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An old logic question goes something like this:

A teacher tells her students on Monday that there will be a test this week, but they will not know on which day till the morning of. The students think about this and come to the conclusion that there will be no test this week. Why?

The accepted answer to this is that the test cannot be on Friday, since that's the last day it could be on, so the students would know the night before it had to be that day. Therefore the test must occur by Thursday. However the test cannot occur on Thursday, since that's the last day it could occur on, so they would know the night before. They apply this logic recursively, eliminating every day this week, and decide therefore the test cannot occur this week.

Suppose the teacher then had the test on Thursday (or Wednesday, or whathaveyou). None of the students would see it coming, so clearly their logic fails at some point.

What is the exact error in their logic?

Joshua Little
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    This is called Unexpected hanging paradox – Momo Jul 08 '18 at 02:00
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    "... there is no consensus on its precise nature and consequently a final correct resolution has not yet been established." Oh dear, there goes my hope for closure. As a side note, this conundrum is why I have absolutely refused to engage in any debate where an entity is assumed to have "perfect" logical reasoning. It's like the logical equivalent of dividing by zero. – Joshua Little Jul 08 '18 at 02:13
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    The argument presumes that the students know that the teacher's announcement is true. If they don't know that, then, even if the exam is scheduled for Friday, on Thursday evening they won't know that the exam is coming on Friday. So what can happen, if the students use your reasoning and then are surprised by the exam on whatever day, is that the teacher's announcement was true but they couldn't know that it's true. A much simpler "paradox" based on the same idea is "There will be an exam next Monday and you can't know that before Monday morning." – Andreas Blass Jul 08 '18 at 02:53
  • @AndreasBlass That isn't the same situation, because knowing the test is next Monday is sufficient to prove the test is next Monday with the information available before Monday. In that case, what the teacher said is simply false. The hangman's paradox comes from the teacher/hangman/drill-instructors prophecy appearing empirically true despite appearing provably false. – DanielV Jul 08 '18 at 08:35
  • @DanielV Since you think that, in my simplified, Monday-only version of the paradox, the teacher's statement is simply false, consider the situation if the students agree with you and decide the teacher is lying. In particular, as far as they're concerned, they have no reason to expect an exam on Monday. Then, if the teacher goes ahead and gives the exam on Monday, the students will not have known about the exam in advance (because they didn't believe the announcement), and so everything the teacher told them turned out to be true. – Andreas Blass Jul 08 '18 at 11:41
  • @AndreasBlass There are 2 assumptions that could go into the proof (the one the teacher denies exists in your example). (1) the assertion of the teacher of when the test can possibly be, and (2) the assertion that the date will not be provable. After seeing your response now I'm not sure which assumptions are assumed in the denied proof in your example, so I'm not sure how to read or respond to your response. Not trying to be antagonistic I am open to insight on this problem, want to make sure I'm reading you correctly. – DanielV Jul 08 '18 at 14:01
  • @DanielV The important assertion here is the teacher's whole statement, the conjunction of the two assertions that you named (1) and (2). It is this conjunction that, though true in the situation I described, cannot be known by the students to be true until the exam happens. The same goes for the assertion in the original 5-day scenario: The assertion "There will be an exam one day this week but you won't know which day before the morning of that day" can turn out to be true (once the exam is given) but not knowable beforehand. – Andreas Blass Jul 08 '18 at 15:13

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