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

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From: Newberry on 25 Feb 2010 09:40

On Feb 25, 3:55 am, Frederick Williams <frederick.willia...(a)tesco.net>
wrote:
> Newberry wrote:
> > If the theory is sound then what it can or cannot prove will depend on
> > the interpretation.
>
> What do you mean by interpretation? If you mean a model plus a function
> that assigns entities in the model to linguistic entities in the theory,
> then that's not so: proof is a syntactic matter and does not depend on
> interpretation (in the sense I use the word).

How do you propose to define soundness?

> > If there are sentences that are neither true nor
> > false then it will not be able to (dis)prove them.- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 25 Feb 2010 09:52

On Feb 25, 4:07 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Feb 24, 7:54=A0am, James Burns <burns...(a)osu.edu> wrote:
> >> 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?
>
> >You mean "true" and "provable"? Isn't it obvious?
>
> No, it's not obvious at all. Especially since it is false.
>
> 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. You are implying that I have said
that it is obvious that provability in any theory equals truth. I did
not say that.

> Now, maybe you want to say that if we choose the correct
> set of axioms to start with, *then* truth and provability
> are the same for that set of axioms. But what set is that,
> and what criterion do you use to choose that set? I don't
> see any criterion for choosing a set of axioms *other*
> than through the belief that they are true (under a
> particular interpretation).

I would agree with this with some qualification. The axioms are
implicit defiitions.

> The notion of truth has to guide your choice of axioms.
> You can't use "provability from the axioms" as your definition
> of truth if you are trying to figure out which axioms to choose.
>
> >How can you arrive at the conclusion that something is true other than
> >by a proof?
>
> 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?

> >BTW, there cannot be any intuition or probability that ZFC
> >or PA are consistent.
>
> Why? I certainly have such an intuition. How do I arrive at
> such an intuition? Let's take PA. I believe it is consistent
> because all of its axioms are true, and you can't derive
> a contradiction from true statements.
>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: Newberry on 25 Feb 2010 10:04

On Feb 25, 4:07 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Feb 24, 7:54=A0am, James Burns <burns...(a)osu.edu> wrote:
> >> 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?
>
> >You mean "true" and "provable"? Isn't it obvious?
>
> No, it's not obvious at all. Especially since it is false.
>
> 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.
>
> Now, maybe you want to say that if we choose the correct
> set of axioms to start with, *then* truth and provability
> are the same for that set of axioms. But what set is that,
> and what criterion do you use to choose that set? I don't
> see any criterion for choosing a set of axioms *other*
> than through the belief that they are true (under a
> particular interpretation).
>
> The notion of truth has to guide your choice of axioms.
> You can't use "provability from the axioms" as your definition
> of truth if you are trying to figure out which axioms to choose.
>
> >How can you arrive at the conclusion that something is true other than
> >by a proof?
>
> 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.
>
> >BTW, there cannot be any intuition or probability that ZFC
> >or PA are consistent.
>
> Why? I certainly have such an intuition.

Then PA, and I would go as far as to say bi-valent logic is not a
model of your thougth process.
> How do I arrive at
> such an intuition? Let's take PA. I believe it is consistent
> because all of its axioms are true, and you can't derive
> a contradiction from true statements.

This is what Torkel Franzen said. Look, Frege also believed that the
axioms in his system were manifestly true. Anyway, the axioms are not
just "true", they are implicit definitions and you still need to prove
that they are consistent. If they seem intuitively true that is
because they follow natural language usage.

The idea that there are some Platonic truths tha we somehow fathom and
inject in our manifestly true axioms ... well, that is basically
Kant's theory that arithmetic is synthetic a priori. It is not
correct.

>
> --
> Daryl McCullough
> Ithaca, NY- Hide quoted text -
>
> - Show quoted text -

From: Frederick Williams on 25 Feb 2010 10:44

Newberry wrote:
>
> On Feb 25, 3:55 am, Frederick Williams <frederick.willia...(a)tesco.net>
> wrote:
> > Newberry wrote:
> > > If the theory is sound then what it can or cannot prove will depend on
> > > the interpretation.
> >
> > What do you mean by interpretation? If you mean a model plus a function
> > that assigns entities in the model to linguistic entities in the theory,
> > then that's not so: proof is a syntactic matter and does not depend on
> > interpretation (in the sense I use the word).
>
> How do you propose to define soundness?

A theory is sound if all its theorem are true in all models (of the
appropriate type).

From: Daryl McCullough on 25 Feb 2010 11:18

Newberry says...
>
>On Feb 25, 4:07=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> How do I arrive at
>> such an intuition? Let's take PA. I believe it is consistent
>> because all of its axioms are true, and you can't derive
>> a contradiction from true statements.
>
>This is what Torkel Franzen said. Look, Frege also believed that the
>axioms in his system were manifestly true.

Well, he was mistaken. The belief that a system is consistent
can be mistaken.

>Anyway, the axioms are not just "true", they are implicit definitions
>and you still need to prove that they are consistent.

You are confusing two different things: one is why do you believe
that something is true, and how can you know for sure that you haven't
made a mistake somewhere. You can never be absolutely, 100% sure that
there is no mistake in your reasoning. But what's supposed to follow
from that?

>The idea that there are some Platonic truths tha we somehow fathom and
>inject in our manifestly true axioms ... well, that is basically
>Kant's theory that arithmetic is synthetic a priori. It is not
>correct.

Then what does truth mean to you? If it's just provability, then
as I asked before, provability from what axioms? What's the criterion
for choosing the axioms?

--
Daryl McCullough
Ithaca, NY