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<Paper uid="P99-1039">
  <Title>Alternating Quantifier Scope in CCG*</Title>
  <Section position="3" start_page="0" end_page="303" type="intro">
    <SectionTitle>
1 Introduction
</SectionTitle>
    <Paragraph position="0"> It is standard to assume that the ambiguity of sentences like (1) is to be accounted for by assigning two logical forms which differ in the scopes assigned to these quantifiers, as in (2a,b): 1  (1) Every boy admires some saxophonist.</Paragraph>
    <Paragraph position="1"> (2) a. Vx.boy' x -+ 3y.saxophonis/ y A admires' yx  b. 3y.saxophonis/ y A Vx.bo/x -+ admires'yx The question then arises of how a grammar/parser can assign all and only the correct interpretations to sentences with multiple quantifiers.</Paragraph>
    <Paragraph position="2"> This process has on occasion been explained in terms of &amp;quot;quantifier movement&amp;quot; or essentially * Early versions of this paper were presented to audiences at Brown U., NYU, and Karlov2~ U. Prague. Thanks to Jason Baldridge, Gann Bierner, Tim Fernando, Kit Fine, Polly Jacobson, Mark Johnson, Aravind Joshi, Richard Kayne, Shalom Lappin, Alex Lascarides, Suresh Manandhar, Jaruslav Peregrin, Jong Park, Anna Szabolcsi, Bonnie Webber, Alistair Willis, and the referees for helpful comments. The work was supported in part by ESRC grant M423284002.</Paragraph>
    <Paragraph position="3"> tThe notation uses juxtaposition fa to indicate application of a functor f to an argument a. Constants are distinguished from variables by a prime, and semantic functors like admires' are assumed to be &amp;quot;Curried&amp;quot;. A convention of &amp;quot;left associativity&amp;quot; is assumed, so that admires'yx is equivalent to (admires'y)x. equivalent computational operations of &amp;quot;quantifying in&amp;quot; or &amp;quot;storage&amp;quot; at the level of logical form. However, such accounts present a problem for monostratal and monotonic theories of grammar like CCG that try to do away with movement or the equivalent in syntax. Having eliminated non-monotonic operations from the syntax, to have to restore them at the level of logical form would be dismaying, given the strong assumptions of transparency between syntax and semantics from which the monotonic theories begin. Given the assumptions of syntactic/semantic transparency and monotonicity that are usual in the Frege-Montague tradition, it is tempting to try to use nothing but the derivational combinatorics of surface grammar to deliver all the readings for ambiguous sentences like (1). Two ways to restore monotonicity have been proposed, namely: enriching the notion of derivation via type-changing operations; or enriching the lexicon and the semantic ontology.</Paragraph>
    <Paragraph position="4"> It is standard in the Frege-Montague tradition to begin by translating expressions like &amp;quot;every boy&amp;quot; and &amp;quot;some saxophonist&amp;quot; into &amp;quot;generalized quantitiers&amp;quot; in effect exchanging the roles of arguments like NPs and functors like verbs by a process of &amp;quot;type-raising&amp;quot; the former. In terms of the notation and assumptions of Combinatory Categorial Grammar (CCG, Steedman 1996) the standard way to incorporate generalized quantifiers into the semantics of CG deterbainers is to transfer type-raising to the lexicon, assig~g the following categories to determiners like every and some, making them functions from nouns to &amp;quot;type-raised&amp;quot; noun-phrases, where the latter are simply the syntactic types corresponding to a generalized quantifier:  (3) every := (T/(T\NP))/N : ~,p,~l.Vx.px -+ qx every := (T\(T/NP))/N : kp.kq.Vx.px --+ qx (4) some := (T/(T\UP))/U:~,p.~l.3x.pxAqx some := (T\(T/NP))/N:Lp.~l.3x.pxAqx  (T is a variable over categories unique to each individual occurrence of the raised categories (3) and (4), abbreviating a finite number of different raised types. We will distinguish such distinct variables as T, T', as necessary.) Because CCG adds rules of function composition to the rules of functional application that are standard in pure Categorial Grammar, the further inclusion of type-raised arguments engenders derivations in which objects command subjects, as well as more traditional ones in which the reverse is true.</Paragraph>
    <Paragraph position="5"> Given the categories in (3) and (4), these alternative derivations will deliver the two distinct logical forms shown in (2), entirely monotonically and without involving structure-changing operations.</Paragraph>
    <Paragraph position="6"> However, linking derivation and scope as simply and directly as this makes the obviously false prediction that in sentences where there is no ambiguity of CCG derivation there should be no scope ambiguity. In particular, object topicalization and object right node raising are derivationally unambiguous in the relevant respects, and force the displaced object to command the rest of the sentence in derivational terms. So they should only have the wide scope reading of the object quantifier. This is not the case: (5) a. Some saxophonist, every boy admires.</Paragraph>
    <Paragraph position="7"> b. Every boy admires, and every girl detests, some saxophonist.</Paragraph>
    <Paragraph position="8"> Both sentences have a narrow scope reading in which every individual has some attitude towards some saxophonist, but not necessarily the same saxophonist. This observation appears to imply that even the relatively free notion of derivation provided by CCG is still too restricted to explain all ambiguities arising from multiple quantifiers.</Paragraph>
    <Paragraph position="9"> Nevertheless, the idea that semantic quantifier scope is limited by syntactic derivational scope has some very attractive features. For example, it immediately explains why scope alternation is both unbounded and sensitive to island constraints. There is a further property of sentence (5b) which was first observed by Geach (1972), and which makes it seem as though scope phenomena are strongly restricted by surface grammar. While the sentence has one reading where all of the boys and girls have strong feelings toward the same saxophonist--say, John Coltrane--and another reading where their feelings are all directed at possibly different saxophonists, it does not have a reading where the saxophonist has wide scope with respect to every boy, but narrow scope with respect to every girl that is, where the boys all admire John Coltrane, but the girls all detest possibly different saxophonists.</Paragraph>
    <Paragraph position="10"> There does not even seem to be a reading involving separate wide-scope saxophonists respectively taking scope over boys and girls--for example where the boys all admire Coltrane and the girls all detest Lester Young.</Paragraph>
    <Paragraph position="11"> These observations are very hard to reconcile with semantic theories that invoke powerful mechanisms like abstraction or &amp;quot;Quantifying In&amp;quot; and its relatives, or &amp;quot;Quantifier Movement.&amp;quot; For example, if quantifiers are mapped from syntactic levels to canonical subject, object etc. position at predicate-argument structure in both conjuncts in (5b), and then migrate up the logical form to take either wide or narrow scope, then it is not clear why some saxophonist should have to take the same scope in both conjuncts. The same applies if quantifiers are generated in situ, then lowered to their surface position. 2 Related observations led Partee and Rooth (1983), and others to propose considerably more general use of type-changing operations than are required in CCG, engendering considerably more flexibility in derivation that seems to be required by the purely syntactic phenomena that have motivated CCG up till now. 3 While the tactic of including such order-preserving type-changing operations in the grammar remains a valid alternative for a monotonic treatment of scope alternation in CCG and related forms of categorial grammar, there is no doubt that it complicates the theory considerably. The type-changing operations necessarily engender infinite sets of categories for each word, requiring heuristics based on (partial) orderings on the operations concerned, and raising questions about completeness and practical parsability. All of these questions have been addressed by Hendriks and others, but the result has been to dramatically raise the ratio of mathematical proofs to sentences analyzed.</Paragraph>
    <Paragraph position="12"> It seems worth exploring an alternative response to these observations concerning interactions of sur2Such observations have been countered by the invocation of a &amp;quot;parallelism condition&amp;quot; on coordinate sentences, a rule of a very expressively powerful &amp;quot;transderivational&amp;quot; kind that one would otherwise wish to avoid.</Paragraph>
    <Paragraph position="13">  reading for sentence (5b), Hendriks (1993), subjects the category of the transitive verb to &amp;quot;argument lifting&amp;quot; to make it a function over a type-raised object type, and the coordination rule must be correspondingly semantically generalized.</Paragraph>
    <Paragraph position="14">  face structure and scope-taking. The present paper follows Fodor (1982), Fodor and Sag (1982), and Park (1995, 1996) in explaining scope ambiguities in terms of a distinction between true generalized quantifiers and other purely referential categories.</Paragraph>
    <Paragraph position="15"> For example, in order to capture the narrow-scope object reading for Geach's right node raised sentence (5b), in whose CCG derivation the object must command everything else, the present paper follows Park in assuming that the narrow scope reading arises from a non-quantificational interpretation of some scecophonist, one which gives rise to a reading indistinguishable from a narrow scope reading when it ends up in the object position at the level of logical form. The obvious candidate for such a non-quantificational interpretation is some kind of referring expression.</Paragraph>
    <Paragraph position="16"> The claim that many noun-phrases which have been assumed to have a single generalized quantifier interpretation are in fact purely referential is not new. Recent literature on the semantics of natural quantifiers has departed considerably from the earlier tendency for semanticists to reduce all semantic distinctions Of nominal meaning such as de dicto/de re, reference/attribution, etc. to distinctions in scope of traditional quantifiers. There is widespread recognition that many such distinctions arise instead from a rich ontology of different types of (collective, distributive, intensional, groupdenoting, arbitrary, etc.) individual to which nominal expressions refer. (See for example Webber 1978, Barwise and Perry 1980, Fodor and Sag 1982, Fodor 1982, Fine 1985, and papers in the recent collection edited by Szabolcsi 1997.) One example of such non-traditional entity types (if an idea that apparently originates with Aristotle can be called non-traditional) is the notion of &amp;quot;arbitrary objects&amp;quot; (Fine 1985). An arbitrary object is an object with which properties can be associated but whose extensional identity in terms of actual objects is unspecified. In this respect, arbitrary objects resemble the Skolem terms that are generated by inference rules like Existential Elimination in proof theories of first-order predicate calculus.</Paragraph>
    <Paragraph position="17"> The rest of the paper will argue that arbitrary objects so interpreted are a necessary element of the ontology for natural language semantics, and that their involvement in CCG explains not only scope alternation (including occasions on which scope alternation is not available), but also certain cases of anomalous scopal binding which are unexplained under any of the alternatives discussed so far.</Paragraph>
    <Paragraph position="18"> 2 Donkeys as Skolem Terms One example of an indefinite that is probably better analyzed as an arbitrary object than as a quantified NP occurs in the following famous sentence, first brought to modern attention by Geach (1962):  (6) Every farmer who owns a donkey/beats it/.</Paragraph>
    <Paragraph position="19">  The pronoun looks as though it might be a variable bound by an existential quantifier associated with a donkey. However, no purely combinatoric analysis in terms of the generalized quantifier categories offered earlier allows this, since the existential cannot both remain within the scope of the universal, and come to c-command the pronoun, as is required for true bound pronominal anaphora, as in:  (7) Every farmer/in the room thinks that she/deserves a subsidy One popular reaction to this observation has been to try to generalize the notion of scope, as in Dynamic Predicate Logic (DPL). Others have pointed out that donkey pronouns in many respects look more like non-bound-variable or discourse-bound pronouns, in examples like the following: (8) Everybody who knows Gilbert/likes him/.</Paragraph>
    <Paragraph position="20">  I shall assume for the sake of argument that &amp;quot;a donkey&amp;quot; translates at predicate-argument structure as something we might write as arb'donkey'. I shall assume that the function arb t yields a Skolem term--that is, a term applying a unique functor to all variables bound by universal quantifiers in whose extent arb'donkey falls. Call it SkdonkeyX in this case, where Skdonkey maps individual instantiations of x-that is, the variable bound by the generalized quantifier every farmer---onto objects with the property donkey in the database. 4 An ordinary discourse-bound pronoun may be bound to this arbitrary object, but unless the pronoun is in the scope of the quantifiers that bind any variables in the Skolem term, it will include a variable that is outside the scope of its binder, and fail to refer.</Paragraph>
    <Paragraph position="21"> This analysis is similar to but distinct from the analyses of Cooper (1979) and Heim (1990), 41 assume that arb p &amp;quot;knows&amp;quot; what scopes it is in by the same mechanism whereby a bound variable pronoun &amp;quot;knows&amp;quot; about its binder. Whatever this mechanism is, it does not have the power of movement, abstraction, or storage. An arbitrary object is deterministically bound to all scoping universals.  who assume that a donkey translates as a quantified expression, and that the entire subject every farmer who owns a donkey establishes a contextually salient function mapping farmers to donkeys, with the donkey/E-type pronoun specifically of the type of such functions. However, by making the pronoun refer instead to a Skolem term or arbitrary object, we free our hands to make the inferences we draw on the basis of such sentences sensitive to world knowledge. For example, if we hear the standard donkey sentence and know that farmers may own more than one donkey, we will probably infer on the basis of knowledge about what makes people beat an arbitrary donkey that she beats all of them. On the other hand, we will not make a parallel inference on the basis of the following sentence (attributed to Jeff Pelletier), and the knowledge that some people have more than one dime in their pocket.</Paragraph>
    <Paragraph position="22"> (9) Everyone who had a dime in their pocket put it in the parking meter.</Paragraph>
    <Paragraph position="23"> The reason is that we know that the reason for putting a dime into a parking meter, unlike the reason for beating a donkey, is voided by the act itself. The proposal to translate indefinites as Skolem term-like discourse entities is anticipated in much early work in Artificial Intelligence and Computational Linguistics, including Kay (1973), Woods (1975 p.76-77), VanLehn (1978), and Webber (1983, p.353, cf. Webber 1978, p.2.52), and also by Chierchia (1995), Schlenker (1998), and in unpublished work by Kratzer. Skolem functors are closely related to, but distinct from, &amp;quot;Choice Functions&amp;quot; (see Reinhart 1997, Winter 1997, Sauerland 1998, and Schlenker 1998 for discussion. Webber's 1978 analysis is essentially a choice functional analysis, as is Fine's.)</Paragraph>
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