pc responds on anaphora, most/many, and other topics

[lojbab (Bob LeChevalier)]

I finally got a response from John Parks-Clifford on some of the issues 
discussed here in April and May.  Following is extracted from his letter.
I will relay any responses and further questions to him, but as you can see,
don;t expect rapid turnaround.

   lojbab
_________Anaphora.  I haven't been keeping up with the latest on this in the
logico-linguistic community (see later), but the last time I looked the
consensus seemed to favor dealing with it (and deixis - the pointing
pronouns) as pointers to offline registers (our memories not our words,
in other words.)  In the semantics of the mess then they get replaced by
appropriate designator dummies (pointers in the computer sense, perhaps)
which then get sorted out to real-world references at the very end of
the process.  I can't tell from the discussion, which uses a confusing
(and confused?) array of terminology, whether this is anyone's position
or a compromise among them.  It is surely different from a literal copy
plan but I don't suppose anyone really held that for more than a
newyorknanosecond.  A quick glance at the soon-to-be-mentioned book
shows that an elaboration of the system I just mentioned is indeed
presented as the hot stuff, with all the problem examples - anaphora
across sentence boundaries and donkey sentences - listed and dealt with.
As usual, the solution is attributed to someone I knew at UCLA, Hans
Kamp in this case (I heard of it from Barbara Partee.)  I just glanced,
so I don't know yet about either the details of the solution nor the
problems it has engendered, of which there are surely some.  But at
first glance and at the syntactic level it looks good enough for Lojban
before the full semantic theory.  The book even mentioned the most
barebones problem sentence:  "If any boy came, he was happy" or a
version thereof.  By the way, I was never very interested in anaphora as
such but for a long time tense was treated as a kind of anaphora problem
(probably still is, since there are at least a lot of good analogies)
and so I got the backwash.  As you know, I think the offline registers
are the way to deal with tense.  The other modals will get the same
treatment, I expect, although there are other devices for these.

Montague Grammar and I go back a long time - to Montague's class in
maybe the Philosophy of Language in the late 50's - early 60's (I have
notes somewhere, maybe even dated.)  I was writing linguistic papers for
him and logic papers for a seminar in linguistic theory in the
linguistics program at the same time, so they get all jumbled in my
head.  I remember that Chaim Gaifman was involved and that I wrote a
paper in one group or the other (probably the linguistic one - Gaifman
would have written for Montague) on Adjuciewicz's cancellation grammar,
which with Chomsky and intentional logic, is the root of Montague's
stuff.  Since then, I've checked in about once a decade on what's
happenin' in the muck.  It was that time again this summer since a
couple of useful books had just come my way from CSLI at U/Chi (maybe
the same as Spackman's home base, though the order of terms is
different.)  So I am reading Logic, Language and Meaning vol 2
"Intensional Logic and Logical Grammar" by LTF Gamut, the name for the
mass of most of the Dutch guys that get mentioned in these discussions:
van Bentham (tense etc. logician) Groenendijk (more so - and informal
logic man as well), de Jongh, Stokhof, and Verkuyl (the latter group I
know less well, though some are also in informal logic of the Dutch
sort, rhetorical/argumentation theory.)  I also have a couple of books
from Barwise (of Barwise and all sorts of people, including especially
Parry, also from my PhD committee) on situation theory, the stripped
down version of model-theoretic semantics that everyone has always
actually used even when they talked whole possible worlds.  I have also
gotten a new anthology on Pragmatics , which I did not bring with me,
but which ties in, since pragmatics was (according to Montague at some
point) just a branch of set theory (as was everything.)

The basis of the whole thing is to give every item a syntactic category
which is essentially described as S/Y, where Y is what you need to have
follow the item in question and X is the category you get when the two
come together.  So all the ordinary rewrite rules collapse to one, W/Y +
Y - > X. This obviously also takes care of grouping and order of
application (if the categories have been defined aright.)  Tidy!
Montague added one and a half things.  He allowed transformations after
the rewrites, so that what was deeply adjoining might end up separated
in one way or another.  And he added a more sophisticated semantics
(intensional logic stuff) so that each syntactic category had a unique
semantic category and the meaning of the whole was composed in the same
way as the parse.  Not just tidy but cute, too.  I forget the details of
the various analyses but the stuff about quantifiers sounds familiar.

The point of all that would be that "all" and "some" are (I forget what
exactly, but roughly) properties of sets, one- place predicates
approximately, while "most" is at least a two-place predicate, a
relation between sets (since how big it is depends upon what the
restricted set is, not directly referring to the universe as a whole)
and "enough" is probably a tree-place relation (since the purpose must
be dealt with as well.)  "Many" is probably ambiguous between an "all"
reading - direct reference to the universe - and a "most" reading -
restricted by the name set :  There are not many tigers, but many of the
ones there are are in zoos (first in the absolute sense, second in the
relative.)  So, if all of "all", "some", "many" and "most" are of the
same syntactic type, that type cannot be correlated with a single
semantic type, as Montague grammars would require.  Of course, we are
not committed to MGs as yet (1) and (2) there is a long tradition that
holds that "all" and "some" are also two place relations (at least) -
indeed, a longer tradition than the other (Aristotle was before Frege by
a good 2 kiloyears.)  So, "enough" aside, it might be that we could save
MGs for Lojban if we really want to.  We do in a couple of places talk
restricted quantification in Lojban (da poi, for example), so the shift
would not be new (and the universal case can be accommodated fairly
easily by referring to the tautological class, things.)  I don't know
what Discourse Representation Theory, Kamp's device, says about
quantification exactly, but, since most of Montague's problems with
anaphora come from treating it as a species of quantification and since
Kamp deals well with anaphora, I assume that he has something useful to
say about quantification as well.

By the way, if you use first order logic with function expressions (as I
- and Montague - always do) you can get a lot closer to saying "most",
since you can specify a bijection (1-1 mapping), which, as someone says,
is the problem with defining "most", lacking cardinality.  "Most Fs are
Gs" comes to something like this "both both for all x, if both Fx and
not Gx then both Ff(x) and Gf(x), and for some y, both both Fy and Gy
and not both Ff(y) and not Gf(y) and both for all z f(f(z)) =z and for w
and v, if f(w) = f(v) then w = v."  I think that will do it, but I may
need another clause or two.  Note that this is not a definition, since
the function f is fixed rather than free floating.  So the "most"
sentence might be true in an interpretation in which this was false (but
the corresponding thing with, say g for f was true.)

Notice that even this will not work in an infinite set, since there is
always a 1-1 mapping between two denumerable sets, say, and also between
one denumerable set and a proper (but denumerable) subset of another.
Thus, we could use this sentence to say that most natural numbers are
not prime, but the exact same technique would also allow us to say that
most are odd - or that most are even.  "Most" in infinite contexts, is a
notion we sometimes want to use, but no one I know of has done a really
convincing (intuitively correct) job on it.  The last I saw was some
attempt at generalizing the notion of local majority - all but a finite
number of cases of primes are surrounded by an outwardly expanding area
of numbers which are not prime, so primes are almost always in a local
minority, and so are in a minority in the limit case.  Needless to say
(I expect), getting all of this both precise and true is not easy to do
even for well-behaved critters like the natural numbers and goes way
beyond the powers of first order logic (or worse - I've not seen a
convincing job within ordinary set theory.)

________
[Note by the way that pc wrote this response while on vacation in Maine,
and hence some of his references are a little vaguer than usual.]