Who is in the barber shop?

Question

Here is the problem:

One day, I was walking on the way to the barber shop. There are three barbers A, B and C, but they don't always stay in the barber shop. In addition, barber A is a famously known coward. A never leaves the barber shop without B. I saw from a distance that the barber shop was still open, indicating that there was at least one barber in it. I like the craft of barber C best, so I hope that C is in the barber's shop at this time.

According to the known conditions and current observations, I am very satisfied to conclude that C must be in the barber shop. My reasoning process is like this: to prove the contrary, suppose C is not in the barber shop. In this case, if A is not in the barber shop, then B must be in the shop, because there is at least one person in the shop; However, if A is not in the barber shop, B should not be in the barber shop, because A will not leave the barber shop without B. Therefore, from "C is not in the barber shop", two contradictory conclusions "if A is not in, B must be in" and "if A is not in, B must not be in".

This shows that the assumption that "C is not in the barber shop" is wrong.

Is my reasoning correct? If not, what is the problem?

Answer 1

The problem is that "C is not in the barbershop" is not the only assumption that was made before you got to the contradiction.

You assumed that C is not in the barbershop, and then started reasoning about the case where A is not in the barbershop. This leads to a contradiction, so at least one of the two assumptions (C not in, A not in) is false. In other words, what you have proved is that at least one of A and C must be present.

Put another way, the two statements "if A is not in, B must be in" and "if A is not in, B must not be in" are not contradictory, but together they imply that A must be in. Those two statements followed from your initial assumption, so you have proved "If C is not in, then A must be in".