liv 



Appendix. 



The last of the three is, as we have seen, distinguishable into two, each of 

 which has the same amount of probability as the first or second alternative. 



One other example we may adduce, which also was miscalculated by a 

 notable personage, the eminent philosopher of our own day, John Stuart Mill. 

 Suppose that a thing, which we shall call T, is a member of the class A, and 

 that of the members of this class just two-thirds have the attribute X ; also, 

 that the same thing T is a member of another class, viz., J9, and that in this 

 class the same attribute X pertains to just three-fourths of the members — the 

 membership in the one class being assumed to be unconnected with that in the 



other : what is the resultant probability that the thing T possesses the 



attribute X ? In the earlier editions of the " System of Logic " this question 

 was answered erroneously ; but subsequent editions gave the correction. The 

 discussion occupies several pages of that work, and is rather abstruse ; but by 

 our having recourse at once to the fundamental principle, that of dividing the 



qual 



Two-thirds of the class 



A is the portion possessing the attribute X : we may, therefore, consider the 

 class A as consisting of sets each composed of three members ; and of each 

 triad let the first two possess the attribute in question, and the third want it. 

 The members of each triad we will designate as A x , A 2 , A3. Similarly, the 

 class B consists of quaternions, in each of which we designate the members as 

 B x , B 2 , B 3 , B 4 ; and of these let the fourth alone be without the attribute X. 

 The whole case, then, consists of the following 7 alternatives : 



(1.) T is Ai and Bi, or 



(2.) T is A 1 and B 2 , or 



(3.) T is A x and B 3 . 

 These are the first three alternatives ; and we have now exhausted A x , because 

 T cannot be A 1 and B 4 ; inasmuch as A a has the attribute X, and B 4 wants it. 

 The remaining alternatives are, therefore, as follows : 



(4.) T is A 2 and B x , or 



(5.) T is A 2 and B 2 , or 



(6.) T is A 2 and B 3 , or else, lastly, 



(7.) T is A 3 and B 4 . 



equal 



six 

 and 



is, 



Mr 



the resultant to be eleven-twelfths, until set right by a mathematical friend. 

 It would occupy too much of our time and attention to exhibit here the 

 manner in which the error was incurred. 



assign 



certain objective quantity, of whatsoever kind it may be. This quantity is 



