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, certainty 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, including 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 this 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 which in its nature admits of determination, though we may practically find it difficult in any particular case to determine it. He considers, in fact, that Probability is a sort of sister science to Formal Logic,[1] speaking of it in the following words: "I cannot understand why the study of the effect, which partial belief of the premises produces with respect to the conclusion, should be separated from that of the consequences of supposing the former to be absolutely true."[2] In other words, there is a science--Formal Logic--which investigates the rules according to which one proposition can be necessarily inferred from another; in close correspondence with this there is a science which investigates the rules according to which the amount of our belief of one proposition varies with the amount of our belief of other propositions with which it is connected. The same view is also supported by another high authority, the late Prof. Donkin, who says (_Phil. Mag._ May, 1851), "It will, I suppose, be generally admitted, and has often been more or less explicitly stated, that the subject-matter of calculation in the mathematical theory of Probabilities is _quantity of belief_."