4. In order to calculate the probability that an event already observed will be repeated any given number of times, the rule is, to divide the number of times the event has been observed, increased by one, by the same number increased by one and the number of times the event is to recur. Thus, if the tide had been observed 9 times, the chance that it would recur ten times more would be +10=(0 ) = 1. "This is the same thing as if each reproduction of the observed event corresponded to putting a white ball in an urn where there were already, before commencing the trials, a white ball and as many black balls as it is supposed that the event observed should re-occur times." 9 20 5. The probability that there exists a cause of the reproduction of any event observed several times in succession is expressed by a fraction which has for its denominator the number 2 multiplied by itself as many times as the event has been observed, and for its numerator the same product minus one. This has been called Bayes' rule, and its validity is not so generally admitted as that of the preceding ones. Thus, supposing the two tides only had been observed, the chance of a cause would be 2×2×2-1 = Where the observations have not all been fa vourable, in order to estimate whether the event will occur once more, the rule is to divide the number of times the event has been observed to hap pen increased by one, by the total number of observations increased by two. Thus, if out of 26 metals known to the chemist, 24 are heavier than water and 2 lighter, the chance that the next discovered, assuming as certain the fact of discovery, will be lighter than water will be 2+1= 3 262 28 ; or 25 to 3. Other examples of these formulæ may readily be found, to make the use of them easy, and to verify their truth. In applying the doctrine of chances to that subject in connexion with which it was invented, - games of chance-the principles of what has been happily termed "moral arithmetic must not be forgotten. Not only would it be difficult for a gamester to find an antagonist on terms, as to fortune and needs, precisely equal, but also it is impossible that with such an equality the advantage of a considerable gain should balance the harm of a serious loss. "If two men," says Buffon, 66 were to determine to play for their whole property, what would be the effect of this agreement? The one would only double his fortune, and the other reduce his to naught. What proportion is there between the loss and the gain? The same that there is between all and nothing. The gain of the one is but a moderate sum, the loss of the other is numerically in finite, and morally so great that the labour of his whole life may not perhaps suffice to restore his property." 99 The theory of chances assists materially in giving a clear conception of modality (p. 306.). A proposition may pass from absolute uncertainty, where there is as much against as for its truth (=) up to absolute certainty (= 1) through an infinite number of deepening shades of probability (,,1%, and so on). These refinements in estimating evidence are little used in ordinary thinking, it is true; and broader lines of distinction suffice. But they seem to justify those who exclude modality from the form of judgments, since otherwise one judgment would seem to be capable of being modified into a hundred, the expression remaining the same, and the evidence only varying. Hume in his "Essay of Miracles" has overlooked one property of highly probable judgments-that the favourable evidence for them not only preponderates over, but utterly expels, the unfavourable, and especially in matters where the moral nature is concerned. The probable evidence that the sun will rise daily for the next ten years is exceedingly strong; and consequently, from "the days of Noah" to the present, people have acted as if the weaker probability had no existence. If a jury find a man guilty, because ten credible witnesses have sworn against him, : and one or two for him, they consider that the testimony of the ten annihilates that of the two; were it otherwise, they must give the prisoner the benefit of their doubt. A son does not estimate the balance in favour of the truth of a father's statement, nor a friend of a friend's because to doubt at all is not to believe. When he asserts that in the case of miracles, "there is a mutual destruction of arguments [for and against them], and the superior only gives us an assurance suitable to that degree of force which remains after deducting the inferior," he neglects the distinction between mathematical and moral subjects; in the one, both favourable and adverse chances must be preserved; in the other, that is, where we have to act on probabilities, adverse arguments must, when once we have made up our minds, be ignored entirely, because to permit them the smallest influence would weaken and fetter our actions. The rest of his argument has been fully refuted. Writers on probabilities have shown how rapidly the scale of belief ascends with the addition of each new independent witness; and Paley has exposed the fallacy of reasoning from what is contrary to one's own experience to what contradicts the universal experience of men. The numerical mode of statement illustrates the operation of the will in moral actions. The action entirely indeterminate, in which there is an exact 100 equilibrium between the motives for and those against a particular course, is represented by (say) = 1: though some maintain that except in the case of the ass of Buridanus, whose "two bundles of hay" are no longer worthy of the dignity of philosophy, so nice a balance cannot occur. The necessary action, where all the motives are on one side, is represented by 100000 1. Between these extremes a vast number of degrees must exist; and though human justice draws a broad line where criminal responsibility begins, its decisions must needs be rough and inac = curate. The application of the doctrine of chances to real cases must be made with great caution. Our illustrations have been drawn for the most part from artificial cases, where causes have been studiously excluded that might have disturbed and complicated the results in nature these are hard to find. : § 123. Syllogisms of Classification, Classification, which enters into all sciences, is the basis of some of them, as Botany, Mineralogy, and Zoology. In every act of classification two steps must be taken; certain marks are to be selected, the possession of which is to be the title to admission into |