On 13 Jun 2014, at 21:53, meekerdb wrote:
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.
OK. Just realize that x denote sentences, not numbers. In logic the
variables are formula or sentences.
But in Fuzzy logic, formula can have truth values in more complex
structures, and here yes, we can say that the attractor is fixed. Hmmm
OK. Should reread Grim for the official definition.
Bruno
Brent
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