Mill’s methods procedures for discovering necessary conditions, sufficient conditions, and necessary and sufficient conditions, where these terms are used as follows: if whenever A then B (e.g., whenever there is a fire then oxygen is present), then B is a necessary (causal) condition for A; and if whenever C then D (e.g., whenever sugar is in water, then it dissolves), then C is a sufficient (causal) condition for D. Method of agreement. Given a pair of hypotheses about necessary conditions, e.g., (1) whenever A then B1 whenever A then B2, then an observation of an individual that is A but not B2 will eliminate the second alternative as false, enabling one to conclude that the uneliminated hypothesis is true. This method for discovering necessary conditions is called the method of agreement. To illustrate the method of agreement, suppose several people have all become ill upon eating potato salad at a restaurant, but have in other respects had quite different meals, some having meat, some vegetables, some desserts. Being ill and not eating meat eliminates the latter as the cause; being ill and not eating dessert eliminates the latter as cause; and so on. It is the condition in which the individuals who are ill agree that is not eliminated. We therefore conclude that this is the cause or necessary condition for the illness. Method of difference. Similarly, with respect to the pair of hypotheses concerning sufficient conditions, e.g., (2) whenever C1 then D whenever C2 then D, an individual that is C1 but not D will eliminate the first hypothesis and enable one to conclude that the second is true. This is the method of difference. A simple change will often yield an example of an inference to a sufficient condition by the method of difference. If something changes from C1 to C2, and also thereupon changes from not- D to D, one can conclude that C2, in respect of which the instances differ, is the cause of D. Thus, Becquerel discovered that burns can be caused by radium, i.e., proximity to radium is a sufficient but not necessary condition for being burned, when he inferred that the radium he carried in a bottle in his pocket was the cause of a burn on his leg by noting that the presence of the radium was the only relevant causal difference between the time when the burn was present and the earlier time when it was not. Clearly, both methods can be generalized to cover any finite number of hypotheses in the set of alternatives. The two methods can be combined in the joint method of agreement and difference to yield the discovery of conditions that are both necessary and sufficient. Sometimes it is possible to eliminate an alternative, not on the basis of observation, but on the basis of previously inferred laws. If we know by previous inductions that no C2 is D, then observation is not needed to eliminate the second hypothesis of (2), and we can infer that what remains, or the residue, gives us the sufficient condition for D. Where an alternative is eliminated by previous inductions, we are said to use the method of residues. The methods may be generalized to cover quantitative laws. A cause of Q may be taken not to be a necessary and sufficient condition, but a factor P on whose magnitude the magnitude of Q functionally depends. If P varies when Q varies, then one can use methods of elimination to infer that P causes Q. This has been called the method of concomitant variation. More complicated methods are needed to infer what precisely is the function that correlates the two magnitudes. Clearly, if we are to conclude that one of (1) is true on the basis of the given data, we need an additional premise to the effect that there is at least one necessary condition for B and it is among the set consisting of A1 and A2. The existence claim here is known as a principle of determinism and the delimited range of alternatives is known as a principle of limited variety. Similar principles are needed for the other methods. Such principles are clearly empirical, and must be given prior inductive support if the methods of elimination are to be conclusive. In practice, generic scientific theories provide these principles to guide the experimenter. Thus, on the basis of the observations that justified Kepler’s laws, Newton was able to eliminate all hypotheses concerning the force that moved the planets about the sun save the inverse square law, provided that he also assumed as applying to this specific sort of system the generic theoretical framework established by his three laws of motion, which asserted that there exists a force accounting for the motion of the planets (determinism) and that this force satisfies certain conditions, e.g., the action-reaction law (limited variety).
The eliminative methods constitute the basic logic of the experimental method in science. They were first elaborated by Francis Bacon (see J. Weinberg, Abstraction, Relation, and Induction, 1965). They were restated by Hume, elaborated by J. F. W. Herschel, and located centrally in scientific methodology by J. S. Mill. Their structure was studied from the perspective of modern developments in logic by Keynes, W. E. Johnson, and especially Broad.
See also CAUSATION, CONFIRMATION, GRUE PARADOX , INDUCTION , PHILOSOPHY OF SCI – ENC. F.W.