From: israel@unixg.ubc.ca (Robert B. Israel)
Newsgroups: soc.culture.jewish
Subject: Re: Belief and science (Einstein)
Date: 13 Nov 92 09:44:26 GMT
Organization: The University of British Columbia
NNTP-Posting-Host: unixg.ubc.ca

In message <1992Nov12.174400.4716@guinness.idbsu.edu>, holmes@garnet.idbsu.edu
   (Randall Holmes) writes:

> 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).  In order to avoid this phenomenon, one has to
> make the expressive capabilities of one's language _weaker_ (making
> the language more expressive does not help!).  In particular, it can
> be applied to ideas in the mind of God, as follows:

> Let a "concept" in the mind of God be an idea of the form "x has
> property P" (where property P may be utterly inconceivable to us -- it
> might be infinitely complex).  Certainly God is aware of all of His
> concepts.  If a is an object and "x has P" is a concept, we say that a
> satisfies "x has P" exactly if a has P.  Examples of concepts: "x
> loves God" is satisfies by Amir if Amir loves God.  "x loves x" is
> satisfied by Amir if Amir loves Amir (note that the variable x can
> appear in more than one place in the definition of a concept).  Now
> consider the following concept in the mind of God: "x is a concept in
> the mind of God and it is not true that x satisfies x".  Call this
> concept C.  Does C satisfy C?  This asks whether the statement which
> results from substituting C for x in the definition of C is true: "C
> is a concept (true!) and it is not true that C satisfies C".  In other
> words, the question of whether C satisfies C is precisely equivalent
> to the question of whether C does _not_ satisfy C.  This is a
> contradiction.  But it certainly seems reasonable that God is familiar
> with concepts in this sense and that God can conceive C.

>         The solution to the paradox in its normal form (which does not
> mention God!) is as follows: an adequate language will be able to
> express the idea of a concept (by talking about sentences in the
> language and substituting names in the language for the variable x)
> and thus the only ingredient of the argument which is open to
> challenge is the idea that we can define truth for sentences of a
> given language in that language (which blocks the definition of C).
> This solution seems unsatisfactory for the theological form of the
> paradox.  God must know the truth or falsehood of any proposition.
            ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
     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.

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

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?

-- 
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