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).
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.
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/
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