Pop Quiz

In school they often give pop quizzes. This shouldn't be possible, but it happens all the time. Why can't we have pop quizzes? Well, because it results in the hangman's paradox.

Here is how it goes:
The teacher says "I am going to give you a pop quiz sometime this week. You'll have a test, but I will only give you the test on a day when it will surprise you."

So, we know we'll be given a test but we don't know the day.

Lets figure out what day it could be.

Well, we know we can't have the test on Friday. Because if the test is on Friday we'd know the test would be that day, because it's the only day left in the week. And since we were told the test would be on a day we wouldn't expect (or know) there would be a test, the test can't be on Friday.

Thursday:
What would we know on Thursday? Well, on Thursday we'd know that we didn't take the test on Monday, Tuesday, and Wednesday, and we'd still know there could be no test on Friday... so the only day we could have the test would be on Thursday, but since we'd know that Thursday was the only day we could have the test, we can't have the test Thursday.

Wednesday:
On Wednesday I'd know that there was no test on Monday and Tuesday (cause I didn't take the test yet), I already have proven that it can't be Thursday or Friday, so the only day it could be is Wednesday, but since I know it's Wednesday there can't be any test that day.

Tuesday and Monday:
The logic can be repeated as it was on Thursday and Wednesday, and it can go on forever. So rather than write it out again, just repeat the previous arguments.

Anyway, we now know that there is no way to give a pop quiz during the week.

But this shouldn't work. There are a lot of articles that have been written about this problem, but no one has solved it. If the argument is formalized it is valid (some people say the paradox comes from the impossibility of formalizing this argument).

What we do know is this shouldn't work. There is either a weakness in our language, in logic, or in formalizing the sentences in logic....cause this argument comes out with the conclusion that there can be no quiz, but it's possible to give a quiz on any given day and surprise someone.

The trick happens when you make the leap from Friday to Thursday. There is something wrong with this leap, something about what you can know at some time and then inferring what you can know at another time. What's strange is that this even works with two days. If you only have Thursday and Friday you can still say "I can't take the test Friday so I have to take the test today, but if I have to take it today I can't take it today."

Anyway, something to think about. Next time you're told about a pop quiz tell the teacher that that's impossible.