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

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From: Nam Nguyen on 20 Mar 2010 13:28

Nam Nguyen wrote:
> Nam Nguyen wrote:
>> Daryl McCullough wrote:
>
>>> It doesn't seem nontrivial to me. It's a one-step proof from
>>> Ay ~(S(y) = 0)
>
>>> It's not the sort of fact about numbers that would lead anybody
>>> to care about number theory. You say you find it interesting,
>>> and I certainly can't know that you don't, but I don't find it
>>> interesting.
>>
>> In summary, your challenge would never make sense, because "interesting"
>> and non-technical "nontrivial" is a *subjective* terminologies, but we
>> were talking about _technical_ merits of the 4 principles.
>
> My guess is that you meant to ask me to give an example of a proof, in a
> reasoning systems under the guidelines of the 4 principles, that's is
> sufficiently complex (i.e. "nontrivial") but is of interest, e.g. GIT.
>
> If that's case, and you have to say if it is or isn't, then an example
> of a "nontrivial" metamathematical theorem would be the following modified
> GIT:
>
> (1) _If_ PA is consistent, then for any consistent formal system T
> sufficient strong to carry out basic notions of arithmetic,
> there's a formula G(T) which is syntactically undecidable in T
> but of which a certain encoded formula, say, encoded(G(T)) is
> provable in PA.
>
> Would (1) be an interesting theorem? To me it's not. But it's a result from
> _valid_ reasoning. Would GIT be an interesting theorem? It might be to
> you, but it's a result from _invalid_ reasoning, as stipulated by the 4
> principles.
>
> My point is good reasoning is a matter of correctly conforming to certain
> clearly stated and defined guidelines, not a matter of relying on mere
> intuition that could turn out to be incorrect or could change be changed
> at will.
>
> *The natural numbers collectively is a mere intuition* that either we can't
> define precisely or we can re-define at any moment. It's not a fixed notion
> and hence can't be used as foundation of reasoning, as we tend to believe
> so post 1931.
>

One can, for example, define a natural-number formal system N as that which
extends Q and in which either GC or "there are infinitely many counter
examples of GC" is provable.

Would such N be inconsistent? Who knows! But if in all reasonings thereafter
we take into account the _fact_ that we don't know and that it might be
impossible to know that, then our reasoning can't be invalid.

Otherwise, we'd not know what we're talking about when we claim a certain
this or that, such as G(PA), is true.

From: Daryl McCullough on 20 Mar 2010 14:54

Nam Nguyen says...
>
>Daryl McCullough wrote:
>> Nam Nguyen says...
>>> MoeBlee wrote:
>>
>>>> What you did is to give an example of an uninteresting theorem in Q.
>>> Apparently you didn't understand the short conversation. First, Daryl
>>> asked me to give an example of a _nontrivial_ theorem. I gave him
>>> just that.
>>
>> The theorem was: ExAy[~(Sy=x)].
>>
>> It doesn't seem nontrivial to me. It's a one-step proof from
>> Ay ~(S(y) = 0)
>
>Whether or not you used "nontrivial" in technical sense, you were
>still wrong on multiple level here. If your "nontrivial" was technical
>then Ay ~(S(y) = 0) is a trivial theorem in Q but ExAy[~(Sy=x)] isn't.

Yes, it certainly is.

>If your "nontrivial" was non-technical then you were asking me for
>an "interesting" example which is *subjective* and as such my example
>satisfied your small challenge here because it was interesting to me

I don't believe you.

--
Daryl McCullough
Ithaca, NY

From: Nam Nguyen on 20 Mar 2010 15:13

Daryl McCullough wrote:
> Nam Nguyen says...
>> Daryl McCullough wrote:
>>> Nam Nguyen says...
>>>> MoeBlee wrote:
>>>>> What you did is to give an example of an uninteresting theorem in Q.
>>>> Apparently you didn't understand the short conversation. First, Daryl
>>>> asked me to give an example of a _nontrivial_ theorem. I gave him
>>>> just that.
>>> The theorem was: ExAy[~(Sy=x)].
>>>
>>> It doesn't seem nontrivial to me. It's a one-step proof from
>>> Ay ~(S(y) = 0)
>> Whether or not you used "nontrivial" in technical sense, you were
>> still wrong on multiple level here. If your "nontrivial" was technical
>> then Ay ~(S(y) = 0) is a trivial theorem in Q but ExAy[~(Sy=x)] isn't.
>
> Yes, it certainly is.

Why, given Ay ~(S(y) = 0) is an an axiom of Q? What do you think the
technical definition of trivial proof be?

>
>> If your "nontrivial" was non-technical then you were asking me for
>> an "interesting" example which is *subjective* and as such my example
>> satisfied your small challenge here because it was interesting to me
>
> I don't believe you.

I don't go by belief in technical argument. If you ask me a _subjective_
opinion and I already gave one (about certain uni-directional flow of
provability) and you don't believe it, that's your problem and not
anybody else's.

For what it's worth, try to be _reasonable_ and stay on technical ground
and not to begin on the unreasonable path about what _subjectively_ is or
isn't interesting. (That would lead your reasoning to go nowhere worth,
naturally).

>
> --
> Daryl McCullough
> Ithaca, NY
>

From: Daryl McCullough on 20 Mar 2010 17:30

Nam Nguyen says...

>I don't go by belief in technical argument.

There is no technical argument at play. You reject standard
mathematics for subjective reasons, and I reject your mathematics
for other subjective reasons: it doesn't sound like it produces
any theorems of interest to anyone.

--
Daryl McCullough
Ithaca, NY

From: Nam Nguyen on 20 Mar 2010 19:01

Daryl McCullough wrote:
> Nam Nguyen says...
>
>> I don't go by belief in technical argument.
>
> There is no technical argument at play.

There is a technical definition of truth in a model. I'm sure you
know that.

> You reject standard
> mathematics for subjective reasons,

I reject that because your "belief" in the natural number is just
that and doesn't conform with your own side's technical definition
of truth in model.

> and I reject your mathematics
> for other subjective reasons: it doesn't sound like it produces
> any theorems of interest to anyone.

Did you happen to heear hear me mentioning something like G2IT
(Godel-Goldbach Incompleteness Theorem of Knowledge) in other
thread sometimes ago. It "sounds" interesting to you isn't it?
No? It's hard to reason with you if you already have some _subjective_
"bad" preconception and shut your mind out of objective analysis
of methods of reasoning. Iow, it could be interesting, Daryl, if
you further objectively investigate it.

You're free of course to reject anything as you wish, including the
reasoning principles that I suspect you don't have valid reasons
to think of them as bad.

All you do is staying unreasonable and attacking the messenger, instead
of technically confronting the 4 principles with technical analysis
and reasons.