1
The method of agreement
Take the cases in which the effect occurred and ask which factor appeared in all of them. A factor missing from even one such case is eliminated.
Lesson · Science, Induction and Causation · Causation
Mill's Methods and the Common Factor
Mill's method of agreement looks for the factor shared by every case with the effect, and his method of difference for the one factor that separates a case with the effect from a case without it. A case table makes the search mechanical: the positive cases keep what they share, the negative cases eliminate what they contain, and the lab lists every one- or two-factor formula that fits. A fit is a candidate for the cause; the table cannot rule out the others that fit, or a factor it does not list.
agreement: shared by every positive case
difference: a well case with it eliminates it
a fit is a candidate, not yet a cause
two causes: no single factor fits
Searching a table for a cause
Core ideas
What this lesson establishes
The earlier lessons asked what evidence does to a hypothesis. This one asks how to find a hypothesis about a cause in a table of cases.
2
The method of difference
Mill's method of difference sets a case with the effect beside one as like it as possible without the effect, and blames the one factor in which they differ. A table rarely holds such a pair, and what it offers instead is elimination: a factor present in a case without the effect cannot be what was sufficient for it.
3
A fit is a candidate
A factor that survives both is the only one the table leaves, among the factors the table lists. Whether it is the cause depends on what the table did not list and on whether one cause is all there is.
The material
Five diners and five dishes
A table first, then the two methods applied to it by hand, then what the lab's search adds.
Five people ate at the same dinner and two of them fell ill. The table records which of five dishes each of them ate, as 1 for eaten and 0 for not, and the verdict, 1 for ill. The dishes are oysters O, soup S, bread B, wine W and cake C.
diner O S B W C | ill
---------------------------------
Ann 1 1 1 0 1 | 1
Ben 1 0 1 1 0 | 1
Cy 0 1 1 1 1 | 0
Di 0 1 0 0 1 | 0
Eve 0 0 1 0 0 | 0
Begin with the method of agreement, which looks only at the ill diners. Ann ate oysters, soup, bread and cake; Ben ate oysters, bread and wine. The dishes both ate are oysters and bread, so soup, wine and cake are eliminated.
Then turn to the well diners. Mill's method of difference would look here for a pair of cases alike in everything but one dish, and the table has none: Ann and Cy, the closest pair, differ in the oysters and in the wine. What the well diners offer instead is elimination. Cy ate bread and did not fall ill, so, if one dish is responsible, bread is not it. No well diner ate oysters. Oysters survive both steps, and nothing else does.
Definition · The joint method
Mill's joint method of agreement and difference applies agreement twice: the cases with the effect agree in having the factor, and the cases without it agree in lacking it. In the lab's terms, it is a definition that agrees with the verdict on every row.
The lab does this search for any table. With the definition O it agrees on all five rows. With the search switched to pairs it also lists every formula built from one factor or two, shows the first one it found, and says how many there are. Two fit: O, and O ∧ B, oysters together with bread.
Theorem · A fit is a candidate
If a formula agrees with the verdict on every row, then the table does not eliminate it. If the table lists every relevant factor, and the effect has a single cause, a formula of one factor that fits is that cause.
Each of those conditions can fail, and the lab shows the first. The formula O ∧ B fits because both ill diners ate bread, even though a well diner did too: Cy ate bread without oysters, so bread alone fails, and bread with oysters has not been excluded. The table has only two ill diners, so it cannot separate oysters from oysters with bread. Its other limits are not visible in it at all: the knife, the water or the oysters' source are not columns, and a table cannot eliminate a factor it does not contain.
The second preset removes the other assumption, that there is one cause. Here prawns P or eggs E made a diner ill, and no diner ate both. The lab starts with the suspect P, which misses Ben. No single factor fits, because prawns miss Ben and eggs miss Ann and Eve, and the agreement method returns nothing common to the ill. The search over pairs finds the disjunction P ∨ E. Mill called this a plurality of causes, and his methods were not designed for it.
The lab's limit is that it checks a formula against the cases it is given. It cannot say whether the table is complete, or whether the cases were chosen fairly, or whether a factor that is always found with the effect is a cause of it or a sign of something else. The last of these is the subject of the next lesson.
Interactive
The method of cases
A definition tested against every case in the table
Method
Searching a case table
Four steps, in this order, for any table of cases and an effect.
1
Apply agreement to the positive cases
Keep only the rows with the effect and list the factors present in every one of them. Everything else is eliminated.
2
Eliminate by the negative cases
Of the factors that remain, remove any that appears in a row without the effect. This is what the method of difference becomes when no two rows are alike in all but one factor, and what is left fits every row.
3
List every formula that fits
Run the search over single factors and pairs, and read the count. More than one fit means the table is too small to say which is operating.
4
Say what the table cannot rule out
Name a factor the table does not list, and a case that would separate the formulas that fit. Those are the next two things to collect.
From a table to a candidate
Score 0 / 0Choose the answer you can justify.
Oysters at the dinner
Five diners, five dishes; Ann and Ben ill, Cy, Di and Eve well.
Ann ate O, S, B, C; Ben ate O, B, W
agreement: dishes in both = O, B
well diners: Cy ate B and stayed well, so B is out
no well diner ate O, so O stays
formulas that fit every row: O, and O ∧ B
The oysters are the candidate. The table does not distinguish oysters from oysters together with bread, since both ill diners ate bread, and it says nothing about a factor it does not list. To tell the two apart you need a case with oysters and no bread.
Common mistakes
1
Thinking the factor common to the cases is the causeIn the five-diner table bread is common to both ill diners and still not what the table points to, since Cy ate it and stayed well. Oysters survive both methods and are still only a candidate: O ∧ B fits the same rows, and neither the knife nor the water is a column.
2
Expecting every effect to have one factor that explains itThe prawns-or-eggs table has no single factor that fits, and the lab's pair search finds P ∨ E. An effect with two independent routes breaks the agreement method without breaking the table.
3
Treating a candidate that survives the well cases as provedAbsence from three well diners eliminates every dish they ate. It does not show that the dish that remains has any effect: one more well diner who ate the oysters would remove it, and the table cannot say that none exists.
Check your understanding
✓
Finish when you can apply both methods to a table and state what the result leaves open.
Given a table of cases with a verdict, you should be able to apply agreement and difference, list the formulas that fit, and name one factor not in the table and one case that would separate the formulas that fit.
Note: The lab searches formulas of at most two factors, from the six conditions it allows. A table that fits only a longer formula will report no candidates, and that is a limit of the search and not a statement that no cause exists.