# Mill's Methods

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Mill's Methods

Mill's Methods are five methods of induction described by philosopher John Stuart Mill in his 1843 book A System of Logic. They are intended to illuminate issues of causation.

## Direct method of agreement

"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 (or effect) of the given phenomenon."

For a property to be a necessary condition it must always be present if the effect is present. Since this is so, then we are interested in looking at cases where the effect is present and taking note of which properties, among those considered to be 'possible necessary conditions' are present and which are absent. Obviously, any properties which are absent when the effect is present cannot be necessary conditions for the effect.

Symbolically, the method of agreement can be represented as:

A B C D occur together with w x y z
A E F G occur together with w t u v
——————————————————
Therefore A is the cause of w.

## 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 effect, or the cause, or an indispensable part of the cause, of the phenomenon.”

A B C D occur together with w x y z

B C D occur together with x y z

——————————————————

Therefore A is the cause, or the effect, or a part of the cause of w.

## 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 effect, or cause, or a necessary part of the cause, of the phenomenon."

Also called simply the "joint method," this principle simply represents the application of the methods of agreement and difference.

Symbolically, the Joint method of agreement and difference can be represented as:

A B C occur together with x y z
A D E occur together with x v w also B C occur with y z
——————————————————
Therefore A is the cause, or the effect, or a part of the cause of x.

## Method of Residue

"Deduct 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."

If a range of factors are believed to cause a range of phenomena, and we have matched all the factors, except one, with all the phenomena, except one, then the remaining phenomenon can be attributed to the remaining factor.

Symbolically, the Method of Residue can be represented as:

A B C occur together with x y z
B is known to be the cause of y
C is known to be the cause of z
——————————————————
Therefore A is the cause or effect of x.

## 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."

If across a range of circumstances leading to a phenomenon, some property of the phenomenon varies in tandem with some factor existing in the circumstances, then the phenomenon can be attributed to that factor. For instance, suppose that various samples of water, each containing both salt and lead, were found to be toxic. If the level of toxicity varied in tandem with the level of lead, one could attribute the toxicity to the presence of lead.

Symbolically, the method of concomitant variation can be represented as (with ± representing a shift):

A B C occur together with x y z
A± B C results in x± y z.
—————————————————————
Therefore A and x are causally connected

Unlike the preceding four inductive methods, the method of concomitant variation doesn't involve the elimination of any circumstance. Changing the magnitude of one factor results in the change in the magnitude of another factor.

## References

• Copi, Irving; Carl Cohen (2001). Introduction to Logic. Prentice Hall. ISBN 0-13-033735-8.
• Ducheyne, Steffen (2008). "J.S. Mill’s Canons of Induction: From true causes to provisional ones". History and Philosophy of Logic 29 (4): 361–376. doi:10.1080/01445340802164377.
• Kreeft, Peter (2009). Socratic Logic, A Logic Text Using Socratic Method, Platonic Questions, and Aristotelian Principles. St. Augustine's Press, South Bend, Indiana. ISBN 1890318892.

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