From reader Neil Taylor comes the request we examine the unexpected hanging paradox. I’m stealing from Wikipedia, which stole a version from Wolfram.
A judge tells a condemned prisoner that he will be hanged at noon on one weekday in the following week but that the execution will be a surprise to the prisoner. He will not know the day of the hanging until the executioner knocks on his cell door at noon that day.
Having reflected on his sentence, the prisoner draws the conclusion that he will escape from the hanging. His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on Friday, as if he hasn’t been hanged by Thursday, there is only one day left – and so it won’t be a surprise if he’s hanged on Friday. Since the judge’s sentence stipulated that the hanging would be a surprise to him, he concludes it cannot occur on Friday.
He then reasons that the surprise hanging cannot be on Thursday either, because Friday has already been eliminated and if he hasn’t been hanged by Wednesday noon, the hanging must occur on Thursday, making a Thursday hanging not a surprise either. By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
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.
Nothing better illustrates the iron-clad fact that all probability is conditional, and that every probability depends on the conditions or premises assumed. In all arguments there are implicit premises about the definitions of the words used and grammar. Many disputes in logic and probability, such as the Monte Hall problem, are resolved with ease once the words are made explicit.
We’re after the truth of different propositions, for the ambiguity starts at the beginning. The first might be A_1 = “The prisoner is hanged next week”; a second A_2 = “The prison is hanged next week, but cannot guess with certainty the day”; a third A_3 = “The prison is hanged next week, but might guess with certainty the day and thus won’t be surprised, and if he does guess the day he won’t be hanged”.
The prisoner in the example was hoping it was A_3, or maybe something like it. He must have been a student in logic (perhaps at Berkeley), hoping that by his mental labors he could outguess the warden, figure the day, and then reason because the warden promised he wouldn’t know the day that therefore his execution would be stayed. But since our condemned con was only a student, he didn’t reckon that the warden would know what the logic student was using logic, would figure the student would reason the way he did, and then would pick a day before Friday and hang that dirty rotten so-and-so.
That is, the student was interpreting the warden’s words with an awful literalness. His interpretation used the same words everybody else gets, but his inference about what those words meant is different for me. An ordinary SOB like Yours Truly, who is as deserving to be hanged as the student, would interpret the words of the warden to mean that I’d be writing my last blog post next week no matter what, even if I got word of the execution date from the grapevine, even if I heard it from the warden himself in a whisper, even if the Governor called to grant a pardon.
On my thinking, the probability of A_1 is 1, and A_2 and A_3 are of no interest to me, even if their probabilities are 0. Why? Because I reason that the warden was speaking loosely, as wardens do when they want to rub it in and make it sting. For A_2 to be true, both conjuncts have to be true. I already figured the first conjunct is certain, therefore the second won’t mean much to me as I make my way to the gallows.
Next consider the student isn’t a student, but a Berkeley professor. He would never take place in riots (not since his knees went) and can’t understand why he’s wearing the stripes. Since he’s a clever fellow, he reasons initially like the student, but then he knows the warden would guess he would reason like the student. At this point he has to take the words of the warden with that extreme literalness, or like an ordinary man. If like an ordinary man, yet still as a Berkeley professor, he’ll make out his will, leaving everything to the local Masonic chapter, and then begin silently weeping.
But suppose he interprets them literally. It makes not one whit, nor even two whits nor three, of difference how the warden actually thinks to the professor’s probability. The warden will do as he may. Probability is a matter of perspective, not necessarily of reality. If the professor thinks the warden is speaking hyper-literally, and will allow the professor to walk free if the professor is genuinely surprised when the headsman walks through the cell door, then it is still up to the professor to figure to what extent the hyper-literal promise means. Whatever the warden will do eventually is immaterial to the probability the professor forms.
Now the prof can figure like the student, reason that the warden would know he would do this, and then the professor has additional implied premises, which is that the warden would think the prof is thinking like a student and not a professor, and so the prof would re-iterate the student’s thinking process, again coming to the conclusion that on no day would he be surprised. But then the professor might think that the warden would know that the professor is a clever fellow, and would figure he’d do the re-iteration, and again surprise the professor on some day.
But then the professor would et cetera. Again, where this stops is irrelevant to what actually happens. Yet it is crucial and decisive to the probability the professor forms. It is he who supplies the implicit premises, and it is his probability. This is true for you, the dear reader, who wants to figure a probability for the professor. You may think the warden is on to the prof, and that the prof in turn is on to the prof, and so on. Your probability would then be different than the professors and the warden’s, and the warden’s would differ from you and the professor.
The answer, then, is that there is no single answer. It all depends.