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.