142 Measurement of Belief. [CHAP. vi. 



to pay Is. if it gives any other number. It will generally be 

 admitted that he ought to give 2s. 6d. for the chance, and 

 that if he does so he will be paying a fair sum. This ex 

 ample only differs from the last in the fact that instead of 

 simple belief in a proposition, we have taken what mathema 

 ticians call the value of the expectation . In other words, 

 we have brought into greater prominence, not merely the be 

 lief, but the conduct which is founded upon the belief. But 

 precisely the same difficulty recurs here. For appealing to the 

 event, the single event, that is, we see that one or other 

 party must lose his money without compensation. In what 

 sense then can such an expectation be said to be a fair one ? 

 20. A possible answer to this, and so far as appears the 

 only possible answer, will be, that what we really mean by 

 saying that we half believe in the occurrence of head is to 

 express our conviction that head will certainly happen on 

 the average every other time. And similarly, in the second 

 example, by calling the sum a fair one it is meant that in 

 the long run neither party will gain or lose. As we shall 

 recur presently to the point raised in this form of answer, the 

 only notice that need be taken of it at this point is to call 

 attention to the fact that it entirely abandons the whole 

 question in dispute, for it admits that this partial belief does 

 not in any strict sense apply to the individual event, since it 

 clearly cannot be justified there. At such a result indeed we 

 cannot be surprised ; at least we cannot on the theory adopted 

 throughout this Essay. For bearing in mind that the em 

 ployment of Probability postulates ignorance of the single 

 event, it is not easy to see how we are to justify any other 

 opinion or statement about the single event than a confes 

 sion of such ignorance. 



21. So far then we do not seem to have made the 

 slightest approximation to a solution of the particular 





