On 9/14/2012 2:56 PM, Bruno Marchal wrote:
On 14 Sep 2012, at 15:41, Stephen P. King wrote:
On 9/14/2012 4:20 AM, Bruno Marchal wrote:
On 13 Sep 2012, at 22:04, Brian Tenneson wrote:
You use B as a predicate symbol for "belief" I think.
I use for the modal unspecified box, in some context (in place of
the more common "").
Then I use it mainly for the box corresponding to Gödel's beweisbar
(provability) arithmetical predicate (definable with the symbols E,
A, &, ->, ~, s, 0 and parentheses.
Thanks to the fact that Bp -> p is not a theorem, it can plays the
role of believability for the ideally correct machines.
What are some properties of B and is there a predicate for
knowing/being aware of that might lead to a definition for
Yes, B and its variants:
B_1 p == Bp & p
B_2 p = Bp & Dt
B_3 p = Bp & Dt & t,
btw, what is a machine and what types of machines are there?
With comp we bet that we are, at some level, digital machine. The
theory is one studied by logicians (Post, Church, Turing, etc.).
Could you elaborate on what your definition of "a digital
Anything Turing emulable.
OK. But you do understand that this assumes an unnecessary
restrictive definition of computation. I define computation as "any
transformation of information" and Information is defined as "the
difference between a pair that makes a difference to a third".
Is it something that can be faithfully represented by a Boolean
Algebra of some sort?
Anything can be represented by Boolean algebra of some sort, even the
quantum logic, despite not being embeddable in Boolean logic.
No, you cannot define a bijective map between a logical
representation of a QM system and a Boolean algebra. The quantum logical
structure (ortho-complete lattice) is not distributive and the Boolean
algebra *is* distributive.
Is there a generic description for a structure (in the math logic
sense) to have a belief or to be aware; something like
A |= "I am the structure A"
Yes, by using the Dx = xx method, you can define a machine having
its integral 3p plan available.
This "3p plan" would be like my internal model of my body that I
have as part of my conscious awareness?
Yes, you can say that.
But the 1p-self, given by Bp & p, does not admit any name. It is the
difference between "I have two legs" and "I have a pain in a leg,
even if a phantom one". G* proves them equivalent (for correct
machines), but G cannot identify them, and they obeys different
logic (G and S4Grz).
This implies, to me, that the 1p-self cannot be defined by an
equivalence class with a fixed equivalence relation. This is
problematic if assumed to be true for all possible 1p-selfs. AFAIK,
your definition would only apply to an machine that is unnameable
infinite such as the totality of all that could exist, aka "God" or
"cosmic intelligence". It reminds me more of theAzathoth of H.P.
Lovecraft's mythos <http://en.wikipedia.org/wiki/Azathoth>.
You ask for proof? I will try. Do you recall our discussion that
"God has no name"? Why did we agree that "god has no name"?
Finally, on a different note, if there is a structure for which all
structures can be 1-1 injected into it, does that in itself imply a
sort of ultimate structure perhaps what Max Tegmark views as the
level IV multiverse?
A 1-1 map is too cheap for that, and the set structure is a too much
I agree, it is just a tautology.
Comp used the simulation, notion, at a non specifiable level
But does not address the computational resource requirement. :_(
It does not solve it, but it address it, like it address all of
physics. I give the tools so that you can ask your question directly
to the machine.
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