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Re: quantifiers and existence

>In message <9509261718.ab10526@punt-4.mail.demon.net> pcliffje@crl.com writes:
 >>         But then the evidence came in that _su'o lo broda_ was going to
>> be a much more commonly used expression than _ro lo broda_ and so -- >> by a
 legislative act, I think -- the implicit external quantifier on _lo_ was
>> changed to _su'o_.

>>That meant that the implicit internal quantifier could
>> no longer be _ro_ -- if that were understood to be without existential
>> import -- at the risk of contradiction.

I think what pc means here is that something of the form:

E(x)[A(x)broda(x)]. has a scope that asserts the second x exists.
So that the A(x) has existential import contrary to the modern
interpretation. But then have been wrong before about what pc means.

>I don't understand this statement.  Obviously {su'o lo ro broda}
>claims the existence of brodas, but that's because of the {su'o},
>not because of the {ro}.  Removing the existential import
>doesn't make it deny existence, so I don't see any contradiction.
>Nor do I see it any differently if either or both quantifiers
>are implicit.
>Iain Alexander                    ia@stryx.demon.co.uk
>                    I.Alexander@bra0125.wins.icl.co.uk

Xorxes has pointed out that all [PA] parse alike so instead of talking
about ro and su'o I will talk about numbers in place of ro and su'o. At
the end I will get back to ro and su'o.

pc has said that "re broda" means the standard  Russell expression
which I take to be:

ExEy[(x\=y & Az(z=x v z=y)) & broda(x) & broda(y)].

One thing stands out about this expression and that is that it claims
existence for x and y.

In English it says something like; There exists at least one x, there
exists at least one y; x is distinct from y; and whatever other object z
you consider, it is really x or y; and x brodas and y brodas.

What does it mean now to say "lo re broda"?  lo by itself claims existence
for the broda it modifies. But we already have a very explicit claim for
existence of the two things, the x and the y. This lo is redundant. It is
like saying "There exist two things and these same two things exist".

lo is also a determiner. It narrows our attention a little and asserts
that we are talking about something objective as opposed to subjective.
But this function could as well be served by words like ti, ta, di'u;
with no loss of quantificational meaning.

Now what does "pa lo re broda" mean? In exploring this we find that the
lo has yet another function. It separates the pa and re from merging
into one number, pare, or 12. But there is no difference between

"pa ti lo ci broda"  and
"pa ti ci broda".

The lo is just redundant here and adds nothing to the meaning.  Each says
"This one, three broda exist." We project our English habits onto the
lo form to get it to mean "this one, taken out of the set of three
broda."  So far no sets have been defined explicitly or implicitly. This
is an important point. The system of inner and outer quantifiers
presupposes a set theoretic view of number.  All the preceeding
analysis has to be thrown out the window if we are going to depart from
the logician's view of number; and set concepts elaborated instead.

Suppose then that "re broda" means, Def:

there exists a set of broda with at least 2 members; and x is a member
and y is a member and x and y are taken from the set.

This is quite a different thing. Now we have
not specified how many brodas there are in all but there must be at
least two.

What does "lo re broda" mean under these assumptions?  Once again "lo"
is asserting existence where it has already been asserted, this time in
the definition, and so lo is again redundant. If we are going to use a
set theoretic basis of number and use cardinals, we should explicitly
say so and use lo'i where we are accustomed to useing "lo".  lo'i is
also redundant in its existence claim; but at least it makes it clear
that we are dealing with sets.  Once that point is established, it is
easy to get subsets.  We should say "lo'i re broda" where we are now
saying lo re broda. The word lo'i at least adds some meaning, the
conversion to a set, to the expression.  "lo re broda" is just "the
three broda" from English projected onto a grammar that does not
accomodate this expression in a matching way.
Well then, what does

" pa lo'i re broda"

mean?  It means "one two-set of broda ". Still not quite right. It
needs to be

"pa da ra'i lo'i re broda" .
"one something, from the set of two broda."

 This says what "pa lo re broda" is claimed to say according to the
 draft grammar, but does not say.  Finally, what does

 "su'o lo ro broda"

 mean?  I don't know. I only know I think it should be abandoned for:

su'o da ra'i lo'i ro broda
at least one something from the set of all broda.

and that a cmavo should be assigned to alias "ra'i lo'i" since it is a
frequent concept. Then it would read:

su'o da xe'o ro broda

and it would be clear that the da exist from the definition of the set.
Please think about it.