quantifiers

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

        Does anyone else find it amusing that the Applied Logician
makes his case by citing the empty set and other arcana of
mathematics while the Pure Logician cites what people actually say
and do in the ordinary world?  The recent history of logic is summed
up in that situation.  Logicians are, of course, quite thankful to
mathematicians for all the help they have given and the new insights
and methods they have made available.  But logicians do occasionally
object when the stubby mathematical tail claims to wag the whole
Great Dane of logic (the mathematical Journal of Symbolic Logic has
published nothing about logic in the 30 years I have subscribed, for
example).
        Logic is, as the name suggests, about people talking -- arguing
usually, but talking above all (including writing and signing, of
course).  So logic ought -- and usually does -- take its guides from
what people do when they are talking.  This is particularly relevant
when logic is being used as a guide to constructing a language to be
used for ordinary discourse, as Lojban is, rather than for a calculus to
carry out some specialized computational task.
        When someone says in the ordinary course of events (well, not
all that ordinary in this traditional example) "All unicorns are white,"
the response "There aren't any unicorns" is neither intended nor
understood as confirming the original claim.  It is a challenge to the
original claim, a contrary claim to it, as much as "Some are blue" is.  It
is this fundamental fact that logic has always taken into account in its
treatment of quatifiers, abetted of course by the fact that we usually do
not talk about what is not, except with conscious care.
        Now, of course, mathematicians are also people and they do
talk and argue a lot.  Indeed, arguing is more important to them than to
anyone else other than philosophers, since they (like philosophers)
have no other way to establish their claims than by argumentation;
there are no observable facts they can point to to make their cases.  So
their usage needs also to be considered, but always remembered as a
very specialized usage within the broad range of argumentation.  For
their special concerns, which do involve talking about what is not (or
seem to at least) we should thus provide an appropriate form of
expression with conscious care.  And, of course, logic -- and Lojban --
do, the universally quantified conditional, the form of subclassification
on the universe of discourse.
        Having said of that, I do feel sorry that Lojban has come to the
situation it is in, where the original distinction between the traditional
and the mathematical universal as simple forms has been lost.  I am
sorry for my part in the slide to this position and I am especially sorry
that it has been me that noticed the results and had to announce it.
After reworking through the stages of the shift, I can find only one step
which seems to me to be open to reconsideration.  That is the
identification of _ro broda_ with _ro lo broda_.  Unlike the existential
import of _ro da poi broda_, which is central to Lojban as a language,
and the particular reading of _lo broda_, which is forced by
compelling claims about usage, this step in the identification chain has
the marks of "we have to put it somewhere" or "we do not want a new
structure except as an abbreviation of an old one."  So far as I can tell,
the thought to make this structure sui generis was never considered or,
if it was, was quickly dismissed (the suggestion to make it _ro da poi
broda_ was considered longer and even made it to some version of
some part of the textbook).
        Or rather, the thought to make it abbreviate a more complex
structure than the just another type of noun phrase.  If we were to let
_ro broda cu brode_ stand for _roda zo'u [if] da broda [then] da brode_
(and make similar accomodations for the other forms of  Q broda_),
then the mathematicians might have their easy form back and the
redundancy of the present system reduced.  I, of course, think it is
unfair to the real world to give the odd notion the shortest form, which
will, therefore, probably get the most use (though perhaps not -- style
seems to prefer the _lo_ forms even when the shorter are equivalent).
Happily, in spite of the short form cutting off the conscious care in use
that should be significant in this case, overuse of the form would
rarely create problems, since even mathematicians usually do talk only
about what is.
        This is a serious proposal and the only vaguely satisfactory way
out of the situation that I see.  I hope that anyone who remembers -- or
comes up with -- serious reasons why _ro broda_ must be tied to _ro lo
broda_ will enter the fray quickly and loudly.  I will be less delighted
to hear again that restricted quantifiers are to be read without import.
(Mathematicians, who deal with things that are not in time -- or space
or existence, for that matter -- tend to have little sense of history, and
so do not notice that this has never been a serious position.  They do
not even notice that "restricted quantification" is a retronym for what
used to be called "quantification" before the restriction to the blank
domain -- "things" or, for some fanatics, "existents" -- came to be the
dominant usage, with all the resulting problems in more specific
restrictions.  Mathematicians, even when they use it -- unconsciously
usually -- think many-sorted quantification, i.e., quantification in the
traditional sense, is suspect.)
        Happily, I think that this proposal will have little effect on the
existing corpus of Lojban.  As I noted, the _ro broda_ form seems
relatively rare and, when it was used, was used either in the belief that
it stood for the import-free version or in a situation where the
distinction does not matter.  Again, contrary evidence is called for -- and
supporting evidence as well.
pc>|83