Theorem: All Cats have nine tails.

Proof:

No cat has eight tails.

One cat has one more tail than no cats.

Therefore, all cats have nine tails.

This is the kind of thing they'll mess with your head with should you chose to do a maths degree. Don't say you weren't warned. It's notable for two reasons- highlighting the ambiguity of the english language (well, definitions and multiple usages of words in general) and the fact that mathematicians are more likely to be cat people than dog people (perhaps a trait of thinkers in general- witness the E2 Feline Conspiracy), which is illustrated by another cat based "proof"-

All Cats are mortal.

Socrates is mortal.

Therefore, Socrates is a cat.

Evidently false, and easier to dissect logically (it's an example of Affirming the consequent) than the first one as the flaw lies in the argument, not the definitions- but if I had put "man" instead of "cat", would you have noticed the error in the reasoning, given that the statements and the conclusion would then all be true?