On 29 May 2013, at 17:47, Quentin Anciaux wrote:
http://en.wikipedia.org/wiki/G%C3%B6del%27s_incompleteness_theorems#Construction_of_a_statement_about_.22provability.22
2013/5/29 John Clark <johnkcl...@gmail.com>
On Wed, May 29, 2013 at 2:37 AM, Bruno Marchal <marc...@ulb.ac.be>
wrote:
>> If you want to communicate why should I need to search at all?
And if even Google doesn't know what the hell Bp&p is then it's
ridiculous to expect your readers to know what you're talking about.
> Come on, John. Search for "true opinion". Bp & p is a formula
using some notation for this,
So when I read your post and you said "Bp & p" I should have said to
myself "obviously if I Google "true opinion" it will tell me what Bp
& p means. Well, that is not obvious to me at all but it doesn't
matter because I just Googled "true opinion" and I still can't find
a damn thing about Bp & p.
Bp = I believe in p, or 'my opinion is that it is the case that p',
or, in the context of ideally correct (and simple machine):
Beweisbar('p').
p, when produced by some system, means, in all books on logic, that p
is true (from the system pov).
So Bp & p is a ay to model true opinion, in some system.
When I write I always ask myself if anybody will understand what I
say, I may not always be successful in making myself clear but at
least I try. You're not even trying.
I have explained this more than one times on this list, to different
people, because once you get it you can't forget.
You have come perhaps too much recently, but you can always ask
question. You should not focus on the formula, but on what it
represents. It is also explained in sane04, and basically, in all my
papers on this subject. Probably with different notations.
Or perhaps you just agree with what Niels Bohr said "I refuse to
speak more clearly than I think".
Bp is for "I believe p", produced by some machinery (machine, formal
system, ...).
In particular, it is an expression in some modal logic. 'Belief' obeys
usually the axioms:
1. B(p->q) -> B(p -> Bq)
2. Bp -> BBp
Bp & p means "(I believe in p) and p". P alone, in the assertative
mode of some entity means "it is the case that p". (independently of
the veracity of p).
For knowledge, we use the axiom:
3. Bp -> p
As Gödel saw in 1933, beweisbar, or provability, does not obey to that
third axiom, and so provability cannot model knowledgeability. Indeed
no consistent machine can prove B('0=1') -> 0=1, which is equivalent
with ~B('0=1'), which is self-consistency.
But it is trivial that the new connector Kp, defined by Bp & p,
verifies the axiom 3. So we get a way to associate a knower to a
machine. But it cannot be defined in arithmetic, as you would need to
define a predicate like B('p') & true('p'), which cannot exist by a
theorem of Tarski saying that true is not definable. We can only
simulate it by the modal trick of Theaetetus, for each arithmetical
formula. For example,
"I know that 1+1=2" can be emulated by B('1+1=2') & 1+1=2. But you
cannot find a general arithmetical predicate for knowledge, and this
makes such kind of knowledge confirming many studies by philosophers
and theologian, in the computer science setting.
Here belief is always a form of rational belief, which is basically
the meaning of the axiom 1 above.
Is is clearer? Ask anything.
I have already given such explanation here, and I will at some point
later explain this again on FOAR. No need to be angry or something,
Bruno
http://iridia.ulb.ac.be/~marchal/
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