Marchal wrote:
> Hi Russell,
> I am glad you borrowed Booloses from a library and that you spent a
> while poring over my thesis. 
> I want just made precise that I have never try to modelise knowledge
> by Bew(|p|).
> This is, actually, a rather sensible point. Most philosopher agree  
> that S4 is a good *axiomatic* of knowledge. Precisely S4 is KT4 + MP,NEC
> or, explicitely (added to the Hilbert Ackerman axioms) :
> AXIOMS      [](A -> B) -> ([]A ->[]B)   K
>             []A -> A                    T
>             []A -> [][]B                4
> RULES       A/[]A    (A & (A->B)) / B    NEC   MP.
> That is, most philosopher (since Plato, but I remember having seen a 
> Buddhist
> similar writing) agree that:
> -if A->B is knowable and if A is knowable, then B is knowable. (K)
> - if A is knowable then A is true.  (T)
> - if A is knowable than that very fact (that A is knowable) is knowable 
> (4)
> Would you agree with that? 4 makes that knowledge somehow introspective.
> Now we will see that if []A represent the formal provability of A, or
> (provability by a sound machine), i.e. Bew(|A|), although 4 and K are 
> verified, we don't have T, that is, we don't have
>                            []A -> A
> provable for all sentence A. Bew(|A|) -> A is not always provable. 
> This entails that formal provability
> cannot and should not be used for the formalisation of knowledge.

Thanks for this extended discussion. It does help a lot, and makes
even more sense if one assumes COMP (which actually I don't, but for
the sake of argument, wil do).

Just one further question. Is it possible for one machine to know p
and another machine to know -p? It seems from the above discussion,
you are only considering consistent machines, which of course
cannot know p and -p simultaneously without being
inconsistent. However, you're not ruling out a society of such
machines who argue over what statements they know to be true (just
like my ardent theists and atheists in Australia - actually this last
example is largely hypothetical - when it comes to religions,
Australians are amongst the most apathetic in the world - an important
fact in us enjoying peace and prosperity).


Dr. Russell Standish                     Director
High Performance Computing Support Unit, Phone 9385 6967                    
UNSW SYDNEY 2052                         Fax   9385 6965                    
Australia                                [EMAIL PROTECTED]             
Room 2075, Red Centre          

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