19. Fallacy of the Undistributed Middle

The fallacy of the undistributed middle is a logical fallacy that occurs when a syllogism’s middle term is undistributed in both of the premises.* The middle term is the term that appears in both of the premises but not in the conclusion.

Example 1:

Major Premise:    All A are B.
Minor Premise:    All C are B.
Conclusion:          Therefore, all C are A.

The argument is fallacious, as follows: From the major premise, a random member of B might or might not be an A (that is, we can’t say), and from the minor premise, a random member of B might or might not be a C. And the answers to these two questions are independent of one another, so B is of no use in determining the relationship between A and C. This means we cannot use these premises to deduce the relationship between A and C.

A similar but concrete example:

Trees are green.
Money is green.
Therefore, money grows on trees.

 

* Note 1: The definition of this fallacy in Wikipedia is wrong. The webpage is: http://en.wikipedia.org/wiki/Fallacy_of_the_undistributed_middle and it says: “The fallacy of the undistributed middle [...] is committed when the middle term in a categorical syllogism is not distributed in the major premise.”[emphasis added]. Looking further afield, I found that other sources contradict that claim. Read the following webpage for a clearer (and consistent) explanation of the fallacy: http://philosophy.lander.edu/logic/middle_fall.html.

To see why Wikipedia’s definition must be incorrect, consider the following argument that complies with their definition and satisfy yourself that the argument is in fact valid (given that there is at least one member of A):

Major Premise:    All A are B.
Minor Premise:    All B are C.
Conclusion:          Therefore, some C are A.

 

Note 2: Wikipedia, in its discussion of the fallacy of the undistributed middle cited above, states “all cases of the fallacy of the undistributed middle are, in fact, examples of affirming the consequent or denying the antecedent.” However, I have been unable to find confirmation of that claim from other, independent sources and I do not know if it is true.

Notice that both of our examples above are examples of affirming the consequent. However, I have been unable to think of a clearcut example that is denying the antecedent. Can you think of such an example?

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Mnemonic Notes:
Note 1: Refer to the notes for fallacy 17 for an explanation of the picture.

Note 2: It might be helpful to think of this fallacy as “Illicit Middle” because of its similarity to Illicit Major and Illicit Minor.

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