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., B, and that in this 

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

 case into equal alternatives, the solution becomes easy. Two-thii'ds 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 Ai, A2, A3. Similarly, the 

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

 Bi, B2, B3, B4 ; 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 Ai and B2, or 



(3.) T is Ai and B3. 

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

 T cannot be Ai and B4 ; inasmuch as A^ has the attribute X, and B4 wants it. 

 The remaining alternatives are, therefore, as follows : — 



(4.) T is A2 and B^, or 



(5.) T is A2 and B2, or 



(6.) T is A2 and B3, or else, lastly, 



(7.) T is A3 and B4. 

 Of these seven equal alternatives, one or other of which must be time, the first 

 six afiirm the proposition in question, viz., that T possesses the attribute X, 

 and the last alternative alone negatives it ; so that the resultant probability of 

 the proposition is, at the most, six-sevenths. Mr. Mill mistakenly inferred 

 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. 



To assign, then, a fraction of probability implies that the case presents a 

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



I 



