Re: quantifiers

[jorge@PHYAST.PITT.EDU]

pc:
> >         ro da poi broda cu brode
> >         For all x that is a broda, x is a brode.
> >
> >         ro da broda nagi'a brode
> >         For all x, if x is broda, then x is a brode.
>
> If there are no brodas, the first is false while the second is true,
> regardless of what brode is.

If the first is false (assuming there are no brodas) then it has
to be true that:

        naku ro da poi broda cu brode
        It is not the case that every broda is a brode.

Now, I would happily transform that sentence to:

        su'o da poi broda naku brode
        For at least one broda, it is not the case that it is brode.

which would have to be true in the absence of brodas. I don't like that
at all. Either that sentence is true, or my transformation is invalid.
If my transformation is invalid, then is it impossible to change the order
of the quantifier and the negation?

> Restricted quantifier sentence are false
> when restricted to an empty set (existential import).

I understand that they are false when using an existential quantifier,
but I don't see the problem with the universal one. Is it false to say
that every element of the empty set is a member of every set? If that
is false, then is it false that the empty set is a subset of every set?
How are subsets defined?


> > However, when we consider {lu'i} and {lu'o} things change, because in
> > this case we are free to select subsets of a certain cardinality
> > from the original set, and that gives rise to many possible individual
> > sets and masses. Thus, there is nothing strange about {re lu'i ci lo plise},
> > two sets of three apples.
>
> Thanx.  As I said, I did not deal with these doubly descriptored forms,
> since I do not really understand them yet.  I must say that what you say
> makes precious little logical sense, since in logic the set would be
> fundamental and here it is getting pretty unclear what, if anything, is
> basic.

I believe it is not essential in logic to bring up sets. In any case, I
don't see why what I say makes little logical sense. Is the concept of two
different sets of three apples illogical? I don't see any problem
with the concept. Or is the notation illogical? Again, I don't see any
problems there, but I admit I may be missing something. Where is the
problem? What's wrong with {re lu'i ci lo plise cu broda} standing for

        E!2 x e {sets of three apples} x broda

> As noted, the fractionators make no sense literally here; does the
> lu'o locution move in he right direction? It is not clear from your
> description.

Again, your problem seems to be with the notation. Why don't they
make sense "literally"? Is there a literal sense other than the
one we agree to by convention? I agree that the notation is not
absolutely transparent, but I think it is the best meaning we could assign
to it. The notational advantage of {lu'o} over {loi} is that {lu'o} lets us
specify the actual number, while {loi} only the number as a fraction
of the total. Since the total is in most cases infinite, concrete
fractions make little sense, but indefinite ones like {pisu'o} or
{piso'u} seem to be fine.

> > You don't say it explicitly, but does this mean that you give your blessing
> > to the thesis that {lo broda} behaves just as {da poi broda} from the
> > logical point of view?
> Let's see: lo broda cu brode= k{broda}<>0 & Ex e {broda}, x brode, which,
> given he general principle that Ax(x e {broda} <=> x broda) should be
> equipollent (at least) to Ex st x broda, x brode, which is da poi broda
> cu brode (via da poi broda zo'u da brode).

Mmm... but you also said that {ro da poi broda cu brode} was different
from {ro da broda nagi'a brode} because the first was of the Ax e {broda}
type. How come {ro da poi broda} is like that but {da poi broda} isn't?

In any case, I agree of course that the equivalence of the two is
unquestionable if the principle that Ax(x e {broda} <=> x broda) holds.

If that principle doesn't hold (which as far as I can see would only be
the case if the set doesn't exist, no problems to me if the set is empty)
then we can still avoid the issue by ignoring the set and getting the
meaning only from the predicate relations.

> Note again that there is still no way given to directly refer to an
> individual in all these forms.

I agree that {lo} doesn't allow that, but {le} does in many cases, and
{la} always.

{la djan} by definition refers directly to an individual.

{le blanu} in most cases refers to an individual, {le pa blanu} makes it
absolutely clear. {le re blanu} refers directly to two individuals, and
condenses two claims into one sentence.

On the other hand {re le mu blanu} again does not directly refer, it only
talks about two of the five blues.

Jorge