here is easily resolved once you realize that "knowing that you will know something in the future" does not imply
that you know it now. This is why there is a time-related aspect of these problems (surprise examination
, unexpected hanging
, egg in a box
It is clear that on the last possible day, you will know that the event has to come. And you may know that you will know this, come the last possible day -- but that does not mean that you know it on the second-to-last possible day. Which sounds confusing, but is true
Let's take a simplified example: the teacher says there will be a pop quiz
-- which will be a surprise -- either Thursday or Friday, and that the students will not know ahead of time which day it will be. Following the logic laid out in the paradox
, the students know it will not be Friday, implying it has to be Thursday, but, since they know this, the teacher has lied. Yet the teacher can give out the pop quiz on Thursday, as we're aware.
The teacher can do this because, while the students on Friday morning will know that the test must be given on Friday, the students Thursday morning do not. Why? Because they are missing a piece of information they do not (and cannot) know: That the quiz was not given out on Thursday. Until they know that, they cannot apply their knowledge
of what will happen on Friday, because it rests on that assumption
So, to rephrase the original situation:
The bright student may reason (correctly) the following:If the quiz is not given on Monday through Thursday, it must be given on Friday.
Continuing, the student may reason (correctly) the following:If the quiz is not given on Monday through Wednesday, it must given on Thursday through Friday.
If the quiz is not given on Monday through Tuesday, it must be given on Wednesday through Friday.
If the quiz is not given on Monday, it must be given on Tuesday through Friday.
Spelled out like this, the flaw in the student's logic becomes apparent. To know
that the quiz must be given out on Friday, you must also know that the quiz was not given out on Monday through Thursday. If it is, for example, Wednesday morning, you do not know this -- you only know that the quiz was not given on Monday through Tuesday. What the bright student is doing is assuming that her proposition is true
, with the following steps:
1) If the quiz hasn't been given out until Friday, then it cannot be given out on Friday.
2) If it hasn't been given out before Thursday, and
it cannot be given out on Friday, then it must be given out on Thursday.
3) If the quiz hasn't been given out before Wednesday, and
it cannot be given out on Friday, and
it cannot be given out on Thursday, it cannot be given out on Wednesday.(etc)
However, the student can't know
that it can't be given out on Thursday until after Wednesday is over, and the student can't know
that it can't be given out on Friday until Thursday is over. Since the student knows none of this on Monday morning, her statement is false.