The several methods have different degrees of cogency. The Methods of Difference and of Concomitant Variations are the most satisfactory; that of Agreement is least satisfactory. But the choice of method is not left entirely to the discretion of the investigator. Each method can only be applied to certain kinds of instances. And when the kind of instances required for the appli cation of the most cogent method cannot be found, or produced, the investigator must perforce resort to the next best method, and so on. Sometimes, indeed, he may even find it necessary to apply rather loosely several of these methods to the same problem, and even then perhaps he may seek confirmation in deductive reasoning from the nature of the case in the light of accepted truths. We shall now formulate and explain each of the five methods sep arately.
If two sets of circumstances are alike in all relevant respects, except that in one of them (the positive instance) a certain antecedent is present and a certain consequent follows, whereas in the other (the negative instance) both are absent, then that antecedent and that consequent are causally connected, that is to say, the consequent may always be expected to follow that antecedent in the absence of counteracting conditions. Using (here and throughout) the early letters of the alphabet to represent antecedents, the later letters to represent consequents, dots to represent the presence of irrelevant cir cumstances, and an arrow (—*) to mean "is followed by," or "is causally connected with," the Method of Difference may be sym bolized as follows : a b c d. . . —> w x y z. . .;abc...-->wxy; .d--> z. The instances compared may be two separate instances; or they may only be successive states of the same group of circumstances from which something is removed in order to obtain the negative instance, or to which something is added in order to obtain the positive instance ; or, lastly, each instance may consist of a group of things. But in all cases the two instances must be as like as possible in all essentials, except in regard to the difference under investigation. The significance of a certain difference between instances can only be determined when the in stances are as like as possible in other respects. The neglect of this precaution (known as the fallacy non ceteris paribus) easily leads to the attribution of an effect to the wrong antecedent (the fallacy of post hoc ergo propter hoc).
If a quantitative change in a certain antecedent is followed by a quantitative change in a certain consequent, although no other relevant circumstance has changed, then the antecedent and the consequent are causally connected. Symbolically, a b c
. . —>w x y zl . . . ; a b c
[or briefly, z is a function of
f(d)] ;.* .d—> z. This method may easily be expressed in a form that will show its kinship with the method of difference, the mere presence of a certain quantitative difference in the antecedent (d1—d2) being followed by a quantitative dif ference in the consequent
But, of course, the method of concomitant variations has no absolute negative instance such as the method of difference has. The tendency of modern science to become increasingly quantitative has given special importance to the method of concomitant variations as the instrument of quantitative induction, that is, for the establishment of precise quantitative correlations between phenomena, even when they are already known to be causally connected, or even when no interest attaches to the question of their causal connection. The concomitant variation may, of course, be either direct or inverse, that is, the consequent may increase with an increase in the ante cedent, and diminish when it diminishes, or the consequent may diminish as the antecedent increases, and increase when it di minishes. And the proportion, in either case, may be quite simple or extremely complicated.
If several instances of a phe nomenon have one relevant antecedent in common, then the com mon antecedent is causally connected with that phenomenon. Symbolically, abcd... --> wxyz.. .;bdfg...—>xzst .; d f k 1 s p r . . ;.• .d--*z. It should be noted
fully that the instances required for the application of this method are instances that are as different from each other as possible ex cept in regard to the common factor or antecedent which is under investigation. Just as the difference between instances is only significant when the instances are very similar in other respects, so the agreement or similarity between instances is significant only when the instances are as different as possible in other re spects. A thousand instances which agree in everything would be no better, no more instructive, than one of them. The main pre cautions to be borne in mind when applying the method of agree ment are the following: (I) To make sure that no relevant cir cumstance is overlooked; (2) Not to regard different consequents as similar, and assign them all to a common antecedent, merely because they may all serve more or less the same practical pur pose; (3) To remember that the antecedent and consequent may have no direct causal connection, but may both be the conse quents of some other antecedents.