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From: Newberry on 14 Feb 2010 10:42
On Feb 14, 3:11 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > I had claimed that if for all a in the range of x
>
> >     (y)Aay                         (1)
>
> > is vacuously true then
>
> >     (x)(y)Aay                    (2)
>
> That's supposed to be (x)(y)Axy, I suppose?

Sorry

> --
> Jesse F. Hughes
> "[I]t's the damndest thing.  There's something wrong with every last
> one of you, and I *never* thought that was a possibility.  But now I
> feel it's the only reasonable conclusion." --JSH sees some sorta light

From: Newberry on 14 Feb 2010 10:52
On Feb 14, 3:18 am, "Jesse F. Hughes" <je...(a)phiwumbda.org> wrote:
> Newberry <newberr...(a)gmail.com> writes:
> > Note that given
>
> >     ~(Ex)(Ey)[(x + y < 6) & (y = 8)]
>
> > in Strawson's logic of presuppositions the formula is neither true nor
> > false for any choice of y. Let y be 8:
>
> >     ~(Ex)[(x + 8 < 6) & (8 = 8)]
> > i.e.
> >     (x)[(x + 8 < 6) -> ~(8 = 8)]
>
> > the subject class is empty and hence the sentence is neither true nor
> > false.
>
> > Let y be anything but 8, say 9:
>
> >     ~(Ex)[(x + 9 < 6) & (9 = 8)]
> > i.e.
> >     (x)[(9 = 8) -> ~(x + 9 < 6)]
>
> > the subject class is empty and hence the sentence is neither true nor
> > false.
>
> Plug in 0 for y and you get
>
>   (x)[(x + 0 < 6) -> ~(0 = 8)]
>
> which is surely true.  Unfortunately, this is equivalent to
>
>   (x)[(0 = 8) -> ~(x + 0 < 6)],
>
> which has an empty subject class[1], and so (according to you) is
> neither true nor false.  Oops!
>
> Footnotes:
> [1]  I think that when you write "(x)(Px -> Qx)" has an empty subject
> class, you mean simply that (x)~Px.  Correct me if I'm mistaken.

Yes, that is what I mean.

Again

~(Ex)[(x + n < 6) & (n = 8)]

is neither true nor false for all n (according to the logic of
presuppositions.) What is wrong with my conclusion that

~(Ex)(Ey)[(x + y < 6) & (y = 8)]

is neither true nor false?

> --
> Jesse F. Hughes
> "Being wrong is easy, knowing when you're right can be hard, but
> actually being right and knowing it, is the hardest thing of all."
>                                                -- James S. Harris- Hide quoted text -
>
> - Show quoted text -

From: Jan Burse on 14 Feb 2010 11:23
Newberry schrieb:

> the subject class is empty and hence the sentence is neither true nor
> false.
>
> Let y be anything but 8, say 9:
>
> ~(Ex)[(x + 9 < 6) & (9 = 8)]
> i.e.
> (x)[(9 = 8) -> ~(x + 9 < 6)]
>
> the subject class is empty and hence the sentence is neither true nor
> false.
>
> Comments appreciated.

In free logic, where the universe can be empty,
quantifiers behave a little bit different than
in logics where we have the assumption that the
universe is non-empty.

Normally we assume that we can derive the
following in non-free logic:

|- forall x px -> exists x px.

But the above is not valid in free logic. What
we could derive is the following:

exist x t |- forall x px -> exists x px.

Of course adding the following axiom to free
logic, gives our non-free logic:

|- exists x t

Bye

Bye




From: Big Red Jeff Rubard on 14 Feb 2010 16:26
On Feb 14, 8:23 am, Jan Burse <janbu...(a)fastmail.fm> wrote:
> Newberry schrieb:
>
> > the subject class is empty and hence the sentence is neither true nor
> > false.
>
> > Let y be anything but 8, say 9:
>
> >     ~(Ex)[(x + 9 < 6) & (9 = 8)]
> > i.e.
> >     (x)[(9 = 8) -> ~(x + 9 < 6)]
>
> > the subject class is empty and hence the sentence is neither true nor
> > false.
>
> > Comments appreciated.
>
> In free logic, where the universe can be empty,
> quantifiers behave a little bit different than
> in logics where we have the assumption that the
> universe is non-empty.
>
> Normally we assume that we can derive the
> following in non-free logic:
>
>    |- forall x px -> exists x px.
>
> But the above is not valid in free logic. What
> we could derive is the following:
>
>     exist x t |- forall x px -> exists x px.
>
> Of course adding the following axiom to free
> logic, gives our non-free logic:
>
>     |- exists x t
>
> Bye
>
> Bye

Ans: *decennium*!
From: Nam Nguyen on 15 Feb 2010 12:20
Aatu Koskensilta wrote:
> calvin <crice5(a)windstream.net> writes:
>
>> The 'continuum hypothesis' is neither true nor false,
>> for example.
>
> This piece of philosophical reflection -- which stands in need of some
> argument -- has no apparent relevance to Newberry's original post.
>

"In need of some argument", yes I'd agree. But not sure about this
post has no relevance to Newberry's original post, when the title
of the thread is "When Are Relations Neither True Nor False".
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