Since it is undeniable that inferences, in the cases examined by Mr. De Morgan, can legitimately be drawn, and that the ordinary theory takes no account of them, I will not say that it was not worth while to show in detail how these also could be reduced to formulæ as rigorous as those of Aristotle. What Mr. De Morgan has done was worth doing once (perhaps more than once, as a school exercise); but I question if its results are worth studying and mastering for any practical purpose. The practical use of technical forms of reasoning is to bar out fallacies: but the fallacies which require to be guarded against in ratiocination properly so called, arise from the incautious use of the common forms of language; and the logician must track the fallacy into that territory, instead of waiting for it on a territory of his own. While he remains among propositions which have acquired the numerical precision of the Calculus of Probabilities, the enemy is left in possession of the only ground on which he can be formidable. And since the propositions (short of universal) on which a thinker has to depend, either for purposes of speculation or of practice, do not, except in a few peculiar cases, admit of any numerical precision; common reasoning cannot be translated into Mr. De Morgan's forms, which therefore cannot serve any purpose as a test of it. Sir William Hamilton's theory of the "quantification of the predicate" (concerning the originality of which in his case there can be no doubt, however Mr. De Morgan may have also, and independently, originated an equivalent doctrine) may be briefly described as follows:-- "Logically" (I quote his own words) "we ought to take into account the quantity, always understood in thought, but usually, for manifest reasons, elided in its expression, not only of the subject, but also of the predicate of a judgment." All A is B, is equivalent to all A is _some_ B. No A is B, to No A is _any_ B. Some A is B, is tantamount to some A is _some_ B. Some A is not B, to Some A is _not any_ B. As in these forms of assertion the predicate is exactly coextensive with the subject, they all admit of simple conversion; and by this we obtain two additional forms--Some B is _all_ A, and No B is _some_ A. We may also make the assertion All A is all B, which will be true if the classes A and B are exactly coextensive. The last three forms, though conveying real assertions, have no place in the ordinary classification of Propositions. All propositions, then, being supposed to be translated into this language, and written each in that one of the preceding forms which answers to its signification, there emerges a new set of syllogistic rules, materially different from the common ones. A general view of the points of difference may be given in the words of Sir W. Hamilton (_Discussions_, 2nd ed. p. 651):--