NUMERICALLY DEFINITE REASONING. 343 



17. Mr. De Morgan's second conclusion,, that the number 

 of not-X's which are not Y's is 



{m-\-n + u — x — y^z) 



or more, may be examined in like manner. By developing 

 the classes numbered in each of these quantities, and stri- 

 king out the redundant terms, we obtain {xyz) — (K.yZ)j 

 in which the term (X?/Z) is wholly undetermined. Here, 

 again, we have as the lower limit of the class xz a quantity 

 indeterminately less than its own part xyz. The number 

 {xz) may accordingly be of any magnitude, while the limit 

 here assigned to it is zero, or even negative. 



Exactly similar remarks may be made concerning the 

 other conclusions which Mr. De Morgan draws. Thus, from 

 rriyiy and nYz (mX's or more are not Y's, and nY'% or more 

 are Z's) he infers 



{m + n — x)xZ and (m + w— •2')X^. 



But it will be found by analysis that the first of these 

 results has the following meaning : — 



{xZ) = {xYZ) - [XYz) ) 



that is to say, the lower limit of the class xZ is a part of 

 itself, xYZ, diminished by the number of another class 

 XY^. 



While believing, however, that Mr. De Morgan's mode 

 of treating the subject admits of improvement, it is im- 

 possible that I should undervalue the extraordinary acute- 

 ness and originality of his writings on this and many other 

 parts of formal logic. Time is required to reveal the wealth 

 of thought which he has embodied in his ' Formal Logic,' 

 and in his Logical Memoirs published by the Cambridge 

 Philosophical Society. 



18. In Mr. De Morgan's third paper on the syllogism ^ 



* Cambridge Phil. Trans, vol. x. part i. p. 8. 



