Method of Agreement Structure
The residue method can be interpreted as a variant of the difference method, in which the negative instance is not observed, but constructed on the basis of already known causal laws. However, in the method of difference, the observation of 1.2 (or, if negations are allowed, those of 2.2) continues to give results, although the conclusions become less complete, that is, the cause is less and less completely specified. For example, in 8.2, where we assume that there is a necessary and sufficient condition for P in F, which may be one of the possible causes, or a negation of one or a combination of possible causes or negations, or a disjunction of possible causes or negations, or conjunctions of possible causes or negations – which in fact makes it possible to construct the real condition in any way from the possible causes – the observation of 2.2 leads to the conclusion that the required condition (A. or. ). that is, it is either A itself, or a conjunction that contains A, or a disjunction in which one of the disjunctions is A itself or a conjunction that contains A. Since any such disjunction is a sufficient condition under a necessary and sufficient condition, this observation, in which the presence of A in I1 is the only potentially relevant difference between I1 and N1, even with the least rigorous way of assuming that A is at least a necessary part of a sufficient condition for P in F – is the sufficient condition (A…). 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 the fact that e.B. BC was the (sufficient or) necessary and sufficient requirement, because if C was not present in N1 and B in N2, then BC as a whole might 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. The use of the above methods is not limited to cases where we start with a question of the form “What is the cause of this or that?” We might as well start with the question: “What is the effect of this or that?” For example: “What effect does it have to apply high voltage to the electrodes of a vacuum tube?” But we are entitled to claim that what is observed is an effect only if the requirements of the corresponding variant of the method of difference are met. So we find that although we have to recognize very different variants of these methods depending on the different types of assumptions used, and although the reasoning that validates the simplest variants fails when it is allowed that different negations and combinations of factors are the root cause, there are still valid demonstrative methods that use even the least rigorous form of hypothesis. that is, which only assume that there is a necessary and sufficient condition for P in F, which somehow consists of a certain limited set of possible causes. But with such a hypothesis, we must be content either to draw (around 8.2) a very incomplete conclusion from the classical observation of differences, or (by 8.12, 8.14, the combination of these two or 8.4) to draw more complete conclusions only from a large number of cases where the possible causes are present or absent in a systematically different way. The difference method, on the other hand (4.2), still requires only the observation of 1.2; This eliminates all possible causes except for A and all disjunctions that do not contain A, either as insufficient because they are present in N1 or as unnecessary because they are missing in I1.
The only disjunctions that are not eliminated are those that occur in I1 but not in N1, and these must contain A. Thus, this observation with this hypothesis shows that a necessary and sufficient condition (A or…) is, that is to say either A itself, or a disjunction containing A, the other disjunctions being possible causes missing in N1. This, of course, means that A itself, the factor thus distinguished, can only be a sufficient condition for P. A less rigorous hypothesis would be that at F, the size of P depends in one way or another completely on the size of one or more factors X, X′, X”, etc., each of the really relevant factors being identical to one of the possible causes A, B, C, D, E. If, given this, we again observe that P varies, while, for example, A varies, but B, C, D, E remain constant, this does not show that B, for example, does not work with X, etc. may be identical; that is, it does not show that variations in B are causally unrelated to P. All it shows is that the size of P does not completely depend on a set of factors that A does not contain, because each of these amounts has remained constant, while P has varied. This leaves open the full cause of P in F could be A itself or a number of factors such as (A, B, D), which also include A and some of the others.
All we know is that List A must be included. So this observation and hypothesis show that a complete cause of P is in F (A, …); this means that A is actually a relevant factor and there may or may not be others. Repeated applications of this method could fill in other factors, but would not close the list. (And yet, another task that needs to be done through another type of survey is to understand how the size of P depends on those factors that have thus proven to be actually relevant.) The common method, with this weaker hypothesis, requires reinforced observation in the same way: that is, each of the possible causes, with the exception of A, must be present in both a positive and a negative case or be absent in a positive case and a negative case, then this variant (2.3) always leads to the conclusion, that A is both necessary and sufficient. In the corresponding variant of the common method (3.3), for the same reason as in 3.14, we need a single negative instance instead of the set of Sn, and the cause is given only in the form (A…). To avoid unnecessary complications, suppose that the conclusion reached by any application of the method of agreement or difference is to have the form “This and that is a cause of this and that type of event or phenomenon”. For a formal study of these methods and the common method, we could consider a cause as a necessary and sufficient effect condition – or in some cases only as a necessary condition or only as a sufficient condition – where to say that X is a necessary condition for Y means only to say that wherever Y is present, X is present, or in short, that all are Y X; and to say that X is a sufficient condition for Y only means to say that wherever X is present, Y is present, or in short, that all X are Y. . .