"716 Miss Wrinch and Dr. H. Jeffreys on some 



possible. In fact it seems that by equally possible he meant 

 equally p?*obable. Thus, as Poincare has pointed out, it seems 

 useless to attempt to make this definition satisfactory; it 

 defines the probability of one proposition in terms of those 

 of a set of others and not in terms of frequency alone, so that 

 the notion Laplace set out to define reappears in the un- 

 defined concept of equally possible. The statement is, in fact, 

 not a definition, but a simple and important principle of 

 probability inference. Nor does it appear that there is any 

 prospect of making any modification of it into a definition of 

 probability ; for there will always be the difficulty of deciding 

 what are to be considered as unit alternatives. It is clear 

 that even if it were possible to avoid introducing the notion 

 of equally probable alternatives, some other way of dis- 

 tinguishing between sets of mutually exclusive and exhaustive 

 alternatives would have to be found, and the immense variety 

 of the circumstances to which it would have to apply seems 

 to indicate that its scope must be at least as wide as that of 

 truth ; and it is very unlikely that a notion so general is 

 capable of definition. 



The view of Venn* is much more complex. He considers 

 that the notion presupposes a series, the terms of which are 

 indefinitely numerous and represent the cases of an attribute 

 (f>. From these one can pick out a smaller class, the members 

 of which possess the further attribute -y\ If? then, we have 

 chosen n members in all and m of them belong to the smaller 

 class, the probability of i|r given (j> is defined as the limit 

 of m/n when n becomes indefinitely great. The form of this 

 definition restricts the field of probability very seriously. 

 In the first place it seems impossible to apply it to any case 

 where the number of members of the first series is finite ; 

 one could attach no meaning to a statement that it is probable 

 that the solar system was formed by the disruptive approach 

 of a star larger than the sun, or that it is improbable that 

 the stellar universe is symmetrical, for the indefinite repetition 

 of entities of such large dimensions is utterly fantastic. Yet 

 such cases as these are the very ones where the notion of 

 probability is particularly valuable in science, and any 

 definition that will not cover them is not satisfactory. 



It may be urged, however, that this theory gives an 

 adequate treatment of probability as applied to the class of 

 cases with which it deals. Serious difficulties nevertheless 

 present themselves. The existence of a probability on this 

 theory requires that a limit shall exist to which a certain 

 ratio tends in the long run ; and one is led to ask what the 

 * ' Logic of Chance/ pp. 162 et seqq. 



