122 Measurement of Belief. [CHAP. vi. 



therefore like to have as clear views as possible about these 

 single cases. Now, the remarks made in the preceding chapters 

 about the subjects which Probability discusses might seem to 

 preclude all enquiries of this kind, for was not ignorance of 

 the individual presupposed to such an extent that even (as 

 will be seen hereafter) causation might be denied, within 

 considerable limits, without affecting our conclusions ? The 

 answer to this enquiry will require us to turn now to the 

 consideration of a totally distinct side of the question, and 

 one which has not yet come before us. Our best introduction 

 to it will be by the discussion of a special example. 



4. Let a penny be tossed up a very great many 

 times ; we may then be supposed to know for certain this- 

 fact (amongst many others) that in the long run head and 

 tail will occur about equally often. But suppose we consider 

 only a moderate number of throws, or fewer still, and so 

 continue limiting the number until we come down to three 

 or two, or even one ? We have, as the extreme cases, cer 

 tainty or something undistinguishably near it, and utter 

 uncertainty. Have we not, between these extremes, all 

 gradations of belief? There is a large body of writers, in 

 cluding some of the most eminent authorities upon this 

 subject, who state or imply that we are distinctly conscious of 

 such a variation of the amount of our belief, and that this 

 state of our minds can be measured and determined with 

 almost the same accuracy as the external events to which 

 they refer. The principal mathematical supporter of thi& 

 view is De Morgan, who has insisted strongly upon it in all 

 his works on the subject. The clearest exposition of his 

 opinions will be found in his Formal Logic, in which work he 

 has made the view which we are now discussing the basis of 

 his system. He holds that we have a certain amount of belief 

 of every proposition which may be set before us, an amount 



