Discussion in 'General' started by pianolady, Dec 14, 2007.
A metronome ...... :roll:
Ohhhh, jeez! Good one! :lol: :lol: :lol:
Let us conclude this problem. We are given three people A.B. and C, one is a knight, one a knave and one a normal. You wish to find out whether there is gold on the island, and you are allowed only two YES/NO questions.Well, the first step is to spot one of the three who is definitely not normal. This can be done in just one question: One way is to use the Nelson Goodman principle and ask A whether he could claim that B is normal. Suppose he answers Yes. Then if he is not normal , B really is normal, regardless of whether A is a knight or a knave. On the other hand , it could be that A is normal, and so we now know that either A or B is normal, and so C is definitely not normal. Thus if A answers YES, then C is definitely not normal. A similar analysis shows that if A answers NO then it must be that B is not normal. Having now spotted a non-normal, you turn to him and ask whether he could claim that there is gold on the island. If he answers YES, then there is gold, regardless of whether he is a knight or a knave, and if he answers NO, then there isn't.That does it!
We spotted a non-normal using the Nelson Goodman principle. There is another way of doing this which is quite elegant! It uses the notion of rank. We recall that Knights are ranked the highest, normals, the middle, and knaves the lowest. You ask A whether B is of higher rank than C.
1--If A answers YES, then who is definitely not normal?
2--If A answers NO, then who is definitely not normal?
1 - If A answers 'yes', then C is definitely not normal.
2 - If A answers 'no', then B is definitely not normal.
Very good Monica, but how did you reason it?
Ok – I’m very embarrassed, but I didn’t reason it at all. I simply looked at the words in your last post. The answers were right there, so I just applied them to these two little questions.
Sorry – I guess that’s cheating. (but it sure saved me a lot of time! :wink: )
What ever made you think that the answers were the same? I myself was surprised that they were!
Oh, boy – I guess now you know how twisted my brain is. I’ll try to explain the reasoning behind my reasoning. The bold text is the part that gave me the answer.
Up above, you said “ask A whether he could claim that B is normal. Suppose he answers Yes. Then if he is not normal, , B really is normal, regardless of whether A is a knight or a knave.”
Ok – so B is normal. Then you go on and say, “On the other hand , it could be that A is normal, and so we now know that either A or B is normal, and so C is definitely not normal. Thus if A answers YES, then C is definitely not normal.
Question 2 - - A similar analysis shows that if A answers NO then it must be that B is not normal.
The ranking of the three types doesn’t matter. I just thought of it as asking the same question “Are you the type who can claim that B is normal?” or in other words, “can you claim that B has a middle rank?”
I hope that makes sense. I may have to come back later and fix all this, because I'm confusing myself, again.
Look, I want to straighten something out once and for all! The question to A: "Could you claim that B is normal? " and the question : Is B of higher rank than C?" though they both lead to the same result, the reasons that they do are completely different! I already explained the reasoning behind the first ( the Nelson Goodman principle) and now I want to explain the second. Suppose A answers YES. He is affirming that B is of higher rank than C.We are to show that if A is not normal then B is( and hence that whether A is normal or not, B cannot be normal). So suppose that A is not normal. Then he is either a knight or a knave. If he is a knight, then B really is of higher rank than C, as A said, which would mean that B must be the normal and C the knave. Now suppose that A is a knave . Then contrary to his false statement, B is not of higher rank than C--it is that C is of higher rank than B, hence C is the knight and B is the normal. Thus whether A is a knight or a knave, in either case, B is normal And so we have proved that If A answers YES to the question, then either A or B is normal, and hence C is not. A similar argument shows that If A answers NO then it is B who is definitely not normal. I hope I have made this clear.
All of that is clear. I do get it - I just can't explain it well. For me, explaining the reasoning is harder than figuring out the solution.
Next we shall visit a very remarkable island called the ISLAND OF
THE SANE AND MAD. In this island, all the inhabitants are
completely truthful-they always state what they really believe, but the
trouble is that half of them are totally mad and all their beliefs are
wrong. All true propositions they believe to be false, and all false
propositions they believe to be true. The sane ones are completely
accurate in their beliefs--everything they believe is correct.
You go to the island to find out whether there is gold there.
1--Suppose a native tells you: " There is gold on this island". Do you
have any way of knowing whether there really is any gold there?
2--Suppose that instead a native tells you: " I believe there is gold on
this island". Do you now have any way of knowing whether there is
The answer to these questions illustrates a very basic principle which
underlies many problems that arise with this strange island!
No - I don't think there is a way to know for sure. What a person believes is what he/she thinks is true.
You have to first find out if the person is sane or mad. If you are allowed two yes/no questions, then my first question would something like, "Is this grass green?" I'd be pointing to some green grass as I ask the question, so if he says yes, then he is sane and then I could ask the question, "Is there gold on the island. But if he answers 'no' to the grass question, then he is mad and I'd still ask him if there was gold on the island, but I would know that however he answers - the truth would be the opposite.
Well,I'll tell you this: On this island of the sane and mad, If an inhabitant says there is gold on the island, there is no way of knowing if there really is, but if he says that he believes there is gold on the island, then one can determine if there is. Can you see why?
Won't what I said before, work?
You first have to determine whether he is sane or mad, and one way to do that is by asking a 'test' question.
Of course if you first determine whether he is sane or mad, that would work, but i am saying that if he says that he believes there is gold n the island, then you can immediately know if there is gold there, without asking any questions! You won't know whether he is sane or mad, but you will know whether or not there is gold! How?
hmmmm. You've stumped me on this one. Maybe after I eat some cold pizza it'll come to me, but I doubt it. Care to offer up a little hint?
no hint - ok - I must be missing something easy with this. Here is another attempt:
If the man who says “I believe there is gold on the island” is sane, then there is gold on the island. If a mad man says this, then since everything he believes is false, I break up the phrase like this: “I believe”, which then really means, “ I don’t believe”. And “there is gold on the island”, really means “there is no gold on the island. All together – “I don’t believe there is no gold on the island” so that means there is gold.
no - that doesn't sound quite right, either. I give up, Raymond. (and I can't stand suspense! :wink: )
Monica, you essentially have it right, but it could be better explained thus: Whatever a mad one says is false, and so if he says that he believes something, it is false that he believes it (he doesn't really believe it, he only believes that he believes it) and so that something must really be true, because if it were false, then he WOULD believe it, which he doesn't! In particular, if he says that he believes there is gold on the island, then there really is gold there. And of course if a sane one says that he believes there is gold there, then there really is. And so let us remember a basic fact about this weird island:
IF AN INHABITANT --MAD OR SANE--SAYS THAT HE BELIEVES SOMETHING, THEN THAT SOMETHING IS REALLY TRUE!
To see if you really understand how the island of the mad and sane works, try the following problem:
An inhabitant of the island once claimed there was gold on the island. When I met him, I asked him whether he believed there was gold on the island, He replied:" Yes"
Was he mad or sane? Why?
If he's sane then it's true, because he's either seen it or knows someone who's seen it, or w/e.
If he's insane, then he believes that it's true and in his eyes it's really there.
Separate names with a comma.