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

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From: Newberry on 31 Mar 2010 00:34

On Mar 30, 6:04 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > 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.
>
> 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

>
> According to you, however, if (2) is true (and I assume we all know
> that (2) was proved by Wiles), then (1) is meaningless. Yet, no one
> here balked at the claim that (1) could be used to prove (2) (once (3)
> was proved). No one here had any trouble understanding what (1)
> means. Everyone in the thread accepted this form of mathematical
> argument as beyond suspicion -- although the claim that Simon actually
> proved (1) is regarded as doubtful.
>
> So, you're just plain wrong. These statements that you call
> meaningless occur in ordinary mathematical reasoning all the time.
>
> Footnotes:
> [1] Message id
> <1917288606.455209.1269716329839.JavaMail.r...(a)gallium.mathforum.org>,
> in the thread "Another Proof of Fermats Last Theorem".
>
> [2] Message id
> <50f09d88-a96b-464c-aec5-be000f0be...(a)x23g2000prd.googlegroups.com>.
>
> [3] Message id
> <e6768d43-7706-41f4-bff8-8e666d693...(a)j21g2000yqh.googlegroups.com>.
>
> --
> "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: Newberry on 31 Mar 2010 00:46

On Mar 30, 3:51 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> 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.

Why are you stating this so categorically? Look at this excerpt from
Gaifman:

QUOTE
Line 1: The sentence on line 1 is not true.
Line 2: The sentence on line 1 is not true.

The standard evaluation rule for a sentence of the form The sentence
written in/on ... is true is roughly this:
(*) Go to ... and evaluate the sentence written there. If that
sentence is true, so is The sentence written in ... is true , else
the latter is false.
END OF QUOTE

If you apply this procedure to 1 it will never terminate, so neither T
nor F will be assigned to 1. If we apply the same procedure to 2,
knowing that 1 is ~(T v F) we obtain that 2 is true.

Another way to see this is that 1 is not expressing any possible state
of affairs. 2 is. It expresses the state of affairs that 1 does not
correspond to an actual state of affairs.

>
> >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: Daryl McCullough on 31 Mar 2010 07:23

Newberry says...
>
>On Mar 30, 3:51=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> 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.
>
>Why are you stating this so categorically?

I'm just stating what truth *means. A sentence makes (or attempts to make)
a claim about something. To understand a sentence means to understand
what is being claimed, and about what.

Look at this excerpt from
>Gaifman:
>
>QUOTE
>Line 1: The sentence on line 1 is not true.
>Line 2: The sentence on line 1 is not true.

Yes, it's a silly notion of truth that gives these two sentences
different truth values.

>The standard evaluation rule for a sentence of the form "The sentence
>written in/on ... is true" is roughly this:
>(*) Go to ... and evaluate the sentence written there. If that
>sentence is true, so is "The sentence written in ... is true", else
>the latter is false.
>END OF QUOTE

To me, the truth of a sentence is determined by what it *says*, not
be the result of an evaluation procedure. Now, of course, you could
use an evaluation procedure to *define* a property of sentences.
That's what proof within a mathematical theory does. It's an evaluation
procedure for sentences. Sentences that pass the evaluation are called
"theorems". If you are proposing a more sophisticated evaluation procedure,
then you're extending the notion of "theorem". But you're not defining
truth.

>Another way to see this is that 1 is not expressing any possible state
>of affairs.

Sure it does. You are using the word "true" to mean "evaluates to true
after applying Gaifman's evaluation procedure". So the meaning of 1
is:

"The sentence on line 1 does not evaluate to true under Gaifman's
evaluation procedure"

That's a perfectly meaningful state of affairs, and it happens to be
the case.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 31 Mar 2010 07:27

Newberry says...
>
>On Mar 30, 3:30=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

>> 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.
>
>In which theory?

It doesn't matter which theory. If it is any consistent
theory in the language of arithmetic such that every true closed
sentence of the form

t1 = t2

(where t1 and t2 are terms built up out of 0, 1, +, *)
is provable, then there are undecidable Diophantine equations.

--
Daryl McCullough
Ithaca, NY

From: Daryl McCullough on 31 Mar 2010 08:12

Newberry says...

>Why are you stating this so categorically? Look at this excerpt from
>Gaifman:
>
>QUOTE
>Line 1: The sentence on line 1 is not true.
>Line 2: The sentence on line 1 is not true.
>
>The standard evaluation rule for a sentence of the form "The sentence
>written in/on ... is true" is roughly this:
>(*) Go to ... and evaluate the sentence written there. If that
>sentence is true, so is "The sentence written in ... is true" , else
>the latter is false.

In general, any *procedure* used to evaluate the truth of sentences
in a self-referential language is incomplete, in the sense that there
are true sentences that are not evaluated as true by the procedure.

This is easy to see: Let P be some procedure to evaluate the truth
of sentences. Then consider the sentence

"When procedure P is applied to this sentence, the result is not true"

Contrary to your claims about sentences that fail to express a possible
state of the world, the above sentence makes a perfectly definite claim:
That a certain procedure applied to a certain sentence does not produce
a certain result. If the procedure is actually applied to that sentence,
and run to completion, then we can just check to see what the result is.

What this shows is that for any sound procedure that purports to evaluate
the truth of sentences, there is a true (in the sense of corresponding to
the facts) sentence that the procedure fails to evaluate as true. So *every*
procedure for evaluating truth of sentences is incomplete. A procedure
can be complete for a certain *class* of sentences, for example, those
expressible in a limited language, but there will always be an extended
language that the procedure fails on.

--
Daryl McCullough
Ithaca, NY