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

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From: Nam Nguyen on 25 Feb 2010 23:17

Frederick Williams wrote:
> Nam Nguyen wrote:
>> [...] And what does your definition have
>> to do with GIT, it being dependent on the naturals as a model of some
>> formal system?
>
> What exactly is GIT here? As I remarked elsewhere [1], Theorem VI [2]
> is entirely syntactic in its statement and proof.

"entirely in its ... proof"? Really? So which formal system is that
"syntactic proof" in? What are the axioms of that particular formal
system? Is that formal system consistent by any chance? How did Godel
"know" that formal system is consistent?

>
> [1] news:4B73F9F6.767498C9(a)tesco.net.
>
> [2] 'On formally undecidable propositions of _Principia Mathematica_ and
> related systems I', pp 596-616 of 'From Frege to G\"odel, a source book
> in mathematical logic, 1879-1931', ed Jean van Heijenoort, Harvard UP.

From: Nam Nguyen on 25 Feb 2010 23:19

Nam Nguyen wrote:
> Frederick Williams wrote:
>> Nam Nguyen wrote:
>>> [...] And what does your definition have
>>> to do with GIT, it being dependent on the naturals as a model of some
>>> formal system?
>>
>> What exactly is GIT here? As I remarked elsewhere [1], Theorem VI [2]
>> is entirely syntactic in its statement and proof.
>
> "entirely in its ... proof"? Really?

I meant: "entirely syntactic in its ... proof"?

> So which formal system is that
> "syntactic proof" in? What are the axioms of that particular formal
> system? Is that formal system consistent by any chance? How did Godel
> "know" that formal system is consistent?

From: Nam Nguyen on 25 Feb 2010 23:33

MoeBlee wrote:
> On Feb 25, 8:52 am, Newberry <newberr...(a)gmail.com> wrote:
>> When we are making logical inferences we are preserving the
>> truth and we are able to formalize the steps syntactically. So when we
>> arrive at the conclusion that G is true how did we fathom it without
>> being able to formalize it?
>
> If you demand formalization, then we can formalize in a meta-theory
> such as Z set theory.

I'm assuming you meant Z set theory is a FOL formal system. (Would my
assumption be correct?)

> That is, e.g., Z set theory proves that the
> Godel sentence for PA is true in the standard model for the language
> of PA, as we also may formally prove (and we can do it in even weaker
> meta-theories than Z set theory) that the the Godel sentence for PA is
> not a theorem of PA.

One perhaps could prove a lot of interesting theorems in the formal system
Z. But the more crucial question here is how would we know Z is consistent
so that the negations of those theorems aren't provable, syntactically
speaking?

From: Newberry on 25 Feb 2010 23:55

On Feb 25, 10:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
> >On Feb 25, 8:25=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> We prove an implication: If PA is consistent, then G. We don't
> >> have a proof (within PA) that PA is consistent, but we believe
> >> it's true. So G follows from the assumption that PA is consistent.
>
> >And we believe that PA is consistent because all its axioms are
> >manifestly true? Is
>
> > "All unicorns have two horns"
>
> >manifestly true?
>
> Which axiom of PA is analogous to "All unicorns have two horns"?

No axiom. But PA uses the same logic. You think that if the axioms are
manifestly true then what the logic itself produces does not have to
be?

And how about

(x)((x # x) -> ( 1 = 2)) ?

Is it manifestly true?

From: Newberry on 25 Feb 2010 23:58

On Feb 25, 9:45 am, MoeBlee <jazzm...(a)hotmail.com> wrote:
> On Feb 25, 11:22 am, Newberry <newberr...(a)gmail.com> wrote:
>
> > And we believe that PA is consistent because all its axioms are
> > manifestly true?
>
> They're clearly true to a lot of people. They seem clearly true to me.
> Which axiom of PA doesn't strike you as true?
>
> Meanwhile, in a theory such as Z, we prove that the axioms of PA are
> true. Though, I don't argue that that, in itself, has much
> epistemological force.
>
> > "All unicorns have two horns"
>
> > manifestly true?
>
> Given ordinary predicate logic, if the sentence is taken as
>
> Ax((x is a horse & x has a horn) -> x has two horns)
>
> then it is clearly true in any model in which there are no horses with
> horns.

"true in any model in which there are no horses with horns" is the
same as "manifestly true"?

>
> What connection do you make with PA?
>
> MoeBlee
>
> That has to do with the logic system you object to. So what?