From jjllambias@xxxxxxx.xxxx Wed Oct 20 14:28:15 1999
X-Digest-Num: 261
Message-ID: <44114.261.1409.959273825@eGroups.com>
Date: Wed, 20 Oct 1999 14:28:15 PDT
From: "Jorge Llambias" >There are many ways in which 3x 2y F(x,y) could have
> >been given meaning. The one chosen is to take it as
> >3x G(x), where G(x) = 2y F(x,y), and there we can see
> >why the scope of the second quantifier is narrower.
>
>It seems to me that if G(x) = 2y F(x, y) then G is a function
>of (x, y) and not (x) alone.
No, G does not depend on any value of y, it is only a
function of x. Replace y in that expression with any other
bound variable and you will see that G only depends on x.
y is not a free variable in the expression 2y: F(x,y)
>"For at least"..."there exists" indicates a dependency of existence. I
>think this fact should be made explicit, and without such a marking, it
>should mean: "There exists exactly 3 dogs, and there exists exactly 2 men,
>such that: each/any dog bites
>each/any man at least once."
That could have been the convention: take all the existentials
first and all the universals later when dealing with more than
one numeric quantifiers.
>This is the symmetrical interpretation, free of the malglico of default
>restricted scope.
I don't see that one interpretation is more or less malglico
than the other. What you gain in symmetry you lose in
the ease of formula reduction. You would also need to
specify what to do when you have for example {ro} and
a number in one expression, {ro} and {su'o} and a number,
etc.
In practical terms, I don't see how it matters much one way
or the other, since we hardly ever will want to say any of
those things. When speaking of groups of things it is much
more common to refer to them collectively, in which case
this problem doesn't even arise.
co'o mi'e xorxes