On 24 Mar 2012, at 21:21, meekerdb wrote:
On 3/24/2012 12:58 PM, Bruno Marchal wrote:
Google on "theaetetus".
Socrates asked to Theaetetus to define "knowledge". Theatetus gives
many definitions that Socrates critizes/refutes, each of them. One
of them consists in defining knowledge by belief, in "modern time"
the mental state, or the computational state of the belief and the
knowledge is the same, and a belief becomes a knowledge only when
it is (whatever the reason or absence of reason) true. Another one
is the justified true belief, which is the one which you can
translate in arithmetic with Gödel's predicate. You can read "Bp &
p" by I can justify p from my previous beliefs AND it is the case
that p. To give you an example, if the snow was blue, a machine
asserting "snow is blue" can be said to know that snow is blue.
Indeed, the machine asserts "the snow is blue", and it is the case
that snow is blue (given the assumption).
The "problem" (for some) with that theory is that it entails that,
when awake, we cannot know if we are dreaming or not, although in
dream we can know that we are dreaming, the same for "being not
correct". It is not a problem for comp which makes that ignorance
unavoidable.
For a machine that "we" know to be arithmetically correct, we know
that Bp and Bp & p are equivalent. Yet, the machine cannot know
that about herself, and the logic of Bp and of "Bp & p" are
different. They obeys to the modal logics G and S4Grz, but I guess
you need to read some book or some web pages to see what I mean here.
I find your explications of knowledge to be confusing. You refer to
Theaetetus who said knowledge = true belief. But in your modal
logic formulation B stands for either provable or proven
(Beweisbar). "Provable" and "believed" are too very different things.
Hmm... I am just using Dennett's intentional stance toward machine. A
machine believes p, really means only that the machine assert p.
I limit myself to machine having their beliefs closed from the modus
ponens rule, and obeying classical logic, and applying classical logic
on their description etc. I limit myself on classically self-
referentially correct machine with respect to some other possible
machines or oracles.
I think that knowledge consists of a belief that is both true and
causally connected to the thing believed (c.f. Gettier's paradox).
That is not incompatible with the Theaetetus' idea. Somehow the whole
problem is there.
Of course belief that is held because the proposition is proven
from some axioms does have a causal connection to the axioms.
Yes. And the axioms can be any successful "memes" in the way to
unravel a difficulty. Axioms which solves many problems and many class
of problems can win local games.
I identify partially a person with the set of its beliefs. More
concrete person can revised beliefs (normally), which is not used
here, given that I restrict myself of the math of the correct machine.
But that is more than just "believed".
Because you talk from the point of view of a much more complex
consistent (let us hope) extension, viewed at some level.
For the interview I limit myself to an apparently simple machine with
beliefs, that we all believe in (I hope).
The problem then arises when you say things like, "We know there are
true but unprovable facts about arithmetic."
No. About the machine. Sometimes that machine is Peano Arithmetic,
because it is the better known Löbian theory. It is an axiomatizable
theory, which means, by a theorem of Craig, a recursively enumerable
set of beliefs.
We only 'know' those things in different meta-system where they do
have a causal connection to other axioms we hypothesize as true.
Of course. The points is that the Löbian machine does that too, and
can even do it for themselves. This gives them ability to climb the
constructive transfinite, and to develop talks and mind tools from
beyond the constructive.
But ultimately we cannot 'know' that axioms are true - as you say we
just bet on them.
I think so. Even unconsciously, when our brain conceptualizes that
there is a reality behind the back of the computer screen. We
extrapolate all the times, and we have partial controls and partial
responsibilities and those kinds of things.
The nice happening here is that by the incompleteness, Bp can't be
confused with Bp & p. By incompleteness Bf does not imply f, so Bf is
not the same as Bf & f. (Beliefs can be wrong. Knowledge can't, by
definition).
This shows that the same set of believed arithmetical sentences can be
seen from many points of views. They provide an arithmetical
interpretation of Plotinus, through what happens when a universal
machine looks inward.
It applies to us, tangentially, as far as we are self-referentially
correct ourselves, tangentially.
Bruno
http://iridia.ulb.ac.be/~marchal/
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