Let us now examine the other assertion—that if the word serpent stands for none but real serpents, the minor premiss (A dragon is a serpent) is false. This is exactly what I have myself said of the premiss, considered as a statement of fact: but it is not false as part of the definition of a dragon; and since the premisses, or one of them, must be false, (the conclusion being so,) the real premiss cannot be the definition, which is true, but the statement of fact, which is false.
Contraries:
All A is B
No A is B
Subtraries:
Some A is B
Some A is not B
Contradictories:
All A is B
Some A is not B
Also contradictories:
No A is B
Some A is B
Respectively subalternate:
All A is B; No A is B
Some A is B; and Some A is not B
His conclusions are, “The first figure is suited to the discovery or proof of the properties of a thing; the second to the discovery or proof of the distinctions between things; the third to the discovery or proof of instances and exceptions; the fourth to the discovery, or exclusion, of the different species of a genus.” The reference of syllogisms in the last three figures to the dictum de omni et nullo is, in Lambert's opinion, strained and unnatural: to each of the three belongs, according to him, a separate axiom, co-ordinate and of equal authority with that dictum, and to which he gives the names of dictum de diverso for the second figure, dictum de exemplo for the third, and dictum de reciproco for the fourth. See part i. or Dianoiologie, chap. iv. § 229 et seqq.
Mr. De Morgan's “Formal Logic, or the Calculus of Inference, Necessary and Probable,” (a work published since the statement in the text was made,) far exceeds in elaborate minuteness Lambert's treatise on the syllogism. Mr. De Morgan's principal object is to bring within strict technical rules the cases in which a conclusion can be drawn from premisses of a form usually classed as particular. He observes, very justly, that from the premisses Most Bs are Cs, most Bs are As, it may be concluded with certainty that some As are Cs, since two portions of the class B, each of them comprising more than half, must necessarily in part consist of the same individuals. Following out this line of thought, it is equally evident that if we knew exactly what proportion the “most” in each of the premisses bear to the entire class B, we could increase in a corresponding degree the definiteness of the conclusion. Thus if 60 per cent of B are included in C, and 70 per cent in A, 30 per cent at least must be common to both; in other words, the number of As which are Cs, and of Cs which are As, must be at least equal to 30 per cent of the class B. Proceeding on this conception of “numerically definite propositions,” and extending it to such forms as these:—“45 Xs (or more) are each of them one of 70 Ys,” or “45 Xs (or more), are no one of them to be found among 70 Ys,” and examining what inferences admit of being drawn from the various combinations which may be made of premisses of this description, Mr. De Morgan establishes universal formulæ for such inferences; creating for that purpose not only a new technical language, but a formidable array of symbols analogous to those of algebra.
Since it is undeniable that inferences, in the cases examined by Mr. De Morgan, can legitimately be drawn, and that the ordinary theory takes no account of them, I will not say that it was not worth while to show in detail how these also could be reduced to formulae as rigorous as those of Aristotle. What Mr. De Morgan has done was worth doing once (perhaps more than once, as a school exercise); but I question if its results are worth studying and mastering for any practical purpose. 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. The “quantification of the predicate,” an invention to which Sir William Hamilton attaches so much importance as to have raised an angry dispute with Mr. De Morgan respecting its authorship, appears to me, I confess, as an accession to the art of Logic, of singularly small value. It is of course true, that “All men are mortal” is equivalent to “Every man is some mortal.” But as mankind certainly will not be persuaded to “quantify” their predicates in common discourse, they want a logic which will teach them to reason correctly with propositions in the usual form, by furnishing them with a type of ratiocination to which propositions can be referred, retaining that form. Not to mention that the quantification of the predicate, instead of being a means of bringing out more clearly the meaning of the proposition, actually leads the mind out of the proposition, into another order of ideas. For when we say, All men are mortal, we simply mean to affirm the attribute mortality of all men; without thinking at all of the class mortal in the concrete, or troubling ourselves about whether it contains any other beings or not. It is only for some artificial purpose that we ever look at the proposition in the aspect in which the predicate also is thought of as a class-name, either including the subject only, or the subject and something more.
Dr. Whewell (Of Induction p. 84) thinks it unreasonable to contend that we know by experience, that our idea of a line exactly resembles a real line. “It does not appear,” he says, “how we can compare our ideas with the realities, since we know the realities only by our ideas.” We know the realities (I conceive) by our eyes. Dr. Whewell surely does not hold the “doctrine of perception by means of ideas,” which Reid gave himself so much trouble to refute.