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

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From: Newberry on 31 Mar 2010 23:57

On Mar 31, 4:23 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> 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.

Indeed. A sentence of the form "The sentence written in/on ... is
true" says that the sentence written at ... is true. Naturally then a
sentence of the form "The sentence written in/on ... is true" is true
If that if the sentence written at ... is true, otherwise the former
is false.

> 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".

I do not know. Yourr definition seems circular.

> 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: Newberry on 1 Apr 2010 00:11

On Mar 31, 5:57 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(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?

It is valid but it makes a silent assumption.

>
> 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?

No.
>
> 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]

No.

> That is, does the act of making an assumption
> change your capacities for imagining stuff?

No.

> If not, then your
> argument above doesn't work.

Why not. Assume there are such things as meaningless sentences. These
sentences are well formed. Well formed sentences can go through
syntactic transformations.

> 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- Hide quoted text -
>
> - Show quoted text -

From: Daryl McCullough on 1 Apr 2010 00:28

Newberry says...
>
>On Mar 31, 4:23=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
>wrote:

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

....

>> 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 procedur=
>e,
>> 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".
>
>I do not know. Your definition seems circular.

It's not *my* definition. It's Gaifman's. He describes a procedure
for evaluating sentences, which either returns "true", "false", or
nothing at all.

The self-referential sentence

"Gaifman's evaluation procedure when applied to this sentence
does not return 'true'"

certainly defines a "possible state of affairs".

--
Daryl McCullough
Ithaca, NY

From: Newberry on 1 Apr 2010 00:49

On Mar 31, 7:30 am, Aatu Koskensilta <aatu.koskensi...(a)uta.fi> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > No.
>
> I see. So you agree that for any formal theory T which is an extension
> of Robinson arithmetic, either directly or through an interpretation,
> and in which we can express statements of the form "the Diophantine
> equation D(x1, ..., xn) = 0 has no solutions", there are infinitely many
> true statements (of the form "the Diophantine equation D(x1, ..., xn) =
> 0 has no solutions") that are unprovable in T if T is consistent?
>

Let me ask you something. Let

~(Ex)(Ey)(Pxy & Qy) (1)

be Goedel's sentence. [Pxy: x is a proof of y, Qy is such that only
one y=m satisfies it & m = #(1)]. You are absolutely convinced that PA
is consistent i.e. (1) is true. That is, the search for x and y will
never terminate. How do you know that the search for x and y will
never terminate?

> > You never answered my question what you ment by "Goedel."
>
> You have asked me what I mean by "Goedel"?
>
> --
> Aatu Koskensilta (aatu.koskensi...(a)uta.fi)
>
> "Wovon man nicht sprechan kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus

From: Daryl McCullough on 1 Apr 2010 06:09

Newberry says...

>Let me ask you something. Let
>
>~(Ex)(Ey)(Pxy & Qy) (1)
>
>be Goedel's sentence. [Pxy: x is a proof of y, Qy is such that only
>one y=m satisfies it & m = #(1)]. You are absolutely convinced that PA
>is consistent i.e. (1) is true. That is, the search for x and y will
>never terminate. How do you know that the search for x and y will
>never terminate?

Suppose you find two natural numbers m and n such that,
P(n,m) & Q(m). Then you can easily prove (Ex) (Ey) (Pxy & Qy).
You can also "decode" n to get a proof of ~(Ex)(Ey)(Pxy & Qy).
So you would, in that case, have a proof of a contradiction.

If your axioms are consistent, the the above case cannot happen.
To say that "the above case cannot happen" is to say that there
are no natural numbers m and n such that P(n,m) & Q(m). Which
is formalized as ~(Ex)(Ey)(Pxy & Qy).

So the assumption that your system is consistent directly leads
to conclusion (1).

--
Daryl McCullough
Ithaca, NY