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

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From: Nam Nguyen on 11 Apr 2010 12:15

Nam Nguyen wrote:
> Aatu Koskensilta wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> So what you're saying is you just _intuit_ PA system be consistent, no
>>> more no less. Of course anyone else could intuit the other way too!
>>
>> Anyone can intuit whatever they want. I said nothing about such matters,
>> I merely noted I have, like any number of logic students over the
>> decades, produced a proof of the consistency of PA in the course of my
>> studies. Intuition has no more and no less to do with it than with any
>> proof in mathematics.
>>
>
> So, is your proof of the consistency of PA a syntactical proof of a
> _consistent_ formal system? (I mean, anyone could syntactically prove
> such proof in an inconsistent formal system). If not then in what sense
> would your proof not require an _intuition about the natural numbers_?

Iow, your notion of "proof" is intuitive (and vague) which is why I mentioned
you "just _intuit_ PA system be consistent".

From: Nam Nguyen on 11 Apr 2010 12:52

Nam Nguyen wrote:
> Aatu Koskensilta wrote:
>> Nam Nguyen <namducnguyen(a)shaw.ca> writes:
>>
>>> 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.
>>
>> But there's nothing clear about this reiteration. What is a "logically
>> acceptable method"?
>>
>
> A method which would conform to definition of syntactical (in)consistency,
> to definition of model of formal systems, to definition of rules of
> inferercne.

For example, one can _actually prove_ this T = {(a=b) /\ ~(a=b)) is inconsistent,
even though one wouldn't be able - or wouldn't care - to intuit the semantic of
each symbol in the axiom.

That's a "logically acceptable" proof.

From: Alan Smaill on 11 Apr 2010 13:58

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?

--
Alan Smaill

From: Marshall on 11 Apr 2010 16:39

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.

Marshall

From: Nam Nguyen on 11 Apr 2010 17:08

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?