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 versâ 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.