On 21 Mar 2015, at 21:38, Telmo Menezes wrote:
It is the p in []p & p, which makes "machine's knowledge" not
definable in term of number and machine. S4Grz formalizable at a
level, what the machine cannot formalize about herself (but can bet
on, ...).
Thanks to incompleteness, the Theaetetus' definition makes sense,
and distinguish the knower from the rational believer for the
machine.
Don't hesitate to ask precision. I am very literal here: the
knower is defined by the true believer. It is a modest definition
of knowledge, and it is not similar with "I know for sure that",
which needs some amount of consistency (like <>t, or <><>t, or
<><><> t, etc.).
What's the difference between <>t and <><>t and so on?
In the Kripke semantics, <>p means that you can, by starting form
the world you are in, access to another world where t is true. It
means that you are not in a cul-de-sac world. It means that you are
"alive", you can access some world. But <>t -> <>[]f. So you despite
being alive, you might be dead in that next world. That next world
might be a cul-de-sac world, where <>t is false, and thus []f is
true. Now, if you are in a world where <><>t is true, then you can
access a world in which <>t is true, so that you are still alive.
<>t = I can access some world (t is true in all worlds), but that
world might be a cul-de-sac world.
<><>t = I can access some world where <>t is true, so from there I
can access some other world.
Put differently:
<>t I am alive (but can die at the next instant)
<><>t, I am alive and I can access to an instant where I am still
alive.
Ok, but it's not obvious to me how temporality (a sequence of
instants) is introduced here.
Here, it is "introduced" simply through the Kripke semantics of the
modal logic involved, G in this case.
Keep in mind that we are in arithmetic. <>t abbreviates ~beweisbar
'~( 2+2=4').
here '~( 2+2=4' denote some number, denoting some falsity. It is
equivalent with the arithmetical
consistent('2+2=4'), which is true, but not provable by the system
itself.
By Gödel's completeness theorem, which applies still on most correct
machine if they follow some recommendation in their way of talking,
consistent('p') is equivalent (from outside) with "there is a model
which satisfy my beliefs and p".
So, literally, <>p means if you add p to my beliefs, it will not leads
to a contradiction (semantically: a cul-de-sac world)
So, there is a sort of interpretation of <>p in "there is some reality
in which p is true", and, as t (true) is true in all worlds, you can
interpret <>t by "there is a world" (or there is a reality, or there
is some truth, or there is some god, or ...).
[0]p = []p, and obeys to G, and fully described by G* (at the
propositional level).
[1]p = []p & p, and obeys to S4Grz,
[2]p = []p & <>t obeys and define the logic Z
[3]p = []p & <>t & p
I definitely don't understand [2]p and [3]p.
[]p & <>t is a weakening of []p & p. Instead of asking p being
true, we ask only for p being consistent
(([]p & <>t) -> ([]p & <>p)).
If you can prove for all p that []p -> p (like with [1]), then you
can prove for all p []p -> <>t or []p & <>p.
So the logic of provability-and-consistency is weaker than the logic
of provability-and-truth.
Provability and consistency avoids the probability one for the
false, which would exist if we take []p as provability. The passage
[]p =====> []p & <>p, approximate the main things for a probability
one.
What does "provability one" mean?
Come on Telmo, it takes me a lot of concentration to write
"probability" instead of "provability", with those damn "b" and "v"
which are so much together. And in our case the distinction is of
importance, and can be made quite clear in arithmetic:
Provability is beweisbar , what is translated in the modal logic of
provability by the box [], usually []p.
Probability is beweisbar-and-consistent, the []p & ~[]~p, or
(equivalent) []p & <>t.
The difference is that with Provability, we can access cul-de-sac
world, in which you have []f. (= ~<>t).
For proVability, we need to add the "by-default hypothesis" that we
will not die during the experience.
When you bet on a coin: the events
{H} or {T}, {H, T} have probability one, because semantically we
abstract from the asteroïds which can smash the playing table, the
dice and the betters.
By forcing <>t on each state (like Theaetetus did with the truth) we
abstract from the cul-de-sac world, by imposing the existence of some
accessible world. Logically, at first sight, it looks like a weakening
of Theaetetus' idea of restricting []p on the truth, but
arithmetically, or mechanicalistically (if I can say), that is by
incompleteness, it appears that replacing "p" by <>p, is an
strengthening, yet dual of some sort of []p.
You might need to revise a little bit the Kripke semantics of modal
logic, and the arithmetical logic of (machine) self-reference. It
works also for machine + oracles.
It models what the guy in Helsinki can be sure if, like drinking a
cup of coffee (in the protocol where he get a cup in both W and M).
It abstract (locally) from the cul-de-sac world. It use the bet on
<>t implicit in the yes-doctor.
I don't know about the cup of coffee protocol either, could you
explain it?
Imagine that you are in Helsinki, and you will be duplicated (cut and
copied) and sent (by waves) to Washington and Moscow, where you are
reconstituted-reincarnated, OK?
But this time, we add in the protocol that both in Moscow, *and* in
Washington, they will offer a cup of coffee to the reconstituted person.
The question (that, btw, I have asked many times to J. Clark without
any answer) is: what is your expectation, when you are still in
Helsinki (of course) of drinking a cup of coffee after the duplication?
Of course, we, including the guy in Helsinki, *assumes*
computationalism. We assume the correctness of the substitution level
chose, and we agree with the default hypotheses (no asteroïds). We
already agree that the expectation of drinking a cup of coffee is the
same (modulo comp and the default hypothesis) after a simple digital
teleportation than with using a plane (assuming it does not crash and
the usual default hypotheses).
So, what do you think?
What the Löbian Universal Machine thinks, is that provability,
knowledge, observable, sensible, obeys different modal logics, and
that most of those modalities splits into a justifiable part and a non
justifiable part.
Then we can test the theory by comparing the machine internally
defined notion of observable ([]p & <>p + p sigma_1) with the
empirically observable. Curiously, at first sight, is that the quantum
only appears on the non justifiable part(*), but that is normal:
physics is first person plural, and that is (arguably) confirmed: QM
(without collapse) makes the splitting of observers contagious to
their colleagues. If you look at the schroedinger cat, the story
differentiates into a story where you see the dead cat, and a story
where you see the cat alive, but the day after, when your colleagues
(in the respective stories) ask you how was the cat, they will split
too at the occasion. To be sure, they would have already splitted/
differentiated, by the natural gossip of the sufficiently hot
environment.
It *does* look like the quantum aspect of reality is explained by the
way the sigma_1 arithmetical truth can look to itself, in the
observable mode.
Bruno
(*) (Z1*, X1*, not X1 nor Z1. To be sure a quantization and a quantum
logic also appear on S4Grz1 which does NOT split. The outer God does
not add anything to what the inner God already knew. And, now, the
inner god has already a foot in matter! This shocked my intuition, as
it makes matter still more on the first person side than I thought,
but it fits better with Plato, Plotinus, East and West Mystics, and
even with the salvia reports and some other reports you can find on
Erowid or on Salvia webpages.
Thanks!
Telmo.
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