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<Paper uid="P95-1028">
  <Title>Quantifier Scope and Constituency</Title>
  <Section position="4" start_page="0" end_page="205" type="metho">
    <SectionTitle>
2 Traditional Approaches
</SectionTitle>
    <Paragraph position="0"> All three paradigms of grammar formalisms introduced earlier share similar linguistic judgments for their grammaticality analyses. This section examines quantifying-in to show (a) that quantifying-in is a powerful device that allows referential NP-interpretations and (b) that quantifying-in is not sufficiently restricted to account for the available readings for quantificational NP-interpretations.</Paragraph>
    <Paragraph position="1"> Quantifying-in is a technique originally introduced to produce appropriate semantic forms for de re interpretations of NPs inside opaque operators  (Montague, 1974). For example, (a) below has two readings, de re and de dicto, depending on the relativity of the existence of such an individual. They are roughly interpretable as (b) and (@2  (1) (a) John believes that a Republican will win.</Paragraph>
    <Paragraph position="2"> (b) 3r.repub(r) A bel(john, uill(uin(r))) (C) bel(john, 3r.repub(r) A uill(uin(r))) (b) has a binder 3 that is quaati.fving a variable r  inside an opaque operator bel, hence the name for the technique. (c) does not have such an intervening operator. Although it is beyond the scope of the present paper to discuss further details of intensionality, it is clear that de re interpretations of NPs are strongly related to referential NP-semantics, in the sense that the de re reading of (a) is about a referred individual and not about an arbitrary such individual. Quantifying-in is designed to make any (possibly embedded) NP take the matrix scope, by leaving a scoped variable in the argument position of the original NP. This would be acceptable for referential NP-semantics.</Paragraph>
    <Paragraph position="3"> Montague also proposed to capture purely extensional scope ambiguities using quantifying-in. For example, wide scope reading of a woman in (a) below is accounted for by quantifying-in (with a meaning postulate), patterned after one for (b).</Paragraph>
    <Paragraph position="4">  (2) (a) Every man loves a woman.</Paragraph>
    <Paragraph position="5"> (b) Every man seeks a white unicorn.</Paragraph>
    <Paragraph position="6">  His suggestion is adopted with various subsequent revisions cited earlier. Since any NP, referential or quantificational, requires quantifying-in to outscope another, quantifying-in consequently confounds referential and quantificational NP-semantics. This causes a problem when there is a distributional difference between referential NPs and non-referential NPs, as Fodor &amp; Sag (1982) have argued, a view which has been followed by the approaches to dynamic interpretation of indefinite NPs cited earlier. It seems hard to reconcile quantifying-in with these observations.</Paragraph>
  </Section>
  <Section position="5" start_page="205" end_page="206" type="metho">
    <SectionTitle>
3 Availability of Readings
</SectionTitle>
    <Paragraph position="0"> This section proposes a way of sharpening our intuition on available readings and re-examines traditional linguistic judgments on grammatical readings.</Paragraph>
    <Paragraph position="1"> While there are undoubted differences in degree of availability among readings dependent upon semantics or discourse preference (Bunt, 1985; Moran, 1988), we will focus on all-or-none structural possibilities afforded by competence grammar. 3 2In this simplistic notation, we gloss over tense analysis, among others.</Paragraph>
    <Paragraph position="2"> 3Moran's preference-based algorithm treats certain readings as &amp;quot;highly unpreferred,&amp;quot; effectively making them structurally unavailable, from those possible sco-Consider the following unambiguous quantification structure in a generalized quantifier format (hereafter oq, Barwise &amp; Cooper, 1981), where quantifier outscopes any quantifiers that may occur in either restriction or body.</Paragraph>
    <Paragraph position="3">  (3) quantifier(variable, restriction, body)  Logical forms as notated this way make explicit the functional dependency between the denotations of two ordered quantificational NPs. For example~ con- null sider (4) (a) (Partee, 1975). (b) shows one way of representing it in a GQ format.</Paragraph>
    <Paragraph position="4"> (4) (a) Three Frenchmen visited five Russians.</Paragraph>
    <Paragraph position="5"> (b) three(f, frenchmen(f), five(r, russians (r), visited(f, r) ) ) We can always argue, by enriching the notation, that (4) (b) represents at least four different readings, de null pending on the particular sense of each involved NP, i.e., group- vs individual-denoting. In every such reading, however, the truth of (4) (b) depends upon finding appropriate individuals (or the group) for f such that each of those individuals (or the group itself) gets associated with appropriate individuals (or a group of individuals) for r via the relation visil;ed. 4 Notice that there is always a functional dependency of individuals denoted by r upon individuals denoted by f. We claim that this explicit functional dependency can be utilized to test availability of readings. 5 First, consider the following sentences without coordination.</Paragraph>
    <Paragraph position="6">  (5) (a) Two representatives of three companies saw most samples.</Paragraph>
    <Paragraph position="7"> (b) Every dealer shows most customers at most three cars.</Paragraph>
    <Paragraph position="8"> (c) Most boys think that every man danced  with two women.</Paragraph>
    <Paragraph position="9"> (a) has three quantifiers, and there are 6 different ways of ordering them. Hobbs &amp; Shieber (1987) show that among these, the reading in which two representatives outscopes most samples which in turn outscopes three companies is not available from the sentence. They attribute the reason to the logical structure of English as in (3), as it is considered unable to afford an unbound variable, a constraint known as the unbound variable constraint (uvc). 6 We should note, however, that there is one reading pings generated by a scheme similar to Hobbs &amp; Shieber (1887). We clash that competence grammax makes even fewer readings available in the first place.</Paragraph>
    <Paragraph position="10"> 4Without losing generality, therefore, we will consider only individual-denoting NPs in this paper.</Paragraph>
    <Paragraph position="11"> SSingular NPs such as a company are not helpful to this task since their denotations do not involve multiple individuals which explicitly induce this functional dependency.</Paragraph>
    <Paragraph position="12"> eThe reading would be represented as follows, which has the first occurrence of the variable c left unbound.  among the remaining five that the uvc allows which in fact does not appear to be available. This is the one in which three companies outscopes most samples which in turn outscopes two representatives (cf. Horn (1972), Fodor (1982)). 7 This suggests that the uvc may not be the only principle under which Hobbs &amp; Shieber's reading is excluded, s The other four readings of (a) are self-evidently available. If we generalize over available readings, they are only those that have no quantifiers which intercalate over NP boundaries. 9 (5) (b) has three quantifiers too, but unlike (5) (a), all the six ways of ordering the quantifiers are available. (5) (c) has only four available readings, where most boys does not intercalate every man and two women. 1deg Consider now sentences including coordination.</Paragraph>
    <Paragraph position="13">  (6) (a) Every girl admired, but most boys detested, one of the saxophonists.</Paragraph>
    <Paragraph position="14"> (b) Most boys think that every man danced  with, but doubt that a few boys talked to, more than two women.</Paragraph>
    <Paragraph position="15"> As Geach (1970) pointed out, (a) has only two grammatical readings, though it has three quantifiers. In reading 1, the same saxophonist was admired and detested at the same time. In reading 2, every girl admired an arbitrary saxophonist and most boys also detested an arbitrary saxophonist. In particular, missing readings include the one in which every girl admired the same saxophonist and most boys detested the same but another saxophonist. (6) (b) rio(r, rep(r) It of(r,c), most(a, samp(s), three(c, comp(c), sag(r,s)))) 7To paraphrase this impossible reading, it is true of a situation under which there were three companies such that there were four samples for each such company such that each such sample was seen by two representatives of that company. Crucially, samples seen by representatives of different companies were not necessarily the same. SThis should not be taken as denying the reality of the uvc itself. For example, as one of the referees pointed out, the uvc is required to explain why, in (a) below, every professor must outscope a friend so as to bind the pronoun his.</Paragraph>
    <Paragraph position="16"> (a) Most students talked to a friend of every professor about his work.</Paragraph>
    <Paragraph position="17">  that (a) and (b) below have two readings each.</Paragraph>
    <Paragraph position="18"> (a) 3ohn thinks that every man danced with two women.</Paragraph>
    <Paragraph position="19"> (b) Most boys think that Bill danced with two  women.</Paragraph>
    <Paragraph position="20"> also has only two grammatical readings. In one, most boys outscopes every man and a few boys which together outscope more than two women. In the other, more than two women outscopes every man and a few boys, which together outscope most boys.</Paragraph>
  </Section>
  <Section position="6" start_page="206" end_page="207" type="metho">
    <SectionTitle>
4 An Account of Availability
</SectionTitle>
    <Paragraph position="0"> This section proposes a generalization at the level of semantics for the phenomena described earlier and considers its apparent counterexamples.</Paragraph>
    <Paragraph position="1"> Consider a language PS for natural language semantics that explicitly represents function-argument relationships (Jackendoff, 1972). Suppose that in PS: the semantic form of a quantified NP is a syntactic argument of the semantic form of a verb or a preposition. (7) through (10) below show well-formed  For instance, of has two arguments three(comp) and two(rep), and show has three arguments.</Paragraph>
    <Paragraph position="2"> /: gives rise to a natural generalization of available readings as summarized below. 12 (11) For a function with n arguments, there are n! ways of successively providing all the arguments to the function.</Paragraph>
    <Paragraph position="3"> This generalization captures the earlier observations about availability of readings. (7), for (4) (a), has two (2!) readings, as viaited has two arguments.</Paragraph>
    <Paragraph position="4"> (8) is an abstraction for four (2!x2!) readings, as both of and maw have two arguments each. (9) is an abstraction for six (3!) readings, as show has three arguments. Likewise, (10) is an abstraction for four readings.</Paragraph>
    <Paragraph position="5"> Coordination gives an interesting constraint on  availability of readings. Geach's observation that (6) (a) has two readings suggests that the scope of the object must be determined before it reduces with the coordinate fragment. Suppose that the non-standard constituent for one of the conjuncts in (6) (a) has a semantic representation shown below.</Paragraph>
    <Paragraph position="6"> (12) ~z adnired(z,svery(girl))  Geach's observation implies that (12) is ambiguous, so that every(girl) can still take wide (or narrow) scope with respect to the unknown argument. A 11The up-operator ^ in (10) takes a term of type t to a term of type e, but a further description of PS is not relevant to the present discussion.</Paragraph>
    <Paragraph position="7"> 12Nan (1991)'s work is based on a related observation, though he does not make use of the distinction between referential and quantificational NP-semantics.</Paragraph>
    <Paragraph position="8">  theory of CCG will be described in the next section to show how to derive scoped logical forms for available readings only.</Paragraph>
    <Paragraph position="9"> But first we must consider some apparent counterexamples to the generalization,  (13) (a) Three hunters shot at five tigers.</Paragraph>
    <Paragraph position="10"> (b) Every representative of a company saw most samples.</Paragraph>
    <Paragraph position="11">  The obvious reading for (a) is called conjunctive or cumulative (Partee, 1975; Webber 1979). In this reading, there are three hunters and five tigers such that shooting events happened between the two parties. Here, arguments are not presented in succession to their function, contrary to the present generalization. Notice, however, that the reading must have two (or more) referential NPs (Higginbotham, 1987). 13 The question is whether our theory should predict this possibility as well. For a precise notion of availability, we claim that we must appeal to the distinction between referential and quantificational NP-semantics, since almost any referential NP can have the appearance of taking the matrix scope, without affecting the rest of scope phenomena. A related example is (b), where in one reading a referen- null tial NP a company arguably outscopes most samples which in turn outscopes every representative (Hobbs &amp; Shieber, 1987). As we have pointed out earlier, the reading does not generalize to quantified NPs in general.</Paragraph>
    <Paragraph position="12"> (14) (a) Some student will investigate two dialects of every language.</Paragraph>
    <Paragraph position="13"> (b) Some student will investigate two dialects of, and collect all interesting examples of coordination in, every language.</Paragraph>
    <Paragraph position="14"> (c) * Two representative of at least three companies touched, but of few universities saw, most samples.</Paragraph>
    <Paragraph position="15"> (a) has a reading in which every language outscopes  some student which in turn outscopes two dialects (May, 1985). In a sense, this has intercalating NP quantifiers, an apparent problem to our generalization. However, the grammaticality of (b) opens up the possibility that the two conjuncts can be represented grammatically as functions of arity two, similar to normal transitive verbs. Notice that the generalization is not at work for the fragment of at least three companies touched in (c), since the conjunct is syntactically ungrammatical. At the end of next section, we show how these finer distinctions are made under the CCG framework (See discussion of Figure 5).</Paragraph>
    <Paragraph position="16"> IZFor example, (a) below lacks such a reading.</Paragraph>
    <Paragraph position="17"> (a) Several men danced with few women.</Paragraph>
  </Section>
  <Section position="7" start_page="207" end_page="208" type="metho">
    <SectionTitle>
5 A CCG Implementation
</SectionTitle>
    <Paragraph position="0"> This section describes a CCG approach to deriving scoped logical forms so that they range over only grammatical readings.</Paragraph>
    <Paragraph position="1"> We will not discuss details of how CCG characterizes natural language syntactically, and refer the interested reader to Steedman (1993). CCGs make use of a limited set of combinators, type raising (T), function composition (B), and function substitution (S), with directionality of combination for syntactic grammaticality. For the examples in this paper, we only need type raising and function composition, along with function application. The following shows rules of derivation that we use. Each rule is associated with a label, such as &gt; or &lt;B etc, shown  at the end.</Paragraph>
    <Paragraph position="2"> (15) (a) x/v ~ =&gt; x (&gt;) (b) Y x\~ =&gt; x (&lt;) (c) x/v Y/Z =&gt; x/z (&gt;a) (d) Y\z x\Y ffi&gt; x\z (&lt;e) (e) np =&gt; T/(T\np) (&gt;T) (f) np =&gt; T\(T/np) (&lt;T)  The mapping from syntax to semantics is usually defined in two different ways. One is to use elementary categories, such as np or s, in encoding both syntactic types and logical forms (Jowsey, 1990; Steedman, 1990; Park, 1992). The other is to associate the entire lexical category with a higher-order expression (Kulick, 1995). In this paper, we take the former alternative to describe a first-order rendering of CCG.</Paragraph>
    <Paragraph position="3"> Some lexical entries for every are shown below.  (16) (s :q-every (X, N, S)/(s : S\np:I) )/n:X'N (17) (s : S/(a : Sknp: s-every(1) ) )/n:W  The information (s/(s\np))/n encodes the syntactic fact that every is a constituent which, when a constituent of category n is provided on its right, returns a constituent of category s/(s\np). q-every(X,li,S) is a term for scoped logical forms. We are using different lexical items, for instance q-every and e-every for every, in order to signify their semantic differences. 14 These lexical entries are just two instances of a general schema for type-raised categories of quantifiers shown below, where T is an arbitrary category.</Paragraph>
    <Paragraph position="4"> (18) (T/(T\np))/na~d (T\(T/np))/n And the semantic part of (16) and (17) is first-order encoding of (19) (a) and (b), respectively. 15 14q-every represents every as a quantifier, and s-every, as a set denoting property. We will use s-every(l^man(X)) and its ~-reduced equivalent s-every(man) interchangeably.</Paragraph>
    <Paragraph position="5"> 1as-quantifier(noun) denotes an arbitrary set N of individuals d such that d has the property noun and that the cardinality of N is determined by quantifier (and  (19) (a) ~n.AP.Vz E s-every(n).P(=) (b) (a) encodes wide scope type raising and (b), narrow.  With standard entries for verbs as in (20), logical forms such as (21) and (22) are po ible.</Paragraph>
    <Paragraph position="6">  (20) saw :- (s:sav(I,Y)\np:X)/np:Y= (21) q-two (X, rep (X), aaw(X, s-f ottr (samp)) ) (22) q-two(X,rep(X) ,q-four(Y,samp(Y),aaw(\]\[,Y=)))  Figure 1 shows different ways of deriving scoped logical forms. In (a), n:I'! unifies with n:X'girl(X), so that Ii gets the value girl(X). This value of !1 is transferred to the expression s:evory(X,li,S) by partial execution (Pereira Shieber, 1987; Steedman, 1990; Park, 1992). (a) shows a derivation for a reading in which object NP takes wide scope and (b) shows a derivation for a reading in which subject NP takes wide scope. There are also other derivations.</Paragraph>
    <Paragraph position="7"> Figure 2 shows logical forms that can be derived in the present framework from Geach's sentence. Notice that the conjunction forces subject NP to be first composed with the verb, so that subject NP must be type-raised and be combined with the semantics of the transitive verb. As noted earlier, the two categories for the object still make both scope possibilities available, as desired. The following category is used for but.</Paragraph>
    <Paragraph position="8"> (23) ((s : and(P ,1~)/np:\]\[)\ (s:P/np:\]\[))/(s :Q/np :\]\[) Readings that involve intercalating quantifiers, such as the one where every girl outscopes one sazophonist, which in turn outscopes most bogs, are correctly excluded.</Paragraph>
    <Paragraph position="9"> Figure 3 shows two different derivations of logical forms for the complex NP two representatives of three companies. (a) shows a derivation for a reading in which the modifying NP takes wide scope and (b) shows the other case. In combination with derivations involving transitive verbs with subject and object NPs, such as ones in Figure 1, this correctly accounts for four grammatical readings for (5) (a). 16 Figure 4 shows a derivation for a reading, among six, in which most customers outscopes every dealer which in turn outscopes three cars. Some of these readings become unavailable when the sentence contains coordinate structure, such as one below. (24) Every dealer shows most customers (at most) three cars but most mechanics every car.</Paragraph>
    <Paragraph position="10"> noun). We conjecture that this can also be made to capture several related NP-semantics, such as collective NP-semantics and/or referential NP-semantics, though we can not discuss further details here.</Paragraph>
    <Paragraph position="11"> lSAs we can see in Figure 3 (a) (b), there m no way quantifiers inside $ can be placed between the two quantifiers two &amp; three, correctly excluding the other two readings.</Paragraph>
    <Paragraph position="12"> In particular, (24) does not have those two readings in which every dealer intercalates most customers and three cars. This is exactly predicted by the present CCG framework, extending Geach's observation regarding (6) (a), since the coordination forces the two NPs, most customers and three cars, to be composed first (Dowty, 1988; Steedman 1990; Park 1992). (25) through (27) show one such derivation, which results in readings where three cars outscopes most customers but every dealer must take either wide or narrow scope with respect to both most customers and three cars.</Paragraph>
    <Paragraph position="13">  Figure 5 shows the relevant derivation for the fragment investigate two dialects of discussed at end of previous section. It is a conjoinable constituent, but since there is no way of using type-raised category for two for a successful derivation, two dialects can not outscope any other NPs, such as subject NP or the modifying NP (Steedman, 1992). This correctly accounts for our intuition that (14) (a) has an apparently intercalating reading and that (14) (b) has only two readings. However, there is no similar derivation for the fragment of three companies touched, as shown below.</Paragraph>
    <Paragraph position="14"> (28) of three companies touched</Paragraph>
    <Paragraph position="16"/>
  </Section>
  <Section position="8" start_page="208" end_page="210" type="metho">
    <SectionTitle>
6 Concluding Remarks
</SectionTitle>
    <Paragraph position="0"> We have shown that the range of grammatical readings allowed by sentences with multiple quantified NPs can be characterized by abstraction at function-argument structure constrained by syntactic adjacency. This result is in principle available to other paradigms that invoke operations like QR at LF or type-lifting, which are essentially equivalent to abstraction. The advantage of CCG's very free notion  every dealer shows host custoners  of surface structure is that it ties abstraction or the equivalent as closely as possible to derivation. Apparent counterexamples to the generalization can be explained by the well-known distinction between referential and quantificational NP-semantics. An implementation of the theory for an English fragment has been written in Prolog, simulating the 2nd order properties.</Paragraph>
    <Paragraph position="1"> There is a question of how the non-standard surface structures of CCG are compatible with well-known conditions on binding and control (including crossover). These conditions are typically stated on standard syntactic dominance relations, but these relations are no longer uniquely derivable once CCG allows non-standard surface structures. We can show, however, that by making use of the obliqueness hierarchy (of. Jackendoff (1972) and much subsequent work) at the level of LF, rather than surface structure, it is possible to state such conditions (Steedman, 1993).</Paragraph>
  </Section>
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