# The Unexpected Hanging Paradox **URL:** \ **Category:** Factual Questions\ **Created:** [September 29, 2009, 4:02pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993 "2009-09-29T16:02:32Z")\ **Posts on this page:** 20\ **Page:** 3
**Author:** ![PlainJain](https://avatars.discourse-cdn.com/v4/letter/p/5daacb/32.png) [@PlainJain](https://boards.straightdope.com/u/PlainJain)\ **Post date:** [October 1, 2009, 5:23pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/41 "2009-10-01T17:23:24Z")
> [@ianzin](#): > > [Ianzin throws in the towel] > > “Well, I _tried_, didn’t I, goddamit? At least I did _that_.” At least I acknowledged your reasoning. Is it really too much to ask for you to explain the flaw in mine? ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [October 1, 2009, 5:26pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/42 "2009-10-01T17:26:07Z")
> [@jtgain](#): > > I’m with you. I still don’t get it either for the reason that you said. The statements are not self-contradictory because they both ended up being true. I think my post #36 explains what **ianzain** is trying to get at (though he should correct me if I’m wrong). ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [October 1, 2009, 5:46pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/43 "2009-10-01T17:46:43Z")
> [@Frylock](#): > > I think my post #36 explains what **ianzain** is trying to get at (though he should correct me if I’m wrong). There’s no flaw in your reasoning–you’re right that the prisoner’s reasoning must be wrong, since it started from true premises but led to a false conclusion. But the question is, _how_ is the prisoner’s reasoning wrong? It looks right! ---
**Author:** ![ianzin](https://avatars.discourse-cdn.com/v4/letter/i/ed655f/32.png) [@ianzin](https://boards.straightdope.com/u/ianzin)\ **Post date:** [October 1, 2009, 5:53pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/44 "2009-10-01T17:53:46Z")
> [@PlainJain](#): > > At least I acknowledged your reasoning. Is it really too much to ask for you to explain the flaw in mine? It isn’t too much to ask. That’s not the point. The point is that I am manifestly not equal to the task. If I try to explain something three times and fail, then I conclude I’m not very good at explaining it, or at least I’m not the right explainer for you. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 1, 2009, 9:20pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/45 "2009-10-01T21:20:40Z")
> [@Frylock](#): > > There’s no flaw in your reasoning–you’re right that the prisoner’s reasoning must be wrong, since it started from true premises but led to a false conclusion. > > But the question is, _how_ is the prisoner’s reasoning wrong? It looks right! It does _not_ follow from true premises. The mistakes are: 1. The judge is rational. 2. The judge has no information that the defendent doesn’t know. (E.g., my earlier example of the hangman going on vacation.) Once you realize that the prisoner doesn’t really know everything, then any “logic” that follows from that is flawed. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [October 1, 2009, 9:51pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/46 "2009-10-01T21:51:08Z")
> [@ftg](#): > > It does _not_ follow from true premises. That’s what I said. I said the reasoning must be flawed, and I said that it starts from true premises. Therefore, the reasoning must not follow from those premises. You’re trying to disagree with me, though, about whether the reasoning _starts_ from true premises. (Starting from premises and following from premises are different matters.) You say that the prisoner starts from the following false premises: > [@](#): > > 1. The judge is rational. > 2. The judge has no information that the defendent doesn’t know. (E.g., my earlier example of the hangman going on vacation.) AFAICT, those are definitely _not_ premises used by the prisoner in his reasoning. It’s fairly easy to check this–the problem lays out the prisoner’s reasoning explicitly. That reasoning includes no claims like those just quoted. What you may be trying to say is that the prisoner’s reasoning requires those two enthymemes _in order_ to follow validly from the prisoner’s explicit premises. In that case, you agree with me that the prisoner uses invalid reasoning to derive a false conclusion from true premises. The reasoning is invalid because it doesn’t logically follow–and moreover, could logically follow only from those premises plus a couple of false enthymemes. Are you right that the prisoner’s reasoning relies on these two hidden presuppositions? I’m not convinced, but I’ll have to get back to you. ---
**Author:** ![Carmady](https://avatars.discourse-cdn.com/v4/letter/c/f04885/32.png) [@Carmady](https://boards.straightdope.com/u/Carmady)\ **Post date:** [October 1, 2009, 11:06pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/47 "2009-10-01T23:06:37Z")
The point is that the prisoner cannot know the judge’s statement is true. He cannot simultaneously know that something will happen and know that it will surprise him. He never even gets to a point where he can begin to reason. In order to reason, he needs to say “the judge’s statement is true, therefore…”. But he can’t do that, because he cannot know the judge’s statement is true. The judge can know his own statement is true. But the prisoner cannot know it is true. ---
**Author:** ![pulykamell](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/pulykamell/32/3166_2.png) [@pulykamell](https://boards.straightdope.com/u/pulykamell)\ **Post date:** [October 1, 2009, 11:53pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/48 "2009-10-01T23:53:12Z")
> [@Carmady](#): > > The point is that the prisoner cannot know the judge’s statement is true. I think the point of the logical/thought exercise is that he does know the judge’s statement is true. I think the logic breakdowns at the very end of the prisoner’s reasoning, when he deduces that he will not be hanged. At this point, this invalidates our premise that a) he will be hanged and b) it will be a surprise. As soon as he deduces he will not be hanged, then at any point he is hanged, it is a surprise, satisfying both conditions. ---
**Author:** ![Frylock](https://avatars.discourse-cdn.com/v4/letter/f/ce7236/32.png) [@Frylock](https://boards.straightdope.com/u/Frylock)\ **Post date:** [October 2, 2009, 12:02am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/49 "2009-10-02T00:02:22Z")
> [@Carmady](#): > > The point is that the prisoner cannot know the judge’s statement is true. > > He cannot simultaneously know that something will happen and know that it will surprise him. > > He never even gets to a point where he can begin to reason. In order to reason, he needs to say “the judge’s statement is true, therefore…”. But he can’t do that, because he cannot know the judge’s statement is true. > > The judge can know his own statement is true. But the prisoner cannot know it is true. I’m not sure who this is addressing, and so I’m not sure of the thrust of your post, but to your point I can mention the following. In order to reason, the prisoner doesn’t need to _know_ anything. He only needs to _believe_ some things. ---
**Author:** ![KellyCriterion](https://avatars.discourse-cdn.com/v4/letter/k/b3f665/32.png) [@KellyCriterion](https://boards.straightdope.com/u/KellyCriterion)\ **Post date:** [October 2, 2009, 11:48am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/50 "2009-10-02T11:48:18Z")
Yep, I remain no closer to solving this paradox. Thanks to everyone who has tried so far to explain it, though. ---
**Author:** ![ftg](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/ftg/32/2801_2.png) [@ftg](https://boards.straightdope.com/u/ftg)\ **Post date:** [October 2, 2009, 11:55am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/51 "2009-10-02T11:55:35Z")
The prisoner starts with false premises _whether he states them or not_. Many examples of flawed logic start with unstated assumptions. ---
**Author:** ![Duhkecco](https://avatars.discourse-cdn.com/v4/letter/d/8491ac/32.png) [@Duhkecco](https://boards.straightdope.com/u/Duhkecco)\ **Post date:** [October 2, 2009, 2:18pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/52 "2009-10-02T14:18:52Z")
> [@Shodan](#): > > The prisoner is correct that he cannot be hanged if he doesn’t receive the knock by Thursday afternoon FWIW, I think that the above statement is false, and what makes it false is the word “cannot”. There’s nothing that the judge says that makes it logically impossible for the prisoner to be hanged after Thursday afternoon. According to the OP: > [@](#): > > 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. There is no explicit “cannot”, but that’s what the prisoner concludes. Also, the judge says that the prisoner “will not know the day of hanging until …” How does the judge know what the prisoner will know or not know? Hey, I don’t know what I know … 😉 ---
**Author:** ![Shodan](https://avatars.discourse-cdn.com/v4/letter/s/9f8e36/32.png) [@Shodan](https://boards.straightdope.com/u/Shodan)\ **Post date:** [October 2, 2009, 3:19pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/53 "2009-10-02T15:19:58Z")
> [@ianzin](#): > > [Ianzin throws in the towel] > > “Well, I _tried_, didn’t I, goddamit? At least I did _that_.” I’m sorry, I didn’t mean to offend you. > [@Carmady](#): > > The point is that the prisoner cannot know the judge’s statement is true. > > He cannot simultaneously know that something will happen and know that it will surprise him. I don’t get this either. “Surprise” in the sense of “I don’t know which day I will be hanged on” is what happens. The prisoner assumes the judge’s statements are true, which they turn out to be, and arrives at a false conclusion - that the statements are contradictory. > [@ftg](#): > > It does not follow from true premises. The mistakes are: > > 1. The judge is rational. > 2. The judge has no information that the defendent doesn’t know. (E.g., my earlier example of the hangman going on vacation.) Again, I don’t see how the premises are untrue. The judge is rational, at least in the sense that he makes correct statements, and I presume the judge knows the day of hanging when he issues the sentence. I am sorry to sound stupid, and I don’t mean to be exasperating, but I don’t get it. If anyone wants to take another crack at it, I would be grateful, but I will also understand if you don’t. Regards, Shodan ---
**Author:** ![Chessic\_Sense](https://avatars.discourse-cdn.com/v4/letter/c/7c8e57/32.png) [@Chessic\_Sense](https://boards.straightdope.com/u/Chessic_Sense)\ **Post date:** [October 2, 2009, 4:00pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/54 "2009-10-02T16:00:24Z")
> [@ianzin](#): > > But if he expects it, then the second part of what the judge said will be false. Ergo, he will **not** be hanged. This is where I think the prisoner errs. There is no statement that if the judge is false that the prisoner won’t be hanged. The prisoner will drop and the judge will be wrong. Incorrect: “If I’m alive on Thursday night, I must be scheduled for Friday. Therefore, I can rule out Friday. So if I’m alive Wednesday night…” Correct: “If I’m alive on Thursday night, I must be scheduled for Friday. Then the judge will be wrong. So if I’m alive on Wednesday night, I know I’m scheduled for Thursday or Friday. The judge will be wrong or right.” From that correct reasoning, there is no regression logic where the prisoner can rule out days for sure. The end result of his logic is that he can guess one day and he’s got a 1/7 shot at getting off the hook (or noose, in this case). That doesn’t seem so paradoxical, does it? I don’t see how this is different than the poison-cup scenario shown in The Princess Bride. “I can’t choose this one. But you know that, so I should choose that one. But you know that, so I shouldn’t…” Also, it’s like the “The next statement is false. The former statement is true.” For those that still don’t understand what I’m saying: When the prisoner rules out Friday and tries to rule out Thursday, he is mistakenly still crossing out Friday. In reality, it should come back into play. This is because the condition is not “will be hung/will not be hung”, it’s “judge is right/wrong”. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [October 4, 2009, 8:46am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/55 "2009-10-04T08:46:08Z")
I’m pretty sure this is logically equivalent to the [Unexpected Egg Paradox](http://www.google.com/url?sa=t&source=web&ct=res&cd=1&url=http%3A%2F%2Fbooks.google.com%2Fbooks%3Fid%3Dorz0SDEakpYC%26pg%3DPA569%26lpg%3DPA569%26dq%3Dunexpected%2Begg%2Bparadox%26source%3Dbl%26ots%3DwHPIlQzdo2%26sig%3DGWMroKHQpyS86xszuHcOtSavbF0%26hl%3Den%26ei%3D1V_ISsf1I5GgsgPVjNihBQ%26sa%3DX%26oi%3Dbook_result%26ct%3Dresult%26resnum%3D1&ei=1V_ISsf1I5GgsgPVjNihBQ&rct=j&q=unexpected+egg+paradox&usg=AFQjCNEThAkzuUJE8oE9L3ql3yAC4juZ9A&sig2=NolNPlazlHSjsSLqZn5Jyw) [Google Book excerpt] which may be a more visual way of reasoning about the puzzle. The excerpt also describes another version of the puzzle. I still have trouble getting my head around any of the versions. ---
**Author:** ![DarrenS](https://avatars.discourse-cdn.com/v4/letter/d/977dab/32.png) [@DarrenS](https://boards.straightdope.com/u/DarrenS)\ **Post date:** [October 4, 2009, 8:55am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/56 "2009-10-04T08:55:31Z")
I also vaguely remember an article on this paradox in Scientific American. It was Ian Stewart, or Martin Gardner, or one of the other recreational mathematicians. The author’s assertion was that the prisoner’s reasoning is equivalent to awakening each day and proclaiming, “I will be hanged today!” But I didn’t get that either. ---
**Author:** ![Aji\_de\_Gallina](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/aji_de_gallina/32/260_2.png) [@Aji\_de\_Gallina](https://boards.straightdope.com/u/Aji_de_Gallina)\ **Post date:** [October 4, 2009, 1:27pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/57 "2009-10-04T13:27:49Z")
The main problem is that he reasons by going backwards day-by-day. This forces his reasoning to go to a specific conclusion rather than being the correct way to interpret reality. ---
**Author:** ![Shodan](https://avatars.discourse-cdn.com/v4/letter/s/9f8e36/32.png) [@Shodan](https://boards.straightdope.com/u/Shodan)\ **Post date:** [October 4, 2009, 2:18pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/58 "2009-10-04T14:18:55Z")
> [@Chessic\_Sense](#): > > This is where I think the prisoner errs. There is no statement that if the judge is false that the prisoner won’t be hanged. The prisoner will drop and the judge will be wrong. > > Incorrect: > “If I’m alive on Thursday night, I must be scheduled for Friday. Therefore, I can rule out Friday. So if I’m alive Wednesday night…” > > Correct: > “If I’m alive on Thursday night, I must be scheduled for Friday. Then the judge will be wrong. So if I’m alive on Wednesday night, I know I’m scheduled for Thursday or Friday. The judge will be wrong or right.” > > From that correct reasoning, there is no regression logic where the prisoner can rule out days for sure. I hope this is right, because it makes sense to me. Thanks very much, **Chessic Sense**. Regards, Shodan ---
**Author:** ![solarnator](https://avatars.discourse-cdn.com/v4/letter/s/74df32/32.png) [@solarnator](https://boards.straightdope.com/u/solarnator)\ **Post date:** [October 4, 2009, 8:10pm UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/59 "2009-10-04T20:10:15Z")
To resolve the paradox, think in reverse. The judge is thinking “I am going to torture with the knowledge your going to die in a week” The prisoner must think each day he is going to die at noon…hence no surprise. The surprise is he lived another day. Just my 2 cents. ---
**Author:** ![PBear42](https://avatars.discourse-cdn.com/v4/letter/p/ecccb3/32.png) [@PBear42](https://boards.straightdope.com/u/PBear42)\ **Post date:** [October 5, 2009, 4:38am UTC](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993/60 "2009-10-05T04:38:57Z")
I agree with **ianzin**. Both conditions cannot be satisfied in all cases. Consider this variation of the paradox. I prefer the surprise exam version, as the element of surprise actually has a purpose (to get the students to prepare for class), so I’ll use that for illustrative purposes. So, a teacher tells her class there will be a surprise quiz one day next week. She takes out five envelopes with one day of the week marked on each and five index cards, on one of which is written “quiz.” She shuffles the index cards face down on her desk, then puts one in each of the envelopes. Each day, she pulls out the appropriate envelope and removes the card. If it’s the “quiz” card, one is given; if not, then not. In this scenario, there’s no question about whether the quiz will occur. The only question is whether it will be a surprise. And the answer is yes, on any day but Friday. The only way to avoid that is to introduce doubt as to whether the quiz will occur at all (e.g., by using a lottery with six cards). Or cheat and, unknown to the students, somehow remove Friday from the lottery. [Previous page](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993.md?page=2) [Next page](https://boards.straightdope.com/t/the-unexpected-hanging-paradox/511993.md?page=4)