AN IMPROPER TREATMENT OF QUANTIFICATION IN ORDINARY ENGLISH 
Jerry R. Hobbs 
SRI International 
Menlo Park, California 
i. The Problem 
Consider the sentence 
In most democratic countries most politicians 
can fool most of the people on almost every 
issue most of the time. 
In the currently standard ways of representing 
quantification in logical form, this sentence has 
120 different readings, or quantifier scopings. 
Moreover, they are truly distinct, in the sense 
that for any two readings, there is a model that 
satisfies one and not the other. With the 
standard logical forms produced by the syntactic 
and semantic translation components of current 
theoretical frameworks and implemented systems, it 
would seem that an inferencing component must 
process each of these 120 readings in turn in 
order to produce a best reading. Yet it is 
obvious that people do not entertain all 120 
possibilities, and people really do understand the 
sentence. The problem is not Just that 
inferencing is required for disamblguation. It is 
that people never do dlsambiguate completely. A 
single quantifier scoping is never chosen. (Van 
Lehn \[1978\] and Bobrow and Webber \[1980\] have also 
made this point.) In the currently standard 
logical notations, it is not clear how this 
vagueness can be represented. 1 
What is needed is a logical form for such 
sentences that is neutral with respect to the 
various scoplng possibilities. It should be a 
notation that can be used easily by an inferenclng 
component. That is, it should be easy to define 
deductive operations on it, and the lo~ical forms 
of typical sentences should not be unwieldy. 
Moreover, when the inferenclng component discovers 
further information about dependencies among sets 
of entities, it should entail only a minor 
modification in the logical form, such as 
conjoining a new proposition, rather than a major 
restructuring. Finally, since the notion of 
"scope" is a powerful tool in semantic analysis, 
there should be a fairly transparent relationship 
between dependency information In the notation and 
standard representations of scope. 
Three possible approaches are ruled out by 
these criteria. 
i. Representing the sentence as a 
disjunction of the various readings. This is 
impossibly unwieldy. 
I Many people feel that most sentences exhibit too 
few quantifier scope ambiguities for much effort 
to be devoted to this problem, but a casual 
inspection of several sentences from any text 
should convince almost everyone otherwise. 
2. Using as the logical notation a triple 
consisting of an expression of the propositional 
content of the sentence, a store of quantifier 
structures (e.g., as in Cooper \[1975\], Woods 
\[19781), and a set of constraints on how the 
quantifier structures could be unstored. This 
would adequately capture the vagueness, but it is 
difficult to imagine defining inference procedures 
that would work on such an object. Indeed, Cooper 
did no inferenclng; Woods did little and chose a 
default reading heuristically before doing so. 
3. Using a set-theoretlc notation like that 
of (I) below, pushing all the universal 
quantifiers to the outside and the existential 
quantifiers to the inside, and replacing the 
existentially quantified variables by Skolem 
functions of all the universally quantlf~ed 
variables. Then when inferencing discovers a 
nondependency, one of the arguments is dropped 
from one of the Skolem functions. One difficulty 
with this is that it yields representations that 
are too general, being satisfied by models that 
correspond to none of the possible intended 
interpretations. Moreover, in sentences in which 
one quantified noun phrase syntactically embeds 
another (what Woods \[1978\] calls "functional 
nesting"), as in 
Every representative of a company arrived. 
no representation that is neutral between the two 
is immediately apparent. With wide scope, "a 
company" is existential, with narrow scope it is 
universal, and a shift in commitment from one to 
the other would involve significant restructuring 
of the logical form. 
The approach taken here uses the notion of 
the "typical element'" of a set, to produce a flat 
logical form of conjoined atomic predications. A 
treatment has been worked out only for monotone 
increasing determiners; this is described in 
Section 2. In Section 3 some ideas about other 
determiners are discussed. An inferenclng 
component, such as that explored in Hobbs \[1976, 
1980\], capable of resolving coreference, doing 
coercions, and refining predicates, will be 
assumed (but not discussed). Thus, translating 
the quantifier scoping problem into one of those 
three processes will count as a solution for the 
purposes of this paper. 
This problem has received little attention in 
linguistics and computational linguistics. Those 
who have investigated the processes by which a 
rich knowledge base is used in interpreting texts 
have largely ignored quantifier ambiguities. 
Those who have studied quantifiers have generally 
noted that inferencing is required for 
57 
disambiguation, without attempting to provide a 
notation that would accommodate this inferencing. 
There are some exceptions. Bobrow and Webber 
\[1980\] discuss many of the issues involved, but it 
is not entirely clear what their proposals are. 
The work of Webber \[1978\] and Melllsh \[1980\] are 
discussed below. 
2. Monotone I~creasin~ Determiners 
2.1. A Set-Theoretic Notation 
Let us represent the pattern of a simple 
intransitive sentence with a quantifier as "Q Ps 
R". In '~ost men work," Q - "most", P = "man", 
and R - "work". Q will be referred to as a 
determiner. A determiner Q is monotone increasing 
if and only if for any RI and R2 such that the 
denotation of R1 is a subset of the denotation of 
R2, "Q Ps RI" implies "Q Ps R2" (Barwlse and 
Cooper \[1981\]). For example, letting RI - "work 
hard" and R2 = "work", since "most men work hard" 
implies "most men work," the determiner "most" is 
monotone increasing. Intuitively, making the verb 
phrase more general doesn't change the truth 
value. Other monotone increasing determiners are 
"every", "some", "many", "several", "'any" and "a 
few". "No" and "few" are not. 
Any noun phrase Q Ps with a monotone 
increasing determiner Q involves two sets, an 
intensionally defined set denoted by the noun 
phrase minus the determiner, the set of all Ps, 
and a nonconstructlvely specified set denoted by 
the entire noun phrase. The determiner Q can be 
viewed as expressing a relation between these two 
sets. Thus the sentence pattern Q Fs R can be 
represented as follows: 
41) (Ts)(Q(s,{x I P(x)}) & (VY)(~s -> R(y))) 
That is, there is a set s which bears the relation 
Q to the set of all Ps, and R is true of every 
element of s. (Barwlse and Cooper call s a 
"witness set".) "Most men work" would be 
represented 
(~ s)(most(s,{x I man(x)}) 
& (~ y)(y~s -> work(y))) 
For collective predicates such as "meet" and 
"agree", R would apply to the set rather than to 
each of its elements. 
(3 s) 0(s,{x I F(x)}) ~ R(s) 
Sometimes with singular noun phrases and 
determiners llke "a", "some" and "any" it will be 
more convenient to treat the determiner as a 
relation between a set and one of its elements. 
(B Y) 0(y,{x I P(x)}) & R(y). 
According to notation (i) there are two 
aspects to quantification. The first, which 
concerns a relation between two sets, is discussed 
in Section 2.2. The second aspect involves a 
predication made about the element~ of one of the 
sets. The approach taken here to this aspect of 
quantification is somewhat more radical, and 
depends on a view of semantics that might be 
called "ontological promiscuity". This is 
described briefly in Section 2.3. Then in Section 
2.4 the scope-neutral representation is presented. 
2.2. Determiners as Relations between Sets 
Expressing determiners as relations between 
sets allows us to express as axioms in a knowledge 
base more refined properties of the determiners 
than can be captured by representing them in terms 
of the standard quantlflers. 
First let us note that, with the proper 
definitions of "every" and "some", 
(V sl,s2) every(sl,s2) <-> sl= s2 
(y x,s2) some(x, s2) <-> x~s2 
formula (I) reduces to the standard notation. 
(This can be seen as explaining why the 
restriction is implicative in universal 
quantification and conjunctive in existential 
quantification.) 
A meaning postulate for "most" that is 
perhaps too mathematical is 
(~sl,s2) most(sl,s2) -> Isll > i/2 Is21 
Next, consider "any". Instead of trying to 
force an interpretation of "any" as a standard 
quantifier, let us take it to mean "a random 
element of". 
(2) (~x,s) any(x,s) ~> x = random(s), 
where "random" is a function that returns a random 
element of a set. This means that the 
prototypical use of "any" is in sentences like 
Pick any card. 
Let me surround this with caveats. This can't be 
right, if for no other reason than that "any" is 
surely a more "primitive" notion in language than 
"random". Nevertheless, mathematics gives us firm 
intuitions about "random" and (2) may thus shed 
light on some linguistic facts. 
Many of the linguistic facts about "any" can 
be subsumed under two broad characterizations: 
i. It requires a "modal" or "nondeflnlte" 
context. For example, "John talks to any woman" 
must be interpreted dispositlonally. If we adopt 
(2), we can see this as deriving from the nature 
of randomness. It simply does not make sense to 
say of an actual entity that it is random. 
2. It normally acts as a universal 
quantifier outside the scope of the most immediate 
modal embedder. This is usually the most natural 
interpretation of "random". 
Moreover, since "any" extracts a single 
element, we can make sense out of cases in which 
"any" fails to act llke "every". 
58 
I'Ii talk to anyone but only to one person. 
* I'Ii talk to everyone but only to one person. 
John wants to marry any Swedish woman. 
* John wants to marry every Swedish woman. 
(The second pair is due to Moore \[1973\].) 
This approach does not, however, seem to 
offer an especially convincing explanation as to 
why "any" functions in questions as an existential 
quantifier. 
2.3. Ontological Promiscuity 
Davidson \[1967\] proposed a treatment of 
action sentences in which events are treated as 
individuals. This facilitated the representation 
of sentences with adverbials. But virtually every 
predication that can be made in natural language 
can be modified adverbially, be specified as to 
time, function as a cause or effect of something 
else, constitute a belief, be nominalized, and be 
referred to pronominally. It is therefore 
convenient to extend Davidson's approach to all 
predications, an approach that might be called 
"ontological promiscuity". One abandons all 
ontological scruples. A similar approach is used 
in many AI systems. 
We will use what might be called a 
"nomlnalization" operator ..... for predicates. 
Corresponding to every n-ary predicate p there 
will be an n+l-ary predicate p" whose first 
argument can be thought of as a condition of p's 
being true of the subsequent arguments. Thus, if 
"see(J,B)" means that John sees Sill, 
"see'(E,J,S)" will mean that E is John's seeing of 
Bill. For the purposes of this paper, we can 
consider that the primed and unprimed predicates 
are related by the following axiom schema: 
(3) (~ x,e) p'(e,x) -> p(x) 
(Vx)(~e) p(x) -> p'(e,x) 
It is beyond the scope of this paper to 
elaborate on the approach further, but it will be 
assumed, and taken to extremes, in the remainder 
of the paper. Let me illustrate the extremes to 
which it will be taken. Frequently we want to 
refer to the condition of two predicates p and q 
holding simultaneously of x. For this we will 
refer to the entity e such that 
and'\[e,el,e2) & p*(el,x) & q'(e2,x) 
Here el is the condition of p being true of x, e2 
is the condition of q being true of X, and e the 
condition of the conjunction being true. 
2.4. The Scope-Neu¢ral Representation 
We will assume that a set has a typical 
element and that the logical form for a plural 
noun phrase will include reference to a set and 
its ~z~ical element. 2 The linguistic intuition 
2 Woods \[1978\] mentions something llke this 
approach, but rejects it because difficulties that 
are worked out here would have to be worked out. 
behind this idea is that one can use singular 
pronouns and definite noun phrases as anaphors for 
plurals. Definite and indefinite generics can 
also be understood as referring to the typical 
element of a set. 
In the spirit of ontological promiscuity, we 
simply assume that typical elements of s~ ~re 
things that exist, and encode in meaning 
postulates the necessary relations between a set's 
typical element and its real elements. This move 
amounts to reifying the universally quantified 
variable. The typical element of s will be 
referred to as ~(s). 
There are two very nearly contradictory 
properties that typical elements must have. The 
first is the equivalent of universal 
instantiation; real elements should inherit the 
properties of the typical element. The second is 
that the typical element cannot itself be an 
element of the set, for that would lead to 
cardinallty problems. The two together would 
imply the set has no elements. 3 
We could get around this problem by positing 
a special set of predicates that apply to typical 
elements and are systematically related to the 
predicates that apply to real elements. This idea 
should be rejected as being ad ho__~c, if aid did not 
come to us from an unexpected quarter -- the 
notion of "grain size". 
When utterances predicate, it is normally at 
some degree of resolution, or "grain". At a 
fairly coarse grain, we might say that John is at 
the post office -- "at(J,PO)". At a more refined 
grain, we have to say that he is at the stamp 
window -- "at(J,SW)'" We normally think of grain 
in terms of distance, but more generally we can 
move from entities at one grain to entities at a 
coarser grain by means of an arbitrary partition. 
Fine-grained entities in the same equivalence 
class are indistinguishable at the coarser grain. 
Given a set S, consider the partition that 
collapses all elements of S into one element and 
leaves everything else unchanged. We can view the 
typical element of S as the set of real elements 
seen at this coarser grain -- a grain at which, 
precisely, the elements of the set are 
indistinguishable. Formally, we can define an 
operator ~ which takes a set and a predicate as 
its arguments and produces what will be referred 
to as an "indexed predicate": 
T, if x=T(s) & (V yes) p(y), 
<;'(s,p)(x) = F, if x=~(s) &~(F y~s) p(y), 
p(x) otherwise. 
We will frequently abbreviate this "P5 " Note 
that predicate indexing gets us out of the above 
3 An alternative approach would be to say that the 
typical element is in fact one of the real 
elements of the set, but that we will never know 
which one, and that furthermore, we will never 
know about the typical element any property that 
is not true of all the elements. This approach 
runs into technical difficulties involving the 
empty set. 
59 
contradiction, for now "~(s) E 5 s" is not only 
true but tautologous. 
We are now in a position to state the 
properties typical elements should have. The 
first implements universal instantiation: 
(4) (Us,y) p$(~(s)) & yes -> p(y) 
(5) (Vs)(\[(¥x~s) p(x)\] -> p~(~s))) 
That is, the properties of the typical element at 
the coarser grain are also the properties of the 
real elements at the finer grain, and the typical 
element has those properties that all the real 
elements have. 
Note that while we can infer a property from 
set membership, we cannot infer set membership 
from a property. That is, the fact that p is 
true of a typical element of a set s and p is true 
of an entity y, does not imply that y is an 
element of s. After all, we will want "three men" 
to refer to a set, and to be able to infer from 
y's being in the set the fact that y is a man. 
But we do not want to infer from y's being a man 
that y is in the set. Nevertheless, we will need 
a notation for expressing this stronger relation 
among a set, a typical element, and a defining 
condition. In particular, we need it for 
representing "every man", Let us develop the 
notation from the standard notation for 
intensionally defined sets, 
(6) s - {x f p<x)}, 
by performing a fairly straightforward, though 
ontologically promiscuous, syntactic translation 
on it. First, instead of viewing x as a 
universally quantified variable, let us treat it 
as the typical element of s. Next, as a way of 
getting a handle on "p(x)", we will use the 
nominalization operator .... to reify it, and refer 
to the condition e of p (or p$) being true of the 
typical element x of s -- "p~ (e,x)". Expression 
(6) can then be translated into the following flat 
predlcate-argument form: 
(7) set(s,x,e) & p~ (e,x) 
This should be read as saying that s is a set 
whose typical element is x and which is defined by 
condition e, which is the condition of p 
(interpreted at the level of the typical element) 
being true of x. The two critical properties of 
the predicate "set" which make (7) equivalent to 
(6) are the following: 
(8) ~s,x,e,y) set(s,x,e) & p~ (e,x) & p(y) -> yes 
(9) (~s,x,e) set(s,x,e) -> x "T(s) 
Axiom schema (8) tells us that if an entity y has 
the defining property p of the set s, then y is an 
element of s. Axiom (9), along with axiom schemas 
(4) and (3), tells us that an element of a set has 
the act's defining property. 
With what we have, we can represent the 
distinction between the distributive and 
collective readings of a sentence like 
(I0) The men lifted the piano. 
For the collective reading the representation 
would include "llft(m)" where m is the set of men. 
For the distributive reading, the representation 
would have "lift(~(m))", where ~(m) is the 
typical element of the set m. To represent the 
ambiguity of (I0), we could use the device 
suggested in Hobbs \[1982 I for prepositional phrase 
and other ambiguities, and wr~te "llft(x) & (x=m v 
x- ~(m) )". 
This approach involves a more thorough use of 
typical elements than two previous approaches. 
Webber \[1978\] admitted both set and prototype (my 
typical element) interpretations of phrases like 
"each man'" in order to have antecedents for both 
"they" and "he", but she maintained a distinction 
between the two. Essentially, she treated "each 
man" as ambiguous, whereas the present approach 
makes both the typical element and the set 
available for subsequent reference. Mellish 
\[1980 1 uses =yplcal elements strictly as an 
intermediate representation that must be resolved 
into more standard notation by the end of 
processing. He can do this because he is working 
in a task domain -- physics problems -- in which 
sets are not just finite but small, and vagueness 
as to their composition must be resolved. Webber 
did not attempt to use typical elements to derive 
a scope-neutral representation; Mellish did so 
only in a limited way. 
Scope dependencies can now be represented as 
relations among typical elements. Consider the 
sentence 
(II) Most men love several women, 
under the reading in which there is a different 
set of women for each man. We can define a 
dependency function f which for each man returns 
the set of women whom that man loves. 
f(m) = {w \[ woman(w) & love(m,w)} 
The relevant parts of the initial logical form, 
produced by a syntactic and semantic translation 
component, for sentence (Ii) will be 
(12) love(~(m),~(w)) & most(m,ml) & manl(~(ml)) 
& several(w) & womanl(~(w)) 
where ml is the set of all men, m the set of most 
of them referred to by the noun phrase "most men", 
and w the set referred to by the noun phrase 
"several women", and where "manl = ~'(ml,man)" and 
"womanl = ~" (w,woman)'. When the inferenclng 
component discovers there is a different set w for 
each element of the set m, w can be viewed as 
refering to the typical element of this set of 
sets: 
w-T({f<x> { x~m}) 
60 
To eliminate the set notation, we can extend the 
definition of the dependency function to the 
typical element of m as follows: 
f(~(m)) -Z({f(x) I x~m}) 
That is, f maps the typical element of a set into 
the typical element of the set of images under f 
of the elements of the set. From here on, we will 
consider all dependency functions so extended to 
the typical elements of their domains. 
The identity "w - f(~(m))" now 
simultaneously encodes the scoplng information and 
involves only existentially quantified variables 
denoting individuals in an (admittedly 
ontologlcally promiscuous) domain. Expressions 
llke (12) are thus the scope-~eutral 
representation, and scoplng information is added 
by conjoining such identities. 
Let us now consider several examples in which 
processes of interpretation result in the 
acquisition of scoplng information. The first 
will involve interpretation against a small model. 
The second will make use of world knowledge, while 
the third illustrates the treatment of embedded 
quantlflers. 
First the simple, and classic, example. 
(13) Every man loves some woman. 
The initial logical form for this sentence 
includes the following: 
lovel(r(ms),w) & manl(~(ms)) & woman(w) 
where "lovel -@(mS,Ax\[love(x,w)\])'" and "manl - 
(ms,man)". Figure i illustrates two small models 
of this sentence. M is the set of men {A,B}, W is 
the set of women {X,Y}, and the arrows signify 
love. Let us assume that the process of 
interpreting this sentence is Just the process of 
identifying the existentially quantified variables 
ms and w and possibly coercing the predicates, in 
a way that makes the sentence true. 4 
M W M W 
A ~ X A ------~ X 
B /Y B ~ Y 
(a) (b) 
Figure I. Two models of sentence (13). 
In Figure l(a), "'love(A,X)" and "love(B,X)" 
are both true, so we can use axiom schema (5) to 
derive "lovel('~(M),X)". Thus, the 
identifications "ms - M'" and "w = X'" result in the 
sentence being true. 
In Figure l(b), "love(A,X)" and "love(B,Y)" 
are both true, but since these predications differ 
4 Bobrow and Webber \[1980\] similarly show scoplng 
information acquired by Interpretatlon against a 
small model. 
in more than one argument, we cannot apply axiom 
schema (5). First we define a dependency function 
f, mapping each man into a woman he loves, 
yielding "love(A,f(A))" and "love(B,f(B))". We 
can now apply axiom schema (5) to derive 
'" love2 ('~ (M), f (~ (M)) ) ", where "love2 = 
~(M,Ax\[love(x,f(x))\])". Thus, we can make the 
sentence true by identifying ms with M and w with 
f(~'(M)), and by coercing "love" to "'love2" and 
"woman" to "~ (W,woman)". , 
In each case we see that the identification 
of w is equivalent to solving the scope ambiguity 
problem. 
In our subsequent examples we will ignore the 
indexing on the predicates, until it must be 
mentioned in the case of embedded quantifiers. 
Next consider an example in which world 
knowledge leads to disamblguatlon: 
Three women had a baby. 
Before inferencing, the scope-neutral 
representation is 
had(~Z~ws),b) & lwsI=3 & woman(~(ws)) & baby(b) 
Let us suppose the inferencing component has 
axioms about the functionality of having a baby -- 
something llke 
(~ x,y) had(x,y) -> x = mother-of(y) 
and that we know about cardlnallty the fact that 
for any function g and set s, 
Ig(s)l ~ fsl 
Then we know the following: 
3 - lwsl = Imother-of(b) I ~ Ibl 
This tells us that b cannot be an individual but 
must be the typical element of some set. Let f be 
a dependency function such that 
wEws & f(w) = x -> had(w,x) 
that is, a function that maps each woman into some 
baby she had. Then we can identify b with 
f('~'(ws)), or equivalently, with 
~({f(w) I w~ ws}), giving us the correct scope. 
Finally, let us return to interpretation with 
respect to small models to see how embedded 
quantiflers are represented. Consider 
(14) Every representative of a company arrived. 
The initial logical form.includes 
arrive(r) & set(rs,r,ea) & and'(ea,er,eo) 
& rep'(er,r) & of'(eo,r,c) & co(c) 
That is, r arrives, where r is the typical element 
of a set rs defined by the conjunction ea of r's 
being a representative and r's being of c, where c 
is a company. We will consider the two models in 
61 
Figure 2. R is the set of representatives 
{A,B,(C)}, K is the set of companies {X,Y,(Z,W)}, 
there is an arrow from the representatives to the 
companies they represent, and the representatives 
who arrived are circled. 
R K R K 
(a) (b) 
Figure 2. Two models of sentence (14). 
In Figure 2(a), "of(A,X)", "of(B,Y)" and "of(B,Z)" 
are true. Define a dependency function f to map A 
into X and B into Y. Then "of(A,f(A))" and 
"of(B,f(B))" are both true, so that 
"of(~(R),f(~(R)))" is also true. Thus we have 
the following identifications: 
c = f(Z(R)) =~({X,Y}), rs = R, r -t(R) 
In Figure 2(b) "of(B~" and "of(C,Y)'" are 
both true, so "'of(~'(Rl),~)is also. Thus we may 
let c be Y and rs be RI, giving us the wide 
reading for "a company". 
In the case where no one represents any 
company and no one arrived, we can let c be 
anything and rs be the empty set. Since, by the 
definition of o" , any predicate indexed by the 
empty set will be true of the typical element of 
the empty set, "arrlve#(~(# ))" will be true, 
and the sentence will be satisfied. 
It is worth pointing out that this approach 
solves the problem of the classic "donkey 
sentences". If in sentence (14) we had had the 
verb phrase "hates it", then "it" would be 
resolved to c, and thus to whatever c was resolved 
to. 
So far the notation of typical elements and 
dependency functions has been introduced; it has 
been shown how scope information can be 
represented by these means; and an example of 
inferential processing acquiring that scope 
information has been given. Now the precise 
relation of this notation to standard notation 
must be specified. This can be done by means of 
an algorithm that takes the inferential notation, 
together with an indication of which proposition 
is asserted by the sentence, and produces In the 
conventional form all of the readings consistent 
with the known dependency information. 
First we must put the sentence into what will 
be called a "bracketed notation". We associate 
with each variable v an indication of the 
corresponding quantifier; this is determined from 
such pieces of the inferential logical form as 
those involving the predicates "set" and "most"; 
in the algorithm below it is refered to as 
"Quant(v)". The translation of the remainder of 
the inferential logical form into bracketed 
notation is best shown by example. For the 
sentence 
A representative of every company saw a sample 
the relevant parts of the inferential logical form 
are 
see(r,s) & rep(r) & of(r,c) & co(c) & sample(s) 
where "see(r,s) '° is asserted. This is translated " 
in a straightforward way into 
(18) see(It I rep(r) & of(r,\[c I co(c)l)\], 
Is I sample(s)\]) 
This may be read "An r such that r is a 
representative and r is of a c such that c is a 
company sees an s such that s is a sample. 
The nondeterministic algorithm below 
generates all the scoplngs from the bracketed 
notation. The function TOPBVS returns a llst of 
all the top-level bracketed variables in Form, 
that is, all the bracketed variables except those 
within the brackets of some other variable -- in 
(18) r and s but not c. BRANCH 
nondetermlnistically generates a separate process 
for each element in a list it is given as 
argument. A four-part notation is used for 
quantifiers (similar to that of Woods \[1978\]) -- 
"(quantifier varlabie restriction body)". 
G(Form) : 
if \[vlRl ~ BRANCH(TOPBVS(Form)) 
then Form ~ (Quant(v) v BRANCH({R,G(R)}) Form~.~ 
if Form is whole sentence 
then Return G(Form) 
else Return BRANCH({Form,G(Form)}) 
else Return Form 
In this algorithm the first BRANCH corresponds to 
the choice in ordering the top-level quantifiers. 
The variable chosen will get the narrowest scope. 
The second BRANCH corresponds to the decision of 
whether or not to give an embedded quantifier a 
wide reading. The choice R corresponds to a wide 
reading, G(R) to a narrow reading. The third 
BRANCH corresponds to the decision of how wide a 
reading to give to an embedded quantifier. 
Dependency constraints can be built into this 
algorithm by restricting the elements of its 
argument that BRANCH can choose. If the variables 
x and y are at the same level and y is dependent 
on x, then the first BRANCH cannot choose x. If y 
is embedded under x and y is dependent on x, then 
the second BRANCH must choose G(R). In the third 
BRANCH, if any top-level bracketed variable in 
Form is dependent on any variable one level of 
recurslon up, then G(Form) must be chosen. 
A fuller explanation of this algorithm and 
several further examples of the use of this 
notation are given in a longer version of this 
paper. 
62 
3. Other Determlners 
The approach of Section 2 will not work for 
monotone decreasing determiners, such as "few" and 
"no". Intuitively, the reason is that the 
sentences they occur in make statements about 
entities other than just those in the sets 
referred to by the noun phrase. Thus, 
Few men work. 
is more a negative statement about all but a few 
of the men than a positive statement about few of 
them. One possible representation would be 
similar to (I), but wlth the implication reversed. 
(Bs)(q(s,{x I P(x)}) 
& (~ y)(P(y) & R(y) -> yes)) 
This is unappealing, however, among other things, 
because the predicate P occurs twice, making the 
relation between sentences and logical forms less 
direct. 
Another approach would take advantage of the 
above intuition about what monotone decreasing 
determiners convey. 
(7 s)(Q(s,{x \[ P(x)}) & (~y)(y£s->-~R(y))) 
That is, we convert the sentence into a negative 
assertion about the complement of the noun phrase, 
reducing this case tO the monotone increasing 
case. For example, "few men work" would be 
represented as follows: 
(~ s)(\[~w(s,{x I man(x)}) 
& (Vy)(y~s ->~work(y))) 5 
(This formulation is equivalent to, but not 
identical with, Barwlse and Cooper's \[1981\] 
witness set condition for monotone decreasing 
determiners.) 
Some determiners are neither monotone 
increasing nor monotone decreasing, but Barwlse 
and Cooper conjecture that it is a linguistic 
universal that all such determiners can be 
expressed as conjunctions of monotone determiners. 
For example, "exactly three" means "at least three 
and at most three". If this is true, then they 
all yield to the approach presented here. 
Moreover, because of redundancy, only two new 
conjuncts would be introduced by this method. 
Acknowledgments 
I have profited considerably in this research 
from discussions with Lauri Kartunnen, Bob Moore, 
Fernando Pereira, Stan Rosenscheln, and Stu 
Shleber, none of whom would necessarily agree with 
what I have written, nor even view it with 
sympathy. This research was supported by the 
Defense Advanced Research Projects Agency under 
Contract No. N00039-82-C-0571, by the National 
Library of Medicine under Grant No. IR01 LM03611- 
5 "~w' is pronounced "few bar". 
01, and by the National Science Foundation under 
Grant No. IST-8209346. 
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63 
