Episode 43: Fallacy of the Undistributed Middle

It was

a dark and smarmy night.  That’s right, around here nights try a little too hard to be your friend, it’s really weird.  I’d completed another play through of my all-time favorite game Box Smasher and I wasn’t quite tired enough to go to bed so I was watching some random videos on the internet.  No, not that kind, I was catching up on some thinking and logic topics.  I came across one person who said he would challenge you and show why your thinking is all wrong.  I thought Hey, this oughta be good!

Undistributed Middle Cropped.png

He started off by saying that we haven’t been taught everything we need to know to think critically.  OK, pretty good so far.  He launched into an example of his new wisdom … All humans need to breathe air, all dancers need to breathe air, therefore all humans are dancers!  Think about it, he said, the premises are both true so the conclusion must follow.  You were never taught how to make these logical leaps to keep your thinking small and constrained.  Well, I did my social media duty and give this a serious down-voting.

This is

a classic case of the Fallacy of the Undistributed Middle.  When I first heard of this fallacy, I thought it had something to do with that box of donuts I’d eaten not going to my mid-section.  But that’s not quite it, or it at all really.  We get to look at a formal fallacy here that deals with an improperly formed syllogism.

Normally, in a syllogism you would say:

If every A is a B                 (Socrates is a person)

And every B is a C            (All people are mortal)

Then every A is a C          (Then Socrates is mortal)

But in our poorly formed syllogism we say:

If every A is a C                 (Socrates likes to eat pizza)

And every B is a C             (All rats like to eat pizza)

Then every A is a B          (Then Socrates is a rat)

The B is the middle term in a syllogism.  In the well-formed version, the B term does not appear in the conclusion.  However, in the poorly formed version, the B term is moved to the conclusion and is, therefore, not distributed properly.  The B term should not appear in the conclusion.  Our media host committed this fallacy and gave us no new knowledge except maybe to finally answer the question “Are we human or are we dancer?”

Often the fallacy of the undistributed middle is very easy to spot.  Consider this: All cars are a method of transportation, all commercial passenger planes are methods of transportation, therefore all cars are commercial passenger planes.  I hope the error here is blatantly obvious – cars are not planes.  At least not yet, but it would be cool!  I’ve seen this fallacy used for comedic effect or as a check to “see if you’re paying attention”.


what if I say something like this: All squirrels like to eat nuts, all rodents like to eat nuts, therefore all squirrels are rodents.  If you paused for a moment to consider this, that’s good.  This is still a Fallacy of the Undistributed Middle but it’s designed to confuse you.  First off, everything here is similar enough that I might have caught you in a “are you paying attention” moment.  You might have just thought, yeah, they both like nuts, all is good.  Plus, squirrels are actually rodents, a member of species Rodentia.  This one is tricky because the conclusion is true, but the conclusion does not follow from its premises (terms).  This is the type you have to watch out for.  Just because you hear a true conclusion, that doesn’t mean it necessarily follows from either it’s premises or it’s logical structure.  You can see that both premises here are actually true, so it’s the structure that makes it a fallacy.

There really aren’t exceptions to this fallacy.  This is what you can often find with formal fallacies – since they are based on formal logic, any time you construct such an argument incorrectly, it is just wrong.  For example in our case with the squirrels, even though the premises and conclusion are correct, the structure is wrong.

This is a type of fallacy that uses symbolic logic.  That is something that would often scare people away, but you just read this and perhaps even found it easy.  So many things we’ve been told are difficult, or thought were out of our reach, are actually easier when broken down into smaller pieces.  The lesson here is: if you hear a syllogism, make sure it follows the form “If A=B, and B=C, then A=C”.   Make sure it’s not actually of the form “If A=C, and B=C, then A=B”.  Sometimes you might have to repeat the premises and conclusions to yourself, call them A, B, C and check.  Don’t let true premises or conclusions fool you either, this one is all about structure.  OK, just one more video about cute puppies getting into trouble them I’m going to bed.