Fundamental Principles of Scientific Inquiry. 383 



between y\—e and y 1 -\-€. ,> The only difference is that the 

 number of different possible observational results is now tt 

 instead of C ; the above process then gives the probability 

 o£ any result of the next trial to be C c /ft 2 -r-C c /N , and 

 this also is indeterminate. The restriction of the possible 

 functions to continuous ones does not affect the result, as 

 the number of alternatives remains the same. Even if the 

 functions involved in physical laws are restricted to being- 

 analytic, a suggestion of Jourdain's that has otherwise much 

 to recommend it, the probability is C/K 2 -^C/tf , which is 

 still indeterminate. 



It seems, therefore, that the hypothesis that all functions 

 of these assigned classes are equally likely to occur in 

 physical laws is incapable by itself of giving any theory of 

 inference. Some further assumption is necessary before 

 any progress can be made, and consequently there is no 

 advantage in adopting the former assumption among our 

 premises, unless it can be shown that the alternative 

 assumption that not all functional relations are equally 

 probable is unworkable. This alternative evidently requires 

 that functions can be arranged in a sequence, so that each 

 is more probable than all that follow it, and such that after 

 any function or set of functions there are one or a finite 

 number of functions whose probability is highest. It may 

 not be necessary that every function shall be either more 

 or less probable than any other : in other words, the 

 sequence may have branches or loops. But it will be 

 necessary that those with the same probability shall be 

 finite in number, for otherwise the objection that inference 

 would be impossible would apply again. 



Thus two assumptions are needed : first, that all functions 

 occurring in physics are capable of being arranged in a 

 sequence with this particular property ; and, second, that 

 this can be achieved by means of the relations more and 

 less probable than. Now the type of arrangement we 

 have just defined has precisely the property called " well- 

 orderednpss " in modern logic. The important characteristic 

 of such a sequence is that, if any set of terms whatever is 

 selected from it, there is at least one member of the sequence 

 that immediately succeeds this set. Consider, for instance, 

 the natural ordinal numbers, and add to them the infinite 

 ordinal numbers ; we then have a particular case of a well- 

 ordered sequence, as follows : — 



1,2,3, . . . n, n + 1, ... o>, co + 1, co + 2, ... 2co. . . . or. ... 



' . . . o) 



