CHAPTER XVI.

THE APPLICATION OF PROBABILITY TO TESTIMONY.

§ 1. On the principles which have been adopted in this work, it becomes questionable whether several classes of problems which may seem to have acquired a prescriptive right to admission, will not have to be excluded from the science of Probability. The most important, perhaps, of these refer to what is commonly called the credibility of testimony, estimated either at first hand and directly, or as influencing a juryman, and so reaching us through his sagacity and trustworthiness. Almost every treatise upon the science contains a discussion of the principles according to which credit is to be attached to combinations of the reports of witnesses of various degrees of trustworthiness, or the verdicts of juries consisting of larger or smaller numbers. A great modern mathematician, Poisson, has written an elaborate treatise expressly upon this subject; whilst a considerable portion of the works of Laplace, De Morgan, and others, is devoted to an examination of similar enquiries. It would be presumptuous to differ from such authorities as these, except upon the strongest grounds; but I confess that the extraordinary ingenuity and mathematical ability which have been devoted to these problems, considered as questions in Probability, fails to convince me that they ought to have been so considered. The following are the principal grounds for this opinion.

§ 2. It will be remembered that in the course of the chapter on Induction we entered into a detailed investigation of the process demanded of us when, instead of the appropriate propositions from which the inference was to be made being set before us, the individual presented himself, and the task was imposed upon us of selecting the requisite groups or series to which to refer him. In other words, instead of calculating the chance of an event from determinate conditions of frequency of its occurrence (these being either obtained by direct experience, or deductively inferred) we have to select the conditions of frequency out of a plurality of more or less suitable ones. When the problem is presented to us at such a stage as this, we may of course assume that the preliminary process of obtaining the statistics which are extended into the proportional propositions has been already performed; we may suppose therefore that we are already in possession of a quantity of such propositions, our principal remaining doubt being as to which of them we should then employ. This selection was shown to be to a certain extent arbitrary; for, owing to the fact of the individual possessing a large number of different properties, he became in consequence a member of different series or groups, which might present different averages. We must now examine, somewhat more fully than we did before, the practical conditions under which any difficulty arising from this source ceases to be of importance.

§ 3. One condition of this kind is very simple and obvious. It is that the different statistics with which we are presented should not in reality offer materially different results, If, for instance, we were enquiring into the probability of a man aged forty dying within the year, we might if we pleased take into account the fact of his having red hair, or his having been born in a certain county or town. Each of these circumstances would serve to specialize the individual, and therefore to restrict the limits of the statistics which were applicable to his case. But the consideration of such qualities as these would either leave the average precisely as it was, or produce such an unimportant alteration in it as no one would think of taking into account. Though we could hardly say with certainty of any conceivable characteristic that it has absolutely no bearing on the result, we may still feel very confident that the bearing of such characteristics as these is utterly insignificant. Of course in the extreme case of the things most perfectly suited to the Calculus of Probability, viz. games of pure chance, these subsidiary characteristics are quite irrelevant. Any further particulars about the characteristics of the cards in a really fair pack, beyond those which are familiar to all the players, would convey no information whatever about the result.

Or again; although the different sets of statistics may not as above give almost identical results, yet they may do what practically comes to very much the same thing, that is, arrange themselves into a small number of groups, all of the statistics in any one group practically coinciding in their results. If for example a consumptive man desired to insure his life, there would be a marked difference in the statistics according as we took his peculiar state of health into account or not. We should here have two sets of statistics, so clearly marked off from one another that they might almost rank with the distinctions of natural kinds, and which would in consequence offer decidedly different results. If we were to specialize still further, by taking into account insignificant qualities like those mentioned in the last paragraph, we might indeed get more limited sets of statistics applicable to persons still more closely resembling the individual in question, but these would not differ sufficiently in their results to make it worth our while to do so. In other words, the different propositions which are applicable to the case in point arrange themselves into a limited number of groups, which, and which only, need be taken into account; whence the range of choice amongst them is very much diminished in practice.

§ 4. The reasons for the conditions above described are not difficult to detect. Where these conditions exist the process of selecting a series or class to which to refer any individual is very simple, and the selection is, for the particular purposes of inference, final. In any case of insurance, for example, the question we have to decide is of the very simple kind; Is A. B. a man of a certain age? If so one in fifty in his circumstances will die in the course of the year. If any further questions have to be decided they would be of the following description. Is A. B. a healthy man? Does he follow a dangerous trade? But here too the classes in question are but few, and the limits by which they are bounded are tolerably precise; so that the reference of an individual to one or other of them is easy. And when we have once chosen our class we remain untroubled by any further considerations; for since no other statistics are supposed to offer a materially different average, we have no occasion to take account of any other properties than those already noticed.

The case of games of chance, already referred to, offers of course an instance of these conditions in an almost ideal state of perfection; the same circumstances which fit them so eminently for the purposes of fair gambling, fitting them equally to become examples in Probability. When a die is to be thrown, all persons alike stand on precisely the same footing of knowledge and of ignorance about the result; the only data to which any one could appeal being that each face turns up on an average once in six times.

§ 5. Let us now examine how far the above conditions are fulfilled in the case of problems which discuss what is called the credibility of testimony. The following would be a fair specimen of one of the elementary enquiries out of which these problems are composed;—Here is a statement made by a witness who lies once in ten times, what am I to conclude about its truth? Objections might fairly be raised against the possibility of thus assigning a man his place upon a graduated scale of mendacity. This however we will pass over, and will assume that the witness goes about the world bearing stamped somehow on his face the appropriate class to which he belongs, and consequently, the degree of credit to which he has a claim on such general grounds. But there are other and stronger reasons against the admissibility of this class of problems.

§ 6. That which has been described in the previous sections as the ‘individual’ which had to be assigned to an appropriate class or series of statistics is, of course, in this case, a statement. In the particular instance in question this individual statement is already assigned to a class, that namely of statements made by a witness of a given degree of veracity; but it is clearly optional with us whether or not we choose to confine our attention to this class in forming our judgment; at least it would be optional whenever we were practically called on to form an opinion. But in the case of this statement, as in that of the mortality of the man whose insurance we were discussing, there are a multitude of other properties observable, besides the one which is supposed to mark the given class. Just as in the latter there were (besides his age), the place of his birth, the nature of his occupation, and so on; so in the former there are (besides its being a statement by a certain kind of witness), the fact of its being uttered at a certain time and place and under certain circumstances. At the time the statement is made all these qualities or attributes of the statement are present to us, and we clearly have a right to take into account as many of them as we please. Now the question at present before us seems to be simply this;—Are the considerations, which we might thus introduce, as immaterial to the result in the case of the truth of a statement of a witness, as the corresponding considerations are in the case of the insurance of a life? There can surely be no hesitation in the reply to such a question. Under ordinary circumstances we soon know all that we can know about the conditions which determine us in judging of the prospect of a man's death, and we therefore rest content with general statistics of mortality; but no one who heard a witness speak would think of simply appealing to his figure of veracity, even supposing that this had been authoritatively communicated to us. The circumstances under which the statement is made instead of being insignificant, are of overwhelming importance. The appearance of the witness, the tone of his voice, the fact of his having objects to gain, together with a countless multitude of other circumstances which would gradually come to light as we reflect upon the matter, would make any sensible man discard the assigned average from his consideration. He would, in fact, no more think of judging in this way than he would of appealing to the Carlisle or Northampton tables of mortality to determine the probable length of life of a soldier who was already in the midst of a battle.

§ 7. It cannot be replied that under these circumstances we still refer the witness to a class, and judge of his veracity by an average of a more limited kind; that we infer, for example, that of men who look and act like him under such circumstances, a much larger proportion, say nine-tenths, are found to lie. There is no appeal to a class in this way at all, there is no immediate reference to statistics of any kind whatever; at least none which we are conscious of using at the time, or to which we should think of resorting for justification afterwards. The decision seems to depend upon the quickness of the observer's senses and of his apprehension generally.

Statistics about the veracity of witnesses seem in fact to be permanently as inappropriate as all other statistics occasionally may be. We may know accurately the percentage of recoveries after amputation of the leg; but what surgeon would think of forming his judgment solely by such tables when he had a case before him? We need not deny, of course, that the opinion he might form about the patient's prospects of recovery might ultimately rest upon the proportions of deaths and recoveries he might have previously witnessed. But if this were the case, these data are lying, as one may say, obscurely in the background. He does not appeal to them directly and immediately in forming his judgment. There has been a far more important intermediate process of apprehension and estimation of what is essential to the case and what is not. Sharp senses, memory, judgment, and practical sagacity have had to be called into play, and there is not therefore the same direct conscious and sole appeal to statistics that there was before. The surgeon may have in his mind two or three instances in which the operation performed was equally severe, but in which the patient's constitution was different; the latter element therefore has to be properly allowed for. There may be other instances in which the constitution was similar, but the operation more severe; and so on. Hence, although the ultimate appeal may be to the statistics, it is not so directly; their value has to be estimated through the somewhat hazy medium of our judgment and memory, which places them under a very different aspect.

§ 8. Any one who knows anything of the game of whist may supply an apposite example of the distinction here insisted on, by recalling to mind the alteration in the nature of our inferences as the game progresses. At the commencement of the game our sole appeal is rightfully made to the theory of Probability. All the rules upon which each player acts, and therefore upon which he infers that the others will act, rest upon the observed frequency (or rather upon the frequency which calculation assures us will be observed) with which such and such combinations of cards are found to occur. Why are we told, if we have more than four trumps, to lead them out at once? Because we are convinced, on pure grounds of probability, capable of being stated in the strictest statistical form, that in a majority of instances we shall draw our opponent's trumps, and therefore be left with the command. Similarly with every other rule which is recognized in the early part of the play.

But as the play progresses all this is changed, and towards its conclusion there is but little reliance upon any rules which either we or others could base upon statistical frequency of occurrence, observed or inferred. A multitude of other considerations have come in; we begin to be influenced partly by our knowledge of the character and practice of our partner and opponents; partly by a rapid combination of a multitude of judgments, founded upon our observation of the actual course of play, the grounds of which we could hardly realize or describe at the time and which may have been forgotten since. That is, the particular combination of cards, now before us, does not readily fall into any well-marked class to which alone it can reasonably be referred by every one who has the facts before him.

§ 9. A criticism somewhat resembling the above has been given by Mill (Logic, Bk. III. Chap. XVIII. § 3) upon the applicability of the theory of Probability to the credibility of witnesses. But he has added other reasons which do not appear to me to be equally valid; he says “common sense would dictate that it is impossible to strike a general average of the veracity, and other qualifications for true testimony, of mankind or any class of them; and if it were possible, such an average would be no guide, the credibility of almost every witness being either below or above the average,” The latter objection would however apply with equal force to estimating the length of a man's life from tables of mortality; for the credibility of different witnesses can scarcely have a wider range of variation than the length of different lives. If statistics of credibility could be obtained, and could be conveniently appealed to when they were obtained, they might furnish us in the long run with as accurate inferences as any other statistics of the same general description. These statistics would however in practice naturally and rightly be neglected, because there can hardly fail to be circumstances in each individual statement which would more appropriately refer it to some new class depending on different statistics, and affording a far better chance of our being right in that particular case. In most instances of the kind in question, indeed, such a change is thus produced in the mode of formation of our opinion, that, as already pointed out, the mental operation ceases to be in any proper sense founded on appeal to statistics.[1]

§ 10. The Chance problems which are concerned with testimony are not altogether confined to such instances as those hitherto referred to. Though we must, as it appears to me, reject all attempts to estimate the credibility of any particular witness, or to refer him to any assigned class in respect of his trustworthiness, and consequently abandon as unsuitable any of the numerous problems which start from such data as ‘a witness who is wrong once in ten times,’ yet it does not follow that testimony may not to a slight extent be treated by our science in a somewhat different manner. We may be quite unable to estimate, except in the roughest possible way, the veracity of any particular witness, and yet it may be possible to form some kind of opinion upon the veracity of certain classes of witnesses; to say, for instance, that Europeans are superior in this way to Orientals. So we might attempt to explain why, and to what extent, an opinion in which the judgments of ten persons, say jurors, concur, is superior to one in which five only concur. Something may also be done towards laying down the principles in accordance with which we are to decide whether, and why, extraordinary stories deserve less credence than ordinary ones, even if we cannot arrive at any precise and definite decision upon the point. This last question is further discussed in the course of the next chapter.

§ 11. The change of view in accordance with which it follows that questions of the kind just mentioned need not be entirely rejected from scientific consideration, presents itself in other directions also. It has, for instance, been already pointed out that the individual characteristics of any sick man's disease would be quite sufficiently important in most cases to prevent any surgeon from judging about his recovery by a genuine and direct appeal to statistics, however such considerations might indirectly operate upon his judgment. But if an opinion had to be formed about a considerable number of cases, say in a large hospital, statistics might again come prominently into play, and be rightly recognized as the principal source of appeal. We should feel able to compare one hospital, or one method of treatment, with another. The ground of the difference is obvious. It arises from the fact that the characteristics of the individuals, which made us so ready to desert the average when we had to judge of them separately, do not produce the same disturbance when were have to judge about a group of cases. The averages then become the most secure and available ground on which to form an opinion, and therefore Probability again becomes applicable.

But although some resort to Probability may be admitted in such cases as these, it nevertheless does not appear to me that they can ever be regarded as particularly appropriate examples to illustrate the methods and resources of the theory. Indeed it is scarcely possible to resist the conviction that the refinements of mathematical calculation have here been pushed to lengths utterly unjustifiable, when we bear in mind the impossibility of obtaining any corresponding degree of accuracy and precision in the data from which we have to start. To cite but one instance. It would be hard to find a case in which love of consistency has prevailed over common sense to such an extent as in the admission of the conclusion that it is unimportant what are the numbers for and against a particular statement, provided the actual majority is the same. That is, the unanimous judgment of a jury of eight is to count for the same as a majority of ten to two in a jury of twelve. And yet this conclusion is admitted by Poisson. The assumptions under which it follows will be indicated in the course of the next chapter.

Again, perfect independence amongst the witnesses or jurors is an almost necessary postulate. But where can this be secured? To say nothing of direct collusion, human beings are in almost all instances greatly under the influence of sympathy in forming their opinions. This influence, under the various names of political bias, class prejudice, local feeling, and so on, always exists to a sufficient degree to induce a cautious person to make many of those individual corrections which we saw to be necessary when we were estimating the trustworthiness, in any given case, of a single witness; that is, they are sufficient to destroy much, if not all, of the confidence with which we resort to statistics and averages in forming our judgment. Since then this Essay is mainly devoted to explaining and establishing the general principles of the science of Probability, we may very fairly be excused from any further treatment of this subject, beyond the brief discussions which are given in the next chapter.

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CHAPTER XVII.

ON THE CREDIBILITY OF EXTRAORDINARY STORIES.

§ 1. It is now time to recur for fuller investigation to an enquiry which has been already briefly touched upon more than once; that is, the validity of testimony to establish, as it is frequently expressed, an otherwise improbable story. It will be remembered that in a previous chapter (the twelfth) we devoted some examination to an assertion by Butler, which seemed to be to some extent countenanced by Mill, that a great improbability before the proof might become but a very small improbability after the proof. In opposition to this it was pointed out that the different estimates which we undoubtedly formed of the credibility of the examples adduced, had nothing to do with the fact of the event being past or future, but arose from a very different cause; that the conception of the event which we entertain at the moment (which is all that is then and there actually present to us, and as to the correctness of which as a representation of facts we have to make up our minds) comes before us in two very different ways. In one instance it was a mere guess of our own which we knew from statistics would be right in a certain proportion of cases; in the other instance it was the assertion of a witness, and therefore the appeal was not now primarily to statistics of the event, but to the trustworthiness of the witness. The conception, or ‘event’ if we will so term it, had in fact passed out of the category of guesses (on statistical grounds), into that of assertions (most likely resting on some specific evidence), and would therefore be naturally regarded in a very different light.

§ 2. But it may seem as if this principle would lead us to somewhat startling conclusions. For, by transferring the appeal from the frequency with which the event occurs to the trustworthiness of the witness who makes the assertion, is it not implied that the probability or improbability of an assertion depends solely upon the veracity of the witness? If so, ought not any story whatever to be believed when it is asserted by a truthful person?

In order to settle this question we must look a little more closely into the circumstances under which such testimony is commonly presented to us. As it is of course necessary, for clearness of exposition, to take a numerical example, let us suppose that a given statement is made by a witness who, on the whole and in the long run, is right in what he says nine times out of ten.[1] Here then is an average given to us, an average veracity that is, which includes all the particular statements which the witness has made or will make.

§ 3. Now it has been abundantly shown in a former chapter (Ch. IX. §§ 14–32) that the mere fact of a particular average having been assigned, is no reason for our being forced invariably to adhere to it, even in those cases in which our most natural and appropriate ground of judgment is found in an appeal to statistics and averages. The general average may constantly have to be corrected in order to meet more accurately the circumstances of particular cases. In statistics of mortality, for instance; instead of resorting to the wider tables furnished by people in general of a given age, we often prefer the narrower tables furnished by men of a particular profession, abode, or mode of life. The reader may however be conveniently reminded here that in so doing we must not suppose that we are able, by any such device, in any special or peculiar way to secure truth. The general average, if persistently adhered to throughout a sufficiently wide and varied experience, would in the long run tend to give us the truth; all the advantage which the more special averages can secure for us is to give us the same tendency to the truth with fewer and slighter aberrations.

§ 4. Returning then to our witness, we know that if we have a very great many statements from him upon all possible subjects, we may feel convinced that in nine out of ten of these he will tell us the truth, and that in the tenth case he will go wrong. This is nothing more than a matter of definition or consistency. But cannot we do better than thus rely upon his general average? Cannot we, in almost any given case, specialize it by attending to various characteristic circumstances in the nature of the statement which he makes; just as we specialize his prospects of mortality by attending to circumstances in his constitution or mode of life?

Undoubtedly we may do this; and in any of the practical contingencies of life, supposing that we were at all guided by considerations of this nature, we should act very foolishly if we did not adopt some such plan. Two methods of thus correcting the average may be suggested: one of them being that which practical sagacity would be most likely to employ, the other that which is almost universally adopted by writers on Probability. The former attempts to make the correction by the following considerations: instead of relying upon the witness' general average, we assign to it a sort of conjectural correction to meet the case before us, founded on our experience or observation; that is, we appeal to experience to establish that stories of such and such a kind are more or less likely to be true, as the case may be, than stories in general. The other proceeds upon a different and somewhat more methodical plan. It is here endeavoured to show, by an analysis of the nature and number of the sources of error in the cases in question, that such and such kinds of stories must be more or less likely to be correctly reported, and this in certain numerical proportions.

§ 5. Before proceeding to a discussion of these methods a distinction must be pointed out to which writers upon the subject have not always attended, or at any rate to which they have not generally sufficiently directed their readers' attention.[2] There are, broadly speaking, two different ways in which we may suppose testimony to be given. It may, in the first place, take the form of a reply to an alternative question, a question, that is, framed to be answered by yes or no. Here, of course, the possible answers are mutually contradictory, so that if one of them is not correct the other must be so:—Has A happened, yes or no? The common mode of illustrating this kind of testimony numerically is by supposing a lottery with a prize and blanks, or a bag of balls of two colours only, the witness knowing that there are only two, or at any rate being confined to naming one or other of them. If they are black and white, and he errs when black is drawn, he must say ‘white,’ The reason for the prominence assigned to examples of this class is, probably, that they correspond to the very important case of verdicts of juries; juries being supposed to have nothing else to do than to say ‘guilty’ or ‘not guilty.’

On the other hand, the testimony may take the form of a more original statement or piece of information. Instead of saying, Did A happen? we may ask, What happened? Here if the witness speaks truth he must be supposed, as before, to have but one way of doing so; for the occurrence of some specific event was of course contemplated. But if he errs he has many ways of going wrong, possibly an infinite number. Ordinarily however his possible false statements are assumed to be limited in number, as must generally be more or less the result in practice. This case is represented numerically by supposing the balls in the bag not to be of two colours only, but to be all distinct from each other; say by their being all numbered successively. It may of course be objected that a large number of the statements that are made in the world are not in any way answers to questions, either of the alternative or of the open kind. For instance, a man simply asserts that he has drawn the seven of spades from a pack of cards; and we do not know perhaps whether he had been asked ‘Has that card been drawn?’ or ‘What card has been drawn?’ or indeed whether he had been asked anything at all. Still more might this be so in the case of any ordinary historical statement.

This objection is quite to the point, and must be recognized as constituting an additional difficulty. All that we can do is to endeavour, as best we may, to ascertain, from the circumstances of the case, what number of alternatives the witness may be supposed to have had before him. When he simply testifies to some matter well known to be in dispute, and does not go much into detail, we may fairly consider that there were practically only the two alternatives before him of saying ‘yes’ or ‘no.’ When, on the other hand, he tells a story of a more original kind, or (what comes to much the same thing) goes into details, we must regard him as having a wide comparative range of alternatives before him.

These two classes of examples, viz. that of the black and white balls, in which only one form of error is possible, and the numbered balls, in which there may be many forms of error, are the only two which we need notice. In practice it would seem that they may gradually merge into each other, according to the varying ways in which we choose to frame our question. Besides asking, Did you see A strike B? and, What did you see? we may introduce any number of intermediate leading questions, as, What did A do? What did he do to B? and so on. In this way we may gradually narrow the possible openings to wrong statement, and so approach to the direct alternative question. But it is clear that all these cases may be represented numerically by a supposed diminution in the number of the balls which are thus distinguished from each other.

§ 6. Of the two plans mentioned in § 4 we will begin with the latter, as it is the only methodical and scientific one which has been proposed. Suppose that there is a bag with 1000 balls, only one of which is white, the rest being all black. A ball is drawn at random, and our witness whose veracity is ⁹/₁₀ reports that the white ball was drawn. Take a great many of his statements upon this particular subject, say 10,000; that is, suppose that 10,000 balls having been successively drawn out of this bag, or bags of exactly the same kind, he makes his report in each case. His 10,000 statements being taken as a fair sample of his general average, we shall find, by supposition, that 9 out of every 10 of them are true and the remaining one false. What will be the nature of these false statements? Under the circumstances in question, he having only one way of going wrong, the answer is easy. In the 10,000 drawings the white ball would come out 10 times, and therefore be rightly asserted 9 times, whilst on the one of these occasions on which he goes wrong he has nothing to say but ‘black.’ So with the 9990 occasions on which black is drawn; he is right and says black on 8991 of them, and is wrong and therefore says white on 999 of them. On the whole, therefore, we conclude that out of every 1008 times on which he says that white is drawn he is wrong 999 times and right only 9 times. That is, his special veracity, as we may term it, for cases of this description, has been reduced from ⁹/₁₀ to ⁹/₁₀₀₈. As it would commonly be expressed, the latter fraction represents the chance that this particular statement of his is true.[3]

§ 7. We will now take the case in which the witness has many ways of going wrong, instead of merely one. Suppose that the balls were all numbered, from 1 to 1,000, and the witness knows this fact. A ball is drawn, and he tells me that it was numbered 25, what are the odds that he is right? Proceeding as before, in 10,000 drawings this ball would be obtained 10 times, and correctly named 9 times. But on the 9990 occasions on which it was not drawn there would be a difference, for the witness has now many openings for error before him. It is, however, generally considered reasonable to assume that his errors will all take the form of announcing wrong numbers; and that, there being no apparent reason why he should choose one number rather than another, he will be likely to announce all the wrong ones equally often. Hence his 999 errors, instead of all leading him now back again to one spot, will be uniformly spread over as many distinct ways of going wrong. On one only of these occasions, therefore, will he mention 25 as having been drawn. It follows therefore that out of every 10 times that he names 25 he is right 9 times; so that in this case his average or general truthfulness applies equally well to the special case in point.

§ 8. With regard to the truth of these conclusions, it must of course be admitted that if we grant the validity of the assumptions about the limits within which the blundering or mendacity of the witness are confined, and the complete impartiality with which his answers are disposed within those limits, the reasoning is perfectly sound. But are not these assumptions extremely arbitrary, that is, are not our lotteries and bags of balls rendered perfectly precise in many respects in which, in ordinary life, the conditions supposed to correspond to them are so vague and uncertain that no such method of reasoning becomes practically available? Suppose that a person whom I have long known, and of whose measure of veracity and judgment I may be supposed therefore to have acquired some knowledge, informs me that there is something to my advantage if I choose to go to certain trouble or expense in order to secure it. As regards the general veracity of the witness, then, there is no difficulty; we suppose that this is determined for us. But as regards his story, difficulty and vagueness emerge at every point. What is the number of balls in the bag here? What in fact are the nature and contents of the bag out of which we suppose the drawing to have been made? It does not seem that the materials for any rational judgment exist here. But if we are to get at any such amended figure of veracity as those attained in the above example, these questions must necessarily be answered with some degree of accuracy; for the main point of the method consists in determining how often the event must be considered not to happen, and thence inferring how often the witness will be led wrongly to assert that it has happened.

It is not of course denied that considerations of the kind in question have some influence upon our decision, but only that this influence could under any ordinary circumstances be submitted to numerical determination. We are doubtless liable to have information given to us that we have come in for some kind of fortune, for instance, when no such good luck has really befallen us; and this not once only but repeatedly. But who can give the faintest intimation of the nature and number of the occasions on which, a blank being thus really drawn, a prize will nevertheless be falsely announced? It appears to me therefore that numerical results of any practical value can seldom, if ever, be looked for from this method of procedure.

§ 9. Our conclusion in the case of the lottery, or, what comes to the same thing, in the case of the bag with black and white balls, has been questioned or objected to[4] on the ground that it is contrary to all experience to suppose that the testimony of a moderately good witness could be so enormously depreciated under such circumstances. I should prefer to base the objection on the ground that experience scarcely ever presents such circumstances as those supposed; but if we postulate their existence the given conclusion seems correct enough. Assume that a man is merely required to say yes or no; assume also a group or succession of cases in which no should rightly be said very much oftener than yes. Then, assuming almost any general truthfulness of the witness, we may easily suppose the rightful occasions for denial to be so much the more frequent that a majority of his affirmative answers will actually occur as false ‘noes’ rather than as correct ‘ayes.’ This of course lowers the average value of his ‘ayes,’ and renders them comparatively untrustworthy.

Consider the following example. I have a gardener whom I trust as to all ordinary matters of fact. If he were to tell me some morning that my dog had run away I should fully believe him. He tells me however that the dog has gone mad. Surely I should accept the statement with much hesitation, and on the grounds indicated above. It is not that he is more likely to be wrong when the dog is mad; but that experience shows that there are other complaints (e.g. fits) which are far more common than madness, and that most of the assertions of madness are erroneous assertions referring to these. This seems a somewhat parallel case to that in which we find that most of the assertions that a white ball had been drawn are really false assertions referring to the drawing of a black ball. Practically I do not think that any one would feel a difficulty in thus exorbitantly discounting some particular assertion of a witness whom in most other respects he fully trusted.

§ 10. There is one particular case which has been regarded as a difficulty in the way of this treatment of the problem, but which seems to me to be a decided confirmation of it; always, be it understood, within the very narrow and artificial limits to which we must suppose ourselves to be confined. This is the case of a witness whose veracity is just one-half; that is, one who, when a mere yes or no is demanded of him, is as often wrong as right. In the case of any other assigned degree of veracity it is extremely difficult to get anything approaching to a confirmation from practical judgment and experience. We are not accustomed to estimate the merits of witnesses in this way, and hardly appreciate what is meant by his numerical degree of truthfulness. But as regards the man whose veracity is one-half, we are (as Mr C. J. Monro has very ingeniously suggested) only too well acquainted with such witnesses, though under a somewhat different name; for this is really nothing else than the case of a person confidently answering a question about a subject-matter of which he knows nothing, and can therefore only give a mere guess.

Now in the case of the lottery with one prize, when the witness whose veracity is one-half tells us that we have gained the prize, we find on calculation that his testimony goes for absolutely nothing; the chances that we have got the prize are just the same as they would be if he had never opened his lips, viz. ¹/₁₀₀₀. But clearly this is what ought to be the result, for the witness who knows nothing about the matter leaves it exactly as he found it. He is indeed, in strictness, scarcely a witness at all; for the natural function of a witness is to examine the matter, and so to add confirmation, more or less, according to his judgment and probity, but at any rate to offer an improvement upon the mere guesser. If, however, we will give heed to his mere guess we are doing just the same thing as if we were to guess ourselves, in which case of course the odds that we are right are simply measured by the frequency of occurrence of the events.

We cannot quite so readily apply the same rule to the other case, namely to that of the numbered balls, for there the witness who is right every other time may really be a very fair, or even excellent, witness. If he has many ways of going wrong, and yet is right in half his statements, it is clear that he must have taken some degree of care, and cannot have merely guessed. In a case of yes or no, any one can be right every other time, but it is different where truth is single and error is manifold. To represent the case of a simply worthless witness when there were 1000 balls and the drawing of one assigned ball was in question, we should have to put his figure of veracity at ¹/₁₀₀₀. If this were done we should of course get a similar result.

§ 11. It deserves notice therefore that the figure of veracity, or fraction representing the general truthfulness of a witness, is in a way relative, not absolute; that is, it depends upon, and varies with, the general character of the answer which he is supposed to give. Two witnesses of equal intrinsic veracity and worth, one of whom confined himself to saying yes and no, whilst the other ventured to make more original assertions, would be represented by different fractions; the former having set himself a much easier task than the latter. The real caution and truthfulness of the witness are only one factor, therefore, in his actual figure of veracity; the other factor consists of the nature of his assertions, as just pointed out. The ordinary plan therefore, in such problems, of assigning an average truthfulness to the witness, and accepting this alike in the case of each of the two kinds of answers, though convenient, seems scarcely sound. This consideration would however be of much more importance were not the discussions upon the subject mainly concerned with only one description of answer, namely that of the ‘yes or no’ kind.

§ 12. So much for the methodical way of treating such a problem. The way in which it would be taken in hand by those who had made no study of Probability is very different. It would, I apprehend, strike them as follows. They would say to themselves, Here is a story related by a witness who tells the truth, say, nine times out of ten. But it is a story of a kind which experience shows to be very generally made untruly, say 99 times out of 100. Having then these opposite inducements to belief, they would attempt in some way to strike a balance between them. Nothing in the nature of a strict rule could be given to enable them to decide how they might escape out of the difficulty. Probably, in so far as they did not judge at haphazard, they would be guided by still further resort to experience, or unconscious recollections of its previous teachings, in order to settle which of the two opposing inductions was better entitled to carry the day in the particular case before them. The reader will readily see that any general solution of the problem, when thus presented, is impossible. It is simply the now familiar case (Chap. IX. §§ 14–32) of an individual which belongs equally to two distinct, or even, in respect of their characteristics, opposing classes. We cannot decide off-hand to which of the two its characteristics most naturally and rightly refer it. A fresh induction is needed in order to settle this point.

§ 13. Rules have indeed been suggested by various writers in order to extricate us from the difficulty. The controversy about miracles has probably been the most fertile occasion for suggestions of this kind on one side or the other. It is to this controversy, presumably, that the phrase is due, so often employed in discussions upon similar subjects, ‘a contest of opposite improbabilities.’ What is meant by such an expression is clearly this: that in forming a judgment upon the truth of certain assertions we may find that they are comprised in two very distinct classes, so that, according as we regarded them as belonging to one or the other of these distinct classes, our opinion as to their truth would be very different. Such an assertion belongs to one class, of course, by its being a statement of a particular witness, or kind of witness; it belongs to the other by its being a particular kind of story, one of what is called an improbable nature. Its belonging to the former class is so far favourable to its truth, its belonging to the latter is so far hostile to its truth. It seems to be assumed, in speaking of a contest of opposite improbabilities, that when these different sources of conviction co-exist together, they would each in some way retain their probative force so as to produce a contest, ending generally in a victory to one or other of them. Hume, for instance, speaks of our deducting one probability from the other, and apportioning our belief to the remainder.[5] Thomson, in his Laws of Thought, speaks of one probability as entirely superseding the other.

§ 14. It does not appear to me that the slightest philosophical value can be attached to any such rules as these. They doubtless may, and indeed will, hold in individual cases, but they cannot lay claim to any generality. Even the notion of a contest, as any necessary ingredient in the case, must be laid aside. For let us refer again to the way in which the perplexity arises, and we shall readily see, as has just been remarked, that it is nothing more than a particular exemplification of a difficulty which has already been recognized as incapable of solution by any general à priori method of treatment. All that we are supposed to have before us is a statement. On this occasion it is made by a witness who lies, say, once in ten times in the long run; that is, who mostly tells the truth. But on the other hand, it is a statement which experience, derived from a variety of witnesses on various occasions, assures us is mostly false; stated numerically it is found, let us suppose, to be false 99 times in a hundred.

Now, as was shown in the chapter on Induction, we are thus brought to a complete dead lock. Our science offers no principles by which we can form an opinion, or attempt to decide the matter one way or the other; for, as we found, there are an indefinite number of conclusions which are all equally possible. For instance, all the witness' extraordinary assertions may be true, or they may all be false, or they may be divided into the true and the false in any proportion whatever. Having gone so far in our appeal to statistics as to recognize that the witness is generally right, but that his story is generally false, we cannot stop there. We ought to make still further appeal to experience, and ascertain how it stands with regard to his stories when they are of that particular nature: or rather, for this would be to make a needlessly narrow reference, how it stands with regard to stories of that kind when advanced by witnesses of his general character, position, sympathies, and so on.[6]

§ 15. That extraordinary stories are in many cases, probably in a great majority of cases, less trustworthy than others must be fully admitted. That is, if we were to make two distinct classes of such stories respectively, we should find that the same witness, or similar witnesses, were proportionally more often wrong when asserting the former than when asserting the latter. But it does not by any means appear to me that this must always be the case. We may well conceive, for instance, that with some people the mere fact of the story being of a very unusual character may make them more careful in what they state, so as actually to add to their veracity. If this were so we might be ready to accept their extraordinary stories with even more readiness than their ordinary ones.

Such a supposition as that just made does not seem to me by any means forced. Put such a case as this: let us suppose that two persons, one of them a man of merely ordinary probity and intelligence, the other a scientific naturalist, make a statement about some common event. We believe them both. Let them now each report some extraordinary lusus naturæ or monstrosity which they profess to have seen. Most persons, we may presume, would receive the statement of the naturalist in this latter case almost as readily as in the former: whereas when the same story came from the unscientific observer it would be received with considerable hesitation. Whence arises the difference? From the conviction that the naturalist will be far more careful, and therefore to the full as accurate, in matters of this kind as in those of the most ordinary description, whereas with the other man we feel by no means the same confidence. Even if any one is not prepared to go this length, he will probably admit that the difference of credit which he would attach to the two kinds of story, respectively, when they came from the naturalist, would be much less than what it would be when they came from the other man.

§ 16. Whilst we are on this part of the subject, it must be pointed out that there is considerable ambiguity and consequent confusion about the use of the term ‘an extraordinary story.’ Within the province of pure Probability it ought to mean simply a story which asserts an unusual event. At least this is the view which has been adopted and maintained, it is hoped consistently, throughout this work. So long as we adhere to this sense we know precisely what we mean by the term. It has a purely objective reference; it simply connotes a very low degree of relative statistical frequency, actual or prospective. Out of a great number of events we suppose a selection of some particular kind to be contemplated, which occurs relatively very seldom, and this is termed an unusual or extraordinary event. It follows, as was abundantly shown in a former chapter, that owing to the rarity of the event we are very little disposed to expect its occurrence in any given case. Our guess about it, in case we thus anticipated it, would very seldom be justified, and we are therefore apt to be much surprised when it does occur. This, I take it, is the only legitimate sense of ‘extraordinary’ so far as Probability is concerned.

But there is another and very different use of the word, which belongs to Induction, or rather to the science of evidence in general, more than to that limited portion of it termed Probability. In this sense the ‘extraordinary,’ and still more the ‘improbable,’ event is not merely one of extreme statistical rarity, which we could not expect to guess aright, but which on moderate evidence we may pretty readily accept; it is rather one which possesses, so to say, an actual evidence-resisting power. It may be something which affects the credibility of the witness at the fountain-head, which makes, that is, his statements upon such a subject essentially inferior to those on other subjects. This is the case, for instance, with anything which excites his prejudices or passions or superstitions. In these cases it would seem unreasonable to attempt to estimate the credibility of the witness by calculating (as in § 6) how often his errors would mislead us through his having been wrongly brought to an affirmation instead of adhering correctly to a negation. We should rather be disposed to put our correction on the witness' average veracity at once.

§ 17. In true Probability, as has just been remarked, every event has its own definitely recognizable degree of frequency of occurrence. It may be excessively rare, rare to any extreme we like to postulate, but still every one who understands and admits the data upon which its occurrence depends will be able to appreciate within what range of experience it may be expected to present itself. We do not expect it in any individual case, nor within any brief range, but we do confidently expect it within an adequately extensive range. How therefore can miraculous stories be similarly taken account of, when the disputants, on one side at least, are not prepared to admit their actual occurrence anywhere or at any time? How can any arrangement of bags and balls, or other mechanical or numerical illustrations of unlikely events, be admitted as fairly illustrative of miraculous occurrences, or indeed of many of those which come under the designation of ‘very extraordinary’ or ‘highly improbable’? Those who contest the occurrence of a particular miracle, as reported by this or that narrator, do not admit that miracles are to be confidently expected sooner or later. It is not a question as to whether what must happen sometimes has happened some particular time, and therefore no illustration of the kind can be regarded as apposite.

How unsuitable these merely rare events, however excessive their rarity may be, are as examples of miraculous events, will be evident from a single consideration. No one, I presume, who admitted the occasional occurrence of an exceedingly unusual combination, would be in much doubt if he considered that he had actually seen it himself.[7] On the other hand, few men of any really scientific turn would readily accept a miracle even if it appeared to happen under their very eyes. They might be staggered at the time, but they would probably soon come to discredit it afterwards, or so explain it as to evacuate it of all that is meant by miraculous.

§ 18. It appears to me therefore, on the whole, that very little can be made of these problems of testimony in the way in which it is generally intended that they should be treated; that is, in obtaining specific rules for the estimation of the testimony under any given circumstances. Assuming that the veracity of the witness can be measured, we encounter the real difficulty in the utter impossibility of determining the limits within which the failures of the event in question are to be considered to lie, and the degree of explicitness with which the witness is supposed to answer the enquiry addressed to him; both of these being characteristics of which it is necessary to have a numerical estimate before we can consider ourselves in possession of the requisite data.

Since therefore the practical resource of most persons, viz. that of putting a direct and immediate correction, of course of a somewhat conjectural nature, upon the general trustworthiness of the witness, by a consideration of the nature of the circumstances under which his statement is made, is essentially unscientific and irreducible to rule; it really seems to me that there is something to be said in favour of the simple plan of trusting in all cases alike to the witness' general veracity.[8] That is, whether his story is ordinary or extraordinary, we may resolve to put it on the same footing of credibility, provided of course that the event is fully recognized as one which does or may occasionally happen. It is true that we shall thus go constantly astray, and may do so to a great extent, so that if there were any rational and precise method of specializing his trustworthiness, according to the nature of his story, we should be on much firmer ground. But at least we may thus know what to expect on the average. Provided we have a sufficient number and variety of statements from him, and always take them at the same constant rate or degree of trustworthiness, we may succeed in balancing and correcting our conduct in the long run so as to avoid any ruinous error.

§ 19. A few words may now be added about the combination of testimony. No new principles are introduced here, though the consequent complication is naturally greater. Let us suppose two witnesses, the veracity of each being ⁹/₁₀. Now suppose 100 statements made by the pair; according to the plan of proceeding adopted before, we should have them both right 81 times and both wrong once, in the remaining 18 cases one being right and the other wrong. But since they are both supposed to give the same account, what we have to compare together are the number of occasions on which they agree and are right, and the total number on which they agree whether right or wrong. The ratio of the former to the latter is the fraction which expresses the trustworthiness of their combination of testimony in the case in question.

In attempting to decide this point the only difficulty is in determining how often they will be found to agree when they are both wrong, for clearly they must agree when they are both right. This enquiry turns of course upon the number of ways in which they can succeed in going wrong. Suppose first the case of a simple yes or no (as in § 6), and take the same example, of a bag with 1000 balls, in which one only is white. Proceeding as before, we should find that out of 100,000 drawings (the number required in order to obtain a complete cycle of all possible occurrences, as well as of all possible reports about them) the two witnesses agree in a correct report of the appearance of white in 81, and agree in a wrong report of it in 999. The Probability therefore of the story when so attested is ⁸¹/₁₀₈₀; the fact therefore of two such witnesses of equal veracity having concurred makes the report nearly 9 times as likely as when it rested upon the authority of only one of them.[9]

§ 20. When however the witnesses have many ways of going wrong, the fact of their agreeing makes the report far more likely to be true. For instance, in the case of the 1000 numbered balls, it is very unlikely that when they both mistake the number they should (without collusion) happen to make the same misstatement. Whereas, in the last case, every combined misstatement necessarily led them both to the assertion that the event in question had happened, we should now find that only once in 999 × 999 times would they both be led to assert that some given number (say, as before, 25) had been drawn. The odds in favour of the event in fact now become ⁸⁰⁹¹⁹/₈₀₉₂₀, which are enormously greater than when there was only one witness.

It appears therefore that when two, and of course still more when many, witnesses agree in a statement in a matter about which they might make many and various errors, the combination of their favourable testimony adds enormously to the likelihood of the event; provided always that there is no chance of collusion. And in the extreme case of the opportunities for error being, as they well may be, practically infinite in number, such combination would produce almost perfect certainty. But then this condition, viz. absence of collusion, very seldom can be secured. Practically our main source of error and suspicion is in the possible existence of some kind of collusion. Since we can seldom entirely get rid of this danger, and when it exists it can never be submitted to numerical calculation, it appears to me that combination of testimony, in regard to detailed accounts, is yet more unfitted for consideration in Probability than even that of single testimony.

§ 21. The impossibility of any adequate or even appropriate consideration of the credibility of miraculous stories by the rules of Probability has been already noticed in § 17. But, since the grounds of this impossibility are often very insufficiently appreciated, a few pages may conveniently be added here with a view to enforcing this point. If it be regarded as a digression, the importance of the subject and the persistency with which various writers have at one time or another attempted to treat it by the rules of our science must be the excuse for entering upon it.

A necessary preliminary will be to decide upon some definition of a miracle. It will, we may suppose, be admitted by most persons that in calling a miracle ‘a suspension of a law of causation,’ we are giving what, though it may not amount to an adequate definition, is at least true as a description. It is true, though it may not be the whole truth. Whatever else the miracle may be, this is its physical aspect: this is the point at which it comes into contact with the subject-matter of science. If it were not considered that any suspension of causation were involved, the event would be regarded merely as an ordinary one to which some special significance was attached, that is, as a type or symbol rather than a miracle. It is this aspect moreover of the miracle which is now exposed to the main brunt of the attack, and in support of which therefore the defence has generally been carried on.

Now it is obvious that this, like most other definitions or descriptions, makes some assumption as to matters of fact, and involves something of a theory. The assumption clearly is, that laws of causation prevail universally, or almost universally, throughout nature, so that infractions of them are marked and exceptional. This assumption is made, but it does not appear that anything more than this is necessarily required; that is, there is nothing which need necessarily make us side with either of the two principal schools which are divided as to the nature of these laws of causation. The definition will serve equally well whether we understand by law nothing more than uniformity of antecedent and consequent, or whether we assert that there is some deeper and more mysterious tie between the events than mere sequence. The use of the term ‘causation’ in this minimum of signification is common to both schools, though the one might consider it inadequate; we may speak, therefore, of ‘suspensions of causation’ without committing ourselves to either.

§ 22. It should be observed that the aspect of the question suggested by this definition is one from which we can hardly escape. Attempts indeed have been sometimes made to avoid the necessity of any assumption as to the universal prevalence of law and order in nature, by defining a miracle from a different point of view. A miracle may be called, for instance, ‘an immediate exertion of creative power,’ ‘a sign of a revelation,’ or, still more vaguely, an ‘extraordinary event.’ But nothing would be gained by adopting any such definitions as these. However they might satisfy the theologian, the student of physical science would not rest content with them for a moment. He would at once assert his own belief, and that of other scientific men, in the existence of universal law, and enquire what was the connection of the definition with this doctrine. An answer would imperatively be demanded to the question, Does the miracle, as you have described it, imply an infraction of one of these laws, or does it not? And an answer must be given, unless indeed we reject his assumption by denying our belief in the existence of this universal law, in which case of course we put ourselves out of the pale of argument with him. The necessity of having to recognize this fact is growing upon men day by day, with the increased study of physical science. And since this aspect of the question has to be met some time or other, it is as well to place it in the front. The difficulty, in its scientific form, is of course a modern one, for the doctrine out of which it arises is modern. But it is only one instance, out of many that might be mentioned, in which the growth of some philosophical conception has gradually affected the nature of the dispute, and at last shifted the position of the battle-ground, in some discussion with which it might not at first have appeared to have any connection whatever.