Aspects of the Theory of Probability. 731 



supposing such a proposition to be safely generalised we hand 

 it over to Probability to say what sort of inferences can be 

 deduced from it." 



The "undefined concept" view of probability can be 

 developed so as to yield a theory of induction adequate for 

 scientific purposes. There are difficulties in the way of 

 obtaining such a theory from the frequency view, and we 

 conclude that the balance is in favour of the " undefined 

 concept " view. 



Summary. 



It is shown that the attempt to give a definition of 

 probability in terms of frequency is unsuccessful. Laplace's 

 definition, apparently in these terms, really involves im- 

 plicitly the concept of probability and is therefore circular 

 in character. Venn's definition in terms of the limit of a 

 series is unsatisfactory because there is no reason to believe 

 that his series do in fact usually tend to a limit ; it is shown 

 that there are many cases where they do not ; and as his 

 process is incapable of being carried out, the existence of 

 such a limit can in any case only be known a priori if at all, 

 so that his method offers no advantage over that of regarding 

 probability as an entity known to exist independently of 

 definition, intelligible without such definition, and perhaps 

 indefinable. 



A set of axioms on which a mathematical theory of 

 probability can be based is then given, which seems to offer 

 certain advantages over the current ones. In particular it is 

 capable of covering cases where the principle of sufficient 

 reason cannot be applied to assess probability. It is also 

 shown that a complete theory of probability must allow for 

 the use of infinitesimals. 



A discussion of sampling induction is given, in which it 

 is shown that when the sample is large enough the prior 

 probabilities of different constitutions of the whole do not 

 usually affect appreciably the probabilities inferred after the 

 samples have been taken. Also the range within which 

 the fractional constitution is as likely as not to lie includes 

 the fractional constitution of the sample, and its extent is 

 inversely proportional to the number of the sample itself, 

 whatever be the number of the whole. 



