On 13 Feb 2014, at 05:38, Russell Standish wrote:
On Wed, Feb 12, 2014 at 12:24:18PM +0100, Bruno Marchal wrote:
On 12 Feb 2014, at 02:02, Russell Standish wrote:
On Tue, Feb 11, 2014 at 07:31:24PM +0100, Bruno Marchal wrote:
You are right, the qualia are in X1* \ X1, like we get quanta in
S4Grz1, Z1*, X1*.
The only thing you can say is that qualia ought to obey the axioms
of
X1*\X1, (and even that supposes that Z captures all observations,
which I think is debatable),
By UDA, "p" to refer to a "physical certainty" needs to
1) UD generated (= sigma_1 arithmetical and true).
2) provable (true in all consistent extensions)
3) and non "trivially" provable (= there must be at least one
consistent extension)
This give the []p & <>t, with p sigma_1.
So the logic of observable certainty should be given by the Z1*
logic.
This is certainly an interesting understanding that I hadn't met in
your writings before.
You worry me a bit, as I think this is explained in all papers and the
thesis. I know that I am concise.
Normally, if everything get clear, you should see that this is what I
am explaining everywhere.
In associating provable with "true in all consistent extensions",
In case of "provable", this is Gödel COMPLEteness result (not
incompleteness!).
In case of an abstract box, in a modal logic having a Kripke
semantics, this is just the semantics of Kripke.
are
you meaning that so long as something (ie proposition) is computed by
all programs instantiating your current state, no matter how far in
the future that calculation might require, then that something is
(sigma_1) provable.
I am not sure. "true in all consistent extensions" is a very general
notion.
What happens is that, in arithmetic, the sigma_1 sentences, when true,
are provable (already by RA).
So they verify the formula A -> []A. (called TRIV for trivial, as
that sentence makes many modal logic collapsing, but not so in the
provability logic, not even in the 1p S4Grz).
In fact a machine is Turing universal iff for all sigma_1 sentences A
we have A -> []A. So "A -> []A" is the Turing universality axiom, when
A is put for any sigma_1 sentence.
G1 is G + A->[]A. Visser proved an equivalent of Solovay theorem for
G1 and G1*. You can find it in Boolos 1993.
It is a way to restrict the logic of the different points of view on
the UD*. "To be a finite piece of computation" is itself given by a
sigma_1 formula, and the sigma-1 sentences model computations.
Then 1&2 gives your hypostase for knowledge, ie S4Grz1.
Only G1 at that stage. To get knowledge, you need to do 1 and 2, but
on []p & p, like to get observation/probability/expectation, you need
to do 1 and 2, but on (3) []p & <>t.
And to get sensible observation, you can mix knowledge ( " & p"), and
"consistency" <>t.
Incompleteness makes all those views obeying to different logic.
It is, of
course the sigma_1 restriction of Theatetus's definition of knowledge,
which both Brent & I share quibbles with, but accept for the "sake of
the argument".
Since Plato, many philosophers quibble on Theaetetus' definition. The
fist quibbler being Socrate, who refuted it.
The magic things happening with comp, is that Socrate's refutation
does no more apply, and the only argument against it which remains, is
the argument put forward by people who believe that they can
distinguish, immediately in the 1p view, simulations or dreams from
reality. But this we have already abandoned when we accept an
artificial brain (like in step 6).
But assuming 3) above is equivalent to assuming the no cul-de-sac
conjecture by fiat.
The beauty is that incompleteness makes sense of that move. In most
modal logic []p -> <>t.
I don't feel comfortable in assuming that axiomatically - I was hoping
for a proof, or even just a better justification for that.
I am not sure what that would mean. Here the proofs is that the move
need to get a probability notion from a provability notion makes
genuine new sense thanks to incompleteness.
When we predict P(head) = 1/2, we also, but *implicitly*, assume <>t
by fiat. Incompleteness gives the opportunity to see that making it
explicit does change the logic, and that is why observation will obeys
to a different logic than knowledge, and that is exactly what we need
to get physics and knowledge, and belief, ... from the same
arithmetical reality accessible by a machine.
A rumor, alluded in the book by Franzen (on the abuse of Gödel!), is
that I define probability by provability, but of course, that is not
the case. Knowledge and probability are intensional nuance of
provability, not provability itself.
Best,
Bruno
http://iridia.ulb.ac.be/~marchal/
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