once causation is admitted as an uni-stract superiority of an estimate of versal law, our expectation of events probability grounded on causes, it is can only be rationally grounded on a fact that in almost all cases in which that law. To a person who recog- chances admit of estimation suffinises that every event depends on ciently precise to render their numecauses, a thing's having happened rical appreciation of any practical once is a reason for expecting it to value, the numerical data are not happen again, only because proving drawn from knowledge of the causes, that there exists, or is liable to exist, but from experience of the events a cause adequate to produce it.* The themselves. The probabilities of life frequency of the particular event, at different ages or in different cliapart from all surmise respecting mates; the probabilities of recovery its cause, can give rise to no other from a particular disease; the chances induction than that per enumerationem of the birth of male or female offsimplicem; and the precarious infer- spring; the chances of the destrucences derived from this are super- tion of houses or other property by seded and disappear from the field, fire; the chances of the loss of a ship as soon as the principle of causation in a particular voyage-are deduced makes its appearance there. from bills of mortality, returns from Notwithstanding, however, the ab-hospitals, registers of births, of ship * "If this be not so, why do we feel so much more probability added by the first instance than by any single subsequent instance? Why, except that the first instance gives us its possibility, (a cause adequate to it,) while every other only gives us the frequency of its conditions? If no reference to a cause be supposed, possibility would have no meaning; yet it is clear that, antecedent to its happening, we might have supposed the event impos sible, i.e., have believed that there was no physical energy really existing in the world equal to producing it. After the first time of happening, which is, then, more important to the whole probability than any other single instance, (because proving the possibility,) the number of times becomes important as an index to the intensity or extent of the cause, and its independence of any particular time. If we took the case of a tremendous leap, for instance, and wished to form an estimate of the probability of its succeeding a certain number of times; the first instance, by showing its possibility, (before doubtful,) is of the most importance; but every succeeding leap shows the power to be more perfectly under control, greater and more invariable, and so increases the probability; and no one would think of reasoning in this case straight from one instance to the next, without referring to the physical energy which each leap indicated. Is it not then clear that we do not ever" (let us rather say, that we do not in an advanced state of our knowledge) "conclude directly from the happening of an event to the probability of its happening again; but that we refer to the cause, regarding the past cases as an index to the cause, and the cause as our guide to the future?"-Prospective Review for February 1850. wrecks, &c., that is, from the observed frequency not of the causes, but of the effects. The reason is, that in all these classes of facts, the causes are either not amenable to direct observation at all, or not with the requisite precision, and we have no means of judging of their frequency except from the empirical law afforded by the frequency of the effects. The inference does not the less depend on causation alone. We reason from an effect to a similar effect by passing through the cause. If the actuary of an insurance office infers from his tables that among a hundred persons now living, of a particular age, five on the average will attain the age of seventy, his inference is legitimate, not for the simple reason that this is the proportion who have lived till seventy in times past, but because the fact of their having so lived shows that this is the proportion existing, at that place and time, between the causes which prolong life to the age of seventy, and those tending to bring it to an earlier close.* * The writer last quoted says that the valuation of chances by comparing the number of cases in which the event occurs with the number in which it does not occur "would generally be wholly erroneous, and "is not the true theory of probability." It is at least that which forms the founda cause, if it existed, would have produced the given effect. Let M be the effect, and A, B, two causes, by either of which it might have been produced. To find the probability that it was produced by the one and not by the other, ascertain which of the two is most likely to have existed, and which of them, if it did exist, was most likely to produce the effect M: the probability sought is a compound of these two probabilities. § 5. From the preceding principles it is easy to deduce the demonstration of that theorem of the doctrine of probabilities which is the foundation of its application to inquiries for ascertaining the occurrence of a given event or the reality of an individual fact. The signs or evidences by which a fact is usually proved are some of its consequences and the inquiry hinges upon determining what cause is most likely to have produced a given effect. The theorem applicable to such investigations is the Sixth CASE I. Let the causes be both Principle in Laplace's Essai Philoso- alike in the second respect; either A phique sur les Probabilités, which is or B, when it exists, being supposed described by him as the "fundamental equally likely (or equally certain) to principle of that branch of the Analy-produce M; but let A be in itself sis of Chances which consists in as twice as likely as B to exist, that is, cending from events to their causes.' twice as frequent a phenomenon. Given an effect to be accounted for, Then it is twice as likely to have and there being several causes which existed in this case, and to have been might have produced it, but of the the cause which produced M. presence of which in the particular case nothing is known; the probability that the effect was produced by any one of these causes is as the antecedent probability of the cause, multiplied by the probability that the tion of insurance, and of all those calcula tions of chances in the business of life which experience so abundantly verifies. The reason which the reviewer gives for rejecting the theory, is that it "would regard an event as certain which had hitherto never failed; which is exceedingly far from the truth, even for a very large number of constant successes." This is not a defect in a particular theory, but in any theory of chances. No principle of evaluation can provide for such a case as that which the reviewer supposes. If an event has never once failed, in a number of trials sufficient to eliminate chance, it really has all the certainty which can be given by an empirical law it is certain during the continuance of the same collocation of causes which existed during the observations. If it ever fails, it is in consequence of some change in that collocation. Now, no theory of chances will enable us to infer the future probability of an event from the past, if the causes in operation capable of influencing the event have intermediately undergone a change. * Pp. 18, 19. The theorem is not stated by Laplace in the exact terms in which I have stated it; but the identity of import of the two modes of expression is easily demonstrable. For, since A exists in nature twice as often as B, in any 300 cases in which one or other existed, A has existed 200 times and B 100. But either A or B must have existed wherever M is produced: therefore in 300 times that M is produced, A B only 100, that is, in the ratio of 2 was the producing cause 200 times, to 1. Thus, then, if the causes are alike in their capacity of producing the effect, the probability as to which actually produced it is in the ratio of their antecedent probabilities. CASE II. Reversing the last hypothesis, let us suppose that the causes are equally frequent, equally likely to have existed, but not equally likely, if they did exist, to produce M: that in three times in which A occurs, it produces that effect twice, while B, in three times, produces it only once. Since the two causes are equally frequent in their occurrence; in every six times that either one or the other exists, A exists three times and B three times. A, of its three times, produces M in two; B, of its three times, produces M in one. the whole six times, M is only proThus, in duced thrice; but of that thrice it is produced twice by A, once only by | B. Consequently, when the antedent probabilities of the causes are equal, the chances that the effect was produced by them are in the ratio of the probabilities that if they did exist they would produce the effect. CASE III. The third case, that in which the causes are unlike in both respects, is solved by what has preceded. For when a quantity depends on two other quantities, in such a manner that while either of them remains constant it is proportional to the other, it must necessarily be proportional to the product of the two quantities, the product being the only function of the two which obeys that law of variation. Therefore the probability that M was produced by either cause is as the antecedent probability of the cause, multiplied by the probability that if it existed it would produce M. Which was to be demonstrated. § 6. It remains to examine the bearing of the doctrine of chances on the peculiar problem which occupied us in the preceding chapter, namely, how to distinguish coincidences which are casual from those which are the result of law-from those in which the facts which accompany or follow one another are somehow connected through causation. The doctrine of chances affords means by which, if we knew the average number of coincidences to be looked for between two phenomena connected only casually, we could determine how often any given deviation from that average will occur by chance. If the probability of any casual coincidence, considered in itself, be, the probability that the same coincidence will be repeated n I mn I I 216 ample, in one throw of a die the probability of ace being, the probabiOr we may prove the third case as we proved the first and second. Let lity of throwing ace twice in succession A be twice as frequent as B; and let will be I divided by the square of 6, or them also be unequally likely, when I For ace is thrown at the first they exist, to produce M; let A pro- 36 duce it twice in four times, B thrice in throw once in six, or six in thirtyfour times. The antecedent probabi-six times, and of those six, the die lity of A is to that of B as 2 to 1; the being cast again, ace will be thrown probabilities of their producing M are but once; being altogether once in as 2 to 3; the product of these ratios thirty-six times. The chance of the is the ratio of 4 to 3; and this will same cast three times successively is, be the ratio of the probabilities that by a similar reasoning, or ; that A or B was the producing cause in the given instance. For, since A is is, the event will happen, on a large twice as frequent as B, out of twelve average, only once in two hundred cases in which one or other exists, A and sixteen throws. exists in 8 and B in 4. But of its eight cases, A, by the supposition, produces M in only 4, while B of its four cases produces M in 3. M, therefore, is only produced at all in seven of the twelve cases; but in four of these it is produced by A, in three by B; hence the probabilities of its being produced by A and by B are as 4 to 3, and are expressed by the fractions and . Which was to be demonstrated. We have thus a rule by which to estimate the probability that any given series of coincidences arises from chance, provided we can measure correctly the probability of a single coincidence. If we can obtain an equally precise expression for the probability that the same series of coincidences arises from causation, we should only have to compare the numbers. This, however, can rarely be done. Let us see what degree of approximation can practically be made to the necessary precision. When, however, the coincidence is one which cannot be accounted for by any known cause, and the connection between the two phenomena, if produced by causation, must be the result of some law of nature hitherto unknown, which is the case we had in view in the last chapter; then, though the probability of a casual coincidence may be capable of ap position, the existence of an undiscovered law of nature, is clearly unsusceptible of even an approximate valuation. In order to have the data which such a case would require, it would be necessary to know what The question falls within Laplace's Sixth Principle, just demonstrated. The given fact, that is to say, the series of coincidences, may have originated either in a casual conjunction of causes or in a law of nature. The probabilities, therefore, that the fact originated in these two modes are, as their antecedent probabilities, multi-preciation, that of the counter-supplied by the probabilities that if they existed they would produce the effect. But the particular combination of chances, if it occurred, or the law of nature if real, would certainly produce the series of coincidences. The probabilities, therefore, that the co-proportion of all the individual seincidences are produced by the two causes in question are as the antecedent probabilities of the causes. One of these, the antecedent probability of the combination of mere chances which would produce the given result, is an appreciable quantity. The antecedent probability of the other supposition may be susceptible of a more or less exact estimation, according to the nature of the case. In some cases the coincidence, supposing it to be the result of causation at all, must be the result of a known cause, as the succession of aces, if not accidental, must arise from the loading of the die. In such cases we may be able to form a conjecture as to the antecedent probability of such a circumstance from the characters of the parties concerned, or other such evidence; but it would be impossible to estimate that probability with any thing like numerical precision. The counter-probability, however, that of the accidental origin of the coincidence, dwindling so rapidly as it does at each new trial; the stage is soon reached at which the chance of unfairness in the die, however small in itself, must be greater than that of a casual coincidence; and on this ground a practical decision can generally be come to without much hesitation, if there be the power of repeating the experiment. quences or co-existences occurring in nature are the result of law, and what proportion are mere casual coincidences. It being evident that we cannot form any plausible conjecture as to this proportion, much less appreciate it numerically, we cannot attempt any precise estimation of the comparative probabilities. But of this we are sure, that the detection of an unknown law of nature-of some previously unrecognised constancy of conjunction among phenomena―is no uncommon event. If, therefore, the number of instances in which a coincidence is observed, over and above that which would arise on the average from the mere concurrence of chances, be such that so great an amount of coincidences from accident alone would be an extremely uncommon event; we have reason to conclude that the coincidence is the effect of causation, and may be received (subject to correction from further experience) as an empirical law. Further than this, in point of precision, we cannot go; nor, in most cases, is greater precision required for the solution of any practical doubt.* *For a fuller treatment of the many interesting questions raised by the theory of probabilities, I may now refer to a recent work by Mr. Venn, Fellow of Caius College, of the most thoughtful and philosophical Cambridge, "The Logic of Chance," one treatises on any subject connected with CHAPTER XIX. OF THE EXTENSION OF DERIVATIVE § 1. WE have had frequent occasion to notice the inferior generality of derivative laws compared with the ultimate laws from which they are derived. This inferiority, which affects not only the extent of the propositions themselves, but their degree of certainty within that extent, is most conspicuous in the uniformities of co-existence and sequence obtaining between effects which depend ultimately on different primeval causes. Such uniformities will only obtain where there exists the same collocation of those primeval causes. If the collocation varies, though the laws themselves remain the same, a totally different set of derivative uniformities may, and generally will, be the result. out of the earth's shadow, consequent on the earth's rotation, and on the illuminating property of the sun. If, therefore, day is ever produced by a different cause or set of causes from this, day will not, or at least may not, be followed by night. On the sun's own surface, for instance, this may be the case. Finally, even when the derivative uniformity is itself a law of causation, (resulting from the combination of several causes,) it is not altogether independent of collocations. If a cause supervenes capable of wholly or partially counteracting the effect of any one of the conjoined causes, the effect will no longer conform to the derivative law. While, therefore, each ultimate law is only liable to frustration from one set of counteracting causes, the derivative law is liable to it from several. Now, the possibility of the occurrence of counteracting causes which do not arise from any of the conditions involved in the law itself depends on the original collocations. Even where the derivative uniformity is between different effects of the same cause, it will by no means It is true that (as we formerly reobtain as universally as the law of the marked) laws of causation, whether cause itself. If a and b accompany ultimate or derivative, are, in most or succeed one another as effects of cases, fulfilled even when counterthe cause A, it by no means follows acted: the cause produces its effect, that A is the only cause which can though that effect is destroyed by produce them, or that if there be something else. That the effect may another cause, as B, capable of pro-be frustrated, is, therefore, no objecducing a, it must produce b like- tion to the universality of laws of wise. The conjunction therefore of a causation. But it is fatal to the uniand b perhaps does not hold univer-versality of the sequences or co-existsally, but only in the instances in which a arises from A. When it is produced by a cause other than A, a and b may be dissevered. Day (for example) is always in our experience followed by night: but day is not the cause of night; both are successive effects of a common cause, the periodical passage of the spectator into and Logic and Evidence which have been pro- very useful to me in revising the corre- obvious to any reader of Mr. Venn's work who is also a reader of this. ences of effects which compose the greater part of the derivative laws flowing from laws of causation. When from the law of a certain combination of causes there results a certain order in the effects, as from the combination of a single sun with the rotation of an opaque body round its axis, there results, on the whole surface of that opaque body, an alternation of day and night; then if we suppose one of the combined causes counteracted, the rotation stopped, the sun extinguished, or a second sun superadded, the truth of that particular law of causation is in no way affected; it is still true that one sun shining on an |