112 



BEASONING. 



would be a more natural mode of 

 stating the argument, and would carry 

 conviction more instantly home, than 

 the same ratiocination strained into 

 the first figure, thus— 



Aristides was virtuous, 

 Some pagan was Aristides, 



therefore 

 Some pagan was virtuous. 

 A German philosopher, Lambert, 

 whose Neues Organon (published in 

 the year 1764) contains among other 

 things one of the most elaborate and 

 complete expositions which had ever 

 been made of the syllogistic doctrine, 

 has expressly examined what sort of 

 arguments fall most naturally and 

 suitably into each of the four figures ; 

 and his investigation is characterised 

 by great ingenuity and clearness of 

 thought.* The argument, however, 

 is one and the same, in whichever 

 figure it is expressed ; since, as we 

 have already seen, the premises of a 

 syllogism in the second, third, or fourth 

 figure, and those of the syllogism in 

 the first figure to which it may be re- 

 duced, are the same premises in every- 

 thing except language, or, at least, as 

 much of them as contributes to the 

 proof of the conclusion is the same. 

 We are therefore at liberty, in confor- 

 mity with the general opinion of logi- 

 cians, to consider the two elementary 

 forms of the first figure as the universal 

 types of all correct ratiocination ; the 

 one, when the conclusion to be proved 



* His conclusions are, " The first figure 

 is suited to the discovery or proof of the 

 properties of the thing ; the second to the 

 discovery or proof of the distinctions be- 

 tween 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 re- 

 ference of syllogismsin the last three figures 

 to the dictum de omni et nullo is, in Lam- 

 bert'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 dictut^ de reci- 

 proco for the fourth. See part i. or Dianoio- 

 logie, chap. iv. § 229 et seq. Mr. Bailey, 

 (Theory of Reasoning, 2d ed. pp. 70-74) 

 takes a similar view of the subject. 



is affirmative, the other, when it is 

 negative ; even though certain argu- 

 ments may have a tendency to clothe 

 themselves in the form of the second, 

 third, and fourth figures ; which, how- 

 ever, cannot possibly happen with the 

 only class of arguments which are of 

 first-rate scientific importance, those 

 in which the conclusion is an universal 

 affirmative, such conclusions being 

 susceptible of proof in the first figure 

 alone.* 



* 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 Mor- 

 gan'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 Philonophy, and 

 at greater length, to his posthumous Lec- 

 tures 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 at- 

 tempt to bring within strict technical 

 rules the cases in which a conclusion can 

 be drawn froiw premises of a form usually 

 classed as particular. Mr. De Morgan ob- 

 serves, very justly, that from the premises 

 Most 13s 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 in- 

 dividuals. 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 beat 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 combina- 

 tions which may be made of premises of 

 this desci-iption, Mr. De Morgan establishes 

 universal formulae 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 tb© cases examined by Mr. De Morgan, 



