148 INDUCTION. 



way of addition, it is easily seen from the theory of 

 probabilities laid down in a former chapter, in what 

 manner each of them adds to the probability of a con- 

 clusion which has the warrant of them all. If two of 

 every three A are B, and three of every four C are B, 

 the probability that something which is both an A and 

 a C is a B, will be more than two in three, or than 

 three in four. Of every twelve things which are A, 

 all except four are B, by the supposition ; and if the 

 whole twelve, and consequently those four, have the 

 characters of C likewise, three more will be B on that 

 ground. Therefore, out of twelve which are both A 

 and C, eleven are B. To state the argument in 

 another way ; a thing which is both A and C, but 

 which is not B, is found in only one of three sections 

 of the class A, and in only one of four sections of the 

 class C ; but this fourth of C being spread over the 

 whole of A indiscriminately, only one- third part of it 

 (or one- twelfth of the whole number) belongs to the 

 third section of A ; therefore a thing which is not B 

 occurs only once, among twelve things which are 

 both A and C. The argument would, in the language 

 of the doctrine of chances, be thus expressed: the 

 chance that an A is not B is |, the chance that a C is 

 not B is J, hence if the thing be both an A and a C 

 the chance is i of J = -jV- 



This argument pre-supposes (as the reader will 

 doubtless have remarked) that the probabilities arising 

 from A and C are independent of one another. 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 have a greater 

 chance of doing so. Else the fourth section of C, 

 instead of being equally distributed over the three 

 sections of A, might be comprised in greater propor- 



