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

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From: Aatu Koskensilta on 24 Feb 2010 12:17

Newberry <newberryxy(a)gmail.com> writes:

> Anyway, whethere GIT suggests that two valued logic is impossible or
> not is not essential to my main point.

Well, I don't really have any interest in your main point. I was just
wondering why the incompleteness theorem suggests to you that "two
valued logic is impossible". Apparently it's impossible to say.

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

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

From: Herman Jurjus on 24 Feb 2010 13:52

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.

--
Cheers,
Herman Jurjus

From: Daryl McCullough on 24 Feb 2010 15:31

Newberry says...
>
>On Feb 24, 7:14=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >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.
>>
>> *Why* does it suggest non-bivalence? If you have a theory that is
>> incapable of proving 0 ~=3D 1 (and also incapable of proving its
>> negation), does that suggest non-bivalence? Not to me; it only
>> suggests that the theories axioms are not strong enough to prove
>> some interesting true statements. At what point would you ever
>> be justified in saying, for any formula Phi, "It's not that our
>> axioms are inadequate to prove Phi or ~Phi, it's that Phi is
>> neither true nor false"?
>
>a) Goedel shows that the axioms CANNOT be made strong enough.

I would say that there is a quantifier ordering ambiguity at
work here:

Godel's theorem says that for every theory T of the right type,
there is a formula Phi such that neither Phi nor ~Phi is provable
in T. It doesn't say: There is a formula Phi such that for any
theory T of the right type, neither Phi nor ~Phi is provable.

In other words, the fact that neither Phi nor ~Phi is provable
in a *particular* theory T does not suggest that Phi is "neither
true nor false", because there will always be some other theory T' in
which it *is* provable.

No formula is *absolutely* unprovable (that is, unprovable by
any sound theory).

--
Daryl McCullough
Ithaca, NY

From: Frederick Williams on 24 Feb 2010 17:13

Daryl McCullough wrote:

> No formula is *absolutely* unprovable (that is, unprovable by
> any sound theory).

Is that what you meant?

From: Frederick Williams on 24 Feb 2010 17:17

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.

If it was a case of neither G nor ~G are true it might.