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

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From: Daryl McCullough on 25 Feb 2010 07:07

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. 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

From: Daryl McCullough on 25 Feb 2010 07:09

Frederick Williams says...

>Daryl McCullough wrote:
>
>> No formula is *absolutely* unprovable (that is, unprovable by
>> any sound theory).
>
>Is that what you meant?

I meant that no formula is absolutely undecidable (neither provable
nor refutable).

--
Daryl McCullough
Ithaca, NY

From: Frederick Williams on 25 Feb 2010 07:12

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.

[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: Daryl McCullough on 25 Feb 2010 07:17

Newberry says...

>On Feb 24, 9:17=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>> Newberry <newberr...(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.
>
>I do not know why you have any interest in knowing why GIT suggests
>that bi-valent logic is impossible, considering that you have already
>made up your mind.

You can be interested in why people *believe* something, without
believing it yourself. GIT does not, in fact, suggest bivalent logic
is impossible, but it might be interesting why you think otherwise.

--
Daryl McCullough
Ithaca, NY

From: Frederick Williams on 25 Feb 2010 07:28

Newberry wrote:

>
> How can you arrive at the conclusion that something is true other than
> by a proof?

If you ever come across a paper of Tarski's called Truth and Proof you
should read it: Scientific American, late sixties. If you want a
precise reference I'll find one for you.

> BTW, there cannot be any intuition or probability that ZFC
> or PA are consistent.

Oh come, come, how do you know what people can intuit?

Why did Dedekind frame Peano's axioms in just the way he did? Because
they squared with his intuitions.