LINGUIST List 5.234

Sun 27 Feb 1994

Disc: Quantifiers

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1. Carpenter, quantifiers

This message was originally constructed in response to David Gil's
posting about why people try to come up with 'complex' theories of
quantification. Hopefully, this will be interesting to a wider
audience than just those who are following Gil's thread. The points
address those raised in Gil's original message.

Point 1)
There is an added complication with 'two men love three women' in that
you have plural noun phrases. This leads to interactions of scope
with distributive and collective readings. The 'standard wisdom' for
these can be found in the non-theory-bound paper of Davies
(_Linguistics and Philosophy_ 1989, 293--324), "Three examiners marked
six scripts", the Heim, Lasnik and May _Linguistic Inquiry_ paper on
reciprocity (1991), containing a GB version, or I could send anyone
who's interested my own unpublished CG account. Davies most clearly
states the possible readings for such a sentence, based on TWO
quantifiers for each noun phrase:

 Exists X, a set of two men
 Exists Y, a set of three women
 ( forall x in X (distributive)
 OR for some x formed of a group of X (collective))
 ( forall y in Y (distributive)
 OR for some y formed of a group of Y (collective))

alternated so that there are no free variables (there are well-founded
mechanisms for guarnteeing this in both Cooper-storage, Montagovian,
GB and HPSG accounts). These possibilities account for the four
readings you suggest, along with the other possibilities, most clearly
seen in cases such as Davies'. These readings (with your
possibilities in brackets):

 Exists X, forall x in X, Exists Y, forall y in Y love(x,y) [I]
 Exists X, forall x in X, Exists Y, some group y of Y love(x,y)
 Exists X, some group x of X, Exists Y, forall y in Y love(x,y)
 Exists X, some group x of X, Exists Y, some group y of Y love(x,y) [IV]

 Exists Y, forall y in Y, Exists X, forall x in X love(x,y) [II]
 Exists Y, forall y in Y, Exists X, some group x of X love(x,y)
 Exists Y, some group y of Y, Exists X, forall x in X love(x,y)

 Exists X, Exists Y, forall x in X, forall y in Y love(x,y) [III]

The other readings, from different scopings, are all logically
equivalent to one of the ones above.

(Note that this assumes the hypothesis of Partee (ms) and later
Roberts (1989) that the cumulative readings of Scha (1984?) are
group-group readings, your reading IV.) Also note that none of these
readings require branching quantifiers, but rather stem from grouping
rather than distributing. Davies used branching quantifiers for the
last case above, your III, as you suggest. But the analyses I just
listed, basically those of Heim et al., show that they aren't
necessary.

Point 2)
What's your empirical evidence that humans generate limited readings
of quantifier scopings as opposed to those proposed in the theoretical
semantics literature? Is it psycholinguistic, or did you
just ask people in the null context?

I'd suggest looking at:

 Howard Kurtzman and Maryellen MacDonald (1993) "Resolution of
 quantifier scope ambiguities", _Cognition_, 48:243--279.

They conclude that scope preferences for readings are NOT structurally
determined, but rather stem from preferences having to do with lexical
semantics of the terms involved. And furthermore, they cite reaction
time and priming evidence which indicates that multiple readings are
computed in parallel on-line.

Point 3)
I didn't see the relevance of branching quantifiers. The only
motivation for branching quantifiers with which I'm familiar (and I
also asked Jon Barwise, albeit he was responding off the top of his
head at a conference) involves the sentences above which Heim et al.
demonstrated don't need that additional device. So as far as I can
tell, branching's inessential.

Point 4)
This is also undercut by the above discussion. As you claim that
there are two equivalent readings, one with branching, and one with
wide-scope existential, I don't see that you can conclude that:
 "(1) should accordingly be represented not with wide scope for the
 existential quantifier (as per alternative (a) Point 3), but rather
 with branching universal and existential quantifiers (as per
 alternative (b) Point 3) -- contrary to Claim B above."

End of response to your arguments. And by the way, Davies does
respond to Kempson and Cormack (1981: Linguistics and Philosophy), who
as far as I can understand Davies' portrayal of K&C and your portrayal
of your own arguments, are making similar points, and Davies also
cites Tennant (Linguistics and Philosophy 1981) as responding to
K&C's claims.

I'm not sure your argument is coherent (reason (a)), because it's not
stated precisely enough (which I guess is your reason (b)). For
instance, what principle do you provide that will do the right thing
for the following two sentences?

 A kid likes every toy.
 Every kid likes a toy.

Here the problem is in the syntax/semantics mapping and making sure
that you get the same scope altenations for both sentences (one
branching, one wide-scope universal, if I understand you correctly).

At this point in time, I'd also say that your argument is fairly
irrelevant, given the analyses of Heim et al., which can be easily
transferred from GB to your favorite syntactic theory.

Note that there is no 'elaborate theoretical edifice' which has been
built to get the above readings. They're very natural -- just look at
the Heim et al. article. And I don't see how you can get around
something similar for quantifier scoping and still get all the possible
readings, even if you do allow branching, which itself goes beyond
classical first-order logic, if that's your base-line for determining
what's an 'elaborate theoretical edifice'.

Furthermore, how do you account for the interaction of quantifiers
with control, unbounded dependencies, extraction islands,
coordination, de-dicto/de-re intensional verbs, negation, adverbs,
adjuncts, embedded sentences, etc.? My own theory, in logical
Categorial Grammar (Lambek style + scope) minus the plurals, can be
gotten via anonymous ftp in compressed format (use binary mode, then
uncompress) on j.gp.cs.cmu.edu by cd-ing to /usr1/carp/ftp/ and
getting the file quants.ps.Z (in binary mode, of course).

I have to admit that I haven't read Aoun and Li's book. I find GB (or
whatever acronymned theory has descended from it lately) pretty
impenetrable at best, and assume it's in that framework. If it's as
good as Heim et al., though, I'll gladly wade through it. I would
assume they at least address the problems stated in the previous
paragraph.

Please send responses to me directly rather than to the list. I'll
summarize and repost if there are multiple comments.

- Bob Carpenter
 Computational Linguistics Program, Philosophy Department
 Carnegie Mellon University, Pittsburgh, PA 15213
 Net: carplcl.cmu.edu Phone: (412) 268-8043 Fax: (412) 268-1440