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

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From: Newberry on 28 Mar 2010 00:46

On Mar 27, 4:59 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
> >On Mar 26, 3:49=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> The Liar sentence is not *expressible* in any standard mathematical
> >> theory (PA or ZFC). So you don't have to do anything to keep the Liar
> >> from spoiling the consistency of those languages.
>
> >Why you think you have to tell me that I do not know. If you lool a
> >few lines above you will see that I was talking about the Liar paradox
> >in the natural language.
>
> So your theory of truth gaps is only for natural language?

No.

> So you
> agree that formal languages such as arithmetic don't require any
> truth gaps?

No.
>
> --
> Daryl McCullough
> Ithaca, NY

From: Newberry on 28 Mar 2010 00:49

On Mar 27, 6:26 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > On Mar 26, 3:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> >> Newberry <newberr...(a)gmail.com> writes:
> >> > On Mar 25, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >> > wrote:
> >> >> Newberry says...
>
> >> >> >Tarski's theorem does not apply to formal systems with gaps. I think
> >> >> >it is preferable.
>
> >> >> If you the way you express Tarski's theorem is like this, then truth
> >> >> gaps don't change anything:
>
> >> >> There is no formula T(x) such that if x is a Godel code of a true
> >> >> sentence, then T(x) is true, and otherwise, ~T(x) is true.
>
> >> >> Anyway, *why* is it preferable to have a formal system for which Tarki's
> >> >> theorem does not apply? Preferable for what purpose?
>
> >> > If truth is expressible then truth can be equivalent to provabilty.
>
> >> So, you'd like to redefine truth (so that vacuously *true* statements
> >> aren't true) and also redefine provability (so trivially provable
> >> statements aren't provable) in such a way that truth is equivalent to
> >> provability.
>
> >> Then what have you accomplished? Hell, I can do that simply by
> >> requiring that nothing is true and nothing is provable. My "fix" is
> >> better than yours, insofar as we can see that it actually "works".
>
> > My theory has some significant advantages over yours. I can go to a
> > grocery store and count how many tomatoes and bananas I have picked.
> > If I have picked 2 small tomatoes and three large tomatoes my theory
> > can prove that I have 5 tomatoes. Also at the checkout counter I can
> > calculate the total price. Can your theory do that?
>
> No. You're right. The classical theory of arithmetic

I thought that we were talking about your theory where "nothing is
true and nothing is provable."

> is incapable of
> proving that 2 + 3 = 5.
>
> I see now that your theory is superior and will alter my brain
> accordingly.
>
> (Honestly, I have no idea what you're talking about. You seem to see
> a disadvantage in classical arithmetic that I simply don't see. Why
> not explain your point?)
>
> --
> Jesse F. Hughes
>
> "I post for many reasons [...] and there's no reason to think that
> I'll stop." -- James S. Harris- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 28 Mar 2010 00:57

On Mar 27, 4:56 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
> >L: ~T(L)
>
> >If v(L) = ~(T v F) then there is no contradiction. L is not true.
>
> But *if* T is a truth predicate, then "L is not true" is formalized
> by the statement ~T(L).
>
> >The argument usually goes "but that is what L says." But L does not
> >say anything.
>
> It says "L is not true".
>
> So your proposed resolution is complete nonsense.

It contains the string "L is not true", but it does not "say" that L
is not true because it is meaningless. Some prefer to say that it does
not express any proposition. Here is some reading for you.

http://www.columbia.edu/~hg17/gaifman6.pdf

From: Newberry on 28 Mar 2010 01:16

On Mar 27, 11:10 am, Alan Smaill <sma...(a)SPAMinf.ed.ac.uk> wrote:
> Nam Nguyen <namducngu...(a)shaw.ca> writes:
> > Daryl McCullough wrote:
> >> Nam Nguyen says...
> >>> Daryl McCullough wrote:
>
> >>>> [G(PA)] is a *relative* truth. It's true in the standard interpretation
> >>>> of the language of PA.
> >>> So you've agreed "G(PA) can be arithmetically false"?
>
> >> It is false in nonstandard models of PA.
>
> > Why don't we make it more precise. When we say F, a formula written
> > in the language of arithmetic, is true or false _by default_ we
> > mean it's being arithmetically true or false: i.e. true or false
> > in the natural numbers. So we're *not* talking about F is being
> > true/false in any general kind of models here.
>
> You haven't (and can't) give me an effective way to use
> this definition to decide truth or falsity in the natural numbers.

But I can. In a system with gaps Tarski's theorem does not apply. We
can then simply equate truth with provability.

> > The point is Alan said he wouldn't know what an absolute truth
> > of G(PA) would mean
> > and I've implicitly defined it for him, and
> > here is the explicit version:
>
> > There's _no other_ context in which the meta statement
> > "G(PA) is arithmetically true in the natural number" would
> > be false.
>
> I'm not obliged to accept any particular definition of yours.
>
> FWIW you can look at Bourbaki's account of Goedel's incompleteness
> theorem to note that they studiously avoid saying that the goedel
> sentence is true (arithaally, absolutely, or in any other way).
>
> > The question I was hoping you'd answer one way or another is
> > whether or not there's a context in which the meta statement:
>
> > "G(PA) is arithmetically true in the natural number"
>
> > would be _false_ ?
>
> > If your answer is "yes", then the [arithmetically-in-the-natural-
> > number] truth of G(PA) is a relative notion. Otherwise it's an
> > absolute notion.
>
> Why should I believe the only answers are no or yes?
> Your realist assumptions are showing.
>
> > Which answer would you have? And perhaps why?
>
> you need a bit more Zen.
>
> --
> Alan Smaill- Hide quoted text -
>
> - Show quoted text -

From: Daryl McCullough on 28 Mar 2010 07:37

Newberry says...
>
>On Mar 27, 4:56=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:
>> Newberry says...
>>
>> >L: ~T(L)
>>
>> >If v(L) =3D ~(T v F) then there is no contradiction. L is not true.
>>
>> But *if* T is a truth predicate, then "L is not true" is formalized
>> by the statement ~T(L).
>>
>> >The argument usually goes "but that is what L says." But L does not
>> >say anything.
>>
>> It says "L is not true".
>>
>> So your proposed resolution is complete nonsense.
>
>It contains the string "L is not true", but it does not "say" that L
>is not true

That's completely silly.

--
Daryl McCullough
Ithaca, NY