On 02 Jul 2014, at 23:35, meekerdb wrote:
On 7/2/2014 9:52 AM, Bruno Marchal wrote:
The classical theory of knowledge, already present in ancien
epistemology is the modal KT theory, or KT4.
K is [](A -> B) -> ([]A -> []B). It is equivalent with ([]A & [](A
-> B)) -> []B. It is a belief in the modus ponens rule.
T is the important thing: the incorrigibility: you know only truth.
[]A -> A. The knowers knows. He might realize latter that he was
wrong, but this disqualify his ancien "knowledge" as knowledge:
admitting being wrong provides the admitting it was a belief, which
can be wrong.
"4" is the formula []A -> [][]A. It is used for more stable
knowledge than an immediate sort of knowledge. You know implies
that you know that you know.
That makes no sense to me. You define knowing and X as "Believing X
and X."
I define "knowing X" as "(believing X) and X". OK.
But then you say the knower might be wrong!?
Yes. But it means that he was not a knower.
This is well reflected by the fact that we will say:
Jules believed that sqrt(2) was a ration of natural numbers, but now
he know better.
And nobody will say:
Jules knew that sqrt(2) was a ratio of natural numbers, but now he
believed better.
Usually, when we believe something, that means that we take it as
true, and so, when very self-confident, we can say "I know that ...".
But we can be wrong, about the "...", making both "..." and "I know
that ..." false. You thought you knew, but realizing the error you
know it was a belief.
You've already assumed he's right as part of defining "knower". So
really you meant "believer" when you wrote "knower"?
It just means that I, or any machine, can use "know" wrongly. With
comp, only God knows when you really know, and are not believing
wrongly.
This is a weak notion of knowledge, which makes its job when we limit
our machine's interview on the self-referentially correct machine,
which are the only one needed to derive the comp "correct" physics.
It is not "knowing for sure", which with comp can be limited to only
the primary raw consciousness. Above that, we seem to know nothing for
sure. Comp is immune to *all* 3p certainty, and seems to get only one
class of 1p-certainty (basically "I exist", "I feel bad", "I feel
good") which are not 3p communicable/rationally-justifiable.
We have (Know p) -> p, but machines can confuse belief and knowledge,
and be wrong in their belief that they know this or that.
A correct Löbian machine will never say "I know that 1+1=2", she just
can't. She might justify correctly both "I believe that 1+1=2", and
"1+1=2", and this will entitle *us* to say that she knows that 1+1=2.
Humans have a non monotonic supplementary layer (itself justifiable
arithmetically, and useful with respect to tractability issue). This
makes the weak notion of knowledge a bit "academic" with respect of
real world epistemological situation, but the monotonic case is still
what we need to get the ideal comp-correct theology which include the
testable comp-correct physics.
'Non monotonic' means that we abandon the rule that we can deduce (A &
B) -> C from (A -> C). We accept that the new information "B" might
prevent C to be true.
For example, in classical logic you can derive "If it snows tomorrow
and I broke my leg today then I will do skying tomorrow" from "if it
snows tomorrow I will do skying tomorrow".
(Note than the quantum logic have such relevant aspect, and its most
"typical" implication does not verify the a posteriori axiom (or the
non relevance axiom): A -> (B -> A).
Monotonic means that the set of theorems grows monotonically with the
addition of axioms, like in math. A new theorem does not jeopardize an
older theorem.
Bruno
Brent
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