# Resolving the unexpected hanging paradox

The un­ex­pected hang­ing para­dox: The war­den tells a pris­oner on death row that he will be ex­e­cuted on some day in the fol­low­ing week (last pos­si­ble day is Fri­day) at noon, and that he will be sur­prised when he gets hanged. The pris­oner re­al­izes that he will not be hanged on Fri­day, be­cause that be­ing the last pos­si­ble day, he would see it com­ing. It fol­lows that Thurs­day is effec­tively the last day that he can be hanged, but by the same rea­son­ing, he would then be un­sur­prised to be hanged on Thurs­day, and Wed­nes­day is the last day he can be hanged. He fol­lows this rea­son­ing all the way back and re­al­izes that he can­not be hanged any day that week at noon with­out him know­ing it in ad­vance. The hang­man comes for him on Wed­nes­day, and he is sur­prised.

Sup­pos­edly, even though the war­den’s state­ment to the pris­oner was para­dox­i­cal, it ended up be­ing true any­way. How­ever, if the pris­oner is no bet­ter at mak­ing in­fer­ences than he is in the prob­lem, the war­den’s state­ment is true and not para­dox­i­cal; the pris­oner was ex­e­cuted at noon within the week, and was sur­prised. This just shows that you can mess with the minds of peo­ple who can’t make in­fer­ences prop­erly. Noth­ing new there.

If the pris­oner can eval­u­ate the war­den’s state­ment prop­erly, then the pris­oner fol­lows the same logic, re­al­izes that he will not be hanged at noon within the week, re­mem­bers that the war­den told him that he would be, and con­cludes that the war­den’s state­ments must be un­re­li­able, and does not use them to pre­dict ac­tual events with con­fi­dence. If the hang­man comes for him at noon any day that week, he will be un­sur­prised, even though he is not con­fi­dent that he will be ex­e­cuted that week at all ei­ther. The war­den’s state­ment is then false and un­para­dox­i­cal. This is similar to the one-day analogue, where the war­den says “You will be ex­e­cuted to­mor­row at noon, and will be sur­prised” and the pris­oner says “wtf?”.

Now let’s as­sume that the pris­oner can make these in­fer­ences, the war­den always tells the truth, and the pris­oner knows this. Well then, yes, that’s a para­dox. But as­sign­ing 100% prob­a­bil­ity to each of two propo­si­tions that con­tra­dict each other com­pletely de­stroys any prob­a­bil­ity dis­tri­bu­tion, mak­ing the pris­oner still un­able to make pre­dic­tions, and thus still not let­ting the war­den’s state­ment be both para­dox­i­cal and cor­rect.

If some­one ac­tu­ally tried the un­ex­pected hang­ing para­dox, the clos­est sim­ple model of what would ac­tu­ally be go­ing on is prob­a­bly that the war­den chose a prob­a­bil­ity dis­tri­bu­tion so that, if the pris­oner knew what the dis­tri­bu­tion was, the pris­oner’s av­er­age ex­pected as­sess­ment of the prob­a­bil­ity that he is about to get ex­e­cuted on the day that he does is min­i­mized. This is a solv­able and un­para­dox­i­cal prob­lem.

• There is an enor­mous (far too enor­mous for its value to the world, in my opinion) liter­a­ture on the un­ex­pected hang­ing para­dox (also known as the sur­prise exam para­dox) in the philos­o­phy and math­e­mat­ics liter­a­ture. The best treat­ments are:

Ti­mothy Y. Chow, The sur­prise ex­am­i­na­tion or un­ex­pected hang­ing para­dox, Amer­i­can Math­e­mat­i­cal Monthly 105 (1998) pp. 41-51. (un­gated)

Elliot Sober, To give a sur­prise exam, use game the­ory, Syn­these 115 (1998) pp. 355-373. (un­gated)

• far too enor­mous for its value to the world, in my opinion

The para­dox ac­tu­ally has prac­ti­cal im­pli­ca­tions. It shows a gen­eral mechanism by which you can “sur­prise” some­one de­spite a pre­dictable out­come. It goes like this:

1) You tell some­one they will be “sur­prised” by an up­com­ing event (e.g., what gift you will buy them).
2) They start to sus­pect it will be one of a num­ber of un­usual out­comes.
3) The event ac­tu­ally has its reg­u­lar, bor­ing, pre­dictable out­come.
4) But the other per­son is still sur­prised, since they did not ex­pect the bor­ing out­come (when be­fore your state­ment, they did)!

I know of a ma­jor case where this rea­son­ing was ap­plied: on one sea­son of the TV show The Ap­pren­tice. (The show where peo­ple try to get a job with Don­ald Trump and one per­son is elimi­nated from con­sid­er­a­tion [“fired”] each week.) Dur­ing the sec­ond epi­sode/​com­pe­ti­tion, one con­tes­tant walked out due to frus­tra­tion, and she didn’t come back un­til eval­u­a­tion time.

Then, in ads for the next epi­sode, they said, “Next time, on The Ap­pren­tice, one can­di­date will quit the com­pe­ti­tion—and you’ll never guess who it is!”

This, of course, prompted spec­u­la­tion that some­one other than the last epi­sode’s quit­ter would be the one to quit … but no, it was the same woman who left, this time per­ma­nently, rather than be­ing fired. Well, it was cer­tainly a suprise by that point!

• Yeah, came here to say the same. Thanks.

When I tried to solve this prob­lem about 10 years ago, I came up with the equiv­a­lent of Fitch’s “Goedelized” solu­tion, de­scribed on pages 5-6 of Chow’s ar­ti­cle. I’m still not sure why many peo­ple con­sider it wrong; it seems to ut­terly dis­solve the “para­dox” for me.

• You know, I just don’t un­der­stand why the pris­oner is so sur­prised when he’s ex­e­cuted on Wed­nes­day—I mean, the ex­e­cu­tioner comes on Wed­nes­day ev­ery sin­gle time I read about this thing, and it’s just not that sur­pris­ing any­more.

• con­cludes that the war­den’s state­ments must be un­re­li­able, and does not use them to pre­dict ac­tual events with con­fi­dence. If the hang­man comes for him at noon any day that week, he will be un­sur­prised, even though he is not con­fi­dent that he will be ex­e­cuted that week at all ei­ther.

“Sur­prised” in this para­dox merely stands for “you will not be able to pre­dict the hang­man’s ap­pear­ance on any given day be­fore the hang­man ap­pears”. So if the pris­oner chooses not to be­lieve the war­den’s state­ment, that still leaves him sur­prised when the hang­man comes.

• True. I was think­ing of “sur­prised” as “as­signed a small prob­a­bil­ity to” rather than “did not as­sign 100% prob­a­bil­ity to”. If we use the lat­ter in­ter­pre­ta­tion, then there is more in­ter­est­ing con­tra­dic­tion. So much for my re­s­olu­tion.

• In­for­ma­tion the­ory has use­ful con­cepts for this situ­a­tion (as usual). It uses the term “sur­prisal” as a way to quan­tify how sur­pris­ing an out­come is. It is equal to the log of the in­verse of the prob­a­bil­ity ( log (1/​p) ) you had as­signed to an event be­fore you learn that it hap­pened. [1]

What sur­prisal value should the judge’s state­ment be in­ter­preted as mean­ing? One first ap­proach would be to say that the judge means the pris­oner will find the re­sult more sur­pris­ing than if he had sim­ply as­sumed an equal prob­a­bil­ity to the seven days. Thus, the judge is say­ing that “the sur­prisal, or in­for­ma­tion gain, from learn­ing your ex­e­cu­tion date will be greater than log(7).”

So, uh, how on earth are you sup­posed to move your prob­a­bil­ity dis­tri­bu­tion over ex­e­cu­tion days upon be­ing given that kind of ev­i­dence? If you (wisely) start from a uniform prob­a­bil­ity dis­tri­bu­tion, you already have, in ex­pec­ta­tion, the max­i­mum sur­prisal value. (En­tropy is equal to the “ex­pected” [i.e., prob­a­bil­ity-weighted] sur­prisal, and a uniform dis­tri­bu­tion is max­i­mum en­tropy.)

No change in prob­a­bil­ity dis­tri­bu­tion can in­crease the ex­pected sur­prisal—un­less, of course, you de­liber­ately skew your PD so that it de­creases the weight on when you “re­ally” ex­pect to be ex­e­cuted. But then that brings up the messy is­sue of what you re­ally be­lieve vs. what you be­lieve you be­lieve.

[1]Con­se­quently, it is equal to how much in­for­ma­tion you get upon ob­serv­ing the event—ob­serv­ing im­prob­a­ble events tells you more than ob­serv­ing prob­a­ble ones. In­tu­itively, do you learn more from when a sus­pect says they’re guilty, or when they claim in­no­cence?

• the is­sue i see here is it is only a para­dox if it re­quires the judge tells the truth and is always cor­rect. but if that is the case then the pris­oner is con­clud­ing that the judge is ly­ing or made a mis­take when he con­cludes he will not be hanged. so since the con­clu­sion he won’t be hanged is a con­tra­dic­tion it­self how can he con­clude it is definitely true? if he can’t con­clude his con­clu­sion is definitely true then he will be un­cer­tain each day

• OK, let’s look at this: The pris­oner re­ceives 2 pieces of in­for­ma­tion from the war­den at the be­gin­ning:

• The first piece of in­for­ma­tion is: He will be kil­led at noon of one day of the next five days.

As­sum­ing that the war­den’s claim is true, there are 5 pos­si­ble out­comes:

Death at noon of Mon, Tue, Wed, Thu or Fri.

As­sum­ing fur­ther­more that the pris­oner has no other in­for­ma­tion that and that he uses prob­a­bil­ity the­ory, he will con­struct the fol­low­ing uniform prob­a­bil­ity dis­tri­bu­tion:

P(Death at noon of X.)=1/​5 where X can be Mon, Tue, Wed, Thu or Fri.

Fur­ther­more he can now also in­fer the con­di­tional probabilities

P(Death at noon of X.|Not dead af­ter noon of Mon.)=1/​4 for X = Tue, Wed, Thu or Fri.

P(Death at noon of X.|Not dead af­ter noon of Tue.)=1/​3 for X = Wed, Thu or Fri.

P(Death at noon of X.|Not dead af­ter noon of Wed.)=1/​2 for X = Thu or Fri.

And fi­nally:

P(Death at noon of Fri.|Not dead af­ter noon of Thu.)=1

Thus the pris­oner will now be not ‘sur­prised’ only by a death at noon of Fri­day. As in: The oc­cur­ring event had P>1/​2.

Is this the proper no­tion of ‘sur­prise’?

I don’t think so. - Sur­prise should be always mea­sured quan­ti­ta­tively.

But ob­serve that in the ‘death at noon of Fri­day’ sce­nario there is no sur­prise what­so­ever. It is qual­i­ta­tively ab­sent un­der the con­di­tion that the war­den speaks the truth:

P(Death at noon of Fri.|Not dead af­ter noon of Thu., The war­den speaks truth.)=1 P(Death at noon of Fri.|Death at noon of Fri.)=1 (duh) There is no up­dat­ing.

The sec­ond piece of in­for­ma­tion from the war­den is:

• The pris­oner will be sur­prised by his death.

What does this mean, any­way? Can we ac­tu­ally al­ter our prob­a­bil­ity dis­tri­bu­tion based on this da­tum?

I highly doubt that, but the pris­oner in the canon­i­cal treat­ment cer­tainly does up­date: As it holds that:

P(Death at noon of Fri.|Not dead af­ter noon of Thu.)=1

He con­cludes:

P(Not dead af­ter noon of Thu. AND The pris­oner will be sur­prised by his death.)=0

And:

P(Death at noon of Fri.|Not dead af­ter noon of Thu., The pris­oner will be sur­prised by his death. )=0

As a spe­cial case of ‘P(Any­thing.|Con­tra­dic­tion.)=0’

He then runs a few iter­a­tions and con­cludes that all out­comes have the P=0 un­der all con­di­tions.

Here the back­ground as­sump­tion is still that the war­den’s words are true. This as­sump­tion how­ever is con­tra­dic­tory if the up­dat­ing pro­ce­dure of the pris­oner is cor­rect. But we can eas­ily see that the new be­lief struc­ture of the pris­oner will be sur­prised by any of the out­comes; ren­der­ing the war­den cor­rect. Thus the para­dox.

The solu­tion is sim­ply that the pris­oner’s up­dat­ing pro­ce­dure is in­cor­rect.

The da­tum ‘The pris­oner will be sur­prised by his death.’ does not war­rant the up­date. The war­den’s state­ments are con­tra­dic­tory if the origi­nal be­lief struc­ture is re­tained and if the only re­main­ing out­come is death at noon of Fri­day. How­ever, af­ter the first change of the be­lief struc­ture by the pris­oner this no longer holds. The fur­ther ‘iter­a­tions’ make even less sense and the whole ‘up­date’ is un­sta­ble, as our sim­ple re­flec­tion shows—now the pris­oner will be sur­prised by any out­come.

So ob­vi­ously, this is not Bayesian up­dat­ing.

The pris­oner tried to rea­son. Con­cluded that he couldn’t be kil­led with­out prov­ing the war­den wrong. Changed his prob­a­bil­ity dis­tri­bu­tion over out­comes to re­flect this. Thus chang­ing the pre­req­ui­sites for his ini­tial con­clu­sion. He did not ex­am­ine the im­pli­ca­tions of the changed pre­req­ui­sites. En­sur­ing that the war­den could always be right.

He thus up­dated wrongly—his be­lieves do not re­flect re­al­ity.

Ob­serve that the out­come ‘The war­den was cor­rect.’ and it’s nega­tion ‘The war­den was in­cor­rect.’ re­gard­ing the propo­si­tion ‘The pris­oner will be sur­prised by his death.‘, given ‘He will be kil­led at noon of one day of the next five days.’ de­pend solely on the be­lief struc­ture of the pris­oner.

Given that a be­lief struc­ture is nor­mally used by an agent to max­i­mize util­ity and yet the pris­oner is not an agent (he lacks a util­ity func­tion), the be­lief struc­ture is in­con­se­quen­tial apart from prov­ing the war­den right or wrong. The shap­ing of the struc­ture is the only choice given to the pris­oner and as such it can be hardly called a struc­ture of be­lief at all.

If there was a non-con­stant util­ity func­tion over ‘The war­den was cor­rect.’ and ‘The war­den was in­cor­rect.‘, this would be a ‘be­lief-de­ter­mined prob­lem’ which is likely an in­con­sis­tent class of prob­lems by it­self—an agent try­ing to max­i­mize such a prob­lem would have to si­mul­ta­neously rep­re­sent the prob­lem and ‘be­lieve’ in things which gen­er­ally con­tra­dict­ing this rep­re­sen­ta­tion in or­der to max­i­mize the pay­off, thus mak­ing the ‘be­lief’ some­thing in­dis­t­in­guish­able from a ‘mere’ de­ci­sion.

Nev­er­the­less, in the canon­i­cal treat­ment the pris­oner en­sured by ‘in­cor­rect’ up­dat­ing that the war­den was always right.

Like­wise, we can con­struct an ‘in­cor­rect’ be­lief struc­ture that en­sures that the war­den will always be in­cor­rect:

P(Death at noon of day #(N).|Not dead af­ter noon of #(N-1).)=1

This struc­ture will be ‘sur­prised’ by any sur­vival, as it ex­pects cer­tain death each day.

Of course, this is to­tal BS from the per­spec­tive of prob­a­bil­ity the­ory, but so is the origi­nal up­dat­ing scheme.

• The main is­sue is how in­tel­li­gent the pris­oner is. As it is, the pris­oner used some clever logic to prove that he will not be ex­e­cuted that week, failing to con­sider the pos­si­bil­ity that the judge will pre­dict that. If he thought about it a bit more, he might re­al­ize that in fact the judge might well be an­ti­ci­pat­ing that, and there­fore, ex­pect the hang­man to come on any given day.

Then, if he kept think­ing, it might oc­cur to him that it is pos­si­ble that the judge pre­dicted that too, and so might not send the hang­man. How­ever, the judge is ca­pa­ble of mak­ing mis­takes. He is hu­man. So, the pris­oner can come to the con­clu­sion that he may well be hanged this week, even though it won’t be a sur­prise, or that the hang­man will not come, and in fact the judge has pre­dicted him perfectly.

This para­dox is only con­fus­ing (from the pris­oner’s stand­point) if you con­sider the judge to be in­fal­lible. He’s not. If the judge were Omega, on the other hand, we might run into some prob­lems.

• Con­sider the sen­tence “AlexMen­nen does not be­lieve this sen­tence”. If you be­lieve it, you’re wrong. If you don’t, you’re wrong.

I think the sen­tence “AlexMen­nen will be hanged to­mor­row but will be­lieve he won’t be hanged” is similar. If you to­mor­row be­lieve it, you be­lieve a con­tra­dic­tion and there­fore you’re wrong. If to­mor­row you don’t be­lieve it, you get hanged and proven wrong.

• I’m to­tally con­fused.

Why would any­body in that situ­a­tion ever be sur­prised?

I mean, they would know that some­body will ex­e­cute them at noon on one of the days (mon­day, tues­day, wednes­day, thurs­day, or fri­day). No mat­ter what day it come on, why would they be sur­prised? If it comes at noon on mon­day, they would think, “Oh, it’s noon on mon­day, and I’m about to die; noth­ing sur­pris­ing here.” If it doesn’t come at noon on mon­day, they would think, “Oh, it’s noon on mon­day, and I’m not about to die; noth­ing sur­pris­ing here (I guess that it will come on one of the other days).” Or what­ever.

(As­sum­ing that the the war­den told the truth, and the pris­oner as­sumed that.)

• This is an old prob­lem, and ap­par­ently it’s a lot harder than it looks. Wikipe­dia says ‘no con­sen­sus on its cor­rect re­s­olu­tion has yet been es­tab­lished.’

My preferred solu­tion, if the ques­tion is posed in vague enough lan­guage, is for the war­den to show up just be­fore noon on Fri­day to hang the pris­oner, while wear­ing a se­quined evening gown and scuba gear in place of his usual uniform. The pris­oner didn’t see that com­ing!

• Perfect!

• For the pur­poses of the prob­lem, to be sur­prised just means that some­thing hap­pened to you which you didn’t pre­dict be­fore­hand.

Its not the usual defi­ni­tion (among other things it im­plies I should be ‘sur­prised’ if a coin I flip comes up heads) but pre­sum­ably who­ever first came up with the para­dox couldn’t think of a bet­ter word to ex­press what­ever they meant.

• For the pur­poses of the prob­lem, to be sur­prised just means that some­thing hap­pened to you which you didn’t pre­dict be­fore­hand.

Okay, I un­der­stand that.

Its not the usual defi­ni­tion (among other things it im­plies I should be ‘sur­prised’ if a coin I flip comes up heads) but pre­sum­ably who­ever first came up with the para­dox couldn’t think of a bet­ter word to ex­press what­ever they meant.

But I don’t un­der­stand that.

I mean, why couldn’t I sim­ply pre­dict that it would be heads or tails?

• Can you ac­cu­rately pre­dict whether a coin will come up heads or tails? I can’t.

• No, but I can ac­cu­rately pre­dict that a coin will come heads or tails.

• Okay, it seems we have hit an­other prob­lem with words.

For the pur­poses of this defi­ni­tion to pre­dict some­thing means to have suffi­cient ev­i­dence to as­sign it a prob­a­bil­ity very close to 1.

• Oh okay.

• Then wouldn’t the pris­oner be sur­prised no mat­ter what?

But, wait, when ex­actly are we judg­ing whether he’s sur­prised?

Let’s say that it’s thurs­day in the af­ter­noon, and he’s sit­ting around say­ing to him­self, “I’m to­tally sur­prised that it’s go­ing to come on fri­day. I was on­line read­ing about this ex­act situ­a­tion, and I thought that it couldn’t come on fri­day, be­cause it would be the last available day, and I would know that it would be com­ing.” Or are we wait­ing for that sur­prise to dis­si­pate, and turn into “well, I guess that I’m go­ing to die to­mor­row”?

From what I can see at this point, I think that the “para­dox” comes of an equiv­o­ca­tion be­tween those two situ­a­tions (be­ing sur­prised right af­ter it doesn’t hap­pen, and then hav­ing that sur­prise dis­si­pate into ex­pec­ta­tion). But I could be wrong.

• The im­proper use of sur­prise is dis­tract­ing you from the main point here, so I sug­gest you ig­nore it.

Allow me to rephrase:

The war­den tells a pris­oner on death row that he will be ex­e­cuted on some day in the fol­low­ing week (last pos­si­ble day is Fri­day) at noon, and that on the day he is hanged he will not know he is go­ing to be hanged un­til he sees the man with the noose at his cell door.

Not as pithy, but that’s the price you some­times pay to avoid am­bi­guity.

• Sounds like a gar­den va­ri­ety con­tra­dic­tion.

(If it’s past noon on thurs­day, ob­vi­ously he would know that it’s go­ing to come the next day at noon; the war­den sim­ply would have been wrong.)

Or am I still mi­s­un­der­stand­ing it?

• If the war­den is wrong, the para­dox is no more. The para­dox de­pends on the as­sump­tion that the war­den tells the truth, and the pris­oner knows for cer­tain that the war­den tells the truth.

• Then here’s an analo­gous “para­dox”:

• There were two men stand­ing in front of me. One said that the ground was red, and the other said that it was blue. Nei­ther of them are ever wrong.

So, yeah, that’s why I said that it sounds like a gar­den va­ri­ety con­tra­dic­tion.

• That’s not ex­actly like that, is it? In the hang­ing para­dox the war­den was cor­rect in the end. There was no con­tra­dic­tion, at least not one easy to pin­point.

• the war­den was cor­rect in the end

Where?

1. The pris­oner will be hanged on one of the five speci­fied oc­ca­sions.

2. The pris­oner will never know for sure when he is go­ing to be hanged be­fore the hang­man comes.

Both state­ments are true. In your “para­dox” at least one man is wrong.

• What if the pris­oner were think­ing about it in the af­ter­noon on thurs­day?

• What about if the pris­oner is still al­ive on thurs­day in the af­ter­noon?

• He is ex­e­cuted on wednes­day. The war­den knew it all along. And even if he didn’t, his state­ments are true.

• Wait, why on wednes­day?

• The hang­man comes for him on Wed­nes­day, and he is sur­prised.

This is how it is de­scribed in the origi­nal post.

(I have a weak feel­ing that you may be mak­ing fun of me. If so, my sense of hu­mour is prob­a­bly in­com­pat­i­ble with yours. If not, please in­clude some ex­pla­na­tion to your ques­tions, I find it hard to guess what ex­actly you dis­agree with and why. Thanks.)

• Wasn’t that line (the one say­ing that the hang­man comes for him on wednes­day) just sup­posed to be an ex­am­ple? I didn’t think that the prob­lem re­quired the hang­man to come on wednes­day; I thought that it left open when he would ac­tu­ally come.

(And, no, I’m definitely not mak­ing fun of you.)

• Sorry for mis­in­ter­pre­ta­tion, then.

I sup­pose Wed­nes­day was not re­quired, but if you ac­cept the story as it is told, then it is coun­ter­fac­tual to ask “what if the pris­oner was still al­ive on Thurs­day evening”. But even if he were, since he de­duced that he couldn’t be hanged, he would be sur­prised even then, af­ter the hang­man ap­peared on Fri­day. (Some in­ter­pre­ta­tions may re­quire the hang­ing to hap­pen sooner than Fri­day to pre­serve para­dox­ness.)

This com­ment links to a good ar­ti­cle by Chow, where he analy­ses the para­dox in de­tail from differ­ent points of view, and shows that there is in­deed a con­tra­dic­tion in one spe­cific (rea­son­able) in­ter­pre­ta­tion of the para­dox, but it isn’t ap­par­ent be­cause the in­ter­pre­ta­tion re­lies on self-refer­en­tial for­mu­la­tion of the prob­lem. It is far less clear than “X says A, Y says not A, both are right”.

• Benel­liott worded it well. Your ob­jec­tions are similar to the ones I was rais­ing.

• Oh, I see.

At first, I missed the sig­nifi­cance of this pas­sage:

The war­den’s state­ment is then false and un­para­dox­i­cal. This is similar to the one-day analogue, where the war­den says “You will be ex­e­cuted to­mor­row at noon, and will be sur­prised” and the pris­oner says “wtf?”.

Does the same rea­son­ing not tell you that no­body is ever sur­prised by their death? I mean, I know I will die on some day, but if I have a fatal heart at­tack right now I’d cer­tainly be sur­prised.

• Sure, you would be sur­prised that you were about to die now.

But would you also be sur­prised that your life didn’t end up be­ing eter­nal? No, be­cause you know that you will die some­day.

But what’s the sig­nifi­cance of this dis­tinc­tion for this prob­lem? Well, I don’t un­der­stand how the pris­oner could think any­thing other than, “I guess that I’m go­ing to end up dead one of these days around noon (mon­day, tues­day, wednes­day, thurs­day, or fri­day).” It’s not like he has any rea­son to think that it would be more likely to hap­pen one of the days rather than an­other. But, in your ex­am­ple, you do have a rea­son for that (dy­ing now would be less likely than dy­ing later).

But, wait, isn’t that the whole is­sue in con­tention (whether he has any rea­son to think that it would be more likely to hap­pen one of the days rather than an­other)? Yeah, so let me get back to that.

Let’s say that the hang­man shows up on the first day at noon (mon­day). Would the pris­oner be “sur­prised” that it was mon­day rather than one of the other days? Why would he? He wouldn’t have any in­for­ma­tion be­sides that it would be on one of those days. Or let’s say that the hang­man shows up on the sec­ond day at noon (tues­day). Would the pris­oner be “sur­prised” that it was tues­day in­stead of one of the other days? I mean, why would he? He wouldn’t have any knowl­edge ex­cept that it would be on one of the next 3 days.

I’m com­pletely con­fused by this “para­dox”.

Maybe you could help me out?