At 16:15 23/07/04 +0200, Jan Harms wrote:
> > (problem 4)
> > You get a native, and asks her ........if Santa Claus exists.
> > The native answers this: "If I am a knight then Santa Claus exists"
> > What can you deduce about the native, and about Santa Claus?
>
>Lets give a name to the sentence:
>S="If I am a knight then Santa Claus exists"
>
>1. If the native is a knight, then S is true. If S is true and the native is
>a knight, then Santa Claus exists. Therefore, Santa Claus exists.
>
>2. If the native is a knave, then S is false. If S is false (1->0), then the
>native must be a knight. So he can't be a knave. So a knave could never
>say S and be consistent.
>
>Conclusion: The native is a knight and Santa Claus exists.
After reading ten times I conclude you got the solution. I have been mislead
by the fact that there are mainly two ways to solve the puzzle (and George
did try the two ways), and your 1. looks like the beginning of the first way,
and your 2. looks like the second way. Also the conclusion of your 1. is that
S is true. You still need your 2. to conclude, of course (as you illustrate).
And then I have been mislead because it looks you assume a knave to be
consistent. But if you ask a knave if 1=0 he answers "yes", so he is certainly
not consistent in the usual sense of the word (which we will define when we
will
introduce the "believe" ...). But I eventually realize you were just meaning
by consistent: "consistent with his status of knave".
So ok you were right.
Let me explain again. I give two independent solutions: A and B.
I recall, you are on the KK island, and a native tells you "If I am a knight
then Santa Claus exists".
A. Let us suppose the native is a knight. Then what he tells us is true.
So if we suppose the native is a knight the *two* following proposition are
true:
-the native is a knight (because if the native is a knight the native
is knight, isn'it?)
-S is true, that is: if the native is a knight then Santa Claus exists.
from which we can conclude that Santa Claus exists.
So we have shown that indeed IF the native is a knight then Santa Claus exists.
(Indeed By supposing that the native is a knight we get the existence of
Santa Klaus).
At this point it is important to see that we have prove only that S is true
(not that
Santa Claus exists).
OK. But the native asserted S, and only a knight can assert true proposition.
So the native is a knight. So know we *know* the two propositions are true:
-the native is a knight
-S is true, that is: if the native is a knight then Santa Claus exists.
From which we can conclude that Santa Claus exists.
From which we can conclude, BTW, that either a knight knave island does
not exist or, if it exists, no native will ever said to you "if I am a knight
Santa Claus exists", or (in case the KK island exists and a native tells you
the S sentence) ... that Santa Claus exists.
B. Let us suppose the native is knave. Then what he said was false.
But he said "if I am a knight then Santa Claus exists". That proposition
can only be false in the case he is a knight and Santa Claus does not exists.
So if he is a knave, he must be a knight and that's a contradiction. So he
cannot be
a knave, and so he is a knight.
But then what he said was true, and giving what he said, Santa Claus exists.
OK? That should give you a smell of Lob. At some point it will be necessary to
understand that in computerland the option of saying the KK island does not
exist
just does not work for hunting away the self-referential proposition. Comp
entails
some Lobian magic around! But I anticipate.
I see you want the next problem.
OK. I hope you all agree that a native, not a tourist, a real native from
the KK
island will never tell you "I am a knave", or, it is equivalent "I am not a
knight".
A knight saying that would be lying, a knave saying that would be telling
the truth.
Now with the next two problems we are doing a big step and a very big step.
So big that we will be forced to work again on 1-4 a little bit more formally.
Big Problem 5:
Could a native tell you "You will never know that I am knight" ?
Very Big Problem 6:
Could a native tell you "You will never believe that I am knight" ?
Bruno
http://iridia.ulb.ac.be/~marchal/
Received on Fri Jul 23 2004 - 13:48:07 PDT