CONNECTION RELATIONS AND QUANTIFIER SCOPE 
Long-in Latecki 
University of Hamburg 
Department of Computer Science 
Bodenstedtstr~ 16, 2000 Hamburg 50, Germany 
e-mail: latecki@rz.informatik.uni-hamburg.dbp.de 
ABSTRACT 
A formalism will be presented in this 
paper which makes it possible to realise the 
idea of assigning only one scope-ambiguous 
representation to a sentence that is ambiguous 
with regard to quantifier scope. The scope 
determination results in extending this 
representation with additional context and 
world knowledge conditions. If there is no 
scope determining information, the formalism 
can work further with this scope-ambiguous 
representation. Thus scope information does 
not have to be completely determined. 
0. INTRODUCTION 
Many natural language sentences have 
more than one possible reading with regard to 
quantifier scope. The most widely used 
methods for scope determination generate all 
possible readings of a sentence with regard to 
quantifier scope by applying all quantifiers 
which occur in the sentence in all 
combinatorically possible sequences. These 
methods do not make use of the inner structure 
and meaning of a quantifier. At best, 
quantifiers are constrained by external 
conditions in order to eliminate some scope 
relations. The best known methods are: 
determination of scope in LF in GB (May 1985), 
Cooper Storage (Cooper 1983, Keller 1988) and 
the algorithm of Hobbs and Shieber 
(Hobbs/Shieber 1987). These methods assign, 
for instance, six possible readings to a sentence 
with three quantifiers. Using these methods, a 
sentence must be disambiguated in order to 
receive a semantic representation. This means 
that a scope-ambiguous sentence necessarily 
has several semantic representations, since the 
formalisms for the representation do not allow 
for scope-ambiguity. 
It is hard to imagine that human beings 
disambiguate scope-ambiguous sentences in the 
same way. The generation of all possible 
combinations of sequences of quantifiers and 
the assignment of these sequences to various 
readings seems to be cognitively inadequate. 
The problem becomes even more complicated 
when natural language quantifiers can be 
interpreted distributively as well as 
collectively, which can also lead to further 
readings. Let us take the following sentence 
from Kempson/Cormack (1981) as an example: 
Two examiners marked six scripts. 
The two quantifying noun phrases can in 
this case be interpreted either distributively 
or collectively. The quantifier two examiners 
can have wide scope over the quantifier six 
scripts, or vice versa, which all in all can lead 
to various readings. Kempson and Cormack 
assign four possible readings to this sentence, 
241 
Davies (1989) even eight. (A detailed 
discussion will follow.) No one, however, will 
make the claim that people will first assign 
all possible representations with regard to the 
scope of the quantifiers and their distribution, 
and will then eliminate certain 
interpretations according to the context; but 
this is today's standard procedure in 
linguistics. In many cases, it is also almost 
impossible to determine a preferred reading. 
The difficulties that people have when they 
are forced to disambiguate such sentences (to 
explicate all possible readings) point to the 
fact that people only assign an under- 
determined scope-ambiguous representation in 
the first place. 
Such a representation of the example 
sentence would only contain the information 
that we are dealing with a marking-relation 
between examiners and scripts, and that we 
are always dealing with two examiners and 
six scripts. This representation does not contain 
any information about scope. On the basis of 
this representation one may in a given context 
derive a representation with a determined 
scope. But it may also be the case that this 
information is sufficient in order to understand 
the sentence if no scope-defining information is 
given in the context, since in many cases human 
beings do not disambiguate such sentences at 
all. They use underdetermined, scopeless 
interpretations, because their knowledge often 
need not be so precise. If a disambiguation is 
carried out, then this process is done in a very 
natural way on the basis of context and world 
knowledge. This points to the assumption that 
scope determination by human beings is 
performed on a semantic level and is deduced 
on the basis of acquired knowledge. 
I will present a formalism which works in 
a similar way. This formalism will also show 
that it is not necessary to work with many 
sequences of quantifiers in order to determine 
the various readings of a sentence with regard 
to quantifier scope. 
Within this formalism it is possible to 
represent an ambiguous sentence with an 
ambiguous representation which need not be 
disambiguated, but can be disambiguated at a 
later stage. The readings can either be 
specified more clearly by giving additional 
conditions, or they can be deduced from the 
basic ambiguous reading by inference. Here, 
the inner structure and the meaning of 
quantifiers play an important role. The process 
of disambiguation can only be performed when 
additional information that restricts the 
number of possible readings is available. As an 
example of such information, I will treat 
anaphoric relations. 
Intuitively speaking, the difference 
between assigning an undertermined 
representation to an ambiguous sentence and 
assigning a disjunction of all possible readings 
to this sentence corresponds to the difference 
between the following statements*: 
"Peter owns between 150 and 200 books." 
and 
"Peter owns 150 or 151 or 152 or ... or 200 books." 
It goes without saying that both 
statements are equivalent, since we can 
understand "150 or 151 or ... or 200" as a precise 
specification of "between 150 and 200". 
Nevertheless, there are procedural differences 
in processing the two pieces of information; 
and there are cognitive differences for human 
beings, since we would never explicitly utter 
the second sentence. If we could represent 
"between 150 and 200" directly by a simple 
formula and not by giving a disjunction of 51 
elements, then we may certainly gain great 
procedural and representational advantages. 
The deduction of readings in semantics does 
not of course exclude a consideration of 
syntactic restrictions. They can be imported 
into the semantics, for example by passing 
syntactic information with special indices, as 
* The comparison stems from Christopher 
Habel. 
242 
described in Latecki (1991). Nevertheless, in 
this paper I will abstain from taking syntactic 
restrictions into consideration. 
1. SCOPE-AMBIGUOUS 
REPRESENTATION AND SCOPE 
DETERMINATION 
The aims of the representation presented 
in this paper are as follows: 
1. Assigning an ambiguous semantic 
representation to an ambiguous sentence (with 
regard to quantifier scope and distributivity), 
from which further readings can later be 
inferred. 
2. The connections between the subject and 
objects of a sentence are explicitly represented 
by relations. The quantifiers (noun phrases) 
constitute restrictions on the domains of these 
relations. 
3. Natural language sentences have more 
than one reading with regard to quantifier 
scope (and distributivity), but these readings 
are not independent of one another. The target 
representation makes the logical dependencies 
of the readings easily discernible. 
4. The construction of complex discourse 
referents for anaphoric processes requires the 
construction of complex sums of existing 
discourse referents. In conventional 
approaches, this can lead to a combinatorical 
explosion (cf. Eschenbach et al. 1989 and 1990). 
In the representation which is presented here, 
the discourse referents are immediately 
available as domains of the relations. 
Therefore, we need not construe any complex 
discourse referents. Sometimes we have to 
specify a discourse referent in more detail, 
which in turn can lead to a reduction in the 
number of possible readings. 
I now present the formalism. 
The representational language used here is 
second-order predicate logic. However, I will 
mainly use set-theoretical notation (which 
can be seen as an abbreviation of the 
corresponding notation of second-order logic). I 
choose this notation because it points to the 
semantic content of the formulas and is thus 
more intuitive. 
Let R ~ XxY be a relation, that means, a 
sub-set of the product of the two sets X and Y. 
The domains of R will be called Dom R and 
Range R, with 
Dom R={x~ X: 3y~ Y R(x,y)} and 
Range R={y~ Y: 3x~ X R(x,y)}. 
I make the explicit assumption here that 
all relations are not empty. (This assumption 
only serves in this paper to make the examples 
simpler.) 
In the formalism, a verb is represented by a 
relation whose domain is defined by the 
arguments of verbs. Determiners constitute 
restrictions on the domains of the relation. 
These restrictions correspond to the role of 
determiners in Barwise's and Cooper's theory 
of generalized quantifiers (Barwise and 
Cooper 1981). This means for the following 
sentence: 
(1.1) Every boy saw a movie. 
that there is a relation of seeing between boys 
and movies. 
In the formal notation of second-order logic 
we can describe this piece of information as 
follows: 
(1.1.a) 3X2 (Vxy (X2(x,y) ~ Saw(x,y) & 
Boy(x) & M0vie(y) )) 
X2 is a second-order variable over the 
domain of the binary predicates; and Saw, 
Boy, and Movie are second-order constants 
which represent a general relation of seeing, 
the set of all boys, and the set of all movies, 
respectively. We will abbreviate the above 
formula by the following set-theoretical 
formula: 
240 
(1.1.b) 3saw (saw ~ Boy x Movie) 
In this formula, we view saw as a sorted 
variable of the sort of the binary seeing- 
relations. The variable saw corresponds to the 
variable X2 in (1.1.a). 
(1.1.b) describes an incomplete semantic 
representation of sentence (1.1). Part of the 
certain knowledge that does not determine 
scope in the case of sentence (1.1) is also the 
information that all boys are involved in the 
relation, which is easily describable as: 
Dom saw=Boy. We obtain this information 
from the denotation of the determiner every. 
In this way we have arrived at the scope- 
ambiguous representation of (1.1): 
(1.1.c) 3saw (saw ~ Boy x Movie & 
Dom saw=Boy) 
It may be that the information presented 
in (1.1.c) is sufficient for the interpretation of 
sentence (1.1). A precise determination of 
quantifier scope need not be important at all, 
since it may be irrelevant whether each boy 
saw a different movie (which corresponds to 
the wide scope of the universal quantifier) or 
whether all boys saw the same movie (which 
corresponds to the wide scope of the 
existential quantifier). 
Classic procedures will in this case 
immediately generate two readings with 
definite scope relations, whose notations in 
predicate logic are given below. 
(1.2.a) Vx(boy(x) --~ 3y(movie(y) & saw(x,y))) 
(1.2.b) 3y(movie(y) & Vx(boy(x) --~ saw(x,y))) 
We can also obtain these representations in 
our formalism by simply adding new conditions 
to (1.1.c), which force the disambigiuation of 
(1.1.c) with regard to quantifier scope. To 
obtain reading (1.2.b), we must come to know 
that there is only one movie, which can be 
formaly writen by I Range saw I =1, where I . I 
denotes the cardinality function. To obtain 
reading (1.2.a) from (1.1.c), we do not need any 
new information, since the two formulas are 
equivalent. This situation is due to the fact 
that (1.2.b) implies (1.2.a), which means that 
(1.2.b) is a special case of (1.2.a). This relation 
can be easly seen by comparing the resulting 
formulas, which correspond to readings (1.2.a) 
and (1.2.b): 
(1.3.a) 3saw (saw c Boy x Movie & 
Dom saw=Boy) 
(1.3.b) 3saw (saw ~ Boy x Movie & 
Dom saw=Boy & I Range saw I =1) 
So, we have (1.3.b) => (1.3.a). 
As I have stated above, however, it is not 
very useful to disambiguate representation 
(1.1.c) immediately. It makes more sense to 
leave representation (1.1.c) unchanged for 
further processing, since it may be that in the 
development a new condition may appear 
which determines the scope. For instance, we 
can obtain the additional condition in (1.3.b), 
when sentence (1.1) is followed by a sentence 
containing a pronoun refering to a movie, as in 
sentence (1.4). 
(1.4) It was "Gone with the Wind". 
Since it refers to a movie, the image of the 
saw-relation (a subset of the set of movies) can 
contain only one element. Thus, the resolution 
of the reference results in an extension of 
representation (1.1.c) by the condition 
I Range saw I = 1. Therefore, we get in this case 
only one reading (1.3.b) as a representation of 
sentence (1.1), which corresponds to wide scope 
of the existential quantifier. Thus in the 
context of (1.4) we have disambiguated 
sentence (1.1) with regard to quantifier scope 
without having first generated all possible 
readings (in our case these were (1.2.a) and 
(1.2.b)). 
244 
Let us now assume that sentence (1.5) 
follows (1.1). 
(1.5) All of them were made by Walt Disney 
Studios. 
Syntactic theories alone are of no help 
here for finding the correct discourse referent 
for them in sentence (1.1), since there is no 
number agreement between them and a movie. 
The plural noun them, however, refers to all 
movies the boys have seen. This causes great 
problems for standard anaphora theories and 
plural theories, since there is no explicit object 
of reference to which them could refer (cf. 
Eschenbach et al. 1990; Link 1986). Thus, the 
usual procedure would be to construe a complex 
reference object as the sum of all movies the 
boys have seen. With my representation, we 
do not need such procedures because the 
discourse referents are always available, 
namely as domains of the relations. In the 
context of (1.1) and (1.5), the pronoun them 
(just as it in (1.4)) refers to the image of the 
relation saw, which additionally serves the 
purpose of determining the quantifier scope. 
Here, just as in the preceding cases, the 
representation (1.1.c) has to be seen as the 
"starting representation" of (1.1). The 
information that them is a plural noun is 
represented by the condition I Range saw I > 1, 
which in turn leads to the following 
representation: 
(1.6) 3saw (saw ~ BOy x Movie & 
Dom saw=Boy & I Range saw I >1) 
The representation (1.6) is not ambiguous 
with regard to quantifier scope. The universal 
quantifier has wide scope over the whole 
sentence, due to the condition I Range saw I > 1. 
The reading presented in (1.6) is a further 
specification of (1.3.a), which at the same 
time excludes reading (1.3.b). Thus (1.6) 
contains more information that formula 
(1.2.a), which is equivalent to (1.3.a). 
A classical scope determining system can 
only choose one of the readings (1.2.a) and 
(1.2.b). However, if it chooses (1.2.a), it will 
not win any new information, since (1.2.b) is a 
special case of (1.2.a). So, quantifier scope can 
not be completely determined by such a system. 
In order to indicate further advantages of 
this representation formalism, let us take a 
look at the following sentence (cf. Link 1986): 
(1.7) Every boy saw a different movie. 
Its representation is generated in the same 
way as that of (1.1), the only difference being 
that the word different carries additional 
information about the relation saw. different 
requires that the relation be injective. 
Therefore, the formula (1.1.c) is extended by 
the condition 'saw is 1-1'. The formula (1.8) 
thus represents the only reading of sentence 
(1.7), in which scope is completely 
determined; the universal quantifier has wide 
scope. 
(1.8) 3saw (saw ~ Boy x Movie & 
Dom saw=Boy & saw is 1-1) 
2. SCOPE-AMBIGUOUS 
REPRESENTATION FOR 
SENTENCES WITH NUMERIC 
QUANTIFIERS 
So far, I have not stated exactly how the 
representation of sentence (1.1) was generated. 
In order to do so, let us take an example 
sentence with numeric quantifiers: 
(2.1) Two examiners marked six scripts. 
It is certainly not a new observation that 
this sentence has many interpretations with 
regard to quantifier scope and distributivity, 
which can be summarized to a few main 
readings. However, their exact number is 
controversial. While Kempson and Cormack 
245 
(1981) assign four readings to this sentence (see 
also Lakoff 1972), Davies (1989) assigns eight 
readings to it. I quote here the readings from 
(Kempson/Cormack 1981): 
Uniformising: 
Replace "(Vx~ Xn)(3Y)" by "(3Y)(Vx~ Xn)" 
10 There were two examiners, and each of 
them marked six scripts (subject noun phrase 
with wide scope). This interpretation could be 
true in a situation with two examiners and 12 
scripts. 
20 There were six scripts, and each of these 
was marked by two examiners (object noun 
phrase with wide scope). This interpretation 
could be true in a situation with twelve 
examiners and six scripts. 
30 The incomplete group interpretation: 
Two examiners as a group marked a group of six 
scripts between them. 
40 The complete group interpretation: Two 
examiners each marked the same set of six 
scripts. 
Kempson and Cormack represent these 
readings with the help of quantifiers over sets 
in the following way: 
10 (3X2)(Vx~ X2)(3S6)(Vs~ S6)Mxs 
20 (3S6)(Vs~ S6)(3X2)(Vx~ X2)Mxs 
30 (3X2)(3S6)(Vx~ X2)(Vs~ S6)Mxs 
40 (3X2)(3S6)(Vx~ X2)(3s~ S6)Mxs & 
(Vs~ $6)(3x~ X2)Mxs 
Here, X 2 is a sorted variable which 
denotes a two-element set of examiners, and S 6 
is a sorted variable that denotes a six-element 
set of scripts. 
Kempson and Cormack derive these 
readings from an initial formula in the 
conventional way by changing the order and 
distributivity of quantifiers. This fact is 
discernible from their derivational rules and 
the following quotation: 
Generalising: 
Replace "(3x~ Xn)" by "(Vx~ Xn)" 
"What we are proposing, then, as an 
alternative to the conventional 
ambiguity account is that all sentences 
of a form corresponding to (42) \[here: 
2.1\] have a single logical form, which 
is then subject to the procedure of 
generalising and uniformising to yield 
the various interpretations of the 
sentence in use." (Kempson/Cormack 
(1981), p. 273) 
Only in reading 40 the relation between 
examiners and scripts is completely 
characterized. For the other formulas there 
are several possible assignments between 
examiners and scripts which make these 
formulas valid. 
At this point I want to make an important 
observation, namely that these four readings 
are not totally independent of one another. I 
am, however, not concerned with logical 
implications between these readings alone, but 
rather with the fact that there is a piece of 
information which is contained in all of these 
readings and which does not necessitate a 
determinated quantifier scope. This is the 
information which - cognitively speaking - can 
be extracted from the sentence by a listener 
without determining the quantifier scope. The 
difficulties which people have when they are 
forced to disambiguate a sentence containing 
numeric quantifiers such as (2.1) without a 
specific context point to the fact that only such 
a scopeless representation is assigned to the 
sentence in the first place. On the basis of this 
representation one can then, within a given 
context, derive a representation with a 
definite scope. We can describe the scopeless 
piece of information of sentence (2.1), which 
all readings have in common, as follows. We 
know that we are dealing with a marking- 
246 
relation between examiners and scripts, and 
that we are always dealing with two 
examiners or with six scripts. In the formalism 
described in this paper this piece of 
information is represented as: 
(2.2) 3mark ( mark c Examiner x Script & 
(IDommarkl=2 v IRangemarkl--6)) 
It may be that this piece of information is 
sufficient in order to understand sentence (2.1). 
If there is no scope-determining information in 
the given context, people can understand the 
sentence just as well. If, for example, we hear 
the following utterance, 
(2.3) In preparation for our workshop, two 
examiners corrected six scripts. 
it may be without any relevance what the 
relation between examiners and scripts is 
exactly like. The only important thing may be 
that the examiners corrected the scripts and 
that we have an idea about the number of 
examiners and the number of scripts. 
Therefore, we have assigned an under- 
determined scope-ambiguous representation 
(2.2) to sentence (2.1), which constitutes the 
maximum scopeless content of information of 
this sentence. The lower line of (2.2) represents 
a scope-neutral part of the information which 
is contained in the meaning of the quantifiers 
two examiners and six scripts. This fact 
indicates that the meaning of a quantifier has 
to be structured internally, since a quantifier 
contains scope-neutral as well as scope- 
determining information. Distributivity is an 
example of scope-determining information. 
Then what happens in a context which 
contains scope-determining information? This 
context just provides restrictions on the 
domains of the relation. These restrictions in 
turn contribute to scope determination. We 
may, for instance, get to know in a given 
context that there were twelve scripts in all, 
which excludes the condition I Range mark I =6 
in the disjunction of (2.2). We then know for 
certain that there were two examiners and 
that each of them marked six different scripts. 
Consequently, the quantifier two examiners 
acquires wide scope, and we are dealing with a 
distributive reading. Thus, in this context we 
have completely disambiguated sentence (2.1) 
with regard to quantifier scope; and that 
simply on the basis of the scopeless, 
incomplete representation (2.2). On the other 
hand, standard procedures (the most 
important were listed at the beginning) first 
have to generate all representations of this 
sentence by considering all combinatorically 
possible scopes together with distributive and 
collective readings. 
3. CONCLUDING REMARKS 
A cognitively adequate method for 
dealing with sentences that are ambiguous 
with regard to quantifier scope has been 
described in this paper. An underdetermined 
scope-ambiguous representation is assigned to 
a scope-ambiguous sentence and then extended 
by additional conditions from context and 
world knowledge, which further specify the 
meaning of the sentence. Scope determination 
in this procedure can be seen as a mere by- 
product. The quantifier scope is completely 
determined when the representation which 
was generated in this way corresponds to an 
interpretation with a fixed scope. Of course, 
this only works if there is scope-determining 
information; if not, one continues to work with 
the scope-ambiguous representation. 
I use the language of second-order 
predicate logic here, but not the whole second- 
order logic, since I need deduction rules for 
scope derivation, but not deduction rules for 
second-order predicate logic (which cannot be 
completely stated). One could even use the 
formalism for scope determination alone and 
then translate the obtained readings into a 
first-order formalism. However, the 
formalism lends itself very easily to 
247 
representation and processing of the derived 
semantic knowledge as well. 
ACKNOWLEDGMENTS 
I would like to thank Christopher Habel, 
Manfred Pinkal and Geoff Simmons. 

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