I see what you’re saying. You really do think that when I say “It’s raining” I really do mean–what I really am trying to say is–not that it’s raining but rather that I believe it’s raining.
Do I have you right?
What if I point out that this means that, in your view, it’s impossible to talk about anything other than one’s own beliefs? Does that mean I misunderstood you? Or is that a bullet you bite?
-Kris
Terminological correction: A statement in predicate logic is not a predicate, but a proposition. A predicate could be thought of as a part of a proposition, but no statement translated into the language of predicate logic would yield a predicate alone.
Here’s a proposition:
Ex(Gx & Hx) <—turn that E backwards of course
In that proposition, here are the predicates:
G, H
Yes to raining question, unless it is provided at the start of the exercise as a statement of fact and not a statement from a human.
I think that within a logical exercise, we can have statements of fact that we are provided, and we can have statements from humans which can’t be given the same true/false status as the other facts provided. Unless we agree by convention that we want to elevate those to fact.
I don’t think this really causes a problem with either the judge or the rain, do you?
Yes, that is correct, it’s been a while.
Having come to understand this about your view, I now see why you would think the things trivial and obvious that you’ve been calling trivial and obvious.
For you, if I understand you right, most statements “X” really only mean “I believe X,” while (I assume you think) a statement “I believe Y” means “I believe Y.”*
If that’s your view, then there is, trivially, a contradiction in saying “X and I don’t believe X.” For in your view, that means (doesn’t just pragmatically signal, but means) “I believe X and I don’t believe X.” Contradiction.
I’m not sure what you think or should think about ianzin’s judge. The judge says “You will be hanged” and “You will be surprised.” In your view, these mean “I believe you will be hanged” and “I believe you will be surprised.” No contradiction there, since the judge is just telling us what he believes. He would not even be contradicting himself if he said “You will be hanged” and “You will not be hanged.” For these actually mean, on your view, “I believe you will be hanged” and “I believe you will not be hanged.” No contradiction there. A weird thing to say, but not, on your view, as far as I understand it, contradictory.
*I assumed you thought that “I believe Y” means “I believe Y” and not “I believe that I believe Y” because if you think the latter, the Moore stuff becomes non-trivial for you after all. For the Moore pair become:
“I believe it’s raining”
“I don’t believe I believe it’s raining.”
No contradiction, but unsayable, and strange in all the same ways Moore’s sentence pair is strange. This can’t be your understanding of “I believe Y” on my hypothesis, since on my hypothesis, your understanding makes Moore-ish sentence pairs trivially and obviously self-contradictory.
Yes to all with the caveat that when it is a spoken statement or a thought.
If, however, it is laid out as a fact by an external party to the logic problem (the teacher, the op, whatever), then it is indeed a fact (or at least a proposition).
And my assumption is that his belief is based on logically sound reasoning, as opposed to random beliefs.
What? How can those not be a contradiction?
A contradiction always has the form A & not-A. “I believe X and I believe not-X” does not have the form A & not-A.
“Believe(I, X) & Believe(I, not-X)” is not a contradiction.
Note that “I believe not-X” is not the same as “I don’t believe X.” The latter denotes a lack of a belief. The former denotes the presence of a belief.
“I believe X and I don’t believe X” would be a contradiction. But that’s a different statement than “I believe X and I believe not-X.”
As requested, I’ve reviewed your post, also the earlier ones and those which followed this one. With all due respect (and I’ve noticed you’re a smart guy and a well-regarded poster), I believe my point was well taken. Nivlac posted to say, in effect, how can you guys still be talking about this? We gave you the correct answer back on Page 1. You said he was justifiably exasperated and that the continuation of the discussion shows he was right that some people just won’t get it. What I’m saying is that is that there are several ways to look at the problem and no one of them is manifestly correct, as evidenced by the fact that professional philosphers are still wrestling with the problem sixty years after if was first propounded.
As for whether the resolution you prefer is epistemelogical, I think so for the reason Frylock stated. What follows from the logical inconsistency as you see it (obvious in the one-day scenario, not obvious and I would argue not true in the five-day one) is that the prisoner simply can’t know what to believe. That’s an epistemelogical proposition. I don’t like this resolution (not saying it’s wrong) because it doesn’t go to why most people find the problem interesting. What’s interesting is that the prisoner’s reasoning seems valid and yet, somehow, the judge’s sentence is carried out as stated. Like the Missing Dollar paradox mentioned upthread, we know the prisoner’s reasoning must be invalid, the question is why. The epistemelogical approach is one answer. To me, that backwards induction doesn’t work (for reasons articulated by many posters in this thread and several of the articles I linked in Post #99) is a better one.
But you could derive a contradiction from that information right? If your answer is no, then clearly the judge’s statement can not be contradictory, by this same reasoning, right?
Here’s the more important question back on topic (it seems like you do, but just wanted to make sure):
Do you believe the judge’s statement is contradictory?
Here’s the problem in my opinion:
“* His reasoning is in several parts. He begins by concluding that the “surprise hanging” can’t be on a Friday, as if he hasn’t been hanged by Thursday”*
On Tuesday (for example) the “if” hasn’t happened yet.
If you mean the statement “You’ll be hanged on one of the five days from Monday to Friday, at noon, and when you find out you are to be hanged on that day, you will be surprised,” then no, I don’t think the statement is contradictory. I think it’s self-defeating in ways I’ve been talking about, but I don’t think that the statement, in and of itself, contains a contradiction.
To everyone who complains that the 5 day scenario isn’t the same as the 1 day scenario, what about the 2 day scenario?
Is the 2 day scenario equivalent to the 5 day scenario? It certainly is.
"You will be hanged on Monday or Tuesday. You will be surprised about the day of the hanging. "
Therefore, if the prisoner is not hanged on Monday, he’ll know he’ll be hanged on Tuesday. But then a hanging on Tuesday would not be a suprise, and therefore can’t be hanged on Tuesday.
Therefore, the prisoner knows he will be hanged on Monday. Therefore, he can’t be hanged on Monday, because he can predict he will be hanged on Monday. Therefore, a hanging on Monday would not be a suprise, and therefore he can’t be hanged on Monday.
Therefore, the 2 day scenario is the same as the 5 day scenario. The only resolution that makes both statements from the Judge true is that, since the prisoner won’t be suprised on either day, he’ll be surprised on either day.
Note that this means that in the 5 day scenario, you can’t rule out Friday either, because the prisoner rules it out, and then when it happens he’s suprised!
And so, since the prisoner can be surprised by a hanging on Friday (since he ruled it out since it wouldn’t be suprise), he can be surprised on ANY DAY.
And this is why the 1 day scenario is the same as the 5 day scenario.
The Judge says, “You will be hanged in 10 minutes. You will be suprised by the hanging.”
The prisoner therefore reasons that since he knows he’ll be hanged in 10 minutes, he won’t be surpised by the hanging, therefore there can’t be a hanging since it wouldn’t be a surprise, therefore when the Judge hangs him 10 minutes later the prisoner is surprised.
Lemur866, the evidence I would offer that the 5 day is not the same as the 1 day, is that in the real world, I can surprise you by choosing a day early in the week (if you haven’t already, take my spoiler test on page 3), but I can’t surprise you with only 1 day. So, there is clearly a difference in the real world.
Also, as a side note, I don’t think the 2 day is the same as the 1 day or the 5 day, but it sure seems closer to the 1 day.
Did you read Pbear’s first link from his/her post with links to various papers on this, and what do you think of it?
I scanned the Ned Block article, and couldn’t digest it on a first reading so would have to go over it again later–something I don’t presently really have time for. Anticipating similar experiences in the other articles, I didn’t read them.
None of Pbear’s summaries, though, mention any of the articles claiming that the judge’s statements are in contradiction. Moreover, I confess I’d be extremely surprised if any of them did. The idea that the statements (the ones I put in quote marks in my previous post) are contradictory goes against my deepest Logic 101 instincts. The statements defeat themselves, but they are not contradictory.
I generally agree with Frylock that there is no inherent contradiction in the judge’s statement. If we take a look at ianzin’s breakdown on the first page:
(bolding added)
I do not think that the bolded section necessarily follows - the judge could be making the statement assuming that the prisoner will NOT believe it to be true. I think there are actually two separate properties that the judge is making a statement on:
It seems to be assumed that the judge saying “You will be hung tomorrow” leads to the prisoner believing “I will be hung tomorrow”, but this doesn’t necessarily follow, does it? In fact, we can see by the prisoner’s logic that for 2), one of the expected results of the judge’s statements are that the prisoner does NOT believe that he will be hung tomorrow (the other expected result is that he expects to be hung, but will not be surprised). I think Frylock is pointing out that since the fact of whether the prisoner will be hanged tomorrow is a separate property from what the prisoner believes, the two statements are not contradictory.
If you skip to page 57 you will at least see his reasoning for why he thinks the student’s logic is invalid.
Not sure why you state this, hopefully I didn’t give you the impression that they did, although other posters did make that claim about the judge’s statements.
I scanned back through the thread and I don’t see how you think the judge’s statements are problematic, can you summarize?
Also, do you believe the 1 day case is the same as the 5 day case? If so, can you provide any kind of logic that takes us from 5 days down to 1 without losing any critical elements of the problem (the most critical being that there are multiple days in the future that can not be eliminated as a possible choice with evidence to support being my spoiler test)? As a side note, in that article, Ned Block also concludes that the 5 day is significantly different than the 1 day.
Thanks, I’ll take a look.
Oh, I said it because in the post where you asked if I’d read the articles, you seemed to be asking that in response to a post of mine which you were quoting–and in that quote, what I’m saying is that the statements aren’t contradictory. So I thought you were asking if I’d checked the articles for arguments against the claim that they’re not contradictory. But I bet you didn’t mean for the relationship between the quote and your question to be that direct. Sorry about that.
In both the five day and the one day case, by saying one thing ("…you will be surprised…") the judge makes it impossible for the hearer to accept the other thing he says ("…you will be hanged…"). (Not that he makes it impossible for the hearer to believe he will be hanged, but rather, he makes it impossible for the hearer to accept the judge’s word that he will be hanged.)
I’m actually very interested in this question now that it’s been brought up. I’d instinctively say they are substantively identical, but I suspect I need to think about it more. Sorry I can’t say more than that.
But what about Indistinguishable’s point that apparently valid reasoning should lead us to be surprised in the one day case as well? “He said it would be a surprise. But if I were to be hanged tomorrow, it wouldn’t be a surprise, since I know about it already. So it must be that I’m not going to be hanged tomorrow after all.” And—surprise! He’s hanged.
That hinges on specific reasoning by the prisoner whereas the 5 day does not.
1 day scenarios:
Prisoner concludes no hanging
In this case he is hanged and surprised, judge is successful
Prisoner concludes no surprise
In this case he is hanged but not surprised, judge failed
Prisoner realizes this is a difficult logic problem and decides not to reason at all
In this case he is hanged and surprised because he didn’t know what to expect, judge is successful
5 day scenarios:
Regardless of what the prisoner reasons, he can not entirely eliminate the “hanged and surprised” result. This has been proven in real world tests. 
That’s the difference.