From: emc@doe.carleton.ca (Eli Chiprout)
Subject: Re: Belief and science (Einstein)
Organization: Dept. of Electronics, Carleton University
Date: Fri, 13 Nov 1992 12:52:58 GMT

israel@unixg.ubc.ca (Robert B. Israel) writes:

>     Even being omnipotent and omniscient doesn't protect you from logical 
>paradoxes.  There are some propositions (of ordinary human language)
>that can't consistently be assigned a truth-value.  The simplest version of
>the Liar Paradox is a statement L that asserts "L is false".  It is
>impossible to assert either that L is true or that L is false without
>contradicting yourself.  This is the case whether you are human or divine.

That is, if you accept logic :-)  It is not a requirement for existing.

>>         Any suggestions?

>     Let's try to do better.  There are some statements that G-d knows.
>I will use GK(A) as an abbreviation for "G-d knows A".  I'll also
>use the following abbreviations for logical operators: ! "not", & "and",
>-> "implies".  The following seem to be a reasonable set of principles:

>For every statement A,
>(a) GK(A) -> GK(GK(A))  (i.e. G-d knows that he knows what he knows).

Which means that GK(GK(A))->GK(GK(GK(A)))->GK^n(A)->...etc.

>(b) !(GK(A) & GK(!A))   (i.e. G-d is consistent).

This sounds fishy!  GK(A) and GK(!A) are not mutually contradictory.
Unless you say GK(A) <-> !GK(!A).  In other words, knowledge of a
statement's truth implies "disknowledge" of its falsehood.  The
problem seems to lie here in that we have not differentiated "knowledge"
of a statement with the assertion of its truthfulness.  If knowledge
implies truth, then we must make that statement.  i.e. 
GK(A) <-> A  and !A <-> GK(!A) <-> !GK(A) which would mean that
since A->!(!A) , GK(A) <-> !GK(!A) as stated above.

>(c) G-d is logical.  If  A follows logically (according
>    to the standard rules of propositional logic) from a set of
>    statements B such that GK(B), then GK(A).

in other words ( GK(B) & B->A ) -> GK(A) or
( GK(A) & A->B ) -> GK(B) or
GK(B) | !GK(A) | (!B & A)        [where | = OR]

>(d) GK((a)) & GK((b))   (i.e. G-d knows those principles).

meaning GK( GK(A)-> GK(GK(A)) )  & GK(!(GK(A) & GK(!A)))

>Unfortunately, these principles can not all be true, for we can derive a
>contradiction.  Let S be a statement that asserts GK(!S).

>(1) S asserts that GK(!S).
>(2) By (a), since this is a statement of the form GK(A), S implies that
>GK(GK(!S)), i.e. that GK(S).
>(3) Thus S implies both GK(!S) and GK(S), which contradicts (b).  
>Therefore we conclude !S.
>(4) Since, as shown above, !S follows logically from (a) and (b),
>and since GK((a)) and GK((b)), by (c) we have GK(!S).
>(5) But that is just what statement S asserts.  So we have deduced 
>a contradiction: S and !S. 

>The cognoscenti will recognize this as a thinly disguised version of
>G\"odel's Second Incompleteness Theorem.  It seems to me that this should
>have an even larger impact on theology than on mathematics.  At least one
>of principles (a) to (d) must be false.  Anyone care to say which one?

See above.

What about working the other way? i.e.  GK(A)->A
If God knows something, then it must be true!  :-)


>-- 
>Robert Israel                            israel@math.ubc.ca
>Department of Mathematics             or israel@unixg.ubc.ca
>University of British Columbia
>Vancouver, BC, Canada V6T 1Y4
--
Eli Chiprout                 
Dept. of Electronics, 
Carleton University, Canada
emc@doe.carleton.ca