It should be observed that the cogency of the proof depends entirely upon its tending to show the unconditionality of the sequence A-_p_, or the indispensability of A as a condition of _p_. That _p_ follows A, even immediately, is nothing by itself: if a man sits down to study and, on the instant, a hand-organ begins under his window, he must not infer malice in the musician: thousands of things follow one another every moment without traceable connection; and this we call "accidental." Even invariable sequence is not enough to prove direct causation; for, in our experience does not night invariable follow day? The proof requires that the instances be such as to show not merely what events _are_ in invariable sequence, but also what _are not_. From among the occasional antecedents of _p_ (or consequents of A) we have to eliminate the accidental ones. And this is done by finding or making "negative instances" in respect of each of them. Thus the instance
A D E _p s t_
is a negative instance of B and C considered as supposable causes of _p_ (and of _q_ and _r_ as supposable effects of A); for it shows that they are absent when _p_ (or A) is present.
To insist upon the cogency of "negative instances" was Bacon"s great contribution to Inductive Logic. If we neglect them, and merely collect examples of the sequence A-_p_, this is "simple enumeration"; and although simple enumeration, when the instances of agreement are numerous enough, may give rise to a strong belief in the connection of phenomena, yet it can never be a methodical or logical proof of causation, since it does not indicate the unconditionalness of the sequence. For simple enumeration of the sequence A-_p_ leaves open the possibility that, besides A, there is always some other antecedent of _p_, say X; and then X may be the cause of _p_. To disprove it, we must find, or make, a negative instance of X--where _p_ occurs, but X is absent.
So far as we recognise the possibility of a plurality of causes, this method of Agreement cannot be quite satisfactory. For then, in such instances as the above, although D is absent in the first, and B in the second, it does not follow that they are not the causes of _p_; for they may be alternative causes: B may have produced _p_ in the first instance, and D in the second; A being in both cases an accidental circ.u.mstance in relation to _p_. To remedy this shortcoming by the method of Agreement itself, the only course is to find more instances of _p_. We may never find a negative instance of A; and, if not, the probability that A is the cause of _p_ increases with the number of instances. But if there be no antecedent that we cannot sometimes exclude, yet the collection of instances will probably give at last all the causes of _p_; and by finding the proportion of instances in which A, B, or X precedes _p_, we may estimate the probability of any one of them being the cause of _p_ in any given case of its occurrence.
But this is not enough. Since there cannot really be vicarious causes, we must define the effect (_p_) more strictly, and examine the cases to find whether there may not be varieties of _p_, with each of which one of the apparent causes is correlated: A with _p_^{1} B with _p_^{11}, X with _p_^{111}. Or, again, it may be that none of the recognised antecedents is effective: as we here depend solely on observation, the true conditions may be so recondite and disguised by other phenomena as to have escaped our scrutiny. This may happen even when we suppose that the chief condition has been isolated: the drinking of foul water was long believed to cause dysentery, because it was a frequent antecedent; whilst observation had overlooked the bacillus, which was the indispensable condition.
Again, though we have a.s.sumed that, in the instances supposed above, immediate sequence is observable, yet in many cases it may not be so, if we rely only on the canon of Agreement; if instances cannot be obtained by experiment, and we have to depend on observation. The phenomena may then be so mixed together that A and _p_ seem to be merely concomitant; so that, though connection of some sort may be rendered highly probable, we may not be able to say which is cause and which is effect. We must then try (as Bain says) to trace the expenditure of energy: if _p_ gains when A loses, the course of events if from A to _p_.
Moreover, where succession cannot be traced, the method of Agreement may point to a connection between two or more facts (perhaps as co-effects of a remote cause) where direct causation seems to be out of the question: e.g., that Negroes, though of different tribes, different localities, customs, etc., are prognathous, woolly-haired and dolichocephalic.
The Method of Agreement, then, cannot by itself prove causation. Its chief use (as Mill says) is to suggest hypotheses as to the cause; which must then be used (if possible) experimentally to try if it produces the given effect. A bacillus, for example, being always found with a certain disease, is probably the chief condition of it: give it to a guinea-pig, and observe whether the disease appears in that animal.
Men often use arguments which, if they knew it, might be shown to conform more or less to this canon; for they collect many instances to show that two events are connected; but usually neglect to bring out the negative side of the proof; so that their arguments only amount to simple enumeration. Thus Ascham in his _Toxophilus_, insisting on the national importance of archery, argues that victory has always depended on superiority in shooting; and, to prove it, he shows how the Parthians checked the Romans, Sesostris conquered a great part of the known world, Tiberius overcame Arminius, the Turks established their empire, and the English defeated the French (with many like examples)--all by superior archery. But having cited these cases to his purpose, he is content; whereas he might have greatly strengthened his proof by showing how one or the other instance excludes other possible causes of success. Thus: the cause was not discipline, for the Romans were better disciplined than the Parthians; nor yet the boasted superiority of a northern habitat, for Sesostris issued from the south; nor better manhood, for here the Germans probably had the advantage of the Romans; nor superior civilisation, for the Turks were less civilised than most of those they conquered; nor numbers, nor even a good cause, for the French were more numerous than the English, and were shamefully attacked by Henry V. on their own soil. Many an argument from simple enumeration may thus be turned into an induction of greater plausibility according to the Canon of Agreement.
Still, in the above case, the effect (victory) is so vaguely conceived, that a plurality of causes must be allowed for: although, e.g., discipline did not enable the Romans to conquer the Parthians, it may have been their chief advantage over the Germans; and it was certainly important to the English under Henry V. in their war with the French.
Here is another argument, somewhat similar to the above, put forward by H. Spencer with a full consciousness of its logical character. States that make war their chief object, he says, a.s.sume a certain type of organisation, involving the growth of the warrior cla.s.s and the treatment of labourers as existing solely to sustain the warriors; the complete subordination of individuals to the will of the despotic soldier-king, their property, liberty and life being at the service of the State; the regimentation of society not only for military but also for civil purposes; the suppression of all private a.s.sociations, etc.
This is the case in Dahomey and in Russia, and it was so at Sparta, in Egypt, and in the empire of the Yncas. But the similarity of organisation in these States cannot have been due to race, for they are all of different races; nor to size, for some are small, some large; nor to climate or other circ.u.mstances of habitat, for here again they differ widely: the one thing they have in common is the military purpose; and this, therefore, must be the cause of their similar organisation.
(_Political Inst.i.tutions._)
By this method, then, to prove that one thing is causally connected with another, say A with _p_, we show, first, that in all instances of _p_, A is present; and, secondly, that any other supposable cause of _p_ may be absent without disturbing _p_. We next come to a method the use of which greatly strengthens the foregoing, by showing that where _p_ is absent A is also absent, and (if possible) that A is the only supposable cause that is always absent along with _p_.
-- 2. THE CANON OF THE JOINT METHOD OF AGREEMENT IN PRESENCE AND IN ABSENCE.
_If_ (1) _two or more instances in which a phenomenon occurs have only one other circ.u.mstance (antecedent or consequent) in common, while_ (2) _two or more instances in which it does not occur (though in important points they resemble the former set of instances) have nothing else in common save the absence of that circ.u.mstance--the circ.u.mstance in which alone the two sets of instances differ throughout (being present in the first set and absent in the second) is probably the effect, or the cause, or an indispensable condition of the phenomenon._
The first clause of this Canon is the same as that of the method of Agreement, and its significance depends upon the same propositions concerning causation. The second clause, relating to instances in which the phenomenon is absent, depends for its probative force upon Prop. II.
(a), and I. (b): its function is to exclude certain circ.u.mstances (whose nature or manner of occurrence gives them some claim to consideration) from the list of possible causes (or effects) of the phenomenon investigated. It might have been better to state this second clause separately as the Canon of the Method of Exclusions.
To prove that A is causally related to _p_, let the two sets of instances be represented as follows:
Instances of Presence. Instances of Absence.
A B C C H F _p q r_ _r x v_ A D E B D K _p s t_ _q y s_ A F G E G M _p u v_ _t f u_
Then A is probably the cause or a condition of _p_, or _p_ is dependent upon A: first, by the Canon of Agreement in Presence, as represented by the first set of instances; and, secondly, by Agreement in Absence in the second set of instances. For there we see that C, H, F, B, D, K, E, G, M occur without the phenomenon _p_, and therefore (by Prop. II. (a)) are not its cause, or not the whole cause, unless they have been counteracted (which is a point for further investigation). We also see that _r, v, q, s, t, u_ occur without A, and therefore are not the effects of A. And, further, if the negative instances represent all possible cases, we see that (according to Prop. I. (b)) A is the cause of _p_, because it cannot be omitted without the cessation of _p_. The inference that A and _p_ are cause and effect, suggested by their being present throughout the first set of instances, is therefore strengthened by their being both absent throughout the second set.
So far as this Double Method, like the Single Method of Agreement, relies on observation, sequence may not be perceptible in the instances observed, and then, direct causation cannot be proved by it, but only the probability of causal connection; and, again, the real cause, though present, may be so obscure as to evade observation. It has, however, one peculiar advantage, namely, that if the second list of instances (in which the phenomenon and its supposed antecedent are both absent) can be made exhaustive, it precludes any hypothesis of a plurality of causes; since all possible antecedents will have been included in this list without producing the phenomenon. Thus, in the above symbolic example, taking the first set of instances, the supposition is left open that B, C, D, E, F, G may, at one time or another, have been a condition of _p_; but, in the second list, these antecedents all occur, here or there, without producing _p_, and therefore (unless counteracted somehow) cannot be a condition of _p_. A, then, stands out as the one thing that is present whenever _p_ is present, and absent whenever _p_ is absent.
Stated in this abstract way, the Double Method may seem very elaborate and difficult; yet, in fact, its use may be very simple. Tyndall, to prove that dispersed light in the air is due to motes, showed by a number of cases (1) that any gas containing motes is luminous; (2) that air in which the motes had been destroyed by heat, and any gas so prepared as to exclude motes, are not luminous. All the instances are of gases, and the result is: motes--luminosity; no motes--no luminosity.
Darwin, to show that cross-fertilisation is favourable to flowers, placed a net about 100 flower-heads, and left 100 others of the same varieties exposed to the bees: the former bore no seed, the latter nearly 3,000. We must a.s.sume that, in Darwin"s judgment, the net did not screen the flowers from light and heat sufficiently to affect the result.
There are instructive applications of this Double Method in Wallace"s _Darwinism_. In chap. viii., on _Colour in Animals_, he observes, that the usefulness of their coloration to animals is shown by the fact that, "as a rule, colour and marking are constant in each species of wild animal, while, in almost every domesticated animal, there arises great variability. We see this in our horses and cattle, our dogs and cats, our pigeons and poultry. Now the essential difference between the conditions of life of domesticated and wild animals is, that the former are protected by man, while the latter have to protect themselves." Wild animals protect themselves by acquiring qualities adapted to their mode of life; and coloration is a very important one, its chief, though not its only use, being concealment. Hence a useful coloration having been established in any species, individuals that occasionally may vary from it, will generally, perish; whilst, among domestic animals, variation of colour or marking is subject to no check except the taste of owners. We have, then, two lists of instances; first, innumerable species of wild animals in which the coloration is constant and which depend upon their own qualities for existence; secondly, several species of domestic animals in which the coloration is _not_ constant, and which do _not_ depend upon their own qualities for existence. In the former list two circ.u.mstances are present together (under all sorts of conditions); in the latter they are absent together. The argument may be further strengthened by adding a third list, parallel to the first, comprising domestic animals in which coloration is approximately constant, but where (as we know) it is made a condition of existence by owners, who only breed from those specimens that come up to a certain standard of coloration.
Wallace goes on to discuss the colouring of arctic animals. In the arctic regions, he says, some animals are wholly white all the year round, such as the polar bear, the American polar hare, the snowy owl and the Greenland falcon: these live amidst almost perpetual snow.
Others, that live where the snow melts in summer, only turn white in winter, such as the arctic hare, the arctic fox, the ermine and the ptarmigan. In all these cases the white colouring is useful, concealing the herbivores from their enemies, and also the carnivores in approaching their prey; this usefulness, therefore, is a condition of the white colouring. Two other explanations have, however, been suggested: first, that the prevalent white of the arctic regions directly colours the animals, either by some photographic or chemical action on the skin, or by a reflex action through vision (as in the chameleon); secondly, that a white skin checks radiation and keeps the animals warm. But there are some exceptions to the rule of white colouring in arctic animals which refute these hypotheses, and confirm the author"s. The sable remains brown throughout the winter; but it frequents trees, with whose bark its colour a.s.similates. The musk-sheep is brown and conspicuous; but it is gregarious, and its safety depends upon its ability to recognise its kind and keep with the herd. The raven is always black; but it fears no enemy and feeds on carrion, and therefore does not need concealment for either defence or attack. The colour of the sable, then, though not white, serves for concealment; the colour of the musk-sheep serves a purpose more important than concealment; the raven needs no concealment. There are thus two sets of instances:--in one set the animals are white (a) all the year, (b) in winter; and white conceals them (a) all the year, (b) in winter; in the other set, the animals are _not_ white, and to them either whiteness would _not_ give concealment, or concealment would _not_ be advantageous. And this second list refutes the rival hypotheses: for the sable, the musk-sheep and the raven are as much exposed to the glare of the snow, and to the cold, as the other animals are.
-- 3. THE CANON OF DIFFERENCE.
_If an instance in which a phenomenon occurs, and an instance in which it does not occur, have every other circ.u.mstance in common save one, that one (whether consequent or antecedent) occurring only in the former; the circ.u.mstance in which alone the two instances differ is the effect, or the cause, or an indispensable condition of the phenomenon._
This follows from Props. I (a) and (b), in chapter xv. -- 7. To prove that A is a condition of _p_, let two instances, such as the Canon requires, be represented thus:
A B C B C _p q r_ _q r_
Then A is the cause or a condition of _p_. For, in the first instance, A being introduced (without further change), _p_ arises (Prop. I. (a)); and, in the second instance, A having been removed (without other change), _p_ disappears (Prop. I. (b)). Similarly we may prove, by the same instances, that _p_ is the effect of A.
The order of the phenomena and the immediacy of their connection is a matter for observation, aided by whatever instruments and methods of inspection and measurement may be available.
As to the invariability of the connection, it may of course be tested by collecting more instances or making more experiments; but it has been maintained, that a single perfect experiment according to this method is sufficient to prove causation, and therefore implies invariability (since causation is uniform), though no other instances should ever be obtainable; because it establishes once for all the unconditionality of the connection
A B C _p q r_.
Now, formally this is true; but in any actual investigation how shall we decide what is a satisfactory or perfect experiment? Such an experiment requires that in the negative instance
B C _q r_,
BC shall be the least a.s.semblage of conditions necessary to co-operate with A in producing _p_; and that it is so cannot be ascertained without either general prior knowledge of the nature of the case or special experiments for the purpose. So that invariability will not really be inferred from a single experiment; besides that every prudent inquirer repeats his experiments, if only to guard against his own liability to error.
The supposed plurality of causes does not affect the method of Difference. In the above symbolic case, A is clearly _one_ cause (or condition) of _p_, whatever other causes may be possible; whereas with the Single Method of Agreement, it remained doubtful (admitting a plurality of causes) whether A, in spite of being always present with _p_, was ever a cause or condition of it.
This method of Difference without our being distinctly aware of it, is oftener than any other the basis of ordinary judgments. That the sun gives light and heat, that food nourishes and fire burns, that a stone breaks a window or kills a bird, that the turning of a tap permits or checks the flow of water or of gas, and thousands of other propositions are known to be true by rough but often emphatic applications of this method in common experience.
The method of Difference may be applied either (1) by observation, on finding two instances (distinct a.s.semblages of conditions) differing only in one phenomenon together with its antecedent or consequent; or (2) by experiment, and then, either (a) by preparing two instances that may be compared side by side, or (b) by taking certain conditions, and then introducing (or subtracting) some agent, supposed to be the cause, to see what happens: in the latter case the "two instances" are the same a.s.semblage of conditions considered before and, again, after, the introduction of the agent. As an example of (a) there is an experiment to show that radium gives off heat: take two gla.s.s tubes, in one put some chloride of radium, in both thermometers, and close them with cotton-wool. Soon the thermometer in the tube along with radium reads 54 F. higher than the other one. The tube without the radium, whose temperature remains unaltered, is called the "control" experiment. Most experiments are of the type (b); and since the Canon, which describes two co-existing instances, does not readily apply to this type, an alternative version may be offered: _Any agent whose introduction into known circ.u.mstances (without further change) is immediately followed by a definite phenomenon is a condition of the occurrence of that phenomenon._
The words _into known circ.u.mstances_ are necessary to emphasise what is required by this Method, namely, that the two instances differ in only one thing; for this cannot be ascertained unless all the other conditions are known; and this further implies that they have been prepared. It is, therefore, not true (as Sigwart a.s.serts) that this method determines only one condition of a phenomenon, and that it is then necessary to inquire into the other conditions. If they were not known they must be investigated; but then the experiment would not have been made upon this method. Practically, experiments have to be made in all degrees of imperfection, and the less perfect they are, that is, the less the circ.u.mstances are known beforehand, the more remains to be done. A common imperfection is delay, or the occurrence of a latent period between the introduction of an agent and the manifestation of its effects; it cannot then be the unconditional cause; though it may be an indispensable remote condition of whatever change occurs. If, feeling out of sorts, you take a drug and some time afterwards feel better, it is not clear on this ground alone that the drug was the cause of recovery, for other curative processes may have been active meanwhile--food, or sleep, or exercise.
Any book of Physics or of Chemistry will furnish scores of examples of the method of Difference: such as Galileo"s experiment to show that air has weight, by first weighing a vessel filled with ordinary air, and then filling it with condensed air and weighing it again; when the increased weight can only be due to the greater quant.i.ty of air contained. The melting-point of solids is determined by heating them until they do melt (as silver at 1000 C., gold at 1250, platinum at 2000); for the only difference between bodies at the time of melting and just before is the addition of so much heat. Similarly with the boiling point of liquids. That the transmission of sound depends upon the continuity of an elastic ponderable medium, is proved by letting a clock strike in a vacuum (under a gla.s.s from which the air has been withdrawn by an air pump), and standing upon a non-elastic pedestal: when the clock be seen to strike, but makes only such a faint sound as may be due to the imperfections of the vacuum and the pedestal.
The experiments by which the chemical a.n.a.lysis or synthesis of various forms of matter is demonstrated are simple or compound applications of this method of Difference, together with the quant.i.tative mark of causation (that cause and effect are equal); since the bodies resulting from an a.n.a.lysis are equal in weight to the body a.n.a.lysed, and the body resulting from a synthesis is equal in weight to the bodies synthesised.
That an electric current resolves water into oxygen and hydrogen may be proved by inserting the poles of a galvanic battery in a vessel of water; when this one change is followed by another, the rise of bubbles from each pole and the very gradual decrease of the water. If the bubbles are caught in receivers placed over them, it can be shown that the joint weight of the two bodies of gas thus formed is equal to the weight of the water that has disappeared; and that the gases are respectively oxygen and hydrogen may then be shown by proving that they have the properties of those gases according to further experiments by the method of Difference; as (e.g.) that one of them is oxygen because it supports combustion, etc.
When water was first decomposed by the electric current, there appeared not only oxygen and hydrogen, but also an acid and an alkali. These products were afterwards traced to impurities of the water and of the operator"s hands. Mill observes that in any experiment the effect, or part of it, may be due, not to the supposed agent, but to the means employed in introducing it. We should know not only the other conditions of an experiment, but that the agent or change introduced is nothing else than what it is supposed to be.
In the more complex sciences the method of Difference is less easily applicable, because of the greater difficulty of being sure that only one circ.u.mstance at a time has altered; still, it is frequently used.
Thus, if by dividing a certain nerve certain muscles are paralysed, it is shown that normally that nerve controls those muscles. That the sense of smell in flies and c.o.c.kroaches is connected with the antennae has been shown by cutting them off: whereupon the insects can no longer find carrion. In his work on _Earthworms_, Darwin shows that, though sensitive to mechanical tremors, they are deaf (or, at least, not sensitive to sonorous vibrations transmitted through the air), by the following experiment. He placed a pot containing a worm that had come to the surface, as usual at night, upon a table, whilst close by a piano was violently played; but the worm took no notice of the noise. He then placed the pot upon the piano, whilst it was being played, when the worm, probably feeling mechanical vibrations, hastily slid back into its burrow.
When, instead of altering one circ.u.mstance in an instance (which we have done our best not otherwise to disturb) and then watching what follows, we try to find two ready-made instances of a phenomenon, which only differ in one other circ.u.mstance, it is, of course, still more difficult to be sure that there is only one other circ.u.mstance in which they differ. It may be worth while, however, to look for such instances.
Thus, that the temperature of ocean currents influences the climate of the sh.o.r.es they wash, seems to be shown by the fact that the average temperature of Newfoundland is lower than that of the Norwegian coast some 15 farther north. Both regions have great continents at their back; and as the mountains of Norway are higher and capped with perennial snow, we might expect a colder climate there: but the sh.o.r.e of Norway is visited by the Gulf Stream, whilst the sh.o.r.e of Newfoundland is traversed by a cold current from Greenland. Again, when in 1841 the railway from Rouen to Paris was being built, gangs of English and gangs of French workmen were employed upon it, and the English got through about one-third more work per man than the French. It was suspected that this difference was due to one other difference, namely, that the English fed better, preferring beef to thin soup. Now, logically, it might have been objected that the evidence was unsatisfactory, seeing that the men differed in other things besides diet--in "race" (say), which explains so much and so easily. But the Frenchmen, having been induced to try the same diet as the English, were, in a few days, able to do as much work: so that the "two instances" were better than they looked. It often happens that evidence, though logically questionable, is good when used by experts, whose familiarity with the subject makes it good.
-- 4. THE CANON OF CONCOMITANT VARIATIONS.