Every one must, I think, feel that such reasoning involves a misuse of the theory of probability. But is that misuse without some justification? The theory, unless I misread it, permits, or rather requires, us to express by the same fraction probabilities based on what is little less than complete knowledge, and probabilities based on what is little more than complete ignorance. To arrive at a clear conclusion, it seems only necessary to apply the “law of sufficient reason” to defined alternatives; and it is apparently a matter of perfect indifference whether we apply this law in its affirmative or its negative shape; whether we say “there is every reason for believing that such and such alternatives happen equally often,” or whether we say “there is no reason for thinking that one alternative happens more often than the other.” I do not criticise this method; still less do I quarrel with it. On the contrary, I am lost in admiration of this instrument of investigation, the quality of whose output seems to depend so little on the sort of raw material with which it is supplied. III My object, indeed, is neither to discuss the basis on which rests the calculus of probabilities—a task for which I own myself totally unfit—nor yet to show that a certain obscurity hangs over the limits within which it may properly be employed. I desire rather to suggest that, wherever those limits are placed, there lies beyond them a kind of probability yet more fundamental, about which the mathematical methods can tell us nothing, though it possesses supreme value as a “guide of life.” Wherein lies the distinction between the two? In this: the doctrine of calculable probability (if I may so call it) has its only application, or its only assured application, within groups whose character is either postulated, or is independently arrived at by inference and observation. These groups, be they natural or conventional, provide a framework, marking out a region wherein prevails the kind of ignorance which is the subjective reflection of objective “randomness.” This is the kind of ignorance which the calculus of probabilities can most successfully transmute into knowledge: and herein lies the reason why the discoverers of the calculus found their original inspiration in the hazards of the gambling-table, and why their successors still find in games of chance its happiest illustrations. For in games of chance the group framework is provided by convention; perfect “randomness” is secured by fitting devices; and he who attempts to modify it is expelled from society as a cheat.