730 Miss Wrinch and Dr. H. Jeffreys on some 



obtain a proof of a proposition superficially the same as the 

 chief theorem of this section ; but let us consider what this 

 result means on the Venn view. It would mean that if we 

 had a large number of classes, each of m members, and from 

 each we had picked out p + q members, of which p were ffs 

 and q not-/3's, then when the number of such classes is in- 

 definitely increased the fraction of them in which the actual 

 number of /3's does not lie within certain limits would tend 

 to zero as a limit. Thus the already hopeless task of pro- 

 ceeding to the limit of an infinite number of observations 

 becomes in this case the still more complex one of repeating 

 similar classes indefinitely. 



Such indefinite repetition of infinite classes is called by 

 Venn the construction of " cross-series " and forms an 

 essential part of his theory of inference. It is necessary, for 

 instance, in giving a meaning to the proposition connecting 

 the probabilities of a proposition referred to different data 

 P (p . q : h)=~P(p : q . k) . P(^:A). For an infinite series is 

 needed to give an account of J*(p : q . h), which is the limit 

 derived from the frequency of the truth of p among entities 

 for which q and h are true. Such entities, however, are only a 

 part of those for which h holds. Thus to establish a meaning 

 for the number P(p . q : li) we must consider all entities 

 satisfying h, whether they satisfy q or not. Thus further 

 series must be constructed which will show how often q is 

 actually true, and this requires, according to Venn, an infinite 

 number of series of entities all satisfying h, so that we can 

 examine in one direction to find the frequency of p given q 

 and li and in the other to find that of q given h. Tims the 

 difficulty of obtaining enough terms, acute in the simple case, 

 is here intensified ; further, there is no more reason to believe 

 in the existence of limits in this case than there was in the 

 other. The difficulties are merely complicated and not re- 

 moved by the use of cross-series. 



There is no evidence that Venn ever attempted to meet 

 these difficulties. Indeed, we may conclude from some 

 passages of his work that they had never suggested them- 

 selves. The following passage, for example, occurs in the 

 third edition of his 'Logic of Chance/ page 208. "The 

 opinion according to which certain inductive formulae are 

 regarded as composing a portion of probability cannot, I 

 think, be maintained. It would be more correct to say .... 

 that induction is quite distinct from probability, yet co- 

 operates in almost all its inferences. By induction we 

 determine for example whether and how far we can safely 

 generalise the proposition that 4 men in 10 live to be 56 : 



