Newsgroups: comp.ai.philosophy
Path: cantaloupe.srv.cs.cmu.edu!das-news2.harvard.edu!news2.near.net!howland.reston.ans.net!pipex!uunet!psinntp!scylla!daryl
From: daryl@oracorp.com (Daryl McCullough)
Subject: Re: Godel, Lucas, Penrose, and Putnam
Message-ID: <1994Dec27.182636.592@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Tue, 27 Dec 1994 18:26:36 GMT
Lines: 87

rickert@cs.niu.edu (Neil Rickert) writes:

>chalmers@bronze.ucs.indiana.edu (David Chalmers) writes:
>
>>Yes, it is interesting.  Sometimes I want to believe that I am sound, but
>>then what do I do when confronted with reasoning of the above sort about my
>>own Godel sentence?  Strange things happen: for example, sometimes I find
>>myself unwilling to go through with modus ponens at a key step, wanting to
>>believe P and P -> Q but not Q.  (e.g. wanting to accept "I am sound" and
>>"I am sound implies G", but not wanting to accept G, because it is my very
>>acceptance of G that gives rise to the problems).  It's enough to give
>>anyone pause.
>
>This reaction is strange.  Surely the correct answer is that you do
>not even know what G actually is. It is probably some horribly
>complex statement involving lots of quantifiers.  Most likely it is
>the kind of proposition that you would not have the least interest in
>proving.

The complexity of G is really beside the point. G is complex in the
language of Peano Arithmetic because of the extremely small number of
primitives. To formalize G, it is necessary first to show how to code
statements as numbers and to formalize "theoremhood" as a predicate on
numbers.

For example, in English, it is easy (with a few definitions) to do the
equivalent of Godel's theorem for the theory of the unassailable
beliefs of Roger Penrose:

We will say that Roger Penrose accepts a string as unassailably true
if he believes the string expresses an unassailably true proposition.
This rules out (1) strings that are not believed by Penrose to be
meaningful, (2) strings that he believes to be meaningful, but false,
and (3) strings that he considers meaningful, but whose truth value
is unknown to him.

Now, define the "diagonalization" of a string S to be the result of
replacing all instances of the string "this sentence" occurring in S
by S itself, enclosed in quotes. For example, the diagonalization of

       this sentence has five words
is
       `this sentence has five words' has five words

Now, let G0 be the following string:

       the diagonalization of this sentence will never
       be accepted as unassailably true by Roger Penrose

Then let G be the diagonalization of G0:

       the diagonalization of `the diagonalization of
       this sentence will never be accepted as unassailably
       true by Roger Penrose' will never be accepted as
        unassailably true by Roger Penrose

The string G expresses the statement that a certain syntactic
operation on strings (taking the diagonalization of the string G0)
produces a string which will never be accepted as unassailably true by
Roger Penrose. It is easy to see that, by definition of
diagonalization, the result of the operation on strings is to
reproduce G itself. Therefore, G expresses the proposition that G
itself will never be accepted as unassailably true by Roger Penrose.

It is easy to see that if Penrose ever *does* accept G as unassailably
true, then his unassailably truths will contain some false statements,
and thus will be unsound. (Since G says that Penrose will never accept
G, it follows that if Penrose *does* accept G, then G must be false.)
Therefore, it follows that *if* Penrose' reasoning is sound, then he
can never accept G. Since G says that Penrose will never accept G, it
follows that if Penrose' reasoning is sound, then G must be true, but
Penrose can never be unassailably convinced that it is true.

Of course, in this sort of formulation, the complexity of G itself is
not at issue, but the complexity of the terms, such as "Penrose
accepts ... as unassailable" is. The only rigorous way to say what
this means is to have a theory of mind in which one can derive what
someone will or will not believe as unassailably true. There is no
such theory of mind, so far, but Penrose hopes that there will be
someday. What this Godelian argument shows is that the possibility of
a rigorous theory of mind seems to be incompatible with Penrose'
conviction that his own reasoning must be sound.

Daryl McCullough
ORA Corp.
Ithaca, NY

