Re: tautology

From: Marchal <marchal.domain.name.hidden>
Date: Wed Dec 8 07:50:46 1999

Russell Standish wrote:

>None of this is in defence of QTI! It merely is to show that your
>measure argument fails - unless you happen to be a Copenhagener :)
>
>As I have mentioned before, in order for QTI to work, there must also be no
>possible "cul-de-sac" branches. In Bruno's model logic, I believe this
>would be expressed as
>
>\forall \alpha \in W, \models_\alpha^W \neg \Box (\Box\top \wedge \Box\bot)
>
>or in slightly more English notation (\top == true, \bot == false,
>\wedge == and)
>
>for every world alpha in the model W, there cannot be a successor
>world that can only access a terminal world
>
>where \Box p is trivially true in a terminal world, regardless of the
>truth table of p, but \Box true and \Box false cannot both be true at
>the same time.

I'm not sure I understand what you mean by ``\models_\alpha^W"

But "BOX TRUE AND BOX FALSE" is indeed true in all, and only in all
terminal worlds. (actually "BOX FALSE" is enough)

The interesting formula to find here is a formula with one
variable "p" which would caracterize Kripke frames with no
terminal worlds. Solution : (BOX p -> DIAMOND p). This formula
is verified, for all the valuation of p, iff the frame is
ideal. There are no cul-de-sac at all.
I agree with you that this is needed for any kind of
``immortality" (quantum immortality, comp immortality, etc.).
It is highly interesting to look at it as a kind of abstract
renormalisation.

>In anycase, I haven't got a clue as to how one might start proving
>this, or working out under what conditions it might hold.

With BOX being read as the Godel's BEWEISBAR
arithmetical provability predicate, it can be show that

1) (BOX p -> DIAMOND p) is true
2) BOX (BOX p -> DIAMOND p) is false

Roughly put : comp immortality is a true but not provable
statement for the self-referentially correct machine!
(remember that true but unprovable statements are still "bettable").

Bruno
Received on Wed Dec 08 1999 - 07:50:46 PST