I feel like the prisoner's explanation is circular logic—akin to reducing something down to 0 = 0:
1. He claims he can't be executed on Friday, because it wouldn't be a surprise.
2. He then argues similarly that he can't be executed any other day.
3. He concludes therefore that he can't be executed.
But if "I won't be executed at all" was a possibility, then he could never even rule out Friday: on Friday, he'll either get executed, or not at all. Both are possible, so he can't claim an execution on Friday would be predictable beforehand.
Great argument. The paradox changes that variable, possibility of no execution, midway through.
#1 If it stays true from the beginning, the first step of the prisoner's logical induction is invalid and the prisoner accepts he will be surprised.
#2 If it stays false from the beginning, the prisoner can make the logical induction, coming to a conclusion that is invalid (contrary to an assumption) so he accepts that he will be surprised.
Without having it thought out carefully I suggest the following. The judge's statement is impossible, i.e. it is not possible that the prisoner will be hanged on any day and is surprised by this. From this impossible statement - similar to a wrong premise - the prisoner draws the false conclusion that he will not be hanged at all and is then obviously surprised by someone knocking on his door.
Well, I don't think you can conclude the judge's statement is impossible, since (at least in the story) it happened. So it had to be at least possible. (The judge doesn't claim that the hanging could happen any day and be a surprise, just that the day it happens it will be a surprise).
Is the last part actually part of the paradox, or just a funny quip at the end. I found that bit quite hilarious, I could imagine that fitting into a Monty Python sketch or something quite well.
>The next week, the executioner knocks on the prisoner's door at noon on Wednesday — which, despite all the above, was an utter surprise to him. Everything the judge said came true.
It's part of the paradox. The prisoner believes he has proven that his execution is impossible—however, he was wrong. The paradox is how a seemingly correct argument leads to a false conclusion.
What's interesting to me about this paradox is that it's (IMO) a clever repackaging of self-referential paradoxes: this statement is false, this statement can't be proven, etc.
In this case, it's "You will die on a day not deducible from this statement", but the self-reference is concealed by asserting it will be a "surprise".
1. He claims he can't be executed on Friday, because it wouldn't be a surprise.
2. He then argues similarly that he can't be executed any other day.
3. He concludes therefore that he can't be executed.
But if "I won't be executed at all" was a possibility, then he could never even rule out Friday: on Friday, he'll either get executed, or not at all. Both are possible, so he can't claim an execution on Friday would be predictable beforehand.