Hi All

Perhaps George Levy is right and I should explain better the strategy before
proceeding.

The role of the logical puzzle is to encourage you to some introspection.
The puzzle are easy but at some point we will give the puzzle to the machine
just for studying its "psychology". You will see John Mikes is somehow right
in the sense that a puzzle or a paradox can be "understand" differently
according to the reasoning capacity of the machine.
And we do this because the UDA has shown that if COMP is correct then
physics is (re)definable as a measure put on the consistent extensions.
If you peruse Smullyan's FU you will see he talks almost everywhere on
consistency, and this is indeed the key notion. We will come back soon.
I am aware that the knight knave island could give the appearance that
such clear-cut logic cannot be serious. As I said to George, there will
be a point, where a very precise link between the KK puzzle and
"machine psychology" will be done, but it's premature now.
Enjoy the puzzle, and if you don't like the puzzle (but are still interested
to see how physics is introspectively deducible by an universal machine)
try just to understand the enunciation and the solution provided.

Here are simple puzzles to familiarize you with the KK island, and to recall
or test your comprehension of the classical connective.

I recall that the natives of the KK island are all either knave or knight. Knaves
always lies, and knight always tells the truth.


Problem 3
a) A visitor to the island would like to know systematically the type of the
natives. He goes to the first house and ask a man his type, and the type of
his spouse. He answered: "We are both knaves". What can you deduce?

b) Next house. Still a couple. He asks: "Are you both knaves?". The
husband answers "At least one of us is". What can you deduce?



The next problem is somehow fundamental. It contains some magic which
will recur again and again, and that magic will crystallize in Lob's theorem,
an unexpected and deeply counter-intuitive generalization of Godel's theorem.
It is fundamental for going to G and G*. Lob is a great Deutsch logician.
I think he is still alive, and there is a rumor saying he lives or lived with a leopard.


I recall you that the classical implication X -> Y, is false only when X is true
and Y is false.


Problem 4:
You have an exam today. The question looks difficult, indeed the
exam question is "Does Santa Claus exist?". You are panicking because
you don't remember that passage of the course. But you are a modern student
so you have a mobile phone, and, after having call your friends and family
without success you decide in a desperate move to call the KK island.
You get a native, and asks her quickly (because it is the end of the exams)
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?

Bruno



http://iridia.ulb.ac.be/~marchal/



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