Saylor.org's Comparative Politics/John Stuart Mill’s Methods of Comparision
Causal Reasoning 
by Garth Kemerling
Another common variety of inductive reasoning is concerned with establishing the presence of causal relationships among events. When we have good reason to believe that events of one sort (the causes) are systematically related to events of some other sort (the effects), it may become possible for us to alter our environment by producing (or by preventing) the occurence of certain kinds of events.
But what constitutes adequate evidence of causation? Although we typically use conditional statements to express our causal beliefs, the logical connective known as material implication seems to capture only a part of what we have in mind. Even regarding the cause as a necessary and sufficient condition for the effect doesn't cover all of our concept of causation. It may be that there is less here than meets the eye; David Hume pointed out that our causal beliefs are unjustifiable, even if they come naturally.
The fundamental presumption behind our concept seems to be that causal connections are lawful (or at least lawlike); they involve some kind of uniformity or reqularity in the natural world. Certainly it is by observing some uniform pattern in the occurrence of events—the regular appearance of the effect following its cause—that we come to expect that the cause will invariably be followed by the effect.
Mill's Methods 
Philosopher John Stuart Mill devised a set of five careful methods (or canons) by means of which to analyze and interpret our observations for the purpose of drawing conclusions about the causal relationships they exhibit.
In order to see how each of the five methods work, let's consider their practical application to a specific situation. Suppose that on an otherwise uneventful afternoon, the College Nurse becomes aware that an unusual number of students are suffering from severe indigestion. Ms. Hayes naturally suspects that this symptom results from something the students ate for lunch, and she would like to find out for sure. The Nurse wants to find evidence that will support a conclusion that "Eating ?xxxx? causes indigestion." Mill's Methods can help.
Method of Agreement 
Suppose that four students come to Ms. Hayes with indigestion, and she questions each about what they had for lunch. The first had pizza, coleslaw, orange juice, and a cookie; the second had a hot dog and french fries, coleslaw, and iced tea; the third ate pizza and coleslaw and drank iced tea; and the fourth ate only french fries, coleslaw, and chocolate cake. Ms. Hayes, of course, concludes that "Eating coleslaw caused the indigestion."
This is an application of Mill's Method of Agreement: investigation of the cases in which the effect occurred revealed only one prior circumstance that all of them shared. Our customary notion here is that similar effects are likely to arise from a similar cause, and since everyone who fell ill had eaten coleslaw, it was probably the cause.
Method of Difference 
On the other hand, suppose that only two students arrive at the Nurse's office. The two are roommates who ate together, but one became ill while the other did not. The first had eaten a hot dog, french fries, coleslaw, chocolate cake, and iced tea, while the other had eaten a hot dog, french fries, chocolate cake, and iced tea. Again, Ms. Hayes concludes that the coleslaw is what made the first roommate ill.
This reasoning applies Mill's Method of Difference: comparison of a case in which the effect occurred and a case in which the effect did not occur revealed that only one prior circumstance was present in the first case but not it the second. In such situations, we commonly suppose that, other things being equal, different effects are likely to arise from different causes, and since only the student who had eaten coleslaw became ill, it was probably the cause.
Joint Method of Agreement and Difference 
Now put these two situations together by assuming that eight students come to Ms. Hayes: four of them suffered from indigestion, and with each of these four there is another who did not. Each pair of students had exactly the same lunch, except that everyone in the first group ate coleslaw and no one in the second group did. The Nurse arrives at the same conclusion.
This situation is an example of Mill's Joint Method of Agreement and Difference: the first four students are evidence that everyone who got ill had eaten coleslaw, and the four matching pairs are evidence that only those who got ill had eaten coleslaw. This is a powerful combination of the first two methods, since it tends to support our notion that genuine causes are necessary and sufficient conditions for their effects.
Method of Concomitant Variation 
Change the situation again. Suppose that the Nurse sees five students: the first ate no coleslaw and feels fine; the second had one bite of coleslaw and felt a little queasy; the third had half a dish of coleslaw and is fairly ill; the fourth ate a whole dish of coleslaw and is violently ill; and the fifth ate two servings of coleslaw and had to be rushed to the hospital. The conclusion is again that coleslaw caused the indigestion.
This is an example of Mill's Method of Concomitant Variation: the evidence appears to show that there is a direct correlation between the degree to which the cause occurred and the degree to which the effect occurred. This conforms to our ordinary supposition that effects are typically proportional to their causes. In effect, this is a sophisticated version of the Joint Method, one in which we notice not just the occurrence or non-occurrence of the causal terms, but the extent to which each of them took place.
Method of Residues 
Finally, suppose that Ms. Hayes, during prior investigations of student illness, has already established that pizza tends to produce a rash and iced tea tends to cause headaches. Today, a student arrives at the Nurse's office complaining of headache, indigestion, and a rash; this student reports having eaten pizza, coleslaw, and iced tea for lunch. Since she can account for most of the student's symptoms as the effects of known causes, Ms. Hayes concludes that the additional effect of indigestion must be caused by the additional circumstance of eating coleslaw.
This pattern of reasoning exemplifies Mill's Method of Residues: many elements of a complex effect are shown to result, by reliable causal beliefs, from several elements of a complex cause; whatever remains of the effect must then have been produced by whatever remains of the cause. Notice that if we suppose the truth of all of the causal relationships involved, this method becomes an application of deductive reasoning.
As a general qualification on the reliability of these Methods, notice that the issue of relevance is again crucial. Our Nurse began with the assumption that what students had eaten for lunch was relevant to their digestive health in the afternoon. That's a reasonable guess, but of course the real cause could have been something else entirely, something about which the Nurse never thought to ask. No matter how much evidence we gather, inductive reasoning cannot achieve perfect certainty.
Discovery or Proof? 
Although Mill's Methods are an important component of serious investigation of natural phenomena, they have significant limitations. Careful application of these methods succeeds only when every relevant antecedent circumstance is taken into account, and that is impossible to guarantee in advance.
If our chief aim in applying Mill's Methods were to discover the unknown cause of some observed event, then they would fail us precisely when we need them most. The Methods show us how to identify the likely cause from among those possibilities that we have considered among the relevant antecedent circumstances of this effect. But the most interesting cases will be those in which the cause we seek lies in some unsuspected source, which we have probably excluded from our analysis of the antecedent circumstances as irrelevant. Thus, Mill's Methods can't help us to discover causes unless we already know (roughly) what those causes are likely to be.
If the goal of using Mill's Methods were to prove that one event is the cause of another, we would fare even worse. Our inability to consider every possible circumstance (even those that seem irrelevant to us) will often lead us to mistake the true cause of an event. What is more, the Methods encourage the identification of single causes, overlooking the fact that many interesting effects may result from some complex combination of partial causes. At best, Mill's Methods can only help us to establish the presence of a correlation between the occurrences of distinct events, leaving any question about the reality of a causal connection unanswered. Used as proof, inductive reasoning generally cannot offer the same certainty that valid deductive reasoning provides.
Perhaps it is best to regard Mill's Methods more modestly, as tools we can employ in our efforts to confirm hypotheses about the natural world. If we have already proposed several specific hypotheses about what may be the cause of an observed event, then using the Methods will be helpful, since that will often enable us to eliminate most of the possible causes we have identified, and this tends to confirm the hypothesis that any remaining circumstance is likely to be the genuine cause.