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

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From: Jesse F. Hughes on 31 Mar 2010 08:23

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

> On Mar 30, 6:07 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Newberry <newberr...(a)gmail.com> writes:
>> > On Mar 28, 9:01 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> >> Newberry <newberr...(a)gmail.com> writes:
>> >> > On Mar 28, 5:54 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> >> >> Newberry <newberr...(a)gmail.com> writes:
>>
>> >> >> > But I can. In a system with gaps Tarski's theorem does not apply. We
>> >> >> > can then simply equate truth with provability.
>>
>> >> >> Your second sentence does not follow. You have to show that you have
>> >> >> a logic in which provability turns out to be equivalent to truth.
>> >> >> Tarski's theorem may not preclude this possibility, but it doesn't
>> >> >> follow that you can then "simply equate truth with provability."
>>
>> >> > Did I say it follows? I meant that it is possible. In classical logic
>> >> > withuot gaps it is impossible. Why did you not interpret what I said
>> >> > this way?
>>
>> >> "We can then simply equate truth with provability."
>>
>> > It does automatically folow but we can nevertheless do that.
>>
>> You have to *show* that this can be done in your system.
>
> And to a reasonable degree I have shown it. And I do NOT mean by
> pointing out that Tarski does not apply.

You have not even given the rules of deduction, so how on earth have
you shown "to a reasonable degree" that provability and truth are the
same?

(For that matter, you have not given the semantics all that clearly,
but instead mentioned various statements that are neither true nor
false.)

--
Jesse F. Hughes
"I think the burden is on those people who think he didn't have
weapons of mass destruction to tell the world where they are."
-- White House spokesman Ari Fleischer

From: Daryl McCullough on 31 Mar 2010 08:39

Jesse F. Hughes says...

>You have not even given the rules of deduction, so how on earth have
>you shown "to a reasonable degree" that provability and truth are the
>same?

The only possible argument for truth and provability being the same is
to claim that the Peano axioms exhaust everything there is to say about
arithmetic; anything that's not provable from them is just not meaningfully
true or false.

I've never seen anyone try to make such an argument.

--
Daryl McCullough
Ithaca, NY

From: Jesse F. Hughes on 31 Mar 2010 08:57

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

> On Mar 30, 6:04 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>
>> You seem to have misrepresented Aatu's claims. Moreover, you're just
>> wrong. I've argued repeatedly that some sentences of the form
>>
>> ~(Ex)(P & Q)
>>
>> occur in ordinary mathematical reasoning (and hence are useful), even
>> when (Ex)P is false. An example occurred in sci.math recently.
>>
>> Simon C. Roberts gave a purported proof of FLT[1], by arguing:
>>
>> ~(En)(Ea,b,c)(a^n + b^n = c^n & a, b, c are pairwise coprime).
>>
>> Let's denote a^n + b^n = c^n by P(a,b,c,n) and a, b, c are pairwise
>> coprime by Q(a,b,c), so that Simon's argument attempts to show that
>>
>> c
>>
>> Of course, I am *not* claiming that he proved what he claims. That's
>> beside my point. A poster named bill replied that (1) is not Fermat's
>> last theorem[2], which has the form
>>
>> ~(En)(Ea,b,c) P(a,b,c,n). (2)
>>
>> Arturo responded[3] by proving
>>
>> (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ). (3)
>>
>> Hence, a proof of (1) yields a proof of (2) by modus tollens.
>
> How about this?
>
> (En)(Ea,b,c) P(a,b,c,n). Assumption
> (En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) ).
> (3)
> ~(En)(Ea,b,c) P(a,b,c,n). Modus Tollens
> ~(En)(Ea,b,c) P(a,b,c,n). RAA
>

How about it? That is certainly valid reasoning classically. I have
no idea whether it's valid in your proposed system, since you haven't
said.

But that is certainly not the argument that was offered in the other
thread. The statement

(En)(Ea,b,c) P(a,b,c,n) -> (En)(Ea,b,c)( P(a,b,c,n) & Q(a,b,c) )

was proved independently of your assumption, and no one balked. Thus,
my question remains: was the argument I gave invalid?

A second question comes to mind: what happened to your imagination
test? You've said that (because FLT is true) you cannot picture
(E a,b,c,n)( P(a,b,c,n) & Q(a,b,c) ). Yet, once you assume (contrary
to fact) that FLT is false, you *can* picture it?

You can't picture a green round triangle, right? What if I say:
assume a round triangle exists. Can you picture a green round
triangle *then*?[1] That is, does the act of making an assumption
change your capacities for imagining stuff? If not, then your
argument above doesn't work. If so, well, then your powers of
imagination are different than mine.

Footnotes:
[1] Green round triangles puzzle me. As far as I can tell, you find
the statement "No green triangles are round" meaningful, but the
statement "No round triangles are green" meaningless.

--
"[I]n mathematics there are two types of integers: primes and
composites. [...] It's like how in the world there are mostly two
kinds of people: male and female [...] and lots of reasons for
interest in the differences." -- JSH on math/biology

From: Tim Golden BandTech.com on 31 Mar 2010 10:02

On Mar 30, 4:29 pm, Transfer Principle <lwal...(a)lausd.net> wrote:
> On Mar 27, 7:02 am, "Tim Golden BandTech.com" <tttppp...(a)yahoo.com>
> wrote:
>
> > On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> > > Any rate, enough talk. Do you have even a single absolute truth you
> > > could show me so that I'd realize I've been wrong all along? Let's
> > > begin with the natural numbers: which formula in the language of
> > > arithmetic could _you_ demonstrate as absolutely true?
> > There is a fairly straightforward construction that can yield both
> > boolean logic and continuous higher forms, and even a lower form that
> > I will call universal.
> > Constrain the real numbers to those values whose magnitude is unity.
> > We see two options
> > +1, -1 .
> > Using polysign numbers extend this system to P3.
>
> Ah yes, the polysign numbers. I still remember Golden's constuction
> of these sets.
>
> > One might initially consider there to be a three verticed logic here,
> > but in the general form we see that the unity values now form a
> > continuous circle.
>
> There was a discussion of alternate-valued logic back when tommy1729
> proposed using three-valued logic (tommy1729 being, of course, one of
> Golden's biggest supporters). But we found out that usually, standard
> theorists object to these alternate forms of logic.
>
> For one thing, standard theorists obviously accept two-valued Boolean
> logic (FOL), and they appear to be open to continuum-valued logic
> (also called "fuzzy" logic). But they tend to object to kappa-valued
> logic, where kappa is a cardinal that is strictly between two and the
> cardinality of the continuum.
>
> > In two dimensions we see that the same procedure yields a continuum of
> > values, though there are arguably those three unique positions
> > -1, +1, *1 .
>
> But notice that Golden does acknowledge a continuum of values. So
> perhaps this could be a form of fuzzy logic that the standard
> theorists
> might accept as well.

Thank you TP.

>
> There's a huge difference between fuzzy logic and Golden's though. For
> fuzzy logic usually considers the values to lie in the interval [0,1],
> with
> 0 being false and 1 being true. Golden's fuzzy logic is decidedly
> _not_
> described by the interval [0,1] at all.
>
> Golden regularly points out that multiplication on the set {-1,+1,*1}
> in
> P3 is isomorphic to addition in the group Z/3Z.

No, P3 are the complex numbers in a different format. I refute the
ring quotient construction method because the polynomial is ill
constructed within abstract algebra's own definition of ring. There is
no addition/multiplication crossover as you suggest. The equivalence
is well beyond isomorphism. These are literally the same numbers in
two different formats. For instance one might use (r,theta) notation
for a complex value or a+bi notation, and now a P3 value can be used
on the same level, but with no further rules created than the real
numbers already contained. P3 and P2 are merely members of the same
family, and their little sibling P1 is quite a simple contraption that
most can barely catch a whiff of from their cartesian base.

> In general, Golden
> wishes to construct an n-valued logic by considering a subset of Pn
> that's isomorphic to addition in the group Z/nZ, which is also
> isomorphic to the multiplicative set of nth roots of unity in C. (As
> was
> discussed in many previous Golden threads, _addition_ in Pn is not
> isomorphic to _addition_ in C, but this current subthread only deals
> with multiplication, not addition.)

No. Addition is easily the same as addition in the complex numbers.
Polysign numbers form a vector space which exposes simplex geometry as
fundamental. P3 are identical to the complex numbers, just in a
different format. Pn are well behaved algebraically; consistent with
the ring definition.

>
> Thus, our continuum-valued logic can be described by the entire unit
> circle in C, not the unit interval [0,1].

Yes. And that this is merely the generalization of the binary logic
when constructed as I've layed it out. Within physics struggles go on
about circular construction within fundamentals. Circular thinking
abounds and perhaps what we have is a stability criterion, for if
A -> B -> C -> A
then we have a stable loop. Whether there is actually a continuum here
could become a linguistic problem. We are using a discrete language
and this may necessarily deny us some purity, yet within this language
we have no upper limit on the ways that we can say something. Many
attempt the shortest possible rendition, yet that can be handicapping.
As the length of the rendition rises then perhaps an attempt at the
continuous form is made even while in the discrete language.

>
> Now we ask ourselves, is such a logic even possible. Back in the
> tommy1729 three-valued logic threads, the standard theorists often
> pointed out that what they needed to see were the laws of inference
> for any proposed logic. Without laws of inference, one can't really
> call
> it a logic at all.

In order to aquire the binary logic from the continuum (and polysign
sx are on the continuum as they use x as a continuous magnitude, s as
discrete sign) one has to constrain that continuum. Rather than using
0 and 1 it makes much more sense to use +1 and -1, for then the sign
product matches the logical behaviors, allowing for the observation
that stanard boolean or binary logic contains a modulo-2 behavior:
Not(Not(A)) = A .
- - A = A .
In that the commutative and distributive principles hold up for
polysign then there is hope for the inference generalization that you
ask for. Bateson finished his days considering the following logical
construction:

Grass dies.
Men die.
Men are grass.

The problem is open Transfer Principle.

>
> I wouldn't mind taking at the laws of inference for fuzzy logic and
> modifying them so that they work for Golden's logic. But of course, I
> don't own a textbook on fuzzy logic or its laws of inference, nor do I
> plan on owning such a book anytime soon.
>
> > By leaving the Euclidean and working the sphere these forms exist
> > naturally.
>
> Hmmm, non-Euclidean geometry and the sphere. This reminds me of
> AP's work as well. I wonder whether the AP-adics might work better
> if we used Golden's logic instead of FOL. (And before Jesse Hughes
> or anyone else protests, I'm fully aware that the link between
> Golden's
> post here and AP's work is even flimsier than that between Newberry
> and Clarke. I'm the one who's trying to unify the so-called "cranks"
> as
> best as I can in order to find a theory that will satisfy at least two
> of
> them, which saves me work from having to find a different theory for
> each and every "crank.")

Well, so far you have not falsified me and instead I have falsified
you twice.
No credentials are required to understand this. Still, thanks for the
review.

The point is TP that the binary logic, when taken as -1 and +1 are the
geometry of the sphere in one dimension. That this generalizes to a
continuous logic in two dimensions does not necessarily rely upon the
polysign numbers, but I am happy to frame the problem within them, for
there is no need of a cartesian product and so a simpler argument is
made. Here we see a nifty continuous/discrete paradigm in action that
is pure geometry, and even a little bit algebra too.

Also, since I'm sure Tommy will love to be talked about I did once
call him a dangerous ally here, but I would point out to you that his
interest in mathematics is very sincere and his innocent egotism is
actually quite refreshing compared to the third order reasoning of
some. To head toward the fundamentals requires first order reasoning,
and I believe that this is lacking in todays accumulation. How nice it
is to discuss such a simple topic as logic and consider the human
limitations of perception and language, even if just tangentially.
What if we simply are not capable of a P3 or P4 language within
existing P2 language? Duality and Universality are thus far accepted
by philosopher and physicist as principles. Can we make it up to
Triality? Thanks for the strong post.

- Tim

From: Tim Golden BandTech.com on 31 Mar 2010 10:04

On Mar 30, 11:52 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> Tim Golden BandTech.com wrote:
> > On Mar 27, 11:25 am, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >> Tim Golden BandTech.com wrote:
> >>> On Mar 26, 5:55 pm, Nam Nguyen <namducngu...(a)shaw.ca> wrote:
> >>>> Nam Nguyen wrote:
> >>>>> Alan Smaill wrote:
> >>>>>> Nam Nguyen <namducngu...(a)shaw.ca> writes:
> >>>>>>>> Seriously, if you could demonstrate a truly absolute abstract truth
> >>>>>>>> in mathematical reasoning, I'd leave the forum never coming back.
> >>>>>>> If you can't (general "you") then I'm sorry: my duty to the Zen council,
> >>>>>>> so to speak, is to see to it that "absolute" truths such as G(PA) is a
> >>>>>>> thing of the past, if not of oblivion.
> >>>>>> one day you will realise that your duty to the Zen council
> >>>>>> is to overcome your feeling of duty to what is purely subjective ...
> >>>>> I'm sure your belief in the "absolute" truth of G(PA) is subjective, which
> >>>>> you'd need to overcome - someday. Each of us (including Godel) coming to
> >>>>> mathematics and reasoning has our own subjective "baggage".
> >>>>> Is it FOL, or FOL=, that you've alluded to? For example.
> >>>> Note how much this physical reality has influenced and shaped our
> >>>> mathematics and mathematical reasonings. Euclidean postulates had their
> >>>> root in our once perception of space. From P(a) we infer Ex[P(x)]
> >>>> wouldn't be an inference if the our physical reality didn't support
> >>>> such at least in some way. And uncertainty in physics is a form
> >>>> relativity.
> >>>> The point is relativity runs deep in reality and you're not fighting
> >>>> with a lone person: you're fighting against your own limitation!
> >>>> Any rate, enough talk. Do you have even a single absolute truth you
> >>>> could show me so that I'd realize I've been wrong all along? Let's
> >>>> begin with the natural numbers: which formula in the language of
> >>>> arithmetic could _you_ demonstrate as absolutely true?
> >>> There is a fairly straightforward construction that can yield both
> >>> boolean logic and continuous higher forms, and even a lower form that
> >>> I will call universal.
> >>> Constrain the real numbers to those values whose magnitude is unity.
> >>> We see two options
> >>> +1, -1 .
> >> It's relative as to how many real numbers one could "constrain". So
> >> "constraint" is a relative notion, not an absolute one.
>
> >> In any rate, in all the below (including the URL) I still couldn't
> >> see an absolute truth. Could you state such truth here?
>
> > By accepting the generalization of sign the existence of dimension
> > follows directly.
> > That is the most absolute truth that I've come up with.
>
> So what would happen if one doesn't accept the "the generalization
> of sign"? Would we get a relative truth, or an absolute falsehood?