72 INDUCTION. 



5. From the preceding principles it is easy to deduce 

 the demonstration of that theorem of the doctrine of proba- 

 bilities, which is the foundation of its application to inquiries 

 for ascertaining the occurrence of a given event, or the reality 

 of an individual fact. The signs or evidences by which a fact 

 is usually proved, are some of its consequences : and the in- 

 quiry hinges upon determining what cause is most likely to 

 have produced a given effect. The theorem applicable to 

 such investigations is the Sixth Principle in Laplace's Essai 

 Philosophique sur les Probabilites, which is described by him 

 as the " fundamental principle of that branch of the Analysis 

 of Chances, which consists in ascending from events to their 

 causes."* 



Given an effect to be accounted for, and there being several 

 causes which might have produced it, but of the presence of 

 which in the particular case nothing is known ; the proba- 

 bility that the effect was produced by any one of these causes 

 is as the antecedent probability of the cause, multiplied by the 

 probability that the cause, if it existed, would have produced 

 the given effect. 



Let M be the effect, and A, B, two causes, by either of 

 which it might have been produced. To find the probability 

 that it was produced by the one and not by the other, ascer- 

 tain which of the two is most likely to have existed, and 

 which of them, if it did exist, was most likely to produce the 

 effect M : the probability sought is a compound of these two 

 probabilities. 



CASE I. Let the causes be both alike in the second 

 respect; either A or B, when it exists, being supposed 

 equally likely (or equally certain) to produce M ; but let A 

 be in itself twice as likely as B to exist, that is, twice as 



continuance of the same collocation of causes which existed during the obser- 

 vations. If it ever fails, it is in consequence of some change in that collocation. 

 Now, no theory of chances will enable us to infer the future probability of an 

 event from the past, if the causes in operation, capable of influencing the 

 event, have intermediately undergone a change. 



* Pp. 18, 19. The theorem is not stated by Laplace in the exact terms in 

 which I have stated it ; but the identity of import of the two modes of expres- 

 sion is easily demonstrable. 



