724 Miss Wrinch and Dr. H. Jeffreys on some 



number M whose reciprocal would be less than A ; but 1/M 

 is the probability of M being 1053 when there are only M 

 possible alternatives, and the probability cannot be increased 

 by increasing the number of alternatives. Hence the pro- 

 bability that x is 1053 is less than any finite number, contrary 

 to what was assumed. It is nevertheless different from zero, 

 for then there would be no means of distinguishing between 

 the probability of this, which is a perfectly possible pro- 

 position on the data, and that of a proposition known to be 

 impossible on the data. Hence this probability is less than 

 any finite number, and yet is different from zero ; in other 

 words, it is an infinitesimal, in the original sense of the term*. 

 Again, we can see that the probability of a particular real 

 number chosen at random being rational is infinitesimal ; so 

 is the probability that a function is analytic, given that all 

 functions are equally probable. Now a complete theory of 

 probability must cover all these cases ; but so long as we are 

 confined to the series of the real numbers that is impossible ; 

 for if this has C members, the number of possible functions 

 whose values are real numbers is O c , which is greater ; hence 

 problems arising in connexion with the probability of func- 

 tions demand the use of a series for comparison whose 

 members are more numerous than the real numbers. Such 

 series are known ; and perhaps one suitable for the purpose 

 may be constructed which will include among its members 

 the real numbers themselves. 



III. On Probability Inference. 



The characteristic feature of the type of inference with 

 which classical logic is primarily concerned is that given the 

 premises it is possible to establish the conclusions with 

 absolute certainty from them. In many cases, however, such 

 a result is unobtainable when it is nevertheless possible to 



* M. E. Borel remarks (Lecons sur .la Theorie des Fonctions, 1914, 

 p. 184) that " there is a true discontinuity between an infinitely small 

 probability, i. e., a variable probability tending towards zero, and a 

 probability equal to zero. However small be the probability of the 

 favourable case, this is possible ; whereas it is impossible if the pro- 

 bability be zero The same is not true of continuous probabilities ; 



the probability that a number taken at random may be rational is ; 

 this must not be considered as equivalent to impossibility." This use of 

 zero to denote the probability of both an impossible alternative and 

 a possible alternative with no finite probability seems likely to lead 

 to confusion. Thft introduction of the conception of a limit does not 

 help matters, for in making a single trial the probability of success 

 is quite definite, and involves no notion of a limit. 



