An explosion, for example, occurs. There are several explosive agencies, capable of causing as much destruction as meets the eye at the first glance. The agent in the case before us may be gunpowder or it may be dynamite. But the two agents are not so alike in their mode of operation as to produce results identical in every circumstance.
The expert inquirer knows by previous observation that when gunpowder acts the objects in the neighbourhood are blackened; and that an explosion of dynamite tears and shatters in a way peculiar to itself. He is thus able to interpret the traces, to make and prove a hypothesis.
A man's body is found dead in water. It may be a question whether death came by drowning or by previous violence. He may have been suffocated and afterwards thrown into the water. But the circumstances will tell the true story. Death by drowning has distinctive symptoms.
If drowning was the cause, water will be found in the stomach and froth in the trachea.
Thus, though there may be a plurality of possible causes, the causation in the given case may be brought home to one by distinctive accompaniments, and it is the business of the scientific inquirer to study these. What is known as the "ripple-mark" in sandstone surfaces may be produced in various ways. The most familiar way is by the action of the tides on the sand of the sea-shore, and the interpreter who knows this way only would ascribe the marks at once to this agency. But ripple-marks are produced also by the winds on drifting sands, by currents of water where no tidal influence is felt, and in fact by any body of water in a state of oscillation. Is it, then, impossible to decide between these alternative possibilities of causation? No: wind-ripples and current-ripples and tidal-ripples have each their own special character and accompanying conditions, and the hypothesis of one rather than another may be made good by means of these. "In rock-formations," Mr. Page says,[1] "there are many things which at first sight seem similar, and yet on more minute examination, differences are detected and conditions discovered which render it impossible that these appearances can have arisen from the same causation."
The truth is that generally when we speak of plurality of causes, of alternative possibilities of causation, we are not thinking of the effect in its individual entirety, but only of some general or abstract aspect of it. When we say, _e.g._, that death may be produced by a great many different causes, poison, gunshot wounds, disease of this or that organ, we are thinking of death in the abstract, not of the particular case under consideration, which as an individual case, has characters so distinctive that only one combination of causes is possible.
The effort of science is to become less and less abstract in this sense, by observing agencies or combinations of agencies apart and studying the special characters of their effects. That knowledge is then applied, on the assumption that where those characters are present, the agent or combination of agencies has been at work. Given an effect to be explained, it is brought home to one out of several possible alternatives by _circumstantial evidence_.
Bacon's phrase, _Instantia Crucis_,[2] or Finger-post Instance, might be conveniently appropriated as a technical name for a circumstance that is decisive between rival hypotheses. This was, in effect, proposed by Sir John Herschel,[3] who drew attention to the importance of these crucial instances, and gave the following example: "A curious example is given by M. Fresnel, as decisive, in his mind, of the question between the two great opinions on the nature of light, which, since the time of Newton and Huyghens, have divided philosophers.
When two very clean glasses are laid one on the other, if they be not perfectly flat, but one or both in an almost imperceptible degree convex or prominent, beautiful and vivid colours will be seen between them; and if these be viewed through a red glass, their appearance will be that of alternate dark and bright stripes.... Now, the coloured stripes thus produced are explicable on both theories, and are appealed to by both as strong confirmatory facts; but there is a difference in one circumstance according as one or the other theory is employed to explain them. In the case of the Huyghenian doctrine, the intervals between the bright stripes ought to appear _absolutely black_; in the other, _half bright_, when viewed [in a particular manner] through a prism. This curious case of difference was tried as soon as the opposing consequences of the two theories were noted by M.
Fresnel, and the result is stated by him to be decisive in favour of that theory which makes light to consist in the vibrations of an elastic medium."
III.--THE PROOF OF A HYPOTHESIS.
The completest proof of a hypothesis is when that which has been hypothetically assumed to exist as a means of accounting for certain phenomena is afterwards actually observed to exist or is proved by descriptive testimony to have existed. Our argument, for example, from internal evidence that Mill in writing his Logic aimed at furnishing a method for social investigations is confirmed by a letter to Miss Caroline Fox, in which he distinctly avowed that object.
The most striking example of this crowning verification in Science is the discovery of the planet Neptune, in which case an agent hypothetically assumed was actually brought under the telescope as calculated. Examples almost equally striking have occurred in the history of the Evolution doctrine. Hypothetical ancestors with certain peculiarities of structure have been assumed as links between living species, and in some cases their fossils have actually been found in the geological register.
Such triumphs of verification are necessarily rare. For the most part the hypothetical method is applied to cases where proof by actual observation is impossible, such as prehistoric conditions of the earth or of life upon the earth, or conditions in the ultimate constitution of matter that are beyond the reach of the strongest microscope.
Indeed, some would confine the word hypothesis to cases of this kind.
This, in fact, was done by Mill: hypothesis, as he defined it, was a conjecture not completely proved, but with a large amount of evidence in its favour. But seeing that the procedure of investigation is the same, namely, conjecture, calculation and comparison of facts with the calculated results, whether the agency assumed can be brought to the test of direct observation or not, it seems better not to restrict the word hypothesis to incompletely proved conjectures, but to apply it simply to a conjecture made at a certain stage in whatever way it may afterwards be verified.
In the absence of direct verification, the proof of a hypothesis is exclusive sufficiency to explain the circumstances. The hypothesis must account for all the circumstances, and there must be no other way of accounting for them. Another requirement was mentioned by Newton in a phrase about the exact meaning of which there has been some contention. The first of his Regulae Philosophandi laid down that the cause assumed must be a _vera causa_. "We are not," the Rule runs, "to admit other causes of natural things than such as both are true, and suffice for explaining their phenomena."[4]
It has been argued that the requirement of "verity" is superfluous; that it is really included in the requirement of sufficiency; that if a cause is sufficient to explain the phenomena it must _ipso facto_ be the true cause. This may be technically arguable, given a sufficient latitude to the word sufficiency: nevertheless, it is convenient to distinguish between mere sufficiency to explain the phenomena in question, and the proof otherwise that the cause assigned really exists _in rerum natura_, or that it operated in the given case. The frequency with which the expression _vera causa_ has been used since Newton's time shows that a need is felt for it, though it may be hard to define "verity" precisely as something apart from "sufficiency". If we examine the common usage of the expression we shall probably find that what is meant by insisting on a _vera causa_ is that we must have some evidence for the cause assigned outside the phenomena in question. In seeking for verification of a hypothesis we must extend our range beyond the limited facts that have engaged our curiosity and that demand explanation.
There can be little doubt that Newton himself aimed his rule at the Cartesian hypothesis of Vortices. This was an attempt to explain the solar system on the hypothesis that cosmic space is filled with a fluid in which the planets are carried round as chips of wood in a whirlpool, or leaves or dust in a whirlwind. Now this is so far a _vera causa_ that the action of fluid vortices is a familiar one: we have only to stir a cup of tea with a bit of stalk in it to get an instance. The agency supposed is sufficient also to account for the revolution of a planet round the sun, given sufficient strength in the fluid to buoy up the planet. But if there were such a fluid in space there would be other phenomena: and in the absence of these other phenomena the hypothesis must be dismissed as imaginary. The fact that comets pass into and out of spaces where the vortices must be assumed to be in action without exhibiting any perturbation is an _instantia crucis_ against the hypothesis.
If by the requirement of a _vera causa_ were meant that the cause assigned must be one directly open to observation, this would undoubtedly be too narrow a limit. It would exclude such causes as the ether which is assumed to fill interstellar space as a medium for the propagation of light. The only evidence for such a medium and its various properties is sufficiency to explain the phenomena. Like suppositions as to the ultimate constitution of bodies, it is of the nature of what Professor Bain calls a "Representative Fiction": the only condition is that it must explain all the phenomena, and that there must be no other way of explaining all. When it is proved that light travels with a finite velocity, we are confined to two alternative ways of conceiving its transmission, a projection of matter from the luminous body and the transference of vibrations through an intervening medium. Either hypothesis would explain many of the facts: our choice must rest with that which best explains all.
But supposing that all the phenomena of light were explained by attributing certain properties to this intervening medium, it would probably be held that the hypothesis of an ether had not been fully verified till other phenomena than those of light had been shown to be incapable of explanation on any other hypothesis. If the properties ascribed to it to explain the phenomena of light sufficed at the same time to explain otherwise inexplicable phenomena connected with Heat, Electricity, or Gravity, the evidence of its reality would be greatly strengthened.
Not only must the circumstances in hand be explained, but other circumstances must be found to be such as we should expect if the cause assigned really operated. Take, for example, the case of Erratic blocks or boulders, huge fragments of rock found at a distance from their parent strata. The lowlands of England, Scotland, and Ireland, and the great central plain of Northern Europe contain many such fragments. Their composition shows indubitably that they once formed part of hills to the northward of their present site. They must somehow have been detached and transported to where we now find them.
How? One old explanation is that they were carried by witches, or that they were themselves witches accidentally dropped and turned into stone. Any such explanation by supernatural means can neither be proved nor disproved. Some logicians would exclude such hypotheses altogether on the ground that they cannot be rendered either more or less probable by subsequent examination.[5] The proper scientific limit, however, is not to the making of hypotheses, but to the proof of them. The more hypotheses the merrier: only if such an agency as witchcraft is suggested, we should expect to find other evidence of its existence in other phenomena that could not otherwise be explained. Again, it has been suggested that the erratic boulders may have been transported by water. Water is so far a _vera causa_ that currents are known to be capable of washing huge blocks to a great distance. But blocks transported in this way have the edges worn off by the friction of their passage: and, besides, currents strong enough to dislodge and force along for miles blocks as big as cottages must have left other marks of their presence. The explanation now received is that glaciers and icebergs were the means of transport. But this explanation was not accepted till multitudes of circumstances were examined all tending to show that glaciers had once been present in the regions where the erratic blocks are found. The minute habits of glaciers have been studied where they still exist: how they slowly move down carrying fragments of rock; how icebergs break off when they reach water, float off with their load, and drop it when they melt; how they grind and smooth the surfaces of rocks over which they pass or that are frozen into them: how they undercut and mark the faces of precipices past which they move; how moraines are formed at the melting ends of them, and so forth. When a district exhibits all the circumstances that are now observed to attend the action of glaciers the proof of the hypothesis that glaciers were once there is complete.
[Footnote 1: Page's _Philosophy of Geology_, p. 38.]
[Footnote 2: Crux in this phrase means a cross erected at the parting of ways, with arms to tell whither each way leads.]
[Footnote 3: _Discourse_, -- 218.]
[Footnote 4: Causas rerum naturalium non plures admitti debere quam quae et veriae sint et carum phenomenis explicandis sufficiant.]
[Footnote 5: See Prof. Fowler on the Conditions of Hypotheses, _Inductive Logic_, pp. 100-115.]
CHAPTER VIII.
SUPPLEMENTARY METHODS OF INVESTIGATION.
I.--THE MAINTENANCE OF AVERAGES.--SUPPLEMENT TO THE METHOD OF DIFFERENCE.
A certain amount of law obtains among events that are usually spoken of as matters of chance or accident in the individual case. Every kind of accident recurs with a certain uniformity. If we take a succession of periods, and divide the total number of any kind of event by the number of periods, we get what is called the average for that period: and it is observed that such averages are maintained from period to period. Over a series of years there is a fixed proportion between good harvests and bad, between wet days and dry: every year nearly the same number of suicides takes place, the same number of crimes, of accidents to life and limb, even of suicides, crimes, or injuries by particular means: every year in a town nearly the same number of children stray from their parents and are restored by the police: every year nearly the same number of persons post letters without putting an address on them.
This maintenance of averages is simple matter of observation, a datum of experience, an empirical law. Once an average for any kind of event has been noted, we may count upon its continuance as we count upon the continuance of any other kind of observed uniformity. Insurance companies proceed upon such empirical laws of average in length of life and immunity from injurious accidents by sea or land: their prosperity is a practical proof of the correctness and completeness of the observed facts and the soundness of their inference to the continuance of the average.
The constancy of averages is thus a guide in practice. But in reasoning upon them in investigations of cause, we make a further assumption than continued uniformity. We assume that the maintenance of the average is due to the permanence of the producing causes. We regard the average as the result of the operation of a limited sum of forces and conditions, incalculable as regards their particular incidence, but always pressing into action, and thus likely to operate a certain number of times within a limited period.
Assuming the correctness of this explanation, it would follow that _any change in the average is due to some change in the producing conditions_; and this derivative law is applied as a help in the observation and explanation of social facts. Statistics are collected and classified: averages are struck: and changes in the average are referred to changes in the concomitant conditions.
With the help of this law, we may make a near approach to the precision of the Method of Difference. A multitude of unknown or unmeasured agents may be at work on a situation, but we may accept the average as the result of their joint operation. If then a new agency is introduced or one of the known agents is changed in degree, and this is at once followed by a change in the average, we may with fair probability refer the change in the result to the change in the antecedents.
The difficulty is to find a situation where only one antecedent has been changed before the appearance of the effect. This difficulty may be diminished in practice by eliminating changes that we have reason to know could not have affected the circumstances in question.
Suppose, for example, our question is whether the Education Act of 1872 had an influence in the decrease of juvenile crime. Such a decrease took place _post hoc_; was it _propter hoc_? We may at once eliminate or put out of account the abolition of Purchase in the Army or the extension of the Franchise as not having possibly exercised any influence on juvenile crime. But with all such eliminations, there may still remain other possible influences, such as an improvement in the organisation of the Police, or an expansion or contraction in employment. "Can you tell me in the face of chronology," a leading statesman once asked, "that the Crimes Act of 1887 did not diminish disorder in Ireland?" But chronological sequence alone is not a proof of causation as long as there are other contemporaneous changes of condition that may also have been influential.
The great source of fallacy is our proneness to eliminate or isolate in accordance with our prejudices. This has led to the gibe that anything can be proved by statistics. Undoubtedly statistics may be made to prove anything if you have a sufficiently low standard of proof and ignore the facts that make against your conclusion. But averages and variations in them are instructive enough if handled with due caution. The remedy for rash conclusions from statistics is not no statistics, but more of them and a sound knowledge of the conditions of reasonable proof.
II.--THE PRESUMPTION FROM EXTRA-CASUAL COINCIDENCE.
We have seen that repeated coincidence raises a presumption of causal connexion between the coinciding events. If we find two events going repeatedly together, either abreast or in sequence, we infer that the two are somehow connected in the way of causation, that there is a reason for the coincidence in the manner of their production. It may not be that the one produces the other, or even that their causes are in any way connected: but at least, if they are independent one of the other, both are tied down to happen at the same place and time,--the coincidence of both with time and place is somehow fixed.
But though this is true in the main, it is not true without qualification. We expect a certain amount of repeated coincidence without supposing causal connexion. If certain events are repeated very often within our experience, if they have great positive frequency, we may observe them happening together more than once without concluding that the coincidence is more than fortuitous.
For example, if we live in a neighbourhood possessed of many black cats, and sally forth to our daily business in the morning, a misfortune in the course of the day might more than once follow upon our meeting a black cat as we went out without raising in our minds any presumption that the one event was the result of the other.
Certain planets are above the horizon at certain periods of the year and below the horizon at certain other periods. All through the year men and women are born who afterwards achieve distinction in various walks of life, in love, in war, in business, at the bar, in the pulpit. We perceive a certain number of coincidences between the ascendancy of certain planets and the birth of distinguished individuals without suspecting that planetary influence was concerned in their superiority.
Marriages take place on all days of the year: the sun shines on a good many days at the ordinary time for such ceremonies; some marriages are happy, some unhappy; but though in the case of many happy marriages the sun has shone upon the bride, we regard the coincidence as merely accidental.
Men often dream of calamities and often suffer calamities in real life: we should expect the coincidence of a dream of calamity followed by a reality to occur more than once as a result of chance. There are thousands of men of different nationalities in business in London, and many fortunes are made: we should expect more than one man of any nationality represented there to make a fortune without arguing any connexion between his nationality and his success.
We allow, then, for a certain amount of repeated coincidence without presuming causal connexion: can any rule be laid down for determining the exact amount?
Prof. Bain has formulated the following rule: "Consider the positive frequency of the phenomena themselves, how great frequency of coincidence must follow from that, supposing there is neither connexion nor repugnance. If there be greater frequency, there is connexion; if less, repugnance."
I do not know that we can go further definite in precept. The number of casual coincidences bears a certain proportion to the positive frequency of the coinciding phenomena: that proportion is to be determined by common-sense in each case. It may be possible, however, to bring out more clearly the principle on which common-sense proceeds in deciding what chance will and will not account for, although our exposition amounts only to making more clear what it is that we mean by chance as distinguished from assignable reason. I would suggest that in deciding what chance will not account for, we make regressive application of a principle which may be called the principle of Equal and Unequal Alternatives, and which may be worded as follows:--
Of a given number of possible alternatives, all equally possible, one of which is bound to occur at a given time, we expect each to have its turn an equal number of times in the long run. If several of the alternatives are of the same kind, we expect an alternative of that kind to recur with a frequency proportioned to their greater number. If any of the alternatives has an advantage, it will recur with a frequency proportioned to the strength of that advantage.
Situations in which alternatives are absolutely equal are rare in nature, but they are artificially created for games "of chance," as in tossing a coin, throwing dice, drawing lots, shuffling and dealing a pack of cards. The essence of all games of chance is to construct a number of equal alternatives, making them as nearly equal as possible, and to make no prearrangement which of the number shall come off. We then say that this is determined by chance. If we ask why we believe that when we go on bringing off one alternative at a time, each will have its turn, part of the answer undoubtedly is that given by De Morgan, namely, that we know no reason why one should be chosen rather than another. This, however, is probably not the whole reason for our belief. The rational belief in the matter is that it is only in the long run or on the average that each of the equal alternatives will have its turn, and this is probably founded on the experience of actual trial. The mere equality of the alternatives, supposing them to be perfectly equal, would justify us as much in expecting that each would have its turn in a single revolution of the series, in one complete cycle of the alternatives. This, indeed, may be described as the natural and primitive expectation which is corrected by experience. Put six balls in a wicker bottle, shake them up, and roll one out: return this one, and repeat the operation: at the end of six draws we might expect each ball to have had its turn of being drawn if we went merely on the abstract equality of the alternatives. But experience shows us that in six successive draws the same ball may come out twice or even three or four times, although when thousands of drawings are made each comes out nearly an equal number of times.
So in tossing a coin, heads may turn up ten or twelve times in succession, though in thousands of tosses heads and tails are nearly equal. Runs of luck are thus within the rational doctrine of chances: it is only in the long run that luck is equalised supposing that the events are pure matter of chance, that is, supposing the fundamental alternatives to be equal.