On 9/14/2012 4:20 AM, Bruno Marchal wrote:
Hi Brian,
On 13 Sep 2012, at 22:04, Brian Tenneson wrote:
Bruno,
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
self-awareness?
Yes, B and its variants:
B_1 p == Bp & p
B_2 p = Bp & Dt
B_3 p = Bp & Dt & t,
and others.
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.).
Dear Bruno,
Could you elaborate on what your definition of "a digital machine"
is? Is it something that can be faithfully represented by a Boolean
Algebra of some sort?
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?
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>.
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
structural flattening.
I agree, it is just a tautology.
Comp used the simulation, notion, at a non specifiable level
substitution.
But does not address the computational resource requirement. :_(
Bruno
http://iridia.ulb.ac.be/~marchal/
--
Onward!
Stephen
http://webpages.charter.net/stephenk1/Outlaw/Outlaw.html
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