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Paradox Day

  1. Sep 10, 2011 #1


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    I got this problem from one of my favorite webcomics:
    Paradox occurs on a random day during the year such that it is unexpected which day it is. This precludes paradox day from occurring on December 31st since on December 30th we would know it had to occur the next day. Likewise, since paradox day cannot occur on December 31st, it cannot occur on December 30th either since on December 29th we would know which day paradox days occurs. Continuing in this fashion we see that the 'holiday' couldn't occur at all.

    To me, this seems like a perfectly legitimate argument; however, a number of people have argued that this isn't a paradox at all, usually on the grounds that you have additional information on the 30th that you don't have on the 29th. This seems fallacious to me since you can know at any calender date that paradox won't occur on the 31st. But I might be thinking about this the wrong way, so I thought I would ask the community here.

    So, is this really a paradox?

    Original Comic: http://spikedmath.com/271.html
  2. jcsd
  3. Sep 10, 2011 #2
    This is quite an old (and perhaps famous) paradox. See http://en.wikipedia.org/wiki/Unexpected_hanging_paradox" [Broken].
    Last edited by a moderator: May 5, 2017
  4. Sep 10, 2011 #3


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    It's a contradictory condition. The condition is that the paradox day occurs at a before or at december 31st such that at any date it would be unexpected. This condition is a conjunction of conditions, each saying that on day X it cannot be known that it happens on day X+1, X+2,...,december 31st. The condition for X = december 30 is a contradiction, since it says that on december 30th, it cannot be known that it happens on december 31st, which it must.
  5. Sep 10, 2011 #4
    Yes, Dec 31 is a contradictory condition. So we eliminated it. Then Dec 30 becomes a contradictory condition! etc. So the paradox must occur earlier than Dec 31. Yes? No?
  6. Sep 10, 2011 #5


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    No, the conjunction of conditions is inherent in the situation that is described. It would be like saying: "This stone is not a stone, and this stone weighs 3 kilograms", and then go on to eliminate the first condition, and simply say that "this stone weighs 3 kilograms". This is not the situation we first described. We just removed the contradictory part. Why did we do that, did we find out more about the stone by removing a part of the condition? How does it matter whether "this stone weighs 3 kilograms" is contradictory or not?

    The condition itself is contradictory as whole if it is a conjunction of one or more contradictory conditions. It isn't "fixed" by removing the contradictions in any way.
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