The Unexpected Hanging Paradox

Factual Questions
121

If anyone is still reading this thread, here’s an experiment that I hope proves the 5 day is not the same as a 1 day choice.

We can re-enact the prisoner’s week by clicking on the spoilers below in sequence. If there exists any person that is surprised by the day that the hanging occurs, then we can infer that the judges statement is not a contradiction for this one example.

Monday

No knock on door

Tuesday

Knock on door, today is the day

Wednesday

No knock on door

Thursday

No knock on door

Friday

No knock on door

If you are surprised and motivated to post, please do. If not, we can either assume that the judge’s statement is a contradiction, or that nobody really cares to respond to this experiment. :slight_smile:

122

You can be surprised even in the 1 day case, if you don’t automatically take the judge’s word as manifest truth (that is, you can be left not sure that there will be any quiz/hanging at all, and thus be surprised to discover that there is one).

Well, why not take the judge’s word as manifest truth? Well, as we’ve seen, this leads to problems. You said before that the student/prisoner could then reason themselves into ruling out Friday. Ok, suppose they rule out Friday. Then, on Friday, the quiz/hanging comes. Surprise! Something went wrong in taking the judge’s word as manifest truth, even though it was actually true. As it happens, the judge is making an assertion which they can correctly make, but you, surprisingly enough, cannot correctly believe.

Anyway, if they could rule out Friday at all on any prior day, then the reasoning that lets them rule out Friday would still be applicable on Thursday night (valid reasoning doesn’t go invalid, as I said before). So, even with just one day left, that reasoning could demonstrate itself to be invalid, and the student could be surprised, and the judge would turn out correct, and we could still ask ourselves what went wrong with the student’s seemingly plausible reasoning. That’s why the 1-day case contains all the essentials.

123

Sorry; by “you” here, I mean the student/prisoner.

It’s just as if I were to say “RaftPeople will never believe the underlined sentence in post #123 of the ‘The Unexpected Hanging Paradox’ thread on the SDMB” (though with less direct self-reference). As long as you avoid believing false things, then I will be correct to have said it; however, in any case, you yourself will never be able to correctly believe it. Just as with the judge’s assertion to the prisoner, once I start making declarations involving your very beliefs, it’s easy for me to make an assertion which is true, but which you cannot believe. That is the nub of the paradox, and it remains whether we use 5 days or 1 day. Even though it looks like there should not be any surprise in the 1 day case, there can be, because the prisoner is unable to correctly believe the entirety of the judge’s assertion (and thus, we are presumably reminded that we oughtn’t maintain that they are bound to believe any of it, including the part that asserts there will be a quiz at all).

124

But note that this is given in the article as one explanation for the unsayability of the sentence, and one that is controversial.

But like I said, it’s not important that you think there’s anything interesting here. Just that you think it’s an example of a sentence that does not contradict itself, and yet can not be said, and can’t be said for some reason that, crudely, feels contradiction-y.

You think it’s trivially obvious that Moore was right and we can’t say the sentence because when we assert something, we imply that we believe it. Others don’t think it’s so trivially obvious that’s what’s going on, or that it’s all that’s going on. (For example, note that it explains the unsayability of the sentence, but not the unbelievability of the sentence!) That’s all right–even if it’s trivially obvious why we can’t say “X and I don’t believe it” or “X will happen and will be a surprise to you”–what I was trying to get across is just that it’s not right to say that what’s going on here is straightforward contradiction. The sentence aren’t self-contradictory. The explanation for their unsayability, or their nonsensical character, or whatever, can’t be said to be that they are “logical contradictions.” They’re not.

125

But anyway, fine. If you’re unhappy with the 1 day case, just view it as a 2 day case. The quiz will be given on either Monday or Tuesday; those of us who are happiest thinking of it as a 1 day case can just think of a quiz given on Tuesday as tantamount to no quiz at all, and those who aren’t, needn’t [just as, with a typical schoolweek, one could think of a quiz given on Saturday as tantamount to no quiz at all, thus viewing “a quiz on one of 6 days, definitely” as equivalent to “a quiz on one of 5 days, or no quiz at all”]. It doesn’t really matter.

ETA: Er, that was to RaftPeople, not Frylock. Anyway, I think Frylock’s been doing an excellent job discussing things so far, while I have been and continue to be at the moment rather lazy and disinclined to do the work necessary for clear communication and patient clarification, so I’ll just let him keep at it instead.

126

I probably read too fast to see that it is controversial, can you summarize?

But I don’t think it just feels contradictory, it is contradictory as long as those two statements are being uttered by a person and that we assume individual statements uttered by people are not deceptions.

If what you mean by “not contradictory” is that the fact that it’s raining and the fact that I believe it is not can both be true at the same time then great, I agree with that.

They are logical contradictions when uttered by a person because that limits the domain from which both statements are being pulled, which is the set of things the person believes to be true. If they were 2 statements from different sources then no problem. (But note that I don’t see the judges statements as being the same as Moore’s paradox).

127

I don’t have time to respond to this the way I would like, need to run out for a while, but did you try my test with the spoilers? Were you surprised (as in could not 100 % predict) which day it was on? If so, how do you reconcile that with a spoiler test with only 1 day which would clearly be no surprise.

128

Yes, I took your quiz and I was surprised. But, like I said, there can also still be a surprise with just 1 day, if you aren’t certain ahead of time whether there will even be a quiz or not.

And if you say “Well, of course I’m certain ahead of time that there will be a quiz. The judge said there would be”, can I not retort “Well, then are you just as certain ahead of time that you will be surprised? The judge said that as well.”?

And if you manage to Moore’s-paradoxically respond “Yes, I treat both parts of the judge’s assertion the same way. I both am certain that there will be a quiz, and am certain that I will be surprised (i.e., I both am certain that there will be a quiz, and am certain that I am not certain that there will be a quiz”)", then, congratulations. You’ve managed to make the judge’s assertion wrong, at least to the extent that the judge was wrong about you being surprised. But only at the cost of you yourself being certain of something wrong, as well.

But, anyway, like I said, fine, if you don’t want to view it as a 1 day case where it’s unclear if you can take the judge at his word that there will be a quiz at all, then it doesn’t really matter, we don’t have to. A 2 day case where it is undeniably clear that there will be a quiz is just as good. So, whatever.

129

Er, I should probably say “I took your test” here…

130

Of course that’s what I’m saying. I’ve said it using pretty much your exact words several times in this thread! :stuck_out_tongue:

[spit take]

But you just said…

[recovers, shuddering]

Whatever you want to call them, they’re not logical contradictions. Logical contradictions can not be true. But the sentences (or sentence pairs) we’re discussing are true. Hence, they are not logical contradictions.

If we want to insist on calling them a kind of contradiction (which I think is misleading at least pedagogically) then call them practical or pragmatic contradictions. Or say that their being uttered involves contradiction, or gives contradictory signals. Please don’t hurt me by calling them–the sentences or sentence pairs themselves–“logical” contradictions. ;):stuck_out_tongue:

Because you don’t think the unexpected hanging can be reduced to the one day problem? Or are you saying you don’t see the analogy even if the reduction is accepted?

131

Let’s put it this way:

Suppose the school rules, set down ex cathedra by the infallible Pope himself and manifestly true to all and sundry, state that there will be one quiz, given on either Monday or Tuesday.

It’s Sunday afternoon. The judge comes up to you and whispers “Tonight, you won’t be certain that the quiz will be on Monday.”

The judge pauses, goes away, has lunch, and then comes back and whispers “Oh, yeah, and another thing. The quiz will be on Monday.”

Now, what do you become certain of, and who manages to avoid saying/being certain of false things (assuming the quiz actually is given on Monday)?

Well
A) You could take the judge at his word on both assertions. In this case, both you and the judge will believe something wrong. The judge will be wrong when he says “Tonight, you won’t be certain that the quiz will be on Monday”. You will also be wrong when you accept that assertion from the judge. So, you won’t be surprised by the quiz, but you will have been wrong about something else.

B) You could refrain from taking the judge at his word on either assertion. In this case, the judge will be correct in both his assertions (that the quiz will be on Monday, and that you will not be certain that the quiz will be on Monday). You also will manage to avoid being certain of false things. So, you won’t be wrong about anything, but you will be surprised by the quiz.

C) You could take the judge at his word on his first assertion but refrain from doing so on his second assertion. In this case, the judge will be correct in both his assertions (that the quiz will be on Monday, and that you will not be certain that the quiz will be on Monday). You also will manage to avoid being certain of false things. So, you won’t be wrong about anything, but you will be surprised by the quiz. [As you see, there isn’t much relevant difference between this case and case B)]

D) You could take the judge at his word on his second assertion but refrain from doing so on his first assertion. In this case, the judge will be wrong when he says “Tonight, you won’t be certain that the quiz will be on Monday”. You, on the other hand, will be correct about everything, and moreover, you will not be surprised by the quiz. Go you!

So, if you want to avoid believing false things, you need to land in cases B), C), or D). The problem with cases C) and D) is that you treat the two judge’s assertions differently: on what principled basis would you accept one as the certain truth but not the other? [Particularly if the judge actually told you both in the form of a single conjunction, instead of, as illustrated here, two separate assertions].

So if you want to believe only true things, and cannot come up with a compelling epistemological principle for why you should accept some assertions from the judge but refrain from accepting other ones, then you must land in case B). In which case, both you and the judge avoid claiming/believing anything false, but you also avoid forming a belief as to the date of the quiz, and so are surprised when it’s given on Monday, despite the judge having told you it would be.

132

If there is 1 day, no.

If there are 5 days, yes.

“am certain that I am not certain that there will be a quiz” this is not correct.

The person is only un-certain of which specific day it will fall on which is the nature of the surprise.

These are the judges 2 statements:

  1. An event will occur
  2. You are unable to predict on which of 5 days the event will occur (that’s the surprise).

Both 1 and 2 are true.

133

I’m human. I focus on some things and then others, can’t remember all the posts.

As uttered by the same person they are not both true at the same time.

I’m fine if we qualify them as a contradiction if uttered, but that is a trivial qualification because they are surrounded by quotes and the entire premise is that they are uttered by a person.

You can’t separate the fact that they were uttered from the possible “interesting-ness” of the statement.

The minute we list them as data points from separate sources there is no more contradiction.

Do you disagree that a statement (meaning it’s been uttered) like “it is raining” is equivalent to “I believe it is raining”?

I have seen nothing to support the fact that the hanging can be reduced to a one day problem.

If it could be shown that 5 days is the same as 1 day, then it seems clear the judges statement would be a contradiction, but first you have to get to 1 day.

134

Ok, I read through it, but this situation is not the same as the one from the op.

In this example, the judge made 2 statements that appear immediately to be in contradiction to each other.

In the op, the judge made 2 statements that do not appear to be in contradiction to each other.
I think the real problem is that the prisoner can’t rule out surprise in the earlier portion of the week by reasoning backwards in time from the last day to the first. When time moves forward, on the first day there are multiple days the judge could have chosen that would be a surprise regardless of any logic used by the prisoner.

But I don’t think the judge’s statements are contradictory and I still don’t even see an attempt to show that a 5 day choice is the same as a 1 day choice.

Do you know, has anyone actually shown this? Is there a link (I only read the wiki, didn’t see it there).

135

In the case of the Judge, I don’t think so. Take the judge’s sentences as discussed by ianzin.

“You will be hanged” ← True
“You will be surprised” ← True

Uttered by one person, on one occasion, and both are true.

What about Moore’s sentence pair? You’re right that it’s not so clear there. If I actually say, honestly:

“It’s raining” <–(Let’s stipulate this is true)

then I do believe it’s raining, and if I were to say

“I don’t believe it’s raining”

then that would be false.

The two propositions can be true at the same time, but I can’t truthfully utter sentences representing those propositions.

But the reason for my kidding about shuddering and you “hurting” me and so on is that logic doesn’t deal with utterances. Logic deals with propositions. (Having said that, on the other hand, I’m certain people have tried to draw up logical systems modeling the pragmatics of actual utterance. But generally when we say something is a “logical” contradiction, we mean “logical” in a fairly traditional sense, involving not utterances but propositions.) And the two propositions can be true simultaneously. And that means there’s no “logical contradiction” between them. There’s something weird about them when taken together and imagined as being uttered by a single person. And there may be some kind of contradiction under that weirdness. But the propositions themselves do not contradict each other. So when ianzin says they are contradictory, he makes his explication of his view difficult to follow. And this isn’t just a quibbling over usage. It’s useful and important to keep track of the difference between a logical contradiction–a relation such that two propositions can’t both be true–and these other kinds of problems that feel “contradiction-y”. The mechanics of them are different. You can do different things with them. The reasons for their deficiency are different. They’re different kinds of things.

They’re not equivalent. (You can do very different things with the two sentences, both taken as propositions and as utterances.) But I do agree that if someone says X, they (generally) thereby signal that they believe X. I think that gives you what you want, right?

136

Well, in what sense do you want them shown to be the same? They’re obviously different in some ways, just as a 7 day case would be different from a 5 day case in some ways (e.g., the one has more days than the other). My assertion is that they’re the same in that the element of the student’s reasoning which leads to a false conclusion is identically present in both cases. I’ve attempted to illustrate this by showing how the student can engage in the same erroneous reasoning in the 1 day case as in the 5 day case, and by illustrating how the 1 day case arises as the special subcase of the 5 day case where the student is sitting around on Thursday night confident that there will be no quiz. What else do you specifically seek a demonstration of?

137

You skipped Monday through Wednesday, which are the interesting days. Thursday night is not the problem. Starting with Thursday night is the same as saying there is 1 day. This I’ve already agreed to.

If there is only 1 day it’s true the judge’s statement is a contradiction. But you didn’t show how to get from 5 days to 1 day, you simply went immediately to 1 day. That doesn’t show how the 5 day can be reduced to 1 day.

In the real world, a spoiler test like the one above would not result in a surprise if there was only 1 choice, but clearly results in a surprise with 5 choices. This tells me there is a difference between the 2 situations.

138

ianzin, I truly marvel at your patience in this discussion and appreciate your comments. To me, the real paradox has become how does one explain the resolution of a paradox on a message board? You can and you can’t.

139

I think the problem you are having is looking at the person’s statement “it is raining” (which is a statement by a person, there is no mistaking that, right?) and you are extracting the information about rain and thinking that is what was just communicated.

That is not what was just communicated.

What was just communicated was a statement about the person’s set of beliefs.

If you want “it is raining” to be a statement about fact then you have to remove it from the statement from a person and make that fact known outside of the human communication process that is being analyzed.

If you are using propositional logic to evaluate then yes you would called them prepositions. I have spent a lot of time (in the past) converting written and spoken word into predicate logic and evaluating the results. In that case we had predicates not propositions.

When you convert that statement into predicate logic, you would be wise to take into account the fact that the statement is being uttered by a human, because that changes the nature of the information you have.

We were not told anywhere that factualy it is raining, we were only told what the person said. Whether it is raining or not is a separate issue, the statement is entirely communicating what is going on in the human brain.

140

So there is 1 resolution? Because the one you yourself quoted contradicted ianizn’s.