xii.] THE INDUCTIVE OR INVERSE METHOD. 243 



le produced by any one of a certain number of different 

 causes, all equally probable a priori, the probabilities of the 

 existence of these causes as inferred from the event, are pro- 

 portional to the probabilities of the event as derived from these 

 causes. In other words, the most probable cause of an 

 event which has happened is that which would most pro- 

 bably lead to the event supposing the cause to exist; but 

 all other possible causes are also to be taken into account 

 with probabilities proportional to the probability that the 

 event would happen if the cause existed. Suppose, to fix 

 our ideas clearly, that E is the event, and G\ C 2 C 8 are the 

 three only conceivable causes. If C exist, the probability 

 is PI that E would follow ; if C 2 or C 3 exist, the like pro- 

 babilities are respectively p 2 and p 3 . Then as p l is to p z , 

 so is the probability of C l being the actual cause to the 

 probability of C 2 being it ; and, similarly, as p 2 is to p z , so 

 is the probability of C 2 being the actual cause to the 

 probability of C 3 being it. By a simple mathematical pro- 

 cess we arrive at the conclusion that the actual probability 

 of Cj being the cause is 



Pi 



Pi + P* + Pz ' 



and the similar probabilities of the existence of C 2 and 

 C 3 are, 



Pz an j P3 



Pl + PZ + PS ' Pl+ Pt + P3 



The sum of these three fractions amounts to unity, which 

 correctly expresses the certainty that one cause or other 

 must be in operation. 



We may thus state the result in general language. 

 If it is certain that one or other of the supposed causes 

 exists, the probability that any one does exist is the proba- 

 bility that if it exists the event happens, divided by the sum 

 of all the similar probabilities. There may seem to be an 

 intricacy in this subject which may prove distasteful to 

 some readers ; but this intricacy is essential to the subject 

 in hand. No one can possibly understand the principles 

 of inductive reasoning, unless he will take the trouble to 

 master the meaning of this rule, by which we recede from 

 an event to the probability of each of its possible causes. 



This rule or principle of the indirect method is that 

 which common sense leads us to adopt almost instinctively, 



R 2 



