Fallacy of the undistributed middle
logical fallacy
The fallacy of the undistributed middle (Latin: non distributio medii) is a formal fallacy that is committed when the middle term in a categorical syllogism is not distributed in either the minor premise or the major premise. It is thus a syllogistic fallacy.
Classical formulation
In classical syllogisms, all statements consist of two terms and are in the form of "A" (all), "E" (none), "I" (some), or "O" (some not). The first term is distributed in A statements; the second is distributed in O statements; both are distributed in "E" statements, and none are distributed in I statements.
The fallacy of the undistributed middle occurs when the term that links the two premises is never distributed.
In this example, distribution is marked in boldface:
All Z is B
All Y is B
Therefore, all Y is Z
B is the common term between the two premises (the middle term) but is never distributed, so this syllogism is invalid. B would be distributed by introducing a premise which states either All B is Z, or No B is Z.
Also, a related rule of logic is that anything distributed in the conclusion must be distributed in at least one premise.
All Z is B
Some Y is Z
Therefore, all Y is B
The middle term—Z—is distributed, but Y is distributed in the conclusion and not in any premise, so this syllogism is invalid.
Pattern
The fallacy of the undistributed middle takes the following form:
All Z is B
Y is B
Therefore, Y is Z
This may be graphically represented as follows:
where the premises are in the green box and the conclusion is indicated above them.
B is the middle term (because it appears in both premises), and it is not distributed in the major premise, "all Z is B".
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