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

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>

On 18 Sep 2013, at 11:43, Telmo Menezes wrote:


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).

Is there any conceivable world where D~t?

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.

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".



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