Aspects of the Theory of Probability. 723 



holds when the principle is not applicable ? Some further 

 assumption is necessary before it can be proved in these cases, 

 and various suggestions could be offered that would bridge 

 the gap without making it necessary to suppose that the 

 relation is known a priori in these cases. There seems, 

 however, to be little or no ground for deciding between them, 

 and the proposition may as well be assumed to hold in general 

 without further discussion. From the propositions so far 

 assumed or proved, with judgments of greater, equal, or less 

 probability in particular cases, the mathematical theory of 

 probability can be developed. 



Another point in connexion with the theory may be briefly 

 mentioned. All that is strictly necessary in order that the 

 notions of probability may be capable of logical treatment is 

 that combinations of propositions and data can be arranged 

 in a series so that whenever a combination A is not more 

 probable than another, B, B shall not precede A in the series. 

 With suitable assumptions regarding the position in the 

 series of a combination, such as the disjunction of two contra- 

 dictory propositions referred to the same data, a theory could 

 be constructed. There is no reason save convenience why 

 the number series should be the one employed for this 

 purpose. So long as we confine ourselves to those cases 

 where a proposition certain on the data can be decomposed 

 into a finite number of equally probable alternatives, and the 

 proposition whose probability is to be estimated is expressible 

 as or equivalent to the disjunction of a class of these, the 

 number series is obviously adequate ; in fact the series of all 

 rational proper fractions in ascending order of magnitude 

 would be adequate. This latter series is, however, at once 

 found to be insufficient when we attempt to deal with cases 

 where the number of equally probable alternatives required 

 to cover the case considered is infinite. This difficulty was 

 thought to be removed by using the series of all the real 

 numbers less than unity instead of that of the rational 

 numbers. But the question that arises now is, whether the 

 series of all the real numbers is itself adequate for the purpose, 

 and the answer seems to be in the negative, for there are 

 evidently cases where the use of infinitesimals is necessary to 

 a complete theory, and the discovery of others, necessitating 

 the introduction of infinitesimals of different orders, is practi- 

 cally certain. For instance, suppose we are given that x is 

 a whole number, and that all whole numbers are equally 

 probable values of x. What is the probability of any 

 particular value of x, say 1053 ? Clearly it is not finitely 

 different from ; for if it were X say, we could find a whole 



