I wish I could take the credit for this, but I got it from Wikipedia.
For those not familiar with the paradox, it’s like this: A prisoner is being sentenced to death by hanging. The judge says “You will be hanged at noon some day next week, and you will not know the day until it comes”. The prisoner smiles, believing that the judge’s statements are inconsistent. If he is to be hung on Friday, he’ll know on Thursay, so Friday’s out. By the same reasoning, Thursday is out. Going like this, he concludes that he can’t be hanged as the judge said. Imagine his surprise when they execute him at noon on Wednesday.
There’s a simpler version that’s worth exploring. Suppose I say to you, “You can’t know that this sentence is true.” If you assume that it’s true, you’ve hit a contradiction, and if you assume that it’s false, the prediction it makes is borne out. So you can’t conclude anything about it. But I can clearly see that you can’t know that it’s true, and I must conclude that it is in fact true.
What’s the rub here? The odd bit is the way that the sentences in question refer to a specific agent. It’s not that no one will be surprised by the date of the hanging, but that the prisoner will be. It’s not that no one can know my claim is true, but that you can’t.
As of right now, the logic of agents and their knowledge is still pretty young. Standard formal logic can’t deal with agents, so it’s the wrong tool to apply to this paradox. That’s where the confusion comes from.
There have been several threads about this topic, and it’s beyond me to follow the logic. But in the example sited it says he will be hanged, and he won’t know what day until it happens. Because of the convoluted reasoning, he was convinced he couldn’t be hanged at all. So he in fact didn’t know what day until it happened.
ultrafilter, you’ve hit the nail right on the head.
With posts like that it’s no wonder I keep coming back to read youse guys.
My take was always that the prisoner’s sentence in the Unexpected Hanging Paradox was a meta-statement, to use Hofstadter’s terminology. Not only is that sentence referring to reality, dates and actions, but also to the logical analysis of same.
Your having pointed out that the paradox only exists for a specific analyser of the logic contained in the prisoner’s sentence is just terrific. My brother and I discussed the paradox a couple of years ago. I now have something fun to trot out at the next family dinner.
(Aside: Was it Raymond Smullyan who first brought this paradox to widespread attention?)
It just shows the danger of extrapolation. Suppose it is Thursday afternoon and the prisoner hasn’t been executed, he can safely conclude that the judge was contradictory. But if that’s only on Thursday morning, he can still be executed on that very day.
Come - oh say 4:00 PM - on thursday, and no Hanging yet, he’ll realize that Friday, being the last day of the week, will have to be the day - hence no surprise when they come for him on Friday at noon… and the logic can be (falsely!) extended to say - OK, so since it can’t be on Friday, the latest is Thursday - so if they don’t come for me on Wednesday at noon… It’ll have to be Thursday (you can see the logic crumbling here, yes?)
Martin Gardner reduced it to its simplest form in his collection of Mathematical Games essays… A guy says to his wife, “I’m getting you an unexpected surprise for your birthday; it’s that diamond bracelet we saw at Tiffany’s.”
The two statements contradict one another: is the gift the diamond bracelet, or is it a surprise? There seems no solution. So, when she opens the package and the bracelet is inside, it’s a surprise, of sorts, even though he told her, truthfully, what it would be.
And with that insight, ultimately, this is little more than a clever variant of the oldest paradox of all: “I am lying.”