Something to Ruin Your Weekend

11 March 2011 at 3:24 pm 6 comments

| Lasse Lien |

On a Monday morning, a professor says to his class, “I will give you a surprise examination someday this week. It may be today, tomorrow, Wednesday, Thursday, or Friday at the latest. On the morning of the examination, when you come to class, you will not know that this is the day of the examination.”

Well, a logic student reasoned as follows: “Obviously I can’t get the exam on the last day, Friday, because if I haven’t gotten the exam by the end of Thursday’s class, then on Friday morning I’ll know that this is the day, and the exam won’t be a surprise. This rules out Friday, so I now know that Thursday is the last possible day. And, if I don’t get the exam by the end of Wednesday, then I’ll know on Thursday morning that this must be the day (because I have already ruled out Friday), hence it won’t be a surprise. So Thursday is also ruled out.”

The student then ruled out Wednesday by the same argument, then Tuesday, and finally Monday, the day on which the professor was speaking. He concluded: “Therefore I cannot get the exam at all; the professor cannot possibly fulfill his statement.” Just then, the professor said: “Now I will give you your exam.” The student was most surprised, but the professor seems to have kept his word.


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6 Comments Add your own

  • 1. ProfDC  |  11 March 2011 at 3:42 pm

    As this version of the paradox is written, there is no paradox.

    The professor says “On the morning of the examination, when you come to class, you will not know that this is the day of the examination” at a time strictly between the beginning of Monday’s class and the end of Monday’s class. The students are already in class when they hear this pronouncement. It is thus a true statement that no student knew “on the morning of the examination, when you come to class” that the exam would be on Monday, since the professor had not made the statement before the beginning of class. Giving the exam on Monday is thus wholly consistent with the statement. Once he’s heard the statement, however, the student should immediately conclude that he will get the exam on Monday, since once Monday’s class is over there are no more morning arrival opportunities when the exam time will be unknowable.

  • 2. Lasse  |  11 March 2011 at 4:37 pm

    True. Good point. But assume instead that the students got the test on tuesday morning, or wednesday for that matter. The students would be surprised, especially the logic student.

  • 3. ProfDC  |  11 March 2011 at 5:14 pm

    I don’t agree.

    Every morning after Monday, the students believe that that day (whatever it is) will be the exam day. It won’t be a surprise because on Tuesday (or any day other than Monday) every student, when he arrives at class, EXPECTS that this is the day. The only way for the professor’s statement to be correct is to NOT give it on that day. The fact that the students are wrong on the days when the exam is not given is consistent with the professor’s statement that they will not KNOW that this is the day, since knowing something implies (among other things) that it’s true.

    Or, put another way (shorter and pithier) — there are many things that I don’t know, but not all of them surprise me.

  • 4. Lasse  |  11 March 2011 at 6:06 pm

    The way I understand it is that the students do not believe that he professor can give them an unexpected exam at any point during the week (barring the monday problem you pointed out above). The students will think so at any point in time. So either he is lying about the test, or he is lying about the claim that they will not know when. Paradoxically, it is exactly this belief that makes it possible for the professor to surprise them.

  • 5. Richard Ebeling  |  12 March 2011 at 9:52 pm

    The logic portrayed as the thinking process of the student reminds me of how too many mainstream economists think.

    If everyone had the same correct knowledge of how the economic system works that I possess — after all, I’m a “modern” macro-economist, and that implies that I know the correct theory of how the economy works.

    And if everyone were to efficiently utilize all information in conjunction with that correct theory, so the only unpredictable events would be the unsystematic unpredictable events, — in a manner just as I use all the statistical economic data that the government so kindly provides me with.

    Then, the economy could never be out of equilibrium except for those random and unsystematic “disturbances.”

    And since, “clearly,” this is the only “rational” way that any person can think and “optimally” choose to use information, there can be no systematic disturbances — just unsystematic, “surprises.” Which once surprised by them, all rational people will never allow themselves to be surprised by them, again.

    And, then, “surprise,” the Fed increases the money supply, distorts interest rates, induces faulty investment decisions, AGAIN, and we are all “surprised” again by the disrupting consequences.

    This all happens when you “rationalize” too much. Because in this hyper-rationality you lose sight of the fact that that is not how real people think, or how events really play out. And sometimes like causes bring about like effects — again, and again, and again, and . . .

    As to why, well, that’s a different story.

    Richard Ebeling

  • 6. ProfDC  |  16 March 2011 at 3:03 pm

    >The way I understand it is that the students do not believe
    >that he professor can give them an unexpected exam
    >at any point during the week…
    >The students will think so at any point in time.

    I agree with this, but I think we are implicitly disagreeing about whether the students’ (in my view, erroneous) conclusion counts as knowledge. I claim that this issue is crucial to the paradox.

    I think we can agree that once they receive the exam, it’s clear that they did not have a justified true belief (much less any stronger knowledge, a la Gettier) upon arrival that they will not get the exam that day — so they certainly didn’t “know” they would. On Monday they can get the exam without any logical problems, as noted above, since upon arrival they did not hold a belief that they would get an exam that day. On each of Tuesday-Friday they can either not expect the exam (i.e., hold a belief that the exam will not be administered that day, and, if the exam comes that day, the professor’s statement will be true on its face) or expect the exam (i.e., hold a belief that the exam will be administered that day) and the professor’s statement will be true because their belief, however justified, was not true.

    I recommend in this vein.

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