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

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From: Nam Nguyen on 25 Mar 2010 21:55

Daryl McCullough wrote:

>
> Ultimately, the people on this newsgroup who object to standard
> mathematics are really objecting to the idea that there can be
> such a thing as a counter-intuitive result.

That's a grossly erroneous over-generalization. It's many of those who
defend standard mathematics who erroneously object to the counter-intuitive
_but real nature_ of mathematics: relativity of truth and provability.

To date they can't cite any absolutely true formula, without the formula
being false in another similar context, and yet they'd believe such
"absolute" truth is intuitive.

> The ultimate logic
> would be one in which it is impossible to prove any result that
> you couldn't already guess was true.

The ultimate logic is one which is relativistic.

From: Transfer Principle on 25 Mar 2010 23:07

On Mar 25, 10:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> I'm just a simple housewife (with somewhat hairy legs).

It would be poetic justice if AP were to quote this sentence in his
..sig, with only the name of its author "Jesse F. Hughes" appearing
in the sig, with neither AP nor any other name appearing anywhere
in the post.

From: Jesse F. Hughes on 25 Mar 2010 23:16

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

> On Mar 25, 10:40 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> I'm just a simple housewife (with somewhat hairy legs).
>
> It would be poetic justice if AP were to quote this sentence in his
> .sig, with only the name of its author "Jesse F. Hughes" appearing
> in the sig, with neither AP nor any other name appearing anywhere
> in the post.

Why should that bother me? He should certainly feel free to do so, so
long as his name appears on the "From" line of the post headers.

--
"But remember, as long as one human being follows the rules of
mathematics, then mathematics as a human discipline survives.
Right now I'm that one human being, so mathematics survives."
-- James S. Harris

From: Transfer Principle on 26 Mar 2010 00:03

On Mar 22, 5:18 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Transfer Principle <lwal...(a)lausd.net> writes:
> > Notice that J. Clarke here is expressing similar ideas regarding the
> > empty set and vacuous truth as the OP of this thread, Newberry.
> No, he's not. He's claiming that the empty set is not a relation. He
> is not claiming that vacuously true statements are neither true nor
> false.
> Why you think this is even remotely related to Newberry's pet project
> is beyond me.

There must be some reason that Clarke claims that the empty set is
not a relation, in contradiction to ZFC and FOL.

Let's go back to a post made by Rotwang:

"And if a physicist accepts all
the axioms of ZFC, as well as the usual rules of FOL, then he must
accept all of the resulting theorems whether he realises it or not."

Applying contapositive to Rotwang's statement, we see that if a
physicist _doesn't_ accept all the theorems of ZFC/FOL, then he
doesn't accept at least one of the axioms of ZFC/FOL.

Of course, we don't know whether Clarke or Newberry are even
physicists, since after all:

> [1] I'm not saying that Newberry or Clarke is untrained in
> mathematics [or physics, for that matter]. I have no idea of their
> backgrounds, of course.

But still, I have yet to see why the contrapositive to Rotwang's
statement can't be generalized to any poster. So any poster who
doesn't accept all of the theorems of ZFC/FOL must reject at least
one of the axioms of ZFC/FOL. And since Clarke rejects "the empty
set is a relation," which is a theorem of ZFC/FOL, then we conclude
that he rejects at least one of the axioms. And I'd like to know which
axiom that is (to allow for the possibility of a theory with takes the
rejected axiom and replace it by, say, its negation -- a theory which
hopefully would be more acceptable to Clarke).

Let's look at an actual proof of "the empty set is a relation." Once
again, we go back to the Metamath proof. This proof consists of
three steps:

http://us.metamath.org/mpegif/rel0.html

We notice that in the list of axioms used by the proof (given at the
bottom of the page), there is only one set theoretic axiom used in
the proof -- the Axiom of Extensionality. Believe it or not, the Empty
Set Axiom isn't used in the proof, despite the theorem actually
referring to "the empty set"! Perhaps this is because even if 0 is
merely a proper class, it would still be a relation anyway. We don't
need 0 to be a set in order for it to be a relation (since there are
many relations, such as =, which clearly aren't sets). All the other
axioms used are actually rules of FOL. Thus, if Clarke accepts the
Axiom of Extensionality, then he must reject some rule of FOL.

Now Step 2 of the proof gives a definition of "relation." We note that
if V is the class of all sets, then VxV is the class of all ordered
pairs, and a relation is defined here as a subclass of VxV. In other
words, a relation is simply a class of ordered pairs (which is the
version of the definition Clarke and Hughes appear to be using).

Step 1 of the proof asserts that 0 is indeed a subclass of VxV --
simply because 0 is a subclass of _any_ class. Step 3 of the
proof uses a form of Modus Ponens -- from "if a class is a subclass
of VxV, then it is a relation" and "0 is a subclass of VxV," conclude
"0 is a relation." QED

We know that Clarke rejects at least one of these steps -- but which
one is it? One could argue that it's step 2 -- since neither he nor
Hughes explicitly refers to subclasses of VxV. But I believe that as
the definition they give is equivalent to Metamath's, my guess is that
step 1 is the objectionable step instead.

So we look at Step 1 in more detail. The proof that 0 is a subclass of
any class is also a three-step proof:

http://us.metamath.org/mpegif/0ss.html

Step 1 looks straightforward -- it merely states that the empty set
has
no elements. Step 3 is a form of the definition of subclass, which
also
looks uncontroversial.

But Step 2 states that any element of the empty set must be an
element of any class -- which sounds like vacuous truth. And when
we click on this step, we see a rule of inference: from ~phi, conclude
phi -> psi -- which sounds just like the Third Paradox of Material
Implication, mentioned elsewhere in this thread:

http://us.metamath.org/mpegif/pm2.21i.html

And why was that Third Paradox mentioned? It was mentioned in the
context of _Newberry_ -- the paradoxes to which Newberry objects.

And thus I claim that there is indeed a connection between Clarke
and Newberry after all -- both of them object to theorems of ZFC/FOL
which use the Third Paradox of Material Implication.

> Why you think this is even remotely related to Newberry's pet project
> is beyond me.

So Hughes doubts my claim -- and we already know that the most
common criticism of me by far is misinterpretation and incorrect
guessing of what others are saying.

So how can I reduce this bad habit of mine, and avoid misinterpreting
what's in Clarke's post. We know that Clarke objects to _some_ axiom
of ZFC/FOL, but how can I find out which one?

I could try asking Clarke directly. But I must do so in a way so as
not
to appear to be a standard theorist or a "bully." I'm not trying to
_interrogate_ Clarke in order to criticize him, but to find out which
axiom
he rejects in order to come up with an alternative theory, without the
standard theorists accusing me of misinterpreting posts.

Also, if I start giving formal symbolic proofs of whichever theorem of
ZFC/FOL is being rejected (such as the proof of "the empty set is a
relation") given above, this also tends to turn off the standard
theorists'
opponents -- since most of them seem to prefer informal English to
formal symbolic language. But I could give a description of the proof
in
informal English -- and indeed, I've already done so in this very
post.

And so I can ask Clarke about his rejection of the empty set as a
relation without interrogating him like a standard theorist or
otherwise
alienating him (lest me find me too hostile, and he killfiles me, just
as
he killfiled Hughes).

So let me do so right now -- this rest of this message is therefore
directed to Clarke. You say that you reject the empty set as a
relation,
yet I've just given a three-step proof in standard theory, ZFC/FOL,
that
the empty set is indeed a relation. Since you reject the conclusion of
the proof, to which of its three steps do you also reject?

From: Marshall on 26 Mar 2010 00:25

On Mar 25, 11:00 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
>
> Ultimately, the people on this newsgroup who object to standard
> mathematics are really objecting to the idea that there can be
> such a thing as a counter-intuitive result. The ultimate logic
> would be one in which it is impossible to prove any result that
> you couldn't already guess was true.

QFT