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: rereRe: The end of god
Message-ID: <1994Nov8.031813.923@oracorp.com>
Organization: Odyssey Research Associates, Inc.
Date: Tue, 8 Nov 1994 03:18:13 GMT
Lines: 155

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

>daryl@oracorp.com (Daryl McCullough) writes:
>
>>...Jim started by saying "Suppose there is a consistent
>>axiomatic system S". Whatever view of truth you are using, surely
>>the truth of "S is consistent" follows from the assumption "S is
>>consistent".
>
>Whatever theory of truth I am using, the proposition "S is
>consistent" can only have a meaning if interpreted according to that
>theory of truth.

Right.


>>I would say that truth of a statement is only meaningful given an
>>*interpretation* of the terms in the statement. However, in the case
>>Jim is talking about, I believe that it is clear that the
>>interpretation of "is consistent" in the statement "S is consistent"
>>is to be the same interpretation of "is consistent" as in the
>>assumption "Suppose that S is a consistent axiomatic system."
>
>Basically, you are accepting a Platonist view

No, I am not. I am just saying that, under the assumption that A is
true, it follows that A is true. Let A be "System S is consistent".

>and insisting that, even if I deny Platonism, the notion of truth must
>still be based on your Platonist view.  I reject that.

Go right ahead. But you're not rejecting what I said, since I didn't
mention Platonism.


>>Formal systems don't determine the truth of statements, they determine
>>the provability of statements.
>
>If one has a theory of truth based on formal systems, then truth has
>no meaning outside a formal system. All you are doing is denying the
>possibility of such a notion of truth. To a mathematical formalist,
>for example, there is no other notion of mathematical truth.

Yes there is. Even a formalist can determine the truth of "There's a
bottle of milk in the refrigerator." I think you are misrepresenting
formalism here, I really don't think that formalism is an attempt to
give a purely formal notion of truth, it is, instead, an approach to
doing mathematics without *any* notion of truth.

But anyway, getting back to the subject, we started out talking about
the issue of the consistency of theories. My point is that
"consistency" is not simply a one-place predicate symbol with an
arbitrary set of axioms. Consistency has an intended interpretation.
It is about the question of whether it is possible to prove "false"
starting from a set of axioms and rules of inference. To answer the
question "Is system S consistent?" with "it depends on what formal
system you are using" makes no more sense than answering the question
"Is there milk in the refrigerator?" with "it depends on what formal
system you are using".

This is not to say that there is a definite, Platonic answer. Maybe
the question of whether there is milk in the refrigerator has no truth
value until I look. Who knows?

>>If S is consistent, then there is no formal system encoding and
>>proving the proposition that S is inconsistent.

>Again, your argument is based on your own assumption which is
>apparently Platonist.

Again, you are wrong. You don't have to believe in Platonism, the
fact that I quote follows from the meanings of the phrases
"S is consistent" and "S' encodes the proposition that S is consistent".
If you want to relativize the truth of these statements, that's fine.
My claim doesn't depend on any notion of absolute Platonic truth.

>>                                                This is a tricky
>>point. Jim started with the assumption:
>
>>    Suppose there is a consistent axiomatic system S with sufficient
>>    expressibility to encode the proposition "S is consistent".
>
>>You need to ask what it *means* to say that one formal system S' can
>>encode the proposition that another formal system S is consistent (S'
>>may be the same system as S or a different one). Usually, that is
>>taken to mean the following:
>
>>     1. There is a way to code statements of S as terms of S'.
>>     2. There is a one-place predicate P(x) definable in S'
>>        such that for every statement A in the language of S,
>>        S' proves P(#A) if and only if S proves A (where #A is the
>>        code of A).
>
>This seems wrong.  You seem to be saying that if S' is an extension
>of S, that a proposition statable in S has no proof in S' unless it
>already has a proof in S.

You are misinterpreting what I said. The formula P(x) is intended to
express the proposition that x is the code of a provable statement of
S. The conditions 1 and 2 above are the criteria under which one system,
S', can encode the notion of provability in another system, S. This
is necessary in order to say that S' can prove that S is consistent.

So, once again, P(x) formalizes provability in S, not provability in
S'.

>If I am correctly interpreting what you
>are claiming, then I believe that your assertion contradicts Goedel.

Nope.

>>These conditions imply that provability in S can be represented in S'.
>>Given these conditions, S' can encode "S is consistent" as:
>
>>        con(S) == not P(#false)
>>where false is any logical contradiction.
>
>>Now, by condition 2. if provability in S is representable in S', then
>>S' proves P(#false) if and only if S proves false. That shows that S'
>>cannot prove S is inconsistent unless S really is inconsistent.
>
>You forget that S' cannot prove that S' is consistent.

No I didn't. That's one of those facts that I never go a single day
without thinking about.

>It might well be that S' is inconsistent.

That's not possible, unless S is also inconsistent. By condition 2, S'
proves P(#A) if and only if S proves A. Since an inconsistent system
proves everything, it follows that S' proves P(#A) for all A, and so
S proves A for all A, so S is inconsistent, as well. That was what I
said: S' cannot prove S is inconsistent unless S really is inconsistent.
 

>Quite apart from this point, your
>argument is flawed because it again assumes something akin to
>Platonism.

No, it doesn't. 

>That is, you make the assertion ".. unless S really is
>inconsistent" and thereby demonstrate that you assume this has a
>meaning outside a formal system.

Look, if you want to relativize the truth, and replace it by something
purely formal, like "provability in ZFC", say, the same thing follows:
You can prove in ZFC that

     S' proves S is inconsistent <-> S is inconsistent

Daryl McCullough
ORA Corp.
Ithaca, NY

