Re: ... cosmology? KNIGHT & KNAVE

From: George Levy <glevy.domain.name.hidden>
Date: Thu, 22 Jul 2004 22:17:34 -0700

Hi Bruno


Bruno Marchal wrote:

>
> 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?


First let's assume that the native is a knight. Since he tells the
truth, then Santa Claus must exist. That's all,... we cannot go any further.

Now let's assume that the native is a knave. Then the statement he made
is false. The corresponding true statement is: "If I am a knight then
Santa Claus does not exist." However we assumed that the native is not a
knight. Therefore the statement does not apply. No information can be
obtained from this statement.

We still don't know if the native is a knight or a knave, and we still
do not know if Santa exists or not.


I compared your version to Smullyan's. The word "believe" is missing:
Paraphrasing Smullyan, I quote: "If you believe I am a knight then Santa
Claus exists."
I read Smullyan's solution on page 124 and 125. It sounds like circular
reasoning. For the rest of the list I am retyping the passage with the
appropriate editing changes in deference to Santa.

Beginning of Smullyan's edited quote
We let k be the proposition that the native is a knight and we let C be
the proposition that Santa exists. At the outset, the reasoner believes
(B) the proposition BC -> C.

The reasoner reasons: "Suppose I ever believe that he is a knight. Then
I'll believe what he says - I'll believe that Bk -> C. Also, if I ever
believe he's a knight. I'll believe that I believe he is a knight - I'll
believe Bk. And so, if I ever believe he is a knight, I'll believe both
Bk and Bk -> C., hence I'll believe C. Thus, If I ever believe he is a
knight, then I'll believe that Santa exists. But if I ever believe that
Santa exists, then Santa exists. And so, If I ever believe he's a
knight, then Santa exists. Well that's exactly what he said. He said
that if I ever believe he's a knight, then Santa exists, and he was
right! Hence he is a knight!

At this point the reasoner believes that the native is a knight and
since the reasoner is normal, he continues: "Now I believe he is a
knight. I have already proved that if I believe he is a knight, then
Santa exists, and since I do believe that he is a knight, Santa exists."

At this point the reasoner believes that Santa exists.....
End of Smullyan's quote.

What if the reasoner had started by supposing that he believes that the
native is a knave?

I am confused by Smullyan. I just see it as circular reasoning. He could
have started by assuming a knave and reached a different conclusion.

George
Received on Fri Jul 23 2004 - 04:16:31 PDT