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

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From: Daryl McCullough on 27 Mar 2010 07:59

Newberry says...
>
>On Mar 26, 3:49=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> The Liar sentence is not *expressible* in any standard mathematical
>> theory (PA or ZFC). So you don't have to do anything to keep the Liar
>> from spoiling the consistency of those languages.
>
>Why you think you have to tell me that I do not know. If you lool a
>few lines above you will see that I was talking about the Liar paradox
>in the natural language.

So your theory of truth gaps is only for natural language? So you
agree that formal languages such as arithmetic don't require any
truth gaps?

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 27 Mar 2010 08:02

Nam Nguyen says...
>
>Alan Smaill wrote:
>
>>> 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".
>>
>> Why on earth do you think I have some belief in the " "absolute" truth "
>> of G(PA) ? I don't even know what that *means* .
>
>OK. Then on the meta level, do you think it's correct to say that
>G(PA) can be arithmetically false?

It is a *relative* truth. It's true in the standard interpretation
of the language of PA.

--
Daryl McCullough
Ithaca, NY

From: Jesse F. Hughes on 27 Mar 2010 09:26

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

> On Mar 26, 3:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Newberry <newberr...(a)gmail.com> writes:
>> > On Mar 25, 3:49 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>> > wrote:
>> >> Newberry says...
>>
>> >> >Tarski's theorem does not apply to formal systems with gaps. I think
>> >> >it is preferable.
>>
>> >> If you the way you express Tarski's theorem is like this, then truth
>> >> gaps don't change anything:
>>
>> >> There is no formula T(x) such that if x is a Godel code of a true
>> >> sentence, then T(x) is true, and otherwise, ~T(x) is true.
>>
>> >> Anyway, *why* is it preferable to have a formal system for which Tarki's
>> >> theorem does not apply? Preferable for what purpose?
>>
>> > If truth is expressible then truth can be equivalent to provabilty.
>>
>> So, you'd like to redefine truth (so that vacuously *true* statements
>> aren't true) and also redefine provability (so trivially provable
>> statements aren't provable) in such a way that truth is equivalent to
>> provability.
>>
>> Then what have you accomplished? Hell, I can do that simply by
>> requiring that nothing is true and nothing is provable. My "fix" is
>> better than yours, insofar as we can see that it actually "works".
>
> My theory has some significant advantages over yours. I can go to a
> grocery store and count how many tomatoes and bananas I have picked.
> If I have picked 2 small tomatoes and three large tomatoes my theory
> can prove that I have 5 tomatoes. Also at the checkout counter I can
> calculate the total price. Can your theory do that?

No. You're right. The classical theory of arithmetic is incapable of
proving that 2 + 3 = 5.

I see now that your theory is superior and will alter my brain
accordingly.

(Honestly, I have no idea what you're talking about. You seem to see
a disadvantage in classical arithmetic that I simply don't see. Why
not explain your point?)

--
Jesse F. Hughes

"I post for many reasons [...] and there's no reason to think that
I'll stop." -- James S. Harris

From: Jesse F. Hughes on 27 Mar 2010 09:27

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

> On Mar 26, 3:56 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> Newberry <newberr...(a)gmail.com> writes:
>> > On Mar 25, 3:05 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> >> MoeBlee <jazzm...(a)hotmail.com> writes:
>> >> > On Mar 25, 1:00 pm, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>> >> > wrote:
>>
>> >> >> Wikipedia has a list of theorems of classical logic that it calls
>> >> >> "paradoxes of material implication":
>>
>> >> >>http://en.wikipedia.org/wiki/Paradoxes_of_material_implication
>>
>> >> >> There's nothing paradoxical about any of them
>>
>> >> > The discussion there about (P&Q) -> R |- (P -> R) v (Q -> R)
>>
>> >> > is at least somewhat interesting.
>>
>> >> Yes, but their example ("If I close switch A and switch B, the light
>> >> will go on. Therefore, it is either true that if I close switch A the
>> >> light will go on, or that if I close switch B the light will go on.")
>> >> is poorly chosen, since P, Q and R stand for propositions, while "I
>> >> close switch A (or B)" is an action. (I'm not sure what type of
>> >> sentence "The light will go on," is -- it's not an action, in the
>> >> sense of dynamic logic, but rather it describes a change in the
>> >> world.)
>>
>> > Do you think that propositions cannot be about actions?
>>
>> Sure, they can, but "I close the switch" is not a proposition. "I am
>> closing the switch" or "I have closed the switch" are propositions.
>
> I am not following.

Well, it is not all that relevant.

--
"If you are a mathematician, then you cannot dispute the result. If
you dispute the result [...] then you are NOT a mathematician. Anyone
who disputes this result [...] is not a mathematician. I am a
mathematician, which is how I could find the result."--James S. Harris

From: Tim Golden BandTech.com on 27 Mar 2010 10:02

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 .

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