Yes, he might be wrong in this case, especially since he himself does not know if there is a flaw. He said his friend told him there was.
Regardless, I would still like to know if there is something wrong. If you don’t think there is a flaw, please address parts A, B, and C of my post.
I also have implicitly assumed what you have assumed.
Yes. Unless the oracle also told the blue-eyed people she/he/it communicated “there is at least 1 blue-eyed person” to all other blue-eyed people on the island. However, that condition wasn’t mentioned in your original post, so for the sake of analysis we must assume it was a public announcement.
I think we are miscommunicating here; I agree with you so far. What I find puzzling is that the public nature is important. By announcing old information pubically, the oracle provides new information. Doesn’t that seem strange to you? Even if there is nothing wrong with it, to me, it seems a bit counterintuitive.
I meant publically, of course.
Or the professor’s logician was right. If the OP here is exactly how the professor posed this problem, he left out the necessary other conditions you point out above. The last 2 conditions seem implicit to me in the problem. However, unless the oracle also states “all blue-eyed islanders must off themselves by jumping into the volcano at exactly midnight of the day they realize they have blue eyes”, if they wait around for the other islanders to off themselves first, this proves a problem. Adding on this condition solves the problem:
“The oracle orders all islanders will meet at the volcano just before midnight each day, and close their eyes. The islanders also bring along a clock that will chime at the instant of midnight. If on that day an islander realizes they have blue eyes, they will jump. One minute after midnight all islanders (or, all surviving islanders) will open their eyes.” If there are 3 blue-eyed islanders, all the brown-eyed islanders will open their eyes at the end of the third day to discover that the blue-eyed islanders had jumped.
It is new information on the possible scenario that there is 1 blue-eyed islander. If there is only one blue-eyed islander, that islander wouldn’t have realized this without the oracles announcement. Only all the rest of the islanders would have been aware there was at least one blue-eyed islander.
Specifically, the information that each blue-eyed person must have in order for the suicides to happen is a set of n-level statements of the form “He knows that he knows . . . that there is at least one blue-eyed person.” Statements at the (n-1)th level won’t do; each blue-eyed person already knows this by looking at all the other blue-eyed people. (I don’t know how to state this more rigorously without making it very cumbrous.) By making a public announcement, the oracle allows the islanders to conclude n-level statements for every n. That is how the public announcement helps. I understand all this, but I still find it strange; perhaps, I am the only one.
Maybe, there is not much more to be gained by further analyzing part B (unless anyone knows of any cryptographic or information theoretic work about how the public nature of knowledge affects things.)
Anyone care to address part A or C?
First, allow me to indulge myself in busting out the logic in the problem, in case anyone’s still trying to figure it out:
N=1
The one blue-eyed islander sees that no one else has blue eyes. As described above, the islander will immediately realize that he must be the one with blue eyes and will throw himself into the volcano at midnight.
N=2
Each blue-eyed islander will see one other islander with blue eyes. Using the logic from the case of N=1, each of those islanders knows that if the other blue-eyed islander is the only one that they would have killed themselves at midnight that first day. Since they see each other still alive the next day, they both realize that the other blue-eyed islander isn’t the only one with blue eyes, which would leave themselves as the other. Hence, they both realize they have blue eyes on the second day and throw themselves into the volcano at midnight.
N=3
In this case, each blue-eyed islander sees two others with blue eyes. Using the logic for N=2, all three will realize they have blue eyes when they see each other still alive on the third day, and all three will throw themselves into the volcano at midnight.
…and so on. This is why all N islanders will jump in the volcano at the end of the Nth day.
I doubt point A was the flaw. Even if something does happen to one of the blue-eyes outside of the normal constraints, everyone still knows how many blue-eyed people they saw on day 1. The only way for any one of the blue-eyes to survive would be if all of the others somehow discovered their eye color through other means on day N-1. That’s a bit of a stretch, particularly because it necessitates something to happen outside of the boundaries given in the problem (someone else would need to break the taboo and tell them they have blue eyes, or they would need to see their reflections).
For point B, the oracle’s only role is to publicly announce that there is at least one blue-eyed islander, hence starting the countdown. Before that, every blue-eyed islander is unaware that they have blue eyes due to the taboo and no way to see their reflection. With the public announcement, each blue-eyed islander will count N-1 others with blue eyes, so they will all wait until day N to see if the others had gone crater-diving the night before.
Even if the oracle herself has blue eyes, the same logic should follow. The only problem here would be if the oracle is the only one with blue eyes. In that case, however, how would she know? Perhaps this is your professor’s friend’s objection?
D
Because she would see know no other islanders with blue eyes, and just kill herself at the volcano at midnight the first day based on the revelation from god there was at least one person with blue eyes on the island.
For the purposes of dicussion, please assume that the oracle is not a member of the tribe. Assume the oracle is a god or some divine entity.
This seems to be intuitively true, but can you prove it? How do you know the blue-eyed islanders don’t all discover that they have blue-eyes on the first day? Maybe, they found some proof that you missed. Prove that if there n blue-eyed islanders, no blue-eyed islander can prove that he has blue-eyed before the nth day.
Point taken, though I liked rfgdxm’s take on it. The irony!
You may be on to something. My elimination of point A is based on the setting given in the problem, specifically:
The blue-eyed islanders have managed to live so far, because to speak of eye color is taboo, and there are no mirrors or reflective surfaces of any kind on this island (assume the water is too murky).
Given this, the only information that they should have is how many others they see with blue eyes and the day the oracle makes the announcement. The proof (as best as I can communicate it) is the solution I gave above. This would require all of the islanders to make the same deduction and wait until day N.
If N is greater than 2, then everyone regardless of eye color will know that they will continue to enjoy the company of their blue-eyed friends until at least day N-1. On day N, though, when each blue-eyed islander sees that the others are still alive, they will all become aware that they have blue eyes and jump into the volcano at midnight, much to the relief of everyone else.
I don’t see any reason for any of the islanders to (correctly and reliably) discover their eye color any other way with what we know they know. If there’s another quicker way, I’ll leave it to someone more skilled than me to show it.
While thinking this through, though, I did come up with couple of other anomalies that would throw this off:
No one enforces the law. Since the law prescribes guilty parties to inflict the punishment upon themselves, everyone who sees N-1 blue-eyed people on day N (or N people on day N+1) assumes that the others just decided not to go through with it instead of deducing that they themselves have blue eyes. Since to call them on it would effectively break taboo, this might be plausible.
N=2. Oracle announces at least one islander has blue eyes. One blue-eyed islander is shocked by the news and immediately dies of a heart attack. Or, you could still have the island raided by pirates who slaughter all but one of the blue-eyed natives.
Of course, in both of these cases, if N=1, he is still doomed.
D
I don’t really think these objections qualify as loopholes.
It should be enough that by day N+1 none of the poor N blue-eyed kids you have observed have killed themselves. The conclusion is inevitable, and you kill yourself, volcano or no volcano, sometime that day. It is not necessary to know exactly when anyone dies, or in fact how many have killed themselves. If the suicide orgy begins at day N, you’re off the hook, otherwise not.
Fair enough, but that’s a practical objection, and not really a hole in the logic.
Well, I don’t know. The problem can be approached either way. If there are N>0 people with blue eyes, N people will kill themselves at day N (if we say the same day is day 1.) If N=0 – what do you think happens next, as long as the islanders really trust the oracle? :eek: (Naughty, naughty oracle 
)
As the problem is stated, I see nothing wrong with it. Of course, in the real world people just wouldn’t go through with it, or could figure out their eye color in alternative ways. But that’s outside the scope of the problem. I can’t see what the mentioned logician is on about.
I am not convinced that anyone will die, unless there is only one blue eyed islander.
Suppose I look around me, after the oracle speaks.
There are seventy one other folks, and six of them have blue eyes. Now, how does this fact tell me anything about my eye color? It doesn’t. It would only work if I saw seventy one other people with brown eyes. Then I would know. But now, I don’t, so I don’t kill myself. Neither do any of the six people I see.
Tomorrow, nothing has changed, nor has it changed on the seventh day. The only case where I kill myself is if I am the only blue eyed person.
Does the oracle tell us the exact number of blue eyed people? If he does, then obviously, we all die the first day, cause we all count and come up one short. But if there can be “at least one” person, there might be only one.
Am I missing something obvious?
Tris
Yep. You’re missing the induction part.
Let’s suppose for a minute that you’re one of the islanders. You and your seventy-one fellows have lived on the island for years. Everyone around you is brown-eyed, except for one other person – let’s call him bitwise, just to pick a name at random. bitwise has blue eyes. Of course, no one ever speaks about eye color, so certainly bitwise doesn’t know what color his eyes are.
Now, along comes the oracle. The oracle states that at least one person has blue eyes. Ah! you think. That’s it for poor old bitwise. Since everyone else has brown eyes, he’ll have to realize the oracle is talking about him. bitwise is perfectly logical after all. Then, of course, he’s obligated to pitch himself in the volcano, stroke of midnight, etc etc.
But the next morning rolls around, and bitwise is still among the land of the living. How can that be? you think. If bitwise looks around and sees only brown eyes, surely he must realize he needs to pitch himself into the volcano! It’s perfectly logical!
Then you, being perfectly logical also, examine that statement. If bitwise looks around and sees only brown eyes, then he must pitch himself into the volcano. He didn’t pitch himself into the volcano. Therefore, when he looked around, he didn’t see only brown eyes.
How is that possible? Everyone else has brown eyes; you can see that. Except, you realize with mounting horror, you didn’t know what color your eyes are. But now you do know: since bitwise didn’t kill himself last night, he must have seen your blue eyes, and thought the oracle was talking about you. And so, with a heavy heart, you trudge toward the volcano…
Only to meet bitwise on the path there. Being perfectly logical, he’s figured out the same thing about his eyes. And you both pitch yourselves into the volcano at midnight.
To clarify even a little bit more the new information the Oracle gives:
One blue-eyed person: The blue-eyed person learns that someone on the island has blue eyes.
Two blue-eyed people: They already know there is at least one blue-eyed person, but now they each also know that the other blue-eyed person knows this. before the Oracle spoke, they couldn’t be sure the other blue-eyed person knew this.
Three blue-eyed people: They already know there is at least one blue-eyed person, and they already know that the other blue-eyed people know this, but now they know that each of the other two blue-eyed people also know that each other knows this.
After this it just gets more convoluted.
The OP’s statement the “However, in and of itself, the oracle’s statment doesn’t tell anyone anything that he did not already know.” isn’t true. The statement extends the chain by one.
Why does it seem as if no one is reading my post carefully? I already mentioned the “chains of knowledge”. My point was the public nature of the statement makes all the difference, not the statement itself. Every islander already knows the statement itself is true.
Here’s a list of logical assumptions and how they change according to events in the precise situation that you have specified. It’s long and repetitive, but I left out a bunch of other assumptions that don’t really change the argument.
I see that persons A, B, C, D, E, and F have blue eyes.
I assume that I have brown eyes.
I assume that everyone else assumes that they have brown eyes.
I assume that A sees 5 people with blue eyes.
I assume that A assumes that B sees 4 people with blue eyes.
I assume that A assumes that B assumes that C sees 3 people with blue eyes.
I assume that A assumes that B assumes that C assumes that D sees 2 people with blue eyes.
I assume that A assumes that B assumes that C assumes that D assumes that E sees 1 person with blue eyes.
I assume that A assumes that B assumes that C assumes that D assumes that E assumes that F sees 0 people with blue eyes.
The oracle makes his announcement!
I assume that A assumes that B assumes that C assumes that D assumes that E assumes that F will kill himself.
The first night passes and F does not kill himself.
I assume that A assumes that B assumes that C assumes that D assumes that E has figured out that he was wrong.
I assume that A assumes that B assumes that C assumes that D assumes that E and F will kill themselves.
The second night passes and E and F do not kill themselves.
I assume that A assumes that B assumes that C assumes that D has figured out that he was wrong.
I assume that A assumes that B assumes that C assumes that D, E and F will kill themselves.
The third night passes and D, E and F do not kill themselves.
I assume that A assumes that B assumes that C has figured out that he was wrong.
I assume that A assumes that B assumes that C, D, E and F will kill themselves.
The fourth night passes and C, D, E and F do not kill themselves.
I assume that A assumes that B has figured out that he was wrong.
I assume that A assumes that B, C, D, E and F will kill themselves.
The fifth night passes and B, C, D, E and F do not kill themselves.
I assume that A has figured out that he was wrong.
I assume that A, B, C, D, E and F will kill themselves.
The sixth night passes and A, B, C, D, E and F do not kill themselves.
I figure out that I was wrong when I assumed I had brown eyes.
Or…
The sixth night passes and A, B, C, D, E and F kill themselves.
I figure out that I was right when I assumed I had brown eyes.
What if the only blue eyed person on the island is bitwise? As bitwise can’t see his own reflection, he wouldn’t know there is at least one blue eyed person on the island. Thus your “Every islander already knows the statement itself is true” is incorrect.
I was implicitly excluding the base case. That is the only case where the statement, “There is at least one blue-eyed person,” is new information to someone (namely, the one and only blue-eyed person). In all other cases, that information is old, but the oracle’s public announcement of the statement provides further information.