At 11:54 06/10/04 -0700, George Levy wrote:
To avoid confusion between my Switch belief function and the one you use,
let me rename my three state switch belief function from B to S.
So now, what I had expressed in an earlier post as qBp becomes qSp, where
q is the switch control line, p is the input and qSp is the output.
So your Bp becomes my 1Sp and your ~Bp becomes my 0Sp.
Note that pSp is never 0. (the control line and the input line carry the
same information.) This reminds me of the anthropic principle.
This seems nice and you could perhaps elaborate.
Now, I don't think, indeed, that your S logic
is related to the B logic, where B is really
an a priori higher level notion (like an affirmation
that some machine stops after having done some search).
Your S is closer to the belief function theory than
the Smullyan/Gödel slef-referential "belief".
To test the degree of resemblance of your S and the
B, you could test S on the original knight/knave problem.
Knights always tell the truth, Knaves always lie. A native
of the Island tells you "You will never believe I am a Knight"
What will you (or your S machine) believe about
the native's nature?
And here is another puzzle, which is not entirely
unrelated with both the KK puzzles and the current probability
discussion: I put three cards, two aces and a jack, on their
face in a row. By using only one yes-no question and pointing
one of the card, you must with certainty find one of the aces.
I know where the cards are, and if you point on a ace, I will
answer truthfully (like a knight), but if you point on the jack, I
will answer completely randomly!
How will you proceed? (The puzzle is one invented by Boolos,
as a subpuzzle of a harder one by Smullyan & McCarthy.
Cf Boolos' book "Logic, logic, and logic". According to
Boolos, it illustrates something nice about the practical
importance of the excluded middle principle. And this is
a hint, perhaps.)