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

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From: Nam Nguyen on 10 Apr 2010 11:38

David Bernier wrote:
> Nam Nguyen wrote:
>> Nam Nguyen wrote:
>>> Jim Burns wrote:
>>>
>>>> I think it would be very useful to me in understanding
>>>> what you are trying to accomplish if you were
>>>> to give a summary of the best arguments AGAINST your
>>>> positions.
>>>
>>> As promised I'll summarize what I think as the best arguments
>>> against my positions.
>>
>>> ***
>>>
>>> Imho, the 3 major and best arguments against my belief, that the
>>> nature of FOL reasoning is that of relativity or of being subjective,
>>> are the following objections:
>>>
>>> (O1) The Universality Objection:
>>>
>>> In this objection, the correctness in reasoning under one logical
>>> framework should be _universally constant_ and shouldn't be a
>>> function of individual subjective beliefs or knowledge. My claiming
>>> on the relativity nature of FOL reasoning seems to violate this
>>> natural and unobjectionable, say, "sanctity".
>>>
>>> (O2) The Philosophy Objection:
>>>
>>> In this objection, the ideas such that there are formulas written
>>> in the language of arithmetic that can be neither arithmetically
>>> true nor false are just philosophical ideas and thus can't be a
>>> basis to attack the current FOL reasoning.
>>>
>>> (O3) The Ordinary Mathematics Objection:
>>>
>>> This objection seems to be a cross-breed between O1 and O2. In this
>>> objection, FOL reasoning is build upon the ordinary mathematical
>>> knowledge that *in principle* should be universally _self evident_
>>> to all who are trained or study mathematics. As such FOL reasoning
>>> should be universally the same and should _not_ be subjective or
>>> relativistic.
>>>
>>> Again, these are only my thoughts of what the arguments against my
>>> position
>>> be. It would certainly be helpful if those who oppose my position could
>>> further clarify in technical clarity what they perceive are problems
>>> in my
>>> positions. I also don't mind in subsequent posts to further defend my
>>> position or to provide more counter-arguments.
>>
>> I'm still hoping to hear some responses, from Jim, my opponents, or
>> from anyone.
>
> In a proof by contradiction, we assume the negation of
> what we want to prove (i.e. not(P) ). Then, with that assumption, we
> arrive
> at "G .and. not(G)" . This is done in an "imaginary world" where P is
> false.
> Because we followed logic, and arrived at "G .and. not(G)", we conclude
> that not(P) is false. So P is true.

But what about the situation where P and not(P) can't be assigned to
a truth value, in an absolute sense (i.e. it must be clear to _all_
that it's true, or false)?

>
> 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.

>
> By the way, I don't remember hearing of the philosopher P. F. Strawson
> before this or a related thread.
>
> Wikipedia on philosopher P. F. Strawson
> < http://en.wikipedia.org/wiki/P._F._Strawson > .
>
> David Bernier

From: Aatu Koskensilta on 11 Apr 2010 11:05

Newberry <newberryxy(a)gmail.com> writes:

> You proved PA consistent?

Sure, as have innumerable others in the course of their logical
studies. But perhaps you can now comment on how you feel about the more
interesting mathematical result, that for any formal theory T extending
Robinson arithmetic, either directly or through an interpretation, there
are infinitely many true statements of the form "the Diophantine
equation D(x1, ..., xn) = 0 has no solutions" which are unprovable in T
if T is consistent?

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 11 Apr 2010 11:12

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.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 11 Apr 2010 11:13

Transfer Principle <lwalke3(a)lausd.net> writes:

> To repeat, what may be intuitive to one poster may be counterintuitive
> to another. And I see no reason to favor one poster's intution over
> another's, no matter what the standard theorists try to say.

I intuit it for you that in general there's no reason to pay much any
notice to our intuitions. I don't recall anyone claiming we should
favour any poster's intuition over that of others.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Aatu Koskensilta on 11 Apr 2010 11:18

Newberry <newberryxy(a)gmail.com> writes:

> If it absolutely certain that PA is consistent why don't we formalize
> the reasoning?

Absolute certainty is irrelevant. Consistency proofs are every bit as
formalizable as other proofs. We can formalize the trivial consistency
proof for PA in the subtheory ACA of second-order arithmetic, in formal
set theory, in Per-Martin L�f's constructive type theory (by a detour
through a double-negation interpretation), and so on.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus