On 12 Feb 2009, at 22:12, russell standish wrote:

>
> On Thu, Feb 12, 2009 at 04:48:22PM +0100, Bruno Marchal wrote:
>>
>> Excellent post Johnatan.
>>
>> Of course those who know a bit of AUDA (which I have already  
>> explained
>> on the list) know that from the third person self-reference views we
>> have cul-de-sac everywhere ("we die all the times", cf the
>> "Papaioannou multiverses"), and this is what forces us, when we  
>> want a
>> theory of observation (which by UDA is a probability or credibilty
>> calculus) to define the probabilities by imposing the absence of cul-
>> de-sac. This is *the* motivation for the new box Bp & Dt.  Dt, by
>> Kripke semantics, is equivalent to imposing the absence of cul-de- 
>> sac.
>> Yet, by incompleteness Dt is not provable by the machine, and after  
>> we
>> make the addition of the "non-cul-de-sac" principle (Dt), we loose  
>> the
>> Kripke semantics. But this is a good news, given that we will have to
>> manage (plausibly) continua of "next observer momen or historiest".
>
> I'm a little confused. Did you mean Dp here? Dp = -B-p


Fair question, given my sometimes poor random typo!

In the so called "normal" modal logic, that is those system of modal  
logic containing the formula K

B(p->q) -> (Bp -> Bq)

and which are closed for both the modus ponens rule (from p and p->q  
you can deduce q)  and the rule of necessitation (from p you can  
deduce Bp) , well, if you remind the definition of the Kripke  
semantics, you can see that

Bp & Dp

is equivalent with

Bp & Dt

Bp is true in the world alpha, means that p is true in all the worlds  
Beta accessible from Alpha.

Dp is indeed equivalent to -B-p, it means that B-p is false, so it is  
false that in all the worlds Beta, accessible from Alpha, -p is true  
in them. So it means that there is world, accessible from Alpha,   
where  -p is false, that means that there exists  a world where p is  
true.

Now if you have in a world, your world if you want,  Bp & Dp, you have  
at least access to a world in which p is true, and thus you have  
access to a world where t is true, given that t is true in all worlds.  
So you have Bp & Dt.
The inverse: now you have in a world Bp & Dt.  Thus you have Bp and  
Dt, of course, and by Dt you have access to a world where t is true.  
But you have also Bp, so you know that in all the accessible world,  
from your world you have p, so you have Dp.

A good and important exercise is to understand that with the Kripke  
semantics,  ~Dt, that is B~t, that is Bf, that is "I prove 0=1", is  
automatically true in all cul-de-sac world. It is important because  
cul-de-sac worlds exists everywhere in the Kripke semantics of the  
self-reference logic G.

If you interpret, if only for the fun, the worlds as state of life,  
then Bf is  really "I am dead".

Bruno







>
>
>>
>> Apology for those who have not follow the (many) old modal posts, but
>> we will soon or later come back to this. Read Boolos book (and
>> mathematical logic books).
>>
>> Bruno
>>
>
> -- 
>
> ----------------------------------------------------------------------------
> Prof Russell Standish                  Phone 0425 253119 (mobile)
> Mathematics                           
> UNSW SYDNEY 2052                       hpco...@hpcoders.com.au
> Australia                                http://www.hpcoders.com.au
> ----------------------------------------------------------------------------
>
> >

http://iridia.ulb.ac.be/~marchal/




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