Here is a short version of a paradox that has intrigued me since I was in college: A condemned prisoner is told by the warden on a Sunday evening that he will be executed at noon on one of the upcoming weekdays, i.e., M-Tu-W-Th or F, but he will not know in advance which day it is until the time comes. The day of his hanging will be a “surprise”. The prisoner reasons that if Thursday noon passes and he has not yet been hung, then he can rule out Friday since that is the only day left and then he would know what day he was to die, in violation of the conditions specified by the warden. But having crossed Friday off the list, he quickly realizes that he can cross Thursday off too, since once Wednesday noon is past Thursday is the only day left (since he has already eliminated Friday). Going on this way, he realizes that he can eliminate all 5 weekdays, one by one, and he is happy to realize that he cannot therefore be hung (let’s say that he strongly believes the warden to be a truthful man). But here is the (sad) paradox…the executioner comes to his cell on (say) that Wednesday, and informs him that he is now to be hung. His last thoughts before dying are that the warden indeed spoke the truth: that he, the prisoner, was indeed going to be hung, and he was certainly surprised by it too. Where did his reasoning go wrong? It seemed so logical that he could not be hung if the warden was telling the truth……and yet, in the end, the warden did tell the truth. Seemingly, a paradox!
Here is my understanding of the resolution of the paradox (and I think I follow, essentially, Martin Gardner’s explanation from his book with the paradox as the title). The prisoners’ reasoning goes wrong on the very first step. He cannot rule out Friday even if Thursday noon has come and gone. In somewhat formal logic terms, I see it as follows. Let the wardens statement to the prisoner be written as C =A + B, where A = “you will be hung”, and B = “you will not know in advance of the day of the deed”. First, suppose A = true