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<Paper uid="P92-1031">
  <Title>CONNECTION RELATIONS AND QUANTIFIER SCOPE</Title>
  <Section position="4" start_page="242" end_page="244" type="metho">
    <SectionTitle>
1. SCOPE-AMBIGUOUS
REPRESENTATION AND SCOPE
DETERMINATION
</SectionTitle>
    <Paragraph position="0"> 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.</Paragraph>
    <Paragraph position="1"> 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.</Paragraph>
    <Paragraph position="2"> 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.</Paragraph>
    <Paragraph position="3"> 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.</Paragraph>
    <Paragraph position="4"> 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.</Paragraph>
    <Paragraph position="5"> I now present the formalism.</Paragraph>
    <Paragraph position="6"> 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.</Paragraph>
    <Paragraph position="7"> 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)}.</Paragraph>
    <Paragraph position="8"> 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.</Paragraph>
    <Paragraph position="9"> that there is a relation of seeing between boys and movies.</Paragraph>
    <Paragraph position="10"> In the formal notation of second-order logic we can describe this piece of information as follows:</Paragraph>
    <Paragraph position="12"> 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:  (1.1.b) 3saw (saw ~ Boy x Movie) In this formula, we view saw as a sorted variable of the sort of the binary seeingrelations. The variable saw corresponds to the variable X2 in (1.1.a).</Paragraph>
    <Paragraph position="13"> (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):</Paragraph>
    <Paragraph position="15"> 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).</Paragraph>
    <Paragraph position="16"> Classic procedures will in this case immediately generate two readings with definite scope relations, whose notations in predicate logic are given below.</Paragraph>
    <Paragraph position="18"> 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):</Paragraph>
    <Paragraph position="20"> So, we have (1.3.b) =&gt; (1.3.a).</Paragraph>
    <Paragraph position="21"> 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).</Paragraph>
    <Paragraph position="22"> (1.4) It was &amp;quot;Gone with the Wind&amp;quot;.</Paragraph>
    <Paragraph position="23"> 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)).</Paragraph>
    <Paragraph position="24">  Let us now assume that sentence (1.5) follows (1.1).</Paragraph>
    <Paragraph position="25"> (1.5) All of them were made by Walt Disney Studios.</Paragraph>
    <Paragraph position="26"> 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.</Paragraph>
    <Paragraph position="27"> 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.</Paragraph>
    <Paragraph position="28"> 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.</Paragraph>
    <Paragraph position="29"> Here, just as in the preceding cases, the representation (1.1.c) has to be seen as the &amp;quot;starting representation&amp;quot; of (1.1). The information that them is a plural noun is represented by the condition I Range saw I &gt; 1, which in turn leads to the following representation: (1.6) 3saw (saw ~ BOy x Movie &amp; Dom saw=Boy &amp; I Range saw I &gt;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 &gt; 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).</Paragraph>
    <Paragraph position="30"> 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.</Paragraph>
    <Paragraph position="31"> 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.</Paragraph>
    <Paragraph position="32"> 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.</Paragraph>
  </Section>
  <Section position="5" start_page="244" end_page="246" type="metho">
    <SectionTitle>
REPRESENTATION FOR
SENTENCES WITH NUMERIC
QUANTIFIERS
</SectionTitle>
    <Paragraph position="0"> So far, I have not stated exactly how the representation of sentence (1.1) was generated.</Paragraph>
    <Paragraph position="1"> In order to do so, let us take an example sentence with numeric quantifiers: (2.1) Two examiners marked six scripts.</Paragraph>
    <Paragraph position="2"> 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  (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 &amp;quot;(Vx~ Xn)(3Y)&amp;quot; by &amp;quot;(3Y)(Vx~ Xn)&amp;quot; 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.</Paragraph>
    <Paragraph position="3"> 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.</Paragraph>
    <Paragraph position="4"> 30 The incomplete group interpretation: Two examiners as a group marked a group of six scripts between them.</Paragraph>
    <Paragraph position="5"> 40 The complete group interpretation: Two examiners each marked the same set of six scripts.</Paragraph>
    <Paragraph position="6"> Kempson and Cormack represent these readings with the help of quantifiers over sets in the following way:  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.</Paragraph>
    <Paragraph position="7"> 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 &amp;quot;(3x~ Xn)&amp;quot; by &amp;quot;(Vx~ Xn)&amp;quot; &amp;quot;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.&amp;quot; (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.</Paragraph>
    <Paragraph position="8"> 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- null 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 &amp; (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.</Paragraph>
    <Paragraph position="9"> 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.</Paragraph>
    <Paragraph position="10"> Therefore, we have assigned an underdetermined 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.</Paragraph>
    <Paragraph position="11"> 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.</Paragraph>
  </Section>
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