In his Institutes of Oratory, the rhetorician Quintilian, in discussing the value of learning logic, mentions the “horn” problem, which evidently was this tricky syllogism:
“You have what you have not lost. You have not lost horns. Therefore you have horns.”
Initially I supposed that this was invalid, until I put it into categorical form:
All things you have not lost are things you have.
All horns are things you have not lost.
∴ All horns are things you have.
This is an AAA-1, and is thus valid. It could just as readily be written as modus ponens:
If you have not lost something then you have it.
You have not lost horns. Therefore you have horns.
How can it be valid, and yet have a false conclusion? Students of logic will realize that in this sort of situation one of the premises must be false. In this case, the false premise is the major premise: There are things that you have not lost that you still do not have. The Eiffel Tower, for instance.
The major premise sounds true because when we speak of not losing something, it is generally assumed that we are referring to what we once had. Thus the premise seems true because of something being assumed but not stated. This is similar to the fallacy of complex question, which makes unstated assumptions that can trip you up if you’re not careful.
So be careful!