Consider an unexpected hanging without the extra days: The judge tells you “I am going to hang you on Monday, but the day of the hanging is not something you will be able to predict.”
The prisoner follows the same reasoning as in the unexpected hanging: the surprise hanging can’t be Monday, because he would then know when it is, and it’s not a surprise. The judge then hangs him on Monday, and of course it’s a surprise. In other words, the unexpected hanging paradox doesn’t require the extra days at all.
If by “surprise” you mean “can’t logically be proven”, then the judge’s statement is equivalent to “X is true, but you cannot prove X”. From that statement, everyone except you can prove X, and you cannot.
Yes, it does. The paradox is that if you accept “A and you cannot prove A” as a premise, and logically derive A, then you arrive at a contradiction: A is true and you can prove it. By the usual laws of logic, this means that you should refute the premise. That is, conclude that the judge is lying.
But if the judge is lying, then what basis do you have for proving that A is true?
Of course, this whole argument may not apply to someone who uses a different basis for reasoning than classical logic with the Law of Excluded Middle.
Consider an unexpected hanging without the extra days: The judge tells you “I am going to hang you on Monday, but the day of the hanging is not something you will be able to predict.”
The prisoner follows the same reasoning as in the unexpected hanging: the surprise hanging can’t be Monday, because he would then know when it is, and it’s not a surprise. The judge then hangs him on Monday, and of course it’s a surprise. In other words, the unexpected hanging paradox doesn’t require the extra days at all.
If by “surprise” you mean “can’t logically be proven”, then the judge’s statement is equivalent to “X is true, but you cannot prove X”. From that statement, everyone except you can prove X, and you cannot.
Doesn’t “A and B” imply A even if B is “you can’t prove A”?
Yes, it does. The paradox is that if you accept “A and you cannot prove A” as a premise, and logically derive A, then you arrive at a contradiction: A is true and you can prove it. By the usual laws of logic, this means that you should refute the premise. That is, conclude that the judge is lying.
But if the judge is lying, then what basis do you have for proving that A is true?
Of course, this whole argument may not apply to someone who uses a different basis for reasoning than classical logic with the Law of Excluded Middle.