Do you mean this question in a general sense, or do you mean to ask it only pertaining to the situation in the puzzle?
But that’s the thing, the Guru’s statement doesn’t stop there. Being a public statement, it is equivalent to “there is at least one blue, and everybody knows it, and everybody knows that everybody knows it, and…” ad infinitum.
Three people on the island, two with blue eyes and a third (the Guru) with green eyes. All have been told that there are 1 or 2 people with blue eyes. In this case however the Guru is bound by the same rules as everyone else.
On the first day there we watch the ferry leave with no one on board. The next day both I and the other blue-eyed guy are on it waving farewell to the Guru. Nothing has been said. The fact that neither of us boarded the boat the day before was enough for us to work out we both had blue eyes. Why would that not work for larger numbers? In other words, does the Guru even need to speak?
How do you get that the Guru’s statement stops at two layers? I can understand someone (incorrectly) thinking it’s only one layer, but if you can get from one layer to two, then you can in the same way get from two layers to three, and from three layers to four, and so on.
What do you mean by “all have been told”? Did someone get us all together in a room before we went to the island and say to everyone at once “There are 1 or 2 blues among you”? In that case, you’re effectively just making someone else the Guru, and the problem proceeds the same as in the original version. Or do you mean that this person took each islander aside, one at a time, and told them this, without saying that they’d told anyone else? In that case, nobody learns anything new, and nobody leaves, since nobody can know on what basis anyone else is reasoning.
I meant it in the puzzle, but I suppose I could mean it more generally, too.
If I know a fact, others might not know it.
If I know that everyone knows it, others might not realize that everyone knows it.
But if everyone knows that everyone knows it, then I don’t see how adding layers is adding to anyone’s information, as we all know that we all know that we all know.
This may be the key that unlocks this thing for you. Unfortunately I can’t type any more right now. Back soon.
Mithras, you seem to have significantly misunderstood what I was trying to say. I wasn’t talking about who would leave when, I was talking about what I know about what other people know just from what I can observe, and specifically how in the scenario described the Guru didn’t say anything anybody didn’t already know.
Here is a simple example
Consider a world where the population is 3: A, B, C
- Everyone knows A likes cookies
- Everyone knows everyone knows A likes cookies
Question: does A know B knows C knows A likes cookies?
Hint: you cannot prove this using any combination of the first 2 axioms, which shows that 2 layers of information is not sufficient to denote common knowledge.
I see my error. I was taking the possible numbers of blue eyes as a given, or rather something the islanders all knew, as I thought the original puzzle did but reading it again that’s not what it says.
Ignorance fought. (What is it about this puzzle that even when I think I’ve got it I keep returning to worry at it like a dog with a bone?)
Well yeah, I think it is extremely hard to intuitively see the difference between
Everyone knows that everyone knows x
and
Everyone knows that everyone knows that everyone knows x
I’m certainly not sure that I can intuitively grasp it. However, logically I accept that the statements are different, and then the whole “top-down” explanation of the puzzle falls into place.
Okay, you said:
What you’re claiming here is that the following statement P is true:
P: If, for some statement X, everyone knows that everyone knows that X, then everyone knows that everyone knows that everyone knows that X.
But P is false. I’ll give you a counterexample to it. Sorry, I’m having trouble coming up with anything particularly mundane, but I think the following example works:
John has a hundred walkie-talkies, and each of them is paired with another walkie-talkie held by an experimental subject in another room. Each subject is in his own room. The subjects believe everything John tells them because they know him to be trustworthy. The subjects all know of each other and their place in the scenario, but can not communicate with each other in any way. They do not know anything about what the others have heard, unless John has told them what the others have heard. They know, also, that John sometimes tells all of them something, and sometimes only tells some of them something.
One by one, he says into each of the walkie-talkies “Ulaanbataar is the capitol of Mongolia.” Call this sentence “X”.
Once John has gone through each of the walkie talkies, the following is true:
Everyone knows X.
None of the subjects at this point knows whether everyone knows X–because they don’t know whether John told X to all of them or only to some of them.
Now, John again goes through each walkie talkie and says “I have now told the person at the other end of all of my walkie talkies that Ulaanbataar is the capitol of Mongolia.”
After John has gone through each of the walkie talkies this time, the following is true:
Everyone knows that everyone knows that X.
But, it is FALSE that:
Everyone knows that everyone knows that everyone knows that X.
Because for everyone to know that, John would have to speak into each walkie talkie, saying “I have now told all 100 walkie-talkie carriers that everyone knows that everyone knows that X.”
I was going to say that I know there is at least 1 person with the letter y in their user name that posted to this thread and if you figure out that you’re the missing one then post immediately but that’d be hard to simulate since everyone everyone really does know their own user name. (wouldn’t be days it would be minutes to wait for).
if anyone posts earlier than the other then i’d say it didn’t work out correctly?
You’re suggesting something like the game offereed by Indistinguishable and Chessic Sense earlier in the thread. I was never sure why no one wanted to play either game.
Mmmm … walkie-talkies.
And cookies and game shows and bridges and binary digits.
Five-letter passwords and four islanders.
Three blue-eyed people.
Two horny people in a bar.
And a guru with no eyes.
Okay, here’s another example of an attempt at demonstrating the solution:
If everyone who knows the solution knows that there’s at least one example that demonstrates the solution, it doesn’t mean that everyone who knows the solution knows that everyone who knows the solution knows that there’s at least one example that demonstrates the solution.
Also, if the example fell in a forest and nobody heard it, would it demonstrate the solution?
And, is the recursive nature of this thread an example of common knowledge, or is like Phil Connors trying to screw Rita?
Man I still don’t get it and it’s not just t hat. I don’t even accept that it can be done. Anyone at least know something like a real live test where the ones who don’t believe can at least observe and see it happen that way I can at LEAST agree it’s true? because after that I would shut up and just admit i’m not smart enough to comprehend.
We could try to run through it, maybe?
You’re in a room with two other people. Each of them has a stand in front of him. On the stand, where you can see it, is a colored card. Neither of them can see their own card, but everyone can see everyone else’s card.
If anyone looks at his own card, he will be executed. So don’t do that.
You can exit the room only if you correctly name the color of your card. You get one guess, and if you’re wrong, you’re executed. So don’t do that either.
What you see is that both the other players (Bob and Carl) have green cards. You have no idea what color card you have.
I won’t make you solve out the whole thing by yourself. Instead I’ll act as part of your brain that leads you to ask the right questions. We’ll see if this helps.
First question: Is it possible that, according to Bob, you have a yellow card?
The ‘Common Knowledge’ link is the explanation that finally convinced me.
I have to admit I have no idea why that would have convinced you if this thread didn’t. But I’m glad in any case.
This, combined with Frylock’s thought-balloon drawing (the improved version), is what finally did it for me.
Thank you both!