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Mill’s methods

John Stuart Mill in his System of Logic (1843) developed five methods of inference for the purpose of understanding the grounds upon which it is appropriate to make judgments about a single cause when there are multiple cases. The five methods include: method of agreement, method of difference, joint method, method of residues, and the method of concomitant variation.

1.      Mill's method 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, and then C is the cause of the effect.

 

The procedure for the method of agreement is:

1. Collect all the cases with the same effect
2. Compare the effect to the rules to create a list of candidate rules
3. Compare the candidate rules to the situation to find a relevant rule
4. Compare the causal side of the rule to each case to create a list of possible causes
5. Compare the possible causes for each case to find one common cause for all cases
6. If there is a single common cause then it is the probable cause

 

For example: Suppose your family went out together for a buffet dinner, but when you got home all of you started feeling sick and experienced stomachaches. How do you determine the cause of the illness?

1. Tom: I have a stomachache.
2. Harry: Who else has a stomachache?
3. Tom: mom, dad, and sis.
4. Harry: Did you all eat at the same place recently?
5. Tom: Yes, we ate a buffet dinner together.
6. Harry: What did you eat?
7. Tom: oyster and salad.
8. Harry: What did mom eat?
9. Tom: oyster, beef, salad and noodles.
10. Harry: What did dad eat?
11. Tom: oyster and noodles.
12. Harry: What did sis eat?
13. Tom: oyster and beef.
14. Harry: since everyone ate oyster, oyster is the probable cause of the stomachache.

Suppose you draw up a table of the food taken by each family member:

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.

 

 2.      The method of difference 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.  

The procedure for the method of difference is:

1. Collect all cases with the same event
2. Compare the effect to rules to create list of candidate rules
3. Compare the candidate rules to the situation to find a relevant rule
4. Compare causal side of rule to recent events to create list of possible causes
5. Compare possible causes for each case
6. If all causes are common except one, and all effects are in common except one then the exception is the probable cause

For example: Suppose your family went out together for a buffet dinner, but when you got home some of you started feeling sick and experienced stomachaches. How do you determine the cause of the illness?

1.  Tom: Some of my family has stomachaches.
2. Harry: Who has a stomachache?
3. Tom: mom, dad, and sis.
4. Harry: Do you have a stomachache?
5. Tom: No.
6. Harry: Did you all eat at the same place recently?
7. Tom: Yes, we ate a buffet dinner together.
8. Harry: What did you eat?
9. Tom: oyster, beef and noodles.
10. Harry: What did mom eat?
11. Tom: oyster, beef, salad and noodles.
12. Harry: What did dad eat?
13. Tom: oyster, beef, salad and noodles.
14. Harry: What did sis eat?
15. Tom: oyster, beef, salad and noodles.
16. Harry: since everyone who ate salad has stomachaches and you did not eat salad and you don’t have a stomachache, salad is the probable cause of the stomachaches.

Now let’s see what the table looks like:

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.

 

 3.              The joint method of agreement and difference combines the two previous methods. 

The procedure for the joint method is:

1.      Collect all cases with the same event

2.      Compare the effect to rules to create list of candidate rules

3.      Compare the candidate rules to the situation to find a relevant rule

4.      Compare causal side of rule to recent events to create list of possible causes

5.      Compare possible causes for each case

6.      If a common cause matches the common effects and that cause is not in common with the negative effect, then the common cause is the probable cause

 

For example: Suppose your family went out together for a buffet dinner, but when you got home some of you started feeling sick and experienced stomachaches and the method of difference failed to give you the cause. How do you determine the cause of the illness?

1. Tom: Some of my family has stomachaches.
2.  Harry: Who has a stomachache?
3. Tom: mom, dad, and sis.
4. Harry: Do you have a stomachache?
5. Tom: No.
6.  Harry: Did you all eat at the same place recently?
7. Tom: Yes, we ate a buffet dinner together.
8.  Harry: What did you eat?
9. Tom: oyster and noodles.
10. Harry: What did mom eat?
11. Tom: oyster, beef, salad and noodles.
12. Harry: What did dad eat?
13. Tom: oyster, beef and noodles.
14. Harry: What did sis eat?
15. Tom: oyster, beef and salad.
16. Harry: since everyone who ate beef has stomachaches and you did not eat beef and you don’t have a stomachache, beef is the probable cause of the stomachaches.

Now let’s see what the table looks like:

The joint method is a matter of applying both the method of agreement and the method of difference, as represented by the table above. So application of the joint method should tell us that it is the beef that is the cause this time.

 

 NOTE: this residues example is incorrect and needs to be fixed.

 4.              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.   

The procedure for the method of residues is:

1.      Compare the effect to rules to find a relevant rule

2.      Compare causal side of rule to this case to create list of possible causes

3.      Collect all cases with counterexamples of each cause

4.      If the counterexamples eliminate all causes but one, then the remaining cause is the probable cause

 

For example: Suppose your family went out together for a buffet dinner, but when you got home you started feeling sick and experienced a stomachache. How do you determine the cause of the illness?

1. Tom: I have a stomachache.
2. Harry: What have you eaten recently?
3. Tom: oyster, beef, salad and noodles.
4. Harry: Who did not get a stomachache?
5. Tom: dad, mom and sis.
6. Harry: What did dad eat?
7. Tom: dad ate oyster.
8. Harry: What did mom eat?
9. Tom: mom ate beef.
10. Harry: What did sis eat?
11. Tom: sis ate salad.
12.  Harry: since you ate oyster, beef, salad and noodles.and you have a stomachache, and others ate oyster, beef, and salad and did not get a stomachache,  then the probable cause of your stomachache is noodles.

Now let’s see what the table looks like:

Application of the method of residues tells us the noodles are the cause this time.

 

 

 5.              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.  

The procedure for the method of concomitant variation is:

1.      Compare the effect to rules to find a relevant rule

2.      Compare causal side of rule to this case to create list of possible causes

3.      Collect all cases with the same possible cause

4.      Measure the effect for each case

5.      Measure the cause for each case

6.      Compare the measurements for each cause and effect

7.      If the amount of change in the effect is relative to the amount of change in cause for each case, then it is the probable cause

 

For example: Suppose your family went out together for a buffet dinner, then you felt somewhat sick having eaten one oyster. How do you determine the cause of the illness?

 

1. Tom: I have a mild stomachache.
2. Harry: What have you eaten recently?
3. Tom: I ate one oyster.
4. Harry: Who else has a stomachache?
5. Tom: dad and sis.
6. Harry: What did dad eat?
7. Tom: dad ate ten oysters.
8. Harry: What did sis eat?
9. Tom: sis ate five oysters.
10. Harry: How bad is dad’s stomachache?
11. Tom: Severe.
12. Harry: How bad is sis’s stomachache?
13. Tom: Moderate..
14. Harry: since everyone ate a different amount of oysters, and each person’s  stomachache pain varied relative to the amount each ate,  then the probable cause of the stomachaches is oysters.

Now let’s see what the table looks like:

Application of the method of concomitant variation tells us oysters are the cause of the stomachaches.

General comments on Mill's methods

One assumption is the list of possible causes and effects come from rules inferred using inductive generalization from prior observations.

 

The other assumption 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.

 

Fault Diagnosis

Abductive reasoning is used for cause identification, and fault diagnosis. Answering questions like “why do you think X”, “what caused X”, or “what could explain X”.The methods used here are for concept understanding, and are not exhaustive, do not deal with complex ambiguity,do not relate the implications of inductive methods,do not deal with universal generalization, anddo not address enthymemes.For more about abduction see: Aristotle, Peirce, Josephson, and Wikipedia.Ultra Hal Assistant developers see plug-in code page.

General Process

1. learn cause - effect relationships (induction)

• a.       implication (A implies B)  
• b.      implication (B implies C)
• c.       implication (A implies D)
• d.      implication (E implies B)
• e.       implication (E implies G)
• f.        etc.

2.       learn contradictions (induction)

• a.       disjunction (F or E)   
• b.      etc.

3.       learn condition - process relationships

4.       create condition expectation

 

5.       Monitor condition expectation

 

6.       detect variance (actual vs expected)

 

7.       specify conditions

• a.       B, C, F, not G

8.       gather candidates (abduction)

• a. reverse modus ponens (A implies B; B therefore    possibly A)
• b.reverse modus ponens (E implies B; B therefore    possibly E)
• c. reverse hypothetical syllogism (A implies B and B    implies C; C therefore possibly A)
• d.candidate = possibly A, E

9.       reduce candidates (deduction)

• a.       disjunctive syllogism (F or E; F therefore not E)
• b.      modus tollens (E implies  G;  not G therefore not E)
• c.       hypothesis = possibly A

10.   explain variance (unexpected observation)

• a.       B, C, F, not G explain A

11.   verify explanation (Test)

• a.       cause is B, C, F, and not G.

 

12.   update cause - effect relationships (induction)

 

13.    correct variances

 

14.   determine root causes

 

15.   update cause - effect relationships (induction)

 

16.   update condition - process relationships

 

17.   prevent root causes

 

 

Example of diagnostic process

Condition statement (the unexpected condition.)

Tom is cranky, Why?

 

situation knowledge (observations)           

 the person is cranky, the person is wet, the person is outside, the shower floor is not wet 

 

experience knowledge

• 1.       if it rained on the person then that person is wet
• 2.       if the person is wet then that person is cranky
• 3.       if the person showered then that person is inside
• 4.       if the person showered then that person is wet
• 5.       if the person showered then the shower floor is wet
• 6.       either the person is outside or the person is inside 

abductive reasoning

• 1.       reverse modus ponens: the person is wet plus  if it rained on the person then the person is wet therefore possibly it rained on the person.
• 2.       reverse modus ponens: the person is wet plus  if a person showered then the person is wet therefore possibly the person showered.
• 3.       reverse hypothetical proposition: the person is cranky plus  if it rained on the person then the person is wet plus  if the person is wet then the person is cranky therefore possibly it rained on the person. 

Candidates:

• 1.       possibly it rained on the person
• 2.       possibly the person showered 

deductive reasoning

• 1.       disjunctive proposition: the person is outside plus  either the person is outside or the person is inside therefore the person is not inside
• 2.       modus tollens: the shower floor is not wet plus  if the person showered then the floor is wet therefore the person showered is not true.
• 3.       modus tollens: the person is not inside plus  if the person showers then the person is inside therefore the person showered is not true. 

Conclusion: the person is cranky, the person is wet, the person is outside, the shower floor is not wet are explained by it rained on the person.

therefore:. Tom is cranky because possibly it rained on him.