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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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Jacob
Wednesday, October 24, 2018 - 04:39:49 PM
@Bo Bennett, PhD:

I can think of another undistributed middle argument which I have actually heard.

George Bush knows how to paint
Adolf Hitler knew how to paint
Therefore George Bush and Hitler are the same person.

I want to write about this but I want to understand this better. In the above example it is pretty obvious to all but the dummest George Bush haters that it is a fallacious argument. This is an argument to the absurd, because the absurdity of the argument makes its fallaciousness self-evident, but how about this argument?

All laps are loops
All lorps are loops
Therefore all laps are lorps

This one is not obviously absurd but it is a fallacy of undistributed middle. My question is...Why is it called the fallacy of undistributed middle? How is the middle term undistributed? I have noticed that in all fallacies of undistributed middle, both middle terms are on one side (the right or left in both premises) and in sound arguments one middle term is on one side and the other middle term is on the other side in the other premise. Why is this? How could I fix the laps/loops/lorps argument so it is sound?

Thanks

-Jacob


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Bo Bennett, PhD
Wednesday, October 24, 2018 - 05:30:27 PM
@Jacob: Also see this: https://www.logicallyfallacious.com/tools/lp/Bo/LogicalFallacies/245/False-Equivalence

You are really testing me here :) Let me sleep on this.

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Bo Bennett, PhD
Friday, October 26, 2018 - 07:04:38 PM
@Jacob: Here we go... Using my example:

All lions are cats.
All cats are animals.
Therefore, all lions are animals.

This is a sound syllogism, and the term "cats" is the middle term that links "lions" with "animals." The middle term (cats) is distributed.

Now the fallacy:

All lions are animals.
All cats are animals.
Therefore, all lions are cats.

No longer is there a middle term that is distributed that connects the two to form a sound syllogism.

Fixing yours would require:

All laps are loops
All loops are lorps
Therefore all laps are lorps

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Jacob
Tuesday, October 30, 2018 - 04:33:37 PM
@Bo Bennett, PhD: Thanks

I was fuzzy on what it meant for a term to be “distributed”. It makes more sense now.

I will try to fix the Hitler example

This is the fallacy of undistributed middle example

George Bush is a painter
Hitler is a painter
Therefore George Bush is Hitler.

Now a valid (but not sound) argument

George Bush Is a painter
Painters are Hitler
Therefore George Bush is Hitler

It would sound better if I wrote it like this

Valid but unsound
George Bush is a painter
Painters are genocidal maniacs
Therefore George Bush is a homocidal maniac

Undistributed middle
George Bush is a painter
A genocidal maniac named Hitler is a painter
Therefore George Bush is a genocidal maniac


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