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

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From: James Burns on 24 Feb 2010 10:54

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.

Jim Burns

From: James Burns on 24 Feb 2010 10:54

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?

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.

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

Jim Burns

From: Aatu Koskensilta on 24 Feb 2010 10:56

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

> If you are not assuming classical logic, why are you writing about
> Goedel's incompleteness theorem? That is classical.

G�del's theorem is perfectly constructive.

> Do you know of a version of GIT 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.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Frederick Williams on 24 Feb 2010 11:26

Newberry wrote:
>
> On Feb 24, 3:22 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> > Newberry <newberr...(a)gmail.com> writes:
> > > On Feb 23, 7:53 am, Frederick Williams <frederick.willia...(a)tesco.net>
> > > wrote:
> > >> Newberry wrote:
> > >> > > Frederick Williams wrote:
> > >> > > > This is one of those threads that causes me to think "would that the
> > >> > > > contributors could find something more interesting to discuss."
> >
> > >> > [...] Goedel's incompeteness theorem suggests that two valued
> > >> > logic is impossible.
> >
> > >> Wow! If that were so I'd withdraw my remark.
> >
> > > What remark?
> >
> > There are two quotes attributed to Frederick in the post you replied
> > to. The second quote was about withdrawing a remark and was probably
> > about a prior remark, don't you think?
> >
> > So, look above and see if you can find a remark made by Frederick.
> > Look carefully! Once you see that remark, chances are good that it's
> > the remark he meant.
> >
>
> Your argument is sound and to the point.
>
> I find it refreshing. You do not see it very often on this board.

Yes, thank you Jesse. As for two valued logic, discussion of GIT brings
to mind recursion theory which in turn brings to mind Kleene's three
valued logic which he finds useful for discussing partial recursive
predicates. Applications to partial other things may be of interest to
you. See Kleene's Introduction to metamathematics for details.

From: Aatu Koskensilta on 24 Feb 2010 12:13

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".

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus