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

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From: Aatu Koskensilta on 28 Feb 2010 13:49

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

> The ideas of Graham Priest are absurd. Please do not compare me with
> him.

Priest's ideas about arithmetical dialatheias are indeed rather
bizarre. I didn't really compare you with Priest, however. I merely
observed, in effect, that it's pointless to exhort people to be more
precise about a distinction that doesn't correspond to anything in our
ordinary mathematical reasoning.

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

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

From: Newberry on 28 Feb 2010 18:43

On Feb 28, 10:49 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > The ideas of Graham Priest are absurd. Please do not compare me with
> > him.
>
> Priest's ideas about arithmetical dialatheias are indeed rather
> bizarre. I didn't really compare you with Priest, however. I merely
> observed, in effect, that it's pointless to exhort people to be more
> precise about a distinction that doesn't correspond to anything in our
> ordinary mathematical reasoning.
>
Yeah?
a) Do you know that my distinction between

~(Ex)(Ey)(P'xy & Q'y)

and

~(Ex)P'xm'

corresponds to Gaifman/Goldsten's solution of the Liar paradox?

b) Is this

(x)((2 > x > 4) -> ~(x < x + 1))

"ordinary mathematical reasoning"? It does not even occur to anybody
until they are thoroughly brainwashed by classes in classical logic
and lectures about empty sets. After that happens they think they have
been enlightened and obtained extraordinary wisdom. Probably no
mathematician believed this before Frege formalized the predicate
calculus.

From: Aatu Koskensilta on 28 Feb 2010 18:54

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

> a) Do you know that my distinction between
>
> ~(Ex)(Ey)(P'xy & Q'y)
>
> and
>
> ~(Ex)P'xm'
>
> corresponds to Gaifman/Goldsten's solution of the Liar paradox?

I'm afraid your formalism is obscure to me. What's the significance of
all these primes?

> b) Is this
>
> (x)((2 > x > 4) -> ~(x < x + 1))
>
> "ordinary mathematical reasoning"?

It isn't reasoning at all. It's a formula that doesn't formalize any
statement we would ordinarily meet in mathematical reasoning.

> It does not even occur to anybody until they are thoroughly
> brainwashed by classes in classical logic and lectures about empty
> sets.

You may of course choose to describe ordinary mathematical education
"brainwashing" if you wish. It remains that in ordinary mathematical
reasoning there is no distinction between

There are no counterexamples to GC less than 27.

and

It's not true that there are counterexamples to GC less than 27.

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

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

From: Daryl McCullough on 28 Feb 2010 20:59

Newberry says...

>On Feb 26, 4:10=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> Unrestricted comprehension, which says if Phi(x) is a formula
>> with one free variable, then there is a set y such that forall
>> x,
>>
>> x is an element of y
>> <->
>> Phi(x)
>>
>> This is clearly not true in the case Phi(x) is the formula
>> "x is an element of x".

That should be "x is not an element of x".

>Looks like you discovered that it was not manifestly true only AFTER
>you derived a contradiction.

Well, the discovery happened long before I was born, so it's hard to
know whether I would have believed Frege's axiom to be "manifestly
true" had I been unaware of Russell's paradox.

--
Daryl McCullough
Ithaca, NY

From: Newberry on 1 Mar 2010 00:04

On Feb 28, 3:54 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > a) Do you know that my distinction between
>
> > ~(Ex)(Ey)(P'xy & Q'y)
>
> > and
>
> > ~(Ex)P'xm'
>
> > corresponds to Gaifman/Goldsten's solution of the Liar paradox?
>
> I'm afraid your formalism is obscure to me. What's the significance of
> all these primes?

Do not wory about the primes. They are irrelevant for our present
purpose.
>
> > b) Is this
>
> > (x)((2 > x > 4) -> ~(x < x + 1))
>
> > "ordinary mathematical reasoning"?
>
> It isn't reasoning at all. It's a formula that doesn't formalize any
> statement we would ordinarily meet in mathematical reasoning.
>
> > It does not even occur to anybody until they are thoroughly
> > brainwashed by classes in classical logic and lectures about empty
> > sets.
>
> You may of course choose to describe ordinary mathematical education
> "brainwashing" if you wish. It remains that in ordinary mathematical
> reasoning there is no distinction between
>
> There are no counterexamples to GC less than 27.
>
> and
>
> It's not true that there are counterexamples to GC less than 27.
>
The way you put it, it sounds like my primary objective is to make a
distinction between the two sentences above for some strange reason.
Actually the distinction is a consequence of outlawing
(x)((2 > x > 4) -> ~(x < x + 1))
and solving the Liar paradox. And I would contend that the necessity
to introduce such a distinction does not do any major harm.

Furthermore I have already indicated that our intution in these cases
is unreliable. There is entire literature on this subject. I have
listed two papers above. The "ordinary mathematical reasoning" is a
mixture of our intutions (incapable of distingushing not-true from
false and the formalism of classical logic. If our intutions are not
capable to distinguish the two and classical logic uses the expression
(x)()x = counterexample to GC) & (x < 27))
then you can claim that it is "ordinary mathematical reasoning." That
does not prove that the classical logic is the right one.