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 can not be translated into Mr. De Morgan’s forms, which therefore can not serve any purpose as a test of it. Sir William Hamilton’s theory of the “quantification of the predicate” may be described as follows: “Logically” (I quote his 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 co-extensive 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 co-extensive. 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, 2d ed., p. 651): “The revocation of the two terms of a Proposition to their true relation; a proposition being always an equation of its subject and its predicate. “The consequent reduction of the Conversion of Propositions from three species to one—that of Simple Conversion. “The reduction of all the General Laws of Categorical Syllogisms to a single Canon. “The evolution from that one canon of all the Species and varieties of Syllogisms. “The abrogation of all the Special Laws of Syllogism.

“A demonstration of the exclusive possibility of Three Syllogistic Figures; and (on new grounds) the scientific and final abolition of the Fourth. “A manifestation that Figure is an unessential variation in syllogistic form; and the consequent absurdity of Reducing the syllogisms of the other figures to the first. “An enouncement of one Organic Principle for each Figure. “A determination of the true number of the Legitimate Moods; with “Their amplification in number (thirty-six); “Their numerical equality under all the figures; and “Their relative equivalence, or virtual identity, throughout every schematic difference. “That, in the second and third figures, the extremes holding both the same relation to the middle term, there is not, as in the first, an opposition and subordination between a term major and a term minor, mutually containing and contained, in the counter wholes of Extension and Comprehension. “Consequently, in the second and third figures, there is no determinate major and minor premises, and there are two indifferent conclusions: whereas in the first the premises are determinate, and there is a single proximate conclusion.” This doctrine, like that of Mr. De Morgan previously noticed, is a real addition to the syllogistic theory; and has moreover this advantage over Mr.