134 INDUCTION. 



since in this calculation seven out of twelve cases seem to 

 have exhausted the possibilities. If T is a B in only six cases 

 of every twelve, and a not-B in only one, what is it in the 

 other five ? The only supposition remaining for those cases 

 is that it is neither a B nor not a B, which is impossible. But 

 this impossibility merely proves that the state of things 

 supposed in the hypothesis does not exist in those cases. 

 They are cases that do not furnish anything which is both an 

 A and a C. 



To make this intelligible, we will substitute for our 

 symbols a concrete case. Let there be two witnesses, M and 

 N, whose probabilities of veracity correspond with the ratios 

 of the preceding example : M speaks truth twice in every 

 thrice, N thrice in every four times. The question is, what 

 is the probability that a statement, in which they both 

 concur, will be true. The cases may be classed as follows. 

 Both the witnesses will speak truly six in every twelve times ; 

 both falsely once in twelve times. Therefore, if they both agree 

 in an assertion, it will be true six times, for once that it will be 

 false. What happens in the remaining cases is here evident ;. 

 there will be five cases in every twelve in which the wit 

 nesses will not agree. M will speak truth and N false 

 hood in two cases of every twelve ; N will speak truth and 

 M falsehood in three cases, making in all five. In these cases, 

 however, the witnesses will not agree in their testimony. But 

 disagreement between them is excluded by the supposition. 

 There are, therefore, only seven cases which are within the 

 conditions of the hypothesis ; of which seven, veracity exists in 

 six, and falsehood in one. Kesuming our former symbols, in 

 five cases out of twelve T is not both an A and a C, but an A 

 only, or a C only. The cases in which it is both are only seven, 

 in six of which it is a B, in one not a B, making the chance 

 six to one, or -&quot; and ^ respectively. 



In this correct, as in the former incorrect computation, it 

 is of course presupposed that the probabilities arising from A 

 and C are independent of each other. There must not be any 

 such connexion between A and C, that when a thing belongs 

 to the one class it will therefore belong to the other, or even 



