Re: quantifiers

[Iain Alexander <ia@stryx.demon.co.uk>]

In message  <9509220034.aa29431@punt-3.mail.demon.net> pcliffje@crl.com writes:
> iain:
> Of course, from my point of view, the alternative you suggest is pretty
> implausible, since a proposition like (3x) (3y) F(x,y) cannot be
> understood by breaking it down into its component parts, e.g. as (3x)
> G(x),
> where G(x) is (3y) F(x,y), but can only be interpreted as a whole.
> pc:
> Remind again why (3x)(3y)Fxy can't be broken down sequentially.
> Now, it is the case that (3x man)(3y dog) x pets y expands in a bit more
> complex fashion, it still seems to me to be complex to component
> expansion.  I suspect  -- from something iain says elsewhere -- that
> problem may be about what there are three of: men (in the first case) or
> men-who-pet-three-dogs.  I take it that the answer is "men" and that
> iain holds with "men who pet three dogs,"  in spite of what the
> componential analysis seems to say.  The first view has some technical
> problems, the second gives the wrong readings some times.  I'll try both
> and see what works out.  Evidence:

Yes, I claim that it is "men who pet three dogs" that there are
three of, which I also claim _is_ what the component analysis says.
If G(x) is (3y dog) pets(x, y), it says that x pets three dogs.
If then we write (3x man) G(man), we have that three is the number
of men that pet three dogs.  When we expand the definition of G
(substitution of equivalents), we get (3x man) (3y dog) pets(x, y).
Or would you write this differently?

> iain:
> The point is that we were discussing multiple quantifiers,
> where the F in Ax(Fx => x=... ) is itself defined as a numerically
> quantified expression, whose leading particular quantifier thus
> comes within the scope of that universal.  You obviously don't
> consider this situation to arise in the case of _consecutive_
> numerics, and it is not clear whether you have some alternative
> means of expressing that situation.
> pc:
> I take it that F in this case is just "is a man" (in the first instance) or
 "is
> a relevant man" or "is a member of [the 3leton declared at the
> beginning], none of which contains any further quantifiers at all.

Your text (not quoted above) did indeed assume that F was something
as simple as that.  My question is what do you do when it is more
complicated, such as e.g. "pets three dogs".

>
> nss:
> > pc:
> > Even if there are no unicorns, what compels us to claim that _ro
> > pavyseljirna cu blanu_ is true?  It is a universal claim, so the
> > minimum
> > truth value of its instances.  It has no instances, so, presumably, it has
> > no
> > truth value.
> At first glance, it may not be obvious what such a value should be.
> But it turns out that the practical way to define the minimum of
> an empty set is the maximum possible value, and conversely for
> the maximum.  This is a fairly well-known trick where I come from.
> It appears counter-intuitive when you first come across it, but
> it works, and you soon get used to it.
> pc:
> The well-known trick is of course commonly used in circles where it is
> ASSUMED that the right result is the conditional one.

What do you mean by "conditional"?

> The trick has no
> other justification and flies in the face of the established rules for all
> the connectives and quantifiers,

(which [rules], as I have already said, are based on arguments which
are only valid when the set is non-empty)

> iain:
> If your question is how to say it in Lojban, my preferred solution
> at the moment would be an explicit {ro}
>         ro ci nanmu cu rapypencu ro ci gerku
> which would be equivalent to
>         ro lo ci lo nanmu cu rapypencu ro lo ci lo gerku
> pc:
> Oh drat!  Is that first one legal?

It's certainly grammatical.  There is however some debate about
what it means.  I hear that the official line is that
{ro ci nanmu} is defined as {ro lo ci nanmu}, meaning
"all men (of which there are [exactly] three)".
I am proposing the alternative above, where the internal ({ci})
quantifiers leap independently and coordinately to an outer scope.

> How is this related to _le ci lo nanmu cu  rapypencu le ci lo
> gerku_, which I can figure out how it might mean what is wanted (if it
> is legal)?

This is simply the difference between {lo} and {le},
veridical vs. non.

>
> nss:
> > Since the first of these
> > was created exactly to have a universal quantifier with existential
> > import,
> Unfortunately, nobody told us that. :-)
> pc:
> Last time I looked (a while ago, admitedly), it was in the commentary,
> where it had been for several years.

"Commentary"?  What's that?
--
Iain Alexander                    ia@stryx.demon.co.uk
                    I.Alexander@bra0125.wins.icl.co.uk