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
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. I think
that knowledge consists of a belief that is both true and causally connected to the thing
believed (c.f. Gettier's paradox). Of course belief that is held because the proposition
is proven from some axioms does have a causal connection to the axioms. But that is more
than just "believed". The problem then arises when you say things like, "We know there
are true but unprovable facts about arithmetic." We only 'know' those things in different
meta-system where they do have a causal connection to other axioms we hypothesize as
true. But ultimately we cannot 'know' that axioms are true - as you say we just bet on them.
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