it is sufficient if, in both the premisses together, its quantification be more than its quantity as a whole (Ultratotal). Therefore, a major part (a more or most), in one premiss, and a half in the other, are sufficient to make it effective. It is enough for a valid syllogism, that the two extreme notions should (or should not), of necessity, partially coincide in the third or middle notion; and this is necessarily shown to be the case, if the one extreme coincide with the middle, to the extent of a half (Dimidiate Quantification); and the other, to the extent of aught more than a half (Ultradimidiate Quantification.) "1 In concrete examples : Three-fourths of the army were French; Three-fourths of the twelve pears were ripe; Therefore some that were ripe were stolen. This form of quantification and reasoning was first suggested by Lambert (Neues Organon, Dianoiologie, § 193 et seq.) It has since been adopted by De Morgan. Hamilton's view of it is, so far, a sound one: "These two quantifications should be taken into account by Logic as authentic forms, but then relegated as of little use in practice, and cumbering the science with a superfluous mass of moods."2 Again, 1 Logic, iv. p. 355. 2 Ibid. he lays down the principles which ought to limit a genuine science of Logic in the following words: "Such quantifications are of no value or application in the one whole (the universal, potential, logical), or, as I would amplify it, in the two correlative and counter wholes (the logical and the formal, actual, metaphysical), with which Logic is conversant. For all that is out of classification, all that has no reference to genus and species, is out of Logic, indeed out of Philosophy; for Philosophy tends always to the universal and necessary. Thus, the highest canons of Deductive Reasoning--the Dicta de Omni et de Nullo-were founded on, and for, the procedure from the universal whole to the subject parts; whilst, conversely, the principle of inductive reasoning was established on, and for, the (real or presumed) collection of all the subject parts as constituting the universal whole. 2°, The integrate or mathematical whole, on the contrary (whether continuous or discrete), the philosophers contemned. For whilst, as Aristotle observes, in mathematics genus and species are of no account, it is, almost exclusively, in the mathematical whole that quantities are compared together, through a middle term, in neither premiss equal to the whole. But this reasoning, in which the middle term is never universal, and the conclusion always particular, is-as vague, partial, and contingent of little or no value in Philosophy. It was accordingly ignored in Logic; and the predesignations more, most, &c., as I have said, referred to universal, or (as was most common) to particular, or to neither, quantity." This is a true insight into the real essence and needs of logical reasoning, as a universal means of thinking, and consequently of logical science. These words hold in themselves the condemnation, scientifically and practically, of the "advances " in Formal Logic, made on geometrical and algebraical lines, of De Morgan and Boole, and even of the more enlightened Jevons. § 541. A reasoning in which the middle term is never definitely known, and in which accordingly we have always a vacillating and particular conclusion, is of no use practically, or in the wide sphere of probable thought. Scientifically, it is a mere tentative,-ending in some is some,- -a mere ap 1 Logic, iv. pp. 353, 354. proach to satisfactory certainty. And even when the premisses are made numerically definite, as with De Morgan, the reasoning is of not the slightest use unless in reference to numbers and a numerical or mathematical whole. It is really of not the smallest consequence, as a rule, that we should know the exact numerical proportion of the middle term to the extremes. We seldom do know it, as a matter of fact, and when we do, we may remit the calculation to arithmetic. § 542. It ought further, I think, to be noted in connection with this form of reasoning, that it readily lends itself to material fallacy, or a conclusion materially untrue. No doubt, in the abstract, if 3 of Y are X, and of Y are Z, some of the Zs are Xs. So if X contains (the part) Y, and Y contains (the part) Z, X contains Z. But this latter formula embodies the law of inference from genus to species, or from whole to part. The other formula does not. It does not tell us in what relation X stands to Y, or Z to Y, whether that of part and whole, or of subject and attribute. Nor do we know, taking X and Z as attributes, whether they are compatible with each other or not. The practical application of the bare formula is therefore of but little use, and readily leads to material error. Thus, if we say : of the potatoes were diseased; was eaten by the crows; Therefore the crows must have eaten some of the diseased; this is correct, because there was not a half left not diseased. If, however, we substitute for diseased, hard as a stone, we should on the same formula have the conclusion that the crows ate some potatoes hard as a stone. There is nothing in the formula itself to prevent us substituting for X and Z incompatible attributes. Thus the following is quite compatible with the formula:: Three-fourths of men are saints; Therefore some who are saints are sinners. Such a formula can thus give a valid and true conclusion only in certain matter,-where the distribution refers to a whole of which the predicates are parts, or in which they are compatible attributes. In fact, the necessary premisses are: Three-fourths of the Ys are Xs; Therefore some of the Zs are Xs. Or, if three-fourths of Y are X, And if X and Z represent things which coexist in the Then some Z is X, or some Z may be thought to be X. 428 CHAPTER XXXII. CATEGORICAL SYLLOGISMS-COMPREHENSIVE REASONING § 543. The Aristotelic Categorical Syllogism proceeds mainly, if not exclusively, in the quantity of Extension. But according to later views, as we have seen, we have reasoning in Comprehension as well. § 544. In the view of Hamilton, every notion has not only an Extensive but an Intensive quantity-breadth and depthand these quantities always stand in an inverse ratio to each other. It would thus seem likely that if notions bear a certain relation to each other in Extension, they must bear a counter-relation to each other in Comprehension. Hence there will be reasoning in Comprehension, as there is reasoning in Extension. In Extension the reasoning runs :— All responsible agents are free-agents (i.e., are contained under the class); Man is a responsible agent (i.e., contained under the class); Therefore man is a free-agent (i..e., contained under the class). In comprehension we necessarily invert the process of this reasoning. The notion free-agent, which in the extensive reasoning is the greatest whole or major term, becomes in comprehension the smallest part or minor term, and the notion man, which is in extension the smallest part or minor term, now becomes the greatest whole or major term. The notion responsible agent remains the middle term in both reasonings; but what was formerly its part is now its whole, and what was formerly its whole is now its part. In Comprehension we reason thus: |