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

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From: Newberry on 26 Feb 2010 00:07

On Feb 25, 11:33 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > And we believe that PA is consistent because all its axioms are
> > manifestly true? Is
>
> > "All unicorns have two horns"
>
> > manifestly true?
>
> It's true if no unicorns exist, obviously.
>
> Let's ask another question, one which you have never answered as far
> as I recall. I do not know whether Goldbach's conjecture is true or
> false, but I do know that there are no counterexamples to GC that are
> less than, say, 27. In fact, this is easy to check. Thus, it seems
> to me perfectly reasonable to say that:
>
> (Ax)( if x is a counterexample to GC, then x >= 27 )
>
> That's a true statement, as far as I'm concerned.

True statement is this

~T[(Ex)((x is a counterexample to GC) & (x < 27))

> How about you? You don't know whether it's true or meaningless unless
> you know whether there are counterexamples to GC.

True.

> Thus, you can't say
> whether my proof is a real proof or not, until you determine whether
> GC is true.

Yes, I can. Please refer to my second order statement above.

> If GC is true, then my proof must not be a proof (since
> its conclusion is meaningless). If GC is false, then my proof is a
> proof after all.
>
> Is this sensible to you? Is my "theorem" true or meaningless or are
> you unable to decide which?
>
> --
> Jesse F. Hughes
> "And hey, if you're moping and miserable because mathematics tests you,
> then maybe, if you think you're a mathematician, you might want to try
> a different field." -- Another James S. Harris self-diagnosis.

From: Nam Nguyen on 26 Feb 2010 00:17

Nam Nguyen wrote:
> MoeBlee wrote:
>> On Feb 25, 11:22 am, Newberry <newberr...(a)gmail.com> wrote:
>>
>>> And we believe that PA is consistent because all its axioms are
>>> manifestly true?
>>
>> They're clearly true to a lot of people. They seem clearly true to me.
>> Which axiom of PA doesn't strike you as true?
>
> How about the very first axiom listed in Shoenfield's book:
>
> N1. Sx =/= 0
>
> which is _clearly false_ to me at this moment, when I'm thinking of
> the (rather "natural") integers!
>
> The moral of the story: "clearly true" is very subjective and has
> no firm basis for deciding what's true or false. But model/relation
> definition conformance would decide what's true or false. Naturally.

In fact, such (model/relation definition) conformance is at the heart
of how modern standard mathematics truth/falsehood is "concretely"
defined, a la Tarski.

As we know, Tarski's concept of truth

http://en.wikipedia.org/wiki/Semantic_theory_of_truth

would go as:

(*) "Snow is white" is true if and only if snow is white.

The long and short of it is a relation-set R in model definition would
play the "concreteness" of the needed fact that snow is white for
"Snow is white" to be true. For example:

(**) "x < y" is true if and only if (x,y) is in R.

In the abstract world of mathematics, (x,y) being in R is as "concrete"
as we could possibly use the meaning of the word in this context.

In summary, no matter how hand-waving with phrases like "the naturals",
"constructively valid", "recursive", "innocuous", etc... in asserting
some thing as true in model theoretical contexts, precise stipulation
of model relations is required, for the "if and only if" part of (*),
or (**).

It's OK that in some cases we might not be able to _completely_ make
such stipulation. But in such cases, we have to be ready to accept
some logical consequence: we simply can't assign the truth values to
certain formulas, simply because the relations are incomplete, and
simply because that's what "if and only if" means in Tarski's concept
of truth/falsehood.

From: Newberry on 26 Feb 2010 00:18

On Feb 25, 8:18 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
> >On Feb 25, 4:07=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> >wrote:
> >> How do I arrive at
> >> such an intuition? Let's take PA. I believe it is consistent
> >> because all of its axioms are true, and you can't derive
> >> a contradiction from true statements.
>
> >This is what Torkel Franzen said. Look, Frege also believed that the
> >axioms in his system were manifestly true.
>
> Well, he was mistaken. The belief that a system is consistent
> can be mistaken.

Which of his axioms was not manifestly true?

> >Anyway, the axioms are not just "true", they are implicit definitions
> >and you still need to prove that they are consistent.
>
> You are confusing two different things: one is why do you believe
> that something is true, and how can you know for sure that you haven't
> made a mistake somewhere. You can never be absolutely, 100% sure that
> there is no mistake in your reasoning. But what's supposed to follow
> from that?
>
> >The idea that there are some Platonic truths tha we somehow fathom and
> >inject in our manifestly true axioms ... well, that is basically
> >Kant's theory that arithmetic is synthetic a priori. It is not
> >correct.
>
> Then what does truth mean to you? If it's just provability, then
> as I asked before, provability from what axioms? What's the criterion
> for choosing the axioms?

0 ~= 1 is true in the same sense as "all bachelors are not married."
That is what it means to me. The short answer to your question is that
we choose the axioms and derivation rules to mimic and formalize the
connectives, quantifiers and arithmentic as we know them from the
natural language. But this is a different issue.

From: Newberry on 26 Feb 2010 00:31

On Feb 25, 7:44 am, Frederick Williams <frederick.willia...(a)tesco.net>
wrote:
> Newberry wrote:
>
> > On Feb 25, 3:55 am, Frederick Williams <frederick.willia...(a)tesco.net>
> > wrote:
> > > Newberry wrote:
> > > > If the theory is sound then what it can or cannot prove will depend on
> > > > the interpretation.
>
> > > What do you mean by interpretation? If you mean a model plus a function
> > > that assigns entities in the model to linguistic entities in the theory,
> > > then that's not so: proof is a syntactic matter and does not depend on
> > > interpretation (in the sense I use the word).
>
> > How do you propose to define soundness?
>
> A theory is sound if all its theorem are true in all models (of the
> appropriate type).

It seems to me that it follows that what a sound theory can prove
depends on what you consider true, does it not?

From: Nam Nguyen on 26 Feb 2010 00:48

MoeBlee wrote:
> On Feb 24, 11:00 pm, Newberry <newberr...(a)gmail.com> wrote:
>
>> How can you arrive at the conclusion that something is true other than
>> by a proof?
>
> We can use the ordinary model theoretic notion of truth, and prove a
> sentence to be true in a particular model but not provable in a
> particular theory. For example in Z set theory we prove that the Godel
> sentence for PA is true in the standard model for the language of PA
> while the Godel sentence for PA is not provable in PA. (And 'true in
> the standard model for the language of PA' is often spoken of as just
> 'true'). And even aside from such formalities, given just ordinary
> methods of mathematical reasoning (ingeniously applied) we may safely
> conclude that the Godel sentence for PA is true.

> I mean, if you doubt
> ordinary arguments that the Godel sentence for PA is true, then you
> might as well doubt a whole range of ordinary mathematical arguments.

I heard this "scare tactic" few times before in "sci.logic". The fact
remains that "ordinary mathematical arguments", such as Axy(x+y=y+x),
can be deduced in both T1 = PA + G(PA) and T2 = PA + G(PA). Even though
both of them could still be inconsistent!

In other words, even in an inconsistent formal systems, there are infinite
theorems that we might "ordinarily" _select_ as "useful", and we just simply
ignore the "bad" theorems. And that doesn't at all mean doubting the
(possible) inconsistency is a bad thing to avoid.