On 30 Dec 2013, at 20:02, Alberto G. Corona wrote:
To summarize, there is no possible pure knowledge,
Why? On the contrary, beliefs can intersect truth, sometimes, and
provably so for simpler machine than us.
What happens is that only God knows when your beliefs are genuine
knowledge. For some, you can know that, but not necessarily in a
justifiable way, or only in a trivial justifiable way, like taking
them as (possibly new) axiom.
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.
Thanks. But I already knew so.
I am not sure you really grasp what I said.
But i the realm of reality, i.e. sensible experience, Edgar is
right here.
The "realm of reality" is what we bet on, and search, not what we
believe already (except for the non justifiable consciousness).
Bruno
2013/12/30 Bruno Marchal <marc...@ulb.ac.be>
On 30 Dec 2013, at 15:25, Alberto G. Corona wrote:
2013/12/30 Bruno Marchal <marc...@ulb.ac.be>
On 30 Dec 2013, at 12:39, Edgar L. Owen wrote:
All,
In response to the discussion of the possibility of a "Final
Theory" I'm starting a new topic on the Nature of Truth since this
is an important and separate issue from previous discussions.
1, it is impossible to directly know the external fundamental
reality, we know external reality only filtered through the
structures of our own minds. What we really know is only our own
mental model of external reality which is provably very very
different than actual external reality.
2, However we can easily prove that we do know external fundamental
reality to an extent sufficient for us to function reasonably
effectively within it. If we didn't have some actual true knowledge
of external reality we could not even function within it and thus
could not exist. So our very existence in actual reality
demonstrates we do have some true knowledge of it. (This true
knowledge consists of snippets of logical structure rather than the
physical world we believe it to be.)
That are belief, not knowledge.
Then, what is knowledge? the one derived from mathematical
deductions based on the belief on + and succ ?
That one is still on the type belief (a consequence of Gödel's
incompleteness).
To know that 1+ 1 = 2, you need to
1) believe that 1 + 1 = 2, but you need also that
2) it is the case that 1 + 1 = 2 (in your "reality")
If you put arithmetical realism on the table, anyone believing that
1 + 1 = 2, knows that 1 + 1 = 2. This needs some "reality"
satisfying the fact that 1+1=2, and we do suspect its existence
indeed, as the structure (N, 0, s, +, *) taught in high school.
Usually "rational belief" in a large sense is axiomatized by the
modal axiom K
B(x -> y) ->(Bx -> By),
with or without the necessitation rule (inferring Bx from x), but
(almost) always with the modus ponens (inferring B from A -> B and A).
Then a form of self-awareness is captured by the possible axioms Bx -
> BBx.
Gödel provability obeys that. That are the K4 reasoners. 4 is the
name (sic) of the formula Bx -> BBx, as it was the main axiom of the
fourth system by Lewis (S4).
S4 is the knowledge theory. It is K4 together with the axiom Bx ->
x. By definition of knowledge, if you know x, x is true. If p were
not true, i.e; if it was not the case that p, you would just be
believing wrongly.
Gödel's provability obeys K4 (indeed K4 + B(Bx->x)->Bx), but does
not obeys Bx -> x, at least from the machine 3p points' of view on
itself.
But the conjunction of Bx & x does obeys S4 (indeed S4 + B(B(x->Bx)-
>x)->x, the Grzegorczyk formula).
Set theoretically, knowledge is the intersection of your beliefs and
truth.
It can be explained that some machine, like PA and ZF, already
understand (prove, or prove from some Dt conditional, or more) that
their *personal* knowledge escape all possible 3p definitions. They
can't believe they are any machine. They still can bet on it, like
"nature" apparently already did.
Bruno
http://iridia.ulb.ac.be/~marchal/
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