Problem in logic

[Yoda]

I have no problem with the recursion stuff, I understand that just fine.

That case is a contradiction due to the guru's statement.

Here is the point where I need some clarification. It's a contradiction due to the content of the guru's statement. That content being that the guru sees someone with blue eyes. Some people are saying that the fact that guru is saying it gives it some extra meaning, which I don't get at all.

The actual content of the Guru's statement can be derived logically from the initial conditions. That is, everyone on the island will see 99 or 100 people with blue eyes, therefore they can conclude that the guru sees at least one person with blue eyes, and all the rest of the logic you laid out would follow from there (and the same logic for brown).

So what am I missing that requires the Guru to actually say something, and what keeps the brown eyed people from leaving?

 

[boobatuba]

ISo what am I missing that requires the Guru to actually say something, and what keeps the brown eyed people from leaving?

The fact that when all the blue-eyed people leave all that the brown-eyed people learn is that their own eyes are not blue. They don't learn that their own eyes are brown.

 

[Yoda]

I never said that. The brown eyed people can deduce they're brown eyed in the exact same way if only the content of the guru's statement matters. They learn they are brown eyed when the rest of the brown eyes don't all leave the day they are expected to if they don't have brown eyes.

 

[boobatuba]

I never said that. The brown eyed people can deduce they're brown eyed in the exact same way if only the content of the guru's statement matters. They learn they are brown eyed when the rest of the brown eyes don't all leave the day they are expected to if they don't have brown eyes.

No, they don't. The last day the blue-eyed people are on the island, they all realize that there was one more person on the island that the guru might have been talking about and therefore also has blue eyes. Themselves.

The brown-eyed people have no such reference. In their minds, they could still have any other color besides blue.

 

[Yoda]

They have that same reference because it follows from the initial conditions that the Guru can see a brown eyed person in the case of 100/100/1.

 

[boobatuba]

They have that same reference because it follows from the initial conditions that the Guru can see a brown eyed person in the case of 100/100/1.

No, they don't have the same reference. The guru says "I see someone with blue eyes." The solution comes from determining, through the process of elimination, how big the group of people containing the one she is talking about is.

Originally, all the blue-eyed people think "it could be any of those 99 blue-eyed people I see." I'll have to wait until the 99th day to see if they all leave because they all know this, too (well, they all either see 98 or 99 people with blue eyes depending on whether or not I do). When they don't, the blue-eyed people all realize that they are also a part of the group that contains the person the guru was talking about and that they therefore, have blue eyes. They can then leave.

The brown-eyed people see 100 blue-eyed people. They're screwed.

 

[Yoda]

Do you agree that the Guru's statement is logically self-evident whether the Guru states it or not? If so what does it matter that the Guru actually states it?

 

[boobatuba]

Do you agree that the Guru's statement is logically self-evident whether the Guru states it or not? If so what does it matter that the Guru actually states it?

It's logically self-evident, yes. The problem is that it's also logically self-evident that someone has brown eyes, too. Lots of things are logically self-evident that don't present a solution to the problem.

What does present a solution to the problem is the binary (she's talking about me/she's talking about someone else) condition the guru introduces when she speaks.

 

[Yoda]

How does stating something someone already knows introduce a condition? The condition would be introduced when the perfect logicians generate the statement as part of their search of the initial conditions, no?

 

[boobatuba]

How does stating something someone already knows introduce a condition? The condition would be introduced when the perfect logicians generate the statement as part of their search of the initial conditions, no?

No. Look at the perspective of a blue-eyed person from the initial conditions. You can see 99 other blue-eyed people on the island. They don't leave on the 99th day. How does this tell you anything about your own eye color? There's no subset of people that include the one the guru was talking about yet.

The same for the brown-eyed people, only they have no subset of brown-eyed people before or after the guru speaks. They only have a subset of non-blue-eyed people. And you can't leave the island just because you know your own eyes aren't blue.

The guru's statement creates two subsets of people...one subset that contains people she could have been talking about (people with blue eyes) and a subset that contains people she could not have been talking about (people with eye colors other than blue).

 

[Yoda]

Initial condition for a person of any eye color. I do not know my eye color. I see blue eyes. I see brown eyes. I see green eyes (if person in question isn't Guru).

Therefore all the following follow

My eyes are blue or not blue.
My eyes are brown or not brown.
My eyes are green or not green.
My eyes are (any eye color that I know exists) or not (that same color).
My eyes are some color I have no knowledge of.

So you can test all those conditions from there.

 

[boobatuba]

Initial condition for a person of any eye color. I do not know my eye color. I see blue eyes. I see brown eyes. I see green eyes (if person in question isn't Guru).

Therefore all the following follow

My eyes are blue or not blue.
My eyes are brown or not brown.
My eyes are green or not green.
My eyes are (any eye color that I know exists) or not (that same color).
My eyes are some color I have no knowledge of.

So you can test all those conditions from there.

But your only test is if other people leave. They don't know any more than you do, so no one can leave. Since no one can leave, no one can learn their own color through such testing. Remember, the initial conditions said you can only leave if you know your own eye color. What test could you possibly do other than watching to see if others leave (which they can never do before the guru speaks since they're all also watching for others to leave).

Again, the guru introduces two new subsets of people that allow the problem to be solved. The natives are solving for the group that contains the person she was talking about...that just happens to also be people with blue eyes. That's the key.

 

[Yoda]

But logically.... one of the things you would consider given being a perfect logician would be exactly that!

The Guru can see a blue eyed person (because I can see a blue eyed person). I am a person. My eyes have a color. I do not know what color my eyes are. People have blue eyes. I may or may not have blue eyes. The Guru can see me. The Guru may or may not see that I have blue eyes.

 

[boobatuba]

But logically.... one of the things you would consider given being a perfect logician would be exactly that!

The Guru can see a blue eyed person (because I can see a blue eyed person). I am a person. My eyes have a color. I do not know what color my eyes are. People have blue eyes. I may or may not have blue eyes. The Guru can see me. The Guru may or may not see that I have blue eyes.

Yes, logically you would conclude all those things. But it would be illogical to assume that everyone could decide independently to solve for only one of the four possibilities (blue eyes, brown eyes, green eyes, some other color). As a perfect logician, you logically conclude that there is no mechanism for everyone to "get on the same page," as it were.

That's why the guru's statement is necessary. It articulates the parameters for the solution.

 

[Scout101]

Doesn't give you anything to solve for until the Guru sets the binary condition, though. Awesome that you can currently see blue, green, and brown, but how does that help you determine your eye color? Everyone on the island (minus the guru) can see those 3 colors too, so how do you determine if you were blue, green, or brown?

By reducing it to Blue or Not-Blue, you can solve it. Otherwise, no. And without another statement, the browns have to stay.

 

[Yoda]

The Guru never states that people should only consider whether they're blue or not.

 

[boobatuba]

The Guru never states that people should only consider whether they're blue or not.

She doesn't have to. Her statement on it's own sets the parameters for solving the problem. They can solve for "not blue," but that still doesn't give an individual his own eye color...only that it's "not blue."

 

[Yoda]

I need sleep. Lindley?!? Where are you? I know you're on their team... but am I just crazy? Do you see my point?!? Lindley? Bueller?

 

[boobatuba]

I need sleep. Lindley?!? Where are you? I know you're on their team... but am I just crazy? Do you see my point?!? Lindley? Bueller?

I see your point. You're not "wrong" in any sense of the word. The only thing we really seem to disagree on is the necessity of the guru's statement.

I'm obviously not very good at explaining why I think it's important, but I appreciate the opportunity to try. These sorts of exercises are great for refining your ability to explain ideas with the written word.

 

[neozeks]

I've thought and read about this some more and I think now I do understand the key point (damn this puzzle, I've spent way too much time on it!). Lindley already explained it and I hope I understood it correctly.

And so on, and so on. It isn't actually the case that there are fewer blues----just that one particular blue, A, is following the logic chain WHAT-IF-I'M-NOT-BLUE, and on every level is assuming that every other randomly chosen blue he can see is also following the same branch of the tree. It's assuming about what others are assuming about what others are assuming about what others are assuming to a fairly mind-boggling level, which is why we need to assume they're perfect logicians. A knows this is false for the others, but he also knows that they *don't* know this, so they'll explore that branch of the decision tree.

Each level of the tree assumes logic at work which relies on fewer blues, until we get down to the 1-blue case. That case is a contradiction due to the guru's statement. At that point, each level of the tree triumphantly determines that I'M-NOT-BLUE is false, cascading all the way back up to A.

The key thing is in the "assuming about what others are assuming about what others are assuming about what others are assuming..." chain. You can solve it for blue because the last person in this "assuming chain", even though in reality it obviously is not true, hypothetically doesn't see any blues. The only way he can infer there's someone with blue eyes (and which then has to be himself) is by guru introducing the blue color. The same logic doesn't work for the browns because the last person in their chain (who hypothetically doesn't see any browns) has no idea there's anyone brown (which would be him) because the brown color hasn't been mentioned by the guru.

Now, I still don't fully "get" this, it still doesn't feel natural for me, but I think that's the logic. Hope someone corrects it if it's wrong. And it would be nice if someone actually worked out in full the though processes in the 4 blues case. All the examples always stop at 3 blues and I have a nagging feeling the logic breaks somewhere at that point but I don't really know why.