Mill's methods

John Stuart Mill (1806-1873) was an English philosopher who wrote on a wide range of topics ranging from language and science to political philosophy. The so-called "Mill's methods" are five rules for investigating causes that he has proposed.
§ S05.1 The Method of Agreement |
The best way to introduce Mill's methods is perhaps through an example. Suppose your family went out together for a buffet dinner, but when you got home all of you started feeling sick and experienced stomach aches. How do you determine the cause of the illness? Suppose you draw up a table of the food taken by each family member : Member / Food taken | Oyster | Beef | Salad | Noodles | Fallen ill? | Mum | Yes | Yes | Yes | Yes | Yes | Dad | Yes | No | No | Yes | Yes | Sister | Yes | Yes | No | No | Yes | You | Yes | No | Yes | No | Yes |
Mill's rule of agreement says that if in all cases where an effect occurs, there is a single prior factor C that is common to all those cases, then C is the cause of the effect. According to the table in this example, the only thing that all of you have eaten is oyster. So applying the rule of agreement we infer that eating oyster is the cause of the illnesses.
§ S05.2 The Method of Difference | |
Now suppose the table had been different in the following way: Member / Food taken | Oyster | Beef | Salad | Noodles | Fallen ill? | Mum | Yes | Yes | Yes | Yes | Yes | Dad | Yes | Yes | Yes | Yes | Yes | Sister | Yes | Yes | Yes | Yes | Yes | You | Yes | Yes | No | Yes | No |
In this particular case you are the only one who did not fall ill. The only difference between you and the others is that you did not take salad. So that is probably the cause of the others' illnesses. This is an application the method of difference. This rule says that where you have one situation that leads to an effect, and another which does not, and the only difference is the presence of a single factor in the first situation, we can infer this factor as the cause of the effect.
§ S05.3 The Joint Method

The joint method is a matter of applying both the method of agreement and the method of difference, as represented by the diagram above. So application of the joint method should tell us that it is the beef which is the cause this time. Member / Food taken | Oyster | Beef | Salad | Noodles | Fallen ill? | Mum | Yes | Yes | Yes | Yes | Yes | Dad | Yes | Yes | No | Yes | Yes | Sister | Yes | Yes | Yes | No | Yes | You | Yes | No | No | Yes | No |
§ S05.4 The Method of Concomitant Variation

The method of concomitant variation says that if across a range of situations that lead to a certain effect, we find a certain property of the effect varying with variation in a factor common to those situations, then we can infer that factor as the cause.
Thus using the same kind of example, we might find that you felt somewhat sick having eaten one oyster, whereas your sister felt rather not well having eaten a few, and your father became critically ill having eaten ten in a row. Since the variation in the number of oysters corresponds to variation in the severity of the illness, it would be rational to infer that the illnesses were caused by the oysters.
§ S05.5 The Method of Residues | |
According to the method of residues, if we have a range of factors believed to be the causes of a range of effects, and we have reason to believe that all the factors, except one factor C, are causes for all the effects, except one, then we should infer that C is the cause of the remaining effect.
§ S05.6 General comments on Mill's methods
Mill's methods should come as no surprise, as these rules articulate some of the principles we use implicitly in causal reasoning in everyday life. But it is important to note the limitations of these rules. * First, the rules presuppose that we have a list of candidate causes to consider. But the rules themselves do not tell us how to come up with such a list. In reality this would depend on our knowledge or informed guesses about likely causes of the effects. * The other assumption presupposed by these methods is that among the list of factors under consideration, only one factor is the unique cause of the effect. But there is no guarantee that this assumption always holds. Also, sometimes the cause might be some complicated combinations of various factors.