From: Chas F Brown <cbrown@cbrownsystems.com>
Subject: Re: Contradiction inherit in the notion of omniscience?
Date: Tue, 29 Aug 2000 20:01:40 -0700
Newsgroups: sci.math
Summary: [missing]

Robert Israel wrote:

<snip>
> The following seem to be a reasonable set of
> principles:
>  For every statement X,
>   (a) AK(X) -> AK(AK(X))  (i.e. A knows that he knows what he
>     knows).
>   (b) !(AK(X) & AK(!X))  (i.e. A is consistent).
>   (c) A is logical.  If X follows logically (according
>     to the standard rules of propositional logic) from a set of
>     statements Y such that AK(Y), then AK(X).
>   (d) AK((a)) & AK((b))  (i.e. A knows those principles).
> 
> Unfortunately, these principles can not all be true, for we can
> derive a contradiction.  Let S be a statement that asserts
> AK(!S).
> 
> (1) S asserts that AK(!S).
> (2) By (a), since this is a statement of the form AK(X), S implies
> that AK(AK(!S)), i.e. that AK(S).
> (3) Thus S implies both AK(!S) and AK(S), which contradicts
> (b). Therefore we conclude !S.
> (4) Since, as shown above, !S follows logically from (a) and (b),
> and since AK((a)) and AK((b)), by (c) we have AK(!S).
> (5) But that is just what statement S asserts.  So we have deduced
> a contradiction: S and !S.
> 

Unfortunately, this argument has nothing to do with omniscience (or
perhaps one should say is independent of omniscience), since we can
append as an axiom:

(e) exists T such that !AK(T) & !AK(!T)

i.e., A is not omniscient, and the same logic follows. Likewise, we can
change (e) to (e'):

(e') !(exists T such that !AK(T) & !AK(!T))

i.e., A is omniscient, and we still get the same contradiction.

Instead the problem appears to be statement (b). Replace statement (b)
with (e') so that we're talking about an omniscient being, and make (d)
read "AK((a)) & AK((e'))"; then assume (b); by the above argument
(discarding for the moment the problem of specifying S asserts AK(!S))
we then get

AK(exists X such that AK(X) & AK(!X))

i.e., that AK(!(b)).

I don't like the conclusion (since I'd tend to *want* (b) from my
omniscient being), but who am I to argue with an omniscient being?

Cheers - Chas

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C Brown Systems Designs
Multimedia Environments for Museums and Theme Parks
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