NUMERICALLY DEFINITE REASONING. 341 



16. It is interesting to compare my mode of treating 

 numerically definite propositions with the earlier mode of 

 Professor De Morgan. Taking X, Y_, and Z to be the 

 three terms of the syllogism^ he adopts^ the following no- 

 tation : — 



?<= whole number of individuals in the universe of 



the problem. 

 .2?= number of X^s. 

 2/= number of Y^s. 

 2r= number of Z's. 



Making m denote any positive number^ mXY means that 

 m or more X^s are Y''s. Similarly uYTi means that u or 

 more Y''s are Z''s. Smaller letters denote the negatives of 

 the larger ones^ somewhat as in my system. Thus m\.y 

 means that m or more X^s are not Y's^ and so on. 



From the two premises 



mXY and ?^YZ, 



Mr. De Morgan draws the two distinct conclusions 



{m-\-n—y)\.Z and {m-^n-{-u—w—y—2)xy. 



Let us consider what results are given by my own notation. 

 The premises may be represented by the equations 



(X Y) = m + m' (YZ) ^n + n\ 



where m and n are the same quantities as in Mr. De 

 Morgan^s system^ and m' and n' two unknown but positive 

 quantities, indicating that the number of XY^s is 7n or more, 

 and the number of YZ^s is n or more. 



The possible combinations of the three terms X_, Y, Ti, 

 and their negatives are eight in number, namely : — 



* Syllabus, p. 27. Mr. De Morgan denotes negative terms bj small 

 Roman letters, for which I have substituted italic letters. 



