Fallacy of the Undistributed Middle

Everyone who takes anabolic steroids has significant muscle mass. All competitive body builders have significant muscle mass, too, so they’re all on anabolic steroids.

The fallacy of the undistributed middle takes this form:

All A are C.
All B are C.
Therefore, all A are B.

Just because two things share a property, it doesn’t mean they are the same.

The middle term is the one that isn’t in the conclusion (C, or “having significant muscle mass” in the previous example).

In logic, a term is distributed if a statement tells us something about all members of that category.

For example, in the phrase “All dogs are mammals,” the term “dogs” is distributed because the statement tells us something about all dogs; however, the term “mammals” is not distributed because the statement tells us nothing about all mammals.

It’s called the fallacy of the undistributed middle because the middle term isn’t distributed, but the argument treats it as if it were.

In the steroid-bodybuilder argument, “having significant muscle mass” is the middle term. No claims are made about all people who have significant muscle mass.

Most examples of this fallacy are very unrealistic, like:

All dogs are mammals.
All cats are mammals.
Therefore, all dogs are cats.

They have absurd conclusions and don’t reflect anything you’d encounter in a real argument.

Here’s a more realistic example with a slightly different structure:

“Margarine shares 27 ingredients with paint. You shouldn’t eat paint, so you shouldn’t eat margarine.”

(Note: The premises aren’t accurate, but let’s set that aside to look at the logic.)

In this case, the speaker isn’t claiming that margarine is paint. Instead, they assume that because margarine and paint share some properties (having these ingredients), they must also share another property (being unsafe to eat).

The middle term is “has these 27 ingredients.” It is undistributed because no claims were made about all products that have these ingredients.

So, while the fallacy is normally defined like this:

All A are C.
All B are C.
Therefore, all A are B.

in real arguments, it often shows up more like this:

A is C.
B is C.
A is D.
Therefore, B is D.

Back to the Formal Fallacy Handbook

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