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

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From: Nam Nguyen on 5 Apr 2010 00:12

Marshall wrote:
> On Apr 4, 6:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> It wouldn't be a surprise if we learn there were those who would
>> defend physics against SR to the bitter end.
>
> This is somewhat off-topic, and I may be misreading you,
> but am I to understand from this sentence that you are
> of the opinion that special relativity is somehow incorrect,
> or not the best currently available theory?

No it's not off the topic: I was using SR as analogy to the
anti-natural-number position some might have held (including
me in this case). I I didn't mean at all SR is incorrect. In

It wouldn't be a surprise if we learn there were those who would
defend physics against SR to the bitter end. I think there are those
today who'd similarly defend "the natural numbers" foundation in
mathematical logic - to the bitter end - even though to no avail.

>
>
> Marshall
>

From: Nam Nguyen on 5 Apr 2010 00:14

Please disregard this post. I accidentally sent it when I was
just editing it. Thanks.

Nam Nguyen wrote:
> Marshall wrote:
>> On Apr 4, 6:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>> It wouldn't be a surprise if we learn there were those who would
>>> defend physics against SR to the bitter end.
>>
>> This is somewhat off-topic, and I may be misreading you,
>> but am I to understand from this sentence that you are
>> of the opinion that special relativity is somehow incorrect,
>> or not the best currently available theory?
>
> No it's not off the topic: I was using SR as analogy to the
> anti-natural-number position some might have held (including
> me in this case). I I didn't mean at all SR is incorrect. In
>
>
> It wouldn't be a surprise if we learn there were those who would
> defend physics against SR to the bitter end. I think there are those
> today who'd similarly defend "the natural numbers" foundation in
> mathematical logic - to the bitter end - even though to no avail.
>
>>
>>
>> Marshall
>>

From: Nam Nguyen on 5 Apr 2010 00:19

Marshall wrote:
> On Apr 4, 6:12 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> It wouldn't be a surprise if we learn there were those who would
>> defend physics against SR to the bitter end.
>
> This is somewhat off-topic, and I may be misreading you,
> but am I to understand from this sentence that you are
> of the opinion that special relativity is somehow incorrect,
> or not the best currently available theory?

No it's not off the topic: I was using SR as analogy to the
anti-natural-number position some might have held (including
me in this case). Also, I didn't mean at all SR is incorrect.
By "defend physics against SR" I meant there were people in the
past who would defend the then "current" physics at the time
when SR was introduced.

From: Nam Nguyen on 5 Apr 2010 01:08

Marshall wrote:
> On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Marshall wrote:
>>> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>>> Let me put to rest the idea we know enough about the natural numbers,
>>>> to prove important thing such as the consistency of PA. I'll do that
>>>> by pointing out the existence of a specific unknown natural number.

>>> Why do you think the existence of a specific unknown number
>>> should have anything to do with consistency?

>> Because they (the syntactical proof of consistency and collectively
>> many formulas about this unknown natural) both connote the same thing
>> in meta level: impossibility of syntactical proof. If you can't prove
>> a certain formula related to this number, you can forget about proving
>> a consistency, syntactically speaking.
>
> Both of these sentences are just a restatement of the position that
> I asked you for support of. My question is not, do you consider
> a proof of consistency of PA and the existence of an unknown natural
> number to be mutually incompatible. You have clearly stated so.

Huh? I'm not quite following you here. First of, you clearly asked me
why the existence of a specific unknown natural number "should have
anything to do with consistency [proof]". And I gave you an answer for
that: their both connoting an "impossibility of [some] syntactical proof"
would be their "having anything to do" with each other.

That's all I said here. And I didn't state anything about them being
"mutually incompatible" or what not!

> Rather, my question is WHY do you see them as mutually
> incompatible.

Again where did I even hint these 2 are "mutually incompatible"?
In fact, who know, some of the formulas about this unknown number
might be logically equivalent to, say, CONT(PA)!

> Is this some theorem of which I am unaware?
> If so, can you provide a reference? Or is it some theorem
> that you yourself have proved? If so, can you provide the
> proof for our inspection?
>
> Is it only *syntactic* consistency proof that are so affected?

Since I didn't say anything about them being incompatible, I obviously
wouldn't able to answer these questions.

> What about model-theoretic proofs of consistency. I seem
> to recall that you do not accept them, but I could be wrong.
>
> Does this mutual incompatibility generalize, or is it something
> specific to the natural numbers?
>
>
>> [Imho, it could be said the the later epitomizes the impossibility of
>> the former].
>
> Can you explain how?

I think I already did: they both connote impossibility of some
syntactical proofs. So we just pick one as a representation
(an epitome so to speak) of the other.

>
>
>> That aside, it's actually my position that it's impossible to
>> to syntactically prove a consistency: simply because the rules
>> of inference won't let us do that; hence it's a _delusion_ that
>> we could have any "sort of thing" that we could "accept as a
>> proof of consistency"!
>
>> [That's why I'd would be surprised if in the past I had said something
>> that has caused you to think there be a criteria to accept a proof
>> of inconsistency].
>
> As far as I am aware, you have not given the conditions under
> which you would consider the consistency of a theory to be
> proven.

You're correct that I haven't, and I never will: because the conditions
would be impossible to be met. That doesn't mean the following isn't
a standard definition of syntactical consistency that you and I and
everyone else would accept:

consistency <-> not being syntactically inconsistent.

> As far as proving INconsistency, I was not under the
> impression that you disagreed with the usual method of
> deriving a contradiction.

That's correct that I've never disagreed: because the method of proving
inconsistency is just a _finite FOL proof_, as I mentioned quite a few
times here and elsewhere.

> For the sake of completeness and
> clarity, and because I suspect you meant "consistency" above
> where you typed "inconsistency" would you clarify if/how you
> consider it possible to prove a theory inconsistent.

If you don't remember or have any doubt about my position, please
allow me to clearly reiterate my position:

It's impossible to have logically acceptable methods for proving a
syntactical consistency.

Hope that I've clarified my answer, position on your question about
proving consistency. If that's what you meant to ask. (As for method
of proving INconsistency, my position is that that's just like proving
a FOL theorem, I've stated above).

From: Marshall on 5 Apr 2010 02:15

On Apr 4, 10:08 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 4, 9:33 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> Marshall wrote:
> >>> On Apr 2, 5:32 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >>>> Let me put to rest the idea we know enough about the natural numbers,
> >>>> to prove important thing such as the consistency of PA. I'll do that
> >>>> by pointing out the existence of a specific unknown natural number.
> >>> Why do you think the existence of a specific unknown number
> >>> should have anything to do with consistency?
> >> Because they (the syntactical proof of consistency and collectively
> >> many formulas about this unknown natural) both connote the same thing
> >> in meta level: impossibility of syntactical proof. If you can't prove
> >> a certain formula related to this number, you can forget about proving
> >> a consistency, syntactically speaking.
>
> > Both of these sentences are just a restatement of the position that
> > I asked you for support of. My question is not, do you consider
> > a proof of consistency of PA and the existence of an unknown natural
> > number to be mutually incompatible. You have clearly stated so.
>
> Huh? I'm not quite following you here. First of, you clearly asked me
> why the existence of a specific unknown natural number "should have
> anything to do with consistency [proof]". And I gave you an answer for
> that: their both connoting an "impossibility of [some] syntactical proof"
> would be their "having anything to do" with each other.
>
> That's all I said here. And I didn't state anything about them being
> "mutually incompatible" or what not!

OK. So you mean that the two things have "something" to do
with each other, but it's just a connotation, not any kind of
logical result.

> > Rather, my question is WHY do you see them as mutually
> > incompatible.
>
> Again where did I even hint these 2 are "mutually incompatible"?

When you said "If you can't prove a certain formula related to this
number, you can forget about proving a consistency" I took that to
mean that being unable to prove a certain formula is incompatible
with a consistency proof. My mistake.

Apparently what you are saying is that "you can forget about
proving a consistency" but at the same time, proving consistency
is still compatible. I guess I don't really understand that.

> I think I already did: they both connote impossibility of some
> syntactical proofs. So we just pick one as a representation
> (an epitome so to speak) of the other.

I have no idea what this is supposed to mean.

> > As far as I am aware, you have not given the conditions under
> > which you would consider the consistency of a theory to be
> > proven.
>
> You're correct that I haven't, and I never will: because the conditions
> would be impossible to be met. That doesn't mean the following isn't
> a standard definition of syntactical consistency that you and I and
> everyone else would accept:
>
> consistency <-> not being syntactically inconsistent.

So you are saying that no theory can ever be proven to be
consistent. (Also, how do you expect to convince me that
the conditions for a consistency proof are impossible to
meet without telling me what those conditions are?)

Would you agree that this position is not one for which
there is any mainstream acceptance?

What about the theory with no axioms; is it impossible to prove
that it is consistent? If it's not consistent, or not possible to
prove it is consistent, what mechanism might admit an
inconsistency? Surely it is not a logical conflict between
some of its axioms? Are you then saying that FOL itself
is not known to be consistent? What about propositional
logic; is it possible to prove it consistent? What about
boolean algebra?

> > As far as proving INconsistency, I was not under the
> > impression that you disagreed with the usual method of
> > deriving a contradiction.
>
> That's correct that I've never disagreed: because the method of proving
> inconsistency is just a _finite FOL proof_, as I mentioned quite a few
> times here and elsewhere.

And what if someone proved that a FOL proof of inconsistency
for some particular theory is impossible? Or are you saying
that it is impossible to prove that something is
impossible to prove? If so, how do you know? Did you prove it?
If you didn't, then how can you claim to know? If you did, I see
a logical problem with your position.

Marshall