quantifiers: three dogs

["John E. Clifford" <pcliffje@CRL.COM>]

        To review on the petting zoo.  It was generally agreed that _ci [da
poi/lo/-] nanmu cu pencu ci[da poi/lo/-] gerku_ required three men and at
least three -- but up to nine -- dogs (three not necessarily the same for each
man).  I took that to mean, in briefest relevant form, "(3x man x)Ay(Iyx
=> (3z dog z)Aw(Iwz => y pets w))" (where "I" just takes care of all the
identities between the various things involved in "3..." and whatever
comes after the "A".  Did I suggest -- sorry about introducing these things -
- _xu'u_ for that?).
        The question then was to find a way of saying that it was the same
three dogs for each of the men, while avoiding saying that it might be up
to nine men, as just changing the order would do.  I suggested "(3 x man
x)(3 y dog y) x pets y," on the ground that, since the order two quantifiers
was irrelevant (i.e., the two orders were equivalent), the choice of
instantiations for one could not depend upon the choice for the other, what
was wanted.  I also suggested that this seemed a natural reading for _ci
[...] nanmu ki ci [] gerku zo'e ny pencu gy_, i.e. the prenex form.
        At this point, xorxes and I got into a confusion worthy of the cross-
talk comedians that killed vaudeville before TV.  When I claimed that the
order of "(3 x man x)" and "(3 y dog y) was irrelevant, he took me to be
talking about the order of _ci[] nanmu_ and _ci [] gerku_.  And when he
insisted that the order of the latter did make a difference (and, further, that
the latter needed to be interpreted in first order logic in a complex way), I
took him to be talking about the former.  (He may also have been trying to
get me to see -- as Iain finally did -- that the form I presented as the
expansion of the formula above was in fact a much stronger form, so that
the simple prenex form is not indeed convertible.)
        That finally got sorted out, I think, but the crux of it was that my
"natural" reading of  the simple prenex form was illegitimate if it were
convertible, for prenexation (the process of moving an expression from
the matrix to the prenex, leaving an anaphoric mark behind, but preserving
order of at least quantified expressions) is specified in the "commentary"
(also "reference grammar" and "notes on grammar") and maybe even in
the draft textbook as being meaning-preserving.  So the form I suggested
as representing the simple logical prenex form  in fact already represented
the same QAQA complex as form with the expressions embedded in the
matrix.  (BTW I think all the embedded forms can be represented
mechanically in that QAQA format, with theorems then reducing some
cases down to simpler forms: if Q=A or E, the following A is lost, since
"I" reduces to a single "=" in those cases.  Other properties also work off
of this, including the possibilities of conversion.)  That is, although we
have a dozen or so relatively simple, first generation, expressions, they are
immediately divided among two equivalence classes, within which they
are -- with the same quantifiers -- all the same (and in a couple of choices
of quantifiers, both A or both E, even the two classes are equivalent).  And
in both these classes, the left quantifier subordinates the right if it can (if
the left is not E nor the right A), that is, different choices of instances for
the left quantifier allow different choices of instances for the right, as the
three different men might each have three different dogs than each other
man.  (I think, by the way, that having all those equivalent forms -- if they
really turn out to be in more complex situations beyond one-sentence
structures -- is a mistake.)
        Xorxes and I then got lost in another failure of communication,
from which we have not yet recovered in one sense.  But I do now
understand his crucial point, if not exactly how he makes it.  Namely, only
in the case where both quantifiers involved are numeric, like the three-
men/three-dog case this began with, is there no form that automatically
gives the two quantifiers independently -- the one requiring exactly three
men and three dogs.  In all other combinations we can get the E left or the
A right, by conversion or raising to the prenex, both non-equivalently, of
course, since we want to make a different claim.  Only with two numerics
will every mere order shift of the quantifiers still leave the right
subordinated to the left.  Nor, short of some way of expressing what I
actually meant by "(3x man x)(3y dog y) x pets y"  -- which there does not
seem to be using quantifier apparatus and the three given predicates alone
-- does there seem to be a way of saying this in the simplest range of
quantifier/descriptor expressions.  Admittedly, the double numeric case is
the most complex one, since none of the added As in the translation into
symbols is absorbed, but it does not seem so much more complex to
expect no solution at this level.  Perhaps it is this failure which drives (as
it does in Iain's proposals) the next levels of complexity, double
quantifiers and double descriptors.
        (For the record, the shortest version I have found which does give
the three-man, three-dog  solution is requires listing men x,y,z, dogs w,v,
u, all different from one another then listing all the pairwise pettings (or
just Ax1 (x1=x,y,z => Ay1(y1=w,v,u => x1 pets y1))) then going into the
"and no other" clauses.  The version using sets is much simpler in several
respects but seems even more remote from the Lojban, unless there is a
difference in the interpretation rules between _da poi_ and _lo_, a
suggestion that has been discouraged in various ways.  Even it would run
afoul of either the meaning preservation of prenxation or the generally
accepted interpretation of the embedded forms.  However, it does seem
that a combination of quantifier and set forms and a corresponding combination
of _da poi_ and _lo_ forms could reduce the redundancy of the present system
and slightly improve its expressive power -- slightly because most of the forms
still fall together, as noted earlier.)
pc>|83