But when the prisoner performed his reasoning, those premises were not true. The status of those premises was unknown.
The prisoner was trying to determine if those premises could both come true given that he has full knowledge of those premises.
But you can have a valid proof that only encompasses a subset of the possibilities surrounding a situation. It would be a mistake to think that the valid conclusion from that proof can be applied to the entire set of possibilities.
It seems that you and moving are on the same page in that the question you want to examine is:
Given that there is zero possibility that either of the judge’s statements turns out to be false, did the prisoner reason correctly?
And for me, it seems like the interesting question is:
Can the judge pull off this trick? Can he make both statements and still have both events happen?
Am I correct about our differences in which question is being analyzed?
Just now noticed your other posts, will read them now.
What I said last time you said this was that if you don’t think statements about the future have truth values, then it should be easy enough to recast the puzzle in terms of sentence that aren’t about the future.
You replied that it’s important that even in such a recasting, the truth value of the sentences involved is unknown to the prisoner.
My response was that it doesn’t matter, when we’re talking about validity, whether statements are known or unknown to various people. It just matters what happens when they’re true. (Not when they’re known but when they are true.)
That the status of the premises is unknown when the prisoner engages in his reasoning is immaterial. The fact is, known or not, they’re true.
You’ve replied by saying they’re not true because they’re unknown, which makes me think, once again, that the sticking point for you is that you don’t think statements about the future have truth values, after all!
Not quite. For one thing, I wouldn’t put it in terms of possibility, mostly for arcane semi-technical reasons.
I’d put it this way: The statements are true. So how can we explain the fact that the prisoner ends up with a false conclusion? Is his argument invalid? Or are there further premises he’s relying on that are false?
My diagnosis is that he’s relying on further premises which are false.
Isn’t the answer to this pretty clear? He does make both statements, and both events do happen. These are just givens in the scenario.
Yes, but when the prisoner is performing his reasoning, the prisoner does not know what will happen.
If our analysis is about the prisoner’s reasoning, then we have to sit in his shoes with the exact same information available to him.
We can certainly analyze the puzzle from a different perspective (that of after the fact with additional information not available to the prisoner, which is also interesting), but when we do that we are not merely analyzing the prisoner’s reasoning.
But why are they true? Because that particular prisoner concluded he would not be hanged. But just because that particular prisoner concluded he won’t be hanged doesn’t mean it was a good/logical/reasonable conclusion.
To me, it seems the question is: What should the prisoner have concluded given the information that he had?
That’s not true. If we’re evaluating the soundness of his reasoning, all we need to know is whether his premises are true, and whether his inferences are licensed by logical rules. This follows from the definition of soundness–a sound argument is a valid argument with true premises.
But maybe by “analysis of the prisoner’s reasoning” you mean something other than figuring out whether it’s sound and so on. For example, I can imagine someone being concerned with the question whether the prisoner is entitled to believe the premises. (He might not be entitled to them even if they are true.) Is something like that what you have in mind?
Okay, but that’s a pretty different question than “Can the judge pull it off?”
My answer to your question is that the prisoner should have realized he can’t assume he’ll think “I’m going to be hanged on Friday” if he survives til Thursday. He can’t assume he’ll think that, because as a result of the reasoning he’s engaged in, he’s actually not going to think he’ll be hanged on Friday if he survives til Thursday. Instead, he’ll be thinking he’s not going to be hanged at all.
So, once he finishes his reasoning, he should go back and review it, as is always prudent, and he should notice that if he accepts the conclusion of the reasoning, then the premise about what he’ll think come Thursday is false. That means if he accepts the conclusion, then the argument–having a false premise as it does–is a bad one, and he shouldn’t accept the conclusion. In short, he should realize that if he accepts the conclusion, he shouldn’t accept the conclusion. That means, basically, that he shouldn’t accept the conclusion. I think.
Given that at the time of his reasoning he has no idea whether the premises are true or not, then we can’t really tell if his argument is sound at the time of his reasoning, right?
To me the more interesting question is what can the prisoner conclude at the time of his reasoning. Checking his logic after the fact is interesting also, but it’s not the same problem.
The prisoner’s reasoning occurred prior to Monday, therefore “analysis of his reasoning” means checking his logic as if we were in the cell with him and we are just trying to make sure he didn’t make a mistake.
From that perspective, the premises (A+B) and the subsequent logic are a smaller part of the overall reasoning that a logical person should perform. Part of the prisoner’s error is the fact that he misses this point and assumes that he has covered all possible situations.
The point I was making in both cases is that, to me, the question is about the prisoner’s reasoning given the information he has available. Both questions are similar in that regard.
Oh dammit. But now that he shouldn’t accept the conclusion, he should accept it. For upon his failing to accept it, things are now such that the premise about what he’ll think on Thursday is true. For if he decides to ignore the reasoning of the argument (i.e. decides not to accept its conclusion) then come Thursday, he will think he’s going to be hanged on Friday.
If he accepts the conclusion, the premise is false and he shouldn’t accept the conclusion.
If he doesn’t accept the conclusion, the premise is true and he should accept the conclusion.
I haven’t had a chance to properly think about what you posted.
Either way, I am interested in seeing what can be concluded on prior days if we make his conclusion about Friday as follows:
“Given that it’s Sunday and I don’t know what will happen, but I’ve reasoned that I can’t accept both of his statements as true and have both events occur on Friday”
Note: In my opinion, the prisoner has correctly stuck in what amounts to a conditional (can’t accept A+B and have both events happen on Friday), in this case he is still leaving open the possibility that both events may not come to pass.
But it’s an incorrect extra step: once the prisoner concludes that a Friday hanging is ruled out, a Friday hanging would be a surprise. Once he shifts from ‘ruling out a surprise Friday hanging’ over to ‘ruling out a Friday hanging’, he can be surprised by the Friday hanging he just ruled out.
If all three are true, then, yes, he can’t be hanged on Tuesday.
Now 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.
Joe is a blithering idiot who drools on himself and falls down a lot.
IMHO, if all three are true, then Joe can be hanged on Tuesday. Do you agree?
No, each of you is leaving aside the other possibility: that the prisoner is a blithering idiot, possibly one who drools on himself and falls down a lot. If such a prisoner makes it to Friday, he might incorrectly conclude that there won’t be a hanging, whereupon a hanging can then occur, which would surprise the heck out of him, in which case the judge was telling the truth about both.
Wait, why do you think I’m leaving aside that possibility? I haven’t said anything at all about the prisoner’s reasoning abilities. I’ve argued sometimes in this thread that the prisoner himself has an opinion about his own reasoning abilities, but I haven’t made a claim about whether he’s right about that or not.
That may be true, but pointing that out doesn’t really get to the bottom of things. Agreed, if the prisoner (somehow) manages to persuade himself that Friday is ruled out, then he indeed stands to be surprised by a Friday hanging! I do agree with that. But what matters here is the “somehow”. How on earth did the prisoner manage to persuade himself (to begin with) that Friday was ruled out? We need to tackle that more directly because if you look at his actual reasons for ruling out Friday, they are not half-bad. Let’s be fair to him?
For instance, as before:
A Friday hanging won’t be a surprise. (You already agree with this.)
But the judge (a man of his word) said that the hanging would be a surprise. (Given.)
Therefore, a Friday hanging is ruled out.
This is one way to lay out his reasoning. There are other ways as well, as shown by Frylock and Indistinguishable. We need to address the prisoner’s reasoning “directly”, if you know what I mean. Yours is an “indirect” way of faulting his reasoning. It’s fine as far as it goes, but it doesn’t go far enough.
Incidentally, are you guys familiar with Raymond Smullyan’s “solution” to this paradox? It’s quite short and I think that some of you will like it because some of the things said on this thread remind me of what Smullyan said. I will dig it up over the weekend and post it here if anyone is interested. Just let me know!
Yes, it’s a bit hard to believe, but I think the fact is that this paradox is really up there with the best of them. It belongs in the top ten paradoxes of all time, in my view, taking into account all relevant factors including entertainment value and difficulty of resolution. The paradox is really very confusing.
The part about no consensus doesn’t mean that everything is up in the air. There does exist some agreement on the rough direction of resolution. The disagreement lies mostly in the details. It is true that no “textbook solution” has yet been agreed upon, of the sort available for the Monty Hall problem, say. In this sense, the paradox has not been fully resolved. But it isn’t true that academics think it “cannot be resolved”. That overstates the case.
So what is the solution to the paradox, even roughly? Unfortunately, it cannot be explained in five minutes or so. At least, not by me. Some degree of antecedent logical skill is necessary to appreciate the solution, e.g., one must already be fluent with concepts like proof by contradiction and begging the question, not to mention such standard distinctions as validity vs. soundness, assumption vs premise, truth vs knowledge.
I believe that anyone can understand the (rough) solution if they are prepared to master the logical apparatus. On the other hand, you could also try hearing the explanation given by Raymond Smullyan, who is good at explaining the crux of things in plain English.
I’ll post Smullyan’s solution to the paradox very soon, unless there is some reason I should not. I take it that this section of SD is devoted towards finding answers to factual questions, and it’s quite possible that you will like Smullyan’s explanation. Personally, I don’t think his explanation gets to the bottom of the paradox, but I do think it forms a central part of the solution. Various bits of his explanation have already been anticipated by many people in this thread, but Smullyan has the enviable gift of explaining exactly what he means very concisely. It also helps that he is highly respected, so people tend to read what he says very carefully.
Anyway … soon … I have to dig it up first. It’s in one of his books and I believe that I have a photocopy somewhere, if only I can find it …
I know this is meant to be a joke, but I’ve been wondering if the sword for this particular Gordian Knot lies in the question: what happens if every day at 11:55 the prisoner sits on his cot staring expectantly at the door and if anyone walks in he says, “I’ve been expecting you.”
Would he be spared?
Forgive me if this has been brought up already, I have no kept up with every post in the thread.
