'When Are Relations Neither True Nor False? [Logic]

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From: James Burns on 24 Feb 2010 19:01

Herman Jurjus wrote:
> James Burns wrote:
>> Newberry wrote:
>>
>>> You are assuming classical logic. I am not.
>>
>> If you are not assuming classical logic, why are you
>> writing about Goedel's incompleteness theorem? That
>> is classical.
>>
>> Do you know of a version of GIT that uses a
>> non-classical logic? If you do, I would be grateful
>> for what information you have about it, references,
>> etc. That sounds interesting.
>
> Do you mean something like this?
>
> T. McCarthy, Self-Reference and Incompleteness
> in a Non-Monotonic setting,
> J. Phil. Logic 23 (1994) 423-449.

Yes, thank you. I did mean something like that.
It looks interesting, although it also looks as
though I'll need some time to make my way through it.

I don't know, but it doesn't look like this article
would help Newberry in his search for sentences
that cannot be proven either true or false because
they /aren't/ either true or false.

Jim Burns

From: James Burns on 24 Feb 2010 19:01

Aatu Koskensilta wrote:
> James Burns <burns.87(a)osu.edu> writes:

>>Do you know of a version of [Goedel's incompleteness
>>theorem] that uses a non-classical logic?
>
> There is no such version. We have, rather, that
> GIT uses only very innocuous logical and
> mathematical principles that are both
> constructively and classically valid.

Herman Jurjus has given a reference what may be
a version of GIT using a non-monotonic logic.
I probably won't understand the paper without a
bit of work, but you, Aatu, will very likely fly
where I must crawl. His reference:
T. McCarthy, Self-Reference and Incompleteness
in a Non-Monotonic setting,
J. Phil. Logic 23 (1994) 423-449.

What are necessary, both axioms and allowed deductive
steps, in order for some version of GIT to be
provable in a particular system?

I vaguely remember someone's opinion that, when using
standard logic, Robinson arithmetic is the weakest
system in which that can be done. Whether or not that
is true, that is the kind of answer I am asking for.

Jim Burns

From: Nam Nguyen on 24 Feb 2010 23:35

Aatu Koskensilta wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> He (Newberry) might have a different idea, but how about "it's impossible
>> to completely define some model relations". Would this be some good details
>> you're looking for?
>
> Not really. "It's impossible to completely define some model relations"
> is just as obscure as "two valued logic is impossible".
>

Model relations are just sets.

Let S0 be the set of all counter examples of GC. Let S1 be the infinite subset
of S0 that containing even numbers with 4 as the last digit. Why do you find
it obscure to say it's impossible to completely define S1? Are you asserting
that the definition of S1 is complete?

Btw, in and of itself there's no intrinsic obscurity in the phrase "two valued
logic is impossible": it just means it's impossible to assert the underlying
statement as true or false (without further qualifying the context where the
assertion is taking place). Two examples:

- Events E1 and E2 happened simultaneously.
- G(PA) is provable.

It's only obscure when the phrase "two valued logic is impossible" is used
without a defined context where it's applicable.

From: Newberry on 24 Feb 2010 23:49

On Feb 24, 2:13 pm, Frederick Williams <frederick.willia...(a)tesco.net>
wrote:
> Daryl McCullough wrote:
> > No formula is *absolutely* unprovable (that is, unprovable by
> > any sound theory).
>
> Is that what you meant?

Me? No.

From: Newberry on 25 Feb 2010 00:00

On Feb 24, 7:54 am, James Burns <burns...(a)osu.edu> wrote:
> Newberry wrote:
> > Goedel's theorem states that there is a sentence G
> > such that neither it nor ~G are provable (in a rather
> > large class of formal systems.) This to me suggests
> > non-bivalence. After all if there are sentences ~(T v F)
> > then it is not surprising that neither them nor their
> > negations are not provable.
>
> The only way the existence of a non-provable and
> non-dis-provable sentence G would suggest non-bivalence
> to me is if I were to ignore the difference between
> "true" and "provably true".
>
> This suggests to me that you are trying to eliminate
> that difference. If you are, why are you?

You mean "true" and "provable"? Isn't it obvious? I do not know what
to say. It is like asking why you want to eliminate contradictions.

How can you arrive at the conclusion that something is true other than
by a proof? BTW, there cannot be any intuition or probability that ZFC
or PA are consistent. Only absolute inability to say. If you do have
such an intuition then you arrived at it somehow.

> By the way, although it is true, as you say, that
> it would not be surprising that any sentences ~(T v F)
> would be unprovable, and their negations as well,
> that has no chance of being some kind of explanation
> for Goedel's theorem, because Goedel used classical
> logic.

Not sure what you mean by "explanation." I was not necessarily
offering an explanation.

> Unless you know of some non-classical version of
> Goedel's theorem?
>
> Jim Burns