III.--MILL'S "JOINT METHOD OF AGREEMENT AND DIFFERENCE".

After examining a variety of instances in which an effect appears, and finding that they all agree in the antecedent presence of some one circumstance, we may proceed to examine instances otherwise similar (_in pari materia_, as Prof. Fowler puts it) where the effect does not appear. If these all agree in the absence of the circumstance that is uniformly present with the effect, we have corroborative evidence that there is causal connexion between this circumstance and the effect.

The principle of this method seems to have been suggested to Mill by Wells's investigations into Dew. Wells exposed a number of polished surfaces of various substances, and compared those in which there was a copious deposit of dew with those in which there was little or none. If he could have got two surfaces, one dewed and the other not, identical in every concomitant but one, he would have attained complete proof on the principle of Single Difference. But this being impracticable, he followed a course which approximated to the method of eliminating every circumstance but one from instances of dew, and every circumstance but one in the instances of no-dew. Mill sums up as follows the results of his experiments: "It appears that the instances in which much dew is deposited, which are very various, agree in this, and, _so far as we are able to observe, in this only_, that they either radiate heat rapidly or conduct it slowly: qualities between which there is no other circumstance of agreement than that by virtue of either, the body tends to lose heat from the surface more rapidly than it can be restored from within. The instances, on the contrary, in which no dew, or but a small quantity of it, is formed, and which are also extremely various, agree (_as far as we can observe_) _in nothing except_ in _not_ having this same property. We seem therefore to have detected the characteristic difference between the substances on which the dew is produced, and those on which it is not produced.

And thus have been realised the requisitions of what we have termed the Indirect Method of Difference, or the Joint Method of Agreement and Difference." The Canon of this Method is accordingly stated by Mill as follows:--

If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance; the circumstance in which alone the two sets of instances differ, is the effect, or the cause, or an indispensable part of the cause, of the phenomenon.

In practice, however, this theoretical standard of proof is never attained. What investigators really proceed upon is the presumption afforded, to use Prof. Bain's terms, by Agreement in Presence combined with Agreement in Absence. When it is found that all substances which have a strong smell agree in being readily oxidisable, and that the marsh gas or carbonetted hydrogen which has no smell is not oxidisable at common temperatures, the presumption that oxidation is one of the causal circumstances in smell is strengthened, even though we have not succeeded in eliminating every circumstance but this one from either the positive or the negative instances. So in the following examples given by Prof. Fowler there is not really a compliance with the theoretical requirements of Mill's Method: there is only an increased presumption from the double agreement. "The Joint Method of Agreement and Difference (or the Indirect Method of Difference, or, as I should prefer to call it, the Double Method of Agreement) is being continually employed by us in the ordinary affairs of life. If when I take a particular kind of food, I find that I invariably suffer from some particular form of illness, whereas, when I leave it off, I cease to suffer, I entertain a double assurance that the food is the cause of my illness. I have observed that a certain plant is invariably plentiful on a particular soil; if, with a wide experience, I fail to find it growing on any other soil, I feel confirmed in my belief that there is in this particular soil some chemical constituent, or some peculiar combination of chemical constituents, which is highly favourable, if not essential, to the growth of the plant."

[Footnote 1: Elimination, or setting aside as being of no concern, must not be confounded with the exclusion of agents practised in applying the Method of Difference. We use the word in its ordinary sense of putting outside the sphere of an argument. By a curious slip, Professor Bain follows Mill in applying the word sometimes to the process of singling out or disentangling a causal circumstance. This is an inadvertent departure from the ordinary usage, according to which elimination means discarding from consideration as being non-essential.]

[Footnote 2: Hirsch's _Geographical and Historical Pathology_, Creighton's translation, vol. ii. pp. 121-202.]

[Footnote 3: The bare titles Difference and Agreement, though they have the advantage of simplicity, are apt to puzzle beginners inasmuch as in the Method of Difference the agreement among the instances is at a maximum, and the difference at a minimum, and _vice versa_ in the Method of Agreement. In both Methods it is really the isolation of the connexion between antecedent and sequent that constitutes the proof.]

[Footnote 4: That rainbows in the sky are produced by the passage of light through minute drops in the clouds was an inference from this observed uniformity.]

CHAPTER VI.

METHODS OF OBSERVATION.--MINOR METHODS.

I.--CONCOMITANT VARIATIONS.

_Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation._

This simple principle is constantly applied by us in connecting and disconnecting phenomena. If we hear a sound which waxes and wanes with the rise and fall of the wind, we at once connect the two phenomena.

We may not know what the causal connexion is, but if they uniformly vary together, there is at once a presumption that the one is causally dependent on the other, or that both are effects of the same cause.

This principle was employed by Wells in his researches into Dew. Some bodies are worse conductors of heat than others, and rough surfaces radiate heat more rapidly than smooth. Wells made observations on conductors and radiators of various degrees, and found that the amount of dew deposited was greater or less according as the objects conducted heat slowly or radiated heat rapidly. He thus established what Herschel called a "scale of intensity" between the conducting and radiating properties of the bodies bedewed, and the amount of the dew deposit. The explanation was that in bad conductors the surface cools more quickly than in good conductors because heat is more slowly supplied from within. Similarly in rough surfaces there is a more rapid cooling because heat is given off more quickly. But whatever the explanation might be, the mere concomitant variation of the dew deposit with these properties showed that there was some causal connexion between them.

It must be remembered that the mere fact of concomitant variation is only an index that some causal connexion exists. The nature of the connexion must be ascertained by other means, and may remain a problem, one of the uses of such observed facts being indeed to suggest problems, for inquiry. Thus a remarkable concomitance has been observed between spots on the sun, displays of Aurora Borealis, and magnetic storms. The probability is that they are causally connected, but science has not yet discovered how. Similarly in the various sciences properties are arranged in scales of intensity, and any correspondence between two scales becomes a subject for investigation on the assumption that it points to a causal connexion. We shall see afterwards how in social investigations concomitant variations in averages furnish material for reasoning.

When two variants can be precisely measured, the ratio of the variation may be ascertained by the Method of Single Difference. We may change an antecedent in degree, and watch the corresponding change in the effect, taking care that no other agent influences the effect in the meantime. Often when we cannot remove an agent altogether, we may remove it in a measurable amount, and observe the result. We cannot remove friction altogether, but the more it is diminished, the further will a body travel under the impulse of the same force.

Until a concomitant variation has been fully explained, it is merely an empirical law, and any inference that it extends at the same rate beyond the limits of observation must be made with due caution.

"Parallel variation," says Professor Bain, "is sometimes interrupted by critical points, as in the expansion of bodies by heat, which suffers a reverse near the point of cooling. Again, the energy of a solution does not always follow the strength; very dilute solutions occasionally exercise a specific power not possessed in any degree by stronger. So, in the animal body, food and stimulants operate proportionally up to a certain point, at which their further operation is checked by the peculiarities in the structure of the living organs.... We cannot always reason from a few steps in a series to the whole series, partly because of the occurrence of critical points, and partly from the development at the extremes of new and unsuspected powers. Sir John Herschel remarks that until very recently 'the formulae empirically deduced for the elasticity of steam, those for the resistance of fluids, and on other similar subjects, have almost invariably failed to support the theoretical structures that have been erected upon them'."[1]

II.--SINGLE RESIDUE.

_Subduct from any phenomenon such part as previous induction has shown to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents._

"Complicated phenomena, in which several causes concurring, opposing, or quite independent of each other, operate at once, so as to produce a compound effect, may be simplified by subducting the effect of all the known causes, as well as the nature of the case permits, either by deductive reasoning or by appeal to experience, and thus leaving as it were a _residual phenomenon_ to be explained. It is by this process, in fact, that science, in its present advanced state, is chiefly promoted. Most of the phenomena which nature presents are very complicated; and when the effects of all known causes are estimated with exactness, and subducted, the residual facts are constantly appearing in the form of phenomena altogether new, and leading to the most important conclusions."[2]

It is obvious that this is not a primary method of observation, but a method that may be employed with great effect to guide observation when a considerable advance has been made in accurate knowledge of agents and their mode of operation. The greatest triumph of the method, the discovery of the planet Neptune, was won some years after the above passage from Herschel's Discourse was written. Certain perturbations were observed in the movements of the planet Uranus: that is to say, its orbit was found not to correspond exactly with what it should be when calculated according to the known influences of the bodies then known to astronomers. These perturbations were a residual phenomenon. It was supposed that they might be due to the action of an unknown planet, and two astronomers, Adams and Le Verrier, simultaneously calculated the position of a body such as would account for the observed deviations. When telescopes were directed to the spot thus indicated, the planet Neptune was discovered. This was in September, 1846: before its actual discovery, Sir John Herschel exulted in the prospect of it in language that strikingly expresses the power of the method. "We see it," he said, "as Columbus saw America from the shores of Spain. Its movements have been felt, trembling along the far-reaching line of our analysis, with a certainty hardly inferior to that of ocular demonstration."[3]

Many of the new elements in Chemistry have been discovered in this way. For example, when distinctive spectrums had been observed for all known substances, then on the assumption that every substance has a distinctive spectrum, the appearance of lines not referable to any known substance indicated the existence of hitherto undiscovered substances and directed search for them. Thus Bunsen in 1860 discovered two new alkaline metals, Caesium and Rubidium. He was examining alkalies left from the evaporation of a large quantity of mineral water from Durkheim. On applying the spectroscope to the flame which this particular salt or mixture of salts gave off, he found that some bright lines were visible which he had never observed before, and which he knew were not produced either by potash or soda. He then set to work to analyse the mixture, and ultimately succeeded in separating two new alkaline substances. When he had succeeded in getting them separate, it was of course by the Method of Difference that he ascertained them to be capable of producing the lines that had excited his curiosity.

[Footnote 1: Bain's _Logic_, vol. ii. p. 64.]

[Footnote 2: Herschel's _Discourse_, -- 158.]

[Footnote 3: De Morgan's _Budget of Paradoxes_, p. 237.]

CHAPTER VII.

THE METHOD OF EXPLANATION.

Given perplexity as to the cause of any phenomenon, what is our natural first step? We may describe it as searching for a clue: we look carefully at the circumstances with a view to finding some means of assimilating what perplexes us to what is already within our knowledge. Our next step is to make a guess, or conjecture, or, in scientific language, a hypothesis. We exercise our Reason or _Nous_, or Imagination, or whatever we choose to call the faculty, and try to conceive some cause that strikes us as sufficient to account for the phenomenon. If it is not at once manifest that this cause has really operated, our third step is to consider what appearances ought to present themselves if it did operate. We then return to the facts in question, and observe whether those appearances do present themselves.

If they do, and if there is no other way of accounting for the effect in all its circumstances, we conclude that our guess is correct, that our hypothesis is proved, that we have reached a satisfactory explanation.

These four steps or stages may be distinguished in most protracted inquiries into cause. They correspond to the four stages of what Mr. Jevons calls the Inductive Method _par excellence_, Preliminary Observation, Hypothesis, Deduction and Verification. Seeing that the word Induction is already an overloaded drudge, perhaps it would be better to call these four stages the Method of Explanation. The word Induction, if we keep near its original and most established meaning, would apply strictly only to the fourth stage, the Verification, the bringing in of the facts to confirm our hypothesis. We might call the method the Newtonian method, for all four stages are marked in the prolonged process by which he made good his theory of Gravitation.

To give the name of Inductive Method simply to all the four stages of an orderly procedure from doubt to a sufficient explanation is to encourage a widespread misapprehension. There could be no greater error than to suppose that only the senses are used in scientific investigation. There is no error that men of science are so apt to resent in the mouths of the non-scientific. Yet they have partly brought it on themselves by their loose use of the word Induction, which they follow Bacon in wresting from the traditional meaning of Induction, using it to cover both Induction or the bringing in of facts--an affair mainly of Observation--and Reasoning, the exercise of Nous, the process of constructing satisfactory hypotheses. In reaction against the popular misconception which Bacon encouraged, it is fashionable now to speak of the use of Imagination in Science. This is well enough polemically. Imagination as commonly understood is akin to the constructive faculty in Science, and it is legitimate warfare to employ the familiar word of high repute to force general recognition of the truth. But in common usage Imagination is appropriated to creative genius in the Fine Arts, and to speak of Imagination in Science is to suggest that Science deals in fictions, and has discarded Newton's declaration _Hypotheses non fingo_. In a fight for popular respect, men of science may be right to claim for themselves Imagination; but in the interests of clear understanding, the logician must deplore that they should defend themselves from a charge due to their abuse of one word by making an equally unwarrantable and confusing extension of another.

Call it what we will, the faculty of likely guessing, of making probable hypotheses, of conceiving in all its circumstances the past situation or the latent and supramicroscopical situation out of which a phenomenon has emerged, is one of the most important of the scientific man's special gifts. It is by virtue of it that the greatest advancements of knowledge have been achieved, the cardinal discoveries in Molar and Molecular Physics, Biology, Geology, and all departments of Science. We must not push the idea of stages in explanatory method too far: the right explanation may be reached in a flash. The idea of stages is really useful mainly in trying to make clear the various difficulties in investigation, and the fact that different men of genius may show different powers in overcoming them.

The right hypothesis may occur in a moment, as if by simple intuition, but it may be tedious to prove, and the gifts that tell in proof, such as Newton's immense mathematical power in calculating what a hypothesis implies, Darwin's patience in verifying, Faraday's ingenuity in devising experiments, are all great gifts, and may be serviceable at different stages. But without originality and fertility in probable hypothesis, nothing can be done.

The dispute between Mill and Whewell as to the place and value of hypotheses in science was in the main a dispute about words. Mill did not really undervalue hypothesis, and he gave a most luminous and accurate account of the conditions of proof. But here and there he incautiously spoke of the "hypothetical method" (by which he meant what we have called the method of Explanation) as if it were a defective kind of proof, a method resorted to by science when the "experimental methods" could not be applied. Whether his language fairly bore this construction is not worth arguing, but this was manifestly the construction that Whewell had in his mind when he retorted, as if in defence of hypotheses, that "the inductive process consists in framing successive hypotheses, the comparison of these with the ascertained facts of nature, and the introduction into them of such modifications as the comparison may render necessary". This is a very fair description of the whole method of explanation. There is nothing really inconsistent with it in Mill's account of his "hypothetical method"; only he erred himself or was the cause of error in others in suggesting, intentionally or unintentionally, that the Experimental Methods were different methods of proof. The "hypothetical method," as he described it, consisting of Induction, Ratiocination, and Verification, really comprehends the principles of all modes of observation, whether naturally or artificially experimental. We see this at once when we ask how the previous knowledge is got in accordance with which hypotheses are framed. The answer must be, by Observation. However profound the calculations, it must be from observed laws, or supposed analogues of them, that we start. And it is always by Observation that the results of these calculations are verified.

Both Mill and Whewell, however, confined themselves too exclusively to the great hypotheses of the Sciences, such as Gravitation and the Undulatory Theory of Light. In the consideration of scientific method, it is a mistake to confine our attention to these great questions, which from the multitude of facts embraced can only be verified by prolonged and intricate inquiry. Attempts at the explanation of the smallest phenomena proceed on the same plan, and the verification of conjectures about them is subject to the same conditions, and the methods of investigation and the conditions of verification can be studied most simply in the smaller cases. Further, I venture to think it a mistake to confine ourselves to scientific inquiry in the narrow sense, meaning thereby inquiry conducted within the pale of the exact sciences. For not merely the exact sciences but all men in the ordinary affairs of life must follow the same methods or at least observe the same principles and conditions, in any satisfactory attempt to explain.

Tares appear among the wheat. Good seed was sown: whence, then, come the tares? "An enemy has done this." If an enemy has actually been observed sowing the tares, his agency can be proved by descriptive testimony. But if he has not been seen in the act, we must resort to what is known in Courts of Law as circumstantial evidence. This is the "hypothetical method" of science. That the tares are the work of an enemy is a hypothesis: we examine all the circumstances of the case in order to prove, by inference from our knowledge of similar cases, that thus, and thus only, can those circumstances be accounted for.

Similarly, when a question is raised as to the authorship of an anonymous book. We first search for a clue by carefully noting the diction, the structure of the sentences, the character and sources of the illustration, the special tracks of thought. We proceed upon the knowledge that every author has characteristic turns of phrase and imagery and favourite veins of thought, and we look out for such internal evidence of authorship in the work before us. Special knowledge and acumen may enable us to detect the authorship at once from the general resemblance to known work. But if we would have clear proof, we must show that the resemblance extends to all the details of phrase, structure and imagery: we must show that our hypothesis of the authorship of XYZ explains all the circumstances. And even this is not sufficient, as many erroneous guesses from internal evidence may convince us. We must establish further that there is no other reasonable way of accounting for the matter and manner of the book; for example, that it is not the work of an imitator. An imitator may reproduce all the superficial peculiarities of an author with such fidelity that the imitation can hardly be distinguished from the original: thus few can distinguish between Fenton's work and Pope's in the translation of the Odyssey. We must take such known facts into account in deciding a hypothesis of authorship. Such hypotheses can seldom be decided on internal evidence alone: other circumstantial evidence--other circumstances that ought to be discoverable if the hypothesis is correct--must be searched for.

The operation of causes that are manifest only in their effects must be proved by the same method as the operation of past causes that have left only their effects behind them. Whether light is caused by a projection of particles from a luminous body or by an agitation communicated through an intervening medium cannot be directly observed. The only proof open is to calculate what should occur on either hypothesis, and observe whether this does occur. In such a case there is room for the utmost calculating power and experimental ingenuity. The mere making of the general hypothesis or guess is simple enough, both modes of transmitting influence, the projection of moving matter and the travelling of an undulation or wave movement, being familiar facts. But it is not so easy to calculate exactly how a given impulse would travel, and what phenomena of ray and shadow, of reflection, refraction and diffraction ought to be visible in its progress. Still, no matter how intricate the calculation, its correspondence with what can be observed is the only legitimate proof of the hypothesis.

II.--OBSTACLES TO EXPLANATION.--PLURALITY OF CAUSES AND INTERMIXTURE OF EFFECTS.

There are two main ways in which explanation may be baffled. There may exist more than one cause singly capable of producing the effect in question, and we may have no means of determining which of the equally sufficient causes has actually been at work. For all that appears the tares in our wheat may be the effect of accident or of malicious design: an anonymous book may be the work of an original author or of an imitator. Again, an effect may be the joint result of several co-operating causes, and it may be impossible to determine their several potencies. The bitter article in the _Quarterly_ may have helped to kill John Keats, but it co-operated with an enfeebled constitution and a naturally over-sensitive temperament, and we cannot assign its exact weight to each of these coefficients. Death may be the result of a combination of causes; organic disease co-operating with exposure, over-fatigue co-operating with the enfeeblement of the system by disease.

The technical names for these difficulties, Plurality of Causes and Intermixture of Effects, are apt to confuse without some clearing up.

In both kinds of difficulty more causes than one are involved: but in the one kind of case there is a plurality of possible or equally probable causes, and we are at a loss to decide which: in the other kind of case there is a plurality of co-operating causes; the effect is the result or product of several causes working conjointly, and we are unable to assign to each its due share.

It is with a view to overcoming these difficulties that Science endeavours to isolate agencies and ascertain what each is capable of singly. Mill and Bain treat Plurality of Causes and Intermixture of Effects in connexion with the Experimental Methods. It is better, perhaps, to regard them simply as obstacles to explanation, and the Experimental Methods as methods of overcoming those obstacles. The whole purpose of the Experimental Methods is to isolate agencies and effects: unless they can be isolated, the Methods are inapplicable.

In situations where the effects observable may be referred with equal probability to more than one cause, you cannot eliminate so as to obtain a single agreement. The Method of Agreement is frustrated. And an investigator can get no light from mixed effects, unless he knows enough of the causes at work to be able to apply the Method of Residues. If he does not, he must simply look out for or devise instances where the agencies are at work separately, and apply the principle of Single Difference.

Great, however, as the difficulties are, the theory of Plurality and Intermixture baldly stated makes them appear greater than they are in practice. There is a consideration that mitigates the complication, and renders the task of unravelling it not altogether hopeless. This is that different causes have distinctive ways of operating, and leave behind them marks of their presence by which their agency in a given case may be recognised.