The Unexpected Hanging
Sitting in his cell, the prisoner noted that if noon passed for him on the sixth day, only one day remained for the hanging to be carried out. But this would be a violation of the sentence, because one condition was that the prisoner could not know the date until 11:00 on the day of the execution. Because the judge may not lie, the execution could not occur on the seventh day. It would be like the judge saying "You will be hanged tomorrow and you will not know what day you will be hanged on until 11:00 tomorrow."
The prisoner then realized he had just reduced his expected life span by half a day with that logic. At first this depressed him until he made the next observation: by eliminating the possibility of being executed on the seventh day, if noon on the fifth day passed without execution, only one day remained for the hanging, the sixth. But then the prisoner would then know the date of his hanging too soon. So the sixth day was eliminated. And the same logic from applied to each preceding day, eliminating all possible days.
That night the prisoner called for his counsel and presented his defense. His counsel agreed and spent rest of the night preparing the argument. At the morning's first light, the judge was awakened and the defense presented its case against the sentence.
The prisoner waited optimistically as the judge and counsel retreated into closed chambers to debate the problem, until eleven o'clock rolled around and the hangman arrived. Much to the prisoner's surprise.
. . .
But the question remains, what was wrong with his defense?
This classic paradox tends to spark much debate when it appears. The judge clearly
was correct: even if the prisoner hadn't been so confident in his defense, he would
have been surprised by the execution. But on the other hand, the defense seems to
be logically well constructed. If on the last eve of the week the prisoner is still
alive, there aren't that many days left for the execution.
Let's simplify the paradox a bit- we don't need a whole week to cause a problem,
two days will work just as well. If the judge tells the prisoner he is to be hanged
either tomorrow or the next day, and he won't know until the day of the execution,
we have a problem. After tomorrow, the prisoner will know the day, which violates
the rule. So it must tomorrow. But if it must be tomorrow, then the prisoner knows
the day. Which violates the rule. Must make the prisonner's head spin.
Which is fortunate for the judge, because it is this paradox he has constructed
which makes the judge's sentence valid. He has created a logical system which traps
the prisoner from being able to know the truth.
This even works if the judge had said: "You will be hanged tomorrow and you will
not know that will be hanged tomorrow." Think about it- from the prisonner's perspective
he cannot know that sentence to be true. If he could, it would be false.
Consider this parallel paradox (insert your name in the blank):
(Your Name Here) cannot correctly know this sentence to be true.
That sentence is true in all cases, yet whoever's name is in the blank cannot
know it. Neat, huh?
If you are interested in these sorts of paradoxes, I strongly recommend
Mind's I, (Hofstadter & Dennett).