My write-up of the "Blue Eyes" solution (SPOILER A

[Pseudonymoniae]

Well, I can certainly agree that this makes logical sense, but it's really a stupid puzzle. Here's my main problem:

It is stated that "They are all perfect logicians...but they cannot otherwise communicate."

The premise being that because these people cannot communicate their task should be virtually impossible. But the answer to the riddle is that, in fact, they can communicate. Why? Because they are all "ideal logicians"... in other words, they all think exactly alike! So yeah, it's a grave disadvantage to be unable to talk to the other people on the island... unless you are able to read the minds of all the other people, in which case talking really isn't all that important.

And yes, I do mean mind reading. If I am an "ideal logician" this is equivalent to saying that in terms of logical operations my brain performs in one way and one way only. This would hold true for all ideal logicians. In fact, one can assume based upon this fact, that ideal logicians would be identical in every respect. If all logicians perform logical operations identically, then each must use logic to answer pretty much every question of interest, which would basically make their personalities and world views identical. In fact, this assumption is pretty much implicit in the wording of the question as it is never mentioned whether all of the logicians even want to leave the island. (This is kind of an important detail, as if even one blue-eyed logician decided to stay he would presumably affect the decisions of all of the other logicians).

Really this gives the whole thing away. The reason why the riddle is "supposed to be difficult" is that it asks us to think about how a group of people would solve a difficult problem. But this never was meant to be a riddle about people, it's a riddle about machines--a bunch of computers running on an identical copy of the same code. Not such an interesting riddle any more is it?

[edit]

And that's another good point. I've decided that this riddle doesn't make sense any more.

Let's put it this way, if all of these logical operators are basically machines running on the same code (sorry, "ideal logicians"), why do they even wait for the Guru to speak up? Any old arbitrary event could set them off on the same set of logical operations. Since they're all basically identical "Clones", why doesn't some guy tripping over a rock and breaking his leg incite them? Maybe a really bad storm comes and they all think, "Hey, it's no longer logical to stay on this island because, well, life sucks here... and since I and all of the other logicians on this island use the exact same reasoning, we will all feel this way." And thus, starts the 99 or 100 day count down to D-Day, at which point all of the people leave. Of course, this would be a problem, as the brown eyed people would presumably feel the same way as the blue-eyed people...

...But then this leads to a key question: What is it about the Guru stating that she sees someone with blue eyes (something that everyone on the island must have been aware of) that specifically sets off blue eyed people to leave? As I point out above, any arbitrary event might do this, except that such an event might incite brown eyed people to try and leave too, which would nullify the strategy. But then, why is it that the Guru's words specifically incite blue-eyed people to leave and not brown-eyed people? What is the logical reason that all of the blue-eyed people say "Hey, I guess all the blue-eyed people and none of the brown eyed people are suddenly going to use this one strategy to leave the island!"

There is no logical reason why this event should set them off. And so, this can go either of two ways. Either, these ideal logicians are so in tune with each other that they are able separately coordinate the exodus of both brown-eyed and blue-eyed individuals using this strategy (in which case, they should be able to do this at just about any point in time; they shouldn't have to wait for the Guru to speak up) or they are not in-tune enough to do so, in which case the Guru's words would be meaningless, as they would be equally likely to incite both brown and blue-eyed people to try and leave... in other words, they would incite no one to leave.

Alternatively, if the Guru were to say that "I see a blue-eyed person. For 100 days, blue-eyed and blue-eyed only people are allow to leave the island, after which time these people are forced to remain here in eternal damnation, during which time they will experience excruciating torture and hellfire." Now, assuming that brown-eyed people aren't total sadistic douchebags who might attempt to trap blue-eyed people on the island by coordinating their own escape, and that the blue-eyed people aren't a bunch of masochists, this would be sufficient to incite the blue-eyed and only the blue-eyed people to leave the island. Personally, I think some change of this sort needs to be added to fix this logic problem. Otherwise, this looks like an arbitrary solution to an "illogic problem".

[Goldstein]

...But then this leads to a key question: What is it about the Guru stating that she sees someone with blue eyes (something that everyone on the island must have been aware of) that specifically sets off blue eyed people to leave? As I point out above, any arbitrary event might do this, except that such an event might incite brown eyed people to try and leave too, which would nullify the strategy. But then, why is it that the Guru's words specifically incite blue-eyed people to leave and not brown-eyed people? What is the logical reason that all of the blue-eyed people say "Hey, I guess all the blue-eyed people and none of the brown eyed people are suddenly going to use this one strategy to leave the island!"

There is no logical reason why this event should set them off.

The writer of the puzzle, myself and everyone who continues to contribute to this thread disagree. In fact this detail is exactly why the puzzle is interesting and is a subtlety that you seem to have missed. For an explanation, read at least a couple of pages of this very long thread.

In fact, this assumption is pretty much implicit in the wording of the question as it is never mentioned whether all of the logicians even want to leave the island. (This is kind of an important detail, as if even one blue-eyed logician decided to stay he would presumably affect the decisions of all of the other logicians).

This isn't an important detail at all. What each individual decides is of no consequence as long as they obey the rules set out in the puzzle. Until you can appreciate why this must be the case - and not talk in terms of meaningless 'count downs' that have nothing to do with logical reasoning - it isn't very fair to judge the puzzle stupid.

[skeptical scientist]

Really this gives the whole thing away. The reason why the riddle is "supposed to be difficult" is that it asks us to think about how a group of people would solve a difficult problem. But this never was meant to be a riddle about people, it's a riddle about machines--a bunch of computers running on an identical copy of the same code. Not such an interesting riddle any more is it?

Why does it matter whether they act like people or machines? It's supposed to be a logic puzzle, not the great American novel. The setup is only interesting insofar as it leads to an interesting puzzle.

There is no logical reason why this event should set them off. And so, this can go either of two ways. Either, these ideal logicians are so in tune with each other that they are able separately coordinate the exodus of both brown-eyed and blue-eyed individuals using this strategy (in which case, they should be able to do this at just about any point in time; they shouldn't have to wait for the Guru to speak up) or they are not in-tune enough to do so, in which case the Guru's words would be meaningless, as they would be equally likely to incite both brown and blue-eyed people to try and leave... in other words, they would incite no one to leave.

No, that shows you still don't fully understand the riddle. Imagine there is the Guru and one other person, who has blue eyes. It should be obvious in this case that a guy tripping over a rock doesn't provide any new information, but the Guru saying that she sees a blue-eyed person immediately tells the other guy he has blue eyes, since he's the only person the Guru could be talking about. With 200 people other than the guru present, the logical inference chain is much more complicated, but it's the same situation. The Guru's statement provides the islanders with new information, which starts the chain of logical inferences eventually leading all blue-eyed islanders to leave.

[Pseudonymoniae]

...But then this leads to a key question: What is it about the Guru stating that she sees someone with blue eyes (something that everyone on the island must have been aware of) that specifically sets off blue eyed people to leave? As I point out above, any arbitrary event might do this, except that such an event might incite brown eyed people to try and leave too, which would nullify the strategy. But then, why is it that the Guru's words specifically incite blue-eyed people to leave and not brown-eyed people? What is the logical reason that all of the blue-eyed people say "Hey, I guess all the blue-eyed people and none of the brown eyed people are suddenly going to use this one strategy to leave the island!"

There is no logical reason why this event should set them off.

The writer of the puzzle, myself and everyone who continues to contribute to this thread disagree. In fact this detail is exactly why the puzzle is interesting and is a subtlety that you seem to have missed. For an explanation, read at least a couple of pages of this very long thread.


Well, I'm a few pages in and I've yet to see anything of relevance other than this infinite hypothetical recursion nonsense. But, since you're not interested in pointing out this brilliant mechanism I'll take a few more minutes to try and find it.

In fact, this assumption is pretty much implicit in the wording of the question as it is never mentioned whether all of the logicians even want to leave the island. (This is kind of an important detail, as if even one blue-eyed logician decided to stay he would presumably affect the decisions of all of the other logicians).

This isn't an important detail at all. What each individual decides is of no consequence as long as they obey the rules set out in the puzzle. Until you can appreciate why this must be the case - and not talk in terms of meaningless 'count downs' that have nothing to do with logical reasoning - it isn't very fair to judge the puzzle stupid.

Alright, I can accept that the decision of each logician might be irrelevant to determining whether others leave, given the many assumptions of this puzzle. So this is something which I won't argue.

[douglasm]

And that's another good point. I've decided that this riddle doesn't make sense any more.

Let's put it this way, if all of these logical operators are basically machines running on the same code (sorry, "ideal logicians"), why do they even wait for the Guru to speak up? Any old arbitrary event could set them off on the same set of logical operations. Since they're all basically identical "Clones", why doesn't some guy tripping over a rock and breaking his leg incite them? Maybe a really bad storm comes and they all think, "Hey, it's no longer logical to stay on this island because, well, life sucks here... and since I and all of the other logicians on this island use the exact same reasoning, we will all feel this way." And thus, starts the 99 or 100 day count down to D-Day, at which point all of the people leave. Of course, this would be a problem, as the brown eyed people would presumably feel the same way as the blue-eyed people...

So, basically, "because we're all identical if I decide on strategy X then all the rest of us will also simultaneously decide on strategy X, thus I can create a strategy based on the assumption that everyone will be following it even without being able to communicate the strategy." This kind of reasoning is called superrationality, and it is not in fact an automatic consequence of perfect rationality as you assume. Sharing the ability to derive all possible logical proofs does not imply sharing the same process of doing so, or the same mental states, or the same random number generation, or any of a bunch of other things that would be required for superrationality to work.

...But then this leads to a key question: What is it about the Guru stating that she sees someone with blue eyes (something that everyone on the island must have been aware of) that specifically sets off blue eyed people to leave? As I point out above, any arbitrary event might do this, except that such an event might incite brown eyed people to try and leave too, which would nullify the strategy. But then, why is it that the Guru's words specifically incite blue-eyed people to leave and not brown-eyed people? What is the logical reason that all of the blue-eyed people say "Hey, I guess all the blue-eyed people and none of the brown eyed people are suddenly going to use this one strategy to leave the island!"

The Guru's statement does, in fact, add additional knowledge specifically about blue eyes to what the people know. Before his statement, everyone knows that there are blue-eyed people on the island, everyone knows that everyone knows that there are blue-eyed people on the island, everyone knows that everyone knows that everyone knows that there are blue-eyed people on the island, and so on, and this "everyone knows that everyone knows" stuff stops precisely one "everyone knows" short of what is necessary to prove that the strategy will work. After the Guru's statement, you could add umpteen zillion repetitions of "everyone knows that" and it would still remain true.

Alright, I can accept that the decision of each logician might be irrelevant to determining whether others leave, given the many assumptions of this puzzle. So this is something which I won't argue.

The logicians don't make any decisions. According to the puzzle statement, anyone who figures out his own eye color leaves. Whether he wants to or not is irrelevant, the puzzle just says that he does in fact leave. Additionally, figuring out his own eye color is not optional, as the way it's phrased in the puzzle statement has all of the logicians automatically deriving every provable result whether they want to or not.

[Pseudonymoniae]

Really this gives the whole thing away. The reason why the riddle is "supposed to be difficult" is that it asks us to think about how a group of people would solve a difficult problem. But this never was meant to be a riddle about people, it's a riddle about machines--a bunch of computers running on an identical copy of the same code. Not such an interesting riddle any more is it?

Why does it matter whether they act like people or machines? It's supposed to be a logic puzzle, not the great American novel. The setup is only interesting insofar as it leads to an interesting puzzle.

There is no logical reason why this event should set them off. And so, this can go either of two ways. Either, these ideal logicians are so in tune with each other that they are able separately coordinate the exodus of both brown-eyed and blue-eyed individuals using this strategy (in which case, they should be able to do this at just about any point in time; they shouldn't have to wait for the Guru to speak up) or they are not in-tune enough to do so, in which case the Guru's words would be meaningless, as they would be equally likely to incite both brown and blue-eyed people to try and leave... in other words, they would incite no one to leave.

No, that shows you still don't fully understand the riddle. Imagine there is the Guru and one other person, who has blue eyes. It should be obvious in this case that a guy tripping over a rock doesn't provide any new information, but the Guru saying that she sees a blue-eyed person immediately tells the other guy he has blue eyes, since he's the only person the Guru could be talking about. With 200 people other than the guru present, the logical inference chain is much more complicated, but it's the same situation. The Guru's statement provides the islanders with new information, which starts the chain of logical inferences eventually leading all blue-eyed islanders to leave.

Sorry, but I disagree. The Guru's statement might provide additional information when there are fewer than 4 individuals on the island, but I disagree that it provides any new information when there are hundreds of them. The logical inference chain is no longer required. And, yes, I've seen all that stuff about infinite recursive hypothetical imaginings of what the others are thinking... and that argument is completely irrelevant. It remains a fact that ALL of the people on the island already know that there are are many people with blue eyes on the island. The Guru simply says it aloud. And I saw some points about "common knowledge" which make little sense: if this was not "common knowledge" prior to the Guru's words, then it clearly will not be so after she has spoken (i.e. if a person cannot already infer that others on the island can see blue eyed people, then that person cannot infer that the Guru's words had any effect on the state of common knowledge).

[douglasm]

Sorry, but I disagree. The Guru's statement might provide additional information when there are fewer than 4 individuals on the island, but I disagree that it provides any new information when there are hundreds of them.

Where's the cutoff? Now work out the behavior just before the cutoff and just after the cutoff and explain why they are different. If you do your reasoning correctly, you will find that your assumption of a cutoff point where it stops working leads to a contradiction no matter what cutoff point you pick. If you don't find such a contradiction, I guarantee someone will find an error in your logic.

[Pseudonymoniae]

And that's another good point. I've decided that this riddle doesn't make sense any more.

Let's put it this way, if all of these logical operators are basically machines running on the same code (sorry, "ideal logicians"), why do they even wait for the Guru to speak up? Any old arbitrary event could set them off on the same set of logical operations. Since they're all basically identical "Clones", why doesn't some guy tripping over a rock and breaking his leg incite them? Maybe a really bad storm comes and they all think, "Hey, it's no longer logical to stay on this island because, well, life sucks here... and since I and all of the other logicians on this island use the exact same reasoning, we will all feel this way." And thus, starts the 99 or 100 day count down to D-Day, at which point all of the people leave. Of course, this would be a problem, as the brown eyed people would presumably feel the same way as the blue-eyed people...

So, basically, "because we're all identical if I decide on strategy X then all the rest of us will also simultaneously decide on strategy X, thus I can create a strategy based on the assumption that everyone will be following it even without being able to communicate the strategy." This kind of reasoning is called superrationality, and it is not in fact an automatic consequence of perfect rationality as you assume. Sharing the ability to derive all possible logical proofs does not imply sharing the same process of doing so, or the same mental states, or the same random number generation, or any of a bunch of other things that would be required for superrationality to work.

I'll just have to take your word on this distinctions between superrationality and perfect rationality. It seems like a pretty fine line.

...But then this leads to a key question: What is it about the Guru stating that she sees someone with blue eyes (something that everyone on the island must have been aware of) that specifically sets off blue eyed people to leave? As I point out above, any arbitrary event might do this, except that such an event might incite brown eyed people to try and leave too, which would nullify the strategy. But then, why is it that the Guru's words specifically incite blue-eyed people to leave and not brown-eyed people? What is the logical reason that all of the blue-eyed people say "Hey, I guess all the blue-eyed people and none of the brown eyed people are suddenly going to use this one strategy to leave the island!"

The Guru's statement does, in fact, add additional knowledge specifically about blue eyes to what the people know. Before his statement, everyone knows that there are blue-eyed people on the island, everyone knows that everyone knows that there are blue-eyed people on the island, everyone knows that everyone knows that everyone knows that there are blue-eyed people on the island, and so on, and this "everyone knows that everyone knows" stuff stops precisely one "everyone knows" short of what is necessary to prove that the strategy will work. After the Guru's statement, you could add umpteen zillion repetitions of "everyone knows that" and it would still remain true.

I disagree that any knowledge is added, except about what the Guru knows, which it is my understanding can already be inferred. This sounds like that same infinite recursion idea which appears to be irrelevant. I will check to see whether there is a better explanation of the concept in a moment, because the way you have described it does not make any sense to me.

[Goldstein]

I'll just have to take your word on this distinctions between superrationality and perfect rationality. It seems like a pretty fine line.

Well... You don't, do you?

[Pseudonymoniae]

Sorry, but I disagree. The Guru's statement might provide additional information when there are fewer than 4 individuals on the island, but I disagree that it provides any new information when there are hundreds of them.

Where's the cutoff? Now work out the behavior just before the cutoff and just after the cutoff and explain why they are different. If you do your reasoning correctly, you will find that your assumption of a cutoff point where it stops working leads to a contradiction no matter what cutoff point you pick. If you don't find such a contradiction, I guarantee someone will find an error in your logic.

If you are referring to the concept that I believe we are discussing, then the cutoff should be around 4. I recall reading this argument on the first page of this forum which suggested that given a group of 3 people with blue eyes, person B with blue eyes would not know whether he has blue eyes and could only be sure that two others have blue eyes. Presuming that he has brown eyes, he might ask the same of person C who has blue eyes. Person C, presuming he has brown eyes might then ask whether person D, who also has blue eyes, might ask that same question. The assumption of this line is reasoning is that while person B can see two people with blue eyes, he thinks that person C might be able to only see one, and that he thinks person C thinks that person D might not be able to see any people with blue eyes. But this is false. Person B can see that both person C and person D have blue eyes, and therefore he knows that both person C and D can see someone with blue eyes. Whether or not he thinks that person C thinks that person D can see anyone with blue eyes should be irrelevant. Why? Because from the perspective of person B, the Guru has not provided any new information!. This will hold true for 99 or 100 people as well. Hence, the infinite recursion idea is irrelevant.

...Then again, perhaps your are referring to something else.

[edit]

I'll just have to take your word on this distinctions between superrationality and perfect rationality. It seems like a pretty fine line.

Well... You don't, do you?

I believe I just did, did I not?

As an aside, you've linked me to "superrationality" and "rationality" but not "perfect rationality". In all, there appears to be a largely semantic distinction between the latter two terms. To be honest, I am not interested in debating the the terminology of logic after I've already admitted that I am not familiar with the exact meaning of "superrationality".

[douglasm]

If you are referring to the concept that I believe we are discussing, then the cutoff should be around 4. I recall reading this argument on the first page of this forum which suggested that given a group of 3 people with blue eyes, person B with blue eyes would not know whether he has blue eyes and could only be sure that two others have blue eyes. Presuming that he has brown eyes, he might ask the same of person C who has blue eyes. Person C, presuming he has brown eyes might then ask whether person D, who also has blue eyes, might ask that same question. The assumption of this line is reasoning is that while person B can see two people with blue eyes, he thinks that person C might be able to only see one, and that he thinks person C thinks that person D might not be able to see any people with blue eyes. But this is false. Person B can see that both person C and person D have blue eyes, and therefore he knows that both person C and D can see someone with blue eyes. Whether or not he thinks that person C thinks that person D can see anyone with blue eyes should be irrelevant. Why? Because from the perspective of person B, the Guru has not provided any new information!. This will hold true for 99 or 100 people as well. Hence, the infinite recursion idea is irrelevant.

...Then again, perhaps your are referring to something else.

So, let's say you think it works with 3 blue-eyed people but not with 4. If there are 3 blue-eyed people, they all leave on day 3. Now suppose you are 1 of 4 blue-eyed people on the island, it is day 4, and no one has left yet. You can determine easily with your own direct observation that there are either 3 or 4 blue-eyed people. You admit that it works with 3 blue-eyes, but it's already past the point where 3 would leave so it can't be 3. It must therefore be 4, and that means your eyes must be blue. Well, what do you know, it works with 4 blue-eyed people after all!

The exact same reasoning applies to any cutoff point you could possibly pick.

Do I understand correctly that you understand the hypothetical recursion idea but just think it's irrelevant? Well, the information the Guru provides is specifically in the recursion. If you go through the chain of reasoning about what A is thinking about what B is thinking about what C is thinking, etc., before the Guru speaks you will eventually reach a point where, within the great depths of nested hypotheticals, someone's imagined thought processes do not include the knowledge that there are blue eyes on the island. After the Guru speaks, it doesn't matter how deep you go, none of the nesting steps will ever eliminate the Guru's announcement and the accompanying knowledge that there are blue eyes on the island. That is the new information the Guru provides, and that knowledge is necessary to give the chain of reasoning a starting point.

[Pseudonymoniae]

There are three critical things that matter here:
1) How many blue eyes a person actually sees establishes a maximum.
2) The Guru's statement establishes a minimum.
3) The passing of time and discrete opportunities to leave the island progressively increases the minimum.

So I've pulled this little point out of the aether. Just to be clear, the point that I disagree with is No.2. I have no problem with the mechanism by which the logicians realize their eye colour on the final day. The issue that I have is this idea that the Guru's words are an appropriate cue to initiate the logical progression. Point number two addresses this. How is it that the Guru's statement establishes the minimum when this minimum is already readily apparent to all the individuals on the island?

[douglasm]

There are three critical things that matter here:
1) How many blue eyes a person actually sees establishes a maximum.
2) The Guru's statement establishes a minimum.
3) The passing of time and discrete opportunities to leave the island progressively increases the minimum.

So I've pulled this little point out of the aether. Just to be clear, the point that I disagree with is No.2. I have no problem with the mechanism by which the logicians realize their eye colour on the final day. The issue that I have is this idea that the Guru's words are an appropriate cue to initiate the logical progression. Point number two addresses this. How is it that the Guru's statement establishes the minimum when this minimum is already readily apparent to all the individuals on the island?

The really important thing is that the Guru establishes a minimum that every last person on the island agrees on. The blue-eyes say, based on their observations, that there are at least 99 blue-eyed people. The brown-eyes say, based on their observations, that there are at least 100 blue-eyed people. Everyone says, based on the Guru's statement, that there is at least 1 blue-eyed person.

[Pseudonymoniae]

If you are referring to the concept that I believe we are discussing, then the cutoff should be around 4. I recall reading this argument on the first page of this forum which suggested that given a group of 3 people with blue eyes, person B with blue eyes would not know whether he has blue eyes and could only be sure that two others have blue eyes. Presuming that he has brown eyes, he might ask the same of person C who has blue eyes. Person C, presuming he has brown eyes might then ask whether person D, who also has blue eyes, might ask that same question. The assumption of this line is reasoning is that while person B can see two people with blue eyes, he thinks that person C might be able to only see one, and that he thinks person C thinks that person D might not be able to see any people with blue eyes. But this is false. Person B can see that both person C and person D have blue eyes, and therefore he knows that both person C and D can see someone with blue eyes. Whether or not he thinks that person C thinks that person D can see anyone with blue eyes should be irrelevant. Why? Because from the perspective of person B, the Guru has not provided any new information!. This will hold true for 99 or 100 people as well. Hence, the infinite recursion idea is irrelevant.

...Then again, perhaps your are referring to something else.

So, let's say you think it works with 3 blue-eyed people but not with 4. If there are 3 blue-eyed people, they all leave on day 3. Now suppose you are 1 of 4 blue-eyed people on the island, it is day 4, and no one has left yet. You can determine easily with your own direct observation that there are either 3 or 4 blue-eyed people. You admit that it works with 3 blue-eyes, but it's already past the point where 3 would leave so it can't be 3. It must therefore be 4, and that means your eyes must be blue. Well, what do you know, it works with 4 blue-eyed people after all!

The exact same reasoning applies to any cutoff point you could possibly pick.

Do I understand correctly that you understand the hypothetical recursion idea but just think it's irrelevant? Well, the information the Guru provides is specifically in the recursion. If you go through the chain of reasoning about what A is thinking about what B is thinking about what C is thinking, etc., before the Guru speaks you will eventually reach a point where, within the great depths of nested hypotheticals, someone's imagined thought processes do not include the knowledge that there are blue eyes on the island. After the Guru speaks, it doesn't matter how deep you go, none of the nesting steps will ever eliminate the Guru's announcement and the accompanying knowledge that there are blue eyes on the island. That is the new information the Guru provides, and that knowledge is necessary to give the chain of reasoning a starting point.

Yes, I have never had any problem with the reasoning used by the logicians to determine whether they have blue eyes. This is simple and straight forward and it should follow all the way up to 99, 100 or one million blue eyed logicians.

My problem is with the Guru's words. Why are they at all relevant? They seem to be an arbitrary factor which is used to incite the logical progression. Moreover, it is not clear to me that any person, whether an ideal logician or not, would logically use this information to initiate the logical chain of reasoning, given that they had not previously initiated this chain of reasoning based upon some other arbitrary factor (I don't know, an outbreak of food poisoning?).

To be specific, this is the statement that I find issue with: "someone's imagined thought processes do not include the knowledge that there are blue eyes on the island". So yes, if you conduct this infinite, hypothetical thought recursion, I can agree that you might come to this conclusion. But, this should be superceded by the fact that A can see that C or D or whoever is at the end of the list must know that there are blue-eyed people on the island. Given that it can be objectively seen that there are 99 blue eyed people who are visible to everyone it does not make sense that the Guru's words have any added impact. In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes. How is this not the case? Are we talking about the same thing?

[edit]

There are three critical things that matter here:
1) How many blue eyes a person actually sees establishes a maximum.
2) The Guru's statement establishes a minimum.
3) The passing of time and discrete opportunities to leave the island progressively increases the minimum.

So I've pulled this little point out of the aether. Just to be clear, the point that I disagree with is No.2. I have no problem with the mechanism by which the logicians realize their eye colour on the final day. The issue that I have is this idea that the Guru's words are an appropriate cue to initiate the logical progression. Point number two addresses this. How is it that the Guru's statement establishes the minimum when this minimum is already readily apparent to all the individuals on the island?

The really important thing is that the Guru establishes a minimum that every last person on the island agrees on. The blue-eyes say, based on their observations, that there are at least 99 blue-eyed people. The brown-eyes say, based on their observations, that there are at least 100 blue-eyed people. Everyone says, based on the Guru's statement, that there is at least 1 blue-eyed person.

And my point is that shouldn't the blue-eyed people's observations establish a minimum?

[douglasm]

The blue-eyed observations does establish a minimum, but it is not a minimum that can be used to derive anything significant.

Look at it this way: someone tripping over a log starts the chain if and only if everyone spontaneously decides to start the chain. The Guru's announcement forces the chain to start independent of anyone's decisions. It is blatantly obvious that 1 lone blue-eyed person would leave immediately after the Guru's announcement. Someone tripping over a log would not force that event.

The Guru's announcement is important not because it provides a marker for when to start counting, but because it provides a marker that forces the count to start.

[phlip]

My problem is with the Guru's words. Why are they at all relevant?

Because if the Guru had said "purple monkey dishwasher" instead, then it would not work in the 1-blue-eyed-person case. And if it doesn't work in the 1-blue-eyed-person case then the 2-blue-eyed-person case breaks down. And if it doesn't work in the 2-blue-eyed-person case, the 3-blue-eyed-person case breaks down.

It's important to recognise that the inductive proof (which you say you do understand) is sufficient to prove that the blue-eyed people do leave. The nested-hypothetical knowledge-theory stuff is only necessary to disprove the "the Guru's statement gives no information" claim... the inductive proof holds strong without it. It can be unsatisfying, but it's valid mathematically.

Moreover, it is not clear to me that any person, whether an ideal logician or not, would logically use this information to initiate the logical chain of reasoning

Note that the definition of "perfect logician" given in the puzzle is that if something can be logically deduced, they will do so immediately. So everyone on the island already deduced every provable statement long before the Guru actually makes the announcement. The possibility that one of the islanders just didn't think of it is ruled out.

In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes.

The point is that just because A knows E can see blue-eyed people, A doesn't know that B knows that C knows that D knows that E can see blue-eyed people. But A does know that B knows that C knows that D knows that E was present at the announcement, because A observed B observing C observing D observing E when the announcement took place.

Just because the statements are hard to say in English and hard to wrap our monkey brains around, doesn't make the concept invalid. Put it symbolically, if you prefer:
K_A(E sees blue-eyed people) ≢ K_A(K_B(K_C(K_D(E sees blue-eyed people))))
K_A(E sees blue-eyed people) is true
K_A(K_B(K_C(K_D(E sees blue-eyed people)))) is false
K_A(K_B(K_C(K_D(K_E(There are blue-eyed people))))) is false before the announcement
K_A(K_B(K_C(K_D(E heard the announcement)))) is true
K_A(K_B(K_C(K_D(K_E(There are blue-eyed people))))) is true after the announcement

[Pseudonymoniae]

In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes.

The point is that just because A knows E can see blue-eyed people, A doesn't know that B knows that C knows that D knows that E can see blue-eyed people. But A does know that B knows that C knows that D knows that E was present at the announcement, because A observed B observing C observing D observing E when the announcement took place.

Aha! And this is exactly the point I am contending. I disagree with you on this. A does know that B knows that C knows that D knows that E can see blue-eyed people. A can look at B, C and D as they all watch E staring into the eyes of a blue-eyed person. Moreover, I find it hard to believe that our ideal logicians cannot find any logical basis to believe that the others are aware that there are blue-eyed people on the island. In fact, it is my understanding that the question specifically states this--what is more it says no trickses! This is why I think that the whole hypothetical mind-bending recursion simulation is irrelevant. Why does A need to listen to the Guru to infer what the others must already know? And, my added point is that if A does not already know that B knows that C knows that D knows that E can see blue-eyed people, how does the Guru's pronouncement demonstrate that this is the case? It seems bizarre to me that on the one hand you can state that A does not know this based upon observation (which would imply either stupidity on A's part, or a very strenuous requirement for claiming that an observation is true) and then go and say that A's observation of the Guru's claims (yes, another observation!) is sufficient to either overcome A's stupidity or this strenuous requirement for truth. You cannot have it both ways. If A can infer based upon the Guru's statement that B knows that C knows that D knows that E knows that there are blue-eyed people on the island, then it must (okay, not must, but within the limits of reason) be the case that A, B, C, and D can all see that E can see that people on the island have blue eyes (and that therefore E knows that people on the island have blue eyes) They are both inferences about the thoughts of other individuals, based upon a limited, but reasonable basis (the same basis that we use in everyday life). And this is the crux of the problem. The Guru's words simply tell us about what the Guru can see. The island's inhabitants must still infer what the other individuals on the island are thinking. In this sense, the information has not been changed based upon the Guru's statement.

Honestly, can you actually demonstrate objectively how these two things differ? To put this bluntly: When the Guru makes his pronouncement that he sees blue-eyed people on this island, this is strong evidence that there are blue-eyed people on the island and that everyone knows it. And, it should follow that this might initiate the logical chain of events that is postulated by this riddle. But, you cannot expect me to believe that any one of those guys on the island is walking around without this knowledge already. Here, let's review the statements of their knowledge:

A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

For you guys to be correct about this, you would have to believe that out of this bunch of perfect logicians who all know that there are at least 98 blue-eyed people on the island, there are at least some who have no idea about whether or not any of the other logicians are aware of this fact. IMHO, this riddle is at worst ambiguous on this point, and at best implicitly implies [strongly states] that this is not the case.

Well, whatever... it's not that important. I would be happy if the puzzle had be written to specifically state that they had absolutely no way of inferring the knowledge of the others prior to the Guru's pronouncement and that this pronouncement unambiguously allowed them to infer what the others knew. Without any such clear statement, the effect of the Guru's statement on the blue-eyed individuals on the island is difficult to determine. That's the only point I'm trying to make and that's why I think it's arbitrary... just as arbitrary as some dude falling over a log.

[edit]

The blue-eyed observations does establish a minimum, but it is not a minimum that can be used to derive anything significant.

Look at it this way: someone tripping over a log starts the chain if and only if everyone spontaneously decides to start the chain. The Guru's announcement forces the chain to start independent of anyone's decisions. It is blatantly obvious that 1 lone blue-eyed person would leave immediately after the Guru's announcement. Someone tripping over a log would not force that event.

The Guru's announcement is important not because it provides a marker for when to start counting, but because it provides a marker that forces the count to start.

As per my last post, I agree that it provides a marker, but I think that the marker is arbitrary, as I explain above.

[skullturf]

To be specific, this is the statement that I find issue with: "someone's imagined thought processes do not include the knowledge that there are blue eyes on the island". So yes, if you conduct this infinite, hypothetical thought recursion, I can agree that you might come to this conclusion. But, this should be superceded by the fact that A can see that C or D or whoever is at the end of the list must know that there are blue-eyed people on the island. Given that it can be objectively seen that there are 99 blue eyed people who are visible to everyone it does not make sense that the Guru's words have any added impact. In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes. How is this not the case? Are we talking about the same thing?

I think this paragraph (also partly quoted by phlip) is the key to why we disagree. (EDIT: You posted while I was composing this.)

It seems like you're okay with the truth of each of the following statements:

(i) A knows there are 99 blue-eyed people.
(ii) A knows that B knows there are 98 blue-eyed people.
(iii) A knows that B knows that C knows there are 97 blue-eyed people.
(iv) A knows that B knows that C knows that D knows there are 96 blue-eyed people.
(v) A knows that B knows that C knows that D knows that E knows there are 95 blue-eyed people.

But it seems like you also want to say that we can "collapse" this in some sense; that we can say "more".

Why is A hypothesizing about E seeing only 95 blue-eyed people, when we know darn well that A is looking right at E along with 98 other blue-eyed people?

Well, the thing is:

-the statement "A knows that E knows there are 98 blue-eyed people" is true,
-the statement "A knows that B knows that C knows that D knows that E knows there are 98 blue-eyed people" is false.

I assume you're okay with the idea that the statement "X is true" is a different statement from "A knows that X is true". The thing is, the same principle holds for longer nested statements. All the statements (i) through (v) above are different from one another.

I know that when there are a large number of blue-eyed people like 100, then subjectively, it seems like "Well, it's just so obvious to everyone that there are a large number of blue-eyed people. Everyone can see it, and everyone can see that everyone else can see it, and so on."

Certainly, from our vantage point outside the island, we know that there are at least 99 blue-eyed people, and that there are at least 5 blue-eyed people, and that there is at least one blue-eyed person. And again, subjectively, it may feel like, "It's just so obvious to everyone on the island that there is at least one blue-eyed person; I mean, everyone knows it, and everyone knows that everyone knows it."

But the thing is, before the Guru's announcement, it's actually NOT possible to construct ARBITRARILY long "nested" true statements of the form "Person 1 knows that Person 2 knows that Person 3 knows that Person 4 knows ...... that there is at least one person with blue eyes."

[douglasm]

Fact 1: E knows that there are blue-eyed people on the island.
Fact 2: D knows fact 1.

1 and 2 are two different facts, and neither of them implies the other.

Fact 3: C knows fact 2.

3 is, again, a different fact from 1 and 2.

Fact 4: B knows fact 3.
Fact 5: A knows fact 4.

Again, these are all distinct and different facts. All of them are false before the Guru's announcement and true after.

Here's another fact that is different from all 5 of the above:
Fact 6: A knows that there are blue-eyed people on the island.

In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes.

A might know that E can see D's blue eyes, but that is not the same thing as the chain of imagined reasoning going through all those people - in that chain, D's eye color is not blue but rather unknown because D's thought processes are part of the chain. Cutting B, C, and D out of the chain changes not only the truth value but also what piece of knowledge you are talking about.

Aha! And this is exactly the point I am contending. I disagree with you on this. A does know that B knows that C knows that D knows that E can see blue-eyed people. A can look at B, C and D as they all watch E staring into the eyes of a blue-eyed person.

And which blue-eyed person is E looking at?
A? Well, A is thinking "I have no idea what color the eyes he's looking at are."
B? Well, B is thinking "I have no idea what color the eyes he's looking at are." A is thinking "B doesn't know what E sees."
C? Well, C is thinking "I have no idea what color the eyes he's looking at are." B is thinking "C doesn't know what E sees." A is thinking "B knows that C is clueless."
D? Well, D is thinking "I have no idea what color the eyes he's looking at are." C is thinking "D doesn't know what E sees." B is thinking "C knows that D is clueless." A is thinking "B knows that C is aware of D's cluelessness."

Moreover, I find it hard to believe that our ideal logicians cannot find any logical basis to believe that the others are aware that there are blue-eyed people on the island.

Actually, the ideal logicians are all quite well aware that everyone knows that there are blue-eyed people on the island.

In fact, it is my understanding that the question specifically states this--what is more it says no trickses!

The 'trickses' referred to are of the circumvent-the-problem type. Drawing in the sand to communicate, using pools of water as mirrors to check your own eye color directly, that sort of thing. Any variety of pure logical analysis that requires nothing but thinking, however convoluted the logic, is fair game.

This is why I think that the whole hypothetical mind-bending recursion simulation is irrelevant. Why does A need to listen to the Guru to infer what the others must already know? And, my added point is that if A does not already know that B knows that C knows that D knows that E can see blue-eyed people, how does the Guru's pronouncement demonstrate that this is the case? It seems bizarre to me that on the one hand you can state that A does not know this based upon observation (which would imply either stupidity on A's part, or a very strenuous requirement for claiming that an observation is true) and then go and say that A's observation of the Guru's claims (yes, another observation!) is sufficient to either overcome A's stupidity or this strenuous requirement for truth. You cannot have it both ways. If A can infer based upon the Guru's statement that B knows that C knows that D knows that E knows that there are blue-eyed people on the island, then it must (okay, not must, but within the limits of reason) be the case that A, B, C, and D can all see that E can see that people on the island have blue eyes (and that therefore E knows that people on the island have blue eyes) They are both inferences about the thoughts of other individuals, based upon a limited, but reasonable basis (the same basis that we use in everyday life). And this is the crux of the problem. The Guru's words simply tell us about what the Guru can see. The island's inhabitants must still infer what the other individuals on the island are thinking. In this sense, the information has not been changed based upon the Guru's statement.

Honestly, can you actually demonstrate objectively how these two things differ? To put this bluntly: When the Guru makes his pronouncement that he sees blue-eyed people on this island, this is strong evidence that there are blue-eyed people on the island and that everyone knows it. And, it should follow that this might initiate the logical chain of events that is postulated by this riddle. But, you cannot expect me to believe that any one of those guys on the island is walking around without this knowledge already. Here, let's review the statements of their knowledge:

A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

For you guys to be correct about this, you would have to believe that out of this bunch of perfect logicians who all know that there are at least 98 blue-eyed people on the island, there are at least some who have no idea about whether or not any of the other logicians are aware of this fact. IMHO, this riddle is at worst ambiguous on this point, and at best implicitly implies [strongly states] that this is not the case.

I think your core problem here is equating two pieces of knowledge that are not actually the same knowledge.

Fact 1: A knows X
Fact 2: B knows fact 1

No matter what X is, facts 1 and 2 are different. Continuing the progression through 100 facts produces 100 distinct facts. You are taking fact 3 or 4 and saying it's the same as fact 100, but they are in fact different.

Well, whatever... it's not that important. I would be happy if the puzzle had be written to specifically state that they had absolutely no way of inferring the knowledge of the others prior to the Guru's pronouncement and that this pronouncement unambiguously allowed them to infer what the others knew. Without any such clear statement, the effect of the Guru's statement on the blue-eyed individuals on the island is difficult to determine. That's the only point I'm trying to make and that's why I think it's arbitrary... just as arbitrary as some dude falling over a log.

Some dude falls over a log: the next day, nothing in particular happens because of it
The Guru makes his announcement: the next day, if there were a single blue-eyed person then he would leave, and everyone can prove this easily

[phlip]

Aha! And this is exactly the point I am contending. I disagree with you on this. A does know that B knows that C knows that D knows that E can see blue-eyed people. A can look at B, C and D as they all watch E staring into the eyes of a blue-eyed person.

A can look at B, C and D as they all watch E staring into the eyes of someone. But A doesn't know if B knows that C knows that D knows that that person has blue eyes. Note that there is also a distinction between "A and B both know X" and "A knows that B knows X". Or, in this case, "A knows that B, C and D know X" and "A knows that B knows that C knows that D knows X".

Say, for the sake of argument, that it was C that E is staring at. The situation is similar for others, but I'll describe it for C.
Now, it it's clearly common knowledge that E can see C. But while D knows that C's eyes are blue, C doesn't know that D knows that C's eyes are blue. Because that would require C to know their own eye colour. And if C doesn't know etc, then B can't know that C does know etc, because being a perfect logician means never knowing something which isn't true (specifically, if logic is consistent, it means never deducing something false from only true information). And, similarly, A can't know that B knows that C knows that D knows that C's eyes are blue.
Generally, any chain of "A knows that B knows that etc that X's eyes are blue", before the announcement, is false, if X appears in the chain of people who are doing the knowing. Which is why when you run that chain through everyone, you end up with "don't know that anyone's eyes are blue".

[edit]

All of them are false before the Guru's announcement and true after.

Actually, no, and this is the point... your Facts 1 through 4 are true before the Guru's announcement, but 5 is false. After the announcement, 5 is also true.

[Pseudonymoniae]

Fact 1: E knows that there are blue-eyed people on the island.
Fact 2: D knows fact 1.

1 and 2 are two different facts, and neither of them implies the other.

Fact 3: C knows fact 2.

3 is, again, a different fact from 1 and 2.

Fact 4: B knows fact 3.
Fact 5: A knows fact 4.

Again, these are all distinct and different facts. All of them are false before the Guru's announcement and true after.

Here's another fact that is different from all 5 of the above:
Fact 6: A knows that there are blue-eyed people on the island.

I mentioned previously that I do not understand upon what basis you can make this point. You seem to be of the opinion that prior to the Guru's announcement the people on the island cannot infer each other's knowledge (I mean the entire chain of A believes B believes C...) based upon observation. Based upon my reading of the premises of this puzzle, this is not the case. I disagree that these facts you have presented are false.

Premise 1: Everyone on the island knows that there are blue-eyed people on the island.
Premise 2: Everyone on the island knows that everyone else on the island knows premise 1.

If this is the case, then then the above facts are all true as they can be taken directly from the premises or can be inferred from the premises. How else could I know that E knows there are blue-eyed people on the island? It's in the premises.

In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes.

A might know that E can see D's blue eyes, but that is not the same thing as the chain of imagined reasoning going through all those people - in that chain, D's eye color is not blue but rather unknown because D's thought processes are part of the chain. Cutting B, C, and D out of the chain changes not only the truth value but also what piece of knowledge you are talking about.

Yes, but each of the individuals can evaluate whether any of the individuals on the island know this one key fact (whether there are blue-eyed people on the island) without going through the chain of reasoning. This is the one key fact that is imparted by the Guru. You seem to believe that because the Guru is telling them this fact in the presence of all, this imparts some special significance that allows this chain of reasoning to exist. I disagree. If the logicians can infer based solely upon the Guru's words (just because the Guru says them does not make them fact) that A knows B knows C knows D... then they should also be able to infer this based upon the premises. Unless you can fundamentally distinguish between the information contained in the premises and that provided by the Guru, I cannot agree with you.

Moreover, I find it hard to believe that our ideal logicians cannot find any logical basis to believe that the others are aware that there are blue-eyed people on the island.

Actually, the ideal logicians are all quite well aware that everyone knows that there are blue-eyed people on the island.

If everyone knows at the outset that everyone else knows there are blue-eyed people on the island, then they all should be able to infer that A knows B knows C knows D knows ... Indeed, if this is false, then by what reasoning do I infer that it is the case? And more importantly, if they cannot infer this based upon observation, then by what reason can they be said to infer this based upon the Guru's statements?

This is why I think that the whole hypothetical mind-bending recursion simulation is irrelevant. Why does A need to listen to the Guru to infer what the others must already know? And, my added point is that if A does not already know that B knows that C knows that D knows that E can see blue-eyed people, how does the Guru's pronouncement demonstrate that this is the case? It seems bizarre to me that on the one hand you can state that A does not know this based upon observation (which would imply either stupidity on A's part, or a very strenuous requirement for claiming that an observation is true) and then go and say that A's observation of the Guru's claims (yes, another observation!) is sufficient to either overcome A's stupidity or this strenuous requirement for truth. You cannot have it both ways. If A can infer based upon the Guru's statement that B knows that C knows that D knows that E knows that there are blue-eyed people on the island, then it must (okay, not must, but within the limits of reason) be the case that A, B, C, and D can all see that E can see that people on the island have blue eyes (and that therefore E knows that people on the island have blue eyes) They are both inferences about the thoughts of other individuals, based upon a limited, but reasonable basis (the same basis that we use in everyday life). And this is the crux of the problem. The Guru's words simply tell us about what the Guru can see. The island's inhabitants must still infer what the other individuals on the island are thinking. In this sense, the information has not been changed based upon the Guru's statement.

Honestly, can you actually demonstrate objectively how these two things differ? To put this bluntly: When the Guru makes his pronouncement that he sees blue-eyed people on this island, this is strong evidence that there are blue-eyed people on the island and that everyone knows it. And, it should follow that this might initiate the logical chain of events that is postulated by this riddle. But, you cannot expect me to believe that any one of those guys on the island is walking around without this knowledge already. Here, let's review the statements of their knowledge:

A group of people with assorted eye colors live on an island. They are all perfect logicians -- if a conclusion can be logically deduced, they will do it instantly. No one knows the color of their eyes. Every night at midnight, a ferry stops at the island. Any islanders who have figured out the color of their own eyes then leave the island, and the rest stay. Everyone can see everyone else at all times and keeps a count of the number of people they see with each eye color (excluding themselves), but they cannot otherwise communicate. Everyone on the island knows all the rules in this paragraph.

For you guys to be correct about this, you would have to believe that out of this bunch of perfect logicians who all know that there are at least 98 blue-eyed people on the island, there are at least some who have no idea about whether or not any of the other logicians are aware of this fact. IMHO, this riddle is at worst ambiguous on this point, and at best implicitly implies [strongly states] that this is not the case.

I think your core problem here is equating two pieces of knowledge that are not actually the same knowledge.

Fact 1: A knows X
Fact 2: B knows fact 1

No matter what X is, facts 1 and 2 are different. Continuing the progression through 100 facts produces 100 distinct facts. You are taking fact 3 or 4 and saying it's the same as fact 100, but they are in fact different.

I'm not sure what you're getting at here. Both A and B should know X. Both A and B should know that each other knows X. And they should be able to infer this all the way down the chain for everyone on the island based upon the premises.

[phlip]

You seem to be of the opinion that prior to the Guru's announcement the people on the island cannot infer each other's knowledge (I mean the entire chain of A believes B believes C...) based upon observation.

The logicians can infer each others' knowledge, but only incompletely. If you know everything I know, and know that I know it, and we're both perfect logicians, you can deduce exactly what I'm able to deduce. However, if you don't know everything I know, you can only deduce part of what I'm able to deduce.

Specifically, in this case, I don't know my own eye colour, but any other islander does know my eye colour. Given that disparity in information, there are things that the other islanders can prove, that I can't know that they know. For instance, if I see 99 blue-eyed people, then I know the true count is either 99 or 100. If it's 99, then I know person A will see 98, and will deduce certain things from that observation. If it's 100, then I know person A will see 99, and will deduce other things from that observation. Since I don't know which one A actually sees, I don't know which deductions they will make. So I have incomplete information about what A can deduce, and there are plenty of statements where "A knows X" and "I don't know if A knows X" can both be true simultaneously.

[Pseudonymoniae]

Aha! And this is exactly the point I am contending. I disagree with you on this. A does know that B knows that C knows that D knows that E can see blue-eyed people. A can look at B, C and D as they all watch E staring into the eyes of a blue-eyed person.

A can look at B, C and D as they all watch E staring into the eyes of someone. But A doesn't know if B knows that C knows that D knows that that person has blue eyes. Note that there is also a distinction between "A and B both know X" and "A knows that B knows X". Or, in this case, "A knows that B, C and D know X" and "A knows that B knows that C knows that D knows X".

[/quote]

Look, these are what I am using as premises based upon the puzzle:

Premise 1: Everyone on the island knows that there are blue-eyed people on the island.
Premise 2: Everyone on the island knows that everyone else on the island knows premise 1.

How can it be said that A cannot infer the entire chain of causality based upon these premises? (I mean honestly, I would like to believe that A could infer this information without the premises, simply based upon the fact that these are reasonable assumptions, but based upon my reading of the premises, A should know these facts.)

And if A cannot infer this causal chain based upon the premises, then how can he do so based upon the Guru's words?

[edit]

You seem to be of the opinion that prior to the Guru's announcement the people on the island cannot infer each other's knowledge (I mean the entire chain of A believes B believes C...) based upon observation.

The logicians can infer each others' knowledge, but only incompletely. If you know everything I know, and know that I know it, and we're both perfect logicians, you can deduce exactly what I'm able to deduce. However, if you don't know everything I know, you can only deduce part of what I'm able to deduce.

Specifically, in this case, I don't know my own eye colour, but any other islander does know my eye colour. Given that disparity in information, there are things that the other islanders can prove, that I can't know that they know. For instance, if I see 99 blue-eyed people, then I know the true count is either 99 or 100. If it's 99, then I know person A will see 98, and will deduce certain things from that observation. If it's 100, then I know person A will see 99, and will deduce other things from that observation. Since I don't know which one A actually sees, I don't know which deductions they will make. So I have incomplete information about what A can deduce, and there are plenty of statements where "A knows X" and "I don't know if A knows X" can both be true simultaneously.

Okay, that's fair. But the Guru's words don't allow us to distinguish between these possibilities. That must be deduced via the causal chain. And I'm saying that the Guru's words are no more able to allow me to deduce the causal chain (everyone knows that everyone knows ad infinitum) than by the premises (everyone on this island knows and everyone knows that everyone else knows), or even simple deduction based upon some kind of probabilistic model (e.g. there's a very high likelihood that everyone on this island knows and everyone knows that everyone knows). And I don't see how given the knowledge that everyone knows that everyone knows that the perfect logician A cannot deduce the causal chain.

[phlip]

[I edited this into my previous post, but it seems that this was after you replied, so splitting it off into a new post]

Premise 1: Everyone on the island knows that there are blue-eyed people on the island.
Premise 2: Everyone on the island knows that everyone else on the island knows premise 1.

These premises don't prove as much as you want them to, though.

Let's try a different situation. I toss a coin, while three people watch. For now, none of them know what the result of the coin toss is. Then, I take each one aside, and I show them the coin. I then tell them "just between you and me, I'll also be showing the coin to the other two. Don't let them know I told you this."

So we have analogs to your premises:
Premise 1: Everyone has seen the coin.
Premise 2: Everyone knows that premise 1 is true.
[edit] I see that your actual premise 2 is "Everyone knows that everyone knows that premise 1 is true", which is slightly longer... however, the pattern is similar... you just need 4 people, and an extra level of nesting in the private information above and the questions below. [/edit]

Now, let's see what people actually know. An easy way to explore this is to see what questions each person is able to answer.

Say I went up to person 1, and asked "So, what was the result of the coin flip?"... they'll be able to answer it. Since they saw the coin.
But say instead I went up to person 2, and asked "If I went up to person 1, and asked 'So, what was the result of the coin flip?', what would they say?"... they'll be able to answer it too. Since I told them I'd shown the coin to person 1.

However, say I instead go up to person 3, and ask "If I went up to person 2, and asked 'If I went up to person 1, and ask "So, what was the result of the coin flip?", what would they say?', what would they say?"... and then ask again, and then write it down so they can figure out the mess that is that question's grammatical structure... they wouldn't be able to answer it. Sure, person 3 knows that person 1 knows the coin... but that's not what we're asking person 3 in this question. And while person 2 could answer their question, they did so by drawing on knowledge that person 3 doesn't know person 2 has. So the only answer person 3 can give to this last question is "I don't know".

However, if I know reveal the coin in plain sight, with all three present at once... it becomes common knowledge. And person 3 would be able to answer that last question.

[Pseudonymoniae]

To be specific, this is the statement that I find issue with: "someone's imagined thought processes do not include the knowledge that there are blue eyes on the island". So yes, if you conduct this infinite, hypothetical thought recursion, I can agree that you might come to this conclusion. But, this should be superceded by the fact that A can see that C or D or whoever is at the end of the list must know that there are blue-eyed people on the island. Given that it can be objectively seen that there are 99 blue eyed people who are visible to everyone it does not make sense that the Guru's words have any added impact. In fact, if it is not "common knowledge" based upon this observable fact, then why should the Guru's words make it so? Do you see my point? A might imagine that B imagines that C imagines that D imagines that E imagines that there are no blue-eyed people on the island. But A can see that E is looking at D's blue eyes. How is this not the case? Are we talking about the same thing?

I think this paragraph (also partly quoted by phlip) is the key to why we disagree. (EDIT: You posted while I was composing this.)

It seems like you're okay with the truth of each of the following statements:

(i) A knows there are 99 blue-eyed people.
(ii) A knows that B knows there are 98 blue-eyed people.
(iii) A knows that B knows that C knows there are 97 blue-eyed people.
(iv) A knows that B knows that C knows that D knows there are 96 blue-eyed people.
(v) A knows that B knows that C knows that D knows that E knows there are 95 blue-eyed people.

But it seems like you also want to say that we can "collapse" this in some sense; that we can say "more".

Why is A hypothesizing about E seeing only 95 blue-eyed people, when we know darn well that A is looking right at E along with 98 other blue-eyed people?

Well, the thing is:

-the statement "A knows that E knows there are 98 blue-eyed people" is true,
-the statement "A knows that B knows that C knows that D knows that E knows there are 98 blue-eyed people" is false.

I assume you're okay with the idea that the statement "X is true" is a different statement from "A knows that X is true". The thing is, the same principle holds for longer nested statements. All the statements (i) through (v) above are different from one another.

I know that when there are a large number of blue-eyed people like 100, then subjectively, it seems like "Well, it's just so obvious to everyone that there are a large number of blue-eyed people. Everyone can see it, and everyone can see that everyone else can see it, and so on."

Certainly, from our vantage point outside the island, we know that there are at least 99 blue-eyed people, and that there are at least 5 blue-eyed people, and that there is at least one blue-eyed person. And again, subjectively, it may feel like, "It's just so obvious to everyone on the island that there is at least one blue-eyed person; I mean, everyone knows it, and everyone knows that everyone knows it."

But the thing is, before the Guru's announcement, it's actually NOT possible to construct ARBITRARILY long "nested" true statements of the form "Person 1 knows that Person 2 knows that Person 3 knows that Person 4 knows ...... that there is at least one person with blue eyes."

This is one of those yeah but's... If this long nested statements cannot be constructed a priori based upon the premises, then how can the Guru's statement possibly provide the requisite information to construct them?

I just don't see how the Guru's pronouncement in the presence of everyone allows for such nested statements, given that it provides observational knowledge about others' thoughts and the fact that the exact same knowledge was not good enough when it was inferred based upon other forms of observation (e.g. seeing the obvious!).

[edit1]

[I edited this into my previous post, but it seems that this was after you replied, so splitting it off into a new post]

Premise 1: Everyone on the island knows that there are blue-eyed people on the island.
Premise 2: Everyone on the island knows that everyone else on the island knows premise 1.

These premises don't prove as much as you want them to, though.

Let's try a different situation. I toss a coin, while three people watch. For now, none of them know what the result of the coin toss is. Then, I take each one aside, and I show them the coin. I then tell them "just between you and me, I'll also be showing the coin to the other two. Don't let them know I told you this."

So we have analogs to your premises:
Premise 1: Everyone has seen the coin.
Premise 2: Everyone knows that premise 1 is true.
[edit] I see that your actual premise 2 is "Everyone knows that everyone knows that premise 1 is true", which is slightly longer... however, the pattern is similar... you just need 4 people, and an extra level of nesting in the private information above and the questions below. [/edit]

Now, let's see what people actually know. An easy way to explore this is to see what questions each person is able to answer.

Say I went up to person 1, and asked "So, what was the result of the coin flip?"... they'll be able to answer it. Since they saw the coin.
But say instead I went up to person 2, and asked "If I went up to person 1, and asked 'So, what was the result of the coin flip?', what would they say?"... they'll be able to answer it too. Since I told them I'd shown the coin to person 1.

However, say I instead go up to person 3, and ask "If I went up to person 2, and asked 'If I went up to person 1, and ask "So, what was the result of the coin flip?", what would they say?', what would they say?"... and then ask again, and then write it down so they can figure out the mess that is that question's grammatical structure... they wouldn't be able to answer it. Sure, person 3 knows that person 1 knows the coin... but that's not what we're asking person 3 in this question. And while person 2 could answer their question, they did so by drawing on knowledge that person 3 doesn't know person 2 has. So the only answer person 3 can give to this last question is "I don't know".

However, if I know reveal the coin in plain sight, with all three present at once... it becomes common knowledge. And person 3 would be able to answer that last question.

See, I'm not sure I even agree with this. It is clear based upon the premises that we all probably know the answer (unless one of us is impaired in some fashion). As person 3, I cannot say for sure what the others know, because I am not privy to their thoughts, but I can infer. And sure, it is more difficult to infer what someone else knows about someone else (indeed, my confidence in making such a statement should decrease) but that doesn't mean that I, as person 3, should not be able to infer that person 2 thinks that person 1 thinks that the coin is heads, given that I know it's heads and I know they both know it is heads. And I wouldn't be making such a big deal about it here, as that is slightly tenuous, except that I don't believe that revealing the coin in plain sight fundamentally changes the knowledge value. Sure, we can call it "common knowledge" now, but that's really a subjective difference. What is "common knowledge" really? I cannot "mind meld" with these guys to see their thoughts and truly know what they know. So then how is it fundamentally different to say that "I saw the coin and I saw them see the coin, therefore I can postulate about what person 1 thinks person 2 saw" than to say that "I saw the coin and I know persons 1 and 2 the coin, so I can postulate what I think person 1 thinks person 2 saw"? And I don't see how adding a fourth person to this changes anything. If I know what everyone knows then I should be able to infer that A knows B knows C.

[edit2]

Look guys this looks like a very technical aspect of formal logic. I understand what the argument being used consists of, but I disagree with how it is constructed. It's not that I can't wrap my head around it... I just disagree. It seems very much like a subjective distinction that's being made that allows the Guru's statement to be considered "common knowledge" (thereby allowing the formulation of these nested statements by a perfect logician), but precludes either the knowledge that can be obtained by visual observation or that provided by the premises of this puzzle from also being considered common knowledge. Indeed, in common parlance, "common knowledge" is considered that which we assume that others know, without any real evidence that this is the case. This would actually be several steps down in terms of truth value (I assume I used that right) compared to that which can be inferred based upon visual observation or the premises in this case. Indeed, if we consider "common knowledge" along a continuum (as opposed to defining it using some arbitrary criterion), then what I am saying makes perfect sense. While the perfect logicians might be able to produce the required chain of nested inferences without the statements of the Guru, these inferences would be slightly weaker. What is more, I should think that even the Guru's pronouncement could be improved upon. Perhaps if we were real sticklers, we could push our benchmark further along the continuum... In this case, we might require that the logicians have mind reading skills, thus allowing them to unambiguously know what the others know.

[phlip]

As person 3, I cannot say for sure what the others know, because I am not privy to their thoughts, but I can infer.

But infer based on what? As person 3, all you know is that (a) The coin is heads (or tails, whichever it actually is). (b) I showed the coin to the other two. Nothing more. You don't know I told person 2 that I showed the coin to person 1. For all you know, person 2 may think they're the only one who even knows what the coin is. That is, the idea that I only showed the coin to person 2, and did nothing else, is consistent with all the information that you know person 2 has. So you can't completely rule out the possibility that person 2 can't completely rule out the possibility that only person 2 has seen the coin. So you can't answer with certainty that person 2 knows that person 1 has seen the coin.

This is not an "arbitrary distinction", this is a real limitation in what person 3 actually knows.

(indeed, my confidence in making such a statement should decrease)

Confidence doesn't enter into it. We're talking logic here... either you can completely prove that it is 100% true with no doubt whatsoever, or it isn't proven. Note that, in the original blue-eyes puzzle, it specifically says that they must prove their own eye colour in order to leave (and if they can prove it, they must leave). Guesses an estimations and "inferences" (in the layman sense of the word) are completely irrelevant. The only thing that matters is inferences in the mathematical sense of the word... that is, things that you can 100% completely logically prove based on your given premises.

[douglasm]

The reason that the nested chain of hypotheticals can't be constructed as known true knowledge in the first place is that every last potential source of the knowledge that blue eyes are present is removed by how the chain nesting is constructed. The Guru provides a source of that knowledge that is unaffected by the chain nesting's construction.

In the bottommost level of the nested hypotheticals, every last pair of blue eyes on the entire island is treated as an unknown color because somewhere in the levels above it there is a person who does not know about that particular pair of blue eyes. Specifically because of that, the bottommost level of the nested hypotheticals does not know that blue eyes are present - it sees 100 pairs of brown eyes and 100 pairs of unknown color. The Guru's announcement exists in all levels, however, so once he makes it then that bottommost level does know that blue eyes are present.

[skullturf]

If I'm not mistaken, an important premise is that it's common knowledge on the island that the Guru is truthful. (I guess that if the Guru isn't known to be truthful, then the Guru's pronouncement really is no different from somebody tripping over a log.)

[Pseudonymoniae]

As person 3, I cannot say for sure what the others know, because I am not privy to their thoughts, but I can infer.

But infer based on what? As person 3, all you know is that (a) The coin is heads (or tails, whichever it actually is). (b) I showed the coin to the other two. Nothing more. You don't know I told person 2 that I showed the coin to person 1. For all you know, person 2 may think they're the only one who even knows what the coin is. That is, the idea that I only showed the coin to person 2, and did nothing else, is consistent with all the information that you know person 2 has. So you can't completely rule out the possibility that person 2 can't completely rule out the possibility that only person 2 has seen the coin. So you can't answer with certainty that person 2 knows that person 1 has seen the coin.

This is not an "arbitrary distinction", this is a real limitation in what person 3 actually knows.

(indeed, my confidence in making such a statement should decrease)

Confidence doesn't enter into it. We're talking logic here... either you can completely prove that it is 100% true with no doubt whatsoever, or it isn't proven. Note that, in the original blue-eyes puzzle, it specifically says that they must prove their own eye colour in order to leave (and if they can prove it, they must leave). Guesses an estimations and "inferences" (in the layman sense of the word) are completely irrelevant. The only thing that matters is inferences in the mathematical sense of the word... that is, things that you can 100% completely logically prove based on your given premises.

Well, hold on, as I understand it we all are aware that each other has been told what the coin is. This might not be the original premises that you postulated, but I had understood it to be the premises which were analogous to those we were discussing (e.g. everyone one the island knows there are blue-eyed people and everyone is aware that the others know as well is analogous to we all know what the coin is and we are all aware that the others know).

[edit1]

Oh, and as to the second point. If we are talking logic in this sort of 100% proof positive framework then I think my point is even more clear. I cannot know 100% that everyone knows what the Guru says is true and understands it as such. This is the whole problem that I have, the entire concept is designed in the context of a reality where these perfect logicians are people. It is not possible, given any of the information from the passage that these guys determine to a 100% certainty their eye colour. And this is the whole point: why is it that the Guru's statements allow 100% certainty even though it still requires them to infer what the others have learned, while their knowledge based upon the premises of the piece and their visual observations do not allow 100% certainty? This is not something that can be proven and it is a flaw of the construct.

[edit2]

The reason that the nested chain of hypotheticals can't be constructed as known true knowledge in the first place is that every last potential source of the knowledge that blue eyes are present is removed by how the chain nesting is constructed. The Guru provides a source of that knowledge that is unaffected by the chain nesting's construction.

In the bottommost level of the nested hypotheticals, every last pair of blue eyes on the entire island is treated as an unknown color because somewhere in the levels above it there is a person who does not know about that particular pair of blue eyes. Specifically because of that, the bottommost level of the nested hypotheticals does not know that blue eyes are present - it sees 100 pairs of brown eyes and 100 pairs of unknown color. The Guru's announcement exists in all levels, however, so once he makes it then that bottommost level does know that blue eyes are present.

Yes, but I still do not agree that this is a property of the Guru's statement which is not already available to the people on the island.

[Gwydion]

Actually, in phlip's constructed coin-flipping world, person 3 is told that persons 1 and 2 will be shown the coin, but not that persons 1 and 2 will know that the other have seen it. That's why:
-all 3 know the result
-all 3 know that the person next to them knows the result
-but all 3 do not know that the person next to them knows that the third person knows the result.

It's important to separate the common parlance "common knowledge" from the formal logical common knowledge. Many people choose to use the term common knowledge to mean "Everybody knows that everybody knows x", when it really means substantially more than that - everybody knows that everybody knows...(as many times as you like)...that everybody knows x. While it seems silly to even have to bring that many cases up, the Blue Eyes puzzle explicitly shows an example of a situation where the former definition is vastly insufficient to explain the logical deduction (or induction) going on. Calling something common knowledge from the perspective of a logic puzzle is not about the likelihood that something is known - it is guaranteed to be known by everyone, and that guarantee is known, and everyone's knowledge of the guarantee is known... ad infinitum.

One of the nicest aspects of this puzzle is the fact that just about every word in the problem statement is both necessary and sufficient to produce the answer. Forcing the Guru to say more might make the puzzle easier to understand, but no less logically valid. Similarly, removing any of the clauses or statements would render the puzzle unsolvable in its current state.

Pseudonymoniae, I'm not really understanding what your hang-up is for the solution as it's currently presented. Why do you disagree with the solutions and explanations presented so far? That could be helpful, as right now, the discussion seems to be going back and forth without making much headway.

As an aside, I think today may have been the most action this thread has seen in a single day since it began.

[Pseudonymoniae]

Here's a summary of the what boils down to my main problems with this question. It's not clear to me that we're really getting anywhere without addressing this directly.

This is the relevant information pertaining to my complaint that is provided to the people on the island:

i)everyone can see everyone else.. and keeps a count of the people of people they see with each colour
ii)everyone knows this rule
iii) "I can see someone who has blue eyes"

Premise 1: everyone knows that there are blue-eyed people on the island
Premise 2: everyone on the island is aware that every other person on the island knows there are people with blue eyes
Addendum: everyone on the island is told that in the presence of all that there are people with blue eyes

My reading of this is that both premise 2 and the addendum cover the same information. If this is the case, then there is no reason why the Guru's words are required to accomplish this feat. Moreover, I disagree with this "common knowledge" explanation. Premise 2 should also be common knowledge. Given either premise 2 or the Guru's addendum, all of the perfect logicians on the island should be able to initiate the logical chain of hypothetical mind inferences.

(I've also tried to point out that if they can deduce premise 2 based upon the Guru's words, then they should also be able to deduce premise 2 based upon visual observation because I do not see a fundamental difference between these two, just a subjective one. Both appear to require some sort of inference which cannot be 100% provable, unless we assume that the rules of the puzzle state that visual observation is inferior to auditory observation... which is hardly mentioned anywhere. Nonetheless, this is not required for the above point.)

[edit]

Actually, in phlip's constructed coin-flipping world, person 3 is told that persons 1 and 2 will be shown the coin, but not that persons 1 and 2 will know that the other have seen it. That's why:
-all 3 know the result
-all 3 know that the person next to them knows the result
-but all 3 do not know that the person next to them knows that the third person knows the result.

It's important to separate the common parlance "common knowledge" from the formal logical common knowledge. Many people choose to use the term common knowledge to mean "Everybody knows that everybody knows x", when it really means substantially more than that - everybody knows that everybody knows...(as many times as you like)...that everybody knows x. While it seems silly to even have to bring that many cases up, the Blue Eyes puzzle explicitly shows an example of a situation where the former definition is vastly insufficient to explain the logical deduction (or induction) going on. Calling something common knowledge from the perspective of a logic puzzle is not about the likelihood that something is known - it is guaranteed to be known by everyone, and that guarantee is known, and everyone's knowledge of the guarantee is known... ad infinitum.

One of the nicest aspects of this puzzle is the fact that just about every word in the problem statement is both necessary and sufficient to produce the answer. Forcing the Guru to say more might make the puzzle easier to understand, but no less logically valid. Similarly, removing any of the clauses or statements would render the puzzle unsolvable in its current state.

Pseudonymoniae, I'm not really understanding what your hang-up is for the solution as it's currently presented. Why do you disagree with the solutions and explanations presented so far? That could be helpful, as right now, the discussion seems to be going back and forth without making much headway.

As an aside, I think today may have been the most action this thread has seen in a single day since it began.

Yeah, I think there was a misunderstanding with the coin flip. As presented, I do not have an issue with the coin flip. However, as was noted, the actual format of my second premise is: "Everyone knows that everyone knows that premise 1 is true", which I think is fundamentally different from the premise presented in the coin flip scenario.

As I just posted above, my issue is that I do not believe that the Guru is providing additional information. I think part of the issue is that much of what I have been arguing is that auditory information provided by the Guru is not < than visually obtained information, or at least is not fundamentally different so as to allow 100% certainty where visual information does not. I would argue that both the Guru's words and visually obtained information are insufficient by this definition.

Although, the broader point that I am attempting to make is that even the premises of the logic puzzle already seem to provide the same information that the Guru provides. There seems to be a distinction between "everyone knows that everyone knows" and the same term used ad infinitum which seems to work really well in mathematical statements but which does not make a lot of sense to me in the context of this puzzle. I suppose the reason is that based upon the information in the puzzle, one must subjectively define a set of logical statements to be used in the equation.

[jestingrabbit]

Premise 1: everyone knows that there are blue-eyed people on the island
Premise 2: everyone on the island is aware that every other person on the island knows there are people with blue eyes
Addendum: everyone on the island is told that in the presence of all that there are people with blue eyes

My reading of this is that both premise 2 and the addendum cover the same information.

They do not. I will demonstrate that they do not on a much smaller island, one in which there are two blue eyed islanders and two brown eyed islanders. Lets call the blue eyed islanders A and B. Certainly, B knows that there is a blue eyed person on the island, namely A. However, B does not know that A knows that there is a blue eyed islander: is is reasonable for B to believe that their eyes are brown, or hazel, or blue, but in all but the last case, A will not see any blue eyed people.

So, in that case the guru's statement conveys the further information to B "A knows that there are blue eyed islander". In fact, the guru adds to the sum total knowledge on the island that "A knows that B knows there are blue eyed islanders" and "B knows that A knows that there are blue eyed islanders".

Moreover, on an island with more blue eyed people, there is a similar introduction of additional facts.

You have resorted to a layman's definition of what common knowledge is several times. In much the same way that I would not rely on a layman's understanding of simultaneity when discussing special relativity, a layman's understanding of what common knowledge is will be similarly misleading here. I strongly urge you to consider that "everyone knows proposition P" and "everyone knows that everyone knows proposition P" and "everyone knows that everyone knows that everyone knows proposition P" are all distinct states of knowledge for a group of people to have. I consider that phlip's example displays that most excellently.

Finally, as a moderator I must ask you to please stop posting multiple times in a row. If you have something to add to what you have already said, and no one else has said anything new, edit your previous post, don't make a new one.

[phlip]

Well, hold on, as I understand it we all are aware that each other has been told what the coin is. This might not be the original premises that you postulated, but I had understood it to be the premises which were analogous to those we were discussing (e.g. everyone one the island knows there are blue-eyed people and everyone is aware that the others know as well is analogous to we all know what the coin is and we are all aware that the others know).

Certainly, you (as person 3) are aware that I've shown the coin to everyone. But you don't know I also told that to the others. That is, as far as you can tell, I may have shown the coin to each of you, and then told you (and only you) that I showed the coin to everyone. In that case, person 2 would not be able to answer the question "what would person 1 say if I asked him what the coin is", because as far as person 2 knows, I never showed the coin to person 1. You know that I actually did, but person 2 may not. So, because you can't rule this possibility out, you can't say for certain that you know that 2 knows that 1 knows the coin's orientation.

Oh, and as to the second point. If we are talking logic in this sort of 100% proof positive framework then I think my point is even more clear. I cannot know 100% that everyone knows what the Guru says is true and understands it as such.

I assume you're touching on the problem that skullturf mentioned above, and which has come up a few times already in the thread so far... which is that the puzzle doesn't explicitly state that it's common knowledge that the Guru's statement is accurate. And they're right that this really should, for clarity, be mentioned in the puzzle statement.

But this is essentially the default state for these sort of logic puzzles - unless stated otherwise, you can treat it as common knowledge that everyone involved is honest, that all communication methods are reliable, that all perfect logicians are also perfect observers, etc, etc. All of this sort of thing is heavily ingrained into the language of logic puzzles.

[Dason]

Have you tried to work out a simple case? Take a case where there are two blue eyed people (call them blue1 and blue2) and one brown eyed person. In this case 'everybody knows that at least one person has blue eyes'. Blue1 can see blue2 so blue1 knows theres at least one person with blue eyes. Blue2 can see blue1 so blue2 knows there is at least one person with blue eyes. Brown can see both so they know there is at least one person with blue eyes. So yes everybody knows there is at least one person with blue eyes. But everybody doesn't know that everybody knows that there is at least one person with blue eyes...

Blue1 doesn't know if blue2 knows that there's at least one person with blue eyes because... Blue1 doesn't know the color of their eyes - so if they pretend for a second that they have brown eyes (which is possible because they don't know the color of their eyes) then in that situation blue2 wouldn't know there is at least one person with blue eyes. So the guru would be giving blue1 some information - mainly that if they have brown eyes that blue2 would be able to deduce that they have blue eyes.

Seriously though have you read this thread? It's a friggin long thread and the problem you're proposing has been suggested before and it has been shown why it is wrong.

[Gwydion]

Premise 1: everyone knows that there are blue-eyed people on the island
Premise 2: everyone on the island is aware that every other person on the island knows there are people with blue eyes
Addendum: everyone on the island is told that in the presence of all that there are people with blue eyes

My reading of this is that both premise 2 and the addendum cover the same information. If this is the case, then there is no reason why the Guru's words are required to accomplish this feat. Moreover, I disagree with this "common knowledge" explanation. Premise 2 should also be common knowledge. Given either premise 2 or the Guru's addendum, all of the perfect logicians on the island should be able to initiate the logical chain of hypothetical mind inferences.

(I've also tried to point out that if they can deduce premise 2 based upon the Guru's words, then they should also be able to deduce premise 2 based upon visual observation because I do not see a fundamental difference between these two, just a subjective one. Both appear to require some sort of inference which cannot be 100% provable, unless we assume that the rules of the puzzle state that visual observation is inferior to auditory observation... which is hardly mentioned anywhere. Nonetheless, this is not required for the above point.)

Actually, that's quite helpful in trying to sort out where the problems arise. The fundamental problem here is that premise 2 is true, but not common knowledge. Of course, neither is premise 1, as you've stated it, as your statements are not logically the same as the original puzzle. Remember that to be common knowledge, not only must something be true, known to be true, and known to be known to be true, but (known to be)^n true. If the premises as you stated them were common knowledge, then I would agree with you that the Guru provides no information, and in fact that would make the puzzle trivial.

As it stands, everyone knows that there are blue eyes on the island, but only because they can see everyone else. The statement "there are blue-eyed islanders" is not common knowledge just from this, however, as was demonstrated earlier with the coins example. Let's try a simplified example - 4 islanders with blue eyes (for the sake of argument, let's assume there are other, brown-eyed islanders too - not needed, but will come up if we try to expand this example beyond 4 islanders).
A sees that B, C, and D have blue eyes, so there are either 3 or 4 blue-eyed islanders.
A knows that B sees C and D with blue eyes, and A knows that B knows there are 2, 3, or 4 blue-eyed islanders.
A knows that B knows that C sees D with blue eyes, so A knows that B knows that C knows there are 1, 2, 3, or 4 blue-eyed islanders. (Obviously, A knows that there are more than 1, but doesn't know that this is common knowledge - it isn't.)
Lastly, A knows that B knows that C knows that D may or may not see anyone with blue eyes. Therefore, A knows... that there are anywhere from 0 to 4 blue-eyed islanders.

The existence of blue eyes is not common knowledge, even though everyone knows that blue eyes exist. The Guru, however, knocks out this last case - after she speaks, the last hypothetical is impossible, and will be ruled out immediately by everyone. Thus, after the Guru speaks, A will know that B knows that C knows that D may or may not see anyone with blue eyes, but if D doesn't see anyone with blue eyes, he would know (by the Guru's statement) that he was the only one, and would leave.
This doesn't happen. Therefore, on night 2, A knows that B knows that C knows that D sees someone with blue eyes, and that there exists at least 2 blue-eyed islanders. If C doesn't see anyone with blue eyes (except for D), he would know that he is one of the 2, and would leave.
This also fails to occur. On night 3, then, A knows that B knows that there are at least 3 blue-eyed islanders, and that both C and D see multiple someones with blue eyes. If B didn't see anyone else (read: if A's eyes were brown), then all 3 would leave that night.
Which brings us to night 4. Everybody is still there, but now, A knows for certain that B sees at least 3 blue-eyed islanders. A sees 2 (not counting B, who can't see himself), and knows that A himself is the third, and all 4 leave.

Do you see where this breaks down without the Guru's statement?

[phlip]

However, as was noted, the actual format of my second premise is: "Everyone knows that everyone knows that premise 1 is true", which I think is fundamentally different from the premise presented in the coin flip scenario.

It's not fundamentally different, it's just harder to explain in English.

For that, consider instead the coin flip scenario with four people. You can be person 4.
After tossing the coin, I take each person aside, and show them the coin. I then say "Yeah, I'm going to show the coin to everyone. Oh, also, by the way, I'm also going to tell everyone 'I'm going to show the coin to everyone'... but don't let anyone else know I told you this extra bit."

Premise 1: Everyone knows which side the coin landed on.
Premise 1.5: Everyone knows that premise 1 is true.
Premise 2: Everyone knows that everyone knows premise 1 is true. AKA Everyone knows that premise 1.5 is true.
Additional: While premise 2 is true, none of the people in the scenario know it. As a person in the scenario, you know premise 1.5 is true, but for all you know, you may be the only person who knows premise 1.5 is true. In order for you to know that premise 2 is true, you'd need to know that other people knew that premise 1.5 is true, which you don't.

Now, call "What side did the coin land on?" question A. If I was to ask anyone this question, they'd be able to answer it - they saw the coin.
Call "If I was to ask the person to your left question A, what would they say?" question B. If I was to ask anyone this question, they'd be able to answer it - I told everyone that everyone saw the coin.
Call "If I was to ask the person to your left question B, what would they say?" question C. If I was to ask you this question, you'd be able to answer it - I told you that I told everyone that everyone saw the coin. However note that to answer this question you have to use information that, as far as you know, I only told to you... you don't know that other people also have that information.
Call "If I was to ask the person to your left question C, what would they say?" question D. If I was to ask you this question, you wouldn't be able to answer it. Because even though the person to your left could in fact answer question C, to do so they'd have to rely on information that you don't know they have. So you wouldn't be able to say that they could answer it.

Now, say I reveal the coin to everyone, together, at once. The actual orientation of the coin will come as a surprise to noone - you all saw the coin, and you all know that everyone saw the coin. But after I do this reveal, you'd be able to answer question D with certainty.

Note that this is identical in form to the previous coin analogy, just with an extra layer.

[skullturf]

There seems to be a distinction between "everyone knows that everyone knows" and the same term used ad infinitum which seems to work really well in mathematical statements but which does not make a lot of sense to me in the context of this puzzle.

(Others posted while I was composing this, but I hope my remarks are still helpful.)

Your quoted remark above may be the key to our disagreement.

I agree that subjectively, this distinction is hard to wrap one's head around.

But the thing is, once you admit that "X is true" and "A knows that X is true" are two different statements, then you have to admit that the following statements are all distinct from one another:

"For all A and B, A knows that B knows that there are 98 blue-eyed people"
"For all A and B and C, A knows that B knows that C knows that there are 98 blue-eyed people"
"For all A and B and C and D, A knows that B knows that C knows that D knows that there are 98 blue-eyed people"
and so on.

Similarly, the following statements are all distinct from one another:

"For all A and B, A knows that B knows that there is at least one blue-eyed person"
"For all A and B and C, A knows that B knows that C knows that there is at least one blue-eyed person"
"For all A and B and C and D, A knows that B knows that C knows that D knows that there is at least one blue-eyed person"
and so on.

---

In a math class I taught, I once asked a question including an expression like "the day after the day after the day before the day after the day after tomorrow". Obviously, that's subjectively hard to follow. But even if psychologically you lose track, surely you must admit that every time you insert the words "the day after", you've changed the meaning.

"Everyone knows that everyone knows that X" is different from "Everyone knows that everyone knows that everyone knows that X", which in turn is different from "Everyone knows that everyone knows that everyone knows that everyone knows that X".

I'm not claiming that there are many practical situations where this distinction is relevant. This is pure logic. If I have some statement P, even if P is a complicated statement and I've subjectively "lost track" of where I am, the statements "P is true" and "Person Y knows that P is true" are not the same statement.

Two layers is different from three layers, five layers is different from six layers, and 99 layers is different from 100 layers.

Remember, of course this is an artificial puzzle. Of course there are no islands full of perfect logicians. Of course in practice, most people on the island would get confused and not be able to follow complicated nested statements. But that's all beside the point. This is a pure logic puzzle.

[douglasm]

Here's a summary of the what boils down to my main problems with this question. It's not clear to me that we're really getting anywhere without addressing this directly.

This is the relevant information pertaining to my complaint that is provided to the people on the island:

i)everyone can see everyone else.. and keeps a count of the people of people they see with each colour
ii)everyone knows this rule
iii) "I can see someone who has blue eyes"

Premise 1: everyone knows that there are blue-eyed people on the island
Premise 2: everyone on the island is aware that every other person on the island knows there are people with blue eyes
Addendum: everyone on the island is told that in the presence of all that there are people with blue eyes

My reading of this is that both premise 2 and the addendum cover the same information. If this is the case, then there is no reason why the Guru's words are required to accomplish this feat. Moreover, I disagree with this "common knowledge" explanation. Premise 2 should also be common knowledge. Given either premise 2 or the Guru's addendum, all of the perfect logicians on the island should be able to initiate the logical chain of hypothetical mind inferences.

(I've also tried to point out that if they can deduce premise 2 based upon the Guru's words, then they should also be able to deduce premise 2 based upon visual observation because I do not see a fundamental difference between these two, just a subjective one. Both appear to require some sort of inference which cannot be 100% provable, unless we assume that the rules of the puzzle state that visual observation is inferior to auditory observation... which is hardly mentioned anywhere. Nonetheless, this is not required for the above point.)

The part I bolded is where you are wrong.

As I just posted above, my issue is that I do not believe that the Guru is providing additional information. I think part of the issue is that much of what I have been arguing is that auditory information provided by the Guru is not < than visually obtained information, or at least is not fundamentally different so as to allow 100% certainty where visual information does not. I would argue that both the Guru's words and visually obtained information are insufficient by this definition.

The importance of the Guru's statement has nothing whatsoever to do with whether it is auditory or visual. It has to do with the fact that all islanders observed it publicly together, and simultaneously observed each other observing it, observed each other observing each other observing it, and so on - and nowhere in any part of any chain of "observing each other observing each other..." is there any knowledge that even a single islander does not have, so that chain can be extended ad infinitum while remaining true.

Although, the broader point that I am attempting to make is that even the premises of the logic puzzle already seem to provide the same information that the Guru provides. There seems to be a distinction between "everyone knows that everyone knows" and the same term used ad infinitum which seems to work really well in mathematical statements but which does not make a lot of sense to me in the context of this puzzle. I suppose the reason is that based upon the information in the puzzle, one must subjectively define a set of logical statements to be used in the equation.

The rules of mathematics and the rules of formal logic are very similar, and regardless of your intuition the number of times "everyone knows that" is repeated is an absolutely critical distinction.

Let's try another approach, addressing why the Guru's announcement is not arbitrary as a marker:
Suppose we designate Day 1 by when someone trips over a log. What can the islanders derive from this event? Nothing whatsoever.
Suppose instead that we designate Day 1 by when the Guru makes his announcement. What can the islanders derive from this event? Every islander instantly derives that if there were only one blue-eyed person he would leave immediately. Every islander can prove this with absolute 100% confidence. It may not seem especially relevant on first consideration, but it is indisputably true. The Guru's announcement therefore adds the statement "if there were only one blue-eyed person he would leave on Day 1" to the knowledge base of the island, and I think it's a lot easier to see how that piece of information was not previously known.

[Pseudonymoniae]

Alright, I've read over the arguments being made. Clearly the disagreement we have been having is related to differences in how we translate the puzzle to mathematical statements. It's been my opinion that the information provided prior to the Guru's speech can allow for the required ad infinitum inference to be made, or at least is no less informative than that information which the Guru provides. It seems that this probably violates some formal logic convention regarding how we define common knowledge (i.e. information which is available to everyone all at once and which everyone can observe everyone else receiving is common, whereas information which I would imagine to be equivalent to this form of knowledge clearly does not follow this convention). Seeing as this is a formal logic problem, I'm willing to throw this one in and agree with the group.

So yes, given this definition of common knowledge, and the explanation that the Guru's words, but not the premises, etc. constitute common knowledge, I can agree that this logic puzzle does work. I retract any slanderous insults I may have made against the puzzle.

Thanks all it's been fun, I've got to get some sleep. Some very good explanations, btw.

[lightvector]

Alright, I've read over the arguments being made. Clearly the disagreement we have been having is related to differences in how we translate the puzzle to mathematical statements. It's been my opinion that the information provided prior to the Guru's speech can allow for the required ad infinitum inference to be made, or at least is no less informative than that information which the Guru provides. It seems that this probably violates some formal logic convention regarding how we define common knowledge (i.e. information which is available to everyone all at once and which everyone can observe everyone else receiving is common, whereas information which I would imagine to be equivalent to this form of knowledge clearly does not follow this convention). Seeing as this is a formal logic problem, I'm willing to throw this one in and agree with the group.

So yes, given this definition of common knowledge, and the explanation that the Guru's words, but not the premises, etc. constitute common knowledge, I can agree that this logic puzzle does work. I retract any slanderous insults I may have made against the puzzle.

Thanks all it's been fun, I've got to get some sleep. Some very good explanations, btw.

I think you still have a lingering confusion about the puzzle then. It's not just a matter of convention that the premises do not provide common knowledge. It's actually the only correct way to interpret the puzzle.

Take 3 people blue-eyed people in a room, A B C, who see each other (and see each other seeing each other, and see each other seeing each other seeing each other... etc). Also let's even assume "everyone in this room is a perfect logician" is common knowledge, but A B and C know nothing else and cannot communicate. Then, "there exists a blue-eyed person" is not common knowledge. In particular, "A knows that B knows that C knows that there exists a blue-eyed person" must be false. Here's why:

It is consistent with all of A's knowledge that there are only 2 blue-eyed people in the room, B and C, because A doesn't know his own eye color. So as far as A knows, his own eyes might be brown, and that B's knowledge might be the knowledge that B has when B sees only one other blue-eyed person, C. In such a case, B might not know that C knows that there were any blue-eyed people. Because in such a case, B would not know his own eye color, so he could not know if C saw any blue eyes at all. So "A knows that B knows that C knows that there exists a blue-eyed person" is false. And it would still be false if 3 blue-eyed people met in real life under these conditions.

The key thing is that nobody knows their own eye color, so that every time we step down into the hypothetical from some person's point of view, we have to assume that the person's eye color might be brown. If we walk down a chain of 2 hypotheticals, like "Does A knows that B knows that...?", we have to assume that both A and B might have brown eyes. When we're talking about A's knowledge, A doesn't know his own eye color, so he has to assume they might be brown, and A's knowledge about B has to take into account B doesn't know his own eye color, so A's knowledge about B's knowledge must assume that both A and B's eyes might be brown. Etc.

The guru's statement is different because there's no information loss as we go down. A knows that B heard the exact same statement, because the guru made it publicly when A and B were both present. A knows that B knows that C heard the exact same statement and so on. So "A knows that B knows that C knows that there exists a blue-eyed person" has now become true. Because A knows that B knows that C heard the guru's statement. Being a public, now-external piece of information, it doesn't diminish every time we step into a new hypothetical.

Of course, we need some other assumptions, like "everyone is not deaf" and "the guru is reliable" and such being common knowledge.

Basically, human intuition sucks when it comes to things like this. In real life, it's not usually necessary to reason deeper that 1-2 levels about other people's knowledge, so humans are not built to do it. Just as for things like Monty Hall, when we trust that intuition too much, we get the wrong answer, and it's not just some valid opinion that happens not to conform to convention, it's actually just wrong.