On Sat, Sep 21, 2013 at 9:43 PM, Bruno Marchal <[email protected]> wrote: > > On 21 Sep 2013, at 15:10, Telmo Menezes wrote: > >> On Fri, Sep 20, 2013 at 3:58 PM, Bruno Marchal <[email protected]> wrote: >>> >>> >>> On 19 Sep 2013, at 16:51, Telmo Menezes wrote: >>> >>>> On Thu, Sep 19, 2013 at 4:31 PM, Bruno Marchal <[email protected]> >>>> wrote: >>>>> >>>>> >>>>> >>>>> On 18 Sep 2013, at 21:45, Telmo Menezes wrote: >>>>> >>>>>> On Wed, Sep 18, 2013 at 6:13 PM, Bruno Marchal <[email protected]> >>>>>> wrote: >>>>>>> >>>>>>> >>>>>>> >>>>>>> >>>>>>> On 18 Sep 2013, at 11:43, Telmo Menezes wrote: >>>>>>> >>>>>>>> <snip> >>>> >>>> >>>> >>>> I know, I meant Dt vs. Dp. Was it a typo? Otherwise what's Dt as opposed >>>> to Dp? >>> >>> >>> >>> OK, sorry. "t" is for the logical constant true. In arithmetic you can >>> interpret it by "1=1". I use for the logical constant false. >>> >>> As the modal logic G has a Kripke semantics (it is a so-called normal >>> modal >>> logic), The intensional nuance Bp & Dp is equivalent with Bp & Dt. "Dt" >>> will >>> just means that there is an accessible world, and by Bp, p will be true >>> in >>> that world. >> >> >> Ok, thanks. >> If there is one or more accessible worlds, why not say []t? (I'm using >> [] for the necessity operator) > > > [] p means that p is true in all accessible worlds. But this makes []p true, > for all p, in the cul-de-sac worlds. We reason in classical logic. "If alpha > is accessible then p is true in alpha" is trivially true, because for any > alpha "alpha is accessible" is false, for a cul-de-sac world. > > And incompleteness makes such cul-de-sac worlds unavoidable (from each > world), in that semantics. In fact [] t is provable in all worlds, but Dt is > provable in none, meaning, in that semantics, that a cul-de-sac world is > always accessible. > > If you interpret "accessing a culd-de-sac world" as dying, the machine told > us that she can die at each instant! (of course there are other > interpretations).
Nice! > > > >> Is there any conceivable world where D~t? > > > No. > But the Z logic can have DDf, like the original (non normal) first modal > logic of Lewis (the S1, S2, S3, less known than S4 (knowlegde) and S5 > (basically Leibniz many-worlds, used by Gödel in his formal "proof of the > existence of God") > > > >> If so, can't we say ~D~t and thus []t? > > > Yes, []t is a theorem, of G and most modal logic, but not of Z! > > > > >> Isn't the only situation where ~Dt the one where this is no world? > > > ~Dt, that is [] f, inconsistency, is the type of the error, dream, lie, and > "near-death", or in-a-cul-de-sac. Thus your interest in near-death experiences? > We should *try* to avoid it, but we can't avoid it without loosing our > universality. > > The consistent machines face the dilemma between security and lack of > freedom-universality. With <>p = ~[] ~p, here are equivalent way to write > it: > > <>t -> ~[]<>t > <>t -> <> [] f > []<>t -> [] f I don't understand how you arrive at this equivalence. > In G (and thus in arithmetic, with [] = beweisbar, and f = "0 = 1", and t = > "1= 1". Thanks! Telmo. > > 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. -- 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.
