deal of what follows in this section is still based on a two-valued logic, while our criticism of the usual independence and consistency proofs is based on an n-valued logic. The necessary alterations are now being worked out by the author.

The controversy which has raged between those who accept Aristotelian Logic and those who reject it, seems to continue. It is, I believe, advisable to consider the arguments on both sides. Modern Logicians for the most part insist that Aristotelian Logic breaks down, i.e. it is not true for all meanings of the terms. On the other hand there are those who insist on the validity of Aristotelian Logic claiming that those values for which it fails are not permissible values. (...) But there is one who stands almost entirely alone in his position—namely Professor H. B. Smith—who maintains that Aristotelian Logic breaks down because and only because its interpretation into Boolean terms has been accomplished by means of a definition that is not forced upon us and which, since it gives up desirable principles, is to be rejected. (shrink)

The keynote of our investigation is suggested by a remark made by Enriques, “the young student of mathematics would indeed look in vain to the classical logic into which he was initiated for an adequate conception of the structure of a deductive science like Geometry let alone for an explanation of the value and meaning of the principles of such a discipline. What are axioms, postulates, definitions? What place do they occupy in the edifice of the theory? What criteria determine (...) their choice and how can we establish whether they are acceptable or not?” These are in general the questions we shall seek to answer in part and then we shall make a special analysis of two of the criteria determining the choice of the axioms. (shrink)

In this study we shall analyze the postulate set into its various elements. In order to carry out our analysis, it is advisable to have before us a typical set.We take the set of five postulates for Boolean Algebras given by H. M. Sheffer in Trans. Am. Math Soc. 14, p. 482 ff.