On 29 May 2013, at 22:46, meekerdb wrote:

On 5/29/2013 9:52 AM, Bruno Marchal wrote:

On 29 May 2013, at 18:37, Bruno Marchal wrote:

On 29 May 2013, at 17:47, Quentin Anciaux wrote:


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.

I'm not sure I understand this. Are you saying we cannot take "(Bp- >p) for all p" as an axiom because it would entail Bf ->f and then ~f->~Bf, and since ~f is true by definition it would entail that the machine is consistent?


More generally p -> f is equivalent with ~p, as you can verify by doing the truth table:

p   ->   f
1    0    0
0    1    0

That's why Löb's theorem B(Bp -> p) -> Bp generalizes Gödel's second incompleteness theorem: just replace p by f. B(~Bf) -> Bf, ~Bf -> ~B(~Bf), Dt -> ~BDt.



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