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

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From: Newberry on 26 Feb 2010 10:00

On Feb 26, 4:12 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
wrote:
> Newberry says...
>
>
>
> >On Feb 25, 7:44=A0am, Frederick Williams <frederick.willia...(a)tesco.net>
> >wrote:
> >> 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?
>
> That's a weird way of putting it. I would say, rather, that
> "It follows that whether a theory is sound or not depends on
> what you consider true".

Let me put it plainly then and give you a concrete example. In the
logic of presuppositions certain sentences that are consdered true in
classical logic would not be true. Therefore the derivation system we
use for classical logic would not be considered sound in the logic of
presuppositions.

From: Frederick Williams on 26 Feb 2010 10:05

Newberry wrote:
>
> On Feb 26, 4:10 am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> wrote:
> > Newberry says...
> >
> >
> >
> > >On Feb 25, 8:18=A0am, stevendaryl3...(a)yahoo.com (Daryl McCullough)
> > >wrote:
> > >> Newberry says...
> > >> >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?
> >
> > Unrestricted comprehension, which says if Phi(x) is a formula
> > with one free variable, then there is a set y such that forall
> > x,
> >
> > x is an element of y
> > <->
> > Phi(x)
> >
> > This is clearly not true in the case Phi(x) is the formula
> > "x is an element of x".
>
> Looks like you discovered that it was not manifestly true only AFTER
> you derived a contradiction.

So what? (The you in question was Russell by the way.)

From: Frederick Williams on 26 Feb 2010 10:09

Nam Nguyen wrote:
>
> Nam Nguyen wrote:
> > Frederick Williams wrote:
> >> Nam Nguyen wrote:
> >>> [...] And what does your definition have
> >>> to do with GIT, it being dependent on the naturals as a model of some
> >>> formal system?
> >>
> >> What exactly is GIT here? As I remarked elsewhere [1], Theorem VI [2]
> >> is entirely syntactic in its statement and proof.
> >
> > "entirely in its ... proof"? Really?
>
> I meant: "entirely syntactic in its ... proof"?

Noted. I'll reply to your questions later today.

From: Jesse F. Hughes on 26 Feb 2010 10:38

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

> On Feb 26, 3:51 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
>> In any case, the proof of (*) is relatively simple and uses no
>> second-order stuff. The sentence (*) is plainly meaningful and I
>> daresay that *you know what it means*.
>
> I know exactly what you want to say: ~T[(Ex)((x is a counterexample to
> GC) & (x < 27)). You are just sloppy in expressing it.

No, it means what I wrote just below. There are no counterexamples to
GC that are less than 27. That's a plain English sentence. Let's
ignore formalisms for now.

I just want to double check the following. Are you honestly telling
me that you're not sure whether the sentence

There is no counterexample to GC that is less than 27. (*)

is meaningful or not? That sentence means the same thing as (is
equivalent to)

Any counterexample to GC is larger than or equal to 27.

As far as you can tell, these two English sentences may be completely
meaningless, but the sentence

It is not true that there is a counterexample to GC less than 27. (**)

is meaningful. Perhaps as well, though you have not said so, the
sentence

It is not false that there is no counterexample to GC less than 27.

is meaningful.

Moreover, you think that when I utter the possibly meaningless (*), I
really *mean* to say (**), but I'm merely sloppy. If I thought hard
about it, I'd realize that (**) is what I meant -- since, although I
didn't notice it, I can't be sure that (*) means anything at all.

If this is honestly your opinion, I guess I have nothing to say. I
know what all these sentences mean. I know what they mean regardless
of whether GC is true or not. It really is that simple: if I know
what something means, then it's not meaningless. If I also know that
it's true, then it's silly to claim that it's neither true nor false
(but its negation is not true).

(The following .sig was randomly chosen, but perfectly suitable --
except I don't regard Newberry as a philosopher.)
--
"How can people [philosophers] talk like that? Acting as if they're
/glad/ they don't know things! Finding out more and more things they
don't know! It's like children proudly coming to show you a full
potty!" -- Terry Pratchett, /Small Gods/

From: Frederick Williams on 26 Feb 2010 12:31

Nam Nguyen wrote:
>
> Frederick Williams wrote:
> > Nam Nguyen wrote:
> >> [...] And what does your definition have
> >> to do with GIT, it being dependent on the naturals as a model of some
> >> formal system?
> >
> > What exactly is GIT here? As I remarked elsewhere [1], Theorem VI [2]
> > is entirely syntactic in its statement and proof.
>
> "entirely in its ... proof"? Really? So which formal system is that
> "syntactic proof" in? What are the axioms of that particular formal
> system?

A good answer would be: read all about in my reference [2]. I call that
a good answer because then you'll get precisely what G\"odel had to say,
well not _precisely_ because [2] is an English translation of a German
work.

On the other hand it's not _that_ good an answer because it seems
impolite, so here's a quotation from [2, p599] (P is the system in
question):

P is essentially the system obtained when the logic of PM is
superposed upon the Peano axioms^{16} (with the numbers as
individuals and the successor relation as primitive notion).

G\"odel's footnote 16 reads:

The addition of the Peano axioms, as well as all other
modifications introduced in the system PM, merely serve to
simplify the proof and is dispensable in principle.

I do not intend to list the axioms and rules but I will remark that the
ramified theory of types is not used.

> Is that formal system consistent by any chance?

One hopes so. But...

> How did Godel
> "know" that formal system is consistent?

.... the facetious answer "you'd better ask him" has some point to it,
for _it doesn't matter_. What G\"odel _assumed_ was that P was omega
consistent. Subsequently Rosser weakened that requirement to ordinary
consistency. Note that omega consistency is syntactically defined.

>
> >
> > [1] news:4B73F9F6.767498C9(a)tesco.net.
> >
> > [2] 'On formally undecidable propositions of _Principia Mathematica_ and
> > related systems I', pp 596-616 of 'From Frege to G\"odel, a source book
> > in mathematical logic, 1879-1931', ed Jean van Heijenoort, Harvard UP.

PS: I have a vague feeling that I M Copi proved (in ZF) that type theory
is consistent, but I cannot find a reference. If anyone knows what I'm
talking about I'd be pleased to be told.

PS the second: One can prove a "theorem VI" for far weaker systems such
as Robinson's.