176 The Rules of Inference in Probability. [CHAP. VIL 



inference in probability, the question at once arises, What is 

 their relation to the great body of formulas which are made 

 use of in treatises upon the science, and in practical applica 

 tions of it ? The reply would be that these formulae, in so 

 far as they properly belong to the science, are nothing else 

 in reality than applications of the above fundamental rules. 

 Such applications may assume any degree of complexity, for 

 owing to the difficulty of particular examples, in the form in 

 which they actually present themselves, recourse must some 

 times be made to the profoundest theorems of mathematics. 

 Still we ought not to regard these theorems as being any 

 thing else than convenient and necessary abbreviations of 

 arithmetical processes, which in practice have become too 

 cumbersome to be otherwise performed. 



This explanation will account for some of the rules as 

 they are ordinarily given, but by no means for all of them. 

 It will account for those which are demonstrable by the cer 

 tain laws of arithmetic, but not for those which in reality 

 rest only upon inductive generalizations. And it can hardly 

 be doubted that many rules of the latter description have 

 become associated with those of the former, so that in popu 

 lar estimation they have been blended into one system, of 

 which all the separate rules are supposed to possess a similar 

 origin and equal certainty. Hints have already been fre 

 quently given of this tendency, but the subject is one of 

 such extreme importance that a separate chapter (that on 

 Induction) must be devoted to its consideration. 



8. In establishing the validity of the above rules, we 

 have taken as the basis of pur investigations, in accordance 

 with the general scheme of this work, the statistical frequency 

 of the events referred to ; but it was also shown that each 

 formula, when established, might with equal propriety be ex 

 pressed in the more familiar form of a fraction representing 



