382 Dr. Dorothy Wrincli and Dr. H. Jeffreys on Certain 



Thus we have the familiar result that a decision between 

 alternative and equally probable theories cannot be made by 

 verifying inferences that can be made from both. The first 

 alternative certainly does not agree with ordinary beliefs 

 about the validity of scientific laws ; however sceptical we 

 may be about the finality of a particular law, we should say 

 that its probability was finite. 



Let us suppose, if possible, that the number of general 

 laws possible is m, and that they all have the prior proba- 

 bility 1/m. Suppose a number of experiments to have been 

 made to test these. Then the only survivors are those 

 which imply the results of these experiments, which may 

 be summed up in the proposition q. Each of them after 

 the series of experiments has the probability l/mY(g :h). 

 Thus every law has the same probability after the expe- 

 riments, and w r e are as far from being able to make any 

 inference by means of them as we were at the start. It 

 therefore appears that, unless there is some difference 

 between the prior probabilities of the alternatives, scientific 

 inference will always be impossible, except in the very 

 special case where only one law out of the class considered 

 admissible fits the observations. 



Instances of the type of assumption that fails in this way 

 to give any workable theory of inference are afforded by all 

 the most obvious assumptions one might make about the 

 extent of the classes of laws that are equally probable. For 

 instance, suppose we are given that all functional relations 

 whatever are equally probable. The prior probability of any 

 one function being right is in this case the infinitesimal 1/C C , 

 where C is the number of points in the continuum. If all 

 the observations were of the exact values of the variables, 

 the prior probability of each would be 1/C, and thus the 

 probability of any law that survived the first trial would 

 be C/C c , which is also infinitesimal. But at the next trial 

 the number of these functional relations that are satisfied by 

 the value observed is C c /C 2 . The product of this by the 

 probability of any one separately gives the probability of 

 this value observed at the second trial, which is therefore 

 C /C 2 x C/C G . This, being the quotient of two equal infinite 

 numbers, is strictly indeterminate. Thus nothing can be 

 inferred about the probabilities of particular values at the 

 next trial. The process may be repeated indefinitely, and 

 we thus see that without some further assumption inference 

 will be impossible. 



The matter is not helped by supposing the observations to 

 be of form " y, for a particular value of x, such as x x , lies 



