HANDLING SCOPE AMBIGUITIES IN ENGLISH 
Sven Hurum 
Department of Computing Science 
615 General Services Building 
University of Alberta 
Edmonton, Canada T6G 2H1 
ABSTRACT 
This paper describes a program for handling "scope 
ambiguities" in individual English sentences. The program 
operates on initial logical translations, generated by a 
parser/translator, in which "unscoped elements" such as 
quantifiers, coordinators and negation are left in place to be 
extracted and positioned by the scoping program. The program 
produces the set of valid scoped readings, omitting logically 
redundant readings, and places the readings in an approximate 
order of preference using a set of domain-independent 
heuristics. The heuristics are based on information about the 
lexical type of each operator and on "structural relations" 
between pairs of operators. The need for such 
domain-independent heuristics is emphasized; in some cases 
they can be decisive and in general they will serve as a guide 
to the use of further heuristics based on domain-specific 
knowledge and on the context of discourse. The emphasis of 
this paper is on discussing several of the more problematic 
aspects of the scoping protocol which wcre encountered during 
the design of the scoping program. 
INTRODUCTION 
Natural languages contain a variety of "logical operators" 
which interact with each other to give rise to different types of 
ambiguity. The logical operators recognized by the scoping 
program include quantifiers, coordinators and negation, which 
are initially "unscoped" and must therefore be moved into 
position by the program, and adverbs, predicates and 
connectives (such as if-then). At the moment, other operators 
such as tense, aspect and modals are left in place and therefore 
assume innermost scope. There is some evidence that the 
handling of the scoping of quantifiers relative to such operators 
may require special treatment (eg. Fodor 1970; Enc 1981; 
S aarinen 1983). 
Three simple examples will illustrate some different 
types of scope ambiguity and their representation in an 
informal first order predicate logic, using restrictions on 
quantifiers and an infix notation for sentential formulas. The 
meanings of the different interpretations should be clear. For 
example, (4) may mean that John didn't meet either Jane or 
Mary (5) or that he didn't meet at least one of them (6). Further 
examples are given in Hurum & Schubert (1986) and Hurum 
(1987). Some alternative proposals for representing scope 
ambiguities are also discussed in the latter. 
(1) Someone loves everyone 
(2) (3x:person (Vy:person \[x loves y\])) 
(3) (Vy:person (3x:person \[x loves y\])) 
(4) John didn't meet Jane or Mary 
(5) --,\[\[John met Jane\] v \[John met Mary\]\] 
(6) \[--,\[John met Jane\] v ~\[John met Mary\]\] 
(7) Someone always comes late 
(8) (3x:person (always \[x comes late\])) 
(9) (always (3x:person \[x comes late\])) 
Until quite recently, designers of natural language 
understanding systems have given little attention to the 
problem of dealing with scope ambiguities. Two of the earliest 
attempts to incorporate quantifier scoping into natural language 
understanding systems in ,an integral way are described in 
Woods (1978) and Dahl (1979). Some more recent scoping 
algorithms are presented in McCord (1981), Warren & Pereira 
(1982), Hobbs (1983), Saint-Dizier (1985) and Hobbs & 
Shieber (1987). 
While each of these algorithms introduces some new 
features, certain problems, such as the scoping of coordinators 
and the use of heuristics to select preferred readings, have 
generally been given little or no treatment. Some of the main 
features of the algorithm being discussed here are: (a) it 
handles ambiguities created by quantifiers, coordinators, 
negation and adverbs, ~ (b) it works bottom-up and left-to-right 
and generates the set of valid scoped readings in one pass, (c) 
it removes logically redundant readings as they are 
encountered during the process of scoping and (d) it uses 
domain-independent heuristics, during the scoping, to arrange 
the readings in an approximate order of preference. 
LOGICAL REPRESENTATION 
The scoping program is designed to be used as an 
extension to a parser/translator which generates initial 
translations in a first order modal logic augmented with certain 
operators (Schubert & Pelletier 1982). The operators being 
used include a generic kind forming operator, g, and the 
Four types of coordinated expression are currently handled: noun 
phrases, noun complements, verbs and verb phrases. At the moment 
adverbs are treated as scoped (unmoved) elements. 
58 
operators ~ and x which form functions and terms, 
respectively, from infix and prefix expressions. For example, 
the operators x I and x 2 map infix and prefix expressions, 
respectively, into terms. 
The syntax of the logical translations has been chosen to 
simplify the mapping from the syntax (using a modified GPSG 
parser). A mixed infix/prefix notation has been used in order to 
keep the logical form as close as possible to the surface form. 
Two examples of the initial logical translations being used are 
shown below. Unscoped operators, which are to be extracted 
and positioned by the scoping program, are placed in angled 
brackets; the square, curly and round brackets signify infix 
(sentential), prefix (predicative) and functional expressions, 
respectively. The suffixes which are attached to each word to 
mark their surface position are not shown here. 
(10) Many people visit Europe every month 
(11) ((at 2 {during <every month>}) 
\[<many person> (PRES {visit Europe})\]) 
(12) That John didn't arrive surprised Jane and Mary 
(13) \[('r 1 \[John <not (PAST {arrive})>\]) 
(PAST {surprise <and Jane Mary>})\] 
A sample of the output from the program is shown in the 
Appendix. The two sentences shown are 
1. All men want to marry Peggy or Sue 
2. Mary (read or told some story to each child) 
The output for each sentence consists of an echo of the input 
formula followed by a list of scoped readings ordered 
according to their average scoping weight (see below). In the 
LISP notation, the prefixes i, p, f, q and c are used to mark 
infix, prefix, functional, quantified and coordinated 
expressions. The first sentence is taken from Schubert & 
Pclletier (1982) which gives a description of the three 
interpretations. The second sentence has been parsed as having 
a verb phrase ambiguity (indicated by the brackets) and the 
input formula therefore contains two duplicated operators. The 
two comparisons made are each~some and each~or. No 
comparison is made between the commutative operators some 
and or. 
COORDINATED EXPRESSIONS 
The scoping of coordinated expressions poses several 
problems. One problem is how to avoid the "vacuous" 
quantification or coordination which may result whenever a 
coordinated expression contains an unscoped operator. For 
example, if the indefinite some blonde in (14) is applied to the 
clause before the coordinator, the subsequent application of the 
latter will result in vacuous quantification (15). 
(14) John met Mary or some blonde 
(1'5) \[(3x: blonde \[John {met Mary}\]) or 
(3x: blonde \[John {met x}\])\] 
Similarly, "vacuous coordination" may result when nested 
coordinators are present. This problem could be avoided by 
scoping in several passes, in each pass scoping only operators 
which are not embedded inside a coordinator. However, this 
would considerably complicate the scoping algorithm and 
would also violate the principle of applying the innermost 
operator first. 
A second problem is how to handle the scoping of 
multiple copies of the same operator which may occur when 
the operator is embedded inside a coordinated expression. This 
problem is unavoidable when it results from the parser; for 
example, (16) may be parsed and initially translated into (17). 
The brackets signify that the sentence has been parsed as 
having a VP coordination. 
(16) John (hopes and intends to buy a boat) 
(17) \[John <and (PRES {hope 
(x2 (INF {buy <a I boat>}))}) 
(PRES {intend 
('r 2 (INF {buy <a 1 boat>}))}) >\] 
Three constraints on a duplicated operator such as a I are that 
(a) it must scope consistently with respect to all other 
operators, (b) it must only be compared once to each other 
operator (for the purposes of computing the preferred scope 
orderings) and (c) if it scopes outside a coordinator which 
initially embeds it, only one copy of the operator can be 
carried up. This poses a problem for bottom-up approaches to 
scoping since some global knowledge is needed to ensure the 
consistency of the scoping of duplicated operators inside the 
different expressions in which they occur. It therefore is 
necessary to use some overhead to keep track of the scope 
relations of operators which are present in multiple copies, and 
to store this information separately for each reading. 
Duplication of operators may also occur during the 
scoping process. For example, the application of one of the 
coordinators in 
(18) John and Bill visited Spain or Morocco 
will result in the duplication of the other. At present, the 
scoping program avoids this problem, as well as the problem 
of vacuous quantification, by using a "branch-trimming" 
function which removes incorrectly embedded operators from 
the different branches of a coordinator at the time of applying 
the coordinator. This function is simple to use but does involve 
some extra overhead. The problem of duplication resulting 
from the parser is handled by labelling readings and by storing 
on the property list of each duplicated operator a list of the 
operators having been scoped inside and outside the operator. 
A third problem is how to treat unscoped operators 
inside "coordinated predicates". In example (16) it seems 
evident that the indefinite a boat cannot have both opaque and 
transparent interpretations in the same reading. That is, 
assuming that the opaque/transparent distinction is to be 
represented in terms of scope, then both copies of the 
indefinite must scope consistently relative to the two 
coordinated predicates hope and intend. Since the two 
predicates are distinct, and therefore should be allowed to 
scope independently with a boat, the current version of the 
59 
program contains a special constraint which forces coordinated 
predicates to scope consistently relative to all duplicated 
operators embedded inside them. This rule could be treated as 
a heuristic rather than as a constraint, but the rule does seem to 
be absolute. 
In contrast, there is a general, but not absolute, 
preference for "symmetric" interpretations whenever 
coordinated expressions contain similar but not identical pairs 
of operators. For example, in (19) one could imagine a context 
in which it is made clear that Sue, but not Mary, has a 
particular hat in mind and in (20)it is possible, though very 
improbable, that the two indefinites have different functional 
dependencies. 
(19) John (knows that Sue wants) and (thinks that Mary hopes) 
to buy a new hat 
(20) Mary read a story to each child or told a story to each 
child 
At present, the program does not adequately handle this 
preference for symmetric readings, which requires some 
non-local heuristic knowledge. 
REDUNDANT READINGS 
A test is made for logically redundant readings whenever 
an unscoped operator is about to be positioned (applied to a 
clausal expression). A reading is considered to be redundant if 
two commutative operators are applied consecutively and the 
suffix of the outer operator is greater than that of the inner one. 
(Suffixes are attached to words by the parser to mark their 
position in the original sentence). If one of the operators is a 
coordinator the criterion used is that the quantifier should 
scope inside the coordinator. Readings will also be removed if 
they contain an ordering of a pair of operators which has a 
scoping weight less than a preset parameter. 
SCOPING WEIGHTS 
In order to quantify scoping preferences, we associate a 
"scoping weight", a value between 0 and 1, with each pair of 
interacting operators. The weight indicates the preference for 
the reading in which the second operator (in surface order) 
scopes outside the first one. For example, the value 0.9 
indicates a strong preference for the reading in which the 
second operator takes wide scope, a preference which might, 
on occasion, be overridden by pragmatics. The weight 
associated with the reverse ordering will automatically be 0.1. 
The value 0.5 indicates an equal preference for both scope 
orderings in a pragmatically neutral context. The following 
examples illustrate how the scoping weights are used. 
(21) Some person on each team was injured .9 
(22) Some person playing on each team was injured .5 
(23) Some person who plays on each team was injured .02 
As the scoping weights indicate, the ability of the embedded 
quantifier each team to widen scope over some person 
decreases as the embedding phrase changes from a 
prepositional phrase (21) to a verb phrase (22) to a full clause 
(23). This "embedding hierarchy" was pointed out by van Lehn 
(1978) and also holds for phrases serving as adverbials or as 
terms. 
The scoping weights used by the program have been 
derived from the examination of a large number of sentences 
such as these. An attempt was made to keep the sentences as 
pragmatically neutral as possible and to try to obtain a 
domain-independent weight for pairs of operators in a given 
"pattern", where a pattern is a combination of two operators of 
given types and in a given structural relation to one another. 
Although the data reflect the intuitve judgements of the author, 
it is likely that there would be a good general agreement in 
cases in which there is a strong preference for one ordering. In 
other cases, the need to include pragmatic knowldege would be 
more important. Some consideration was also given to the 
empirical data on scoping preferences described previously 
(eg. Ioup 1975, van Lehn 1978, Gil 1982). 
Given that we can determine scoping weights for pairs of 
operators, it is still necessary to combine these to arrive at an 
overall rating of a reading. This involves two separate 
problems: how to select pairs of operators for comparison and 
then how to combine the weights obtained. There appears to be 
no obvious solution to either of these problems. There are at 
least three different choices which need to be made when 
picking a strategy for selecting pairs of operators for 
comparison, none of which is clearcut. For example, if a 
sentence contains three quantifiers at the same level, such as a 
subject and two objects, should all three pairs of quantifiers be 
compared or should the results of each comparison made be 
used to reduce the number of further comparisons needed? 
There is also no obviou.s way to combine the scoping 
weights obtained. A probabilistic treatment is not feasible, in 
part because different readings of a sentence may involve 
different numbers of comparisons. The simplest method is to 
order the readings according to their average scoping weight 
and this appears to give quite good results. The major 
drawback to this method is that it tends to smooth out the 
effect of very low individual weights. However, there are ways 
to minimize this problem. At present, a parameter is used to 
specify the minimal acceptable scoping weight so that readings 
with very low pairwise orderings can be removed. 
Alternatively, readings could be tagged with their lowest 
weights and some readings later be set aside or some more 
complex function could be used for combining the scoping 
weights. These problems are discussed in Hurum (1987). 
HEURISTICS BASED ON LEXICAL TYPES 
The domain-independent heuristics are based on two 
types of information: the lexical type of each operator and 
structural relations between pairs of operators. Some heuristics 
are def'med for individual lexical types, such as each, some and 
or, and others for classes of individual types, such as universal 
or existential quantifiers. Most of the heuristics used by the 
program are stored in a table of scoping weights. To minimize 
the amount of data, universal and existential quantifiers are 
sometimes represented by the "standards" each and some and 
60 
other members of these classes are then related to the standards 
by ratios. Most, but not all, of the heuristics described here are 
currently being used by the program. 
The universal quantifiers may be arranged in the 
hierarchy each > every > all in terms of the tendency to tak~ 
wide scope. This hierarchy has been mentioned by both Ioup 
and van Lehn and a number of people have commented that 
the function of each in English may partly be to indicate the 
distributive (ie. wide scope) reading. Universal quantifiers have 
a surpringly marked tendency to scope inside a negation 
(24-26) given their usual tendency, with the exception of both, 
to take wide scope: 
(24) All people aren't happy 
(25) John didn't win every race 
(26) John didn't win both races 
.6 
.2 
.1 
Non-universal quantifiers in the subject position of a negated 
sentence seldom scope inside the negation. 
Few and no have very little ability to widen scope over a 
preceding operator but, in contrast, have a strong tendency to 
trap subsequent operators. Therefore, a distinction needs to be 
made between the ability to widen scope over a preceding 
operator and to trap subsequent operators. The following 
examples show the scoping of few and no relative to 
quantifiers (27,28), the negation operator (29) and temporal 
adverbs (30). By comparison, some does not create a strong 
trap for always (31). 
(27) Nobody read every article 
(28) Someone read no articles 
.02 
.02 
(29) Few people weren't surprised .01 
(30) Few people always come late 
(31) Someone always comes late 
.01 
.5 
There appear to be some sentences, typically containing 
two no or few quantifiers, which are used in a sense which 
does not appear to correspond to any straightforward ordering 
of the quantifiers. Instead, the total quantity of predications 
being made seems to be emphasized. An example is given in 
(35). One possible way of representing such sentences might 
be to use branching quantification (Hintikka 1974). 
(35) Few boys kissed few girls 
Sentences containing operators which create negated 
contexts (eg. few, no, not, never) are often disambiguated by 
the presence of any, ever ("at any time") or neither-nor. For 
example, after few or no the adverb sometimes is usually 
replaced by ever (32,33) and the wide-scope reading of never 
in (34) is best obtained by replacing the or with and or by 
using neither-nor and ever (35). 
(32) ?Few people sometimes come late 
(33) Few people ever come late 
.1 
.0 
(34) (Either) John or Bill never comes late 
(35) Neither John nor Bill ever comes late 
.05 
.0 
The singular indefinite a is quite consistently more likely 
than some to take narrow scope. For example, it would be 
more natural to use (36) and (38) than (37) and (39) to indicate 
the narrow scope existential reading. Also, (40) is acceptable 
but (41) is not. (The scoping weights given for (40) and (41) 
have not been adjusted to take into account the effect of the 
modifier different). 
(36) Each person grabbed a chair .3 
(37) Each person grabbed some chair .5 
(38) John didn't find a chair .3 
(39) John didn't find some chair .6 
(40) A different person brought each chair (.7) 
(41) *Some different person brought each chair (.5) 
The scoping of sentences containing the determiner a 
may be complicated by the presence of generic interpretations. 
For example, in (42) the non-specific reading could be 
obtained either by giving never wide scope or by treating a 
guest as a quasi-universal quantifier (derived from the generic 
interpretation via meaning postulates). Assuming that the 
generic reading is present, the standard interpretation in which 
never has wide scope must be treated as being either absent or 
logically redundant. (This is an oversimplified view; some 
attempts are currently being made to give a uniform 
interpretation to indefinites which would avoid this problem of 
redundancy). A somewhat similar problem arises when 
indefinites which may have generic interpretations are present 
inside the antecedent clause of an if-then sentence (as in certain 
donkey sentences). Note that there is no comparable reading 
when a is replaced by some, which does not receive a generic 
interpretation (43). 
(42) An old sailor never gets seasick 
(43) Some old sailor never gets seasick 
.5? 
.01 
Plural indefinites can be placed in an approximate 
hierarchy in terms of their ability to receive collective 
interpretations: some > three > several > many. This correlates 
with their ability to be given "specific" interpretations and 
therefore with their ability to widen scope from strong clausal 
scope traps (44,45) and perhaps also, to a lesser extent, relative 
to the negation operator (46,47). The scoping weights shown 
are associated with the scoping of the existentially quantified 
collections. 
(44) If three people show up then I will come 
(45) If many people show up then I will come 
.3 
.05 
(46) John didn't find three chairs 
(47) John didn't find many chairs 
.4 
.2 
Plural indefinites may have implicit universal partitives 
associated with them (see Hurum & Schubert 1986) and, when 
present, these must be scoped separately. While the existential 
61 
quantifiers associated with indefinites are free to scope to any 
position, in the absence of pragmatic information, there are 
considerable restrictions on the ability of plural indefinites to 
distribute over preceding operators. For example, plural 
indefinites in the object position almost never distribute over 
quantifiers in the subject position unless preceded by an 
explicit partitive. 
HEURISTICS BASED ON STRUCTURAL RELATIONS 
Scoping preferences are strongly influenced by 
"structural relations", that is, the relations between pairs of 
operators in the initial logical translations (or, approximately, 
in the parse tree). Structural relations may be loosely classified 
as "horizontal", an example being the subject-object relation, 
or "vertical", an example being the relation between a noun 
phrase determiner and an operator inside the noun 
complement. Although this distinction is not always clearcut, 
the scoping program makes considerable use of it and separate 
heuristics are used for horizontal and vertical relations. 
As a general rule in English, scope order tends to follow 
surface order, although there axe some exceptions such as in 
the case of postposed adverbials. The effect of surface order is 
strengthened considerably by "shifting", where shifting is used 
here in a general sense to include the preposing of adverbials, 
topicaiization and perhaps the dative shift. For example, it is 
much more likely that (48) refers to a different set of people 
each year than (49) and the distributive reading is more likely 
in (50) than in (51). 
(48) Every year many people visit Europe 
(49) Many people visit Europe every year 
.02 
.5 
(50) Every sailor gave flowers to two girls 
(51) To two girls, every sailor gave flowers 
It should be pointed out that Ioup (1975) has presented 
evidence that in a wide range of languages "grammatical 
function" (eg. subject, direct object .... ) may be a more 
important determiner of scope than surface order. (Ioup 
considers "topic" to be a grammatical category rather than a 
result of shifting.) It happens that in English there is a close 
correlation between surface order and scope order. However, it 
would always be possible, if necessary, to reinterpret some of 
the heuristics shown here in terms of grammatical relations 
rather than in terms of surface order. 
The effect of surface order and shifting also appears to 
hold for temporal adverbs, although the interaction of 
quantifiers with such adverbs can sometimes be quite complex. 
In the case of negated quantifiers (eg. no, few) and not the 
effect of surface order is again quite decisive, with the 
exception of certain postposed adverbs (see below): 
(52) Often, nobody is late for lunch 
(53) Nobody is often late for lunch 
.01 
.0 
The effect of shifting can also be seen with existential 
quantifiers. The following examples show the scoping of the 
existential quantifier associated with many relative to often in 
preposed, medial and postposed positions. The effect of adverb 
placement is clear, although the scoping of postposed adverbs 
will be radically different depending on such factors as the 
pronounciation or the presence or absence of a comma (56,57). 
(54) Often, many people are late for lunch 
(55) Many people are often late for lunch 
(56) Many people are late for lunch often 
(57) Many people axe late for lunch, often 
.02 
.5 
.1 
.98 
The principal ambiguity in these sentences is related to 
whether or not the same group of people is being referred to in 
each situation (we may loosely interpret often as quantifying 
over instances of a type of situation, in this case a lunch 
setting). This ambiguity can be represented by scoping the 
existential quantifier associated with many relative to often. It 
is very unlikely that we would give many wide scope in (54) 
although this would be more likely with indefinites which can 
more easily receive specific interpretations, such as some, three 
and several. 
There is also an optional universal partitive associated 
with plural indefinites such as many and this must also be 
scoped. The interaction of universal quantifiers with temporal 
adverbs involves some quite subtle ambiguities which axe 
related to whether or not all members of some collection axe 
involved in the same situation. However, the effect of surface 
position is still notable: 
(58) Often, everyone is late for lunch 
(59) Everyone is often late for lunch 
.02 
.5 
Different types of embedding construct form quite 
consistent traps for quantifiers and other unscoped operators. 
Operators inside prepositional phrases generally widen scope 
over the head quantifier, those inside full clauses almost never 
do (with the exception of specific indefinites) and those inside 
bare verb phrases have an intermediate tendency to do so (see 
(21)-(23)). Verb phrases serving as noun complements form 
considerably weaker traps than do those serving as 
nominalized arguments. Preposed antecedent clauses of 
connective sentences such as if-then sentences appear to form 
absolute traps for distributive quantifiers, in contrast to 
consequent or postposed antecedent clauses, and for connective 
clauses in general the ordering of the antecedent and 
consequent clauses needs to be considered. 
The effect of structural relations on the scoping of 
quantifiers generally holds for coordinators as well. Some 
examples will illustrate the effect of the surface position of NP 
coordinators relative to negation (60,61) and to quantifiers 
(62,63). The presence of either, by emphasizing the 
disjunction, tends to widen the scope of or somewhat. 
(60) (Either) Sue or Mary didn't dance with John .2 
(61) John didn't dance with Sue or Mary .2 
(62) Few people danced with Sue or Mary .2 
(63) (Either) Sue or Mary danced with few people .2 
62 
Verb coordinators usually scope inside quantifiers in the 
subject and object positions. For quantifiers in the subject 
position this is clearly a structural constraint; in both (64) and 
(65) the subject presumably scopes outside the coordinator and 
it is difficult to reverse this ordering by passivization or by 
replacing the subject with someone different. By contrast, the 
examples show that the seoping of a direct object relative to a 
verb coordinator is largely dependent on pragmaties. 
(64) Someone wrote and mailed a letter 
(65) Someone wrote and received a letter 
However, there is probably some bias, which might be 
considered structural, for scoping an object outside a verb 
coordinator, and this bias is stronger for prepositional objects 
and for some (66). It is always possible for or to take wide 
scope, both relative to subject and object quantifiers, although 
the latter is more likely (67,68). This is the "speaker's 
uncertainty" reading. Although always present, it is particularly 
difficult to get this reading with few or no in the object 
position. 
(66) John drove and flew to some resort .8 
(67) John flew or drove to each resort 
(68) Each person drove or flew to the resort 
.7 
.1 
The interaction of plural quantifiers with verb 
conjunction is more complex and we make a distinction 
between primary and secondary scope dependencies: the 
former involves the scoping of the collection formed from the 
plural quantifier and the latter the details of the predications of 
individual members of the collection. For example, in (69) 
there is presumably only one set of two people, meaning that 
the collection formed from the subject scopes outside the 
coordinator. The details of the individual predications can be 
specified later. In general, some members of the set might be 
involved in both predications and some in just one. This type 
of interaction between sets is similar to that between two plural 
quantifiers. The conjoined subject in (70) could also intitially 
be treated as a collection, or the ambiguity might in this case 
be handled directly by the parser. 
(69) Two different people painted and redecorated the 
apartment 
(70) John and Fred, respectively, fixed and upholstered the 
chair 
The scoping of noun coordinators is somewhat similar to 
that of verb coordinators, although there is evidence that the 
wide scope and reading is elliptical for a NP coordination and 
should therfore be handled by the parser. Therefore, the 
scoping weights for (71)-(74) have been placed in brackets. 
The "scoping" of and relative to a singular indefinite is again 
largely dependent on pragmatics (71,72), although there is 
probably a statistical bias in favour of the wide scope 
(elliptical) and reading. Again, this reading is less likely when 
a is replaced with (singular) some. 
(71) A man and woman came to help 
(72) A friend and colleague came to help 
(.5) 
(.5) 
Plural indefinites also display two levels of interaction 
with coordinated nouns. The initial scope (or syntactic) 
ambiguities of (73) and (74) again are related to whether there 
is one collection or two: the latter (meaning a wide scope and) 
is pragmatically more likely in (73) simply because we 
wouldn't use this wording to refer to a man and a woman. The 
details of applying the predicates to members of the collection 
can again be postponed until later. (A scoping program clearly 
needs to have some specialized knowledge for handling 
interactions between sets of objects and predicates.) 
(73) Two men and women arrived 
(74) Twenty men and women arrived 
(.5) 
(.5) 
Again, or tends to be trapped (75) and the trap is 
especially strong with no and few (76). 
(75) Every freshman or sophomore finished the course .2 
(76) Few freshmen or sophomores finished the course .05 
Coordinators are treated as forming complete scope traps 
except for existential quantifiers and or. This useful rule 
removes the three unwanted readings of (77) in which either or 
both of the universal quantifiers take wide scope. It may 
prevent the anaphoric binding of pronouns, as in (78), but this 
is part of a more general problem for which there is no 
satisfactory theory at the moment (see Lepore & Garson 1983; 
Schubert & Pelletier 1987a,1987b; Hobbs & Shieber 1987). 
(77) Every man or every woman arrived late 
(78) Every man or some friend of his arrived late 
Like universal quantifiers, and tends to be trapped by 
clausal embedding whereas or, though less easily than 
existential quantifiers, can generally widen scope from strong 
scope traps such as "scope islands". For example, there is no 
reading of (79) in which and scopes outside someone, that is in 
which there is a different person for Sue and Mary, but in (80) 
there is a reading, probably the preferred one, in which or must 
scope outside the nested clause, meaning that each person has 
either heard that his aunt or that his uncle is arriving. There is 
also a reading of (80), perhaps not obvious at first, in which or 
has maximally wide scope, meaning that the speaker is not 
sure whether it was his aunt or his uncle that each person heard 
was arriving. 
(79) Someone heard the news that Sue and Mary were arriving 
(80) Each person heard the news that his aunt or his uncle was 
arriving 
IMPLEMENTATION OF HEURISTICS 
Each formula (given an input list of sentential formulas) 
is traversed in a bottom-up left-to-right order with different 
types of expression, such as infix, prefix and coordinated 
expressions, being scoped by separate procedures. As each 
unscoped operator is encountered, its structural category is 
stored on its property list and this information is later used to 
63 
determine the structural relation between a given pair of 
operators. Vertical relations are passed as parameters to 
subordinate procedures; in the case of if-then sentences the 
parameters also contain information about the position and 
type of clause being scoped (eg. "preposed antecedent clause"). 
The scoping weight for a pair of operators can then be 
determined from a table of weights which is indexed according 
to structural relations and operator types. The table has been 
kept as small as possible by the use of default values and 
"standard" operator types (such as some and each for 
existential and universal quantifiers). Although the use of a 
table of weights does not in itself have much psychological 
plausibility, the rules on which the table is based, such as those 
described above, are generally quite simple and it is hoped that 
rules such as these can eventually be incorporated into a more 
comprehensive model of the grammatical biases which underly 
scope preferences. 
PRAGMATICS 
The most obvious place to try to combine pragmatic and 
domain-independent information is at the level of determining 
the pairwise scoping weights. The problem of how these 
weights should then be combined still remains but this 
approach does seem to be worth pursuing. Since properly 
applied pragmatic knowledge will often result in strong, if not 
absolute, preferences for certain scope orderings, the chances 
of selecting the best overall reading of a sentence will be 
improved when pragmatic heuristics are added. The ability of 
pragmatic knowledge to veto certain scope orderings can quite 
easily be implemented by setting the appropriate scoping 
weight below the value of the rain-weight parameter which will 
automatically disallow any readings containing such orderings. 
CONCLUSION 
This paper has described some features of a program 
designed to handle scope ambiguities in English. Some of the 
more problematic issues which were encountered during the 
designing of the program were selected for discussion: the 
choice of logical representation, the seeping of coordinated 
expressions, the choice of a strategy for selecting preferred 
scope orderings and the determination of a set of 
domain-independent heuristics. The program is currently being 
extended to include a wider range of lexical types and input 
expressions and the heuristics are being improved. Following 
this, it is hoped to incorporate some simple types of domain- 
and discourse-dependent knowledge into the program, in 
particular knowledge about expected relations among objects 
in a given domain and a simple discourse focus structure. 
The selection of preferred scope orderings depends on 
the complex interaction of linguistic and context-dependent 
knowldege. It would be a considerable advantage to be able to 
factor out the contributions of different types of knowledge 
required and then at some later time to combine them. One 
conclusion of this work is that there is a body of largely 
domain-independent knowledge which can play an important, 
and at times decisive, role in the disambiguation of scope. 
Such knowledge is most useful when it indicates a very strong 
or absolute preference for one reading. 
Absolute preferences typically occur with operators such 
as any or both and with distributive quantifiers or and inside 
strong clausal trap or inside a coordinator. Very strong 
preferences may occur with operators such as few, no or each, 
with preposed or topicalized operators and with operators 
inside prepositional phrases. When the domain-independent 
heuristics do not provide a strong preference for one reading, 
they may still serve as a useful guide guide for the later 
application of pragmatic knowledge. This is commonly the 
case when indefinites are present, as the "specificity" of 
indefinites is mainly context-dependent. 
A number of problems have not been discussed here 
because they remain unresolved. These include: the scoping of 
quantifiers relative to tense and opaque operators, the logical 
representation and scoping of generics, the treatment of 
pronouns not embedded within their quantifier antecedents, 
non-local problems such as the preference for "symmetric" 
readings, the use of stray words, such as together and both (as 
an adverb), which provide important clues for preferred scope 
relations and the difficult problem of combining linguistic and 
context-dependent heuristic knowledge. 
ACKNOWLEDGEMENTS 
I would like to thank Dr. Len Schubert for originally 
suggesting this project and for his advice and many helpful 
comments throughout the course of this work. I would also like 
to thank the members of my thesis committee and the members 
of the Logical Grammar Study Group at the University of 
Alberta for their comments on parts of this work. This work 
was supported in part by NSERC Operating Grant A8818. 

APPENDIX: OL~I'PL~I " FROM SCOPING PROGRAM 
Sentence 1 
(i (q alll man2) 
(f PRES (p want3 (TAU2 (f INrF (p marry4 (c or6 Peggy5 Sue"/))))))) 
l. The average weight is 0.7 based on t comparison 
(q alll y5 
(i y5 man2) 
(i y5 
(f PRES (p want3 
(TAU2 
(I yl0 
(i (i yI0 (f EN'F (p marry4 Peggy5))) 
or6 
(i yl0 (f L\'F (p marry4 Sue7)))))))))) 
2. The average weight is 0.5 based on 2 comparisons 
(q allt 
y5 (i y5 man2) 
(i (i y5 (f PRES (p want3 (TAU2 (f 1ANF (p marry4 Peggy5)))))) or6 
(i y5 (f PRES (p want3 (TAU2 (f EN'F (p marry4 Sue7)))))))) 
3. The average weight is 0.3 based on 2 comparisons 
(i (q alll 
y5 (i y5 man2) 
(i y5 (f PRES (p want3 (TAU2 (f IN'F (p marry4 Peggy5))))))) or6 
(q alll 
y5 (i y5 man2) 
(i y5 (f PRES (p want3 (TAU2 (f L-NF (p marry4 Sue*'/)))))))) 
time used = 308 msecs. 
Sentence 2 
(i M'ary l 
(c or4 
(f PAST (p read3 (q some6 story7) (q each8 child9))) 
(f PAST (p tell5 (q some6 story7) (q each8 child9))))) 
1. The average weight is 0.82 based on 2 comparisons 
(i (q each8 
y17 
(i y17 child9) 
(q some6 yl5 (i yl5 story7) (i Maryl (f PAST (p read3 y15 yl7))))) 
or4 (q each8 
yl7 
(i y17 child9) 
(q some6 yl5 (i y15 story7) (i Maryl (f PAST (p tell5 y15 y17)))))) 
2. The average weight is 0.67 based on 2 comparisons 
(i (q some6 
yl5 
(i y15 story7) 
(q each8 y17 (i y17 child9) (i Maryl (f I'AST (p read3 yl5 y17))))) or4 
(q some6 
y15 
(i yl5 story7) 
(q each8 yl7 (i y17 child9) (i Maryl (f PAST (p tell5 y15 y17)))))) 
time used = 386 msecs. 
65 

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