From: israel@unixg.ubc.ca (Robert B. Israel)
Newsgroups: soc.culture.jewish
Subject: Re: Belief and science (Einstein)
Date: 17 Nov 92 18:48:47 GMT
Organization: The University of British Columbia
NNTP-Posting-Host: unixg.ubc.ca


In <1992Nov13.173612.21953@guinness.idbsu.edu>, holmes@garnet.idbsu.edu (Randall Holmes) writes:

> Robert Israel has left out the crucial step of demonstrating that
> there is such a proposition.  This can be adapted from my posting:

> Let Px be a statement with a variable in it, and define "Px holds of
> a" as Pa.  Now consider "GK(x does not hold of x) holds of GK(x does
> not hold of x)" Carrying out the appropriate substitutions reveals
> that this sentence asserts that God knows that it is not true.

This is one way to get a "purely logical" version of the statement S.  In
the context of a formal language, where you have no external props, this
is indeed crucial.  But in ordinary discourse, I don't think that it's 
really needed.  Self-referential statements occur all the time.  For example,
in the Torah: "and these words that I command to you this day..."  If you
want more precision than just "G-d knows the negation of this statement",
try this one:
  "G-d knows the negation of the statement in <israel.722026127@unixg.ubc.ca>
that contains the string 'QQQXYZ'"
  
In <emc.721659178@tristan>, emc@doe.carleton.ca (Eli Chiprout) writes:

>>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.

This could be a problem for a human, since you might argue that we only have
the capacity to "know" a finite number of bits of information at a time.  But
surely G-d would have no problem with it.

>>(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.
  
To say GK(A)&GK(!A) is not a contradiction for _us_, but it would mean that
G-d's "knowledge" contains a contradiction.  In particular, being able to
make logical inferences from the statements A and !A, G-d would "know" every
statement, true or false (A, therefore !A -> B, but !A, therefore B).  So
with (c) but not (b), GK would not be a very useful predicate.  
  
I would hope that GK(A) -> A, i.e. everything G-d knows is true, but that's 
not necessary for the argument.  I certainly don't want to assume 
GK(A) <-> A, because as Randall Holmes points out in <1992Nov12.174400.4716@guinness.idbsu.edu>,

>It is mathematically provable that the class of all true sentences in
> a reasonable language is impossible to define in that language
> (Tarski's Paradox).
   
-- 
Robert Israel                            israel@math.ubc.ca
Department of Mathematics             or israel@unixg.ubc.ca
University of British Columbia
Vancouver, BC, Canada V6T 1Y4
Newsgroups: soc.culture.jewish