Le 14-juin-05, ā 00:35, George Levy a écrit :
Bruno Marchal wrote:
~Bf -> ~B(~Bf),
which is equivalent to B(Bf -> f) -> Bf,
Just a little aside a la Descartes + Godel: (assume that "think" and
"believe" are synonymous and that f = "you are")
All right. Of course this follows that for any p in the language of the
machine, we have indeed that the machine can prove
B(Bp -> p) -> Bp
That is: the machine does prove its Lob's theorem. (in my post to
Brent f was the constant "FALSE").
B(Bf -> f) -> Bf can be rendered as:
If you believe that "if you think that you are therefore you are",
then you think you are.
Nice! This makes a relation between Lob's theorem (which generalizes
Godel's second incompleteness theorem) and Descartes systematic
doubting procedure. The link exists already with Godel's theorem. If
you look at the "arithmetical placebo phenomenon" (in my SANE paper),
you are relating Descartes and the Placebo. Quite cute!
That's what Descartes thought!
I agree essentially. See Slezak for a pionering and readable paper
relating Godel and Descartes:
SLEZAK P., 1983, Descartes 's Diagonal Deduction, Brit. J. Phil.
Sci. 34, pp. 13-36.
And this is related also with the debate on Godel and Mechanism
(against Penrose and Lucas), on which Slezak wrote a paper, which could
be needed for the reading of its Godelian reading of Descartes.
SLEZAK P., 1982, Gödel's Theorem and the Mind, Brit. J. Phil. Sci.
33, pp. 41-52.