[pieguy]

Here's my take. The prisoner considers the statement "I will be hanged, and it will be a surprise", and after "proving" it false, takes its negation "Either I won't be hanged, or it won't be a surprise". But this is not correct. The prisoner actually disproves the statement "I will be hanged, and it will be a surprise no matter which day it happens". When you negate this statement, you obtain "Either I won't be hanged, or there is a day that it would not be a surprise", which is consistent with the outcome.

[p1esk]

The paradox is because the prisoner does indeed disprove the statement "I will be hanged, and it will be a surprise no matter which day it happens", but the statement turns out to be true (he is both hanged, and surprised no matter what day he's hanged).

[pieguy]

No, it would not have been a surprise if he were hanged on Friday. The prisoner disproves the existence of a strategy for the judge that guarantees surprise, but does not disprove the existence of a strategy that gives the possibility of surprise.

[p1esk]

It would have been a surprise if he were hanged on Friday, because he believes he wouldn't be hanged at all.

[pieguy]

He cannot logically conclude that. He can only conclude that either he won't be hanged or his hanging might not be a surprise.

[p1esk]

He concludes that because he assumes what the judge said is true. If we assume that, we must reject the scenario where he gets executed without surprise.

[pieguy]

> he believes he wouldn't be hanged at all

> he assumes what the judge said is true

The judge says he will be hanged. Your comments are inconsistent.