frequently in actual controversies it is assumed that the denial of what is taken for an A proposition carries with it the truth of its contrary E. The fact simply calls attention to the arbitrary assumption that "all propositions can be interpreted according to the relations of quantity and quality expressed in the accepted theory of Opposition, and what I wish to show here is either the existence of propositions to which formal rules do not apply, or the need of multiplying the theoretical principles of Opposition in order to meet a variety of thought relations not hitherto reckoned with in logic.
In the current theory the relations of contradiction, contrariety, and subalternation are based upon the assumptions, not stated by the logician, that A and E are taken distributively, and I and O unambiguously; that is, the sign "some" denotes a part, and it may or may not be the whole. The least modification or variation from these assumptions in practice must make the rules perfectly useless, at least as mechanical agencies for testing the truth of certain assertions under any conditions. Thus, if my propositions are collective in their import, the rules for subalternation are absurd, and we know that considerable of our actual reasoning turns upon this very function of judgment. Again, if the word "some," or the thought involved without using the term, mean only a part and not all, the relation of subalternation is again modified, and neither A nor E can be true when I or O is true. Perhaps this duplex use of "some" in practice is more common than the one on which the current rules are founded. Similar exceptions occur with the use of what are called "general," and perhaps should be called "generic" propositions, which I shall notice after showing the complications involved in the variations from the assumptions upon which rests the current theory.
To show what a limited application to actual practice the usual rules have, I have only to call attention to the ambiguities of the term "All." First, this has a distributive, second, a collective, and third, a duplex or implicative import. The third meaning is that of all and no less, and is connected with
