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224 F. Y. EDGEWOETH : controverted points the present writer in the main agrees, while yet attempting to show that Mr. Venn's logical scep- ticism has often carried him too far from the position held by the majority of previous writers upon Chance. Thus he seems to go too far when he insists that quantity of belief is a " mere appendage " of the science. That Probability should be concerned only with certainty is surely a little paradoxical, and challenges inquiry. Our first question then is : How far are gradations of belief a subject of science ? Mr. Venn's negative criticism has great force ; but it does not seem to hit, perhaps it was not aimed at, the moderate position here taken. Gradations of belief, it is submitted, can be discerned with some pre- cision in simple cases, such as those which the mathemati- cians posit. Suppose there is an urn containing a hundred balls, black and white. The felt probability, the quantity of belief, that a white ball will be obtained at a single drawing continually increases, as we increase, other things being con- stant, the proportion of white to black balls. This measure- ment of a subjective feeling is like the measurement of felt heat by the thermometer. It is very like the Fechnerian measurements of sensation. Like the Fechnerian measure- ments it is directly applicable only to simple cases ; and yet, as Fechner's law has some analogical bearing upon the higher problem of Hedonics, and seems to be the basis of the law of diminishing utility, so belief about the simple events of games of chance has fruitful analogy with belief about the more complicated events of real life. It must be frankly admitted, however, that the practical outcome of this subjective view is slight ; that little can be worked by it which cannot be worked without it. There is only one class of practical problems to which the subjective view is ex- clusively applicable ; those actions which cannot be regarded as forming part of a ' series ' in Mr. Venn's sense ; a class which with the increase of providence and sympathy is likely to disappear. The principal arguments adduced by Mr. Venn in his Logic of Chance, ch. 5, against ' gradations of belief ' must now be passed in review. (1) He doubts " that our belief of every proposition is a thing which we can be strictly said to measure". The answer is suggested by Mr. Venn when he supposes " that it were possible to strike a sort of average " of a fluctuating belief. Doubtless our feeling of quantity of belief is, even in the simpler cases, a somewhat vague presentation, like our feeling of quantity of time when, without having looked at the clock for some time, we try to think what o'clock it is.