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

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From: Marshall on 11 Apr 2010 18:37

On Apr 11, 2:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Marshall wrote:
> > On Apr 11, 8:25 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> >> Marshall <marshall.spi...(a)gmail.com> writes:
> >>> I don't see any reason to pay much attention to anyone's
> >>> intuition, my own included.
> >> This is a sober attitude.
>
> > I am mostly a sober person, in that I am drunk less
> > than half the time.
>
> >>> "Intuition" is just a fancy word for "hunch."
> >> "Intuition" can mean pretty much anything, from a vague hunch to
> >> something very specific, as in e.g. Kant's thought.
>
> > Indeed so, which is exactly what makes it a poor choice
> > when used in contexts such as this newsgroup.
>
> Then, it also looks like a poor choice of using the _intuition_ about
> the naturals as a foundation of reasoning, as the school of thought AK,
> TF seem to have subscribed to, would suggest.

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

Marshall

From: Nam Nguyen on 11 Apr 2010 18:51

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.

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.

From: Nam Nguyen on 11 Apr 2010 18:54

Marshall wrote:
> On Apr 11, 2:46 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
>> Marshall wrote:
>>> On Apr 11, 8:25 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>>>> Marshall <marshall.spi...(a)gmail.com> writes:
>>>>> I don't see any reason to pay much attention to anyone's
>>>>> intuition, my own included.
>>>> This is a sober attitude.
>>> I am mostly a sober person, in that I am drunk less
>>> than half the time.
>>>>> "Intuition" is just a fancy word for "hunch."
>>>> "Intuition" can mean pretty much anything, from a vague hunch to
>>>> something very specific, as in e.g. Kant's thought.
>>> Indeed so, which is exactly what makes it a poor choice
>>> when used in contexts such as this newsgroup.
>> Then, it also looks like a poor choice of using the _intuition_ about
>> the naturals as a foundation of reasoning, as the school of thought AK,
>> TF seem to have subscribed to, would suggest.
>
> 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?

From: Alan Smaill on 11 Apr 2010 19:10

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?

--
Alan Smaill

From: Marshall on 11 Apr 2010 19:22

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.

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

Marshall