To summarize, there is no possible pure knowledge, only rules to extract knowledge from assumed beliefs. Thanks. But I already knew so.
But i the realm of reality, i.e. sensible experience, Edgar is right here. 2013/12/30 Bruno Marchal <[email protected]> > > On 30 Dec 2013, at 15:25, Alberto G. Corona wrote: > > > > > 2013/12/30 Bruno Marchal <[email protected]> > >> >> On 30 Dec 2013, at 12:39, Edgar L. Owen wrote: >> >> All, >>> >>> In response to the discussion of the possibility of a "Final Theory" I'm >>> starting a new topic on the Nature of Truth since this is an important and >>> separate issue from previous discussions. >>> >>> >>> 1, it is impossible to directly know the external fundamental reality, >>> we know external reality only filtered through the structures of our own >>> minds. What we really know is only our own mental model of external reality >>> which is provably very very different than actual external reality. >>> >>> 2, However we can easily prove that we do know external fundamental >>> reality to an extent sufficient for us to function reasonably effectively >>> within it. If we didn't have some actual true knowledge of external reality >>> we could not even function within it and thus could not exist. So our very >>> existence in actual reality demonstrates we do have some true knowledge of >>> it. (This true knowledge consists of snippets of logical structure rather >>> than the physical world we believe it to be.) >>> >> >> That are belief, not knowledge. >> >> Then, what is knowledge? the one derived from mathematical deductions > based on the belief on + and succ ? > > > > That one is still on the type belief (a consequence of Gödel's > incompleteness). > > To know that 1+ 1 = 2, you need to > > 1) believe that 1 + 1 = 2, but you need also that > > 2) it is the case that 1 + 1 = 2 (in your "reality") > > If you put arithmetical realism on the table, anyone believing that 1 + 1 > = 2, knows that 1 + 1 = 2. This needs some "reality" satisfying the fact > that 1+1=2, and we do suspect its existence indeed, as the structure (N, 0, > s, +, *) taught in high school. > > Usually "rational belief" in a large sense is axiomatized by the modal > axiom K > > B(x -> y) ->(Bx -> By), > > with or without the necessitation rule (inferring Bx from x), but (almost) > always with the modus ponens (inferring B from A -> B and A). > > Then a form of self-awareness is captured by the possible axioms Bx -> BBx. > > Gödel provability obeys that. That are the K4 reasoners. 4 is the name > (sic) of the formula Bx -> BBx, as it was the main axiom of the fourth > system by Lewis (S4). > > S4 is the knowledge theory. It is K4 together with the axiom Bx -> x. By > definition of knowledge, if you know x, x is true. If p were not true, i.e; > if it was not the case that p, you would just be believing wrongly. > > Gödel's provability obeys K4 (indeed K4 + B(Bx->x)->Bx), but does not > obeys Bx -> x, at least from the machine 3p points' of view on itself. > > But the conjunction of Bx & x does obeys S4 (indeed S4 + > B(B(x->Bx)->x)->x, the Grzegorczyk formula). > > Set theoretically, knowledge is the intersection of your beliefs and truth. > > It can be explained that some machine, like PA and ZF, already understand > (prove, or prove from some Dt conditional, or more) that their *personal* > knowledge escape all possible 3p definitions. They can't believe they are > any machine. They still can bet on it, like "nature" apparently already did. > > Bruno > > > > 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 [email protected]. > To post to this group, send email to [email protected]. > Visit this group at http://groups.google.com/group/everything-list. > For more options, visit https://groups.google.com/groups/opt_out. > -- Alberto. -- 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 [email protected]. To post to this group, send email to [email protected]. Visit this group at http://groups.google.com/group/everything-list. For more options, visit https://groups.google.com/groups/opt_out.
