On 6/13/2014 9:23 AM, Bruno Marchal wrote:
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.
Ok, so it's some chaotic attractor that is "fixed", not a point. I understood a fixed
point to be the the value of f(x) where x=f(x) when the value exists in the sense of
convergence in the limit of iterating f.
Brent
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