Aspects of the Theory of Probability. 729 



former case the inference becomes tautologous, and we are 

 concerned only with the latter. But it has been shown above 

 that if n be the number of black crows in the world, n/m is 

 not likely to deviate from p/(p + q) by more than a quantity 

 of the order of (p-\-q)~* \ and in this case, as p is great and 

 q is zero, we are justified in inferring that the number of 

 crows that are not black is a small fraction of the whole, 

 which is all that is inferred in practice ; for the possibility in 

 exceptional cases of sport, albinos, and so on is well known. 

 The other type of general law is one that is held to be true 

 in every instance of the entities to which it is held to apply. 

 Such a law cannot be derived by means of probability in- 

 ference, for it deals only with certainties. Here Mr. Broad's 

 argument is valid, and no such law can derive a reasonable 

 probability from experience alone ; some further datum is 

 required. One way of arriving at such laws may be sug- 

 gested here. Suppose we have an a priori belief that either 

 every x has the property cf) or every x has the property y]r. 

 If then a single x, say c, is found to satisfy but not ty, we 

 can infer deductively the universal proposition that all #'s 

 satisfy <£. Such cases are fairly frequent : if for instance 

 we consider that either Einstein's or Silberstein's form of the 

 principle of general relativity is true, a single fact contra- 

 dictory to one would amount to a proof of the other in every 

 case. 



Before leaving the important question of induction, we 

 propose to consider it in relation to the Venn view. If 

 Venn's definition of probability be adopted the existence of a 

 numerical estimate of probability depends on the possibility 

 (at least imagined) of indefinite repetition of the data, the 

 truth or falsehood of the proposition whose probability 

 relative to the data is to be estimated being recorded at each 

 repetition. The probability is then the limit of the ratio of 

 the number of favourable cases to the number of all cases. 

 Now on this basis it is never possible, by what has been said 

 already, to prove that in any given case such a limit will 

 exist. All the axioms of the " undefined concept " theory are 

 therefore indemonstrable, and must be assumed a priori in 

 the same way. Even in the simple case of picking indis- 

 tinguishable balls out of a bag the probability of picking 

 any particular individual cannot be assessed without some 

 hypothesis about the limit of the results obtained by making 

 an indefinite number of selections, eaeh ball being replaced 

 after being drawn. In the problem of sampling induction 

 we can therefore by making enough assumptions of this 

 character, which there seems little or no reason to believe. 



