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

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From: Nam Nguyen on 11 Apr 2010 19:52

Alan Smaill wrote:
> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>
>> Alan Smaill wrote:
>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>>
>>>> Nam Nguyen wrote:
>>>>> Alan Smaill wrote:
>>>>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>>>>>
>>>>>>> David Bernier wrote:
>>>>>>>> Do you see problems with starting with a false premise, not(P) ?
>>>>>>> Yes. I'd break "the Principle of Symmetry": if we could start with
>>>>>>> not(P),
>>>>>>> we could start with P.
>>>>>> And so you could, so that's not a problem.
>>>>>>
>>>>>> But what would you prove?
>>>>>>
>>>>> It was a typo, I meant "It'd the Principle of Symmetry".
>>>>> Would you still have any question then?
>>>> I meant "It'd break the Principle of Symmetry".
>>> How?
>>>
>>> You *can* start by supposing P;
>>> and if you can derive a contradiction, you get a proof
>>> on "not P".
>>>
>>> What's the symmetry that's broken, according to you?
>> Let's recall this conversation started by David Bernier's suggestion
>> to a way of obtaining an absolute truth, in responding to my suggestion
>> the nature of mathematical reasoning is subjective and relative.
>
> I'm not addressing that question.
>
>> Let's also recall that the Principle as suggested in this thread
>> to safeguard against _incorrect assumption_ of an absolute truth of
>> ANY non-tautologous, non-contradictory formulas.
>>
>> Choosing a formula to be absolutely true is to break this Principle's
>> safeguard of correcting reasoning.
>
> What I'm querying is your claim that your objection to reductio ad
> absurdum is based on a failure of *symmetry*. You say "if we could
> start with not(P), we could start with P", as though there is a problem
> with starting with P in particular.
>
> If "P" is problematic (non tautology, non contradictory),
> then I expect "not P" to be likewise problematic, by symmetry,
> and your principle of symmetry is not broken --
> you reject both sorts of argument, I take it.
>
> So my question remains: what symmetry do you think is broken?

I don't think you've paid attention to the conversation.
Whatever I had said here that your question was about has a context:
my responding to DB's alluded a formula being _absolutely true_.

And my point here is that he (or any of us) could forget about a formula
[neg(P) or P] being absolutely true, since that would break symmetry: the
Principle would stipulate both formulas must necessarily be of relative
truth, to be symmetrical in reasoning! To claim some formula being
absolutely true (or false) is to destroy the notion of such symmetry.

From: Nam Nguyen on 11 Apr 2010 20:09

Marshall wrote:
> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>> I reject your claim that intuition, of whatever kind, plays
>>> any part in our thinking about the naturals.
>> So is the naturals collectively a finite syntactical notion *to you*,
>> since you'd reject the idea they're an intuition notion?
>
> I reject the idea that the naturals are anything particular *to me*
> that they are not to anyone else.

The question was simply asking you to explain from your mathematical
knowledge what the natural number be. That's all it was asked of you.
Whether your concept of the naturals is the same as mine, or TF's,
or anybody else' isn't relevant to the question.

Can you share with us your resounding, firm, answer to that simple
question? After all, you seem to have said you know the naturals
number well enough to reject the idea they're of an intuition notion!

>
> Other than that, and as much as I hesitate to accede to any
> formula of yours given how unreliably you use terminology,
> my answer is "yes." We can completely capture enough
> about the naturals to uniquely characterize them up to
> isomorphism. For example, we have many syntactic
> representations of natural numbers available to us that
> can represent any natural up to resource limits, and we
> have simple algorithms on those representations that
> can compute successor, addition, etc. of any naturals
> up to resource limits. These things are entirely
> mechanical, and free of any vague or intuitive aspects.

I'm sorry: "up to resource limits" isn't an intuition or technical
notion of the natural numbers. So you don't seem to understand
the notion of the natural numbers, despite your having claimed
otherwise.

>
> I am under no delusions that you will agree with me,
> but please spare me the umpteenth repetition of your
> GC counterargument.

Until you demonstrate precisely what you meant by the naturals,
while asserting they're not of intuition notion, I wouldn't
have a reason not to repeat my GC counter argument.

From: Alan Smaill on 11 Apr 2010 20:15

Nam Nguyen <namducnguyen(a)shaw.ca> writes:

> Alan Smaill wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> Alan Smaill wrote:
>>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>>>
>>>>> Nam Nguyen wrote:
>>>>>> Alan Smaill wrote:
>>>>>>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>>>>>>
>>>>>>>> David Bernier wrote:
>>>>>>>>> Do you see problems with starting with a false premise, not(P) ?
>>>>>>>> Yes. I'd break "the Principle of Symmetry": if we could start with
>>>>>>>> not(P),
>>>>>>>> we could start with P.
>>>>>>> And so you could, so that's not a problem.
>>>>>>>
>>>>>>> But what would you prove?
>>>>>>>
>>>>>> It was a typo, I meant "It'd the Principle of Symmetry".
>>>>>> Would you still have any question then?
>>>>> I meant "It'd break the Principle of Symmetry".
>>>> How?
>>>>
>>>> You *can* start by supposing P;
>>>> and if you can derive a contradiction, you get a proof
>>>> on "not P".
>>>>
>>>> What's the symmetry that's broken, according to you?
>>> Let's recall this conversation started by David Bernier's suggestion
>>> to a way of obtaining an absolute truth, in responding to my suggestion
>>> the nature of mathematical reasoning is subjective and relative.
>>
>> I'm not addressing that question.
>>
>>> Let's also recall that the Principle as suggested in this thread
>>> to safeguard against _incorrect assumption_ of an absolute truth of
>>> ANY non-tautologous, non-contradictory formulas.
>>>
>>> Choosing a formula to be absolutely true is to break this Principle's
>>> safeguard of correcting reasoning.
>>
>> What I'm querying is your claim that your objection to reductio ad
>> absurdum is based on a failure of *symmetry*. You say "if we could
>> start with not(P), we could start with P", as though there is a problem
>> with starting with P in particular.
>>
>> If "P" is problematic (non tautology, non contradictory),
>> then I expect "not P" to be likewise problematic, by symmetry,
>> and your principle of symmetry is not broken --
>> you reject both sorts of argument, I take it.
>>
>> So my question remains: what symmetry do you think is broken?
>
> I don't think you've paid attention to the conversation.
> Whatever I had said here that your question was about has a context:
> my responding to DB's alluded a formula being _absolutely true_.

I'm simply not addressing the whole conversation.

> And my point here is that he (or any of us) could forget about a formula
> [neg(P) or P] being absolutely true, since that would break symmetry: the
> Principle would stipulate both formulas must necessarily be of relative
> truth, to be symmetrical in reasoning!

I just agreed in my previous post with the point that symmetry says both
are of relative truth, by symmetry.

> To claim some formula being
> absolutely true (or false) is to destroy the notion of such symmetry.

Do you agree that symmetry is only
broken if the cases "P" and "not P" are treated differently?

--
Alan Smaill

From: Marshall on 11 Apr 2010 20:46

On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >>> I reject your claim that intuition, of whatever kind, plays
> >>> any part in our thinking about the naturals.
> >> So is the naturals collectively a finite syntactical notion *to you*,
> >> since you'd reject the idea they're an intuition notion?
>
> > I reject the idea that the naturals are anything particular *to me*
> > that they are not to anyone else.
>
> The question was simply asking you to explain from your mathematical
> knowledge what the natural number be. That's all it was asked of you.
> Whether your concept of the naturals is the same as mine, or TF's,
> or anybody else' isn't relevant to the question.

Exactly. The "*to you*" part of your question was irrelevant, so
I rejected it. Fortunately we both agree it is irrelevant.

> Can you share with us your resounding, firm, answer to that simple
> question?

You mean, again? You want me to say "yes" again? Okay:

Yes.

> > Other than that, and as much as I hesitate to accede to any
> > formula of yours given how unreliably you use terminology,
> > my answer is "yes." We can completely capture enough
> > about the naturals to uniquely characterize them up to
> > isomorphism. For example, we have many syntactic
> > representations of natural numbers available to us that
> > can represent any natural up to resource limits, and we
> > have simple algorithms on those representations that
> > can compute successor, addition, etc. of any naturals
> > up to resource limits. These things are entirely
> > mechanical, and free of any vague or intuitive aspects.
>
> I'm sorry: "up to resource limits" isn't an intuition or technical
> notion of the natural numbers. So you don't seem to understand
> the notion of the natural numbers, despite your having claimed
> otherwise.

Of course it's not an intuition. I've already made it clear that
I'm not talking about that. But it certainly *is* a technical
notion when discussing any "entirely mechanical" implementation
of an algorithm.

> > I am under no delusions that you will agree with me,
> > but please spare me the umpteenth repetition of your
> > GC counterargument.
>
> Until you demonstrate precisely what you meant by the naturals,
> while asserting they're not of intuition notion, I wouldn't
> have a reason not to repeat my GC counter argument.

I am pretty sure a marginally competent ten year old
can explain what a base ten natural number is (whether
or not his school uses those exact terms) and can
explain how to find the successor, how to add two
naturals, multiply, etc. I could certainly supply you
with my own version of that explanation, but if you
can't already do it yourself, then I propose you
ought to first go learn that. If you can, then there
is no point in my retyping it.

Marshall

From: Nam Nguyen on 11 Apr 2010 22:51

Marshall wrote:
> On Apr 11, 5:09 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Marshall wrote:
>>> On Apr 11, 3:54 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>>>>> I reject your claim that intuition, of whatever kind, plays
>>>>> any part in our thinking about the naturals.
>>>> So is the naturals collectively a finite syntactical notion *to you*,
>>>> since you'd reject the idea they're an intuition notion?
>
>> Can you share with us your resounding, firm, answer to that simple
>> question?
>
> You mean, again? You want me to say "yes" again? Okay:
>
> Yes.

So your resounding answer is "yes" that the notion of the naturals
is based on an intuition but is a notion of syntactical formal system.
How would you know though such a formal system is syntactical
consistent?

>
>
>>> Other than that, and as much as I hesitate to accede to any
>>> formula of yours given how unreliably you use terminology,
>>> my answer is "yes." We can completely capture enough
>>> about the naturals to uniquely characterize them up to
>>> isomorphism. For example, we have many syntactic
>>> representations of natural numbers available to us that
>>> can represent any natural up to resource limits, and we
>>> have simple algorithms on those representations that
>>> can compute successor, addition, etc. of any naturals
>>> up to resource limits. These things are entirely
>>> mechanical, and free of any vague or intuitive aspects.
>> I'm sorry: "up to resource limits" isn't an intuition or technical
>> notion of the natural numbers. So you don't seem to understand
>> the notion of the natural numbers, despite your having claimed
>> otherwise.
>
> Of course it's not an intuition. I've already made it clear that
> I'm not talking about that. But it certainly *is* a technical
> notion when discussing any "entirely mechanical" implementation
> of an algorithm.

"Entirely mechanical" is not a technical definition, just in case
you're not aware. If the naturals are of syntactical notion then
they must be described by a formal system. So far you keep saying
it's syntactical but giving no hint what such natural-number formal
system be!

>
> I am pretty sure a marginally competent ten year old
> can explain what a base ten natural number is (whether
> or not his school uses those exact terms) and can
> explain how to find the successor, how to add two
> naturals, multiply, etc.

But that's what we'd call _intuition_ and the 10-year old
would most likely never claim he could use such an intuition
to prove the consistency of PA, right? But AK would do
so and you'd do the same or at least support such a claim.

> I could certainly supply you
> with my own version of that explanation, but if you
> can't already do it yourself, then I propose you
> ought to first go learn that. If you can, then there
> is no point in my retyping it.

Oh I learnt the naturals numbers the same way mathematicians
would learn: it's an intuitive notion!

It's you who'd go against the mainstream of mathematicians
and made the following odd notion:

>>>>> I reject your claim that intuition, of whatever kind, plays
>>>>> any part in our thinking about the naturals.

[What textbook's author would even hint such a bogus an idea?]