CHAP. XX.] PROBLEMS ON CAUSES. 361 



exists a cause which under given circumstances would always 

 produce it, then the fact that that phenomenon has ever been no- 

 ticed under those circumstances, renders certain its re-appearance 

 under the same.* 



4th. As n increases, the expression approaches in value to 

 unity. This indicates that the probability of the recurrence of 

 the event increases with the frequency of its successive appear- 

 ances, a result agreeable to the natural laws of expectation. 



SECOND CASE. We are now to seek the probability d pos- 

 teriori of the existence of a permanent cause of the phenomenon. 

 This requires that we ascertain the value of the fraction 



aO^*. . x n 



Prob. Xi # 2 . . x n 9 



the denominator of which has already been determined. 

 To determine the numerator assume 



tX l X z . . X n = W, 



then proceeding as before, we obtain for the logical develop- 

 ment, 



w = tXi x% . . x n + (1 - t). 



Whence, passing from Logic to Algebra, we have at once 



Prob. w = , 



a result which might have been anticipated. Substituting then 

 for the numerator and denominator of the above fraction their 

 values, we have for the d posteriori probability of a permanent 

 cause, the expression 



* As we can neither re-enter nor recall the. state of infancy, we are unable to 

 say how far such results as the above serve to explain the confidence with which 

 young children connect events whose association they have once perceived. 

 But we may conjecture, generally, that the strength of their expectations is 

 due to the necessity of inferring (as a part of their rational nature), and the 

 narrow but impressive experience upon which the faculty is exercised. Hence 

 the reference of every kind of sequence to that of cause and effect. A little 

 friend of the author's, on being put to bed, was heard to ask his brother the 

 pertinent question, " Why does going to sleep at night make it light in the 

 morning?" The brother, who was a year older, was able to reply, that it 

 would be light in the morning even if little boys did not go to sleep at night. 



