On 09 Jan 2014, at 05:55, meekerdb wrote:
Bruno writes Bp & p, where "Bp" ambiguously means "Proves
p" (Beweisbar?) and "Believes p".
What is ambiguous? I said that I limit the interview to Platonist
*correct* machine, believing in arithmetic or in recursively
enumerable extension of arithmetic. And the fact that the machine
cannot prove Bp -> p for all p, suggest that provability obeys to the
axioms I gave for belief, and not for knowledge (where Bp->p is not
just true but believed as well).
"Believes p and P" is then a belief that is "true".
OK. That's correct.
I put scare quotes around "true" because I think it just means "is a
consequence of some (Peano's) axioms", which is not necessarily the
same as "expresses a fact".
At the meta-level (G*), that is true, but the machine does not know
that, and for correct machine, this change nothing. We have Bp -> p
(as a theorem of G*, not of G).
Bruno
Brent
On 1/8/2014 2:11 PM, John Mikes wrote:
Bruno and Brent:
did you agree whether "TRUE BELIEF" means in your sentences
1. one's belief that is TRUE, (not likely), or
2. the TRUTH that one believes in it (a maybe)?
(none of the two may be 'true').
JM
On Wed, Jan 1, 2014 at 5:50 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
On 31 Dec 2013, at 21:09, meekerdb wrote:
On 12/31/2013 1:07 AM, Bruno Marchal wrote:
only rules to extract knowledge from assumed beliefs.
?
I answered "no" to your question. Knowledge is not extracted in
any way from belief (assumed or not). knowledge *is* belief, when
or in the world those beliefs are true, but this you can never
know as such.
Since your theory to an infinite number of semi-classical worlds
with different events (and even different physics) it seems that
"true belief" is not a very useful concept.
It is, because by incompleteness, we will have that Bp & p (true
belief) obeys a different logic (an epistemic intuitionist logic)
despite G* knows that it is the same machine, having the same
action. The machine just dont know that, although it can infer it
from comp + a sort of faith in herself.
Every belief is going to have probability zero of being true.
neither Bp nor Bp & p is a priori related to probability. For this
you need []p -> <>p, which is ocrrect for Bp & p, though, and
indeed a physics appears already there, but that is a sort of
anomaly (which confirms what I took as an anomaly in Plotinus, but
the machine agrees with him).
Now, Bp, when present in the nuances, gives the logic of the
corresponding "certainty", so it is trivially a probability one. We
need to extract the logic, and the probability different from 1 are
handled by the mathematics, and is related to the Dp (not Bp). The
probability bears on the accessible "worlds".
The interesting concept is the probability of future events
relative to one's current state.
That's exactly why we need to go from Bp to Bp & Dt (or Bp & Dt &
p, or actually Bp & p). This gives the relevant notion of relative
consistency together with some temporal interpretation.
Bruno
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