QUANTIFICATION OF TIVE PREDICATE, a phrase belonging to logic. and introduced' by sir W. Hamilton to express the characteristic of certain logical doctrines of his. respecting the proposition and the syllogism.
According to theAristotelian logic. propositions are divided. according to their quality, into affirmative and negative (" the sun has set," " the sun has not set"); and, accord ing to their QUANTITY, into universal and particular ("all men are mortal." "some mem live eighty years"). If we combine the two divisions, we obtain four kinds of proposi tions—affirmative universal (" all men are mortal "), affirmative particular (" some men live to eighty"), negative universal (" no men are omninotent"), negative particular ("some men are•not wise").
Now, it is remarked by sir W. Hamilton, that the statement of the QUANTITY of these various propositions is left incomplete; only the subject of each has its quantity expressed (all men, some men, no men); while there is implied or understood in every case a certain quantity of the predicate. Thus, "all men are mortal," is not fully stated; the meaning is, that all men are a part of mortal things. there being (possibly and probably) other mortal things besides .men. Let this meaning be expressed, and we have a complete proposition to this effect: " all men are some (or part of) mortals," where quantity is as signed, not only to the subject, but also to the predicate. It might be that the predicate contained under it only the subject, as in the proposition: " all matter gravitates." There is no other thing in the universe except matter that obeys the law of gravitation. Knowing this, we might quantify the predicate accordingly: "all matter is all gravitat ing things," a kind of proposition not recognized in the old logic. Another original form of proposition, brought out by supplying the quantity of the predicate, is, "some A is all B;" "some men are all Englishmen." So that, instead of two kinds of propositions under affirmation, sir IV. Hamilton's system gives four. In the same way, lie increases the number of negative propositions. 1. For "no man is omnipotent," he writes, gum-.
tifying the predicate, " any man is not any omnipotent;" or, "all men are out of all om nipotent things." 2. " Some men are not young" is fully quantified; "some men are not any young things;" "some men are hut of all young things." These two (in their unquantified shape) are the ordinarily recjignized propositions of the negative class. To them sir W. Hamilton adds: 3. "All men are not some animals," "all men are excluded
from a certain division of the class animal;" and 4. "Some animals are not some men;' " a portion of the animals is not included in a portion of men." The first result, therefore, of completing the statement of a proposition by inserting what Hamilton considers as implied in the thought—namely, the quantity of the predi cate—is to give eight kinds of propositions instead of four. The next result is to modify the process called the conversion of propositions. See CONVERSE. The kind of con version called limitation (all A is B, some B is A) is resolved into simple conversion, or mere transposition of premises without further change. "All A is some B;" "some is all A." The multiplication of varieties of propositions is 'attended with the further conse quence of greatly increasing the number of syllogisms, or forms of deductive reasoning_ See SYI.LOGISNI. In the scholastic logic, as usually expounded, there are nineteen such forms, distributed under four figures (four in the first, four in the second, six in the third, five in the fourth). By ringing the changes on eight sorts of propositions, instead. of the old number, four, thirty-six valid syllogisms can be formed in the first figure_ Whether the increase serves any practical object, is another question.
Sir W. Hamilton also considers that he has been led, by the new system, to a simpli fication of the fundamental laws of the syllogism, or, as he expresses it, " the reduction of all the gen,era!, laws of categorical syllogisms to a single canon.".
Professor De Morgan, in his elaborate system of Formal Logic, has also invented and carried out into great detail a plan of expressing the quantity of the predicate; but he does not admit the whole of Hamilton's eight propositional forms, rejecting in particular the last-mentioned in the above enumeration. He also increases the number of valid -syllogisms as compared with the old logic. Not content with indicating that the pred icate has quantity as well as the subject, he supposes the possibility of a numerical esti mate of quantity in both terms of the proposition, and from this draws a new set of in ferences. Thus, if 60 per cent of B are included in C, and 70 per cent in A, 30 per cent at least of B must be found both in A and in C.—See sir W. Hamilton's Discussions; Spencer Baynes's New Analytic of Logical Forms; De Morgan's Formal Logic; Mill's Logic, under the syllogism; and his Examination of Sir IV. Hamilton's Philosophy.