appear to be a necessary relation between universals may be purely subjective, and due to our inability to dissociate ideas that have been cemented together by custom. The third view does not by itself seem to explain the necessary character of the axioms, but it is a very safe view to hold, because the most important principles of knowledge are those that are applicable to the existing world; and, moreover, no proposition claiming to be an axiom is so far beyond the possibility of error that it can afford to despise the support of experience. The three views are not, indeed, mutually exclusive. Without any inconsistency, we may hold that the genuine axioms are those ultimate or indemonstrable propositions which are presupposed by a body of doctrines; that they do express relations based on the intrinsic natures of their terms, thus rendering it impossible to imagine them false; and that they are always corroborated, and never contradicted by experience.
We have now seen how universal concepts originate, and how the necessary judgments that express their relations are derived and validated. It remains to discuss the second or more objective aspect of the conflict between rationalism and empiricism, which involves the question, not as to how we derive universal terms and judgments, but as to what kind of ontological status they possess, and the extent of their applicability to the world of existence.
The Hyper-Syllogism: a Reduction of the Syllogisms of Formal Logic to Special Cases of the General Logic of Relatives.
We have seen that if two classes imply each other they are identical; and that whether we define implication as partial identity, or define identity as mutual or reversible implication, is merely a matter of convenience. I prefer the former, however, just because there are many cases in which relations other than those of implication are combined in reasoning. As, for example, when I say with De Morgan, "This is the head of an animal, because it is the head of a horse, and a horse is an animal"; or, "John is the nephew of James,