The Unexpected Hanging Paradox

[The_Other_Waldo_Pepper]

@Frylock

But I think you’re right that the problem is in line 4. There’s nothing in the argument to justify the conclusion that Joe will know he’s going to be hung on T. There’s a hidden premise snuck in there. You’re saying the premise is “Joe is not an idiot.” You’re at least this right: “Joe is not an idiot” would justify line 4.

It looks to me like what you’re saying is pretty similar to what I’m saying in a way. You think there’s a hidden premise that Joe isn’t stupid. I think there’s a hidden premise that (or premises justifying a notion that) if Joe makes it through Monday, then he’ll know he’s to be hung on Tuesday. In a way, my hidden premise is a way of spelling out what it means for Joe not to be “an idiot” in this scenario. A reasonably intelligent person would generally reason that if you have two possibilities, and one’s been eliminated, then the other one is certain to be true.

Where we differ more widely is in our respective bases for calling the hidden premises. You’re saying he is an idiot because he doesn’t see that there is some alternative reasoning that would have led him to conclude he’d be hung but not surprised.

Well . . . as with the judge’s statement, I’m actually saying two things.

First off, I’m saying that the prisoner may or may not be an idiot (or however you want to put it). We’re not explicitly told, in the OP’s set-up (or in Hall’s set-up, or in Smullyan’s set-up) that the guy in question is a good reasoner; I therefore reject your hidden premise not because he’s actually an idiot, but (as you say) because there’s “nothing in the argument to justify the conclusion that Joe will know he’s going to be hung on T.” We’re not told that he’s a good reasoner; maybe he’s not; since it’s possible that he’s not, you shouldn’t add Line 4.

Now, some folks in this thread take issue with that; they say it’s not an interesting problem if we don’t assume he’s a good reasoner. This strikes me as utterly irrelevant, but I reluctantly address it with a second statement: that, as it turns out, we’re not really in a “maybe” situation; rather, it’s an “in fact, he’s an idiot” situation. I only lean on this to emphasize the possibility that he’s an idiot; we shouldn’t need evidence of that possibility, since it’s not ruled out in the set-up – but if anyone does need some, the best evidence of possibility would be, uh, actuality.

One thing wrong with what you’re saying here is the fact that what actually happens is that he is both hung and surprised. His reasoning that he wouldn’t be hung turns out to be faulty. But had he reasoned (as you recommend) that he wouldn’t be surprised, that too would have been faulty.

No, it wouldn’t; if he (a) reasoned that he wouldn’t be surprised by a Friday hanging, and then (b) got hung on Friday right when he expected it, then his reasoning wouldn’t have been faulty. There would likewise be nothing faulty in (a) reasoning that he would be surprised by a Tuesday hanging or a Wednesday hanging or whatever, and then (b) getting surprised when it happens on, say, Tuesday.

It’d be uninteresting, but accurate.

So it’s not clear to me what the basis for your recommendation is. Why should he have reasoned as you recommend, and why is he an idiot for doing otherwise, given that both his reasoning and your recommended reasoning lead to false conclusions?

He’s an idiot for reaching a false conclusion. I don’t see that my recommended reasoning leads to false conclusions.

Meanwhile, I show my hidden premise is false by a direct observation about the scenario. The premise says that once he gets through Monday, he’ll know he’s to be hung on Tuesday. But in fact, if you look to see what actually happens, you can see that the premise is false. He gets through Monday, and continues to believe (as a result of his reasoning) that he won’t be hung at all. I’ve spotted a hidden premise which can be straightforwardly shown false simply by direct appeal to facts given in the scenario. That seems adequate to me. Why not to you?

You show the hidden premise is false by making a direct appeal to the facts shown in the scenario. I never included the hidden premise in the first place, because it wasn’t among the facts of the scenario to begin with. If you want to insert two extra steps, then by all means suggest a premise before showing why it’s false; I simply don’t bother with the premise at all, thereby solving the problem…

…until and unless someone like Indistinguishable starts in about how “in the scenario as generally presented, there’s no reason for the prisoner to fail to draw the belief that he will be hung (on the model of the rational prisoner I have adopted; one who draws all logical consequences of the judge’s assertions as his beliefs)”, at which point I need to spell out that, no, see, first off, we’re not told that the prisoner will draw that belief, and so you shouldn’t adopt such an assumption in that regard – and, for bonus points, as it happens, the prisoner doesn’t in fact draw that belief.

But my argument is correct even before we get to any of that other stuff; so long as we don’t insert a “hidden” premise in the first place, then we can simply note that he rules out a hanging rather than ruling out a surprise hanging, whereupon the hanging both occurs and surprises him.