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Edit: it looks like I'm wrong. The first argument is true; the second has the logical flaw. The flaw is assuming that induction can continue despite additiona

Edit: it looks like I'm wrong. The first argument is true; the second has the logical flaw. The flaw is assuming that induction can continue despite additional pre-knowledge available when there are greater numbers of blue-eyed people. The statement will have no effect when the number of blue-eyed people is 3 or more: When the number of blue-eyed people is 0, the foreigner is lying, and if the tribe believes him, everyone commits ritual suicide. When the number of blue-eyed people is 1, the blue-eyed person did not know there were any blue-eyed people in the tribe. Knowledge is added by the statement, and the blue-eyed person commits ritual suicide. When the number of blue-eyed people is 2, the blue-eyed people knew that there was a blue-eyed person but did not know that the blue-eyed person knew that the tribe had any blue-eyed people. Knowledge is added by the statement, and the blue-eyed people commit ritual suicide. When the number of blue-eyed people is 3 or more, the blue-eyed people knew that there were blue-eyed people and knew that the blue-eyed people knew that there were blue-eyed people. No knowledge is added by the statement, and no one commits ritual suicide. Edit: clarity.

thaumasiotes 13y ago
I read an essay once remarking that the ancient Greeks really wanted to square the circle (you can think of the problem as being to construct a line of length pi using only a reference line of length 1 and a compass and straightedge). Despite not knowing that this couldn't be done, no reference survives to any Greek claiming that it could. Today, we know perfectly well that the problem is impossible. But we get dozens of papers a year claiming to have solved it. The essay reflected that something has gone wrong in our culture. Please don't contribute to that problem by spouting off about problems you don't understand.
- vytasgd 13y ago
  Pardon? The parent's comment is correct. Edit: my comment is still correct thanks to parent's 2nd edit.
  - thaumasiotes 13y ago
    Parent makes this very clear statement: > The statement will have no effect when the number of blue-eyed people is 3 or more This is wrong, and is known to be wrong. Again, if you cannot understand a problem yourself, you might at least look into it before assuming you can.
  - thaumasiotes 13y ago
    > Edit: my comment is still correct thanks to parent's 2nd edit. If you're playing the retroactive-history game, let me point out that that edit had not been made when you made your comment. If party A issues a statement, I point out that it's nonsense, you say that you agree with party A, and then A recants, that doesn't make you right by virtue of the wording you used to agree with him. It makes you foolish.
- backlava 13y ago
  Oh please. The only things that have changed are access to such ideas and the volume of our records. He's already corrected his position. No harm in being wrong on the internet. Politeness is more important than rightness as a rule to enforce. Rightness sorts itself out among able and interested parties.
  - thaumasiotes 13y ago
    > The only things that have changed are access to such ideas and the volume of our records. Both of those definitely changed; unlike some of the other things you say, this is not ridiculous. However, I can easily imagine finding in, say, the personal correspondence of Archimedes, a statement like "this idiot tried to show me another squaring-the-circle proof again". We do preserve personal correspondence in ancient records, and that sort of thing doesn't require a lot of access. > No harm in being wrong on the internet. http://www.mamapedia.com/article/mmr-vaccination http://www.mamapedia.com/article/mmr-vaccination Money quote: "I have really close friends who swear that within 24 hours of the MMR their child had symptoms of autism" > Politeness is more important than rightness as a rule to enforce. It's better to be right. Enforcement of politeness greatly retards, and can prevent, the discovery / acknowledgement of rightness. > Rightness sorts itself out among able and interested parties. This can be tautologically true, depending on your definition of "able". It's very clear that rightness does not sort itself out among interested parties, and they can be harmed thereby.
    - bonobo 13y ago
      You missed what I think is baklava's main point: "The only things that have changed are access to such ideas and the volume of our records." Your argument that "The essay reflected that something has gone wrong in our culture" is a pretty bold statement that doesn't take into consideration the volume of and access to information we have today. To me it seems you're contradicting yourself. What does this bold statement of yours tells you about trying to contribute to a problem one does not understand?
      - thaumasiotes 13y ago
jader201 13y ago
I agree with you. Additionally: > When the number of blue-eyed people is 1, the blue-eyed person did not know there were any blue-eyed people in the tribe. Knowledge is added by the statement. Wouldn't the blue-eyed person think either: a) The visitor is lying, and wonder why everyone else doesn't think they are blue-eyed, thus committing suicide? b) Since they know no one else is blue-eyed, deduce that they are the sole blue-eyed person and commit suicide? > When the number of blue-eyed people is 2, the blue-eyed people knew that there was a blue-eyed person but did not know that the blue-eyed person knew that the tribe had any blue-eyed people. Knowledge is added by the statement. Wouldn't these two people wonder why the other blue-eyed person isn't committing suicide, and deduce that they too must also be blue-eyed?
- Mindless2112 13y ago
  Yes, I edited to clarify that a bit.
dnautics 13y ago
the error is in sloppy induction. The inductive step attempts to assume the lesser n condition by mapping it onto itself (which is a fallacious situation).
MereInterest 13y ago
If n=3, then each of the blue-eyed people know that there are blue-eyed people and know that the other blue-eyed people know. However, they don't know that all the blue-eyed people know that the blue-eyed people know. This is the piece of information that is learned by the statement given. In general, if there are N blue-eyed people, then it is the Nth abstraction of "he knows that I know that he knows that I know that..." that is learned by the statement.
- voyou 13y ago
  "they don't know that all the blue-eyed people know that the blue-eyed people know" Yes, they do. In the three-person case, a blue-eyed person can see two other blue-eyed people, A and B, and they know that A can see B, and vice-versa, so they know that A and B both know that there are blue-eyed people, and they know that both A and B would be able to us the same logic they used, so they also know that A and B know that A and B know that there are blue-eyed people.
- vytasgd 13y ago
  Can you go into further detail about the nth abstraction of "he knows that I know..." and what effect that has on the population? The way I see it is that as soon as everybody knows that there are blue-eyed ppl and everybody knows that everybody else knows there are blue-eyed ppl the countdown begins... additionally, at that point you can already get to an infinite number of abstractions of the "he knows that I know that he.." statement because you both know that the other knows blue-eyed ppl exist, so you both know that the other knows that you know that blue-eyed ppl exist, so you both know....
- Mindless2112 13y ago
  At n = 3, every blue-eyed person knows everything every other blue-eyed person knows and that every blue-eyed person knows it, so you can already construct an infinitely-long chain of "I know that he knows that I know that he knows..." at that point. (Of course, I am assuming that all the people think/reason exactly the same and that they all know that they think/reason exactly the same, which are assumptions of the original problem.)
DannoHung 13y ago
I think you've got it wrong. The information added is that a blue eyed person has been positively identified. In the three person case, each blue eyed person can see two people and is internally modeling their logic about the two person scenario. Once the logic for a two person scenario falls through, they can infer that there are not two people with blue eyes. The brown eyed people are however modeling an N+1 person case.
- Mindless2112 13y ago
  Hmm. That seems right. With 3 blue-eyed people, a blue-eyed person observing that the 2 blue-eyed people did nothing does add the additional knowledge that there are a total of 3 blue-eyed people. So the additional knowledge added for 3 or more blue-eyed people is a lower bound on the total number of blue-eyed people (which grows over time), but it's not exactly clear to me why the foreigner's statement triggers this.
  - vytasgd 13y ago
    I've been with you on this, but I think we are wrong. Here's where I'm at: (without foreigner's statement) 1 blue eyed person- He know's nothing, nothing happens 2 blue eyed people- Each thinks the other could be the only blue but they don't know it... deadlock. 3 blue eyed people- Each thinks the other two are caught in the 2 person deadlock scenario knowing nothing. 4 blue eyed people- Each thinks the other 3 are caught in the 3 person scenario etc... Once the foreigner adds the knowledge: 1- he'd leave the first day 2- the second would recognize the first didn't leave, they'd both leave on day 2 3- the third would recognize the first two didn't leave on day 2, etc, etc, we were wrong...bollocks.
  - pavelrub 13y ago
    The new information (common knowledge) is added immediately to everybody, not after 2 days. Using this information (that everybody knows that everybody knows that everybody knows...), it is then possible to decide whether you have blue eyes or not after 2 days.
  - DannoHung 13y ago
pavelrub 13y ago
No. There is nothing wrong with the induction and the second argument is the correct one. This is a well known puzzle that became very popular on Terrence Tao's blog (http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/ http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islan...) - where you can also find the solution (and various wrong attempts) in the comments. With 3 blue-eyed people: Everybody knows that there are other people with blue eyes. Everybody knows that everybody knows that there are other people with blue eyes. However, prior to the visitor's statement, nobody knows that everybody knows that everybody knows that there are other people with blue eyes. Because, consider the case where there are 3 blue eyed people named A,B,C. A knows that B knows that C has blue eyes. However A cannot possibly know that B knows that C knows that somebody has blue-eyes, because that somebody can either be B - in which case B has no way of knowing that, and therefore no way of knowing that C knows that, A - in which case A has no way of knowing that , and therefore no way of knowing that B knows that C knows that, or C, in which case C has no way of knowing that, and therefore B has no way of knowing that C knows that. Following the visitor's comment - A does know that, and eventually everybody kills himself. Formally this new information is knows as "common knowledge": http://en.wikipedia.org/wiki/Common_knowledge_(logic) http://en.wikipedia.org/wiki/Common_knowledge_(logic)

thaumasiotes 13y ago

I read an essay once remarking that the ancient Greeks really wanted to square the circle (you can think of the problem as being to construct a line of length pi using only a reference line of length 1 and a compass and straightedge). Despite not knowing that this couldn't be done, no reference survives to any Greek claiming that it could. Today, we know perfectly well that the problem is impossible. But we get dozens of papers a year claiming to have solved it. The essay reflected that something has gone wrong in our culture. Please don't contribute to that problem by spouting off about problems you don't understand.

vytasgd 13y ago
Pardon? The parent's comment is correct. Edit: my comment is still correct thanks to parent's 2nd edit.
- thaumasiotes 13y ago
  Parent makes this very clear statement: > The statement will have no effect when the number of blue-eyed people is 3 or more This is wrong, and is known to be wrong. Again, if you cannot understand a problem yourself, you might at least look into it before assuming you can.
- thaumasiotes 13y ago
  > Edit: my comment is still correct thanks to parent's 2nd edit. If you're playing the retroactive-history game, let me point out that that edit had not been made when you made your comment. If party A issues a statement, I point out that it's nonsense, you say that you agree with party A, and then A recants, that doesn't make you right by virtue of the wording you used to agree with him. It makes you foolish.
backlava 13y ago
Oh please. The only things that have changed are access to such ideas and the volume of our records. He's already corrected his position. No harm in being wrong on the internet. Politeness is more important than rightness as a rule to enforce. Rightness sorts itself out among able and interested parties.
- thaumasiotes 13y ago
  > The only things that have changed are access to such ideas and the volume of our records. Both of those definitely changed; unlike some of the other things you say, this is not ridiculous. However, I can easily imagine finding in, say, the personal correspondence of Archimedes, a statement like "this idiot tried to show me another squaring-the-circle proof again". We do preserve personal correspondence in ancient records, and that sort of thing doesn't require a lot of access. > No harm in being wrong on the internet. http://www.mamapedia.com/article/mmr-vaccination http://www.mamapedia.com/article/mmr-vaccination Money quote: "I have really close friends who swear that within 24 hours of the MMR their child had symptoms of autism" > Politeness is more important than rightness as a rule to enforce. It's better to be right. Enforcement of politeness greatly retards, and can prevent, the discovery / acknowledgement of rightness. > Rightness sorts itself out among able and interested parties. This can be tautologically true, depending on your definition of "able". It's very clear that rightness does not sort itself out among interested parties, and they can be harmed thereby.
  - bonobo 13y ago
    You missed what I think is baklava's main point: "The only things that have changed are access to such ideas and the volume of our records." Your argument that "The essay reflected that something has gone wrong in our culture" is a pretty bold statement that doesn't take into consideration the volume of and access to information we have today. To me it seems you're contradicting yourself. What does this bold statement of yours tells you about trying to contribute to a problem one does not understand?
    - thaumasiotes 13y ago

jader201 13y ago

I agree with you. Additionally: > When the number of blue-eyed people is 1, the blue-eyed person did not know there were any blue-eyed people in the tribe. Knowledge is added by the statement. Wouldn't the blue-eyed person think either: a) The visitor is lying, and wonder why everyone else doesn't think they are blue-eyed, thus committing suicide? b) Since they know no one else is blue-eyed, deduce that they are the sole blue-eyed person and commit suicide? > When the number of blue-eyed people is 2, the blue-eyed people knew that there was a blue-eyed person but did not know that the blue-eyed person knew that the tribe had any blue-eyed people. Knowledge is added by the statement. Wouldn't these two people wonder why the other blue-eyed person isn't committing suicide, and deduce that they too must also be blue-eyed?

Mindless2112 13y ago
Yes, I edited to clarify that a bit.

dnautics 13y ago

the error is in sloppy induction. The inductive step attempts to assume the lesser n condition by mapping it onto itself (which is a fallacious situation).

MereInterest 13y ago

If n=3, then each of the blue-eyed people know that there are blue-eyed people and know that the other blue-eyed people know. However, they don't know that all the blue-eyed people know that the blue-eyed people know. This is the piece of information that is learned by the statement given. In general, if there are N blue-eyed people, then it is the Nth abstraction of "he knows that I know that he knows that I know that..." that is learned by the statement.

voyou 13y ago
"they don't know that all the blue-eyed people know that the blue-eyed people know" Yes, they do. In the three-person case, a blue-eyed person can see two other blue-eyed people, A and B, and they know that A can see B, and vice-versa, so they know that A and B both know that there are blue-eyed people, and they know that both A and B would be able to us the same logic they used, so they also know that A and B know that A and B know that there are blue-eyed people.
vytasgd 13y ago
Can you go into further detail about the nth abstraction of "he knows that I know..." and what effect that has on the population? The way I see it is that as soon as everybody knows that there are blue-eyed ppl and everybody knows that everybody else knows there are blue-eyed ppl the countdown begins... additionally, at that point you can already get to an infinite number of abstractions of the "he knows that I know that he.." statement because you both know that the other knows blue-eyed ppl exist, so you both know that the other knows that you know that blue-eyed ppl exist, so you both know....
Mindless2112 13y ago
At n = 3, every blue-eyed person knows everything every other blue-eyed person knows and that every blue-eyed person knows it, so you can already construct an infinitely-long chain of "I know that he knows that I know that he knows..." at that point. (Of course, I am assuming that all the people think/reason exactly the same and that they all know that they think/reason exactly the same, which are assumptions of the original problem.)

DannoHung 13y ago

I think you've got it wrong. The information added is that a blue eyed person has been positively identified. In the three person case, each blue eyed person can see two people and is internally modeling their logic about the two person scenario. Once the logic for a two person scenario falls through, they can infer that there are not two people with blue eyes. The brown eyed people are however modeling an N+1 person case.

Mindless2112 13y ago
Hmm. That seems right. With 3 blue-eyed people, a blue-eyed person observing that the 2 blue-eyed people did nothing does add the additional knowledge that there are a total of 3 blue-eyed people. So the additional knowledge added for 3 or more blue-eyed people is a lower bound on the total number of blue-eyed people (which grows over time), but it's not exactly clear to me why the foreigner's statement triggers this.
- vytasgd 13y ago
  I've been with you on this, but I think we are wrong. Here's where I'm at: (without foreigner's statement) 1 blue eyed person- He know's nothing, nothing happens 2 blue eyed people- Each thinks the other could be the only blue but they don't know it... deadlock. 3 blue eyed people- Each thinks the other two are caught in the 2 person deadlock scenario knowing nothing. 4 blue eyed people- Each thinks the other 3 are caught in the 3 person scenario etc... Once the foreigner adds the knowledge: 1- he'd leave the first day 2- the second would recognize the first didn't leave, they'd both leave on day 2 3- the third would recognize the first two didn't leave on day 2, etc, etc, we were wrong...bollocks.
- pavelrub 13y ago
  The new information (common knowledge) is added immediately to everybody, not after 2 days. Using this information (that everybody knows that everybody knows that everybody knows...), it is then possible to decide whether you have blue eyes or not after 2 days.
- DannoHung 13y ago

pavelrub 13y ago

No. There is nothing wrong with the induction and the second argument is the correct one. This is a well known puzzle that became very popular on Terrence Tao's blog (http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islanders-puzzle/ http://terrytao.wordpress.com/2008/02/05/the-blue-eyed-islan...) - where you can also find the solution (and various wrong attempts) in the comments. With 3 blue-eyed people: Everybody knows that there are other people with blue eyes. Everybody knows that everybody knows that there are other people with blue eyes. However, prior to the visitor's statement, nobody knows that everybody knows that everybody knows that there are other people with blue eyes. Because, consider the case where there are 3 blue eyed people named A,B,C. A knows that B knows that C has blue eyes. However A cannot possibly know that B knows that C knows that somebody has blue-eyes, because that somebody can either be B - in which case B has no way of knowing that, and therefore no way of knowing that C knows that, A - in which case A has no way of knowing that , and therefore no way of knowing that B knows that C knows that, or C, in which case C has no way of knowing that, and therefore B has no way of knowing that C knows that. Following the visitor's comment - A does know that, and eventually everybody kills himself. Formally this new information is knows as "common knowledge": http://en.wikipedia.org/wiki/Common_knowledge_(logic) http://en.wikipedia.org/wiki/Common_knowledge_(logic)