On 21 Sep 2013, at 15:10, Telmo Menezes wrote:

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On Fri, Sep 20, 2013 at 3:58 PM, Bruno Marchal <marc...@ulb.ac.be>wrote:On 19 Sep 2013, at 16:51, Telmo Menezes wrote:On Thu, Sep 19, 2013 at 4:31 PM, Bruno Marchal <marc...@ulb.ac.be>wrote:On 18 Sep 2013, at 21:45, Telmo Menezes wrote:On Wed, Sep 18, 2013 at 6:13 PM, Bruno Marchal <marc...@ulb.ac.be> 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 asopposedto Dp?OK, sorry. "t" is for the logical constant true. In arithmetic youcaninterpret it by "1=1". I use for the logical constant false.As the modal logic G has a Kripke semantics (it is a so-callednormal modallogic), The intensional nuance Bp & Dp is equivalent with Bp & Dt."Dt" willjust means that there is an accessible world, and by Bp, p will betrue inthat 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).`

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 GĂ¶del 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.`

`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

`In G (and thus in arithmetic, with [] = beweisbar, and f = "0 = 1",`

`and t = "1= 1".`

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 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.