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Philosophy 103: Introduction to Logic
Fallacy of the Undistributed Middle Term

Abstract:  The Fallacy of the Undistributed Middle Term is discussed and illustrated.

I. We continue our study of the syllogistic fallacies with a second common fallacy.
A. Note, how in the following argument, about the only persons likely to be sympathetic are those who dislike Senator Jones.  (Notice that singular statements are treated as universal affirmative propositions.)
All [Communists] are [believers in heavy taxes].
[Senator Jones] is a [believer in heavy taxes].
[
Senator Jones] is a [Communist].
The Venn Diagram would be sketched like this: Diagram of AAA-2, Undistributed Middle Fallacy
B. It is fairly evident that for the conclusion to follow logically, one would have to presuppose instead that "All believers in heavy taxes are Communists," not "All Communists are believers in heavy taxes."  Notice that the former statement  would distribute the term "believers in heavy taxes." But this distribution is not what is asserted in the original argument.   In the original argument, the middle term is undistributed in both premisses.
C. Fallacy of the Undistributed Middle Term occurs when the middle term is undistributed in both premisses.
1. Rule: In a valid standard form categorical syllogism, the middle term must be distributed in at least one premiss.
2. Reason: for the two terms of the conclusion to be connected through the third, as in the mechanism sketched below, at least one of them must be related to the whole of the class designated by the middle term. Otherwise, the connection might be with different parts of the middle term, as illustrated below, and no connection can be made. Diagram of Undistributed Middle Term
3.  Note:  Remember for the Fallacy of the Undistributed Middle Term to occur, the middle term must be undistributed in both premisses, not just one premiss.

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