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. Bailey (Theory of Reasoning, 2d ed., pp. 70-74) takes a similar view of the subject.
Since this chapter was written, two treatises have appeared (or rather a treatise and a fragment of a treatise), which aim at a further improvement in the theory of the forms of ratiocination: Mr. De Morgan’s “Formal Logic; or, the Calculus of Inference, Necessary and Probable;” and the “New Analytic of Logical Forms,” attached as an Appendix to Sir William Hamilton’s Discussions on Philosophy, and at greater length, to his posthumous Lectures on Logic. In Mr. De Morgan’s volume—abounding, in its more popular parts, with valuable observations felicitously expressed—the principal feature of originality is an attempt to bring within strict technical rules the cases in which a conclusion can be drawn from premises of a form usually classed as particular. Mr. De Morgan observes, very justly, that from the premises 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 premises 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 premises 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 formulæ 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.