On 03 Dec 2012, at 00:04, meekerdb wrote:
On 12/2/2012 7:27 AM, Bruno Marchal wrote:
The 1p truth of the machine is not coded in the machine. Some
actual machines knows already that, and can justified that If there
are machine (and from outside we can know this to correct) then the
1p-truth is not codable. The 1p truth are more related to the
relation between belief and reality (not necessarily physical
reality, except for observation and sensation).
Even the simple, and apparently formal Bp & p is NOT codable.
Most truth about machine, including some that they can know, are
not codable.
Many things true about us is not codable either.
Let me see if I understand that. I think you are saying that p,
i.e. that "p" describes a fact about the world, a meta-level above
the coding of a machine.
No, p is for some statement at the base level, like 1+1 = 2.
That the Mars Rover believes it is south of it's landing point is
implicit in its state and might be inferred from its behavior, but
there is no part of the state corresponding to "I *believe* I am
south of my landing point."
Then Mars Rover is not Löbian. But I am not even sure that Mars Rover
is Turing universal, or that it exploits its Turing universality.
But PA and ZF can represent "I believe". So we can study the logic of
a new 'knowledge" operator defined (at the meta level, for each
arithmetical proposition) by Bp & p. For example if p is "1+1=2", it is
Believe"("1+1=2") & 1+1 = 2.
We cannot define such operator in arithmetic. We would need something
like Believe"("1+1=2") & True("1+1 = 2"), but True, in general cannot
defined in arithmetic. Yet, we can metadefine it and study its logic,
which obeys a soprt of temporal intuionistic logic (interpreting the
S4Grz logic obtained).
One could include such second-level states (which one might want to
communicate to Pasadena) but then that state would be just another
first-level state. Right?
Not sure I see what you mean. The meta, available by the machine is in
the "I believe". It is the 3-I. The presentation of myself to myself.
The 1-I will be the non definable operator above. We connect the
believer to the truth. It is easy to do for the sound correct machine.
Bruno
Brent
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