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

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From: Michael Stemper on 23 Mar 2010 13:42

In article <87hbo7nrd7.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes:
>mstemper(a)walkabout.empros.com (Michael Stemper) writes:
>> In article <87iq8op9if.fsf(a)dialatheia.truth.invalid>, Aatu Koskensilta <aatu.koskensilta(a)uta.fi> writes:

>>> On the set theoretic conception, most relations don't establish any
>>> rule in the ordinary sense of the word.
>>
>> When you say this, do you mean "a rule that can be expressed more
>> simply than by just listing all of the members of the set"?
>
>In general we can't list the elements of a set at all.

Quite true. Which leads back to my previous question. What do you
mean when you say "most relations don't establish any rule in the
ordinary sense of the word"?

If a relation is neither finite, nor expressible in a finte set of
words (such as "aRb iff (a,b in Z and a==b mod 17) or (a<b)" ), is
that when you say that it no longer establishes a rule?

--
Michael F. Stemper
#include <Standard_Disclaimer>
Why doesn't anybody care about apathy?

From: Transfer Principle on 23 Mar 2010 23:53

On Mar 22, 5:58 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> MoeBlee <jazzm...(a)hotmail.com> writes:
> > Okay, but still Transfer Principle's proposal wouldn't block all
> > instances of what you just described, right?
> I've no idea what Walker's doing.

OK then, here's what I'm trying to do in this thread. I'm trying to
find a theory T such that:

1. T isn't known to be inconsistent.
2. T proves phi, even though ZFC proves ~phi, where phi is some
formula that someone who disagrees with the standard theorists
(in this thread, that includes Clarke, Newberry, and Nguyen)
believes to be true.
3. The theory T is acceptable to the standard theorists based on
at least one criterion that they mentioned (in this thread, MoeBlee
mentioned axiomatization for the sciences, Spight mentioned
power and ease of use, and so on.)

In short, I want to find a theory T on which at least one standard
theorist and one of their opponents can agree. So perhaps I can
find a theory which agrees with Clarke's views about the empty
set, yet axiomatizes math for the sciences to the extent that
MoeBlee would be satisfied. Or maybe a theory that exists via
Nguyen's Principle of Symmetry, yet is powerful and easy enough
to use to satisfy Spight.

And if 1-3 are mutually exclusive for the standard theorists and
their opponents, then I should move to another thread with a
debate between these two sides and try again.

From: Newberry on 24 Mar 2010 00:21

On Mar 23, 4:55 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > It is meaningless only if ~(Ex)Px is necessarly true. If ~(Ex)Px is
> > contingent then
>
> > (Ax)(Px -> Qx)
>
> > is merely neither true nor false. In arithmetic it would indeed be
> > meaningless.
>
> Okay. I don't recall you mentioning this distinction previously (and
> I may well forget it hence).
>
> Thus. the sentence
>
> ~(E a,b)( a^2/b^2 = 2 and gcd(a,b) = 1 )
>
> is meaningless, right? You don't have any idea what it means?
> Despite the fact that the classical proof that sqrt(2) is irrational
> proceeds thus:
>
> |- (E a,b)( a^2/b^2 = 2 ) -> (E a,b)( a^2/b^2 = 2 and gcd(a,b) = 1 )
> |- ~(E a,b)( a^2/b^2 = 2 and gcd(a,b) = 1 )
> --------------------------------------------------------------------
> So, ~(E a,b)( a^2/b^2 = 2 )
>
I will look into this when I have time.

> --
> "I AM serious about this being a short route to a Ph.d for some of
> you, but just remember, I'm the guy who proved Fermat's Last Theorem
> in just a bit over 6 years [...] My standards are kind of high."
> --James Harris, founding a new mathematical school

From: Nam Nguyen on 24 Mar 2010 00:22

Transfer Principle wrote:
> On Mar 22, 5:58 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> MoeBlee <jazzm...(a)hotmail.com> writes:
>>> Okay, but still Transfer Principle's proposal wouldn't block all
>>> instances of what you just described, right?
>> I've no idea what Walker's doing.
>
> OK then, here's what I'm trying to do in this thread. I'm trying to
> find a theory T such that:
>
> 1. T isn't known to be inconsistent.
> 2. T proves phi, even though ZFC proves ~phi, where phi is some
> formula that someone who disagrees with the standard theorists
> (in this thread, that includes Clarke, Newberry, and Nguyen)
> believes to be true.

First, what did you mean for Phi to be true here? True in a model?
What model? Secondly, _why_ was my name mentioned in this context?
[I don't think I've ever said in this thread anything about any belief
of mine that a particular formula Phi in L(ZF) should be true or false!]

> 3. The theory T is acceptable to the standard theorists based on
> at least one criterion that they mentioned (in this thread, MoeBlee
> mentioned axiomatization for the sciences, Spight mentioned
> power and ease of use, and so on.)
>
> In short, I want to find a theory T on which at least one standard
> theorist and one of their opponents can agree. So perhaps I can
> find a theory which agrees with Clarke's views about the empty
> set, yet axiomatizes math for the sciences to the extent that
> MoeBlee would be satisfied.

> Or maybe a theory that exists via
> Nguyen's Principle of Symmetry, yet is powerful and easy enough
> to use to satisfy Spight.

My proposed principles have nothing to do with the "existence" of any
theory at all. Other than human being's specifying, formalizing formal
systems (theories), the phrase "theory that exists" doesn't make sense
at all!

>
> And if 1-3 are mutually exclusive for the standard theorists and
> their opponents, then I should move to another thread with a
> debate between these two sides and try again.

From: Newberry on 24 Mar 2010 00:45

On Mar 23, 5:58 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Aatu Koskensilta <aatu.koskensi...(a)uta.fi> writes:
> > "Jesse F. Hughes" <je...(a)phiwumbda.org> writes:
>
> >> Thus. the sentence
>
> >> ~(E a,b)( a^2/b^2 = 2 and gcd(a,b) = 1 )
>
> >> is meaningless, right?
>
> > Newberry said that (x)(Px --> Qx) is meaningless if ~(Ex)Px is
> > necessarily true. How do you get from this the meaninglessness of
>
> > ~(E a,b)( a^2/b^2 = 2 and gcd(a,b) = 1 )
>
> > which is not of the form (x)(Px --> Qx)?
>
> He also says that
>
> ~(Ex)(Px & Qx)
>
> is meaningless in exactly the same situations that
>
> (Ax)(Px -> Qx)
>
> is meaningless. He wants the two formulas to remain equivalent.
>
> (Surely, you're not taking issue with the fact that I've used two
> existential statements rather than one?)
>
> As usual, Newberry can correct me if I'm mistaken on his claims.

Yes, this is what I am proposing. Not sure what two existential
statements you are referring though.

>
> --
> "We are happy that you agree that customers need to know that Open
> Source is legal and stable, and we heartily agree with that sentence
> of your letter. The others don't seem to make as much sense, but we
> find the dialogue refreshing." -- Linus Torvalds to Darl McBride