If three out of six balls are of the same colour, we expect a ball of that colour to come out three times as often as any other colour on the average of a long succession of tries. This illustrates the second clause of our principle. The third is illustrated by a loaded coin or die.
By making regressive application of the principle thus ascertained by experience, we often obtain a clue to special causal connexion. We are at least enabled to isolate a problem for investigation. If we find one of a number of alternatives recurring more frequently than the others, we are entitled to presume that they are not equally possible, that there is some inequality in their conditions.
The inequality may simply lie in the greater possible frequency of one of the coinciding events, as when there are three black balls in a bottle of six. We must therefore discount the positive frequency before looking for any other cause. Suppose, for example, we find that the ascendancy of Jupiter coincides more frequently with the birth of men afterwards distinguished in business than with the birth of men otherwise distinguished, say in war, or at the bar, or in scholarship.
We are not at liberty to conclude planetary influence till we have compared the positive frequency of the different modes of distinction.
The explanation of the more frequently repeated coincidence may simply be that more men altogether are successful in business than in war or law or scholarship. If so, we say that chance accounts for the coincidence, that is to say, that the coincidence is casual as far as planetary influence is concerned.
So in epidemics of fever, if we find on taking a long average that more cases occur in some streets of a town than in others, we are not warranted in concluding that the cause lies in the sanitary conditions of those streets or in any special liability to infection without first taking into account the number of families in the different streets. If one street showed on the average ten times as many cases as another, the coincidence might still be judged casual if there were ten times as many families in it.
Apart from the fallacy of overlooking the positive frequency, certain other fallacies or liabilities to error in applying this doctrine of chances may be specified.
1. We are apt, under the influence of prepossession or prejudice, to remember certain coincidences better than others, and so to imagine extra-casual coincidence where none exists. This bias works in confirming all kinds of established beliefs, superstitious and other, beliefs in dreams, omens, retributions, telepathic communications, and so forth. Many people believe that nobody who thwarts them ever comes to good, and can produce numerous instances from experience in support of this belief.
2. We are apt, after proving that there is a residuum beyond what chance will account for on due allowance made for positive frequency, to take for granted that we have proved some particular cause for this residuum. Now we have not really explained the residuum by the application of the principle of chances: we have only isolated a problem for explanation. There may be more than chance will account for: yet the cause may not be the cause that we assign off-hand. Take, for example, the coincidence that has been remarked between race and different forms of Christianity in Europe. If the distribution of religious systems were entirely independent of race, it might be said that you would expect one system to coincide equally often with different races in proportion to the positive number of their communities. But the Greek system is found almost solely among Slavonic peoples, the Roman among Celtic, and the Protestant among Teutonic. The coincidence is greater than chance will account for. Is the explanation then to be found in some special adaptability of the religious system to the character of the people? This may be the right explanation, but we have not proved it by merely discounting chance.
To prove this we must show that there was no other cause at work, that character was the only operative condition in the choice of system, that political combinations, for example, had nothing to do with it.
The presumption from extra-casual coincidence is only that there is a special cause: in determining what that is we must conform to the ordinary conditions of explanation.
So coincidence between membership of the Government and a classical education may be greater than chance would account for, and yet the circumstance of having been taught Latin and Greek at school may have had no special influence in qualifying the members for their duties.
The proportion of classically educated in the Government may be greater than the proportion of them in the House of Commons, and yet their eminence may be in no way due to their education. Men of a certain social position have an advantage in the competition for office, and all those men have been taught Latin and Greek as a matter of course. Technically speaking, the coinciding phenomena may be independent effects of the same cause.
3. Where the alternative possibilities are very numerous, we are apt not to make due allowance for the number, sometimes overrating it, sometimes underrating it.
The fallacy of underrating the number is often seen in games of chance, where the object is to create a vast number of alternatives, all equally possible, equally open to the player, without his being able to affect the advent of one more than another. In whist, for example, there are some six billions of possible hands. Yet it is a common impression that, one night with another, in the course of a year, a player will have dealt to him about an equal number of good and bad hands. This is a fallacy. A very much longer time is required to exhaust the possible combinations. Suppose a player to have 2000 hands in the course of a year: this is only one "set," one combination, out of thousands of millions of such sets possible. Among those millions of sets, if there is nothing but chance in the matter, there ought to be all proportions of good and bad, some sets all good, some all bad, as well as some equally divided between good and bad.[1]
Sometimes, however, the number of possible alternatives is overrated.
Thus, visitors to London often remark that they never go there without meeting somebody from their own locality, and they are surprised at this as if they had the same chance of meeting their fellow-visitors and any other of the four millions of the metropolis. But really the possible alternatives of rencounter are far less numerous. The places frequented by visitors to London are filled by much more limited numbers: the possible rencounters are to be counted by thousands rather than by millions.
[Footnote 1: See De Morgan's _Essay on Probabilities_, c. vi., "On Common Notions of Probability".]
CHAPTER IX.
PROBABLE INFERENCE TO PARTICULARS--THE MEASUREMENT OF PROBABILITY.
Undoubtedly there are degrees of probability. Not only do we expect some events with more confidence than others: we may do so, and our confidence may be misplaced: but we have reason to expect some with more confidence than others. There are different degrees of rational expectation. Can those degrees be measured numerically?
The question has come into Logic from the mathematicians. The calculation of Probabilities is a branch of Mathematics. We have seen how it may be applied to guide investigation by eliminating what is due to chance, and it has been vaguely conceived by logicians that what is called the calculus of probabilities might be found useful also in determining by exact numerical measurement the probability of single events. Dr. Venn, who has written a separate treatise on the Logic of Chance, mentions "accurate quantitative apportionment of our belief" as one of the goals which Logic should strive to attain. The following passage will show his drift.[1]
A man in good health would doubtless like to know whether he will be alive this time next year. The fact will be settled one way or the other in due time, if he can afford to wait, but if he wants a present decision, Statistics and the Theory of Probability can alone give him any information. He learns that the odds are, say five to one that he will survive, and this is an answer to his question as far as any answer can be given. Statisticians are gradually accumulating a vast mass of data of this general character. What they may be said to aim at is to place us in the position of being able to say, in any given time or place, what are the odds for or against any at present indeterminable fact which belongs to a class admitting of statistical treatment.
Again, outside the regions of statistics proper--which deal, broadly speaking, with events which can be numbered or measured, and which occur with some frequency--there is still a large field as to which some better approach to a reasoned intensity of belief can be acquired. What will be the issue of a coming war? Which party will win in the next election? Will a patient in the crisis of a given disease recover or not?
That statistics are lying here in the background, and are thus indirectly efficient in producing and graduating our belief, I fully hold; but there is such a large intermediate process of estimating, and such scope for the exercise of a practised judgment, that no direct appeal to statistics in the common sense can directly help us. In sketching out therefore the claims of an Ideal condition of knowledge, we ought clearly to include a due apportionment of belief to every event of such a class as this. It is an obvious defect that one man should regard as almost certain what another man regards as almost impossible. Short, therefore, of certain prevision of the future, we want complete agreement as to the degree of probability of every future event: and for that matter of every past event as well.
Technically speaking, if we extend the name Modality (see p. 78) to any qualification of the certainty of a statement of belief, what Dr.
Venn here desiderates, as he has himself suggested, is a more exact measurement of the Modality of propositions. We speak of things as being certain, possible, impossible, probable, extremely probable, faintly probable, and so forth: taking certainty as the highest degree of probability[2] shading gradually down to the zero of the impossible, can we obtain an exact numerical measure for the gradations of assurance?
To examine the principles of all the cases in which chances for and against an occurrence have been calculated from real or hypothetical data, would be to trespass into the province of Mathematics, but a few simple cases will serve to show what it is that the calculus attempts to measure, and what is the practical value of the measurement as applied to the probability of a single event.
Suppose there are 100 balls in a box, 30 white and 70 black, all being alike except in respect of colour, we say that the chances of drawing a black ball as against a white are as 7 to 3, and the probability of drawing black is measured by the fraction 7/10. In believing this we proceed on the principle already explained (p. 356) of Proportional Chances. We do not know for certain whether black or white will emerge, but knowing the antecedent situation we expect black rather than white with a degree of assurance corresponding to the proportions of the two in the box. It is our degree of rational assurance that we measure by this fraction, and the rationality of it depends on the objective condition of the facts, and is the same for all men, however much their actual degree of confidence may vary with individual temperament. That black will be drawn seven times out of every ten on an average if we go on drawing to infinity, is as certain as any empirical law: it is the probability of a single draw that we measure by the fraction 7/10.
When we build expectations of single events on statistics of observed proportions of events of that kind, it is ultimately on the same principle that rational expectation rests. That the proportion will obtain on the average we regard as certain: the ratio of favourable cases to the whole number of possible alternatives is the measure of rational expectation or probability in regard to a particular occurrence. If every year five per cent. of the children of a town stray from their guardians, the probability of this or that child's going astray is 1/20. The ratio is a correct measure only on the assumption that the average is maintained from year to year.
Without going into the combination of probabilities, we are now in a position to see the practical value of such a calculus as applied to particular cases. There has been some misunderstanding among logicians on the point. Mr. Jevons rebuked Mill for speaking disrespectfully of the calculus, eulogised it as one of the noblest creations of the human intellect, and quoted Butler's saying that "Probability is the guide of life". But when Butler uttered this famous saying he was probably not thinking of the mathematical calculus of probabilities as applied to particular cases, and it was this special application to which Mill attached comparatively little value.
The truth is that we seldom calculate or have any occasion to calculate individual chances except as a matter of curiosity. It is true that insurance offices calculate probabilities, but it is not the probability of this or that man dying at a particular age. The precise shade of probability for the individual, in so far as this depends on vital statistics, is a matter of indifference to the company as long as the average is maintained. Our expectations about any individual life cannot be measured by a calculation of the chances because a variety of other elements affect those expectations. We form beliefs about individual cases, but we try to get surer grounds for them than the chances as calculable from statistical data. Suppose a person were to institute a home for lost dogs, he would doubtless try to ascertain how many dogs were likely to go astray, and in so doing would be guided by statistics. But in judging of the probability of the straying of a particular dog, he would pay little heed to statistics as determining the chances, but would proceed upon empirical knowledge of the character of the dog and his master. Even in betting on the field against a particular horse, the bookmaker does not calculate from numerical data such as the number of horses entered or the number of times the favourite has been beaten: he tries to get at the pedigree and previous performances of the various horses in the running. We proceed by calculation of chances only when we cannot do better.
[Footnote 1: _Empirical Logic_, p. 556.]
[Footnote 2: Mr. Jevons held that all inference is merely probable and that no inference is certain. But this is a purposeless repudiation of common meaning, which he cannot himself consistently adhere to. We find him saying that if a penny is tossed into the air it will _certainly_ come down on one side or the other, on which side being a matter of probability. In common speech probability is applied to a degree of belief short of certainty, but to say that certainty is the highest degree of probability does no violence to the common meaning.]
CHAPTER X.
INFERENCE FROM ANALOGY.
The word Analogy was appropriated by Mill, in accordance with the usage of the eighteenth century, to designate a ground of inference distinct from that on which we proceed in extending a law, empirical or scientific, to a new case. But it is used in various other senses, more or less similar, and in order to make clear the exact logical sense, it is well to specify some of these. The original word [Greek: analogia], as employed by Aristotle, corresponds to the word Proportion in Arithmetic: it signified an equality of ratios, [Greek: isotes logon]: two compared with four is analogous to four compared with eight. There is something of the same meaning in the technical use of the word in Physiology, where it is used to signify similarity of function as distinguished from similarity of structure, which is called homology: thus the tail of a whale is analogous to the tail of a fish, inasmuch as it is similarly used for motion, but it is homologous with the hind legs of a quadruped; a man's arms are homologous with a horse's fore legs, but they are not analogous inasmuch as they are not used for progression. Apart from these technical employments, the word is loosely used in common speech for any kind of resemblance. Thus De Quincey speaks of the "analogical"
power in memory, meaning thereby the power of recalling things by their inherent likeness as distinguished from their casual connexions or their order in a series. But even in common speech, there is a trace of the original meaning: generally when we speak of analogy we have in our minds more than one pair of things, and what we call the analogy is some resemblance between the different pairs. This is probably what Whately had in view when he defined analogy as "resemblance of relations".
In a strict logical sense, however, as defined by Mill, sanctioned by the previous usage of Butler and Kant, analogy means more than a resemblance of relations. It means a preponderating resemblance between two things such as to warrant us in inferring that the resemblance extends further. This is a species of argument distinct from the extension of an empirical law. In the extension of an empirical law, the ground of inference is a coincidence frequently repeated within our experience, and the inference is that it has occurred or will occur beyond that experience: in the argument from analogy, the ground of inference is the resemblance between two individual objects or kinds of objects in a certain number of points, and the inference is that they resemble one another in some other point, known to belong to the one, but not known to belong to the other. "Two things go together in many cases, therefore in all, including this one," is the argument in extending a generalisation: "Two things agree in many respects, therefore in this other," is the argument from analogy.
The example given by Reid in his _Intellectual Powers_ has become the standard illustration of the peculiar argument from analogy.
We may observe a very great similitude between this earth which we inhabit, and the other planets, Saturn, Jupiter, Mars, Venus and Mercury. They all revolve round the sun, as the earth does, although at different distances and in different periods. They borrow all their light from the sun, as the earth does. Several of them are known to revolve round their axis like the earth, and by that means have like succession of day and night. Some of them have moons, that serve to give them light in the absence of the sun, as our moon does to us. They are all, in their motions, subject to the same law of gravitation as the earth is. From all this similitude it is not unreasonable to think that these planets may, like our earth, be the habitation of various orders of living creatures. There is some probability in this conclusion from analogy.[1]
The argument from analogy is sometimes said to range through all degrees of probability from certainty to zero. But this is true only if we take the word analogy in its loosest sense for any kind of resemblance. If we do this, we may call any kind of argument an argument from analogy, for all inferences turn upon resemblance. I believe that if I throw my pen in the air it will come down again, because it is like other ponderable bodies. But if we use the word in its limited logical sense, the degree of probability is much nearer zero than certainty. This is apparent from the conditions that logicians have formulated of a strict argument from analogy.
1. The resemblance must be preponderating. In estimating the value of an argument from analogy, we must reckon the points of difference as counting against the conclusion, and also the points in regard to which we do not know whether the two objects agree or differ. The numerical measure of value is the ratio of the points of resemblance to the points of difference _plus_ the unknown points. Thus, in the argument that the planets are inhabited because they resemble the earth in some respects and the earth is inhabited, the force of the analogy is weakened by the fact that we know very little about the surface of the planets.
2. In a numerical estimate all circumstances that hang together as effects of one cause must be reckoned as one. Otherwise, we might make a fallaciously imposing array of points of resemblance. Thus in Reid's enumeration of the agreements between the earth and the planets, their revolution round the sun and their obedience to the law of gravitation should count as one point of resemblance. If two objects agree in _a_, _b_, _c_, _d_, _e_, but _b_ follows from _a_, and _d_ and _e_ from _c_, the five points count only as two.
3. If the object to which we infer is known to possess some property incompatible with the property inferred, the general resemblance counts for nothing. The moon has no atmosphere, and we know that air is an indispensable condition of life. Hence, however much the moon may resemble the earth, we are debarred from concluding that there are living creatures on the moon such as we know to exist on the earth. We know also that life such as it is on the earth is possible only within certain limits of temperature, and that Mercury is too hot for life, and Saturn too cold, no matter how great the resemblance to the earth in other respects.
4. If the property inferred is known or presumed to be a concomitant of one or more of the points of resemblance, any argument from analogy is superfluous. This is, in effect, to say that we have no occasion to argue from general resemblance when we have reason to believe that a property follows from something that an object is known to possess.
If we knew that any one of the planets possessed all the conditions, positive and negative, of life, we should not require to reckon up all the respects in which it resembles the earth in order to create a presumption that it is inhabited. We should be able to draw the conclusion on other grounds than those of analogy. Newton's famous inference that the diamond is combustible is sometimes quoted as an argument from analogy. But, technically speaking, it was rather, as Professor Bain has pointed out, of the nature of an extended generalisation. Comparing bodies in respect of their densities and refracting powers, he observed that combustible bodies refract more than others of the same density; and observing the exceptionally high refracting power of the diamond, he inferred from this that it was combustible, an inference afterwards confirmed by experiment. "The concurrence of high refracting power with inflammability was an empirical law; and Newton, perceiving the law, extended it to the adjacent case of the diamond. The remark is made by Brewster that had Newton known the refractive powers of the minerals _greenockite_ and _octohedrite_, he would have extended the inference to them, and would have been mistaken."[2]
From these conditions it will be seen that we cannot conclude with any high degree of probability from analogy alone. This is not to deny, as Mr. Jevons seems to suppose, that analogies, in the sense of general resemblances, are often useful in directing investigation. When we find two things very much alike, and ascertain that one of them possesses a certain property, the presumption that the other has the same is strong enough to make it worth while trying whether as a matter of fact it has. It is said that a general resemblance of the hills near Ballarat in Australia to the Californian hills where gold had been found suggested the idea of digging for gold at Ballarat.
This was a lucky issue to an argument from analogy, but doubtless many have dug for gold on similar general resemblances without finding that the resemblance extended to that particular. Similarly, many of the extensions of the Pharmacopeia have proceeded upon general resemblances, the fact that one drug resembles another in certain properties being a sufficient reason for trying whether the resemblance goes further. The lucky guesses of what is known as natural sagacity are often analogical. A man of wide experience in any subject-matter such as the weather, or the conduct of men in war, in business, or in politics, may conclude to the case in hand from some previous case that bears a general resemblance to it, and very often his conclusions may be perfectly sound though he has not made a numerical estimate of the data.
The chief source of fallacy in analogical argument is ignoring the number of points of difference. It often happens that an amount of resemblance only sufficient for a rhetorical simile is made to do duty as a solid argument. Thus the resemblance between a living body and the body politic is sometimes used to support inferences from successful therapeutic treatment to State policy. The advocates of annual Parliaments in the time of the Commonwealth based their case on the serpent's habit of annually casting its skin.
Wisest of beasts the serpent see, Just emblem of eternity, And of a State's duration; Each year an annual skin he takes, And with fresh life and vigour wakes At every renovation.
Britain! that serpent imitate.
Thy Commons House, that skin of State, By annual choice restore; So choosing thou shall live secure, And freedom to thy sons inure, Till Time shall be no more.