On 10/21/2013 7:25 AM, Bruno Marchal wrote:

On 21 Oct 2013, at 05:09, meekerdb wrote:On 10/20/2013 2:15 PM, Russell Standish wrote:On Sun, Oct 20, 2013 at 06:22:15PM +0200, Bruno Marchal wrote:On 20 Oct 2013, at 12:01, Russell Standish wrote:On Sun, Oct 20, 2013 at 08:52:41AM +0200, Bruno Marchal wrote:We have always that [o]p -> [o][o]p (like we have also always that []p -> [][]p)There may be things we can prove, but about which we are in fact mistaken, ie []p & -pThat is consistent. (Shit happens, we became unsound).Consistency is []p & ~[]~p. I was saying []p & ~p, ie mistaken belief.ISTM that Bruno equivocates and [] sometimes means "believes" and sometimes "provable".But I am allowed to do that, because []p -> p is not a theorem (for some p, byincompleteness)

`Isn't incompleteness ~(p->[]p) ? And you assume the machine is consistent, so doesn't`

`that entail []p->p ?`

and thus (rational formal) provability behaves like believability.A mathematician told me that I was dead mad by saying this, but that is standard inmathematical logic (ignored by most mathematicians). It is counter-intuitive. Mostpeople believes that formal proof guaranties truth, when starting from true theorem (andthat is true for the ideally correct machine, but no machine can know she is correct,and her probability does behave like a believability, indeed one on Which theapplication of Theaetus' definition leads to the classical knowledge logic (the modallogic S4).

`That still seems like equivocation to me. Even if they "obey the same modal" logic, that`

`only means they have the axioms and rules of inference. It doesn't follow that the true`

`but unprovable propositions are the same propositions as the true but not-believable ones.`

The hypostases will work for any correct machine whose beliefs extend soundly thebeliefs in elementary arithmetic.The key is in the mathematical trick to limit us to correct machine, with enough beliefs(about machines or numbers) so that they are under the spell of the secondincompleteness theorem, or Löb, and can prove that.

`But doesn't limiting us to a correct machine mean the []p->p ; isn't that what "correct"`

`means?`

Brent

In the literature such machine/theories are qualified as being "sufficiently rich", but"Löbian" is shorter, (and then the Löb formula also characterize their provability logic).BrunoBrentObviously, one cannot prove []p & p, for very many statements, ie[]p & p does not entail []([o]p)[]p -> [][]p OK?Why? This is not obvious. It translates as being able to prove that you can prove stuff when you can prove it. If this were a theorem of G, then it suggests G does not capture the nature of proof. Oh, I see that you are just restating axiom "4". But how can you prove that you've proven something? How does Boolos justify that?(and []p -> []p, and p -> p) + ([](p & p) <-> []p & []q) (derivable in G)Did you mean [](p&q) <-> []p & []q? That theorem at least sounds plausable as being about proof.so []p & p -> [][]p & ([]p & p) -> []([]p & p) & ([]p & p), thus ([]p & p) -> [][o]p (& [o]p : thus [o]p -> [o][o]p)Therefore, it cannot be that [o]p -> [o]([o]p) ??? Something must be wrong...I hope I am not too short above, (and that there is not to much typo!) BrunoAnd thus you've proven that for everything you know, you can know that you know it. This seems wrong, as the 4 colour theorem indicates. We can prove the 4 colour theorem by means of a computer program, and it may indeed be correct, so that we Theatetically know the 4 colour theorem is true, but we cannot prove the proof is correct (at least at this stage, proving program correctness is practically impossible).--You received this message because you are subscribed to the Google Groups "EverythingList" group.To unsubscribe from this group and stop receiving emails from it, send an email toeverything-list+unsubscr...@googlegroups.com.To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.http://iridia.ulb.ac.be/~marchal/

-- You received this message because you are subscribed to the Google Groups "Everything List" group. To unsubscribe from this group and stop receiving emails from it, send an email to everything-list+unsubscr...@googlegroups.com. To post to this group, send email to everything-list@googlegroups.com. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.