§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]