Logic, Inductive and Deductive

by William Minto

When two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree is the cause (or effect) of the given phenomenon.

Herschel's statement, on which this canon is founded, runs as follows: "Any circumstance in which all the facts without exception agree, may be the cause in question, or if not, at least a collateral effect of the same cause: if there be but one such point of agreement, the possibility becomes a certainty".

All the instances examined must agree in one circumstance—hence the title Method of Agreement. But it is not in the agreement merely that the proof consists, but the agreement in one circumstance combined with difference in all the other circumstances, when we are certain that every circumstance has come within our observation. It is the singleness of the agreement that constitutes the proof just as it is the singleness of the difference in the Method of Difference.[³]

It has been said that Mill's Method of Agreement amounts after all only to an uncontradicted Inductio per enumerationem simplicem, which he himself stigmatised as Induction improperly so called. But this is not strictly correct. It is a misunderstanding probably caused by calling the method that of agreement simply, instead of calling it the Method of Single Agreement, so as to lay stress upon the process of elimination by which the singleness is established. It is true that in the course of our observations we do perform an induction by simple enumeration. In eliminating, we at the same time generalise. That is to say, in multiplying instances for the elimination of non-causes, we necessarily at the same time multiply instances where the true causal antecedent, if there is only one possible, is present. An antecedent containing the true cause must always be there when the phenomenon appears, and thus we may establish by our eliminating observations a uniformity of connexion between two facts.

Take, for example, Roger Bacon's inquiry into the cause of the colours of the rainbow. His first notion seems to have been to connect the phenomenon with the substance crystal, probably from his thinking of the crystal firmament then supposed to encircle the universe. He found the rainbow colours produced by the passage of light through hexagonal crystals. But on extending his observations, he found that the passage of light through other transparent mediums was also attended by the phenomenon. He found it in dewdrops, in the spray of waterfalls, in drops shaken from the oar in rowing. He thus eliminated the substance crystal, and at the same time established the empirical law that the passage of light through transparent mediums of a globular or prismatic shape was a causal antecedent of the rainbow colours.[⁴]

Ascertainment of invariable antecedents may thus proceed side by side with that of variable antecedents, the use of the elimination being simply to narrow the scope of the inquiry. But the proof set forth in Mill's Canon does not depend merely on one antecedent or concomitant being invariably present, but also on the assumption that all the influential circumstances have been within our observation. Then only can we be sure that the instances have only one circumstance in common.

The truth is that owing to the difficulty of fulfilling this condition, proof of causation in accordance with Mill's Canon is practically all but impossible. It is not attained in any of the examples commonly given. The want of conclusiveness is disguised by the fact that both elimination and positive observation of mere agreement or uniform concomitance are useful and suggestive in the search for causes, though they do not amount to complete proof such as the Canon describes. Thus in the inquiry into the cause of goitre, the elimination serves some purpose though the result is purely negative. When the inquirer is satisfied that goitre is not originated by any directly observable local conditions, altitude, temperature, climate, soil, water, social circumstances, habits of exertion, his search is profitably limited. And mere frequency, much more constancy of concomitance, raises a presumption of causal connexion, and looking out for it is valuable as a mode of reconnoitring. The first thing that an inquirer naturally asks when confronted by numerous instances of a phenomenon is, What have they in common? And if he finds that they have some one circumstance invariably or even frequently present, although he cannot prove that they have no other circumstance in common as the Cannon of Single Agreement requires, the presumption of causal connexion is strong enough to furnish good ground for further inquiry. If an inquirer finds an illness with marked symptoms in a number of different households, and finds also that all the households get their milk supply from the same source, this is not conclusive proof of causation, but it is a sufficient presumption to warrant him in examining whether there is any virulent ingredient in the milk.

Thus varying the circumstances so as to bring out a common antecedent, though it does not end in exact proof, may indicate causal connexion though it does not prove what the nature of the connexion is. Roger Bacon's observations indicated that the production of rainbow colours was connected with the passage of light through a transparent globe or prism. It was reserved for Newton to prove by other methods that white light was composed of rays, and that those rays were differently refracted in passing through the transparent medium. We have another example of how far mere agreement, revealed by varying the circumstances, carries us towards discovery of the cause, in Wells's investigation of the cause of dew. Comparing the numerous instances of dew appearing without visible fall of moisture, Wells found that they all agreed in the comparative coldness of the surface dewed. This was all the agreement that he established by observation; he did not carry observation to the point of determining that there was absolutely no other common circumstance: when he had simply discovered dewed surfaces, he tried next to show by reasoning from other knows facts how the coldness of the surface affected the aqueous vapour of the neighbouring air. He did not establish his Theory of Dew by the Method of Agreement: but the observation of an agreement or common feature in a number of instances was a stage in the process by which he reached his theory.