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):--
"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.