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

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From: Daryl McCullough on 30 Mar 2010 06:30

Newberry says...
>
>On Mar 29, 10:17=A0pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:

>> How are we to apply your ideas about vacuity, meaningfulness, truth,
>> proof, what not, in context of the following mathematical observation:
>> for any consistent theory T extending Robinson arithmetic, either
>> directly or through an interpretation, in which statements of the form
>> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions" can
>> be expressed, there are infinitely many Diophantine equations
>> D(x1, ...,xn) = 0 that have no solutions but for which
>> "the Diophantine equation D(x1, ..., xn) = 0 has no solutions"
>> is not provable in T.

>> On an ordinary understanding, a statement of the form "the Diophantine
>> equation D(x1, ..., xn) = 0 has no solutions" is true just in case
>> D(x1 ..., xn) = 0 has no solutions. According to your account some
>> such statements are neither true nor false
>
>No.

What do you mean, no? You are proposing to equate truth and provability.
Some sentences of the form "the Diophantine equation D(x1, ..., xn) = 0
has no solutions" are neither provable nor refutable. It follows from
your equating of truth and provability that they are neither true nor
false.

The reasoning that there are statements that are true, but unprovable
goes like this:

1. Let D(x1,...,xn) be a polynomial in the variables x1, ..., xn.

2. If m1, ..., mn are n integers such that D(m1,...,mn) = 0,
then the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0"
is provable. We can easily prove this by plugging in m1, ..., mn and
checking to see if the result is 0.

3. If the statement "There exists x1, ..., xn such that D(x1,...,xn) = 0"
is not provable, then there are no integers m1, ..., mn such that
D(m1, ..., mn) = 0. This follows immediately from 2.

4. Note that if Phi is the formula
"There exists x1, ..., xn such that D(x1,...,xn) = 0",
then 3. has the form: "If Phi is not provable, then ~Phi".
In other words, if Phi is not provable, then the negation of Phi
holds.

5. Therefore, if Phi is neither provable nor refutable, then
the negation of Phi holds. So if Phi is neither provable nor
refutable, then Phi is false. ("Phi is false" means the same
thing as "The negation of Phi holds").

6. Therefore, if Phi is neither provable nor refutable, then
there is a statement, Phi, that is false, but not provably false.
There is another statement, ~Phi that is true, but not provable.

7. Therefore, if there is a statement Phi (of the appropriate
form) that is neither provable nor refutable, then provability
and truth are not the same.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 30 Mar 2010 06:34

Newberry says...

>Indeed. But if we leave out all the vacuous sentences we can still do
>all the useful arithmetic as we know it.

The undecidable statements of a theory are *not* vacuous. If you
have a statement of the form "there is no solution to the
polynomial equation D(x1, x2, ..., xn) = 0", it's not vacuous.
It tells us something very specific, namely that a search for
solutions to the polynomial equation will never succeed. Such a
statement may be undecidable in a particular theory, but it's
not vacuous.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 30 Mar 2010 06:51

Newberry says...

>There are two issues here.
>a) The two tokens have the same subject and the same predicate.
>b) The resolution can be seemingly defeated by forcing all tokens into
>one type.
>
>Not sure why you think they are related.

Any theory of truth that is worth considering, if two sentence
tokens have the same subject and same predicate, then they have
the same truth value. Otherwise, your notion of truth is unconnected
with the meaning of sentences.

>Let's take a) first. Gaifman's evaluation procedure is such that if
>two tokens have the same subjects and predicates one can nevertheless
>be true and the other neither true nor false.
>
>Now b):
> This sentence is not truthy.
> "This sentence is not truthy" is not truthy.
>
>These two sentences have the same subjects and predicates. The former
>is self-referential the latter is not.

Using Godel coding, you can eliminate direct self-reference and thereby
make the two sentences identical. Then it is a contradiction to say that
one is truthy and the other is not.

--
Daryl McCullough
Ithaca, NY

From: Tim Golden BandTech.com on 30 Mar 2010 09:06

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.

The logical concept of true/false is a binary logic. It is discrete.
One could treat the signs which are already discrete as the basis for
boolean logic, but I find the spherical construction that I gave more
interesting. From that breed we see the binary option's next stage is
on a continuum rather than a discrete triplet. The spherical logic is
likewise interesting to physical concepts, where working in a 4D space
which is constrained to unity magnitude elements we see our ordinary
3D world locally and we see an upper limit on universe size.

This spherical paradigm allows the definition of translation as
rotation, whereas the Euclidean version will have us always defining
rotation in terms of translation. This is an important feature but I
admit it is askance to the OP's problem.

I do find a continuum logic acceptable. In philosophical problems when
one feels uncertain of a conclusion then we should not assume that the
individual's interpretation is flawed. Qualities of many sorts can
fall on a continuum, and the dimensionality of the problem leaves us
with the obvious option of constraining some of those qualities in
order to make a true statement. Each different wording of a statement
makes subtle changes in sensibility and to even admit that there is a
best wording is to accept that the lesser wordings were on a continuum
of truth. To accept that in the future a finer wording might exist
leaves open all interpretations to superior replacements. When we cast
our individual belief into the socially accepted version we should
only do so partially, if we hope to find superior replacements.

Nam, you are too cryptic. You have already presupposed some conclusion
that you have not stated clearly. I suppose that the only statement
you will find acceptable as absolutely true is:
true is true;
but then there will be no productive work to do. If you refute the
existence of absolute truth then I believe you are in support of a
truth continuum.

- Tim

>
>
>
> > Using polysign numbers extend this system to P3.
> > (http://bandtechnology.com/PolySigned)
> > 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.
> > This is a nice exercise in the continuous/discrete paradigms of
> > throught. In one dimension we see a discrete type, not unlike charge.
> > 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 .
>
> > Inspecting the product logic back in P2 (the boolean or constrained
> > real number case)
> > - + = -
> > + - = -
> > + + = +
> > - - = +
> > and likewise in the three signed case (overlooking the above
> > redundancy)
> > - - = +
> > - + = *
> > - * = -
> > + + = -
> > + * = +
> > * * = *
>
> > Does a false false yield a true? The english language discourages the
> > usage of double negatives, yet their use does exist within in it with
> > such phrases as
> > 'I am not an atheist.'
> > Back in ordinary logic it is no problem to see that the math holds up
> > in P2 so that
> > Not(Not(A)) = A.
> > The meaning of false and true cannot be reused in P3 and it is a nice
> > human puzzle to consider that we and our dualistic thought patterns
> > have artificially limited us. The P3 language is not sensible to the
> > human mind, yet it may be entirely accurate.
>
> > Treading on P1 is difficult for most, but there we see just one
> > instance within this logical paradigm
> > -1 .
> > Thus the polysign allow for a universal but fairly inanimate form at
> > the bottom of the hierarchy
> > universality
> > duality
> > triality (not to be confused with Clifford form)
> > ...
> > By leaving the Euclidean and working the sphere these forms exist
> > naturally.
>
> > - Tim

From: Jesse F. Hughes on 30 Mar 2010 09:07

Newberry <newberryxy(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, indeed, the word "equate" is still misleading, since it suggests
that define true to mean "provable". That can certainly be done. I
can say that, hereafter, when I say that a statement of PA is true, I
mean that there is a proof of P in PA. Of course, such semantic play
is unsatisfactory.

--
"After years of arguing I realize that your intellects are too limited
to fully grasp my work. [...] Still, no matter how child-like your
minds are, [...] since you have language, [...] there's a chance that
I'll be able to find something that your minds can handle." --JSH