Problem in logic

[Holdfast]

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?

OK, try looking at it this way: before the guru speaks, there is no shared frame of reference with regard to eye colour at all. Your eyes could be any colour under the sun, right?

And everyone else has to think that about their eyes too. They don't know what everyone else knows, so can't determine that some people are blue-eyed, some are brown-eyed, and that there are no other colours. You can't know that you might have blue eyes.

If that's the case, how can you possibly determine your eyes are one colour or another if you don't communicate with each other? The non-cheating solution (ie not breaking the constraints of the riddle by using human abilities like philosophical induction; nor using lethal methods to narrow the set very rapidly on day one after the guru speaks) is impossible to implement on day 100 (or any day). There is no common frame of reference to compare the outcome against.

It's only when the guru mentions a specific colour, that it becomes possible to define two groups into existence: blue and not-blue. Not blue could still include any colour under the sun, so they can never leave. Blue is a now known and defined colour that could include you, so given enough time, determining whether you have blue eyes or not is solvable (if everyone now uses deductive logic from this point outwards).

I was coming back into the thread to reply to Jadzia, explaining the same thing, but you've highlighted the same issue (more or less) she has so...

... on that note, I still feel I haven't explained my thinking clearly, but I'm going to bed.

 

[boobatuba]

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. 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 they might be one of them even before he speaks.

Because before the guru speaks, the parameters for solving the riddle are blue eyes, brown eyes, green eyes, and possibly another color. After she speaks, the parameters are blue eyes and "not blue eyes." The guru makes the decision for everyone what color to "solve for."

ETA: Yeah, what Holdfast said.

 

[neozeks]

I'm not completely sure you guys are using the same argument... argh, the headache.

Because before the guru speaks, the parameters for solving the riddle are blue eyes, brown eyes, green eyes, and possibly another color. After she speaks, the parameters are blue eyes and "not blue eyes." The guru makes the decision for everyone what color to "solve for."

Isn't "brown eyes, green eyes and posibly another color" essentially the same as "not blue eyes"? Why do they need the guru to "group" it for them? Can't they do it for themselves?
Why do they follow the guru's "decision"? They also know that she's seeing a brown-eyed person and that person might be them, even if the guru doesn't mention it. Why don't they use brown and "not brown"?

And everyone else has to think that about their eyes too. They don't know what everyone else knows, so can't determine that some people are blue-eyed, some are brown-eyed, and that there are no other colours. You can't know that you might have blue eyes.

But why? You see every other person. You see that they have brown, blue or green eyes. You know you have brown, blue, green or some other color. You know you have blue or "not blue". You know every other person also sees every other person on the island. You know they see every color you see plus your own and minus their own (assuming you and they have a color that no one else has). The guru speaking changes none of those options.

 

[boobatuba]

Isn't "brown eyes, green eyes and posibly another color" essentially the same as "not blue eyes"? Why do they need the guru to "group" it for them? Can't they do it for themselves?

No, they cant. Reduce the problem to only four people on the island...two with blue eyes and two with brown eyes. Show me logically how they can solve this problem themselves without the guru's statement.

Why do they follow the guru's "decision"? They also know that she's seeing a brown-eyed person and that person might be them, even if the guru doesn't mention it. Why don't they use brown and "not brown"?

Because she says she sees someone with blue eyes. No one knows who she's speaking about. That's the information they use to generate new probability trees and solve the problem. The fact that the guru might be talking about them is the key to the riddle that each person on the island uses to solve the riddle.

ETA: This chart should help:

Using logic, everyone on the island will now know for a fact that there are only two possibilities for the number of people with blue eyes. People who do have blue eyes (and make the chart on the right) will know that since blue-eyed people either see 98 or 99 other people with blue eyes, there must be either 99 or 100 people total with blue eyes. People who don't have blue eyes (and make the chart on the left) will know that since blue-eyed people either see 99 or 100 other people with blue eyes, there must be either 100 or 101 people total with blue eyes.

Of course, 100 people with blue eyes is the correct total, but no one can know that for sure until 99 people don't leave on the 99th day. This confirms to all the blue-eyed people that all of them do indeed see 99 other people with blue eyes. They all leave the next day. That act of leaving confirms for all the others that there were 100 blue-eyed people and not 101.

I could show a similar chart that would be made by everyone before the guru's statement, but it's got many more branches since other colors must be allowed for.

 

[Ha'kiv]

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)

You cannot possibly know who the guru was talking about if there is more than one blue-eyed person, it makes no sense to say that.

 

[Ha'kiv]

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.

Of course, all of these things make it much easier to deal with certain problems. I suppose I'm not doing a very good job explaining myself. Wave-particle duality, quantum tunneling, electron clouds - we have no problem explaining these concepts, using them, and so on. But I assure you, no sane man can honestly imagine this. Concepts like infinity and nothing - sure, no problem. Something with no end and the lack of something. But imagining this? Out of the question.

Also, you needn't reply unless you are honestly interested in debating on this subject. I think my head will explode if I see the word "guru" and "different way of thinking" again.

 

[boobatuba]

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)

You cannot possibly know who the guru was talking about if there is more than one blue-eyed person, it makes no sense to say that.

Yeah, I stated that incorrectly. It should read "if you only see one blue-eyed person and they don't leave on the first night, you can logically deduce that you also have blue eyes since the other two people you can both see have brown eyes and if you didn't have blue eyes the person you can see with blue eyes would have seen no one else with blue eyes. He could therefore deduce on the first day that the guru was talking about him and would have left the first night."

If there's two blue-eyed people, the guru obviously could have seen either one of them.

This is why the guru is important. Without the guru's statement, there would be no reason for you to assume the person you can see with blue eyes would have left the first night if he sees no one with blue eyes. That person would have no reason to without relying on the guru's information.

So, was the chart helpful?

 

[sojourner]

Of course, all of these things make it much easier to deal with certain problems. I suppose I'm not doing a very good job explaining myself. Wave-particle duality, quantum tunneling, electron clouds - we have no problem explaining these concepts, using them, and so on. But I assure you, no sane man can honestly imagine this. Concepts like infinity and nothing - sure, no problem. Something with no end and the lack of something. But imagining this? Out of the question.

^Someone had to imagine it before they could start to work on how to explain it. Where do you think new ideas come from?

Edit to add: with a clear explanation, of which there have been several in this thread, the answer makes sense to me.

 

[Ha'kiv]

Er.. No, actually, I'm afraid that's not how science always works. To quote a great man, "If you think you understand quantum mechanics, you don't understand quantum mechanics".

Believe me, no one actually went "Hey, wait a minute. What if in some cases, lets say, hm, on a very small scale, the cause and effect concept which has been nailed into my brain throughout the thousands of years and makes it quite impossible for me to make sense of anything which is not based on that concept - isn't actually true? Yes! I will now go and write an equation for this wonderful, clear idea!"

 

[boobatuba]

Er.. No, actually, I'm afraid that's not how science always works.

Actually, it is. It's called a hypothesis. Someone imagining how something works. Then they go about the process trying to prove it scientifically.

So, was the chart helpful?

 

[Ha'kiv]

Er.. No, actually, I'm afraid that's not how science always works.

Actually, it is. It's called a hypothesis. Someone imagining how something works. Then they go about the process trying to prove it scientifically.

So, was the chart helpful?

It is all always based on some kind of observation. In quantum mechanics, even more so, because the results, to us, make absolutely no sense. Or would you care to enlighten my foolish self along with other, should I say, less foolish scientific minds?

Chart?

 

[Ha'kiv]

This is why the guru is important. Without the guru's statement, there would be no reason for you to assume the person you can see with blue eyes would have left the first night if he sees no one with blue eyes. That person would have no reason to without relying on the guru's information.

But if that person sees more than one other person with blue eyes? When you are aware of the fact that each person can see many people with blue eyes?

 

[sojourner]

Believe me, no one actually went "Hey, wait a minute. What if in some cases, lets say, hm, on a very small scale, the cause and effect concept which has been nailed into my brain throughout the thousands of years and makes it quite impossible for me to make sense of anything which is not based on that concept - isn't actually true? Yes! I will now go and write an equation for this wonderful, clear idea!"

Um, yeah, actually that is how it happens much of the time.

There is this thing called "ideas". People have them all the time. A lot of ideas turn out bad. Some turn out good and worth researching/developing, thus knowledge grows.

By your logic there is no such thing as original thought and we should all still be living in caves.

To turn your example on it's head: "hmmm, I'm hungry, should I go out and kill something? or go to a restaurant?". The instinct to hunt and gather has been nailed into our brains for millions of years, so why did we develop all of this technology for food production?

Oh, and the biggest problem with your example: the question begins with "what if", which is the root question mankind has always asked. Hell, without "what if" you don't have any of the imagined worlds of science fiction. And all science fact started out as science fiction.

 

[boobatuba]

It is all always based on some kind of observation. In quantum mechanics, even more so, because the results, to us, make absolutely no sense. Or would you care to enlighten my foolish self along with other, should I say, less foolish scientific minds?

Certainly. Atomic theory certainly wasn't based on observation first. It began in ancient Greece as a philosophical concept. John Dalton was able to estimate atomic weights in the very early 19th century even though he couldn't observe individual atoms. There were many other ideas about the nature of the atom before it was actually "observed" with the scanning tunneling microscope.

Chart?

Here ya go. I'm not sure how you could have missed it.

But if that person sees more than one other person with blue eyes? When you are aware of the fact that each person can see many people with blue eyes?

Read my explanation under the chart. It's on page 7 of the thread.

 

[Ha'kiv]

Believe me, no one actually went "Hey, wait a minute. What if in some cases, lets say, hm, on a very small scale, the cause and effect concept which has been nailed into my brain throughout the thousands of years and makes it quite impossible for me to make sense of anything which is not based on that concept - isn't actually true? Yes! I will now go and write an equation for this wonderful, clear idea!"

Um, yeah, actually that is how it happens much of the time.

There is this thing called "ideas". People have them all the time. A lot of ideas turn out bad. Some turn out good and worth researching/developing, thus knowledge grows.

By your logic there is no such thing as original thought and we should all still be living in caves.

*sigh* Do you understand how different "ideas" can be? Do you understand the difference between "Perhaps this animal has more fur because her ancestors lived in cold areas" and "Perhaps a solid particle, on a very small scale, simply goes through a barrier - something that all of our knowledge, classical laws, and common sense opposes."

Unless you are a complete lunatic, completely out of touch with reality, with a lot of good luck - you're not going to simply come up with an idea like that (with it actually being correct). You will observe, observe, observe, observe, observe, and after removing all other possibilities, you will conclude that it apparently must be so, however impossible and strange it may seem to us. -This- is not an idea which pops up one day during lunch. It is not something you can imagine, because to our mind, it is impossible, and you -cannot- imagine something which to us seems impossible without breaking any laws of physics.

 

[Ha'kiv]

Certainly. Atomic theory certainly wasn't based on observation first. It began in ancient Greece as a philosophical concept. John Dalton was able to estimate atomic weights in the very early 19th century even though he couldn't observe individual atoms. There were many other ideas about the nature of the atom before it was actually "observed" with the scanning tunneling microscope.

I'm afraid I'm a complete empiricist when it comes to philosophy. Everything is based on observation or interaction with the environment, as was the so called philosophical concept of the atom.


As for the chart, I don't have problems with the riddle, I have problems with how the guru is giving new information. I am fairly certain that I'm just not getting it, so don't get this the wrong way. Honestly, I see no new information. More like an idea, a common ground.

 

[sojourner]

and you -cannot- imagine something which to us seems impossible without breaking any laws of physics.

Did you really post this response on a forum where people debate the imaginary mechanics of warp drive????

If you want to say that you cannot imagine impossible things and then do research to see if they may be true, Go ahead. But don't include me or the rest of the human race in your limited view.

And on that note, good night.

 

[boobatuba]

*sigh* Do you understand how different "ideas" can be? Do you understand the difference between "Perhaps this animal has more fur because her ancestors lived in cold areas" and "Perhaps a solid particle, on a very small scale, simply goes through a barrier - something that all of our knowledge, classical laws, and common sense opposes."

Unless you are a complete lunatic, completely out of touch with reality, with a lot of good luck - you're not going to simply come up with an idea like that (with it actually being correct). You will observe, observe, observe, observe, observe, and after removing all other possibilities, you will conclude that it apparently must be so, however impossible and strange it may seem to us. -This- is not an idea which pops up one day during lunch. It is not something you can imagine, because to our mind, it is impossible, and you -cannot- imagine something which to us seems impossible without breaking any laws of physics.

You'll have to look up Democritus for an example of that. Somehow he was able to "come up" with the idea of atoms during lunch. He certainly couldn't have observed anything to lead him to that hypothesis with the instruments available in 400 BC.

As for the chart, I don't have problems with the riddle, I have problems with how the guru is giving new information. I am fairly certain that I'm just not getting it, so don't get this the wrong way. Honestly, I see no new information. More like an idea, a common ground.

The new information is that someone specific the guru was looking at has blue eyes. That gives everyone new probability charts which lead to a solution (see chart). They can logically eliminate or include themselves among those who have blue eyes based on their observation of everyone else's behavior.

Yes, they could have done this sooner if everyone had known to "solve for blue." They could have done it for brown as well, or even green. But until the guru spoke, there was no way for everyone to know what to do.

 

[Ha'kiv]

and you -cannot- imagine something which to us seems impossible without breaking any laws of physics.

Did you really post this response on a forum where people debate the imaginary mechanics of warp drive????

Good point. However we're not discussing science fiction at the moment, and I assume you people are capable of being realistic.

If you want to say that you cannot imagine impossible things and then do research to see if they may be true, Go ahead. But don't include me or the rest of the human race in your limited view.

You missed my point completely. I see no reason for explaining it again to you.

 

[Ha'kiv]

You'll have to look up Democritus for an example of that. Somehow he was able to "come up" with the idea of atoms during lunch. He certainly couldn't have observed anything to lead him to that hypothesis with the instruments available in 400 BC.

A piece of bread, breadcrumbs, smaller breadcrumbs. Such cases throughout life. The environment provided him with the logic necessary to come up with such an idea. The idea of a building block of all matter is a reasonable idea. The "idea" that particles can pass through things they, so to say, could not possibly pass through, is not a reasonable idea.

Yes, they could have done this sooner if everyone had known to "solve for blue." They could have done it for brown as well, or even green. But until the guru spoke, there was no way for everyone to know what to do.

Exactly. That's what the guru is doing - telling them to solve for blue. That's the only "new information" I see.