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

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From: Newberry on 25 Feb 2010 12:22

On Feb 25, 8:25 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
> >On Feb 25, 4:07=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> Look, consider the theory with no axioms. Can you prove
> >> 0 ~= 1? No, you can't. Is it true? Of course, it is. So
> >> truth and provability are not the same. That's obvious.
>
> >The question was why I was trying to eliminate the duality. And I
> >thought that the answer was obvious.
>
> No, it seems obvious to me that there is a distinction between
> truth and provability.
>
> >> As I said, we know that 0 ~= 1 prior to any notion of proof. We
> >> choose our axioms and our proof theory so that true statements
> >> are provable.
>
> >Indeed. 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?
>
> 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?

From: MoeBlee on 25 Feb 2010 12:45

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.

What connection do you make with PA?

MoeBlee

That has to do with the logic system you object to. So what?

From: Daryl McCullough on 25 Feb 2010 13:49

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"?

--
Daryl McCullough
Ithaca, NY

From: Jesse F. Hughes on 25 Feb 2010 14:33

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

> And we believe that PA is consistent because all its axioms are
> manifestly true? Is
>
> "All unicorns have two horns"
>
> manifestly true?
>

It's true if no unicorns exist, obviously.

Let's ask another question, one which you have never answered as far
as I recall. I do not know whether Goldbach's conjecture is true or
false, but I do know that there are no counterexamples to GC that are
less than, say, 27. In fact, this is easy to check. Thus, it seems
to me perfectly reasonable to say that:

(Ax)( if x is a counterexample to GC, then x >= 27 )

That's a true statement, as far as I'm concerned.

How about you? You don't know whether it's true or meaningless unless
you know whether there are counterexamples to GC. Thus, you can't say
whether my proof is a real proof or not, until you determine whether
GC is true. If GC is true, then my proof must not be a proof (since
its conclusion is meaningless). If GC is false, then my proof is a
proof after all.

Is this sensible to you? Is my "theorem" true or meaningless or are
you unable to decide which?

--
Jesse F. Hughes
"And hey, if you're moping and miserable because mathematics tests you,
then maybe, if you think you're a mathematician, you might want to try
a different field." -- Another James S. Harris self-diagnosis.

From: Nam Nguyen on 25 Feb 2010 22:39

MoeBlee 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?

How about the very first axiom listed in Shoenfield's book:

N1. Sx =/= 0

which is _clearly false_ to me at this moment, when I'm thinking of
the (rather "natural") integers!

The moral of the story: "clearly true" is very subjective and has
no firm basis for deciding what's true or false. But model/relation
definition conformance would decide what's true or false. Naturally.