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

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From: Newberry on 30 Mar 2010 00:21

On Mar 29, 3:30 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
>
>
>
>
> >On Mar 28, 4:37=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> Newberry says...
>
> >> >On Mar 27, 4:56=3DA0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >> >wrote:
> >> >> Newberry says...
>
> >> >> >L: ~T(L)
>
> >> >> >If v(L) =3D3D ~(T v F) then there is no contradiction. L is not true.
>
> >> >> But *if* T is a truth predicate, then "L is not true" is formalized
> >> >> by the statement ~T(L).
>
> >> >> >The argument usually goes "but that is what L says." But L does not
> >> >> >say anything.
>
> >> >> It says "L is not true".
>
> >> >> So your proposed resolution is complete nonsense.
>
> >> >It contains the string "L is not true", but it does not "say" that L
> >> >is not true
>
> >> That's completely silly.
>
> >Did you read this?
> >http://www.columbia.edu/~hg17/gaifman6.pdf
>
> Yes, and I think it's silly. He wants to associate truth
> with sentence tokens (occurrences of sentences, rather
> than sentences themselves), so that two identical sentences
> may differ in truth values. There is a sense in which that
> is necessary, when referring expressions are used (for example,
> if I say "That is a cat", obviously some tokens are true,
> when I'm pointing to a cat, and some tokens are false, when
> I'm pointing to a dog).
>
> However, if you have two sentence tokens, and they have
> the same subject, and the same predicate, it's silly to
> call one true and one false (or more generally, not true).
> It's a silly resolution that doesn't resolve anything. If
> "true" is a predicate applying to sentence *tokens*, then
> we can invent a second predicate, "truthy" applying to
> sentences:
>
> A sentence X is truthy if at least one of its tokens is true.
>
> Then you can form a new Liar paradox:
>
> This sentence is not truthy.
>
There are two issues here.
a) The two tokens have the same subject and the same predicate.
b) The resolution can be semingly defeated by forcing all tokens into
one type.

Not sure why you think they are related. Maybe they are but it is not
immediately obvious to me.

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. The former is ~(T v F), the
latter is T.

From: Newberry on 30 Mar 2010 00:41

On Mar 28, 9:04 pm, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > On Mar 28, 5:50 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> >> Anyway, I won't really defend my theory. My point is: you claim that
> >> your approach may yield a theory in which truth and provability are
> >> equivalent. Ignoring the fact that this is wishful thinking thus far,
> >> so what? You do so only by redefining what truth means, so that
> >> vacuously true statements are not true. I don't see any advantage to
> >> that.
>
> > Do you agree that Tarki's theorem does not apply to systems with
> > gaps?
>
> You keep saying so. Although I haven't looked up the reference, I
> assume that you're not mistaken.

"The proof by Goedel and Tarski that a language cannot contain its
own
semantics applied only to languages without truth gaps."

Outline of a Theory of Truth
Saul Kripke
The Journal of Philosophy, Vol. 72, No. 19, Seventy-Second Annual
Meeting American Philosophical Association, Eastern Division. (Nov.
6,
1975), p. 714.

>
> > DO you agree that if we say that the vacuous sentences are neither
> > true nor false that we will have gaps?
>
> Sure. I also think that if we simply say "1 + 1 = 2" is neither true
> nor false (while every other formula is interpreted in the standard
> way), then we have a theory with gaps. It does not follow, of course,
> that truth and provability are the same in this theory, nor that my
> new and improved notion of truth is sensible.

Indeed. But if we leave out all the vacuous sentences we can still do
all the useful arithmetic as we know it. Although all the people on
this board believe that such sentences are true nobody argued that
they were useful. Aatu even said that they did not belong in ordinary
mathematical reasoning. Furthermore there is a reason to think that
they are neither true nor false. I cannot think of any good reason for
claiming that 1 + 1 = 2 is not true.
>
> --
> "There's lots of things in this old world to take a poor boy down.
> If you leave them be, you can save yourself some pain.
> You don't have to live in fear, but you best have some respect,
> For rattlesnakes, painted ladies and cocaine." -- Bob Childers

From: Aatu Koskensilta on 30 Mar 2010 01:17

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

> But if we leave out all the vacuous sentences we can still do all the
> useful arithmetic as we know it. Although all the people on this board
> believe that such sentences are true nobody argued that they were
> useful. Aatu even said that they did not belong in ordinary
> mathematical reasoning.

You're imagining things.

> Furthermore there is a reason to think that they are neither true nor
> false. I cannot think of any good reason for claiming that 1 + 1 = 2
> is not true.

Well, perhaps you might be moved to answer the following query, which I
have now presented on several occasions:

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 -- owing, so I gather, to your
eager zeal for equating formal provability with truth, contra G�del and
his lackwit lackeys -- regardless of whether the corresponding
Diophantine equations are soluble or not. What are we to make of this?
In what way, and just how, do your musing connect to our mathematical
experience or reasoning? You may of course introduce whatever technical
terminology you wish, defining truth and falsity whichever way you see
fit, to suit your whim and fancy, but unless answers to questions like I
have presented are forthcoming your fiddling will be of no apparent
interest in the wider scheme of things in mathematics, in the philosophy
of mathematics, in our everyday arithmetical reasoning, thinking,
reflection.

It's presumptuous of me, but I nonetheless surmise it is at least in
part a hope of yours, in these your endless mumblings Usenetical on
matters logical, the liar, vacuity, what not, that others come to
recognise the wonderful clarity of your theory of meaning, your take on
logic, your this and that. On this surmise, I can only suggest you make
some attempt to demonstrate the value of your insight to the skeptic
masses by showing how some conundrum can be exorcised, some problematic
enigma eradicated, by way of transparent and compelling reasoning
involving your novel insights, some traditional baffler dissected with
particular cunning made possible by the notions you champion, and so
on. You could do well to emulate Dummett or Kreisel, neither of whom I'm
an unreserved fan, but who never fail to stimulate and inspire, even if
it is only to cry out in philosophical exasperation.

--
Aatu Koskensilta (aatu.koskensilta(a)uta.fi)

"Wovon man nicht sprechan kann, dar�ber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Nam Nguyen on 30 Mar 2010 01:47

Jim Burns wrote:

> I think it would be very useful to me in understanding
> what you are trying to accomplish if you were
> to give a summary of the best arguments AGAINST your
> positions.

As promised I'll summarize what I think as the best arguments
against my positions. The caveats however are a) these are my
own opinions and might not reflect what they might have actually
thought and b) it seems to me a lot of arguments went back and
forth and then became "dangling" in the sense that key questions
weren't answered, terminologies were quite agreed, definitions
weren't accepted, et...; consequently my discerning the nature
of their arguments might not be 100% accurate.

So, I could only do my best in the summarization, by extrapolating
and triangulating from what were said in this thread or in the past
ones that are directly relevant to the arguments.

***

Imho, the 3 major and best arguments against my belief, that the
nature of FOL reasoning is that of relativity or of being subjective,
are the following objections:

(O1) The Universality Objection:

In this objection, the correctness in reasoning under one logical
framework should be _universally constant_ and shouldn't be a
function of individual subjective beliefs or knowledge. My claiming
on the relativity nature of FOL reasoning seems to violate this natural
and unobjectionable, say, "sanctity".

(O2) The Philosophy Objection:

In this objection, the ideas such that there are formulas written
in the language of arithmetic that can be neither arithmetically
true nor false are just philosophical ideas and thus can't be a
basis to attack the current FOL reasoning.

(O3) The Ordinary Mathematics Objection:

This objection seems to be a cross-breed between O1 and O2. In this
objection, FOL reasoning is build upon the ordinary mathematical
knowledge that *in principle* should be universally _self evident_
to all who are trained or study mathematics. As such FOL reasoning
should be universally the same and should _not_ be subjective or
relativistic.

On O1, perhaps the following conversation between DCU and me on Dec 22,
2005 would best reflect the objection (the argument) against my
"reasoning relativity" position. It's in the thread "About Consistency
in 1st Order Theories" where DCU gave some critique comments on my desire
of changing FOL to make it conform to relativity of reasoning.

NN: Specifically I'd like to change FOL in such a way that mathematical
reasoning is no longer absolute: the reasoning agent's limited
(finite) reasoning knowledge, and his/her freedom to make certain
interpretation on certain levels of introspection, would all affect
the results of the reasoning.

DCU: If you want people to help you with this you might start by
trying to convince people that there's a _need_ for this
radical new version of logic.

Exactly what the objective is is not clear to me. It seems
possible that you might want to call it something other than
"logic".

Because whatever it is, it seems that in the thing you're
looking for the "logic" is going to vary from person to person,
and I suspect it's going to seem to a lot of people like the
whole point to _logic_ is to study _correct_ reasoning, which
will _not_ vary from person to person.

Now, the class of mathematical facts that a given individual
is actually able to prove certainly varies from person to person.
If you want to study that somehow fine, but that seems more
a topic in something like psychology than pure logic. If I'm
correct in thinking that in the system you have in mind
the _definition_ of correct reasoning is going to vary
from person to person that seems even less like "logic".

That's about 5 years ago and DCU isn't participating in the current
thread and I don't know if he has changed his opinion on the subject
of relativity in reasoning I've have been pursuing since. But apparent
from his comments above are counter arguments against my position of
mathematical relativity in reasoning.

On O2, and O3, I believe some of AK's conversations in this thread
and others in the past and some of TF's writings would reflect these
2 objections in various degrees. I have to admit though I don't have
right off specifics in what AK said and might need more time to search
for some samples. But let me excerpt some of what TF wrote about ordinary
mathematical truth that should be universally accepted and "there is
no need to assume that we are introducing any problematic philosophical
notions".

In TF's Chapter 2, "G�del�s Theorem" one would see:

In a mathematical context, on the other hand, mathematicians easily
speak of truth. If the generalized Riemann hypothesis is true...,
There are strong grounds for believing that Goldbach�s conjecture is
true..., If the twin prime conjecture is true, there are infinitely
many counterexamples.... In such contexts, the assumption that an
arithmetical statement is true is not an assumption about what can be
proved in any formal system, or about what can be seen to be true, and
nor is it an assumption presupposing any dubious metaphysics. Rather,
the assumption that Goldbach�s conjecture is true is exactly equivalent
to the assumption that every even number greater than 2 is the sum of
two primes, the assumption that the twin prime conjecture is true means
no more and no less than the assumption that there are infinitely many
primes p such that p+2 is also a prime, and so on. In other words the
twin prime conjecture is true is simply another way of saying exactly
what the twin prime conjecture says. It is a mathematical statement,
not a statement about what can be known or proved, or about any relation
between language and a mathematical reality. Similarly, when we talk
about arithmetical statements being true but undecidable in PA, there
is no need to assume that we are introducing any problematic philosophical
notions. That the twin prime conjecture may be true although undecidable
in PA means simply that it may be the case that there are infinitely many
primes p such that p+2 is also a prime, even though this is undecidable
in PA.

Again, these are only my thoughts of what the arguments against my position
be. It would certainly be helpful if those who oppose my position could
further clarify in technical clarity what they perceive are problems in my
positions. I also don't mind in subsequent posts to further defend my position
or to provide more counter-arguments.

Cheers,

-Nam Nguyen

From: Newberry on 30 Mar 2010 01:55

On Mar 29, 10:17 pm, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > But if we leave out all the vacuous sentences we can still do all the
> > useful arithmetic as we know it. Although all the people on this board
> > believe that such sentences are true nobody argued that they were
> > useful. Aatu even said that they did not belong in ordinary
> > mathematical reasoning.
>
> You're imagining things.
>
> > Furthermore there is a reason to think that they are neither true nor
> > false. I cannot think of any good reason for claiming that 1 + 1 = 2
> > is not true.
>
> Well, perhaps you might be moved to answer the following query, which I
> have now presented on several occasions:
>
> 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.

> -- owing, so I gather, to your
> eager zeal for equating formal provability with truth, contra G del and
> his lackwit lackeys --

You never answered my question what you ment by "Goedel."
a) Every system capable of arithmetic is syntactically incomplete
b) In every system capable of arithmetic there are true but unprovable
formulae.

> regardless of whether the corresponding
> Diophantine equations are soluble or not. What are we to make of this?
> In what way, and just how, do your musing connect to our mathematical
> experience or reasoning? You may of course introduce whatever technical
> terminology you wish, defining truth and falsity whichever way you see
> fit, to suit your whim and fancy, but unless answers to questions like I
> have presented are forthcoming your fiddling will be of no apparent
> interest in the wider scheme of things in mathematics, in the philosophy
> of mathematics, in our everyday arithmetical reasoning, thinking,
> reflection.
>
> It's presumptuous of me, but I nonetheless surmise it is at least in
> part a hope of yours, in these your endless mumblings Usenetical on
> matters logical, the liar, vacuity, what not, that others come to
> recognise the wonderful clarity of your theory of meaning, your take on
> logic, your this and that. On this surmise, I can only suggest you make
> some attempt to demonstrate the value of your insight to the skeptic
> masses by showing how some conundrum can be exorcised, some problematic
> enigma eradicated, by way of transparent and compelling reasoning
> involving your novel insights, some traditional baffler dissected with
> particular cunning made possible by the notions you champion, and so
> on.

This will be impossible if the skeptic is not willing to discuss
anything specific I have written and only utters general derogatory
epithets.

> You could do well to emulate Dummett or Kreisel, neither of whom I'm
> an unreserved fan, but who never fail to stimulate and inspire, even if
> it is only to cry out in philosophical exasperation.
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, dar ber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus