Also simply called the “common method”, this principle simply represents the application of the methods of agreement and difference. We should expect to find analogs of concomitant variation of the agreement method and the difference method, i.e. ways of arguing for a causal relationship between P and, say, A, both from the observation of cases where P remains constant while A remains constant, but all other potentially relevant factors vary, and case observation, where P varies, while A varies, but all other potentially relevant factors remain constant. And indeed, there are methods of both types, but those of the second type, the analogues of the method of difference, are more important. So there will not be one method of agreement, one method of difference and one common method, but a series of variants of each. A complete overview of all possible methods of this type could be carried out, numbered as follows: A number from 1 to 8 before a decimal point indicates the type of hypothesis. Therefore, it is believed that there is a real cause (What Mill and his followers describe as the common method may not be this indirect method of difference, but a double method of agreement, in which a set of positive instances identifies a necessary condition and a number of negative instances a sufficient condition. Such a combination is redundant with a hypothesis of one of the first two species, but not if the hypothesis is further relaxed.) The simplest variants of the agreement method can only be used in the event that our prior knowledge narrowly limits the possible causes and justifies the belief that they will work individually. For example, if the nature of a disease indicates that it is of bacterial origin, the microorganism responsible can be identified by discovering that in a number of cases of the disease, there is only one species of microorganisms that are not yet known to be innocent.
Otherwise, the observation of what seems to be the only common factor in a number of cases of a phenomenon can only be used very provisionally to propose a hypothesis that must be tested in a different way. We obtain a variant of the agreement method (1.12) by combining with this hypothesis the following observation: A set of one or more positive instances, so that a possible cause, say A, is present in each case, but for every other possible cause there is an instance of which this cause is missing. This leads to the conclusion that A for P in F is necessary and sufficient. Next, we consider a third-type assumption that the requirement is either a possible cause or a combination of possible causes. (The latter possibility seems to be at least part of what Mill meant by “a mixture of effects.”) This possibility does not affect the positive method of agreement, because if a conjunction is necessary, each of its conjunctions is necessary, and therefore the candidates can be eliminated as before. But since conjunctions in a necessary and sufficient state may not be sufficient individually, the negative method of agreement, as explained above, will not work. The observation of (1.13 or) 1.14 would now leave open this e.B. BC was the (sufficient or) necessary and sufficient requirement, because if C in N1 and B in N2 were not present, then BC as a whole could still be sufficient: it would not be eliminated by any of these cases. This method now needs (in 3.14) a stronger observation, namely a single negative instance N1, in which a possible cause, say A, is missing, but all other possible causes are present.
This will show that no possible cause or conjunction of possible causes that does not contain A is sufficient for P in F. But even this does not show that the requirement is A itself, but only that it is either A itself or a conjunction in which A is a conjunction. We can express this by saying that the cause is (A…), where the dots indicate that other conjunctions may be part of the condition, and the points are underlined while A should not indicate that A must appear in the formula of the actual cause, but that other conjunctions may or may not occur. Repeating the sequence – for example, immersing another similar piece of litmus paper in the liquid – confirms the idea that the liquid caused the color change. But it is not true that we use the agreement method in this case; The repetition only makes it less likely that another change occurred to cause the color change at the same time as each of the two dives, which confirms our belief that instances are what using the difference method would require. .