Re: ... cosmology? KNIGHT & KNAVE

[George Levy]

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