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Lecture 6 Mill’s Methods. WANG Huaping Philosophy Department, Shandong University. 3. 3. Overview. 1. 3. Mill’s Five Methods. 2. Problems. Contents. Overview. In order to identify the cause of some event or condition, we will use Mill’s Methods.

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Lecture 6Mill’s Methods

WANG Huaping

Philosophy Department, Shandong University







Mill’s Five Methods






  • In order to identify the cause of some event or condition, we will use Mill’s Methods.
  • Mill’s Methods are five rules of causal reasoning in inductive logic.
  • These rules are the bases of contemporary scientific methodology emphasizing careful observation and (controlled) experiments.
  • Attractions of Mill’s Methods:
  • Nicely captures the reasoning in controlled experiments and everyday causal reasoning.
  • Describes how our background beliefs constrain our inferences.


Start with

empirical data

Look for



Hypothesis =





Key concepts:

  • Necessary conditions: C is a necessary condition for E when it is the case that if C does not occur, then E will not occur, and whenever E occurs, C has also occurred. (But C can occur when E does not.)
  • Sufficient conditions: C is a sufficient condition for E when it is the case that if C is present, then E has to be present. (But E may be present when C is absent.)


(1) Igniting a mixture of nitroglycerin and sawdust is a __________ condition for creating an explosion.

(2) A fuel source is a __________ condition for an engine to run.

(3) Placing a block of ice in a bucket of warm water is a __________ condition for melting it.

(4) Throwing a brick through a glass window pane is a __________ condition for breaking the glass.


The Method of Agreement

Mill states his first method, the method of agreement, this way:

If two or more instances of the phenomenon under investigation have only one circumstance in common, the circumstance in which alone all the instances agree, is the cause … of the given phenomenon. (1874, p. 280)


The Method of Agreement

As an example, consider the following situation. Jack, Jill, and Joan have lunch at a local diner, and afterwards all three have food poisoning. For lunch, they variously had (A) iced tea, (B) water, (C) chicken, (D) french fries, and (E) soup.


The Method of Agreement

In the 1960 more than 100,000 young turkeys on poultry farms in England died in the course of a few months from an apparently new disease that was termed “Turkey X disease” . It was soon found that the difficulty was not limited to turkeys. Ducklings and young pheasants were also affected and heavy mortality was experienced .

A careful survey of the early outbreaks showed that they were all associated with feeds, namely Brazilian peanut meal . An intensive investigation of the suspect peanut meal was undertaken and it was quickly found that this peanut meal was highly toxic to poultry and ducklings with symptoms typical of Turkey X disease.


The Method of Agreement

Later, coats, cats and pigeons feed with the meal were soon found dead.

Speculations made during 1960 regarding the nature of the toxin suggested that it might be of fungal origin and was given the name Aflatoxin.

Further experiments made with rats, ferrets and fishes showed that aflatoxin causes serious liver damage and leads to liver tumors.

This conclusion is made in light of the Method of Agreement.


The Method of Difference

If an instance in which the phenomenon under investigation occurs, and an instance in which it does not occur, have every circumstance in common save one, that one occurring only in the former; the circumstance in which alone the two instances differ is … the cause, or an indispensable part of the cause of the phenomenon. (1874, p. 280)


The Method of Difference

According to the ancient thought, life could be formed from a mixture of sea water, mud and sunlight. The idea that living microorganisms could be spontaneously generated from air persisted until the middle of the 18th century. In the 1750s, John Needham, a Scottish clergyman and naturalist, claimed to have proved that spontaneous generation does occur when he showed that microorganisms flourished in certain foods such as soup broth, even after they had been briefly boiled and covered.


The Method of Difference

Several years later, the Italian abbot and biologist Lazzaro Spallanzani, boiled soup broth for over an hour and then placed bowls of this soup in different conditions, sealing some and leaving others exposed to air. Spallanzani found that microorganisms grew in the soup exposed to air but were absent from the sealed soup. He therefore challenged Needham’s conclusions and hypothesized that microorganisms suspended in air settled onto the exposed soup but not the sealed soup, and rejected the idea of spontaneous generation.


The Method of Difference

The debate over the spontaneous generation theory of life became so heated that in 1860, the French Academy of Sciences established the Alhumbert prize of 2,500 francs to the first person who could conclusively resolve the conflict.

In 1864, Louis Pasteur achieved that result with a series of well-controlled experiments and in doing so claimed the Alhumbert prize.

Pasteur repeated Spallanzani’s method of boiling soup broth, but he divided the broth into portions and exposed these portions to different controlled conditions.


The Method of Difference

Some broth was placed in flasks that had straight necks that were open to the air, some broth was placed in sealed flasks that were not open to the air, and some broth was placed into a specially designed set of swan-necked flasks, in which the broth would be open to the air but the air would have to travel a curved path before

reaching the broth, thus preventing anything that might be present in the air from simply settling onto the soup.


The Method of Difference

Pasteur then observed the response of the dependent variable in response to the independent variable.

If spontaneous generation did indeed occur upon exposure to air, Pasteur hypothesized, microorganisms would be found in both the swan-neck flasks and the straight-necked flasks, but not in the sealed flasks. Instead, Pasteur found that microorganisms appeared in the straight necked flasks, but not in the sealed flasks or the swan-necked flasks.

Pasteur’s work helped refute the theory of spontaneous generation - his experiments showed that air alone was not the cause of bacterial growth in the flask, and his research supported the hypothesis that live microorganisms suspended in air could settle onto the broth in open-necked flasks via gravity.


The Joint Method of Agreement and Difference

If two or more instances in which the phenomenon occurs have only one circumstance in common, while two or more instances in which it does not occur have nothing in common save the absence of that circumstance, the circumstance in which alone the two sets of instances differ, is … the cause, or an indispensable part of the cause, of the phenomenon. (1874, p. 284)

The joint method of agreement and difference extends the method of difference to the point that the method of agreement is also being employed. Here, the probable cause is the one antecedent circumstance that is always present when the phenomenon is present and always absent when the phenomenon is absent.


The Method of Agreement

Pasteur puzzled by the fact that hens were refractory to anthrax, he had wondered whether this might not be explained by their body temperature, which is higher than that of animals susceptible to this disease. To test his hypothesis, he inoculated hens with anthrax bacilli and placed them in a cold bath to lower their body temperature. Animals so treated died the next day, showing numerous bacilli in their blood and organs.


The Method of Agreement

Another hen, similarly infected and maintained in the cold bath until the disease was in full progress, was then taken out of the water, dried, wrapped, and placed under conditions that allowed rapid return to normal body temperature. Mirabile dictu, this hen made a complete recovery. Thus, a mere fall of a few degrees in body temperature was sufficient to render birds almost as receptive to anthrax as were rabbits or guinea pigs.


The Method of Agreement

  • Agreement
  • Rabbits have a normally low temperature and are susceptible to anthrax.
  • Guinea pigs have a normally low temperature and are susceptible to anthrax.
  • Hens whose body temperature has been lowered (by cold bath) to a below normal temperature are susceptible to anthrax.
  • Therefore, “a mere fall of a few degrees in body temperature was sufficient to render birds almost as receptive to anthrax as were rabbits or guinea pigs”.

The Method of Agreement

  • Difference
  • Hens whose body temperature has been lowered to a below normal temperature and contract anthrax will die.
  • Hens whose body temperature has been lowered to a below normal temperature and contract anthrax and are then warmed to their normal temperature recover from anthrax.
  • Therefore, “a mere fall of a few degrees in body temperature was sufficient to render birds almost as receptive to anthrax as were rabbits or guinea pigs.

The Method of Concomitant Variations

Whatever phenomenon varies in any manner whenever another phenomenon varies in some particular manner, is either a cause or an effect of that phenomenon, or is connected with it through some fact of causation. (1874, p. 287)

This method is similar to the method of agreement except that here the agreement is quantitative. The cause and effect are not just always present together, they also vary together—typically, they rise and fall together. This method is useful when two phenomena are, at different times, present at different levels, but neither is ever entirely absent.


The Method of Concomitant Variations

For example, the U.S. economy is always present (as long as the United States is in existence) and likewise for suicide rates. And although neither can ever be suspended, both increase and decrease. Hence, the method of concomitant variations is used to put together the following argument:

[P1] Suicide rates among 25 to 64 year olds were highest in 1932 during the Great Depression.

[P2] Suicide rates among 25 to 64 year olds were also high during the 1973–1975 oil crisis and the 1980–1982 recession.

[P3] Suicide rates among 25 to 64 year olds were lowest in 2000 at the end of the dot-com boom.


The Method of Concomitant Variations

[P4] Suicide rates among 25 to 64 year olds were also low when the economy was growing during World War II and from 1991–2001.

[C] Therefore, the economy and suicide rates are probably causally related.

While this method can be used to determine that two events are causally related, it is not designed to indicate which one is the cause and which is the effect. With the other methods, the occurrence of the antecedent circumstance prior to the occurrence of the phenomenon indicates which one is the cause.


The Method of Residue

Subtract from any phenomenon such part as is known by previous inductions to be the effect of certain antecedents, and the residue of the phenomenon is the effect of the remaining antecedents (p. 285).

For this method to be used, multiple sufficient causes for a certain type of phenomenon must already be known, and some of the occurrences of the phenomenon must be attributable to specific causes. Then, using this method, it is inferred that the remaining occurrences of the phenomenon were caused by the remaining cause.


The Method of Residue

One famous application of the method of residue is the discovery, by Marie Curie (1867 – 1934), of the elements polonium and radium in 1898. One year earlier, Curie had begun investigating radioactivity, a newly discovered property that was initially only known to belong to uranium. Using a new procedure for measuring levels of radioactivity, she examined pitchblende, a mineral mostly composed of uranium. But Curie found that pitchblende was several times more radioactive than pure uranium. She concluded that pitchblende must contain a radioactive element besides the uranium. Written out, Curie’s argument looks like this:


The Method of Residue

[P1] Pitchblende, which contains uranium, produces x amount of radiation.

[P2] Pure uranium produces y amount of radiation.

[C] Therefore, another element in pitchblende must produce x minus y amount of radiation.

Further work by her and her husband, Pierre Curie, revealed two previously unknown elements: polonium (named for M. Curie’s native Poland) and radium.



#1: Too strong of assumptions

  • Agreement assumes there is only one similarity between perfectly diverse cases
  • Difference assumes there is onlyone difference between perfectly similar cases
  • Residues & Concomitant Variation assume that all other contributing causes are known


#2: Scientific Drinker Problem

  • Possible solution to the Strong Assumption Problem: All relevant similarities, differences, etc. are known.
  • Problem: Relevance cannot be determined by Mill’s Methods alone:
  • The rules presuppose that we have a list of candidate causes to consider, but they themselves do not tell us how to come up with such a list. This list would depend on our knowledge or informed guesses about likely causes of the effects.
  • Thus, Mill’s Methods are at best derivative, not fundamental canons of scientific reasoning.


#3: Not methods of discovery

  • Mill’s Methods presuppose, but do not prescribe a criterion of relevance, namely CAUSAL relevance. (Scientific Drinker Problem)
  • Thus, Mill's Methods can't help us to discover causes unless we already know (roughly) what those causes are likely to be.


#4: Not methods of proof


  • It is always possible that we have overlooked some observable cause (Strong Assumption Problem)
  • Therefore, any hypothesis inferred via Mill’s Methods might be more a function of our ignorance than a genuine gain in causal knowledge.


#5: Limits of observation

  • We always apply Mill’s Methods in the light of limited observed data.
  • Therefore, if subsequent data reveals a previously unknown similarity, difference, etc., our original causal hypotheses will be false.
  • Reply 1: This is actually a virtue of Mill’s Methods, as it captures the dynamics of scientific reasoning.
  • Reply 2: It is also a problem for any model of scientific reasoning that puts a strong emphasis on observation; so Mill’s Methods are not alone in this regard.


#6: Interaction effects

  • Methods assume no interaction effects, i.e., causes of an effect x are unaffected by one another.
  • Related to the No Discovery Problem: if methods presuppose causal hypotheses, then, because interaction effects are part of a causal hypothesis, they must presuppose that the interaction effects have already been worked out.


#6: Vertical inferences

  • Mill’s Methods only deal with horizontal inferences:
  • Horizontal inference: from observable to observable
  • Vertical inference: from observable to unobservable
  • Many scientific hypotheses posit unobservable or only indirectly observable causes, e.g., anything caused by an electron or smaller.
  • Therefore, Mill’s Methods cannot provide any justification to many important scientific hypotheses.


Dr Sun Simiao (AD 581-682) was a very famous doctor in Tang Dynasty.

There were several rich people (named them Li, Qin and Cheng) living in Chang’an got a strange disease at the same time. Their symptoms included leg edema, muscle pain, fatigue and tiredness.

All the famous doctors could not treat the diseases.



Dr. Sun was invited to treat one of the patients (suppose he was Li). He went to the kitchen of Li’s. The cook told him that his master did not like to eat too much meat and fish but he liked to eat polished rice (white rice which had been polished several times). Dr. Sun also paid visits to Qin and Cheng who suffered from the same diseases. He found that these patients had the same habit of eating polished rice.



Strangely, there are no poor people suffering this disease.

Sun investigated several poor people (named them Sun, Zhao and Zhu) on their diet. These poor people all ate coarse rice which had not been polished.

Base on these investigation, Sun knew the cause of his patients.



1. Create a table that contains the information in the passage. Based on this information, what conclusion can be drawn?

2. Which of Mill’s methods is used to draw this conclusion?