6 Mh. ELLIS, ON THE FOUNDATIONS OF THE THEORY OF PROBABILITIES. 



(17.) Thus, it appears this result is based upon some thing equivalent to the following as- 

 sumption : There are an infinity of events whose simple probability a priori is x, and another 



infinite number for which it is of. These two infinities bear to one another the definite ratio of 

 equality, (.c and w may represent any quantity from to 1.) Now in reality, as we have seen, 

 these numbers are not only infinite, but in rerum natura indeterminate, and therefore the assump- 

 tion that they bear to one another a definite ratio is illusory. 



And this assumption runs through all the applications of the theory to events whose causes 

 are unknown. 



This position could be directly proved only by an analysis of the various ways in which this 

 part of the subject has been considered, which would require a good deal of detail. Those who 

 take an interest in the question, may without much difficulty satisfy themselves, whether the 

 view I have taken (which at least avoids the manifest contradictions of the received results) is 

 correct. 



(18.) I will add only one remark. If in (16) instead of taking one event from each of the 

 trials there specified, we had taken p in succession, and kept account only of those sequences 

 of p events each, which contained none but events of the kind sought ; we should have had 

 of such sequences 



1 2" 



'^^-V^^m'-' 



') 



of which 



'•{(^.^r^ ■! 



ii 



these two expressions is ultimately 



would have belonged to trials wliere the simple a -priori probability was > - : the ratio of 



y;Wr 



-G)' 



This is the expression applied to determine the probability of a common cause among similar 

 phenomena, as in the case already mentioned of the planets. 



But this application is founded on a petitio principii : we assume that all the phenomena 

 are allied : that they are the results of repetitions of the same trial, that they have the same 

 simple probability ; all that, setting other objections aside, we really determine, is the probability, 



1 



that this simple probability common to all these allied phenomena is > 



But how docs this determine the force of the presumption that the phenomena are allied, 

 or to use Condorcet's illustration, that they all come out of the same infinite lottery ? 



(19.) The object of this little essay being to call attention to the subject rather than 

 fully to discuss it, I have omitted several questions which entered into my original design. 



The principle on which the whole depends, is the necessity of recognizing the tendency 

 of a series of trials towards regularity, as the basis of the theory of probabilities. 



I have also attempted to show that the estimates furnished by what is called the theory 

 a posteriori of the force of inductive results are illusory. 



If these two positions were satisfactorily established, the theory would cease to be, what 

 I cannot avoid thinking it now is, in opposition to a philosophy of science which recognizes 

 ideal elements of knowledge, and which makes the process of induction depend on them. 



