Re: tautology

From: Russell Standish <>
Date: Thu, 9 Dec 1999 10:02:04 +1100 (EST)

> 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"

I'm using LaTeX symbols. In English, the above phrase reads "is a
theorem in world alpha of model W".

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

Consider the following diagram:

               /------ [False]
   [\Box False]{--- [False]

I was reading this as saying "False is a true statement", therefore
\Box False is true is the predecsessor world. Does one rule out
statements like "False is a true statement" from the picture utterly?

> 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.

True, but idealism is a sufficient, but not necessary condition for
absence of cul-de-sacs. One can have a model in which every terminal
world is preceded by a world that acesses non-terminal states. Such a
model is required for QTI to hold.

> 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

Dr. Russell Standish Director
High Performance Computing Support Unit,
University of NSW Phone 9385 6967
Sydney 2052 Fax 9385 6965
Room 2075, Red Centre
Received on Wed Dec 08 1999 - 15:02:23 PST

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