That’s all fine. You can tack on an implicit “Assuming the judge’s statements are true” to every line of the argument. That comes to nothing, since as it turns out in the scenario described, the judge’s statements are true.
Assuming the judge’s statements are true (which, incidentally, it turns out they are) the prisoner should still not count on his conclusion being correct, since he should realize he shouldn’t count on premise C being correct.
Things are often true while remaining unknown, aren’t they?
If questions about the truth or falsehood of future contingents is a sticking point for you, just re-imagine the scenario to involve cards stuffed into envelopes, all already present.
The future is an unknown from the prisoner’s perspective. Do you seriously think it is valid to use future results in the prisoner’s reasoning prior to the events occurring?
If so, you are really losing me with that line of thinking.
Not sure exactly what you’re asking. In my last couple of posts, I’ve come around to the view that the prisoner shouldn’t assume that, come Thursday, he’ll believe the hanging’s on Friday.
But if you are asking whether I think it’s generally okay to use statements about the future in one’s present reasoning, then of course I think that’s generally okay. Otherwise I’d have no basis for reasoning about the future at all. I couldn’t make plans of any kind in that case.
Not statements about the future, you are using facts gained from a future time period about future events to help your reasoning now about those very same future events.
I simply mean that, upon supposing your X (“I am to be hanged on Friday”), he can either suppose A (“When I find out I am about to be hanged, I will be surprised”) or not-A (which is to say, ‘When I find out I am about to be hanged, I won’t be surprised’). That’s not a contradiction; it’s like supposing the next President will be male, and then supposing either that he’ll be bald or have a full head of hair.
It’d be a contradiction if we had to assume both A and not-A. It’d be a contradiction if we had to assume a President who was both bald and not bald – or a bachelor and married, or whatever. It’d be a contradiction if we had to suppose X and A and not-A, but we don’t so long as we’re just supposing X. As long as we’re just supposing X, we can suppose A or not-A; we don’t need to suppose A and not-A.
Upon introducing Y, we suddenly need to suppose “Y and not-X, or X and not-Y” – because supposing Y would mean that supposing X requires us to suppose both A and not-A. So if it’s Y, then not-X – but if it’s X, then not-Y. Someone who can draw basic inferences would thus conclude “Y and not-X, or X and not-Y”. The prisoner fails to draw this basic inference, instead concluding “Y and not-X, doop de doo, doop de doop de doo”.
I don’t insist he must do a “suppose A” at this point. Before introducing Y, he can either suppose or reject A.
But when inferences are drawn, he invalidly concludes “Y and not-X” rather than “Y and not-X, or X and not-Y”. Doing so proves that he lacks the ability to draw basic inferences, because he bungled it upon trying to make one.
IMHO, it’s a quick bit of equivocation. Coming back to the OP: "He begins by concluding that the ‘surprise hanging’ can’t be on a Friday … he concludes it cannot occur on Friday. He then reasons that the surprise hanging cannot be on Thursday either … By similar reasoning he concludes that the hanging can also not occur on Wednesday, Tuesday or Monday. Joyfully he retires to his cell confident that the hanging will not occur at all.
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So we start off with it being referred to as a “surprise hanging” that can’t occur on Friday. It then shifts to saying “it cannot occur on Friday”, using “it” as shorthand for “surprise hanging” – and, just for emphasis, it’s again phrased as “surprise hanging”. But then it’s just phrased as “the hanging can also not occur”, where “surprise hanging” is easily implied – or easily dropped, as when it’s then called “the hanging” while falsely concluding that “the hanging will not occur at all”.
AFAICT, that shift is what’s doing all the work; it’s what makes the reasoning look plausible, since he could legitimately rule out a surprise hanging on Friday (by realizing there won’t be a hanging or it won’t be a surprise), and so can rule out “the hanging” if he means ‘the hanging in question, a “surprise hanging”, all in quotes just like that’. But he can’t rightly conclude that “the hanging won’t occur at all” for Friday if he’s quietly dropped the part about it being “a surprise hanging”.
If the prisoner can show that a Friday hanging won’t be a surprise, then it’s just one extra step to show that a Friday hanging won’t occur at all. He can just reason:
A Friday hanging won’t be a surprise.
But the judge said that the hanging would be a surprise.
Therefore, a Friday hanging is ruled out.
There is no “shift” going on, just an extra step in the logical reasoning. This was quite explicit in the OP, as highlighted in red here:
I’m losing the thread of conversation. Let me ask you this. Suppose the following are all true:
Joe is going to be hanged on Monday or Tuesday.
Joe will not have a belief about when he is going to be hanged until he’s just about to be hanged.
If Joe is not hanged on Monday, he will believe he is going to be hanged on Tuesday.
I think that if these are all three true then Joe must be hanged on Monday, and can’t be hanged on Tuesday. But do you disagree?
Here it is on Friday and I haven’t been hanged yet
Therefore, if there’s a hanging today it won’t be a suprise.
Therefore, 1 must be wrong, and there will be no hanging.
However, the prisoner could alternatively conclude:
Therefore, 2 must be wrong, and there will be a hanging but it won’t be a suprise.
If the prisoner makes it to Friday, he knows that one of the things the Judge told him was wrong. But he has no way of telling whether 1 was the lie, or whether 2 was the lie. He knows at least one must be wrong, but he can’t predict which one.
Nothing in the three sentences I listed out say anything about what Joe does or doesn’t know. They don’t even say whether he knows anything about the possibility of an impending hanging at all. Sorry I didn’t highlight that fact. I should have.
When determining what the prisoner can conclude prior to the events occurring, you can’t jump into the future, look at the results then come back to the present and say “well, we can use the fact that you will definitely get hung to help us determine if the hanging will happen on Friday.”
You can’t discard the conditional and replace it with a guarantee because there is no guarantee prior to the events actually occurring.
We are trying to determine what the prisoner can logically conclude prior to the events occurring, right? Everything he can logically conclude must include the possibility that one or both of the judge’s statements are not going to end up being true.
He doesn’t have to predict which one, so long as he sees that if he makes it to Friday, the judge will have broken his word (either way). But the judge is known to be a man of his word. So the prisoner may conclude that he won’t make it to Friday.
RaftPeople, apologies for jumping in here, but I disagree with this remark of yours.
Actually, I believe it is part of the problem that the judge is known to be a man of his word, otherwise the problem would have no interest. If the prisoner had no faith in the judge’s word, then of course the judge could surprise him with a hanging on any day of the week, even Friday. That would be trivial! The problem only has bite when the prisoner is allowed to take the judge at his word. Assuming that the prisoner can take the judge at his word, is he entitled to conclude that Friday cannot be the hanging day? This is the question.
The question was, is the argument valid. If the premises are true but the conclusion false, the argument is invalid, by definition. My point was that the premises are true. It doesn’t matter whether the prisoner can look into the future and see whether they are true or not. It’s just that they are true, independently of his epistemological relation to them. Why was I insisting they are true? Because I am asking about the argument’s validity, which turns in part on the possibility of finding an argument with that logical form which has true premises. Validity doesn’t involve the proof-giver’s being justified in holding the premises to be true. Rather, it involves what happens when the premises simply are true.
I understand your point. You are saying the interesting analysis is if we decide up front that the judge’s two statements are rock solid and guaranteed, absolute fact. The events will both come true.
To me it seems like the question is: “Can the judge pull this off?” and it’s possible the answer is no. If this is the question, then you first perform the analysis assuming your position, but you remember that it is just one analysis out of multiple that cover all possibilities. Once all analysis is complete, then you review the different conclusions and arrive at a final aggregate conclusion.
Not speaking for moving here, but to me, it seems like this is the clear intention of the puzzle. It is stated, as part of the scenario, that the prisoner is hung, and that he is surprised. It’s not that I as a reader am making a prediction and taking the prediction to be a solid one. It’s just a given. It’s given in the text of the puzzle that those two premises are true. If you’re talking about a scenario where these aren’t, or might turn out not, to be true, then you’re simply talking about a different scenario than the one people are puzzling about when they think of The Unexpected Hanging.
It looks like he starts from premises which incidentally are true even if he can’t justifiedly believe them, and he ends with a conclusion that is false. There must be something wrong with his argument, then. The trick is trying to figure out what that error is.
My own diagnosis, right now, is that he actually doesn’t start with true premises. It’s true he’ll be hung, and it’s true he’ll be surprised, but it’s not true that if he survives til Thursday he’ll believe the hanging will occur on Friday. He needs that to get his argument going, and it turns out, it’s false. In the scenario described, if he makes it to Thursday, he won’t believe the hanging will take place on Friday*, because he will already have convinced himself there’s to be no hanging.
*The astute reader will have noticed that “If X then not-Y” is not the negation of “If X then Y.” Something funny is going on there, I agree, but I’m pretty sure it can be fixed up, I’ll figure it out later. I’m trying my darnedest to avoid adding modal stuff into the argument if possible, but I may just have to put it in there. The premise may have to be something like “Necessarily, if he makes it to Thursday, he’ll believe he’ll be hanged on Friday,” for some value of “necessarily.” Then the contradicting statement would be “It’s possible for him to make it to Thursday yet not believe he’ll be hanged on Friday.”
(In case anyone cares, I still intend to come back and do the promised part 2 of my above analysis of some particular formalization in some particular modal logic of the problem [what happens once student-soundness is student-provable]. However, I will probably wait until I’m done with my qualifying exam tomorrow.)
I’m trying my darnedest to avoid adding modal stuff into the argument if possible, but I may just have to put it in there. The premise may have to be something like “Necessarily, if he makes it to Thursday, he’ll believe he’ll be hanged on Friday,” for some value of “necessarily.” Then the contradicting statement would be “It’s possible for him to make it to Thursday yet not believe he’ll be hanged on Friday.”