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 of Y are X, and 3 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;

Three-fourths of men are sinners ;

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;

Three-fourths of the Ys are [also] Zs ;

Therefore some of the Zs are Xs.

Or, if three-fourths of Y are X,

And if three-fourths of Y are Z,

And if X and Z represent things which coexist in the same (or are compatible),

Then some Z is X, or some Z may be thought to be X.

428

CHAPTER XXXII.

CATEGORICAL SYLLOGISMS-COMPREHENSIVE REASONING—
THE FIVE SYLLOGISTIC FORMS.

§ 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 Comiprehension we reason thus :

:

The notion man comprehends in it the notion responsible agent; The notion responsible agent comprehends in it the notion freeagent;

Therefore the notion man comprehends in it the notion freeagent.

In Extension.

B is A;
C is B;

.. C is A.

In Intension.

Cis B;

B is A;

.. C is A.

Thus, by reversing the order of the premisses and the meaning of the copula, we can always change a categorical syllogism of Extension into one of Intension, and vice versa. The reasoning in Comprehension has been generally overlooked by logicians; but it is genuine, and it is prior to extensive reasoning in the order both of nature and knowledge. Aristotle gives a definition of the middle term, which applies to the comprehensive reasoning.1

§ 545. Hamilton holds broadly that whatever mood and figure is valid in the one quantity is valid in the other, and every anomaly is equally an anomaly in both. The rules of Extensive reasoning are equally applicable to the Comprehensive reasoning, with the single proviso that all that is said of the sumption (major premiss) in extension is to be understood of the subsumption (minor premiss) in comprehension, and vice versâ.

§ 546. Of course the mere transposition of the premisses does not constitute the difference between reasoning in Comprehension and in Extension; that depends on the inner relation of the subject and predicate of the propositions as whole and part, or as part and whole. The transposition of the premisses in Extension or in Comprehension might, as Hamilton elsewhere remarks, be made without changing the essential character of the reasoning. It would not be natural, but it would not affect the reasoning as a mental process. But the position of the premisses as indicated is the natural way of showing when we reason in Comprehension or in Extension. course, it is hardly necessary to say in passing that Hamilton does not, as Mill states, make the distinction of Comprehension and Extension depend merely on the transposition of the premisses.2

1 Logic, L. xvi. p. 299, and above, p. 407.

2 Examination, p. 505.

Of

$547. The quantities of Breadth and Depth are explicitly held by Hamilton to be merely views of the same relation from opposite points, not things in themselves different.1 He combats the view that the reading a proposition in depth in contrast to its reading in breadth is "not another reading of the same proposition, but another proposition derived inferentially, though not syllogistically."

He holds very distinctly that Breadth and Depth, though named quantities, are really one and the same quantity, viewed in counter-relations and from opposite ends. Nothing is the one which is not, pro tanto, the other. Though different in the order of thought (ratione), the two quantities are identical in the nature of things (re). In effect it is precisely the same reasoning, whether we argue in Depth or in Breadth. Thus, in Depth, we may argue the individual Z is (or contains in it attribute) some Y; all Y is some U; all U is some O; all 0 is some I; all I is some E; all E is some A; therefore Z is some A. (Take Socrates, Athenian, Greek, European, Man, Mammal, Animal.)

In Breadth, the argument would be the same: Some A (i.e., as class contains under it the subject part) is all E; some E is all I; some I is all O; some O is all U; some U is all Y; some Y is Z; therefore some A is Z. (Reverse the concrete concepts already given.) Hamilton adds that as the proposition in either quantity is only an equation, only an affirmation of identity or its negation, the substantive verb is or is not expresses the relation more accurately, than containing and contained,-whether in or under. We are told, also, that in syllogisms the contrast of the two quantities is abolished, and the differences of figure, major and minor, premiss and term, likewise disappear.

§ 548. It has been objected to this that "the two modes of reading propositions in Depth and Breadth are not convertible; the extensive mode gives the intensive, but not vice versâ in all cases." "In the affirmative, any portion of the intension of the predicate may be affirmed of the subject; in the negative, it is not true that any portion of the intension of the predicate may be denied of the subject. Thus, 'No

1 See Discussions, p. 697. Hamilton gives the fullest and most explicit account of his views on Breadth and Depth in Reasoning in connection with the Table figured in Discussions, p. 699.