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

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From: Newberry on 3 Apr 2010 00:05

On Apr 2, 3:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough) wrote:
> Newberry says...
>
> >What I was getting at is how we know that the system is consistent.
>
> In the case of PA, it's because we know that everything it says
> about the natural numbers is true.
>
> Basically, the axioms of PA consist of:
> 1. Axioms that recursively define plus and times in terms of successor.
> 2. Axioms that say that zero is the smallest natural natural number,
> and that successor is 1-1.
> 3. The induction axioms, which basically say that every natural
> number is obtained from zero by repeatedly applying the successor
> function.

If it absolutely certain that PA is consistent why don't we formalize
the reasoning? We have done it in English so we should be able to do
it in a formal language. For example why don't we simply add PA is
consistent as another axiom?

>
> --
> Daryl McCullough
> Ithaca, NY

From: Newberry on 3 Apr 2010 00:16

On Apr 2, 3:16 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > I do not know how it will turn out. I forgot who proved that the
> > square root of 2 was irrational and what his proof looked like. Maybe
> > your version is something concocted by the modern mathematicians who
> > take classical logic for granted.
>
> It was due to nameless Pythagorean. It was a geometric proof, rather
> than the more familiar algebraic proof, but I don't know the details.
>
> > Maybe it will turn invalid, maybe valid with some modifications or
> > added assumptions. Mind you the Greeks did not have the concept that
> > the vacuous sentences were true. The traditional syllogism
> > presupposes that the subject class is non- empty.
>
> I'll betcha that the mathematical proofs of, say, Euclid, do not
> follow the logical restrictions of Aristotle's categorical logic.
> But, again, I'm no expert on this by any stretch of the imagination.
>
> I would've thought that you had certain aims for your logic. I
> would've thought that, for instance, you would want that, if a set of
> sentences T entails P in your logic, then that same set of sentences
> entails P in classical logic. Roughly, that classical logic makes
> more things true, but doesn't make different things true.

This is not an a priori goal. I have certain aims but this is not one
of them.

> If so, of
> course, you'd have to drop the claim (recently made) that
>
> ~(Ex)(Px & Qx) -> (Ex)Px.
>
> Similarly, of course, I would expect that if your logic proves P, then
> so does classical logic.
>
> Right now, I'm not sure whether you've considered questions like
> this. If not, you prob'ly oughta.

I have considered that my logic can prove different things. For
example it can potentially prove

~(Ex)Pxm

>
> --
> Jesse F. Hughes
>
> "When you try to kiss a girl, it's hard not to get spit on the girl."
> -- Quincy P. Hughes, age 3 (almost 4)

From: Newberry on 3 Apr 2010 00:18

On Apr 2, 7:00 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Daryl McCullough wrote:
> > Newberry says...
> >> On Apr 1, 5:59=A0am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
>
> >>> How do I know that Peano arithmetic is consistent? I know it the way I
> >>> know any mathematical theorem I have personally proved.
> >> You proved PA consistent?
>
> > It's easy to prove in ZF.
>
> Is ZF _syntactically_ consistent?

Such a proof in ZF that PA is consistent is obviously wothless. We had
a long discussion about this a while ago.

From: Tim Golden BandTech.com on 3 Apr 2010 02:23

On Mar 20, 11:34 pm, Newberry <newberr...(a)gmail.com> wrote:
> On Mar 20, 2:44 pm, r...(a)zedat.fu-berlin.de (Stefan Ram) wrote:
>
> > Newberry <newberr...(a)gmail.com> writes:
> > >When Are Relations Neither True Nor False?
>
> > (I am late to this thread, so please excuse me if I
> > should repeat something that was already written.)
>
> > Always. A relation is a set of pairs, and a set is
> > never true nor false. Therefore, every relation is
> > neither true nor false.
>
> Your point is not well taken. From the context we see that I mean
> "sentences with more than one variable." But since "when are sentences
> with more than one variable neither true nor false" sounds awkward I
> instead use the expression above.

Here you are Newberry caught in the dimensional paradigm.

When one rises to the two dimensional problem is it not apparent that
the logic might rise to a continuous state?

Let's go slowly, and backtrack to the one dimensional logic, which
alotted a true/false value. This is consistent with the sphere in one
dimension, which will give just two values:
+1, -1 .
Upon generalizing to two dimensions should we consider the spherical
concept? Perhaps a realistic example should be played out. Let's just
suppose that we have a criterion as a functional measure upon a given
statement:
Truth(A)
where A is the statement and Truth(A) then is the boolean measure of
the statement A. Is the two dimensional version of this statement
Truth( A + B )
accurate where
Truth( A + B ) = Truth( A ) + Truth( B ) ?
Now we have attempted to formalize the logic of a dimensional
construction, yet insofar as we have the following qualms
+ 1 + 1 = + 2 ,
+ 1 - 1 = 0 ,
- 1 - 1 = - 2
then we have entered a tritype. We are left with a conditional as
sensible, so that if either A or B is not true, then the result is not
true. This is neither a product nor a sum, and instead requires a
logic operation AND.

Is this any better? At least it has only two values. Yet if we entered
a multidimensional realm, even just a two dimensional realm, then we
are forced to address the problem more cleanly. I suggest heading back
toward the spherical implication, which causes from geometrical
interpretation a continuum logic, directly from the spherical
constraint.

The issue of independence is apparent here. When two concepts (or
logic) are independent then what right do we have to intertwine them?
In effect the logic is challenging the cartesian product. I believe
that this is appropriate. The problem then becomes one of
multidimensional logic, and we should admit that a two dimensional
logic is starkly different than a one dimensional logic, or two one
dimensional logics for that matter.
I believe that the correct answer is that a two dimensional logic is a
continuous logic, consistent with the spherical interpretation, and so
appropriate to the subject of this thread. Yet, the perception of this
two-dimensional logic may actually be beyond us in modernity, even
while we hem and haw on the truth of complex issues in a continuous
way.

I believe that the human inherently does practice a continuous truth
process, though we are caught in discrete language. This is an issue
for Chomsky and his ilk, as much as it is for the namby-pamby
usenetters, myself included.

The independence problem that I mention above applies to geometry in
general, and in hindsight of the polysign system we have the option of
disavowing the cartesian product, which leaves a two-dimensional realm
unique from a one dimensional realm, or two one dimensional realms.
The implication of such a construction is not light hearted at all. We
may be lacking the final calculus that sets things straight, or rayed,
as the unidirectional quality of polysign geometry would have me
believe:
http://bandtechnology.com/PolySigned/Lattice/Lattice.html

In that the AND statement and any such combinatorial logic must marry
independent values do we have a cause?

Can this cause be taken to the two dimension spherical continuum?

I honestly do not know, but I think that this could be approched as an
open problem.
Whether there is any productive work here...
You tell me.

- Tim

From: Daryl McCullough on 3 Apr 2010 09:54

Newberry says...

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

It has been. It's easily formalized in ZFC.

--
Daryl McCullough
Ithaca, NY