Problem in logic

[smiki]

I'm with Jadzia on this one. But first, I must ask a few questions (just to make sure). Does the Guru tell each day:"There is someone with blue eyes", or just on the first day, and if he says it every day, does he on every new day speak about another person, as in: "There is another one with blue eyes".

If it's just once, or just: "I see someone with blue eyes", I know what is the fault with the "solution"(I hope) and I will try to explain it, and then as I understand it, nobody will ever leave.

But I'll wait for the answer first, in order not to type in vain.

Edited to add (Ahh... what the hell ):
I can't bother with reading 5 pages at the moment, so I hope this is the accepted "solution":

Exactly.

Each of the blue eyed-people knows that there are 99 blue-eyed people on the island (plus themselves but they don't know that).

If there's only one blue-eyed person on the island then he sees nobody with blue-eyes. Which means he can leave on the first night.

But that's not the case.

If there are 2 blue-eyed persons each of them sees 1 blue-eyed person on the island. Since they both still see one they can't be sure of their own colour. So neither of them leaves.

If neither of them leaves they know that they must both have blue eyes because nobody else does. They leave on the 2nd night.

If there are three blue-eyed people each one of them can see 2 blue-eyed people. If those two blue-eyed people haven't left on the 2nd day then the 3rd person knows that he must have blue eyes, too. They all leave on the third day.

Rinse and repeat. After 99 days everybody will know if he or she has blue eyes because nobody has left by then. They will know that in addition to the 99 blue-eyed people each of them sees there must be a 100th person. So they all leave on the 100th day.

I resisted the urge to google it but I think it should be pretty correct.

The problem with the bolded part is the following. It carries the assumtion that even in the three case scenario the two people "look at each other"-thinking still stands ground, but it doesn't 'cause it was based on the fact there were only two of them, and now there aren't. I mean, it's obvious. Now there are three of them on the first night. Each one of them sees two blue eyed people. Now the statement:"There is someone with blue eyes", means nothing they already don't know, and the fact nobody leaves doesn't tell them anything new either. Now they can't conlude: if the other guy hasn't left it's because I have blue eyes as well, so he thought the guru spoke about me, because there are now two guys other than me, and the whole reasoning falls flat on its head.
I believe the riddle is flawed.

 

[Jadzia]

To know that the other parties know other people have blue eyes requires a leap of faith (it's inductive, not deductive). Humans like you or I can do this, but as Kommander jokingly - but correctly - pointed out, these people on the island are not human like you or I; they work solely on the basis of fact - perfect strong logic - and can't make that inductive leap of faith. Therefore it's only when actively provided with the absolute factual knowledge that other people know that other people have blue eyes, that then can solve the problem using their perfect deductive logic.

THAT'S why they need the Guru under the original solution; to provide 100% truthful data which they cannot gather objectively via any other means. Without that fact, the problem ends up being insoluble as they won't induct it for themselves. You or I (being capable of induction) would not require the guru, you're correct... but we could be wrong in our inductive logic

No. The puzzle states they are perfect logicians, which implies perfect memory. Also, they are perfect observers, because we are to assume they do in fact look at everyone else's eyes.

Furthermore, the puzzle suggests indirectly that each person knows that the others are perfect logicians, knows they have perfect memory and knows they make perfect observations. The inductive 'solution' presented in this thread actually relies on the islanders having this mutual knowing.

So N=10, each islander will be able to see someone with blue eyes, and each will know that everyone can see someone with blue eyes. They will know it because it is a simple consequence of knowing that all islanders are perfect observers with perfect memory. No inductive/leap-of-faith/could-be-wrong reasoning is required for that deduction.

I've worked out what is wrong with the inductive step too. But I shall leave that for another post.

 

[Lindley]

The fact that the inductive step requires someone else to behave as if you (the newest case) do *not* have blue eyes beforehand is somewhat worrying. I'm not sure if it's a problem though.

 

[boobatuba]

I don't see why this is so hard for some to grasp. Of course everyone on the island knows that some people have blue eyes. That's not the additional information that the guru imparts. What they find out is that one specific person the guru sees has blue eyes...and it might be them.

Getting back to the 2 blue-eyed, 2 brown-eyed, and the guru scenario...if you only see one blue-eyed person and they don't leave on the first night, you can logically deduce that the guru was talking about you and that you, therefore, have blue eyes. Since the person you see with blue eyes DIDN'T leave, they must also see someone with blue eyes (because if they didn't they would naturally KNOW that the guru must have been talking about them).

The scenario is no different if N people have blue eyes except that it will take N days to determine for everyone if they have blue eyes or not.

In the 100-100 scenario, assume you have blue eyes. You see 99 people with blue eyes and 100 with brown eyes. You have to assume that either you have blue eyes or something else. Either 99 people do or 100 do. Everyone else is able to deduce this as well. Since 99 people DON'T leave on the 99th day, there must be 100 people with blue eyes and the 100th must be you (since you can see it's not any of the people you see with brown eyes).

Again, without the guru's statement, you would have had no way to deduce your own eye color on the 99th day when no one leaves. You didn't have a guru talking about someone and couldn't deduce that the someone was you.

The guru introduces a binary situation that didn't exist before. That is, either she saw me with blue eyes or saw someone else with blue eyes. This introduces a series of possibilities that can be charted and solved by observing when/if the blue-eyed group leaves.

 

[Lindley]

I don't see why this is so hard for some to grasp. Of course everyone on the island knows that some people have blue eyes. That's not the additional information that the guru imparts. What they find out is that one specific person the guru sees has blue eyes...and it might be them.

That's a more confusing way of saying it, IMO, but essentially it's the same as pointing out the guru is necessary for the induction base case. Of course, you have to understand inductive reasoning (which has nothing to do with a "leap of faith", contrary to what Holdfast said earlier) for that to be the simpler explanation...

 

[Scout101]

Yeah, I don't really see the problem. Boobatuba explained it simply enough. And the people with brown eyes never have to worry about it, because their count is always one higher than someone with blue eyes would have, so everyone gets up and leaves before the brown eyed people have to make a decision...

 

[Lindley]

No, the brown eyed people will never leave simply because the guru didn't say anything about brown eyes. Therefore, the entire induction sequence can't begin, and the brown-eyed people can make no determinations about their own eyes.

Going back to the explanation:

If there's only one blue-eyed person on the island then he sees nobody with blue-eyes. Which means he can leave on the first night.

This is only true because of what the guru said about blue eyes. Without that part of the puzzle, the brown-eyed people are screwed.

 

[Scout101]

Think your logic is flawed. The brown people don't know they are brown and shouldn't participate. They have to play the same game, but their count is always 1 higher than a blue would have, so they never have to make a decision to stay or go. If there are 2 of each, the brown can see both blues, so doesn't have to leave unless both blues fuck up their logic and stay, in which case the problem falls apart.

They don't know their own color either, so they ARE in the sequence, just one number higher than the blues.

 

[Lindley]

They don't have a base case. There is no sequence for browns, because there's no reason why, if there were only a single brown, he would know that he could leave on the first night.

 

[Scout101]

Read what I wrote. Why wouldn't they be trying to figure out if they have blue eyes or not, like everyone else is trying to do? They won't ever LEAVE, but they have to play the game of figuring it out.

For a blue, if they see 2 blues, and they don't leave, he must be #3, and the all 3 leave. For a brown, he saw 3 the whole time, so doesn't get to make a decision unless those 3 all stayed (through poor logic).

The browns have to have a sequence, because they don't know they have brown eyes and shouldn't be counting. For all they know, they are another blue...

 

[Lindley]

Oh, I see what you're saying now. Yes, that makes sense----they're always 1 step behind, and then they give up once the blues all leave since they know that means they're not blue.

My point was that, despite being able to leave after figuring out their own color *regardless* of whether it's blue or not, blue is the only color anyone will (possibly) ever be able to identify their own eyes as due to the guru's statement.

 

[boobatuba]

Think your logic is flawed. The brown people don't know they are brown and shouldn't participate. They have to play the same game, but their count is always 1 higher than a blue would have, so they never have to make a decision to stay or go. If there are 2 of each, the brown can see both blues, so doesn't have to leave unless both blues fuck up their logic and stay, in which case the problem falls apart.

They don't know their own color either, so they ARE in the sequence, just one number higher than the blues.

This is well said.

Every blue-eyed person on the island can see 99 people with blue eyes. After the guru's statement, they know for a fact that either the guru was talking about them and there are 100 people on the island with blue eyes or there are 99 people on the island with blue eyes and she was talking about someone else. They find out on the 99th day when no one leaves that there are 100.

Every brown-eyed person on the island can see 100 people with blue eyes. After the guru's statement, they know for a fact that either the guru was talking about them and there are 101 people on the island with blue eyes or there are 100 people on the island with blue eyes and she was talking about someone else. They find out on the 100th day when everyone with blue eyes leaves that they don't have blue eyes.

 

[Holdfast]

Of course, you have to understand inductive reasoning (which has nothing to do with a "leap of faith", contrary to what Holdfast said earlier) for that to be the simpler explanation...

Actually, I think we're using different terminology, to describe different things.

Philosophical induction is different to mathematical induction.

To be really specific about this: the islanders cannot do philosophical induction, but can do mathematical induction.

Mathematical induction is actually, philosophically speaking, deductive in nature. Despite its name, it is NOT inductive (to a philosopher).

Philosophical induction requires an intuitive step when reasoning from the general to the specific, mathematical induction does not. The islanders cannot do philosophical inductive reasoning, so cannot project their knowledge onto others without the guru absolutely confirming the starting point. Once the guru does confirm it, they can do mathematical induction (what I would call deduction) to solve the problem.

Humans can do both, so can solve the problem without the guru. The islanders can only do mathematical induction, so require the guru to define the starting point.

Apologies if we were talking in cross purposes; hope this clarifies it to your satisfaction.

Jadzia, I'll address your comment on my post separately in just a sec.

 

[Lindley]

Can you imagine logarithms, wave functions, infinity?

Logarithms actually make it much easier to deal with extremely small or extremely large numbers. I'm not sure what I'm supposed to be "imagining", but they don't cause me any confusion.

Wave functions....are you referring to periodic functions like sin, or to electron orbitals and the like? The latter are certainly much less intuitive.

Infinity is difficult to imagine as a reality, but very easy as a mathematical construct.

 

[boobatuba]

Humans can do both, so can solve the problem without the guru. The islanders can only do mathematical induction, so require the guru to define the starting point.

No, they can't solve the problem without the guru's statement.

Starting on the first day when the rules are set, they can still see either 99 blues and 100 browns or 100 blues and 99 browns. But they can't make any inference about their own eye color at all. On the 100th day, the possibilities still exist that there are 99/101, 100/100, or 99/100/something else.

The guru's statement introduces a new element. She was either talking about me or she wasn't. From that, a new set of probability trees can be constructed and then solved. The parameters go from "blue, brown, and possibly something else" to "blue and not blue." On the 100th day, the possibilities are only 99/101 or 100/100, and the fact that 99 blue-eyed people didn't leave the day before solves that.

And the reason the brown-eyed people can't leave that same day is that all they know for sure about their own eye color is that it's "not blue."

 

[Holdfast]

Humans can do both, so can solve the problem without the guru. The islanders can only do mathematical induction, so require the guru to define the starting point.

No, they can't solve the problem without the guru's statement.

Starting on the first day when the rules are set, they can still see either 99 blues and 100 browns or 100 blues and 99 browns. But they can't make any inference about their own eye color at all. On the 100th day, the possibilities still exist that there are 99/101, 100/100, or 99/100/something else.

Humans CAN make that inference.

But it's not a deductive one; it's intuitive (philosophically inductive).

They can say that it's "most likely" that I either have blue eyes or brown, because the 199 other people on this island fall into one of those two camps. Moving from the general (the other 199) to the specific (me), on the basis of intuition because of how common the two other sets are.

That's why the question goes out of its way to point out that the islanders can only use deductive logic. If you only use rigorous deductive logic (including mathematical induction), you NEED the guru to define the starting point in the way you've explained.

Humans can reach the correct starting point by philosophical induction, by reasoning intuitively based on how common the two groups of blue and brown are. They cannot be certain they are right, because they are using philosophical induction, but they can still get to the right answer through this method. They cannot prove it rigorously, but they can still correctly state their eye colour to the ferryman by making the assumption that only blue and brown eye colours exist. In fact, human beings would also get the second batch of 100 people off the island on day 101, using the same principle.

Again proving why humans are best off not relying on deductive logic only in solving problems. They can use philosophical induction: the "educated guess".

The original islanders in the question don't have that option, of course, so need the guru in the way you've described.

 

[Lindley]

Intuitive and inductive are not the same thing.

 

[boobatuba]

Humans can do both, so can solve the problem without the guru. The islanders can only do mathematical induction, so require the guru to define the starting point.

No, they can't solve the problem without the guru's statement.

Starting on the first day when the rules are set, they can still see either 99 blues and 100 browns or 100 blues and 99 browns. But they can't make any inference about their own eye color at all. On the 100th day, the possibilities still exist that there are 99/101, 100/100, or 99/100/something else.

Humans CAN make that inference.

But it's not a deductive one; it's intuitive (philosophically inductive).

They can say that it's "most likely" that I either have blue eyes or brown, because the 199 other people on this island fall into one of those two camps. Moving from the general (the other 199) to the specific (me), on the basis of intuition because of how common the two other sets are.

That's why the question goes out of its way to point out that the islanders can only use deductive logic. If you only use rigorous deductive logic (including mathematical induction), you NEED the guru to define the starting point in the way you've explained.

Humans can reach the correct starting point by philosophical induction, by reasoning intuitively based on how common the two groups of blue and brown are. They cannot be certain they are right, because they are using philosophical induction, but they can still get to the right answer through this method. They cannot prove it rigorously, but they can still correctly state their eye colour to the ferryman by making the assumption that only blue and brown eye colours exist. In fact, human beings would also get the second batch of 100 people off the island on day 101, using the same principle.

Again proving why humans are best off not relying on deductive logic only in solving problems.

The original islanders in the question don't have that option, of course, so need the guru in the way you've described.

No. "Most likely" is not the same thing as "provably true." There's always the possibility that you are the exception and have red eyes or some other color. They can never, without a doubt, correctly state their own eye color to the ferryman by making an assumption that it's either blue or brown.

What you are saying is that humans can guess based on the probability that they have one of the same eye colors as 199 of their fellow island dwellers. I don't disagree. But that's hardly the point of the riddle.

 

[Holdfast]

What you are saying is that humans can guess based on the probability that they have one of the same eye colors as 199 of their fellow island dwellers. But that's hardly the point of the riddle.

Exactly, I agree.

But your post in response to mine earlier was about why I thought humans could solve the problem without the guru whereas the islanders need the guru. Humans are not limited by the constraints of the riddle. Your reply to me was that humans would also need the guru. That's what I was explaining; they don't because they aren't bound by the riddle in the way the islanders are. They can use philosophical induction.

Lindley: David Hume does indeed accept that intuition is part of the inductive reasoning process in his Enquiry Concerning Human Understanding. He very specifically argues that philosophical induction conflates these two statements into one:

"I have found that such an object has always been attended with such an effect"
"I foresee that other objects which are in appearance similar will be attended with similar effects".

There is no deductive reason why one follows the other automatically. Again, this is different to the deductive processes used in mathematical induction.

(in fact, Hume goes on to point out that this is why induction is essentially flawed. But as it works more often than not, we should still use it.)

 

[neozeks]

I'm rather tired and I may be just having a dumb moment, but something doesn't feel quite right to me either...

I don't see why this is so hard for some to grasp. Of course everyone on the island knows that some people have blue eyes. That's not the additional information that the guru imparts. What they find out is that one specific person the guru sees has blue eyes...and it might be them.

But don't they know that even before the guru speaks? They see everything the guru sees, except themselves. They know that the guru sees 100 brown-eyed, 99 blue-eyed and them. They know that they themselves either do have or don't have blue eyes. If they themselves don't have blue eyes, the guru sees 99 blue eyes. If they themselves do have blue eyes, the guru sees 100 blue-eyes. So even before the guru speaks, they know he's seeing a blue-eyed person, and it might or might not be them. They know what the guru will say. They know that every other person knows what the guru will say. In fact, they also know he's seeing a brown-eyed person as well, and that also might or might not be them. Why does the guru even need to say anything?

I was speaking from a blue-eyed perspective there. It's the same from a brown-eyed perspective, just with slightly different numbers. The brown-eyes also know he's seeing blue-eyed people (and brown-eyed people) and they might be one of them even before he speaks.

The only thing they don't know is whether the guru also also sees a third colour - which would have to be them. If the guru said "I see someone with red eyes" they would immediately know it was them. But if he doesn't mention a third colour, they're in the same position as before he said anything.

Since 99 people DON'T leave on the 99th day, there must be 100 people with blue eyes and the 100th must be you (since you can see it's not any of the people you see with brown eyes).

I understand the basic logic behind it. But there's just some kind of a nagging feeling that this "scaling" from 2 people to 100 people doesn't work quite as it should... I could be wrong, of course...