non distributio medii ≍ “undistributed middle” fallacy

• This fallacy is committed when the middle term in a categorical syllogism is not distributed in either the minor premise or the major premise…

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…

1. All 𝚾 is B
2. All 𝚼 is B
3. Therefore, all 𝚼 is 𝚾

• 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 𝚾, or Some B is 𝚾.
• Also, a related rule of logic is that anything distributed in the conclusion must be distributed in at least one premise.

1. All 𝚾 is B
2. Some Y is 𝚾
3. Therefore, all 𝚼 is B

• The middle term “𝚾” is distributed, but Y is distributed in the conclusion and not in any premise, so this syllogism is invalid!

EXAMPLES:

1. All liberals are politically “Left” of center.
2. All Socialists are politically “Left” of center.
3. Therefore, all liberals are Socialists.

• The fallacy of the undistributed middle is referenced in Edgar Allan Poe’s detective story, “The Purloined Letter”…

“This functionary, however, has been thoroughly mystified; and the remote source of his defeat lies in the supposition that the Minister is a fool because he has acquired renown as a poet. All fools are poets; this the Prefect feels, and he is merely guilty of a non distributio medii  in thence inferring that all poets are fools.”