Re: quantifiers

[jorge@PHYAST.PITT.EDU]

pc:
> _da_ is a variable, so it ranges over certain things, deals
> distributively with them.  These things are everything in the universe of
> discourse (or that exists, but that is another fight) unless we
> explicitly restrict to some other things.  This restriction we do in
> Lojban with _poi_ followed by the predicate expression that defines the
> new range (_voi_ also works and, I expect, some other things as well).

Of course, I don't disagree with that. I just want to allow extreme
restriction to the empty class.

> 1a ro broda cu brode
> 1b ro da poi broda cu brode
> 1c ro da broda nagi'a brode
> 1d su'o da broda ije ro da broda nagi'a brode

> Actually, I would not really take 1a and 1b to mean the same as 1d
> although they turn out to be true on the same occasions (assuming,
> to avoid much more muddling matters, that all the brodas are in the domain
> of discourse or that restriction is to a subdomain).

That's what I meant by "meaning the same". I am not claiming that they
would not have different connotations. What I meant was that in a logical
argument I could jump from any one to the other.

> > O: Some S is not P (xu'o?)
> If the default for the complement of {da'a} was "at least 1", rather
> than 1, I think that would be it.
> pc:
> I don't recognize da'a, but the complement of xu'o would not be "at least
> one."

da'a means "all but", and if not followed by anything, then by default it
is taken to be "all but one". If instead of that default, the default was
"all but at least one" then that would be "xu'o".

> Why is duality so important but existential import not?

Well, for one thing, it allows to write all of A, E, I and O in terms
of a single quantifier. For instance in terms of {ro}:

A: ro broda cu brode
E: ro broda naku cu brode
I: naku ro broda naku cu brode
O: naku ro broda cu brode

In terms of {su'o}:

A: naku su'o broda naku cu brode
E: naku su'o broda cu brode
I: su'o broda cu brode
O: su'o broda naku cu brode

In terms of {no}:

A: no broda naku cu brode
E: no broda cu brode
I: naku no broda cu brode
O: naku no broda naku cu brode

In terms of {da'a} (taking it to mean "all but at least one"):

A: naku da'a broda cu brode
E: naku da'a broda naku cu brode
I: da'a broda naku cu brode
O: da'a broda cu brode

Not all of those are correct with existential import, of course.

> Duality is,
> after all, only a pleasant technical trick in some logic systems,
> existential import is a psychologically significant claim about
> the world.

But then what is so basic about A, E, I and O taken with existential
import? They lose their simplicity in that case. They are no longer
the different negation complements of each other.

> And, given the pains taken in Lojban to get rid of negation
> problems, even the technical trick is relatively unused.

I would say that with existantial import the manipulation of negation
becomes much more complicated. It becomes almost impossible to change
the position of the negator to make the claim more readable.

> I am not even
> sure what to do with the duality of ro and su'o in Lojban, since naku ro
> da naku broda seems a pretty implausible thing to want to say. And it IS
> equivalent to suo da broda anyhow-- other cases are more complex.

It is an implausible thing to say, that's why it is useful to be able to
drive it away. For example, in making a DeMorgan type expansion you may
end up with something like that. For example:

        naku ro broda cu brode nagi'a brodi

Can be expanded to:

        naku ro da poi broda zo'u da brode inaja da brodi

Without existential import, it can be simplified to:

        su'o da poi broda zo'u da brode ije da na brodi

Which is also:

        su'o broda cu brode gi'enai brodi

But with existential import it becomes more complicated.

> xorxes:
> I guess it's just a matter of aesthetics. To me these two should be
> exactly equivalent:
>         re da poi nanmu cu pencu re de poi gerku
>         re da poi nanmu ku re de poi gerku zo'u: da pencu de

> pc: I don't
> see the aesthetic point.  The assumption is that a change in an expression
> signals some change.  And the case I was discussing involved also re da
> poi nanmu re da poi gerku cu pencu (which I have to admit does seem to me
> to mean the same -- except for emphasis, perhaps -- as the first case).
> In this case certainly the change is not a superficial one (as it is in
> the parenthetically mentioned case) but a profound one that alters the
> whole underlying structure of the sentence, syntactically and logically.

If you define it like that. My point is that there is no need to.

> Syntactically (one story anyhow, others are parallel) the sentence shifts
> from one with a predicate (pencu) head to one with a quantification head.

Yes, but that may just be two different notations for the same thing.

> Logically, the scopes of the quantifiers are changed (at least -- I think
> rather more is involved).

The simplest definition would be that they not be changed.

> In any case, simplicity -- an aesthetic virtue
> -- and coherence -- another -- would suggest that such a change meant
> something.  And it does.  The question is only whether it is enough to
> carry the freight I claim for it.

That it may have different connotations (such as emphasis) I don't argue.
But for more basic differences, I think that simplicity and coherence
require that they mean the same thing.

> I would read it as "there are exactly two men for which there are exactly
> two dogs such that...", i.e. for each of the men.
> pc:
> I do not see why you want to read the "for which" in there, since it is
> not in there.

To make the English clearer.

> I do not have to define it as having equal scope, since I
> can prove that the two scope-readings are equipollent.

There are three scope readings: Two dogs for each man, two men for
each dog, or two men and two dogs in all. The last is the equal scope
case, and the other two are the ones for which one has scope over the
other. I'm not sure what you mean by proving anything, since it is
a matter of definition which notation expresses which claim.

> So far as I can
> recall, the only agreement that has actually been expessed is that the
> re nanmu cu pencu re gerku version could involve up to four dogs, two for
> each man.

Right. And now you are saying that writing it as {re nanmu re gerku zo'u
ny pencu gy} changes things. I am saying that I would rather prefer
it didn't. It can be defined consistently either way.

> The present task is to find a way to say some of the other
> things, particularly (in this case) the two dogs total version (the
> others take other quantifier orders or, as you say -- so this is another
> agreement -- get involved in masses).

Yes. Whatever way we find with the present language, it will not be general.
A general solution will require something along the lines of what And
proposed.

Jorge