190 REASONING, 



elusion is the same. We are therefore at liberty, in con- 

 formity with the general opinion of logicians, 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 is affirmative, the other, when it is negative ; 

 even though certain arguments may have a tendency to clothe 

 themselves in the forms of the second, third, and fourth 

 figures ; which, however, cannot possibly happen with the 

 only class of arguments which are of first-rate scientific im- 

 portance, those in which the conclusion is an universal affirma- 

 tive, 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 theoiy 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 Hamil- 

 ton's Discussions on Philosophy, 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 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 neces- 

 sarily 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 examin- 

 ing what inferences admit of being drawn from the various combinations which 

 may be made of premises of this description, Mr. De Morgan establishes uni- 

 versal 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 the cases examined by Mr. De 

 Morgan, can legitimately be drawn, and that the ordinary theory takes no 



