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In Harrie de Swart book Philosophical and Mathematical Logic, Quine's classification of paradoxes to falsidical, veridical and antinomies paradoxes is explained. Then in exercise 2.70 we are asked to classify the unexpected hanging paradox to one of these categories:

Is the following paradox an antinomy, a veridical or a falsidical one? A judge tells a condemned prisoner that he will be hanged either on Monday, Tues- day, Wednesday, Thursday or Friday of the next week, but that the day of the hang- ing will come as a surprise: he will not know until the last moment that he is going to be hanged on that day. The prisoner reasons that if the first four days go by with- out the hanging, he will know on Friday, that he is due to be hanged that day. So it cannot be on Friday that he will be hanged. But now with Friday eliminated, if the first three days go by without the hanging, he will know on Thursday that he is due to be hanged that day, and it would not be a surprise. So it cannot be Thursday. In the same way he rules out Wednesday, Tuesday and Monday, and convinces himself that he cannot be hanged at all. But he is very surprised on Wednesday when the executioner arrives at his cell. (See also Exercise 6.12 and its solution.)

In the solution in the book, this paradox is explained from the epistemic point of view, but they don't actually specify which type of paradox is this. I know there isn't always a definite answer to this question, but I wanted to get feedback about my answer:

One could say that this paradox is a falsidical paradox in relation to the conclusion of the prisoner (before he discovered he was wrong). The fallacy in the thinking of the prisoner is that he doesn't really know that what the judge told him is true, so if he is not hanged until friday it is possible that either he will not be hanged and the judged lied, or he will be hanged on friday and the judge told the truth.

Michael Novak
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  • The operative idea here is that the prisoner knows when they’ll be executed only if it’s the actual day of the execution. A prediction of the right day isn’t enough; the prisoner has to actually know when the execution will take place. So, if the prisoner reasons as in the example, then any day the executioner chooses will suffice the prisoner’s ignorance. If the prisoner doesn’t reason about it, but blindly guesses, then the prisoner doesn’t actually know when the execution will take place. So, the prisoner will be without knowledge either way if “is surprised” means “doesn’t know.” – PW_246 Apr 24 '23 at 18:08
  • I have a minority view of this question. See this answer. Based on my viewpoint, there is no paradox. – user2661923 Apr 24 '23 at 19:49

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One could say that this paradox is a falsidical paradox in relation to the conclusion of the prisoner (before he discovered he was wrong). The fallacy in the thinking of the prisoner is that he doesn't really know that what the judge told him is true, so if he is not hanged until friday it is possible that either he will not be hanged and the judged lied, or he will be hanged on friday and the judge told the truth.

Yes, I would agree that it is falsidical and for the very reason you provide here. In general, a falsidical paradox 'establishes' a result that not only appears false but actually is false, due to a fallacy in the demonstration. That is exactly what is going here: We have a seemingly logically compelling argument that the prisoner is not going to be hanged, despite our judgment that it should of course be perfectly possible for the prisoner to be hanged at some day. Thus, the argument seems to establish a result that is seemingly false ... and in fact is false, because the prisoner is being hanged without knowing beforehand which day. Also, the argument draws the wrong conclusion because there is a flaw ('fallacy') in the argument: the argument starts out with the assumption that the prisoner knows that he is going to be hanged, but the prisoner does not know this. In fact, even if the prisoner would be hanged on Friday, the prisoner would still not know this on Friday morning: maybe the judge was lying, like you say, or maybe a very untimely asteroid impact prevents the hanging, or .... It is only at the moment that the prisoner gets hanged that the prisoner knows that they get hanged. Not knowing this beforehand, the logic of the argument breaks down.

So, this case has all the markings of a falsidical paradox. It is like Zeno's argument: we have a logically compelling argument that a faster thing cannot pass a slower thing (or that all motion is impossible!), but of course the conclusion of that argument seems to be false, is in fact false, and is based on a fallacy in the argument ... although people are still disagreeing what exactly the fallacy is. Quine himself thought the fallacy was that an infinite number of points of time does not constitute all of time, but others point the finger at the assumption of the argument that space and time are infinitely divisible, and question that assumption. But regardless of where exactly the fallacy is, everyone agree that the argument must be fallacious somehow since the conclusion is obviously false. And Zeno's argument was classified by Quine himself as a falsidical paradox.

Bram28
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  • See my comment, following the posted question. In summary, I disagree because (in my view) there is no paradox at all. – user2661923 Apr 24 '23 at 19:52
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    @user2661923 Thanks. I like your analysis! Yes, you can definitely hold the position that the logic of the argument is perfectly good. So the problem with the argument is with its assumptions., not its logic. And that's where I put my finger as well: while the logic of the prisoner is perfectly valid, the prisoner starts with the assumption that they know that they will be hanged ... and I would argue that's a bad assumption. – Bram28 Apr 24 '23 at 19:56
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    It can't be a bad assumption, since it is one of the premises of the problem. To use your approach as an escape hatch would require that as a matter of protocol, any premises that involve someone's state of mind would have to be classified as meaningless premises that must be ignored. – user2661923 Apr 24 '23 at 19:58
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    @user2661923 In general I would agree with you: when we are told something as part of a problem set-up, we should be able to take that as a premise, yes. However, this particular paradox hinges exactly on the prisoner's knowledge .. or lack thereof. – Bram28 Apr 24 '23 at 20:48
  • That's right. So, if there is a paradox, then the paradox is caused by the prisoner relying on the premise concerning his state of mind. Therefore, you have two choices. One choice is to decide that therefore, the prisoner should not have relied on the premise, because the reliance caused a paradox. The other (arguably unsatisfying) choice is to presume that the prisoner's reliance is still justified, and that there is no paradox, even though contradictory premises both became simultaneously true. – user2661923 Apr 24 '23 at 20:53