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Fallacy of (the) Undistributed Middle

(also known as: maldistributed middle, undistributed middle term)

Description: A formal fallacy in a categorical syllogism where the middle term, or the term that does not appear in the conclusion, is not distributed to the other two terms.

Logical Form:

All A's are C's.

All B's are C's.

Therefore, all A’s are B’s.

Example #1:

All lions are animals.

All cats are animals.

Therefore, all lions are cats.

Explanation: We are tricked because the conclusion makes sense, so out of laziness we accept the argument, but the argument is invalid, and by plugging in new terms, like in the next example, we can see why.

Example #2:

All ghosts are imaginary.

All unicorns are imaginary.

Therefore, all ghosts are unicorns.

Explanation: While there may be ghosts that are unicorns, it does not follow from the premises: the only thing the premises tell us about ghosts and unicorns is that they are both imaginary -- we have no information how they are related to each other.

References:

Goodman, M. F. (1993). First Logic. University Press of America.



Registered User Comments

Jacob
Monday, September 17, 2018 - 02:31:54 PM
I just saw a video about listerine which said it was originally used as a floor cleaner. They imply that floor cleaner is toxic and should not be put in your mouth so listerine should not be put in your mouth either. I don't know what was in the original listerine, lets say it was alcohol just for the sake of argument. Alcohol can be used as a disinfectant, a floor cleaner, also as a drink, and a myriad of other things. Is this the fallacy of undistributed middle?

Alcohol is in a toxic floor cleaner
Alcohol is in a Vodka Tonic
Therefore Vodka Tonics are toxic.

I have seen this same logic in other places. A few years ago several people told me that you should not eat subway sandwiches because there is an unspecified and presumably dangerous ingredient which is used both to make yoga mats and the bread in subway sandwiches. Even before I knew about fallacies my mind balked at this idea. Of course you are going to find some of the same things in both yoga mats and bread: carbon for instance.

Ingredient is in Yoga Mats
Same ingredient is in subway bread
Therefore subway bread is yoga mats

One more... An argument against fluoride in the water is that the fluoride used to fluoridate water was "industrial waste". But anything, even pure water, could be called industrial waste.

There is fluoride in industrial waste
There is fluoride in toothpaste
Therefore toothpaste is industrial waste.

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Bo Bennett, PhD
Monday, September 17, 2018 - 05:17:34 PM
This is a good example of this. Before I finished your post I thought of "Food Babe's" Subway / yoga mat "argument". Steve Novella has a classic take down of this here: https://theness.com/neurologicablog/index.php/eating-yoga-mats/.

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