At 22:17 22/07/04 -0700, George Levy 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?
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.
Does everybody agree with George? Well, if everybody agree then ... everybody should think twice!
(It *is* a little bit tricky, but that trickiness is really what we need
to understand Godel, Lob, and then, as you can
suspect, the derivation of physics from logic/arithmetic through Godel, Lob, ...).
For the fun, I let everybody think twice!
(Actually you really need to think "twice". This is a hint.
Another hint: the reasoning toward the solution will look a little bit
circular, but only "look"; appearance can be deceiving.
Things will be more tricky when we will introduce the modal "know" or "believe",
but that's premature. We are still in pure PC. (PC = Propositional Calculus).
George, thanks for your attempt. There is no shame to be wrong of course, on the contrary it is the *only* way to learn. Note that some professional mathematician have criticize my thesis by doing similar error!!! (most acknowledge at time, but not all!!!). It is my revelation of the last ten years, logic is not well known even by "scientist". ... and the problem 4 is somehow tricky, I let you enjoy thinking twice. If nobody solves the problem 4, I will give the solution tomorrow, unless someone asks for having more time ...