Every islander can see either 99 or 100 Blue in the scenario as written, and I can conclude that everyone else can either see 99 or 100 because there is only three possible numerical outcomes as I spelt out above. The setup that they are perfect logicians and don’t realise that the minimum is 99 blue eyes (and that everyone else has concluded the same) doesn’t make any sense.
coremelt, consider the case where there are 3 islanders (only because I am too lazy to write out the repetition for 100 islanders), Bob, Fred, and Joe, all with blue eyes:
Yes, each can see two blue eyes. Yes, everyone knows that everyone knows that there is at least one person with blue eyes.
BUT:
Since Bob does not know his own eye color, Bob thinks he might have brown eyes. Because Bob thinks he might have brown eyes, Bob thinks Fred might see Bob with brown eyes. Bob also knows that Fred doesn’t know his own eye color. So Bob thinks it might be the case that Fred thinks it might be the case that both Bob and Fred have brown eyes. Which is to say, Bob thinks it might be the case that Fred thinks it might be the case that Joe sees only brown eyes. Which is to say, Bob thinks it might be the case that Fred thinks it might be the case that Joe thinks it might be the case that everyone has brown eyes.
So it isn’t the case that Bob knows that Fred knows that Joe knows that someone has blue eyes. Don’t think, but look!
After the Guru makes his announcement, though, the existence of blue eyes becomes common knowledge. After the announcement, Bob knows that Fred knows that Joe knows that at least one person has blue eyes.
Originally, there is a path of length 3 from island (1, 1, 1) to an island with no blue eyes. So it isn’t the case that everyone knows that (iterated 3 times) there are blue eyes. The Guru’s public announcement serves the important role of cutting off the island with no blue eyes; he makes the existence of blue eyes common knowledge, where it wasn’t before. It’s not that what the Guru is saying is news to anyone; it’s the fact that he’s saying it publicly that has an effect.
In your case if they don’t all realise that the only possibilities are (2 Blue, 1 Unknown) or (3 Blue) AND that everyone has already come to that conclusion then they are not perfect logicians.
The solution to the puzzle requires that they are perfect logicians about inductive logic but stupidly ignorant about the possible set combinations that everyone else can see given what they know.
coremelt, do you agree that Bob thinks it’s possible that Fred sees Bob with brown eyes?
Do you agree that if Fred were to see Bob with brown eyes, Fred would think it’s possible that Joe sees only brown eyes?
Do you agree that if Joe were to only see brown eyes, Joe would think it’s possible that everyone has brown eyes?
I know Bob doesn’t actually have brown eyes. I know that. You know that. But Bob doesn’t know that.
Bob knows that Fred doesn’t actually have brown eyes. I know that, you know that, Bob knows that. But Fred doesn’t know that. And Bob knows that Fred doesn’t know that.
Bob thinks it’s possible that Bob has brown eyes. Bob knows Fred thinks it’s possible that Fred has brown eyes. Putting these together, Bob thinks it’s possible that Fred thinks it’s possible that Bob and Fred both have brown eyes.
Which means Bob thinks it’s possible that Fred thinks it’s possible that Joe sees only brown eyes. Bob himself knows that Joe doesn’t see only brown eyes, but Bob thinks it’s possible (in the case where Bob has brown eyes, which Bob can’t rule out) that Fred thinks it’s possible (in the case where Fred has brown eyes, which Fred can’t rule out) that Joe sees only brown eyes.
For what it’s worth, avoiding confusion about this sort of thing is the whole reason I brought up bridges in the first place… Could you read post #27 and tell me how you feel about that situation, coremelt?
Another way of thinking about it is that the “Bob knows that” operator works like this: Bob knows that some property of the island configuration holds just in case that property would still hold even if Bob’s eye color were changed around arbitrarily. And similarly for “Fred knows that”, and “Joe knows that”.
So to say “Joe knows that there is at least one blue eyed islander” means “Even if you changed Joe’s eye color, there would still be at least one blue eyed islander afterwards”.
To say “Fred knows that Joe knows that there is at least one blue eyed islander” means “Even if you changed Fred’s eye color, even if you then changed Joe’s eye color, there would still be at least one blue eyed islander afterwards.”
To say “Bob knows that Fred knows that Joe knows that there is at least one blue eyed islander” means “Even if you changed Bob’s eye color, even if you then changed Fred’s eye color, even if you then changed Joe’s eye color, there would still be at least one blue eyed islander afterwards”.
But if you were to change everyone’s eye color, there wouldn’t be any blue eyed islanders left!
The proposition “X knows that Y knows that Z knows that … P” means “P is true, regardless of the eye colors of X, Y, Z, …”. If X, Y, Z, …, covers everyone, then this means “P is true regardless of everyone’s eye color”. Which of course isn’t the case for P = “At least one person has blue eyes”.
I don’t believe that the Bridge situation is the same, the guards have more clearly defined rules about how they act.
In the scenario as written is inconsistent, they can’t both be perfect logicians and not realise what the possible set combinations are (and what everyone else knows they are) at the start of the scenario, namely in the scenario as written, everyone knows that everyone else can see either 99 or 100 pairs of blue eyes because they can all work out the only possible outcomes.
coremelt, I propose you actually play this game:
You, Ximenean, and Little Nemo are on an island.
I know what everyone’s eye colors are, but I won’t tell everyone everything just yet. I will tell you what you personally can see:
(Note: This information is for coremelt only)
Ximenean and Little Nemo both have blue eyes
Would you be willing to bet twenty bucks that Ximenean has enough information to prove that Little Nemo sees blue eyes? Are you that confident that Ximenean knows Little Nemo can see blue eyes?
Ok, now, when Ximenean returns, I will tell them what they personally can see. Then I will offer them a bet: if Little Nemo can see blue eyes, I will give Ximenean twenty bucks. Otherwise, Ximenean will give me a hundred bucks. We’ll see how confident Ximenean is in their chances of winning this bet (and also ask them why they feel confident or unconfident).
Twenty bucks? I’m in.
Er, um, even before I tell you what you can see? I was hoping you’d be a bit more risk-averse… You could lose a hundred bucks!
Why such premature confidence? [What if I told you I’ve selected all the eye colors by random coin flip?]
I mean, I will consider taking the bet after you tell me what I can see.
Ah, excellent. Alright.
This information is for Ximenean only. If you are not Ximenean and you read this, you are cheating:
You can see that coremelt has brown eyes and Little Nemo has blue eyes
Alright, Ximenean, here’s the offer: if you like, we can make a bet where you win twenty bucks if Little Nemo can see blue eyes, but you owe me a hundred bucks otherwise. (And your house. And a car. And firstborns of each gender.)
Let me know if that bet sounds appealing to you or not, and explain your reasoning as well (keeping in mind that I’ve chosen these eye colors by random coin flip). Take as much time as you like to make your decision; once you do, please post it in a spoiler box, which no one should read until I tell them to.
In the meanwhile, I’ll throw things back to coremelt… coremelt, I previously asked you (within your spoiler box) a question, which I would like to see your answer to.
I wasn’t sure what you meant here–were you saying to “look” at the reasoning you typed out?
Yes, and also making a Wittgenstein allusion for no reason.
Oops, I didn’t recognize it. Bad philosopher. BAD philosopher.
Look I do get the logic behind it and what you’re trying to prove. I’m challenging the premises.
a) The gurus statement doesn’t change anything, as each person could see either 99 or 100 blue eyes and everyone on the island knew that. It doesn’t regress back to thinking it’s possible that someone can’t see any blue eyes. It’s only possible for someone to think there is one less than what you can see. It stops there.
b) on day 100 no one would leave, even if you accept the inductive logic, they are all waiting to see if anyone else moves to know if they have blue eyes or not. It’s a deadlock situation in programming terms, no actor can act without knowing what the others will do, but they are all in that state, so no one leaves.
You seem to be copping out of playing the game. Just play the game. If you play the game, you’ll understand what the rest of us are saying. If you don’t play the game, it’s hard to claim you understand better the way the game plays out.
Just tell me, are you confident, from the information you have in your spoiler box, that Ximenean will take the bet offered to them or not? Why or why not?
I assume you are self-chiding here. 
OK, here is my decision:
[spoiler]I decline the bet. Since **coremelt **has brown eyes, I don’t know if Little Nemo can see any blue eyes. I’ll only win if my own eyes are blue, and that appears to be a 50% chance.
(I hope I’m doing this right and not messing up your illustration!)
[/spoiler]