On 13 Jun 2014, at 01:00, meekerdb wrote:
On 6/12/2014 6:33 AM, Bruno Marchal wrote:
Actually Grim and another guy studied version of Gödel and Löb
theorem in fuzzy logic (meaning that they use the closed interval
[0, 1] has set of truth values. They illustrate that the truth
values of most fixed points in self-reference logic describe
chaotic trajectories (in the set of truth value).
I don't understand what they a "fixed points" of, if not truth value?
In the (classical) self-reference logic, they are sentences, and they
are fixed point in the sense of being a solution of a self-reference.
The self-reference x <-> ~[]x has solution the sentence <>f
(beweisbar("0=1")). (Gödel 1931)
The self-reference x <-> []x has solution the sentence t (or
"0=0") (Löb 1955)
The self-reference x <-> []~x has solution the sentence []f
(beweisbar("0=0")) (Jeroslow, Smullyan)
The self-reference x <-> ~[]~x has solution the sentence f (or
"0=1"). (Gödel)
But in fuzzy logic, some of those "fixed points" are not fixed, and
moves in the truth set in a chaotic way, with a variety of attractors.
Bruno
Brent
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